diff --git a/00_use-case/auxfuns/mpc_data.m b/00_use-case/auxfuns/mpc_data.m
index 2dde7f7..349491a 100644
--- a/00_use-case/auxfuns/mpc_data.m
+++ b/00_use-case/auxfuns/mpc_data.m
@@ -152,5 +152,16 @@
                                 2 4  64  10;
                                 2 5  837  8;
                            ];
+    elseif strcmp(casefile, '18') 
+
+        mpc.trans  = ext2int(loadcase('case9'));
+        mpc.dist = { ext2int(loadcase('case9'))};
+                            % region 1 - region 2
+        mpc.connection_array = [
+                            % region 1 - region 2
+                            1 2 2 1
+
+                            ]; 
     end
+    
 end
\ No newline at end of file
diff --git a/00_use-case/getting_started_opf.m b/00_use-case/getting_started_opf.m
index dd854aa..e65fb92 100644
--- a/00_use-case/getting_started_opf.m
+++ b/00_use-case/getting_started_opf.m
@@ -14,16 +14,35 @@
 mpc.fields_to_merge = {'bus', 'gen', 'branch', 'gencost'};
 
 mpc_temp = loadcase('case5');
-mpc_temp.gencost(:, 4) = 3;
-mpc_temp.gencost(:, 6:7) = rand();
-%mpc.fields_to_merge = {'bus', 'gen', 'branch'};
-mpc.trans = mpc_temp;
-mpc.dist = { mpc_temp;
-             mpc_temp
-                    };
+% mpc_temp.gencost(:, 4) = 3;
+% mpc_temp.gencost(:, 6:7) = rand();
+
+% remove limits in casefile
+% generator limits
+% upper bounds
 
-mpc.connection_array = [ 1 2 1 5;
-                         2 3 4 1];
+% line flow limits
+% upper bounds
+% mpc_temp.branch(:, 6) = 0;
+
+%mpc.fields_to_merge = {'bus', 'gen', 'branch'};
+% mpc.trans = mpc_temp;
+% mpc.dist = { mpc_temp;
+%              mpc_temp
+%                     };
+% 
+% mpc.connection_array = [ 1 2 1 5;
+%                          2 3 4 1];
+%                      
+
+        mpc.trans  = ext2int(loadcase('case9'));
+        mpc.dist = { ext2int(loadcase('case9'))};
+                            % region 1 - region 2
+        mpc.connection_array = [
+                            % region 1 - region 2
+                            1 2 2 1
+
+                            ]; 
 
 % compatibility tests
 % Opf data is provided ??
@@ -90,16 +109,86 @@
 %% setup distributed opf
 % start values from pf
 
+%% solve pf
+
+ mpc_split = run_case_file_splitter(mpc_merge, conn, names);
+ problem_type = 'feasibility';
+ % generate distributed problem
+ problem = generate_distributed_pf_for_aladin(mpc_split, names, problem_type);
+ problem.solver = 'fmincon';
+
+
+    opts = struct('maxiter',50, 'solveQP','quadprog');
+    opts.reg ='false';
+    opts.rho0= 1e2;
+    % 
+    % % opts.regParam = 1e-12;
+ %   [xsol_aladin, xsol_stack_aladin, mpc_sol_aladin, logg] = solve_distributed_problem_with_aladin(mpc_split, problem, names, opts);
+    %%
+%    comparison_aladin    = compare_results(xval, xsol_aladin)
+%    violation            = compare_constraints_violation(problem, logg);
+    %%
+%    [a,b,c] = compare_power_flow_between_regions(mpc_sol_aladin, mpc_merge.connections, mpc_split.regions, conn(:,1:2));
+    %% 
+%    deviation = deviation_violation_iter_plot(mpc_split, xval, logg, names, xsol_aladin);
+
+
+%% solve opf problem
+
+
+
 
 % generate distributed problem
 problem = generate_distributed_opf_for_aladin(mpc_split, names, 'feasibility');
+
+% 
+% option           = AladinOption;
+% option.iter_max  = 15;
+% option.tol       = 1e-8;
+% option.mu0       = 1e3;
+% option.rho0      = 1e2;
+% option.nlp       = NLPoption;
+% option.nlp.solver = 'casadi';
+% option.nlp.iter_display = true;
+% option.nlp.active_set = true;
+% option.qp        = QPoption;
+% % option.qp.regularization_hess = true;
+% % option.qp.solver = 'lsqminnorm';
+% % option.qp.solver = 'lsqlin';
+% option.qp.solver = 'casadi';
+% 
+% 
+% 
+% 
+% for i = 1 : length(problem.AA)
+%     local_funs = originalFuns(problem.locFuns.ffi{i}, [], [], problem.AA{i}, [], [], problem.locFuns.ggi{i}, [], problem.locFuns.hhi{i}, []);
+%     nlps(i)    = localNLP(local_funs,option.nlp,problem.llbx{i},problem.uubx{i});
+% end
+% [xopt,logg] = run_aladin_algorithm(nlps,problem.zz0,problem.lam0,horzcat(problem.AA(:)),problem.b,option);
+% 
+% 
+% 
+% 
+% for i = 1:Nregion
+% 
+%     local_funs = originalFuns(fi{i}, [], [], AA{i}, [], [], con_eq{i}, [], con_ineq{i}, []);
+% 
+%     nlps(i)    = localNLP(local_funs,option.nlp,lbx{i},ubx{i});
+% 
+% end
+% 
+% [xopt,logg] = run_aladin_algorithm(nlps,x0,lam0,A,b,option); 
+
+
+
 problem.solver = 'fmincon';
 % [xval, xval_stacked] = validate_distributed_problem_formulation(problem, mpc_split, names);
 
 % solve distributed ALADIN
 
-opts = struct('maxiter',50);
+opts = struct('maxiter',50, 'solveQP','quadprog');
 opts.reg ='false';
 opts.rho0= 1e2;
+opts.maxiter = 50;
     
 [xsol_aladin, xsol_stack_aladin, mpc_sol_aladin, logg] = solve_distributed_problem_with_aladin(mpc_split, problem, names, opts);
\ No newline at end of file
diff --git a/00_use-case/mini_example.m b/00_use-case/mini_example.m
new file mode 100644
index 0000000..ceb942f
--- /dev/null
+++ b/00_use-case/mini_example.m
@@ -0,0 +1,34 @@
+mpc = loadcase('case3_personalized');
+results_pf = runpf('case3_personalized');
+results_opf = runopf('case3_personalized');
+
+[f, df, d2f] = opf_costfcn(results_opf.x, results_opf.om);
+[h, g, dh, dg] = opf_consfcn(results_opf.x, results_opf.om);
+
+names                = generate_name_struct();
+mpc.fields_to_merge = {'bus', 'gen', 'branch', 'gencost'};
+
+mpc_temp = loadcase('case3_personalized');
+mpc.trans = mpc_temp;
+mpc.dist = { mpc_temp;};
+
+mpc.connection_array = [ 1 2 3 3];
+
+fields_to_merge      = mpc.fields_to_merge;
+connection_array     = mpc.connection_array;
+
+
+trafo_params.r = 0;
+trafo_params.x = 0.00623;
+trafo_params.b = 0;
+trafo_params.ratio = 0.985;
+trafo_params.angle = 0;
+
+conn = build_connection_table(connection_array, trafo_params);
+Nconnections = height(conn);
+
+%% main
+% case-file-generator
+mpc_merge = run_case_file_generator(mpc.trans, mpc.dist, conn, fields_to_merge, names);
+mpc_split = run_case_file_splitter(mpc_merge, conn, names);
+
diff --git a/00_use-case/mpc_merge.m b/00_use-case/mpc_merge.m
index 9fb853c..28bfbe4 100644
--- a/00_use-case/mpc_merge.m
+++ b/00_use-case/mpc_merge.m
@@ -20,12 +20,12 @@
 	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	10	1	0	0	0	0	1	1.1	0	230	1	1.1	0.9;
-	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,39 +35,40 @@
 	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	5	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	15	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
 %	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
 mpc.branch = [
-	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	2	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
-	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	12	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	14	15	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -84,7 +85,8 @@
 	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
-	2	0	0	2	10	0;
 ];
diff --git a/00_use-case/mpc_merge_split.m b/00_use-case/mpc_merge_split.m
index d911eeb..47dc320 100644
--- a/00_use-case/mpc_merge_split.m
+++ b/00_use-case/mpc_merge_split.m
@@ -20,12 +20,12 @@
 	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	10	1	0	0	0	0	1	1.1	0	230	1	1.1	0.9;
-	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,39 +35,40 @@
 	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	5	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	15	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
 %	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
 mpc.branch = [
-	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	2	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
-	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	12	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	14	15	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -84,7 +85,8 @@
 	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
-	2	0	0	2	10	0;
 ];
diff --git a/00_use-case/test_functions.m b/00_use-case/test_functions.m
new file mode 100644
index 0000000..9e22208
--- /dev/null
+++ b/00_use-case/test_functions.m
@@ -0,0 +1,37 @@
+results_pf = runpf('case3_personalized');
+results_opf = runopf('case3_personalized');
+
+[f, df, d2f] = opf_costfcn(results_opf.x, results_opf.om);
+
+mpc   = loadcase('case3_personalized');
+names = generate_name_struct();
+
+[mpc_opf, om, local_buses_to_remove, mpopt] = prepare_case_file(mpc, names);
+
+
+
+
+
+[constraint_function, ~] = build_local_constraint_function(mpc_opf, om, mpopt);
+%% cost function + cost gradient
+[cost, grad_cost, hess_cost] = build_local_cost_function(om);
+%% equalities + Jacobian
+eq = @(x)get_eq_cons(x, constraint_function, inds);
+eq_jac = @(x)get_eq_cons_jacobian(x, constraint_function, inds);
+%% inequalities + Jacobian
+[ineq, ineq_jac] = build_local_inequalities(constraint_function);
+
+pf = eq(results_opf.x);
+h = ineq(results_opf.x);
+
+%% equalities
+function g = get_eq_cons(x, gh_fcn, inds)
+    [~,g,~,~] = gh_fcn(x);
+    % remove power flow equations for all copy buses
+    g(inds) = [];
+end
+
+function dg = get_eq_cons_jacobian(x, gh_fcn, inds)
+    [~,~,~,dg] = gh_fcn(x);
+    dg(:, inds) = [];
+end
diff --git a/00_use-case/test_merged_5bus_systems.m b/00_use-case/test_merged_5bus_systems.m
new file mode 100644
index 0000000..5ed4267
--- /dev/null
+++ b/00_use-case/test_merged_5bus_systems.m
@@ -0,0 +1,207 @@
+
+clc
+clear
+close all
+
+% branch idx
+[F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, ...
+TAP, SHIFT, BR_STATUS, PF, QF, PT, QT, MU_SF, MU_ST, ...
+ANGMIN, ANGMAX, MU_ANGMIN, MU_ANGMAX] = idx_brch;
+% gen idx
+
+names                = generate_name_struct();
+mpc.fields_to_merge = {'bus', 'gen', 'branch', 'gencost'};
+
+mpc_temp = loadcase('case5');
+
+%  mpc_temp.branch(:, RATE_A) = 0;
+%  mpc_temp.branch(:, RATE_B) = 0;
+%  mpc_temp.branch(:, RATE_C) = 0;
+
+mpc.trans = mpc_temp;
+mpc.dist = { mpc_temp};
+
+mpc.connection_array = [ 1 2 1 5];
+% 
+ mpc.trans = mpc_temp;
+ mpc.dist = { mpc_temp;
+              mpc_temp
+                     };
+ 
+  mpc.connection_array = [ 1 2 1 5;
+                         2 3 1 5];
+                     
+                     
+%  
+%         mpc.trans  = ext2int(loadcase('case9'));
+%         mpc.dist = { ext2int(loadcase('case9'))};
+%                             % region 1 - region 2
+%         mpc.connection_array = [
+%                             % region 1 - region 2
+%                             1 2 2 1
+% 
+%                             ];      
+
+% 
+% 
+%         mpc.trans  = ext2int(loadcase('case9'));
+%         mpc.dist = { ext2int(loadcase('case9'));
+%              ext2int(loadcase('case9'));
+%              ext2int(loadcase('case9'))};
+%                             % region 1 - region 2
+%         mpc.connection_array = [
+%                             % region 1 - region 2
+%                             1 2 2 1
+%                             1 3 1 1
+%                             1 4 3 1
+% 
+%                             ];    
+
+% connected a from generator to a to generator with much higher Pmax 
+%% 
+
+% [mpc_trans,mpc_dist] = gen_shift_key(mpc, gsk); % P = P * 0.2
+fields_to_merge      = mpc.fields_to_merge;
+connection_array     = mpc.connection_array;
+
+
+trafo_params.r = 0;
+trafo_params.x = 0.00623;
+trafo_params.b = 0;
+trafo_params.ratio = 0.985;
+trafo_params.angle = 0;
+
+conn = build_connection_table(connection_array, trafo_params);
+Nconnections = height(conn);
+
+%% main
+% case-file-generator
+mpc_merge = run_case_file_generator(mpc.trans, mpc.dist, conn, fields_to_merge, names);
+mpc_split = run_case_file_splitter(mpc_merge, conn, names);
+
+split_file_1  = mpc_split.split_case_files{1, 1};
+split_file_2 = mpc_split.split_case_files{2, 1};
+
+runopf(mpc_merge);
+
+problem = generate_distributed_opf_for_aladin(mpc_split, names, 'feasibility');
+
+mpopt = mpoption('out.lim.all', 2, 'opf.return_raw_der', 1);
+result_opf = runopf(mpc_merge, mpopt);
+% [f, df, d2f] = opf_costfcn(result_opf.x, result_opf.om);
+% [h, g, dh, dg] = opf_consfcn(result_opf.x, result_opf.om);
+% 
+% x = result_opf.x;
+% x1 = vertcat(x(1:5), x(10), x(11:15), x(20), x(21:25), x(30:34));
+% x1_test = get_local_variable_from_global_objective_variable(x, 1, mpc_split, names);
+% x2 = vertcat(x(6:10), x(1), x(16:20), x(11), x(26:29), x(35:38));
+% x2_test = get_local_variable_from_global_objective_variable(x, 2, mpc_split, names);
+% x3_test = get_local_variable_from_global_objective_variable(x, 3, mpc_split, names);
+% 
+% fx1 = problem.locFuns.ffi{1}(x1_test);
+% fx2 = problem.locFuns.ffi{2}(x2_test);
+% fx3 = problem.locFuns.ffi{3}(x3_test);
+% 
+% diff_f = result_opf.f - (fx1 + fx2 + fx3);
+% 
+% gx1 = problem.locFuns.ggi{1}(x1_test);
+% gx2 = problem.locFuns.ggi{2}(x2_test);
+% gx3 = problem.locFuns.ggi{3}(x3_test);
+% 
+% hx1 = problem.locFuns.hhi{1}(x1_test);
+% hx2 = problem.locFuns.hhi{2}(x2_test);
+% hx3 = problem.locFuns.hhi{3}(x3_test);
+% 
+% consensus = problem.AA{1}*x1_test + problem.AA{2}*x2_test + problem.AA{3}*x3_test;
+%% ALADIN-alpha
+% 
+% problem.solver = 'fmincon';
+% % [xval, xval_stacked] = validate_distributed_problem_formulation(problem, mpc_split, names);
+% 
+% % solve distributed ALADIN
+% 
+% opts = struct('maxiter',50, 'solveQP','ipopt');
+% opts.reg ='false';
+% opts.rho0= 1e2;
+% opts.maxiter = 20;
+% [xsol_aladin, xsol_stack_aladin, mpc_sol_aladin, logg] = solve_distributed_problem_with_aladin(mpc_split, problem, names, opts);
+%% ALADIN-OOP
+option           = AladinOption;
+option.iter_max  = 20;
+option.tol       = 1e-8;
+option.mu0       = 1e4;
+option.rho0      = 1e2;
+option.nlp       = NLPoption;
+option.nlp.solver = 'fmincon';
+option.nlp.iter_display = true;
+option.nlp.active_set = true;
+option.qp        = QPoption;
+% option.qp.regularization_hess = true;
+% option.qp.solver = 'lsqminnorm';
+% option.qp.solver = 'lsqlin';
+option.qp.solver = 'casadi';
+
+
+
+
+for i = 1 : length(problem.AA)
+    local_funs = originalFuns(problem.locFuns.ffi{i}, problem.sens.gg{i}, problem.sens.HH{i}, problem.AA{i}, [], [], problem.locFuns.ggi{i}, problem.sens.JJac_eq{i}, problem.locFuns.hhi{i}, problem.sens.JJac_ineq{i});
+    nlps(i)    = localNLP(local_funs,option.nlp,problem.llbx{i},problem.uubx{i});
+end
+
+[xopt,logg] = run_aladin_algorithm(nlps,problem.zz0,problem.lam0,horzcat(problem.AA{:}),problem.b,option);
+%% validation
+
+% compare ALADIN_Alex with ALADIN_OOP
+%diff_aladin_alpha_oop = xopt - xsol_aladin;
+
+% compare ALADIN_Alex with run_opf solution
+n_regions = size(mpc_merge.regions, 2);
+x = cell(n_regions, 1);
+x_ref = [];
+for i = 1 : n_regions
+    x{i, 1} = get_local_variable_from_global_objective_variable(result_opf.x, i, mpc_split, names);
+    x_ref = vertcat(x_ref, x{i, 1});
+end
+% diff_aladin_alpha_runopf = x_ref - xsol_aladin;
+
+% compare ALADIN_OOP with run_opf solution
+diff_aladin_oop_runopf = x_ref - xopt;
+
+
+%% validation old
+opts = mpoption;
+% fix deviation
+opts.opf.violation = 1e-8;
+mpc_merge = runopf(mpc_merge,opts);
+% initialize local NLP problem by extracting data from rapidPF problem
+Nregion = n_regions;
+baseMVA = 100;
+nlps(Nregion,1)     = localNLP;
+mpc_split = run_case_file_splitter(mpc_merge, conn, names);
+mpc = mpc_split;
+GEN_BUS = 1;
+VA = 9;
+VM =8;
+PG=2;
+QG=3;
+for i = 1:Nregion
+    mpc_local = mpc_split.split_case_files{i};
+    gen_idx             =   ismember(mpc_local.regions,mpc_local.gen(:,GEN_BUS));
+    gen_bus_entries     =   find(gen_idx);     % entries of bus data
+
+    gen_bus_global      =   mpc_local.bus(gen_bus_entries,GEN_BUS);
+    gencost_idx_global = find(ismember(mpc.gen(:,GEN_BUS), gen_bus_global));
+    Vang_opt = mpc_local.bus(:,VA)/180*pi;
+    Vmag_opt = mpc_local.bus(:,VM);
+%     gencost haven't been splitted - complicated - need to be simplified
+    Pg_opt   = mpc.gen(gencost_idx_global,PG)/baseMVA;
+    Qg_opt   = mpc.gen(gencost_idx_global,QG)/baseMVA;
+    xsol{i} = stack_state(Vang_opt,Vmag_opt,Pg_opt,Qg_opt);
+end
+XOPT = vertcat(xsol{:});
+logg.plot_distance(XOPT);
+dx = xopt-XOPT
+dx_norm = norm(dx,inf)
+
+% [xsol_aladin, xsol_stack_aladin, mpc_sol_aladin, logg] = solve_distributed_problem_with_aladin(mpc_split, problem, names, opts);
\ No newline at end of file
diff --git a/00_use-case/testscript.m b/00_use-case/testscript.m
new file mode 100644
index 0000000..607114e
--- /dev/null
+++ b/00_use-case/testscript.m
@@ -0,0 +1,80 @@
+mpc = loadcase('case3_personalized');
+results_pf = runpf('case3_personalized');
+results_opf = runopf('case3_personalized');
+
+[f, df, d2f] = opf_costfcn(results_opf.x, results_opf.om);
+[h, g, dh, dg] = opf_consfcn(results_opf.x, results_opf.om);
+
+
+%%%%%%%%%%%%%%
+
+mpc = loadcase('case5');
+results_pf = runpf('case5');
+results_opf = runopf('case5');
+
+[f, df, d2f] = opf_costfcn(results_opf.x, results_opf.om);
+[g, h, dg, dh] = opf_consfcn(results_opf.x, results_opf.om);
+
+%%%%%%%%%%%%%%%
+
+%% setup
+names                = generate_name_struct();
+mpc.fields_to_merge = {'bus', 'gen', 'branch', 'gencost'};
+
+mpc_temp = loadcase('case5');
+
+mpc.trans = mpc_temp;
+mpc.dist = { mpc_temp};
+
+mpc.connection_array = [ 1 2 1 5];
+
+
+% connected a from generator to a to generator with much higher Pmax 
+%% 
+
+% [mpc_trans,mpc_dist] = gen_shift_key(mpc, gsk); % P = P * 0.2
+fields_to_merge      = mpc.fields_to_merge;
+connection_array     = mpc.connection_array;
+
+
+trafo_params.r = 0;
+trafo_params.x = 0.00623;
+trafo_params.b = 0;
+trafo_params.ratio = 0.985;
+trafo_params.angle = 0;
+
+conn = build_connection_table(connection_array, trafo_params);
+Nconnections = height(conn);
+
+%% main
+% case-file-generator
+mpc_merge = run_case_file_generator(mpc.trans, mpc.dist, conn, fields_to_merge, names);
+mpc_split = run_case_file_splitter(mpc_merge, conn, names);
+
+split_file_1  = mpc_split.split_case_files{1, 1};
+split_file_2 = mpc_split.split_case_files{2, 1};
+
+problem = generate_distributed_opf_for_aladin(mpc_split, names, 'feasibility');
+
+mpopt = mpoption('out.lim.all', 2, 'opf.return_raw_der', 1);
+result_opf = runopf(mpc_merge, mpopt);
+[f, df, d2f] = opf_costfcn(result_opf.x, results_simple.om);
+[h, g, dh, dg] = opf_consfcn(results_opf.x, results_simple.om);
+
+x = result_opf.x;
+x1 = vertcat(x(1:5), x(10), x(11:15), x(20), x(21:25), x(30:34));
+x2 = vertcat(x(1), x(6:10), x(11), x(16:20), x(26:29), x(35:38));
+
+fx1 = problem.locFuns.ffi{1}(x1);
+fx2 = problem.locFuns.ffi{2}(x2);
+
+diff_f = result_opf.f - (fx1 + fx2);
+
+gx1 = problem.locFuns.ggi{1}(x1);
+gx2 = problem.locFuns.ggi{2}(x2);
+
+mpopt = mpoption('out.lim.all', 2, 'opf.return_raw_der', 1);
+results_simple = runopf('case5', mpopt);
+
+[f, df, d2f] = opf_costfcn(results_simple.x, results_simple.om);
+[h, g, dh, dg] = opf_consfcn(results_simple.x, results_simple.om);
diff --git a/01_generator/mpc_merge.m b/01_generator/mpc_merge.m
new file mode 100644
index 0000000..eb5c13a
--- /dev/null
+++ b/01_generator/mpc_merge.m
@@ -0,0 +1,90 @@
+function mpc = mpc_merge
+%MPC_MERGE
+
+%% MATPOWER Case Format : Version 2
+mpc.version = '2';
+
+%%-----  Power Flow Data  -----%%
+%% system MVA base
+mpc.baseMVA = 100;
+
+%% bus data
+%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
+mpc.bus = [
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+];
+
+%% generator data
+%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
+mpc.gen = [
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	15	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+];
+
+%% branch data
+%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
+mpc.branch = [
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	3	14	0.278498219	0.278498219;
+	2	0	0	3	15	0.278498219	0.278498219;
+	2	0	0	3	30	0.278498219	0.278498219;
+	2	0	0	3	40	0.278498219	0.278498219;
+	2	0	0	3	10	0.278498219	0.278498219;
+	2	0	0	3	14	0.278498219	0.278498219;
+	2	0	0	3	15	0.278498219	0.278498219;
+	2	0	0	3	30	0.278498219	0.278498219;
+	2	0	0	3	40	0.278498219	0.278498219;
+	2	0	0	3	30	0.278498219	0.278498219;
+	2	0	0	3	40	0.278498219	0.278498219;
+	2	0	0	3	10	0.278498219	0.278498219;
+];
diff --git a/01_generator/mpc_merge_split.m b/01_generator/mpc_merge_split.m
new file mode 100644
index 0000000..6c6319d
--- /dev/null
+++ b/01_generator/mpc_merge_split.m
@@ -0,0 +1,90 @@
+function mpc = mpc_merge_split
+%MPC_MERGE_SPLIT
+
+%% MATPOWER Case Format : Version 2
+mpc.version = '2';
+
+%%-----  Power Flow Data  -----%%
+%% system MVA base
+mpc.baseMVA = 100;
+
+%% bus data
+%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
+mpc.bus = [
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+];
+
+%% generator data
+%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
+mpc.gen = [
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	15	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+];
+
+%% branch data
+%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
+mpc.branch = [
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	3	14	0.278498219	0.278498219;
+	2	0	0	3	15	0.278498219	0.278498219;
+	2	0	0	3	30	0.278498219	0.278498219;
+	2	0	0	3	40	0.278498219	0.278498219;
+	2	0	0	3	10	0.278498219	0.278498219;
+	2	0	0	3	14	0.278498219	0.278498219;
+	2	0	0	3	15	0.278498219	0.278498219;
+	2	0	0	3	30	0.278498219	0.278498219;
+	2	0	0	3	40	0.278498219	0.278498219;
+	2	0	0	3	30	0.278498219	0.278498219;
+	2	0	0	3	40	0.278498219	0.278498219;
+	2	0	0	3	10	0.278498219	0.278498219;
+];
diff --git a/02_splitter/mpc_merge.m b/02_splitter/mpc_merge.m
new file mode 100644
index 0000000..00aa94c
--- /dev/null
+++ b/02_splitter/mpc_merge.m
@@ -0,0 +1,72 @@
+function mpc = mpc_merge
+%MPC_MERGE
+
+%% MATPOWER Case Format : Version 2
+mpc.version = '2';
+
+%%-----  Power Flow Data  -----%%
+%% system MVA base
+mpc.baseMVA = 100;
+
+%% bus data
+%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
+mpc.bus = [
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+];
+
+%% generator data
+%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
+mpc.gen = [
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+];
+
+%% branch data
+%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
+mpc.branch = [
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+	2	0	0	2	10	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+];
diff --git a/02_splitter/mpc_merge_split.m b/02_splitter/mpc_merge_split.m
new file mode 100644
index 0000000..303a74f
--- /dev/null
+++ b/02_splitter/mpc_merge_split.m
@@ -0,0 +1,72 @@
+function mpc = mpc_merge_split
+%MPC_MERGE_SPLIT
+
+%% MATPOWER Case Format : Version 2
+mpc.version = '2';
+
+%%-----  Power Flow Data  -----%%
+%% system MVA base
+mpc.baseMVA = 100;
+
+%% bus data
+%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
+mpc.bus = [
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+];
+
+%% generator data
+%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
+mpc.gen = [
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+];
+
+%% branch data
+%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
+mpc.branch = [
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+	2	0	0	2	10	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+];
diff --git a/03_parser/auxfuns/create_consensus_matrices_opf.m b/03_parser/auxfuns/create_consensus_matrices_opf.m
index 36038eb..54c91e1 100644
--- a/03_parser/auxfuns/create_consensus_matrices_opf.m
+++ b/03_parser/auxfuns/create_consensus_matrices_opf.m
@@ -1,21 +1,18 @@
 function A = create_consensus_matrices_opf(tab, number_of_buses_in_region, number_of_generators_in_region)
 % create_consensus_matrices_opf
 %
-%   `copy the declaration of the function in here (leave the ticks unchanged)`
+%   `A = create_consensus_matrices_opf(tab, number_of_buses_in_region, number_of_generators_in_region)`
 %
-%   _describe what the function does in the following line_
+%   _creates optimal power flow consensus matrix for distributed optization_
 %
-%   # Markdown formatting is supported
-%   Equations are possible to, e.g $a^2 + b^2 = c^2$.
-%   So are lists:
-%   - item 1
-%   - item 2
-%   ```matlab
-%   function y = square(x)
-%       x^2
-%   end
-%   ```
-%   See also: [run_case_file_splitter](run_case_file_splitter.md)
+%   INPUT:
+%   - $\texttt{tab}$ connection table
+%   - $\texttt{number\_of\_buses\_in\_region}$
+%   - $\texttt{number\_of\_generators\_in\_region}$
+%
+%  OUTPUT:
+%  - $\texttt{A}$ cell with consensus matrices
+
     assert(istable(tab), 'expecting tab to be a table.');
     assert(mod(height(tab), 2) == 0, 'inconsistent number of consensus restrictions.')
     assert(numel(number_of_buses_in_region) == numel(number_of_generators_in_region), 'inconsistent dimensions');
diff --git a/03_parser/auxfuns/create_state_mp.m b/03_parser/auxfuns/create_state_mp.m
index 7e234ab..80d0c43 100644
--- a/03_parser/auxfuns/create_state_mp.m
+++ b/03_parser/auxfuns/create_state_mp.m
@@ -1,21 +1,21 @@
 function [vang, vmag, pg, qg] = create_state_mp(postfix, Nbus, Ngen)
 % create_state
 %
-%   `copy the declaration of the function in here (leave the ticks unchanged)`
+%   `[vang, vmag, pg, qg] = create_state_mp(postfix, Nbus, Ngen)`
 %
-%   _describe what the function does in the following line_
+%   _gives an abstract representation of the optimization variable used in MATPOWER_
 %
-%   # Markdown formatting is supported
-%   Equations are possible to, e.g $a^2 + b^2 = c^2$.
-%   So are lists:
-%   - item 1
-%   - item 2
-%   ```matlab
-%   function y = square(x)
-%       x^2
-%   end
-%   ```
-%   See also: [run_case_file_splitter](run_case_file_splitter.md)
+%   Input
+%   - $\ŧexttt{postfix}$ either $\textt{core}$ or $\texttt{copy}$ postfix
+%   for symbolic variable
+%   - $\texttt{Nbus}$ number of $\texttt{postfix}$ - buses in splitted
+%   system
+%   - $\texttt{Ngen}$ number of generators in splitted system
+%  
+%   Output
+%   - $\texttt{[vang, vm, Pq, Qg]}$ symbolic voltage entries for Nbus bus entries
+%   and real and reactive power entries for Ngen generators
+
     vang = sym(strcat('Va_', postfix, '_'), [Nbus 1], 'real');
     vmag = sym(strcat('Vm_', postfix, '_'), [Nbus 1], 'real');
     pg = sym(strcat('Pg_', postfix, '_'), [Ngen 1], 'real');
diff --git a/03_parser/auxfuns/modelfuns/build_local_opf.m b/03_parser/auxfuns/modelfuns/build_local_opf.m
index fc5081e..4621c6d 100644
--- a/03_parser/auxfuns/modelfuns/build_local_opf.m
+++ b/03_parser/auxfuns/modelfuns/build_local_opf.m
@@ -1,42 +1,53 @@
 function [cost, ineq, eq, x0, grad_cost, eq_jac, ineq_jac, lagrangian_hessian, state, dims, lb, ub] = build_local_opf(mpc, names, postfix)
 % build_local_opf
 %
-%   `copy the declaration of the function in here (leave the ticks unchanged)`
+%   `[cost, ineq, eq, x0, grad_cost, eq_jac, ineq_jac, lagrangian_hessian, state, dims, lb, ub] = build_local_opf(mpc, names, postfix)`
 %
-%   _describe what the function does in the following line_
+%   _extracts the local functions and information needed for opf_
 %
-%   # Markdown formatting is supported
-%   Equations are possible to, e.g $a^2 + b^2 = c^2$.
-%   So are lists:
-%   - item 1
-%   - item 2
-%   ```matlab
-%   function y = square(x)
-%       x^2
-%   end
-%   ```
-%   See also: [run_case_file_splitter](run_case_file_splitter.md)
+%   INPUT: 
+%          - $\texttt{mpc}$ splitted full casefile
+%          - $\texttt{names}$ struct of names corresponding to the fields
+%          of mpc
+%          - $\ŧexttt{postfix}$ index of subsystem
+%   OUTPUT:
+%          - $\texttt{cost}$ scalar local cost function
+%          - $\texttt{ineq}$ vector of equality constraint functions
+%          - $\texttt{eq}$ vector of inequality constraint functions
+%          - $\texttt{x0}$ intial condition of x
+%          - $\texttt{grad_cost}$ gradient of cost function
+%          - $\texttt{eq_jac}$ jacobian of eqality contraints
+%          - $\texttt{ineq_jac}$ jacobian of inequality constraints
+%          - $\texttt{lagrangian_hessian}$ hesisan of reduced lagrange
+%          function
+%          - $\texttt{state}$ representation of objective ariable
+%          - $\texttt{dims}$ struct of local dimensions
+%          - $\texttt{lb}$ lower bound of x
+%          - $\texttt{ub}$ upper bond of x
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%% The following code (implicitly) assumes that the copy buses are
 %%% always at the end of the bus numbering.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
     %% preparation
     [mpc_opf, om, local_buses_to_remove, mpopt] = prepare_case_file(mpc, names);
-    [constraint_function, ~] = build_local_constraint_function(mpc_opf, om, mpopt);
+    [constraint_function, Lxx] = build_local_constraint_function(mpc_opf, om, mpopt);
     %% cost function + cost gradient
     [cost, grad_cost, hess_cost] = build_local_cost_function(om);
     %% equalities + Jacobian
-    [eq, eq_jac] = build_local_equalities(constraint_function, local_buses_to_remove);
+    total_number_of_nodes = length(mpc.connections_with_aux_nodes);
+    [eq, eq_jac] = build_local_equalities(constraint_function, local_buses_to_remove, total_number_of_nodes);
     %% inequalities + Jacobian
     [ineq, ineq_jac] = build_local_inequalities(constraint_function);
     %% symbolic state
     state = build_local_state(mpc_opf, names, postfix);
     %% hessian of Lagrangian
-    [~, ~, lagrangian_hessian] = build_local_lagrangian_function(cost, grad_cost, eq, eq_jac, ineq, ineq_jac);
+    [~, ~, lagrangian_hessian] = build_local_lagrangian_function(cost, grad_cost, eq, eq_jac, ineq, ineq_jac, state);
     %% initial conditions
     x0 = build_local_initial_conditions(om);
     %% lower and upper bounds
     [lb, ub] = build_local_bounds(om);
     %% dimensions of state, equalities, inequalities
-    dims = build_local_dimensions(mpc_opf, eq, ineq);
+    dims = build_local_dimensions(mpc_opf, eq, ineq, local_buses_to_remove);
+    %% for testing
+    dims.ineq = [];
 end
diff --git a/03_parser/auxfuns/modelfuns/mpc_merge.m b/03_parser/auxfuns/modelfuns/mpc_merge.m
index 00aa94c..330a5f2 100644
--- a/03_parser/auxfuns/modelfuns/mpc_merge.m
+++ b/03_parser/auxfuns/modelfuns/mpc_merge.m
@@ -21,6 +21,11 @@
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
 	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,6 +40,9 @@
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	15	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
@@ -53,6 +61,13 @@
 	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
 	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -60,13 +75,16 @@
 %	1	startup	shutdown	n	x1	y1	...	xn	yn
 %	2	startup	shutdown	n	c(n-1)	...	c0
 mpc.gencost = [
-	2	0	0	2	14	0;
-	2	0	0	2	15	0;
-	2	0	0	2	30	0;
-	2	0	0	2	40	0;
-	2	0	0	2	10	0;
-	2	0	0	2	14	0;
-	2	0	0	2	15	0;
-	2	0	0	2	30	0;
-	2	0	0	2	40	0;
+	2	0	0	3	14	0.157613082	0.157613082;
+	2	0	0	3	15	0.157613082	0.157613082;
+	2	0	0	3	30	0.157613082	0.157613082;
+	2	0	0	3	40	0.157613082	0.157613082;
+	2	0	0	3	10	0.157613082	0.157613082;
+	2	0	0	3	14	0.157613082	0.157613082;
+	2	0	0	3	15	0.157613082	0.157613082;
+	2	0	0	3	30	0.157613082	0.157613082;
+	2	0	0	3	40	0.157613082	0.157613082;
+	2	0	0	3	30	0.157613082	0.157613082;
+	2	0	0	3	40	0.157613082	0.157613082;
+	2	0	0	3	10	0.157613082	0.157613082;
 ];
diff --git a/03_parser/auxfuns/modelfuns/mpc_merge_split.m b/03_parser/auxfuns/modelfuns/mpc_merge_split.m
index 303a74f..6b47d6a 100644
--- a/03_parser/auxfuns/modelfuns/mpc_merge_split.m
+++ b/03_parser/auxfuns/modelfuns/mpc_merge_split.m
@@ -21,6 +21,11 @@
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
 	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,6 +40,9 @@
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	15	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
@@ -53,6 +61,13 @@
 	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
 	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -60,13 +75,16 @@
 %	1	startup	shutdown	n	x1	y1	...	xn	yn
 %	2	startup	shutdown	n	c(n-1)	...	c0
 mpc.gencost = [
-	2	0	0	2	14	0;
-	2	0	0	2	15	0;
-	2	0	0	2	30	0;
-	2	0	0	2	40	0;
-	2	0	0	2	10	0;
-	2	0	0	2	14	0;
-	2	0	0	2	15	0;
-	2	0	0	2	30	0;
-	2	0	0	2	40	0;
+	2	0	0	3	14	0.157613082	0.157613082;
+	2	0	0	3	15	0.157613082	0.157613082;
+	2	0	0	3	30	0.157613082	0.157613082;
+	2	0	0	3	40	0.157613082	0.157613082;
+	2	0	0	3	10	0.157613082	0.157613082;
+	2	0	0	3	14	0.157613082	0.157613082;
+	2	0	0	3	15	0.157613082	0.157613082;
+	2	0	0	3	30	0.157613082	0.157613082;
+	2	0	0	3	40	0.157613082	0.157613082;
+	2	0	0	3	30	0.157613082	0.157613082;
+	2	0	0	3	40	0.157613082	0.157613082;
+	2	0	0	3	10	0.157613082	0.157613082;
 ];
diff --git a/03_parser/auxfuns/mpc_merge.m b/03_parser/auxfuns/mpc_merge.m
new file mode 100644
index 0000000..7fa34d0
--- /dev/null
+++ b/03_parser/auxfuns/mpc_merge.m
@@ -0,0 +1,92 @@
+function mpc = mpc_merge
+%MPC_MERGE
+
+%% MATPOWER Case Format : Version 2
+mpc.version = '2';
+
+%%-----  Power Flow Data  -----%%
+%% system MVA base
+mpc.baseMVA = 100;
+
+%% bus data
+%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
+mpc.bus = [
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+];
+
+%% generator data
+%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
+mpc.gen = [
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+];
+
+%% branch data
+%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
+mpc.branch = [
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+	2	0	0	2	10	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+];
diff --git a/03_parser/auxfuns/mpc_merge_split.m b/03_parser/auxfuns/mpc_merge_split.m
new file mode 100644
index 0000000..209b807
--- /dev/null
+++ b/03_parser/auxfuns/mpc_merge_split.m
@@ -0,0 +1,92 @@
+function mpc = mpc_merge_split
+%MPC_MERGE_SPLIT
+
+%% MATPOWER Case Format : Version 2
+mpc.version = '2';
+
+%%-----  Power Flow Data  -----%%
+%% system MVA base
+mpc.baseMVA = 100;
+
+%% bus data
+%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
+mpc.bus = [
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+];
+
+%% generator data
+%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
+mpc.gen = [
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+];
+
+%% branch data
+%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
+mpc.branch = [
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+	2	0	0	2	10	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
+	2	0	0	2	30	0;
+	2	0	0	2	40	0;
+];
diff --git a/03_parser/generate_distributed_opf.m b/03_parser/generate_distributed_opf.m
index 2a455da..569d90b 100644
--- a/03_parser/generate_distributed_opf.m
+++ b/03_parser/generate_distributed_opf.m
@@ -1,24 +1,31 @@
-function problem = generate_distributed_opf(mpc, names, problem_type)
+function problem = generate_distributed_opf(mpc, names, ~)
 % generate_distributed_opf
 %
-%   `copy the declaration of the function in here (leave the ticks unchanged)`
+%   `problem = generate_distributed_opf(mpc, names, ~)`
 %
-%   _describe what the function does in the following line_
+%   _extracts the information from the splitted case files to run a distributed optimization with ALADIN_
 %
-%   # Markdown formatting is supported
-%   Equations are possible to, e.g $a^2 + b^2 = c^2$.
-%   So are lists:
-%   - item 1
-%   - item 2
-%   ```matlab
-%   function y = square(x)
-%       x^2
-%   end
-%   ```
-%   See also: [run_case_file_splitter](run_case_file_splitter.md)
-    % extract Data from casefile
+%   INPUT:
+%         - $\texttt{mpc}$ struct with splitted case files
+%         - $\texttt{names}$ struct containing the names of the fields of
+%         $\texttt{mpc}
+%  Output:
+%         - $\texttt{problem}$ struct with the following fields
+%              - $\texttt{locFuns}$ struct with fields of cells for local costs,
+%              equality constraints, inequality constraints and dimensions
+%              - $\texttt{sens}$ struct with fields of cells for gradient of costs,
+%              jacobian of equality and inequality and the Hessian of the
+%              Lagrangian
+%              - $\texttt{zz0}$ cell of initial conditions
+%              - $\texttt{AA}$ cell of consensus matrices
+%              - $\texttt{state}$ cell that contans the local states
+%              - $\texttt{llbx}$ cell that contains the local lower bounds
+%              - $\texttt{uubx}$ cell that cotains the local upper bounds
+
+
+% extract Data from casefile
     [N_regions, N_buses_in_regions, N_copy_buses_in_regions, ~] = get_relevant_information(mpc, names);
-    [costs,  inequalities, equalities, xx0, grads, Jacs, Hessians, states, dims, lbs, ubs] = deal(cell(N_regions,1));
+    [costs,  inequalities, equalities, xx0, grads, Jacs, Jacs_eq, Jacs_ineq, Hessians, states, dims, lbs, ubs] = deal(cell(N_regions,1));
     connection_table = mpc.(names.consensus);
     % set up the Ai's
 
@@ -28,8 +35,16 @@
         fprintf('Creating power flow problem for system %i...', i);
         [cost, inequality, equality, x0, grad, eq_jac, ineq_jac, Hessian, state, dim, lb, ub] = build_local_opf(mpc.(names.split){i}, names, num2str(i));
         % combine Jacobians of inequalities and equalities in single Jacobian
+        % test i problem are the inqalitites
         Jac = @(x)[eq_jac(x), ineq_jac(x)]';
-        [costs{i},  inequalities{i}, equalities{i}, xx0{i}, grads{i}, Jacs{i}, Hessians{i}, states{i}, dims{i}, lbs{i}, ubs{i}] = deal(cost, inequality, equality, x0, grad, Jac, Hessian, state, dim, lb, ub);
+        Jac_eq = @(x)eq_jac(x)';
+        Jac_ineq = @(x)ineq_jac(x)';
+%        %% only for testing
+%        Jac = @(x) [eq_jac(x)'];
+%        inequality = @(x) [];
+%        cost = @(x) 0;
+        %%
+        [costs{i},  inequalities{i}, equalities{i}, xx0{i}, grads{i}, Jacs{i}, Jacs_eq{i}, Jacs_ineq{i}, Hessians{i}, states{i}, dims{i}, lbs{i}, ubs{i}] = deal(cost, inequality, equality, x0, grad, Jac, Jac_eq, Jac_ineq, Hessian, state, dim, lb, ub);
         fprintf('done.\n')
     end
     
@@ -44,6 +59,8 @@
     
     problem.sens.gg = grads;
     problem.sens.JJac = Jacs;
+    problem.sens.JJac_eq = Jacs_eq;
+    problem.sens.JJac_ineq = Jacs_ineq;
     problem.sens.HH = Hessians;
 
     problem.zz0 = xx0;
diff --git a/03_parser/mpc_merge.m b/03_parser/mpc_merge.m
index 00aa94c..06ce239 100644
--- a/03_parser/mpc_merge.m
+++ b/03_parser/mpc_merge.m
@@ -40,18 +40,18 @@
 %% branch data
 %	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
 mpc.branch = [
-	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	2	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360;
 	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
 	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
 	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
diff --git a/03_parser/mpc_merge_split.m b/03_parser/mpc_merge_split.m
index 303a74f..39f565b 100644
--- a/03_parser/mpc_merge_split.m
+++ b/03_parser/mpc_merge_split.m
@@ -9,50 +9,50 @@
 mpc.baseMVA = 100;
 
 %% bus data
-%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
+%	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin	lam_P	lam_Q	mu_Vmax	mu_Vmin
 mpc.bus = [
-	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
-	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
-	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
-	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
-	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
-	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
-	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
-	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	1	2	0	0	0	0	1	1.08492247	2.09607558	230	1	1.1	0.9	29.8339	0.0000	0.0000	0.0000;
+	2	1	300	98.61	0	0	1	1.07444025	-0.161333044	230	1	1.1	0.9	30.0757	0.0430	0.0000	0.0000;
+	3	2	300	98.61	0	0	1	1.08358895	0.513085573	230	1	1.1	0.9	30.0000	0.0000	0.0000	0.0000;
+	4	3	400	131.47	0	0	1	1.08262907	0	230	1	1.1	0.9	30.0534	0.0000	0.0000	0.0000;
+	5	2	0	0	0	0	1	1.08791679	3.25443171	230	1	1.1	0.9	29.7135	0.0000	0.0000	0.0000;
+	6	2	0	0	0	0	1	1.0982527	0.992065913	230	1	1.1	0.9	29.8648	0.0000	0.0000	0.0000;
+	7	1	300	98.61	0	0	1	1.08797688	-1.02594578	230	1	1.1	0.9	30.0812	0.0416	0.0000	0.0000;
+	8	2	300	98.61	0	0	1	1.09699403	-0.29719505	230	1	1.1	0.9	30.0000	0.0000	0.0000	0.0000;
+	9	2	400	131.47	0	0	1	1.09479564	-1.24884322	230	1	1.1	0.9	30.0995	0.0000	0.0000	0.0000;
+	10	1	0	0	0	0	1	1.1	1.28086191	230	1	1.1	0.9	29.8337	-0.0108	320.3592	0.0000;
 ];
 
 %% generator data
-%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
+%	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf	mu_Pmax	mu_Pmin	mu_Qmax	mu_Qmin
 mpc.gen = [
-	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
-	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
-	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
-	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
-	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
-	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
-	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
-	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	40	3.43105871	30	-30	1.08492247	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0	15.8339	0.0000	0.0000	0.0000;
+	1	170	35.5624823	127.5	-127.5	1.08492247	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0	14.8339	0.0000	0.0000	0.0000;
+	3	470.177037	176.3076	390	-390	1.08358895	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0	0.0000	0.0000	0.0000	0.0000;
+	4	4.27028697e-08	148.49222	150	-150	1.08262907	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0	0.0000	9.9466	0.0000	0.0000;
+	5	600	18.1841424	450	-450	1.08791679	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0	19.7135	0.0000	0.0000	0.0000;
+	6	40	1.75262358	30	-30	1.0982527	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0	15.8648	0.0000	0.0000	0.0000;
+	6	170	3.10464255	127.5	-127.5	1.0982527	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0	14.8648	0.0000	0.0000	0.0000;
+	8	515.674602	176.684679	390	-390	1.09699403	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0	0.0000	0.0000	0.0000	0.0000;
+	9	4.29015966e-08	138.064008	150	-150	1.09479564	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0	0.0000	9.9005	0.0000	0.0000;
 ];
 
 %% branch data
-%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
+%	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax	Pf	Qf	Pt	Qt	mu_Sf	mu_St	mu_angmin	mu_angmax
 mpc.branch = [
-	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
-	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
-	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
-	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
-	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
-	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
-	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
-	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
-	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
-	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	1	2	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360	166.1071	26.6607	-165.4309	-20.7287	0.0000	0.0000	0.0000	0.0000;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360	140.9835	-3.7157	-140.4698	8.0791	0.0000	0.0000	0.0000	0.0000;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360	-373.7856	-11.4518	374.5457	15.3639	0.0000	0.0000	0.0000	0.0000;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360	-134.5691	-77.8813	134.7937	77.9712	0.0000	0.0000	0.0000	0.0000;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360	35.3834	-0.2736	-35.3517	-0.2004	0.0000	0.0000	0.0000	0.0000;
+	4	5	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360	-224.1785	9.1435	225.4543	2.8203	0.0000	0.0000	0.0000	0.0000;
+	6	7	0.00281	0.0281	0.00712	0	0	0	0	0	1	-360	360	152.4912	27.1204	-151.9318	-22.3770	0.0000	0.0000	0.0000	0.0000;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360	154.6558	-0.3485	-154.0530	5.5857	0.0000	0.0000	0.0000	0.0000;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360	-97.1470	-21.9147	97.1992	18.6603	0.0000	0.0000	0.0000	0.0000;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360	-148.0682	-76.2330	148.3198	76.5380	0.0000	0.0000	0.0000	0.0000;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360	67.3548	1.5367	-67.2428	-1.2255	0.0000	0.0000	0.0000	0.0000;
+	9	10	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360	-178.7043	2.2338	179.4958	4.8696	0.0000	0.0000	0.0000	0.0000;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0	276.6950	27.5002	-276.6950	-23.5298	0.0000	0.0000	0.0000	0.0000;
 ];
 
 %%-----  OPF Data  -----%%
diff --git a/03_parser/solvefuns/solve_distributed_problem_with_aladin.m b/03_parser/solvefuns/solve_distributed_problem_with_aladin.m
index 1feee97..6eb7830 100644
--- a/03_parser/solvefuns/solve_distributed_problem_with_aladin.m
+++ b/03_parser/solvefuns/solve_distributed_problem_with_aladin.m
@@ -21,15 +21,17 @@
     end
     sol = run_ALADINnew(problem, opts);
     xsol = vertcat(sol.xxOpt{:});
-    [xsol, xsol_stacked] = deal_solution(xsol, mpc, names); 
-    
-    %% numerical solution back to matpower casefile
-    timers = sol.timers;
-    iter          =  sol.iter.i - 1; % number of iteration
-    elapsed_time  =  timers.totTime - timers.setupT;
-    alg           =  'ALADIN';
+    xsol_stacked = 0;
+    mpc_sol = 0;
+%     [xsol, xsol_stacked] = deal_solution(xsol, mpc, names); 
+%     
+%     %% numerical solution back to matpower casefile
+%     timers = sol.timers;
+%     iter          =  sol.iter.i - 1; % number of iteration
+%     elapsed_time  =  timers.totTime - timers.setupT;
+%     alg           =  'ALADIN';
     logg.X        =  sol.iter.logg.Y;
     logg.iter     =  sol.iter.i - 1;
     logg.cons_violations = sol.iter.logg.consViol;
-    mpc_sol       =  back_to_mpc(mpc, xsol, elapsed_time, iter, alg);
+%     mpc_sol       =  back_to_mpc(mpc, xsol, elapsed_time, iter, alg);
 end
\ No newline at end of file
diff --git a/05_UI_test/mpc_merge.m b/05_UI_test/mpc_merge.m
index d675ba9..6c751a3 100644
--- a/05_UI_test/mpc_merge.m
+++ b/05_UI_test/mpc_merge.m
@@ -11,151 +11,80 @@
 %% bus data
 %	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
 mpc.bus = [
-	1	3	0	0	0	0	1	1.06	0	0	1	1.06	0.94;
-	2	2	21.7	12.7	0	0	1	1.045	-4.98	0	1	1.06	0.94;
-	3	2	94.2	19	0	0	1	1.01	-12.72	0	1	1.06	0.94;
-	4	1	47.8	-3.9	0	0	1	1.019	-10.33	0	1	1.06	0.94;
-	5	1	7.6	1.6	0	0	1	1.02	-8.78	0	1	1.06	0.94;
-	6	2	11.2	7.5	0	0	1	1.07	-14.22	0	1	1.06	0.94;
-	7	1	0	0	0	0	1	1.062	-13.37	0	1	1.06	0.94;
-	8	2	0	0	0	0	1	1.09	-13.36	0	1	1.06	0.94;
-	9	1	29.5	16.6	0	19	1	1.056	-14.94	0	1	1.06	0.94;
-	10	1	9	5.8	0	0	1	1.051	-15.1	0	1	1.06	0.94;
-	11	1	3.5	1.8	0	0	1	1.057	-14.79	0	1	1.06	0.94;
-	12	1	6.1	1.6	0	0	1	1.055	-15.07	0	1	1.06	0.94;
-	13	1	13.5	5.8	0	0	1	1.05	-15.16	0	1	1.06	0.94;
-	14	1	14.9	5	0	0	1	1.036	-16.04	0	1	1.06	0.94;
-	15	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	16	2	21.7	12.7	0	0	1	1	0	135	1	1.1	0.95;
-	17	1	2.4	1.2	0	0	1	1	0	135	1	1.05	0.95;
-	18	1	7.6	1.6	0	0	1	1	0	135	1	1.05	0.95;
-	19	1	0	0	0	0.19	1	1	0	135	1	1.05	0.95;
-	20	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	21	1	22.8	10.9	0	0	1	1	0	135	1	1.05	0.95;
-	22	1	30	30	0	0	1	1	0	135	1	1.05	0.95;
-	23	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	24	1	5.8	2	0	0	3	1	0	135	1	1.05	0.95;
-	25	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	26	1	11.2	7.5	0	0	2	1	0	135	1	1.05	0.95;
-	27	2	0	0	0	0	2	1	0	135	1	1.1	0.95;
-	28	1	6.2	1.6	0	0	2	1	0	135	1	1.05	0.95;
-	29	1	8.2	2.5	0	0	2	1	0	135	1	1.05	0.95;
-	30	1	3.5	1.8	0	0	2	1	0	135	1	1.05	0.95;
-	31	1	9	5.8	0	0	2	1	0	135	1	1.05	0.95;
-	32	1	3.2	0.9	0	0	2	1	0	135	1	1.05	0.95;
-	33	1	9.5	3.4	0	0	2	1	0	135	1	1.05	0.95;
-	34	1	2.2	0.7	0	0	2	1	0	135	1	1.05	0.95;
-	35	1	17.5	11.2	0	0	3	1	0	135	1	1.05	0.95;
-	36	2	0	0	0	0	3	1	0	135	1	1.1	0.95;
-	37	2	3.2	1.6	0	0	2	1	0	135	1	1.1	0.95;
-	38	1	8.7	6.7	0	0.04	3	1	0	135	1	1.05	0.95;
-	39	1	0	0	0	0	3	1	0	135	1	1.05	0.95;
-	40	1	3.5	2.3	0	0	3	1	0	135	1	1.05	0.95;
-	41	2	0	0	0	0	3	1	0	135	1	1.1	0.95;
-	42	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	43	1	2.4	0.9	0	0	3	1	0	135	1	1.05	0.95;
-	44	1	10.6	1.9	0	0	3	1	0	135	1	1.05	0.95;
-	45	1	0	0	0	0	1	1.04	0	345	1	1.1	0.9;
-	46	2	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	47	1	0	0	0	0	1	1.025	0	345	1	1.1	0.9;
-	48	1	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	49	1	90	30	0	0	1	1	0	345	1	1.1	0.9;
-	50	1	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	51	1	100	35	0	0	1	1	0	345	1	1.1	0.9;
-	52	1	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	53	1	125	50	0	0	1	1	0	345	1	1.1	0.9;
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
 %	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
 mpc.gen = [
-	1	232.4	-16.9	10	0	1.06	100	1	332.4	0	0	0	0	0	0	0	0	0	0	0	0;
-	2	40	42.4	50	-40	1.045	100	1	140	0	0	0	0	0	0	0	0	0	0	0	0;
-	3	0	23.4	40	0	1.01	100	1	100	0	0	0	0	0	0	0	0	0	0	0	0;
-	6	0	12.2	24	-6	1.07	100	1	100	0	0	0	0	0	0	0	0	0	0	0	0;
-	8	0	17.4	24	-6	1.09	100	1	100	0	0	0	0	0	0	0	0	0	0	0	0;
-	16	60.97	0	60	-20	1	100	1	80	0	0	0	0	0	0	0	0	0	0	0	0;
-	36	21.59	0	62.5	-15	1	100	1	50	0	0	0	0	0	0	0	0	0	0	0	0;
-	41	26.91	0	48.7	-15	1	100	1	55	0	0	0	0	0	0	0	0	0	0	0	0;
-	37	19.2	0	40	-10	1	100	1	30	0	0	0	0	0	0	0	0	0	0	0	0;
-	27	37	0	44.7	-15	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
-	46	163	6.54	300	-300	1.025	100	1	300	10	0	0	0	0	0	0	0	0	0	0	0;
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	15	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
 %	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
 mpc.branch = [
-	1	2	0.01938	0.05917	0.0528	0	0	0	0	0	1	-360	360;
-	1	5	0.05403	0.22304	0.0492	0	0	0	0	0	1	-360	360;
-	2	3	0.04699	0.19797	0.0438	0	0	0	0	0	1	-360	360;
-	2	4	0.05811	0.17632	0.034	0	0	0	0	0	1	-360	360;
-	2	5	0.05695	0.17388	0.0346	0	0	0	0	0	1	-360	360;
-	3	4	0.06701	0.17103	0.0128	0	0	0	0	0	1	-360	360;
-	4	5	0.01335	0.04211	0	0	0	0	0	0	1	-360	360;
-	4	7	0	0.20912	0	0	0	0	0.978	0	1	-360	360;
-	4	9	0	0.55618	0	0	0	0	0.969	0	1	-360	360;
-	5	6	0	0.25202	0	0	0	0	0.932	0	1	-360	360;
-	6	11	0.09498	0.1989	0	0	0	0	0	0	1	-360	360;
-	6	12	0.12291	0.25581	0	0	0	0	0	0	1	-360	360;
-	6	13	0.06615	0.13027	0	0	0	0	0	0	1	-360	360;
-	7	8	0	0.17615	0	0	0	0	0	0	1	-360	360;
-	7	9	0	0.11001	0	0	0	0	0	0	1	-360	360;
-	9	10	0.03181	0.0845	0	0	0	0	0	0	1	-360	360;
-	9	14	0.12711	0.27038	0	0	0	0	0	0	1	-360	360;
-	10	11	0.08205	0.19207	0	0	0	0	0	0	1	-360	360;
-	12	13	0.22092	0.19988	0	0	0	0	0	0	1	-360	360;
-	13	14	0.17093	0.34802	0	0	0	0	0	0	1	-360	360;
-	15	16	0.02	0.06	0.03	130	130	130	0	0	1	-360	360;
-	15	17	0.05	0.19	0.02	130	130	130	0	0	1	-360	360;
-	16	18	0.06	0.17	0.02	65	65	65	0	0	1	-360	360;
-	17	18	0.01	0.04	0	130	130	130	0	0	1	-360	360;
-	16	19	0.05	0.2	0.02	130	130	130	0	0	1	-360	360;
-	16	20	0.06	0.18	0.02	65	65	65	0	0	1	-360	360;
-	18	20	0.01	0.04	0	90	90	90	0	0	1	-360	360;
-	19	21	0.05	0.12	0.01	70	70	70	0	0	1	-360	360;
-	20	21	0.03	0.08	0.01	130	130	130	0	0	1	-360	360;
-	20	22	0.01	0.04	0	32	32	32	0	0	1	-360	360;
-	20	23	0	0.21	0	65	65	65	0	0	1	-360	360;
-	20	24	0	0.56	0	32	32	32	0	0	1	-360	360;
-	23	25	0	0.21	0	65	65	65	0	0	1	-360	360;
-	23	24	0	0.11	0	65	65	65	0	0	1	-360	360;
-	18	26	0	0.26	0	65	65	65	0	0	1	-360	360;
-	26	27	0	0.14	0	65	65	65	0	0	1	-360	360;
-	26	28	0.12	0.26	0	32	32	32	0	0	1	-360	360;
-	26	29	0.07	0.13	0	32	32	32	0	0	1	-360	360;
-	26	30	0.09	0.2	0	32	32	32	0	0	1	-360	360;
-	28	29	0.22	0.2	0	16	16	16	0	0	1	-360	360;
-	30	31	0.08	0.19	0	16	16	16	0	0	1	-360	360;
-	29	32	0.11	0.22	0	16	16	16	0	0	1	-360	360;
-	32	33	0.06	0.13	0	16	16	16	0	0	1	-360	360;
-	33	34	0.03	0.07	0	32	32	32	0	0	1	-360	360;
-	24	34	0.09	0.21	0	32	32	32	0	0	1	-360	360;
-	24	31	0.03	0.08	0	32	32	32	0	0	1	-360	360;
-	24	35	0.03	0.07	0	32	32	32	0	0	1	-360	360;
-	24	36	0.07	0.15	0	32	32	32	0	0	1	-360	360;
-	35	36	0.01	0.02	0	32	32	32	0	0	1	-360	360;
-	29	37	0.1	0.2	0	16	16	16	0	0	1	-360	360;
-	36	38	0.12	0.18	0	16	16	16	0	0	1	-360	360;
-	37	38	0.13	0.27	0	16	16	16	0	0	1	-360	360;
-	38	39	0.19	0.33	0	16	16	16	0	0	1	-360	360;
-	39	40	0.25	0.38	0	16	16	16	0	0	1	-360	360;
-	39	41	0.11	0.21	0	16	16	16	0	0	1	-360	360;
-	42	41	0	0.4	0	65	65	65	0	0	1	-360	360;
-	41	43	0.22	0.42	0	16	16	16	0	0	1	-360	360;
-	41	44	0.32	0.6	0	16	16	16	0	0	1	-360	360;
-	43	44	0.24	0.45	0	16	16	16	0	0	1	-360	360;
-	22	42	0.06	0.2	0.02	32	32	32	0	0	1	-360	360;
-	20	42	0.02	0.06	0.01	32	32	32	0	0	1	-360	360;
-	2	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
-	45	48	0	0.0576	0	250	250	250	0	0	1	-360	360;
-	48	49	0.017	0.092	0.158	250	250	250	0	0	1	-360	360;
-	49	50	0.039	0.17	0.358	150	150	150	0	0	1	-360	360;
-	47	50	0	0.0586	0	300	300	300	0	0	1	-360	360;
-	50	51	0.0119	0.1008	0.209	150	150	150	0	0	1	-360	360;
-	51	52	0.0085	0.072	0.149	250	250	250	0	0	1	-360	360;
-	52	46	0	0.0625	0	250	250	250	0	0	1	-360	360;
-	52	53	0.032	0.161	0.306	250	250	250	0	0	1	-360	360;
-	53	48	0.01	0.085	0.176	250	250	250	0	0	1	-360	360;
-	16	47	0	0.00623	0	0	0	0	0.985	0	1	0	0;
-	27	45	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	3	14	0.546881519	0.546881519;
+	2	0	0	3	15	0.546881519	0.546881519;
+	2	0	0	3	30	0.546881519	0.546881519;
+	2	0	0	3	40	0.546881519	0.546881519;
+	2	0	0	3	10	0.546881519	0.546881519;
+	2	0	0	3	14	0.546881519	0.546881519;
+	2	0	0	3	15	0.546881519	0.546881519;
+	2	0	0	3	30	0.546881519	0.546881519;
+	2	0	0	3	40	0.546881519	0.546881519;
+	2	0	0	3	30	0.546881519	0.546881519;
+	2	0	0	3	40	0.546881519	0.546881519;
+	2	0	0	3	10	0.546881519	0.546881519;
 ];
diff --git a/05_UI_test/mpc_merge_split.m b/05_UI_test/mpc_merge_split.m
index 4e74008..11bb937 100644
--- a/05_UI_test/mpc_merge_split.m
+++ b/05_UI_test/mpc_merge_split.m
@@ -11,151 +11,80 @@
 %% bus data
 %	bus_i	type	Pd	Qd	Gs	Bs	area	Vm	Va	baseKV	zone	Vmax	Vmin
 mpc.bus = [
-	1	3	0	0	0	0	1	1.06	0	0	1	1.06	0.94;
-	2	2	21.7	12.7	0	0	1	1.045	-4.98	0	1	1.06	0.94;
-	3	2	94.2	19	0	0	1	1.01	-12.72	0	1	1.06	0.94;
-	4	1	47.8	-3.9	0	0	1	1.019	-10.33	0	1	1.06	0.94;
-	5	1	7.6	1.6	0	0	1	1.02	-8.78	0	1	1.06	0.94;
-	6	2	11.2	7.5	0	0	1	1.07	-14.22	0	1	1.06	0.94;
-	7	1	0	0	0	0	1	1.062	-13.37	0	1	1.06	0.94;
-	8	2	0	0	0	0	1	1.09	-13.36	0	1	1.06	0.94;
-	9	1	29.5	16.6	0	19	1	1.056	-14.94	0	1	1.06	0.94;
-	10	1	9	5.8	0	0	1	1.051	-15.1	0	1	1.06	0.94;
-	11	1	3.5	1.8	0	0	1	1.057	-14.79	0	1	1.06	0.94;
-	12	1	6.1	1.6	0	0	1	1.055	-15.07	0	1	1.06	0.94;
-	13	1	13.5	5.8	0	0	1	1.05	-15.16	0	1	1.06	0.94;
-	14	1	14.9	5	0	0	1	1.036	-16.04	0	1	1.06	0.94;
-	15	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	16	2	21.7	12.7	0	0	1	1	0	135	1	1.1	0.95;
-	17	1	2.4	1.2	0	0	1	1	0	135	1	1.05	0.95;
-	18	1	7.6	1.6	0	0	1	1	0	135	1	1.05	0.95;
-	19	1	0	0	0	0.19	1	1	0	135	1	1.05	0.95;
-	20	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	21	1	22.8	10.9	0	0	1	1	0	135	1	1.05	0.95;
-	22	1	30	30	0	0	1	1	0	135	1	1.05	0.95;
-	23	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	24	1	5.8	2	0	0	3	1	0	135	1	1.05	0.95;
-	25	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	26	1	11.2	7.5	0	0	2	1	0	135	1	1.05	0.95;
-	27	2	0	0	0	0	2	1	0	135	1	1.1	0.95;
-	28	1	6.2	1.6	0	0	2	1	0	135	1	1.05	0.95;
-	29	1	8.2	2.5	0	0	2	1	0	135	1	1.05	0.95;
-	30	1	3.5	1.8	0	0	2	1	0	135	1	1.05	0.95;
-	31	1	9	5.8	0	0	2	1	0	135	1	1.05	0.95;
-	32	1	3.2	0.9	0	0	2	1	0	135	1	1.05	0.95;
-	33	1	9.5	3.4	0	0	2	1	0	135	1	1.05	0.95;
-	34	1	2.2	0.7	0	0	2	1	0	135	1	1.05	0.95;
-	35	1	17.5	11.2	0	0	3	1	0	135	1	1.05	0.95;
-	36	2	0	0	0	0	3	1	0	135	1	1.1	0.95;
-	37	2	3.2	1.6	0	0	2	1	0	135	1	1.1	0.95;
-	38	1	8.7	6.7	0	0.04	3	1	0	135	1	1.05	0.95;
-	39	1	0	0	0	0	3	1	0	135	1	1.05	0.95;
-	40	1	3.5	2.3	0	0	3	1	0	135	1	1.05	0.95;
-	41	2	0	0	0	0	3	1	0	135	1	1.1	0.95;
-	42	1	0	0	0	0	1	1	0	135	1	1.05	0.95;
-	43	1	2.4	0.9	0	0	3	1	0	135	1	1.05	0.95;
-	44	1	10.6	1.9	0	0	3	1	0	135	1	1.05	0.95;
-	45	1	0	0	0	0	1	1.04	0	345	1	1.1	0.9;
-	46	2	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	47	1	0	0	0	0	1	1.025	0	345	1	1.1	0.9;
-	48	1	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	49	1	90	30	0	0	1	1	0	345	1	1.1	0.9;
-	50	1	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	51	1	100	35	0	0	1	1	0	345	1	1.1	0.9;
-	52	1	0	0	0	0	1	1	0	345	1	1.1	0.9;
-	53	1	125	50	0	0	1	1	0	345	1	1.1	0.9;
+	1	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	2	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	3	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	4	3	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	5	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	6	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
+	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
+	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
 %	bus	Pg	Qg	Qmax	Qmin	Vg	mBase	status	Pmax	Pmin	Pc1	Pc2	Qc1min	Qc1max	Qc2min	Qc2max	ramp_agc	ramp_10	ramp_30	ramp_q	apf
 mpc.gen = [
-	1	232.4	-16.9	10	0	1.06	100	1	332.4	0	0	0	0	0	0	0	0	0	0	0	0;
-	2	40	42.4	50	-40	1.045	100	1	140	0	0	0	0	0	0	0	0	0	0	0	0;
-	3	0	23.4	40	0	1.01	100	1	100	0	0	0	0	0	0	0	0	0	0	0	0;
-	6	0	12.2	24	-6	1.07	100	1	100	0	0	0	0	0	0	0	0	0	0	0	0;
-	8	0	17.4	24	-6	1.09	100	1	100	0	0	0	0	0	0	0	0	0	0	0	0;
-	16	60.97	0	60	-20	1	100	1	80	0	0	0	0	0	0	0	0	0	0	0	0;
-	36	21.59	0	62.5	-15	1	100	1	50	0	0	0	0	0	0	0	0	0	0	0	0;
-	41	26.91	0	48.7	-15	1	100	1	55	0	0	0	0	0	0	0	0	0	0	0	0;
-	37	19.2	0	40	-10	1	100	1	30	0	0	0	0	0	0	0	0	0	0	0	0;
-	27	37	0	44.7	-15	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
-	46	163	6.54	300	-300	1.025	100	1	300	10	0	0	0	0	0	0	0	0	0	0	0;
+	1	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
+	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
+	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	15	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
 %	fbus	tbus	r	x	b	rateA	rateB	rateC	ratio	angle	status	angmin	angmax
 mpc.branch = [
-	1	2	0.01938	0.05917	0.0528	0	0	0	0	0	1	-360	360;
-	1	5	0.05403	0.22304	0.0492	0	0	0	0	0	1	-360	360;
-	2	3	0.04699	0.19797	0.0438	0	0	0	0	0	1	-360	360;
-	2	4	0.05811	0.17632	0.034	0	0	0	0	0	1	-360	360;
-	2	5	0.05695	0.17388	0.0346	0	0	0	0	0	1	-360	360;
-	3	4	0.06701	0.17103	0.0128	0	0	0	0	0	1	-360	360;
-	4	5	0.01335	0.04211	0	0	0	0	0	0	1	-360	360;
-	4	7	0	0.20912	0	0	0	0	0.978	0	1	-360	360;
-	4	9	0	0.55618	0	0	0	0	0.969	0	1	-360	360;
-	5	6	0	0.25202	0	0	0	0	0.932	0	1	-360	360;
-	6	11	0.09498	0.1989	0	0	0	0	0	0	1	-360	360;
-	6	12	0.12291	0.25581	0	0	0	0	0	0	1	-360	360;
-	6	13	0.06615	0.13027	0	0	0	0	0	0	1	-360	360;
-	7	8	0	0.17615	0	0	0	0	0	0	1	-360	360;
-	7	9	0	0.11001	0	0	0	0	0	0	1	-360	360;
-	9	10	0.03181	0.0845	0	0	0	0	0	0	1	-360	360;
-	9	14	0.12711	0.27038	0	0	0	0	0	0	1	-360	360;
-	10	11	0.08205	0.19207	0	0	0	0	0	0	1	-360	360;
-	12	13	0.22092	0.19988	0	0	0	0	0	0	1	-360	360;
-	13	14	0.17093	0.34802	0	0	0	0	0	0	1	-360	360;
-	15	16	0.02	0.06	0.03	130	130	130	0	0	1	-360	360;
-	15	17	0.05	0.19	0.02	130	130	130	0	0	1	-360	360;
-	16	18	0.06	0.17	0.02	65	65	65	0	0	1	-360	360;
-	17	18	0.01	0.04	0	130	130	130	0	0	1	-360	360;
-	16	19	0.05	0.2	0.02	130	130	130	0	0	1	-360	360;
-	16	20	0.06	0.18	0.02	65	65	65	0	0	1	-360	360;
-	18	20	0.01	0.04	0	90	90	90	0	0	1	-360	360;
-	19	21	0.05	0.12	0.01	70	70	70	0	0	1	-360	360;
-	20	21	0.03	0.08	0.01	130	130	130	0	0	1	-360	360;
-	20	22	0.01	0.04	0	32	32	32	0	0	1	-360	360;
-	20	23	0	0.21	0	65	65	65	0	0	1	-360	360;
-	20	24	0	0.56	0	32	32	32	0	0	1	-360	360;
-	23	25	0	0.21	0	65	65	65	0	0	1	-360	360;
-	23	24	0	0.11	0	65	65	65	0	0	1	-360	360;
-	18	26	0	0.26	0	65	65	65	0	0	1	-360	360;
-	26	27	0	0.14	0	65	65	65	0	0	1	-360	360;
-	26	28	0.12	0.26	0	32	32	32	0	0	1	-360	360;
-	26	29	0.07	0.13	0	32	32	32	0	0	1	-360	360;
-	26	30	0.09	0.2	0	32	32	32	0	0	1	-360	360;
-	28	29	0.22	0.2	0	16	16	16	0	0	1	-360	360;
-	30	31	0.08	0.19	0	16	16	16	0	0	1	-360	360;
-	29	32	0.11	0.22	0	16	16	16	0	0	1	-360	360;
-	32	33	0.06	0.13	0	16	16	16	0	0	1	-360	360;
-	33	34	0.03	0.07	0	32	32	32	0	0	1	-360	360;
-	24	34	0.09	0.21	0	32	32	32	0	0	1	-360	360;
-	24	31	0.03	0.08	0	32	32	32	0	0	1	-360	360;
-	24	35	0.03	0.07	0	32	32	32	0	0	1	-360	360;
-	24	36	0.07	0.15	0	32	32	32	0	0	1	-360	360;
-	35	36	0.01	0.02	0	32	32	32	0	0	1	-360	360;
-	29	37	0.1	0.2	0	16	16	16	0	0	1	-360	360;
-	36	38	0.12	0.18	0	16	16	16	0	0	1	-360	360;
-	37	38	0.13	0.27	0	16	16	16	0	0	1	-360	360;
-	38	39	0.19	0.33	0	16	16	16	0	0	1	-360	360;
-	39	40	0.25	0.38	0	16	16	16	0	0	1	-360	360;
-	39	41	0.11	0.21	0	16	16	16	0	0	1	-360	360;
-	42	41	0	0.4	0	65	65	65	0	0	1	-360	360;
-	41	43	0.22	0.42	0	16	16	16	0	0	1	-360	360;
-	41	44	0.32	0.6	0	16	16	16	0	0	1	-360	360;
-	43	44	0.24	0.45	0	16	16	16	0	0	1	-360	360;
-	22	42	0.06	0.2	0.02	32	32	32	0	0	1	-360	360;
-	20	42	0.02	0.06	0.01	32	32	32	0	0	1	-360	360;
-	2	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
-	45	48	0	0.0576	0	250	250	250	0	0	1	-360	360;
-	48	49	0.017	0.092	0.158	250	250	250	0	0	1	-360	360;
-	49	50	0.039	0.17	0.358	150	150	150	0	0	1	-360	360;
-	47	50	0	0.0586	0	300	300	300	0	0	1	-360	360;
-	50	51	0.0119	0.1008	0.209	150	150	150	0	0	1	-360	360;
-	51	52	0.0085	0.072	0.149	250	250	250	0	0	1	-360	360;
-	52	46	0	0.0625	0	250	250	250	0	0	1	-360	360;
-	52	53	0.032	0.161	0.306	250	250	250	0	0	1	-360	360;
-	53	48	0.01	0.085	0.176	250	250	250	0	0	1	-360	360;
-	16	47	0	0.00623	0	0	0	0	0.985	0	1	0	0;
-	27	45	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	1	2	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	1	4	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	1	5	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	2	3	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	3	4	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	4	5	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	6	7	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	6	9	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	6	10	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	7	8	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	8	9	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	9	10	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	1	10	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	11	12	0.00281	0.0281	0.00712	400	400	400	0	0	1	-360	360;
+	11	14	0.00304	0.0304	0.00658	0	0	0	0	0	1	-360	360;
+	11	15	0.00064	0.0064	0.03126	0	0	0	0	0	1	-360	360;
+	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
+	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
+	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
+	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+];
+
+%%-----  OPF Data  -----%%
+%% generator cost data
+%	1	startup	shutdown	n	x1	y1	...	xn	yn
+%	2	startup	shutdown	n	c(n-1)	...	c0
+mpc.gencost = [
+	2	0	0	3	14	0.546881519	0.546881519;
+	2	0	0	3	15	0.546881519	0.546881519;
+	2	0	0	3	30	0.546881519	0.546881519;
+	2	0	0	3	40	0.546881519	0.546881519;
+	2	0	0	3	10	0.546881519	0.546881519;
+	2	0	0	3	14	0.546881519	0.546881519;
+	2	0	0	3	15	0.546881519	0.546881519;
+	2	0	0	3	30	0.546881519	0.546881519;
+	2	0	0	3	40	0.546881519	0.546881519;
+	2	0	0	3	30	0.546881519	0.546881519;
+	2	0	0	3	40	0.546881519	0.546881519;
+	2	0	0	3	10	0.546881519	0.546881519;
 ];
diff --git a/06_opf_extension/build_local_bounds.m b/06_opf_extension/build_local_bounds.m
index 386159e..ffad94d 100644
--- a/06_opf_extension/build_local_bounds.m
+++ b/06_opf_extension/build_local_bounds.m
@@ -1,3 +1,14 @@
 function [lb, ub] = build_local_bounds(om)
+% BUILD_LOACL_BOUNDS
+%
+%   `[lb, ub] = build_local_bounds(om)`
+%
+%   _extracts local bounds for x from MATPOWER_
+%
+%   INPUT:
+%   - $\texttt{om}$ optimization model of reduced splitted case file
+%
+%  OUTPUT:
+%  - $\texttt{[lb, ub]}$ lower and upper bond for objective variable $x$ of model $\texttt{om}$ of subsystem $i$    
     [~, lb, ub] = om.params_var();
 end
\ No newline at end of file
diff --git a/06_opf_extension/build_local_constraint_function.m b/06_opf_extension/build_local_constraint_function.m
index cd6e245..33d48a5 100644
--- a/06_opf_extension/build_local_constraint_function.m
+++ b/06_opf_extension/build_local_constraint_function.m
@@ -1,4 +1,11 @@
 function [constraint_function, Lxx] = build_local_constraint_function(mpc_opf, om, mpopt)
+    %BUILD_LOCAL_CONSTRAINT_FUNCTION extracts matpower equality and
+    %inequality constraints and their jacobian to rapidOPF
+    %
+    %Output: - constraint_function @(x)[h, dh, g, dh]
+    %        - Lxx Lagrangian function of complete system in MATPOWER. It
+    %          is not usable by rapidOPF but taken for completeness
+    %       
     [Ybus, Yf, Yt] = makeYbus(mpc_opf);
     il = find(mpc_opf.branch(:, 6) ~= 0 & mpc_opf.branch(:, 6) < 1e10);
     
diff --git a/06_opf_extension/build_local_cost_function.m b/06_opf_extension/build_local_cost_function.m
index d2e2009..0a72d8e 100644
--- a/06_opf_extension/build_local_cost_function.m
+++ b/06_opf_extension/build_local_cost_function.m
@@ -1,4 +1,14 @@
 function [cost, grad, hess] = build_local_cost_function(om)
+% BUILD_LOCAL_COST_FUNCTION
+%
+%    `[mpc_opf, om, copy_buses_local, mpopt] = prepare_case_file(mpc, names))`
+%
+% INPUT:  - om MATPOWER optimization model
+% OUTPUT: - cost function handle for costs
+%         - grad function handle for gradient of costs
+%         - hess function handle for hessian of cost
+
+
     f = @(x)opf_costfcn(x, om);
     cost = @(x)get_cost(x, f);
     grad = @(x)get_cost_gradient(x, f);
@@ -7,11 +17,11 @@
 
 % helper functions
 function f = get_cost(x, f_fcn)
-    [f,~] = f_fcn(x);
+    [f,~,~] = f_fcn(x);
 end
 
 function df = get_cost_gradient(x, f_fcn)
-    [~, df] = f_fcn(x);
+    [~, df, ~] = f_fcn(x);
 end
 
 function d2f = get_cost_hess(x, f_fcn)
diff --git a/06_opf_extension/build_local_dimensions.m b/06_opf_extension/build_local_dimensions.m
index b9114a9..f06e7eb 100644
--- a/06_opf_extension/build_local_dimensions.m
+++ b/06_opf_extension/build_local_dimensions.m
@@ -1,15 +1,28 @@
-function dims = build_local_dimensions(mpc_opf, eq, ineq)
-    % this file assumes that all generators at the copy busses have been turned off
-    % i.e. prepare_case_file() was called already!
+function dims = build_local_dimensions(mpc_opf, eq, ineq, local_buses_to_remove)
+% DIMS 
+%
+%   `dims = build_local_dimensions(mpc_opf, eq, ineq, local_buses_to_remove)`
+%
+%   _creates a field of dimensions as how they should look like and use it for testing_
+%
+%   INPUT:  - mpc_opf splitted case files
+%           - eq power flow equations for core nodes plus their jacbian'
+%           - ineq flow limits inequality constraints plus their jacobian'
+%           - local_buses_to_remove copy nodes
+%   OUTPUT: - dims struct containing dimensions
+    
     Nbus = size(mpc_opf.bus, 1);
     Ngen = size(mpc_opf.gen, 1);
     dims.state = 2 * (Nbus + Ngen);
-    dims.eq = 2 * Nbus;
+    dims.eq = 2 * (Nbus - length(local_buses_to_remove));
+    
+    il = find(mpc_opf.branch(:, 6) ~= 0 & mpc_opf.branch(:, 6) < 1e10);
+    dims.ineq = 2 * length(il); % one inequality for each from and to bus
+        
     dims.n.bus = Nbus;
     dims.n.gen = Ngen;
-    %% ToDo!
-    % verification missing!!!
+    %% Test dimensions
     x = rand(dims.state, 1);
-    dims.ineq = numel(ineq(x));
-    dims.eq = numel(eq(x));
+ %   assert (dims.ineq == numel(ineq(x)), 'Error during dimension check of number of inequality constraints'); 
+    assert (dims.eq == numel(eq(x)), 'Error during dimension check of number of equality constraints');
 end
\ No newline at end of file
diff --git a/06_opf_extension/build_local_equalities.m b/06_opf_extension/build_local_equalities.m
index 312eac7..6e8fbfa 100644
--- a/06_opf_extension/build_local_equalities.m
+++ b/06_opf_extension/build_local_equalities.m
@@ -1,5 +1,11 @@
-function [eq, eq_jac] = build_local_equalities(constraint_function, local_buses_to_remove)
-    inds = [local_buses_to_remove; 2 * local_buses_to_remove];
+function [eq, eq_jac] = build_local_equalities(constraint_function, local_buses_to_remove, total_number_of_nodes)
+% BUILD_LOCAL_EQUALITIES
+%
+%   `[eq, eq_jac] = build_local_equalities(constraint_function, local_buses_to_remove)`
+%
+%   _extracts the relevant power flow equation for the core buses. Power flow equations of the copy buses are removed_
+%   
+    inds = [local_buses_to_remove; local_buses_to_remove + total_number_of_nodes];
     eq = @(x)get_eq_cons(x, constraint_function, inds);
     eq_jac = @(x)get_eq_cons_jacobian(x, constraint_function, inds);
 end
diff --git a/06_opf_extension/build_local_inequalities.m b/06_opf_extension/build_local_inequalities.m
index 090bafe..7f031ee 100644
--- a/06_opf_extension/build_local_inequalities.m
+++ b/06_opf_extension/build_local_inequalities.m
@@ -1,4 +1,10 @@
 function [ineq, ineq_jac] = build_local_inequalities(constraint_function)
+% BUILD_LOCAL_INEQUALITIES
+%
+%   `[ineq, ineq_jac] = build_local_inequalities(constraint_function, local_buses_to_remove)`
+%
+%   _extracts all inequalitites from opf_consfcn.m_
+%   
     ineq = @(x)get_ineq_cons(x, constraint_function);
     ineq_jac = @(x)get_ineq_cons_jacobian(x, constraint_function);
 end
diff --git a/06_opf_extension/build_local_initial_conditions.m b/06_opf_extension/build_local_initial_conditions.m
index 4d3f143..5ff345d 100644
--- a/06_opf_extension/build_local_initial_conditions.m
+++ b/06_opf_extension/build_local_initial_conditions.m
@@ -1,3 +1,14 @@
 function x0 = build_local_initial_conditions(om)
-    x0 = om.params_var();
+% BUILD_LOACL_INITIAL_CONDITION
+%
+%   `x0 = build_local_initial_conditions(om)`
+%
+%   _extracts initial condition for x from MATPOWER_
+%
+%   INPUT:
+%   - $\texttt{om}$ optimization model of reduced splitted case file
+%
+%  OUTPUT:
+%  - $\texttt{x0}$ initial condition for objective variable $x$ of model $\texttt{om}$ of subsystem $i$    
+x0 = om.params_var();
 end
\ No newline at end of file
diff --git a/06_opf_extension/build_local_lagrangian_function.m b/06_opf_extension/build_local_lagrangian_function.m
index fc6c312..9976610 100644
--- a/06_opf_extension/build_local_lagrangian_function.m
+++ b/06_opf_extension/build_local_lagrangian_function.m
@@ -1,38 +1,52 @@
-function [L, dLdx, d2Ldx] = build_local_lagrangian_function(f, df, g, dg, h, dh)
-%BUILD_LOCAL_LAGRANGIAN_FUNCTION Bulds the Lagrangian of the local mpc file
-%   kappa is [lambda; mu] with lambda, mu are column vectors
-% dg is the transposed of the jacobian of g, i.e. dimension nx_x \times n_eq 
-% dh is the transposed jacobian of h
-% L = f + lambda'g + mu'h
-% dL = grad_f + lambda'(Jac_g)' + mu'(Jac_h)'
-
-    if nargin == 4
-        L = @(x, kappa, Neq) f(x) + kappa(1:Neq)'*g(x);
-        dLdx = @(x, kappa, Neq) df(x) ...
-            + get_grad_lambda_g(kappa(1:Neq), dg, x);
-        d2Ldx = @(x, kappa, rho, Neq)  get_Hess(dLdx, x, kappa, Neq);
-    elseif nargin == 6
-        L = @(x, kappa, Neq) f(x) + kappa(1:Neq)'*g(x) + kappa(Neq+1:end)'*h(x);
-        dLdx = @(x, kappa, Neq) df(x)  + ...
-            get_grad_lambda_g(kappa(1:Neq), dg, x) + ...
-            get_grad_mu_h(kappa(Neq+1:end), dh, x);
-        d2Ldx = @(x, kappa, rho, Neq) get_Hess(dLdx, x, kappa, Neq);
-    end
+function [L, dLdx, d2Ldx] = build_local_lagrangian_function(f, df, g, dg, h, dh, state)
+%BUILD_LOCAL_LAGRANGIAN_FUNCTION 
+%
+%  `Builds the Lagrangian of the local mpc file`
+% 
+%  INPUT: 
+%         - $\texttt{f}$ scalar valued cost function
+%         - $\texttt{df}$ gradient of $\textt{f}$ 
+%         - $\texttt{g}$ equality constraints
+%         - $\texttt{dg}$ transposed jacobian of the 'reduced' equality constraints
+%         - $\ŧexttt{h}$ inequality constraints
+%         - $\texttt{dh}$ transposed jacobian of the inequality constraints
+% 
+%  OUTPUT:
+%         - $\texttt{L}$ 'reduced' Lagrangefunction
+%         - $\texttt{dLdx}$ grandient of the Lagrangian
+%         - $\ŧexttt{d2Ldx}$ hessian of the Lagrangian
+%
+%  REMARK:
+%   mu is the langrange multiplier for the 
+%   kappa = [lambda; mu] (with lambda, mu column vectors)
+%   L = f + lambda'g + mu'h
+%   dL = grad_f + \sum lambda_i (Jac_g)'(:, i) + \sum mu_i(Jac_h)'(:, i)
 
+x = zeros(length(state), 1);
+Neq = numel(g(x));
+Nineq = numel(h(x));
+    
+if Neq > 0 && Nineq == 0
+        L       = @(x, kappa) f(x) * kappa'*g(x);
+        dLdx    = @(x, kappa) df(x) + get_grad_lambda_g(kappa, dg, x);
+        d2Ldx   = @(x, kappa, rho, Neq)get_Hess(dLdx, x, kappa);
+elseif Neq == 0 && Nineq > 0 
+        L       = @(x, kappa) f(x) * kappa'*h(x);
+        dLdx    = @(x, kappa) df(x) + get_grad_mu_h(kappa, dh, x);
+        d2Ldx   = @(x, kappa, rho, Neq)get_Hess(dLdx, x, kappa);
+elseif Neq > 0 && Nineq > 0 
+        L       = @(x, kappa) f(x) * kappa(1:Neq)'*g(x) + kappa(Neq + 1:end)'*h(x);
+        dLdx    = @(x, kappa) df(x) + get_grad_lambda_g(kappa(1:Neq), dg, x) + get_grad_mu_h(kappa(Neq+1:end), dh, x);
+        d2Ldx   = @(x, kappa, rho, Neq)get_Hess(dLdx, x, kappa);
+end
 
-% L = @(x, kappa, Neq) f(x) + kappa(1:Neq)'*g(x) + kappa(Neq+1:end)'*h(x);
-% dLdx = @(x, kappa, Neq) df(x)  + ...
-%    get_grad_lambda_g(kappa(1:Neq), dg, x) + ...
-%    get_grad_mu_h(kappa(Neq+1:end), dh, x);
-% d2Ldx = @(x, kappa, rho) get_Hess(dLdx, x, kappa, Neq);
 
-end
 
 %% get gradient of lambda'*g
 function grad_lambda_g = get_grad_lambda_g(lambda, dg, x) 
     grad_lambda_g = sparse(length(x), 1);
     dg_at_x = dg(x);
-    for i = 1 : length(lambda)
+    for i = 1 : size(dg_at_x, 2)
         grad_lambda_g = grad_lambda_g + lambda(i)*dg_at_x(:, i);
     end
 end
@@ -46,14 +60,16 @@
 end
 
 %% get Hessian Function
-function d2Ldx = get_Hess(dLdx, x, kappa, Neq)
-epsilon = 1e-10;
-epsilon_inv = 1/epsilon;
-nx = length(x);
-d2Ldx = zeros(nx, nx);
-for i = 1 : nx
-    dx_i = [ zeros(i-1, 1); 1; zeros(nx-i, 1)];
-    d2Ldx(:, i) = (dLdx(x + epsilon*dx_i, kappa, Neq) ...
-        - dLdx(x - epsilon*dx_i, kappa, Neq)).*0.5.*epsilon_inv;
+function d2Ldx = get_Hess(dLdx, x, kappa)
+    epsilon = 1e-10;
+    epsilon_inv = 1/epsilon;
+    nx = length(x);
+    d2Ldx = zeros(nx, nx);
+    for i = 1 : nx
+        dx_i = [ zeros(i-1, 1); 1; zeros(nx-i, 1)];
+        d2Ldx(:, i) = (dLdx(x + epsilon*dx_i, kappa) ...
+            - dLdx(x - epsilon*dx_i, kappa)).*0.5.*epsilon_inv;
+    end
 end
+
 end
diff --git a/06_opf_extension/build_local_lagrangian_function_old.m b/06_opf_extension/build_local_lagrangian_function_old.m
new file mode 100644
index 0000000..761013b
--- /dev/null
+++ b/06_opf_extension/build_local_lagrangian_function_old.m
@@ -0,0 +1,34 @@
+function [L, dLdx, d2Ldx] = build_local_lagrangian_function_old(f, df, g, dg, h, dh)
+%BUILD_LOCAL_LAGRANGIAN_FUNCTION Bulds the Lagrangian of the local mpc file
+%   kappa is [lambda; mu] with lambda, mu are column vectors
+% dg is the transposed of g, i.e. dimension nx_x \times n_eq 
+% dh is the transposed jacobian of h
+% L = f + lambda'g + mu'h
+% dL = grad_f + lambda'(dg)' + mu'(dh)'
+    if nargin == 4
+        L = @(x, kappa, Neq) f(x) + kappa(1:Neq)'*g(x);
+        dLdx = @(x, kappa, Neq) df(x)  + (kappa(1:Neq)'*dg(x)')';
+        d2Ldx = @(x, kappa, rho, Neq) get_Hess(dLdx, x, kappa(1:Neq), kappa(Neq + 1: end));
+    elseif nargin == 6
+        L = @(x, kappa, Neq) f(x) + kappa(1:Neq)'*g(x) + kappa(Neq+1:end)'*h(x);
+        dLdx = @(x, kappa, Neq) df(x)  + (kappa(1:Neq)'*dg(x)')' + (kappa(Neq+1:end)'*dh(x)')';
+        d2Ldx = @(x, kappa, rho, Neq) get_Hess(dLdx, x, kappa(1:Neq), kappa(Neq + 1: end));
+    end
+end
+
+%% get Hessian Function
+function d2Ldx = get_Hess(dLdx, x, lambda, mu)
+    epsilon = 1e-10;
+    epsilon_inv = 1/epsilon;
+    nx = length(x);
+    d2Ldx = sparse(nx, nx);
+    
+    kappa = [lambda; mu];
+    Neq = numel(lambda);
+    
+    for i = 1 : nx
+        dx_i = [ zeros(i-1, 1); 1; zeros(nx-i, 1)];
+        d2Ldx(:, i) = (dLdx(x + epsilon*dx_i, kappa, Neq) ...
+            - dLdx(x - epsilon*dx_i, kappa, Neq)).*0.5.*epsilon_inv;
+    end
+end
diff --git a/06_opf_extension/build_local_state.m b/06_opf_extension/build_local_state.m
index 349807e..17e8b20 100644
--- a/06_opf_extension/build_local_state.m
+++ b/06_opf_extension/build_local_state.m
@@ -1,4 +1,29 @@
 function state = build_local_state(mpc, names, postfix)
+% BUILD_LOCAL_STATE
+%
+%   `state = build_local_state(mpc, names, postfix)`
+%
+%   _returns symbolic representation of optimization variable_
+%
+%   # Formate
+% Input:
+%
+%   - $\texttt{mpc}$ splitted casefile
+%   - $\texttt{names}$ specific names of mpc struct fields
+%   - $\texttt{copy}$ either $\texttt{core}$ or $\texttt{copy}$
+% Output
+%   - $\texttŧ{state}$ symbolic state
+%
+% Final formate:
+% state = (Vang; Vm; Pg; Qg) (column vector) with 
+% - Vang = (Vang_1; ... ; Vang_{#Nbuses})
+% - Vm = (Vm_1; ... ; Vm_{#Nbuses})
+% - Pg = (Pg_1; ... ; Pg_{#switched_on_generators})
+% - Qg = (Qg_1; ... ; Qg_{#switched_on_generators})
+
+    assert (size(mpc.gen, 1) == size(mpc.gen(:, 8) == 1, 1), ...
+        'mpc file in build_local_state should only contain generators that are switched on');
+    
     Ngen_on = size(mpc.gen, 1);
     Ncopy = numel(mpc.(names.copy_buses.local));
     Ncore = size(mpc.bus, 1) - Ncopy;
diff --git a/06_opf_extension/get_local_variable_from_global_objective_variable.m b/06_opf_extension/get_local_variable_from_global_objective_variable.m
new file mode 100644
index 0000000..533b0e5
--- /dev/null
+++ b/06_opf_extension/get_local_variable_from_global_objective_variable.m
@@ -0,0 +1,47 @@
+function x_local = get_local_variable_from_global_objective_variable(x_global, i_subsystem, mpc_split, names)
+% get local objective variable of subsystem i from global variable of
+% merged system.
+
+% To be debugged!!!
+
+N_subsystems = length(mpc_split.regions);
+
+assert(i_subsystem <= N_subsystems, 'index of subsystem is larger then total number of subsystem');
+
+total_number_of_nodes   = length(horzcat(mpc_split.regions{:}));
+total_number_of_gens    = 0;
+
+for i = 1 : N_subsystems
+    mpc_split.split_case_files{i, 1} = prepare_case_file(mpc_split.split_case_files{i, 1}, names);
+    N_gens{i} = sum(mpc_split.split_case_files{i,1}.gen(:, 8));
+    total_number_of_gens = total_number_of_gens + N_gens{i};
+end
+
+% initizalize raw vectors
+theta_local             = zeros(1, length(mpc_split.connections_with_aux_nodes{i_subsystem}));
+V_local                 = zeros(1, length(mpc_split.connections_with_aux_nodes{i_subsystem}));
+Pg_local                = zeros(1, N_gens{i_subsystem});
+Qg_local                = zeros(1, N_gens{i_subsystem});
+
+gens_before           = 0;
+for i = 1 : i_subsystem - 1
+    gens_before  = gens_before + N_gens{i};
+end
+
+theta_local = x_global(mpc_split.connections_with_aux_nodes{i_subsystem})';
+V_local = x_global(mpc_split.connections_with_aux_nodes{i_subsystem} + total_number_of_nodes)';
+
+% get indices corresponding to Pg
+for i = 1 : length(Pg_local)
+    Pg_local(i) = x_global(i + gens_before + 2*total_number_of_nodes);
+end
+
+% get indices corresponding to Qg
+for i = 1 : length(Qg_local)
+   Qg_local(i) = x_global(i +  gens_before + total_number_of_gens + 2*total_number_of_nodes);
+end
+
+x_local = vertcat(theta_local', V_local', Pg_local', Qg_local');
+
+end
+
diff --git a/06_opf_extension/mpc_merge.m b/06_opf_extension/mpc_merge.m
index 9fb853c..7fa34d0 100644
--- a/06_opf_extension/mpc_merge.m
+++ b/06_opf_extension/mpc_merge.m
@@ -20,12 +20,12 @@
 	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	10	1	0	0	0	0	1	1.1	0	230	1	1.1	0.9;
-	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,14 +35,15 @@
 	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	5	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	15	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
@@ -67,7 +68,7 @@
 	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -84,7 +85,8 @@
 	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
-	2	0	0	2	10	0;
 ];
diff --git a/06_opf_extension/mpc_merge_split.m b/06_opf_extension/mpc_merge_split.m
index d911eeb..209b807 100644
--- a/06_opf_extension/mpc_merge_split.m
+++ b/06_opf_extension/mpc_merge_split.m
@@ -20,12 +20,12 @@
 	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	10	1	0	0	0	0	1	1.1	0	230	1	1.1	0.9;
-	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,14 +35,15 @@
 	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	5	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	15	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
@@ -67,7 +68,7 @@
 	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -84,7 +85,8 @@
 	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
-	2	0	0	2	10	0;
 ];
diff --git a/06_opf_extension/prepare_case_file.m b/06_opf_extension/prepare_case_file.m
index 430066f..82115d8 100644
--- a/06_opf_extension/prepare_case_file.m
+++ b/06_opf_extension/prepare_case_file.m
@@ -4,6 +4,13 @@
 %   `[mpc_opf, om, copy_buses_local, mpopt] = prepare_case_file(mpc, names))`
 %
 %   _Prepares the splitted case file to such that MATPOWER opf functions and methodes can be applied_
+%
+%   INPUT:  - mpc splitted case files
+%           - names names of mpc struct fields
+%   OUTPUT: - mpc_opf cleaned up matpower casefile (deleted generators) in MATPOWER internal numering
+%           - om MATPOWER optimization model of mpc_opf
+%           - copy_buses_local local copy buses
+%           - mpopt MATPOWER optimization parameters
 
     [GEN_BUS, PG, QG, QMAX, QMIN, VG, MBASE, GEN_STATUS, PMAX, PMIN, ...
             MU_PMAX, MU_PMIN, MU_QMAX, MU_QMIN, PC1, PC2, QC1MIN, QC1MAX, ...
@@ -27,10 +34,9 @@
     
     %% we changed the case file after it was switched to internal indexing
     % we need to account for that
-    
-    
+    % ??? option in line 40 safer??? 
     mpc.order.state = 'e';
-   % mpc = int2ext(mpc);
+    % mpc = int2ext(mpc);
     %% return values
     [mpc_opf, mpopt] = opf_args(mpc);
     mpc_opf = ext2int(mpc_opf);
diff --git a/docs/extension_to_opf.html b/docs/extension_to_opf.html
index 6412848..47e4801 100644
--- a/docs/extension_to_opf.html
+++ b/docs/extension_to_opf.html
@@ -10,7 +10,7 @@
 
 </head>
 <body class="markdown-body">
-<h1 id="rapidopf---extension-to-optimal-power-flow" data-line="0" class="code-line">RapidOPF - Extension to Optimal Power Flow</h1>
+<h1 id="rapidopf---extension-of-rapidpf-to-optimal-power-flow" data-line="0" class="code-line">RapidOPF - Extension of RapidPF to Optimal Power Flow</h1>
 <!--!!! warning "Extension to optimal power flow = work in progress"
     __The documentation and code is currently being written.
     It's work in progress.__-->
@@ -47,86 +47,482 @@ <h2 id="subsumption-in-rapidpf" data-line="27" class="code-line">Subsumption in
 <h2 id="extensions-to-existing-code" data-line="38" class="code-line">Extensions to existing Code</h2>
 <p data-line="39" class="code-line">New files:</p>
 <ul>
-<li data-line="41" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">generate_distributed_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{generate\_distributed\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">generate_distributed_opf.m</span></span></span></span></span></li>
-<li data-line="42" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_consensus_matrices_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_consensus\_matrices\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">create_consensus_matrices_opf.m</span></span></span></span></span></li>
-<li data-line="43" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_opf.m</span></span></span></span></span></li>
-<li data-line="44" class="code-line"><a href="#textttpreparecasefilem" data-href="#textttpreparecasefilem"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">prepare_case_file.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{prepare\_case\_file.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">prepare_case_file.m</span></span></span></span></span></a></li>
-<li data-line="45" class="code-line"><a href="#local-decision-variable" data-href="#local-decision-variable"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_state.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_state.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_state.m</span></span></span></span></span></a></li>
-<li data-line="46" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_state_mp</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_state\_mp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">create_state_mp</span></span></span></span></span></li>
-<li data-line="47" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_dims.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_dims.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_dims.m</span></span></span></span></span></li>
-<li data-line="48" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_cost_functon.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_cost\_functon.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_cost_functon.m</span></span></span></span></span></li>
-<li data-line="49" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_equalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_equalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_equalities.m</span></span></span></span></span></li>
-<li data-line="50" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_inequalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_inequalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_inequalities.m</span></span></span></span></span></li>
-<li data-line="51" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_lagrangian_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_lagrangian\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_lagrangian_function.m</span></span></span></span></span></li>
-<li data-line="52" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_initial_condition.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_initial\_condition.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_initial_condition.m</span></span></span></span></span></li>
-<li data-line="53" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_bounds.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_bounds.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_bounds.m</span></span></span></span></span></li>
-</ul>
-<h2 id="preparation-of-splitted-casefile" data-line="56" class="code-line">Preparation of splitted Casefile</h2>
-<p data-line="57" class="code-line">The RapidPF splitted casefiles consist of the following fields:</p>
-<ul>
-<li data-line="58" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">baseMVA</mtext></mrow><annotation encoding="application/x-tex">\texttt{baseMVA}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">baseMVA</span></span></span></span></span></li>
-<li data-line="59" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">bus</mtext></mrow><annotation encoding="application/x-tex">\texttt{bus}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">bus</span></span></span></span></span></li>
-<li data-line="60" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gen</mtext></mrow><annotation encoding="application/x-tex">\texttt{gen}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gen</span></span></span></span></span></li>
-<li data-line="61" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">branch</mtext></mrow><annotation encoding="application/x-tex">\texttt{branch}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">branch</span></span></span></span></span></li>
-<li data-line="62" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">regions</mtext></mrow><annotation encoding="application/x-tex">\texttt{regions}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">regions</span></span></span></span></span></li>
-<li data-line="63" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">connections_with_aux_nodes</mtext></mrow><annotation encoding="application/x-tex">\texttt{connections\_with\_aux\_nodes}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">connections_with_aux_nodes</span></span></span></span></span></li>
-<li data-line="64" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_bues_global</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_bues\_global}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_bues_global</span></span></span></span></span></li>
-<li data-line="65" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_buses_local</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_buses\_local}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_buses_local</span></span></span></span></span></li>
-</ul>
-<p data-line="67" class="code-line">Additionally to these fields, the field - if existing in the original case file -</p>
-<ul>
-<li data-line="68" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gencost</mtext></mrow><annotation encoding="application/x-tex">\texttt{gencost}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gencost</span></span></span></span></span></li>
-</ul>
-<p data-line="70" class="code-line">is transferred by an extension located in the file <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">split_case_file.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{split\_case\_file.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">split_case_file.m</span></span></span></span></span>. The field contains costs for any generator that occurs in the field <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gen</mtext></mrow><annotation encoding="application/x-tex">\texttt{gen}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gen</span></span></span></span></span>.</p>
-<p data-line="72" class="code-line">To use the functions provided by <a href="https://matpower.org/docs/manual.pdf" data-href="https://matpower.org/docs/manual.pdf">MATPOWER</a> we need to know its structure of its objective variables. MATPOWER's optimization vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> for the standard AC OPF proble consists of the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>b</mi></msub><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_b \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> vectors of voltage angles <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Θ</span></span></span></span> and magnitudes <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">V_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>g</mi></msub><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_g \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> vectors of generator real and reactive injections <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">P_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">Q_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>, i.e.</p>
-<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">Θ</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>V</mi><mi>m</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>P</mi><mi>g</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Q</mi><mi>g</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">x = \left[ \begin{array}{c}
-\Theta \\ V_m \\ P_g \\Q_g
-\end{array}\right] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:4.80303em;vertical-align:-2.15003em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Θ</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
-<p data-line="76" class="code-line">Here, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">n_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> corresponds to the number of buses in the system and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">n_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> corresponds to the number of generators in the system.</p>
-<p data-line="78" class="code-line">At this point we need to be careful as the fields <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gen</mtext></mrow><annotation encoding="application/x-tex">\texttt{gen}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gen</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gencost</mtext></mrow><annotation encoding="application/x-tex">\texttt{gencost}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gencost</span></span></span></span></span> also do contain the generators at the copy nodes that shall not have any impact on the local objectives. Switching them of in the casefile as it is done in the function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">prepare_case_file</mtext></mrow><annotation encoding="application/x-tex">\texttt{prepare\_case\_file}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">prepare_case_file</span></span></span></span></span> leads them in MATPOWER to be treated as non existing, thus giving us the wanted effect.</p>
-<h3 id="textttpreparecasefilem" data-line="80" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">prepare_case_file.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{prepare\_case\_file.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">prepare_case_file.m</span></span></span></span></span></h3>
-<p data-line="81" class="code-line">Function to prepare the rapidPF splitted case for OPF. Generators at the copy nodes are switched off and gen entries and gencost entries are deleted from the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> file.
-Input:</p>
+<li data-line="41" class="code-line"><a href="#rapidopf---extension-of-rapidpf-to-optimal-power-flow" data-href="#rapidopf---extension-of-rapidpf-to-optimal-power-flow">RapidOPF - Extension of RapidPF to Optimal Power Flow</a>
+<ul>
+<li data-line="42" class="code-line"><a href="#opf-problem-formulation" data-href="#opf-problem-formulation">OPF Problem Formulation</a></li>
+<li data-line="43" class="code-line"><a href="#subsumption-in-rapidpf" data-href="#subsumption-in-rapidpf">Subsumption in RapidPf</a></li>
+<li data-line="44" class="code-line"><a href="#extensions-to-existing-code" data-href="#extensions-to-existing-code">Extensions to existing Code</a></li>
+<li data-line="45" class="code-line"><a href="#generation-of-distributed-opf" data-href="#generation-of-distributed-opf">Generation of distributed opf</a></li>
+<li data-line="46" class="code-line"><a href="#preparation-of-splitted-casefile" data-href="#preparation-of-splitted-casefile">Preparation of splitted Casefile</a>
+<ul>
+<li data-line="47" class="code-line"><a href="#corresponding-functions" data-href="#corresponding-functions">Corresponding functions:</a></li>
+</ul>
+</li>
+<li data-line="48" class="code-line"><a href="#local-decision-variable" data-href="#local-decision-variable">Local Decision Variable</a>
+<ul>
+<li data-line="49" class="code-line"><a href="#corresponding-functions-1" data-href="#corresponding-functions-1">Corresponding functions</a></li>
+</ul>
+</li>
+<li data-line="50" class="code-line"><a href="#local-cost-function" data-href="#local-cost-function">Local cost function</a>
+<ul>
+<li data-line="51" class="code-line"><a href="#corresponding-functions-2" data-href="#corresponding-functions-2">Corresponding functions</a></li>
+</ul>
+</li>
+<li data-line="52" class="code-line"><a href="#local-equality-and-inequality-constraints" data-href="#local-equality-and-inequality-constraints">Local equality and inequality constraints</a>
+<ul>
+<li data-line="53" class="code-line"><a href="#corresponding-functions-3" data-href="#corresponding-functions-3">Corresponding functions</a></li>
+</ul>
+</li>
+<li data-line="54" class="code-line"><a href="#lagrange-function" data-href="#lagrange-function">Lagrange function</a>
+<ul>
+<li data-line="55" class="code-line"><a href="#corresponding-functions-4" data-href="#corresponding-functions-4">Corresponding functions</a></li>
+</ul>
+</li>
+<li data-line="56" class="code-line"><a href="#consensus-matrices" data-href="#consensus-matrices">Consensus Matrices</a>
 <ul>
-<li data-line="83" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc</span></span></span></span></span> - splitted rapidPF case file</li>
-<li data-line="84" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">names</mtext></mrow><annotation encoding="application/x-tex">\texttt{names}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">names</span></span></span></span></span> - struct that contains the names of the mpc file</li>
+<li data-line="57" class="code-line"><a href="#corresponding-functions-5" data-href="#corresponding-functions-5">Corresponding functions</a></li>
 </ul>
-<p data-line="86" class="code-line">Output:</p>
+</li>
+<li data-line="58" class="code-line"><a href="#dimensions-check-up" data-href="#dimensions-check-up">Dimensions check up</a>
 <ul>
-<li data-line="87" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> - splitted rapidOPF case file</li>
-<li data-line="88" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> - MATPOWER optimization model of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> that is needed to obtain the optimization functions</li>
-<li data-line="89" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> - prepared rapiOPF casefile</li>
-<li data-line="90" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_buses_local</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_buses\_local}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_buses_local</span></span></span></span></span> - vector of local copy buses</li>
-<li data-line="91" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpopt</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpopt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpopt</span></span></span></span></span> - standard MATPOWER opf options</li>
+<li data-line="59" class="code-line"><a href="#corresponding-functions-6" data-href="#corresponding-functions-6">Corresponding functions</a></li>
 </ul>
-<p data-line="93" class="code-line">Tests:</p>
+</li>
+<li data-line="60" class="code-line"><a href="#initial-conditions-and-bounds" data-href="#initial-conditions-and-bounds">Initial conditions and bounds</a>
+<ul>
+<li data-line="61" class="code-line"><a href="#corresponding-functions-7" data-href="#corresponding-functions-7">Corresponding functions</a></li>
+</ul>
+</li>
+<li data-line="62" class="code-line"><a href="#detailed-documentation-of-new-functions" data-href="#detailed-documentation-of-new-functions">Detailed documentation of new functions</a>
+<ul>
+<li data-line="63" class="code-line"><a href="#textttpreparecasefilem" data-href="#textttpreparecasefilem"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">prepare_case_file.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{prepare\_case\_file.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">prepare_case_file.m</span></span></span></span></span></a></li>
+<li data-line="64" class="code-line"><a href="#textttbuildlocalstatem" data-href="#textttbuildlocalstatem"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_state.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_state.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_state.m</span></span></span></span></span></a></li>
+<li data-line="65" class="code-line"><a href="#textttcreatestatempm" data-href="#textttcreatestatempm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_state_mp.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_state\_mp.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">create_state_mp.m</span></span></span></span></span></a></li>
+<li data-line="66" class="code-line"><a href="#textttbuildlocalcostfunctionm" data-href="#textttbuildlocalcostfunctionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_cost_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_cost\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_cost_function.m</span></span></span></span></span></a></li>
+<li data-line="67" class="code-line"><a href="#textttbuildlocalconstraintfunctionm" data-href="#textttbuildlocalconstraintfunctionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_constraint_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_constraint\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_constraint_function.m</span></span></span></span></span></a></li>
+<li data-line="68" class="code-line"><a href="#textttbuildlocalequalitiesm" data-href="#textttbuildlocalequalitiesm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_equalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_equalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_equalities.m</span></span></span></span></span></a></li>
+<li data-line="69" class="code-line"><a href="#textttbuildlocalinequalitiesm" data-href="#textttbuildlocalinequalitiesm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_inequalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_inequalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_inequalities.m</span></span></span></span></span></a></li>
+<li data-line="70" class="code-line"><a href="#textttbuildlocallagrangianfunctionm" data-href="#textttbuildlocallagrangianfunctionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_lagrangian_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_lagrangian\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_lagrangian_function.m</span></span></span></span></span></a></li>
+<li data-line="71" class="code-line"><a href="#textttbuildlocalinitialcondistionm" data-href="#textttbuildlocalinitialcondistionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_initial_condistion.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_initial\_condistion.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_initial_condistion.m</span></span></span></span></span></a></li>
+<li data-line="72" class="code-line"><a href="#textttbuildlocalboundsm" data-href="#textttbuildlocalboundsm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_bounds.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_bounds.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_bounds.m</span></span></span></span></span></a></li>
+<li data-line="73" class="code-line"><a href="#textttbuildlocaldimensionsm" data-href="#textttbuildlocaldimensionsm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_dimensions.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_dimensions.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_dimensions.m</span></span></span></span></span></a></li>
+<li data-line="74" class="code-line"><a href="#textttcreateconsensusmatricesm" data-href="#textttcreateconsensusmatricesm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_consensus_matrices.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_consensus\_matrices.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">create_consensus_matrices.m</span></span></span></span></span></a></li>
+<li data-line="75" class="code-line"><a href="#textttgeneratedistributedopfm" data-href="#textttgeneratedistributedopfm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">generate_distributed_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{generate\_distributed\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">generate_distributed_opf.m</span></span></span></span></span></a></li>
+<li data-line="76" class="code-line"><a href="#textttbuildlocalopfm" data-href="#textttbuildlocalopfm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_opf.m</span></span></span></span></span></a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+<h2 id="generation-of-distributed-opf" data-line="80" class="code-line">Generation of distributed opf</h2>
+<p data-line="81" class="code-line">From the casefiles, all information needed for the distributed solver need to be extracted from the casefile of splitted casefiles. To do so, several functions are called within the fuctions</p>
+<ul>
+<li data-line="82" class="code-line"><a href="#textttgeneratedistributedopfm" data-href="#textttgeneratedistributedopfm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">generate_distributed_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{generate\_distributed\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">generate_distributed_opf.m</span></span></span></span></span></a></li>
+<li data-line="83" class="code-line"><a href="" data-href=""><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_opf.m</span></span></span></span></span></a></li>
+</ul>
+<h2 id="preparation-of-splitted-casefile" data-line="84" class="code-line">Preparation of splitted Casefile</h2>
+<p data-line="85" class="code-line">The RapidPF splitted casefiles consist of the following fields:</p>
+<ul>
+<li data-line="86" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">baseMVA</mtext></mrow><annotation encoding="application/x-tex">\texttt{baseMVA}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">baseMVA</span></span></span></span></span></li>
+<li data-line="87" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">bus</mtext></mrow><annotation encoding="application/x-tex">\texttt{bus}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">bus</span></span></span></span></span></li>
+<li data-line="88" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gen</mtext></mrow><annotation encoding="application/x-tex">\texttt{gen}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gen</span></span></span></span></span></li>
+<li data-line="89" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">branch</mtext></mrow><annotation encoding="application/x-tex">\texttt{branch}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">branch</span></span></span></span></span></li>
+<li data-line="90" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">regions</mtext></mrow><annotation encoding="application/x-tex">\texttt{regions}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">regions</span></span></span></span></span></li>
+<li data-line="91" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">connections_with_aux_nodes</mtext></mrow><annotation encoding="application/x-tex">\texttt{connections\_with\_aux\_nodes}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">connections_with_aux_nodes</span></span></span></span></span></li>
+<li data-line="92" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_bues_global</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_bues\_global}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_bues_global</span></span></span></span></span></li>
+<li data-line="93" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_buses_local</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_buses\_local}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_buses_local</span></span></span></span></span></li>
+</ul>
+<p data-line="95" class="code-line">Additionally to these fields, the field - if existing in the original case file -</p>
+<ul>
+<li data-line="96" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gencost</mtext></mrow><annotation encoding="application/x-tex">\texttt{gencost}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gencost</span></span></span></span></span></li>
+</ul>
+<p data-line="98" class="code-line">is transferred by an extension located in the file <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">split_case_file.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{split\_case\_file.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">split_case_file.m</span></span></span></span></span>. The field contains costs for any generator that occurs in the field <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gen</mtext></mrow><annotation encoding="application/x-tex">\texttt{gen}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gen</span></span></span></span></span>.</p>
+<p data-line="100" class="code-line">To use the functions provided by <a href="https://matpower.org/docs/manual.pdf" data-href="https://matpower.org/docs/manual.pdf">MATPOWER</a> we need to know its structure of its objective variables. MATPOWER's optimization vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> for the standard AC OPF proble consists of the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>b</mi></msub><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_b \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> vectors of voltage angles <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Θ</span></span></span></span> and magnitudes <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">V_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>g</mi></msub><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_g \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> vectors of generator real and reactive injections <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">P_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">Q_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>, i.e.</p>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">Θ</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>V</mi><mi>m</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>P</mi><mi>g</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Q</mi><mi>g</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">x = \left[ \begin{array}{c}
+\Theta \\ V_m \\ P_g \\Q_g
+\end{array}\right] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:4.80303em;vertical-align:-2.15003em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Θ</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
+<p data-line="104" class="code-line">Here, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">n_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> corresponds to the number of buses in the system and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mi>g</mi></msub></mrow><annotation encoding="application/x-tex">n_g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> corresponds to the number of generators in the system.</p>
+<p data-line="106" class="code-line">At this point we need to be careful as the fields <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gen</mtext></mrow><annotation encoding="application/x-tex">\texttt{gen}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gen</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">gencost</mtext></mrow><annotation encoding="application/x-tex">\texttt{gencost}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">gencost</span></span></span></span></span> also do contain the generators at the copy nodes that shall not have any impact on the local objectives. Switching them of in the casefile as it is done in the function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">prepare_case_file</mtext></mrow><annotation encoding="application/x-tex">\texttt{prepare\_case\_file}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">prepare_case_file</span></span></span></span></span> leads them in MATPOWER to be treated as non existing, thus giving us the wanted effect.</p>
+<h3 id="corresponding-functions" data-line="108" class="code-line">Corresponding functions:</h3>
 <ul>
-<li data-line="94" class="code-line">added sample test to 05_UI_test</li>
-<li data-line="95" class="code-line">added assert to check whether there are still gen entries left in local <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> case file.</li>
+<li data-line="109" class="code-line"><a href="#textttpreparecasefilem" data-href="#textttpreparecasefilem"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">prepare_case_file.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{prepare\_case\_file.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">prepare_case_file.m</span></span></span></span></span></a></li>
 </ul>
-<h2 id="local-decision-variable" data-line="97" class="code-line">Local Decision Variable</h2>
-<p data-line="98" class="code-line">Each system has a local decision variable. We need the following values:</p>
+<h2 id="local-decision-variable" data-line="112" class="code-line">Local Decision Variable</h2>
+<p data-line="113" class="code-line">Each system has a local decision variable. We need the following values:</p>
 <ul>
-<li data-line="99" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">N_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> number of buses in splitted system</li>
-<li data-line="100" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">N_{b-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span> number of core-buses in splitted system</li>
-<li data-line="101" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">N_{b-copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> number of copy-busus in splitted system</li>
-<li data-line="102" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">N_{g-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> number of core-generators in spliited system</li>
+<li data-line="114" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">N_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> number of buses in splitted system</li>
+<li data-line="115" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">N_{b-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span> number of core-buses in splitted system</li>
+<li data-line="116" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">N_{b-copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> number of copy-busus in splitted system</li>
+<li data-line="117" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">N_{g-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span> number of core-generators in spliited system</li>
 </ul>
-<p data-line="104" class="code-line">Then <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> writes as</p>
+<p data-line="119" class="code-line">Then <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> writes as</p>
 <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup><mo separator="true">,</mo><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mo separator="true">,</mo><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup><mo separator="true">,</mo><mi>P</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mo separator="true">,</mo><mi>Q</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><msup><mo stretchy="false">)</mo><mi mathvariant="normal">⊤</mi></msup></mrow><annotation encoding="application/x-tex">x_i = (\theta_i^{core}, \theta_i^{copy}, Vm_i^{core}, Vm_i^{copy}, Pg_i^{core}, Qg_i^{core})^\top</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1759719999999998em;vertical-align:-0.276864em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">Q</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">⊤</span></span></span></span></span></span></span></span></span></span></span></span></p>
-<p data-line="106" class="code-line">with</p>
+<p data-line="121" class="code-line">with</p>
 <ul>
-<li data-line="107" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.953104em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span>  voltage angles at the core buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">| \theta_i^{core} | = N_{b-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></li>
-<li data-line="108" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span></span>  voltage angles at the copy buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">| \theta_i^{copy} | = N_{b-copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
-<li data-line="109" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Vm_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.941994em;vertical-align:-0.258664em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> voltage magnitudes at the core buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Vm_i^{core}| =  N_{b-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></li>
-<li data-line="110" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Vm_i^{copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span></span> voltage magnitudes at the copy buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Vm_i^{copy}| =  N_{b-copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
-<li data-line="111" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Pg_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.941994em;vertical-align:-0.258664em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> active powers at all generators placed at the core nodes with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>P</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Pg_i^{core}| =  N_{g-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
-<li data-line="112" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Qg_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.941994em;vertical-align:-0.258664em;"></span><span class="mord mathdefault">Q</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> reactive powers at all generators placed at the core nodes with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>Q</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Qg_i^{core}| =  N_{g-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord mathdefault">Q</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li data-line="122" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.953104em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span>  voltage angles at the core buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">| \theta_i^{core} | = N_{b-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li data-line="123" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span></span>  voltage angles at the copy buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">| \theta_i^{copy} | = N_{b-copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li data-line="124" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Vm_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.941994em;vertical-align:-0.258664em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> voltage magnitudes at the core buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Vm_i^{core}| =  N_{b-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li data-line="125" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Vm_i^{copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span></span> voltage magnitudes at the copy buses with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>V</mi><msubsup><mi>m</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Vm_i^{copy}| =  N_{b-copy}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em;"></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li data-line="126" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Pg_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.941994em;vertical-align:-0.258664em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> active powers at all generators placed at the core nodes with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>P</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Pg_i^{core}| =  N_{g-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
+<li data-line="127" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Qg_i^{core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.941994em;vertical-align:-0.258664em;"></span><span class="mord mathdefault">Q</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span> reactive powers at all generators placed at the core nodes with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>Q</mi><msubsup><mi>g</mi><mi>i</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msubsup><mi mathvariant="normal">∣</mi><mo>=</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">|Qg_i^{core}| =  N_{g-core}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008664em;vertical-align:-0.258664em;"></span><span class="mord">∣</span><span class="mord mathdefault">Q</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></li>
 </ul>
-<p data-line="114" class="code-line">Thus the total dimension of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is</p>
+<p data-line="129" class="code-line">Thus the total dimension of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is</p>
 <p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>dim</mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub><mo>+</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>b</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub><mo>+</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mi>b</mi></msub><mo>+</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} 
 \text{dim}(x_i) &amp; = 2\cdot N_{b-core} + 2\cdot N_{b-copy} + 2\cdot N_{g-core} \\ &amp; = 2\cdot N_b + 2 \cdot N_{g-core}
 \end{aligned}
 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000000000000004em;vertical-align:-1.2500000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">dim</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
+<h3 id="corresponding-functions-1" data-line="135" class="code-line">Corresponding functions</h3>
+<ul>
+<li data-line="136" class="code-line"><a href="#textttbuildlocalstatem" data-href="#textttbuildlocalstatem"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_state.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_state.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_state.m</span></span></span></span></span></a></li>
+<li data-line="137" class="code-line"><a href="#textttcreatestatempm" data-href="#textttcreatestatempm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_state_mp.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_state\_mp.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">create_state_mp.m</span></span></span></span></span></a></li>
+</ul>
+<h2 id="local-cost-function" data-line="139" class="code-line">Local cost function</h2>
+<p data-line="140" class="code-line">To calculate the cost function for rapidOPF, the MATPOWER function <a href="https://matpower.org/docs/ref/matpower5.0/opf_costfcn.html" data-href="https://matpower.org/docs/ref/matpower5.0/opf_costfcn.html"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">opf_costfcn.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{opf\_costfcn.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">opf_costfcn.m</span></span></span></span></span></a>. The call of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">opf_costfcn(x,om)</mtext></mrow><annotation encoding="application/x-tex">\texttt{opf\_costfcn(x,om)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">opf_costfcn(x,om)</span></span></span></span></span> returns</p>
+<ul>
+<li data-line="141" class="code-line">the evaluation of the cost function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> ( dimension <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>)</li>
+<li data-line="142" class="code-line">the evaluation of the gradient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">\nabla f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∇</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span> of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> (dimension <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>length</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{length}(x)\times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">length</span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>) and</li>
+<li data-line="143" class="code-line">the evaluation of the hessian matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi></mrow><annotation encoding="application/x-tex">\nabla^2 f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span> of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> (dimension <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>length</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>×</mo><mtext>length</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{length}(x) \times \text{length}(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">length</span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">length</span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>)</li>
+</ul>
+<p data-line="145" class="code-line">with respect to the structure of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> illustrated above.</p>
+<p data-line="147" class="code-line">The results are one to one used by rapidOPF</p>
+<h3 id="corresponding-functions-2" data-line="149" class="code-line">Corresponding functions</h3>
+<ul>
+<li data-line="150" class="code-line"><a href="#textttbuildlocalcostfunctionm" data-href="#textttbuildlocalcostfunctionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_cost_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_cost\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_cost_function.m</span></span></span></span></span></a></li>
+</ul>
+<h2 id="local-equality-and-inequality-constraints" data-line="153" class="code-line">Local equality and inequality constraints</h2>
+<p data-line="154" class="code-line">To calculate the local equality and inequality constraints for rapidOPF, the MATPOWER function <a href="https://matpower.org/docs/ref/matpower5.0/opf_consfcn.html" data-href="https://matpower.org/docs/ref/matpower5.0/opf_consfcn.html"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">opf_consfcn.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{opf\_consfcn.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">opf_consfcn.m</span></span></span></span></span></a> is used. We recall the output formate of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">opf_consfcn(x,</mtext><mtext> </mtext><mtext mathvariant="monospace">om,</mtext><mtext> </mtext><mtext mathvariant="monospace">Ybus,</mtext><mtext> </mtext><mtext mathvariant="monospace">Yf(il,:),</mtext><mtext> </mtext><mtext mathvariant="monospace">Yt(il,:),</mtext><mtext> </mtext><mtext mathvariant="monospace">mpopt,</mtext><mtext> </mtext><mtext mathvariant="monospace">il)</mtext></mrow><annotation encoding="application/x-tex">\texttt{opf\_consfcn(x, om, Ybus, Yf(il,:), Yt(il,:), mpopt, il)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">opf_consfcn(x, om, Ybus, Yf(il,:), Yt(il,:), mpopt, il)</span></span></span></span></span>:</p>
+<ul>
+<li data-line="155" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">h(x)</mtext></mrow><annotation encoding="application/x-tex">\texttt{h(x)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord text"><span class="mord texttt">h(x)</span></span></span></span></span> inequality constraints at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> of dim Nineq <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">×</span><span class="mord">1</span></span></span></span></li>
+<li data-line="156" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">g(x)</mtext></mrow><annotation encoding="application/x-tex">\texttt{g(x)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">g(x)</span></span></span></span></span> equality constraints at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> of dim Neq <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">×</span><span class="mord">1</span></span></span></span></li>
+<li data-line="157" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dh(x)</mtext></mrow><annotation encoding="application/x-tex">\texttt{dh(x)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord text"><span class="mord texttt">dh(x)</span></span></span></span></span> transposed of the jacobian of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">h</span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> of dimension dim Nx <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">×</span></span></span></span> dim Nineq, i.e.</li>
+</ul>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>h</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>h</mi><mrow><mi>n</mi><mi>g</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>h</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mrow><mi>n</mi><mi>x</mi></mrow></msub></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>h</mi><mrow><mi>n</mi><mi>g</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mrow><mi>n</mi><mi>x</mi></mrow></msub></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">
+\left[ \begin{array}{ccc}
+\frac{\partial h_1}{\partial x_1} &amp; \cdots &amp; \frac{\partial h_{ng}}{\partial x_{1}} \\ \vdots &amp; \ddots &amp; \vdots \\ 
+\frac{\partial h_{1}}{\partial x_{nx}} &amp; \cdots &amp; \frac{\partial h_{ng}}{\partial x_{nx}}
+\end{array}\right] 
+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.80303em;vertical-align:-2.15003em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.618528em;"><span style="top:-5.3126em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8962079999999999em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.3674999999999997em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.0140719999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8962079999999999em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1185280000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.618528em;"><span style="top:-5.1251em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.1799999999999997em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.8265719999999996em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1185280000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.618528em;"><span style="top:-5.3126em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.993428em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.50732em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285716em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.3674999999999997em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.0140719999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.993428em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.50732em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285716em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1185280000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
+<ul>
+<li data-line="164" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dg(x)</mtext></mrow><annotation encoding="application/x-tex">\texttt{dg(x)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">dg(x)</span></span></span></span></span> transposed of jacobian of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span></span></span></span> at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> of dimension dim Nx <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">×</span></span></span></span> dim Neq, i.e.</li>
+</ul>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>g</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>g</mi><mrow><mi>n</mi><mi>g</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>g</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mrow><mi>n</mi><mi>x</mi></mrow></msub></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>g</mi><mrow><mi>n</mi><mi>g</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mrow><mi>n</mi><mi>x</mi></mrow></msub></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">
+\left[ \begin{array}{ccc}
+\frac{\partial g_1}{\partial x_1} &amp; \cdots &amp; \frac{\partial g_{ng}}{\partial x_{1}} \\ \vdots &amp; \ddots &amp; \vdots \\ 
+\frac{\partial g_{1}}{\partial x_{nx}} &amp; \cdots &amp; \frac{\partial g_{ng}}{\partial x_{nx}}
+\end{array}\right] 
+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.80303em;vertical-align:-2.15003em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.618528em;"><span style="top:-5.3126em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322159999999999em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.446108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.3674999999999997em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.0140719999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322159999999999em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.446108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1185280000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.618528em;"><span style="top:-5.1251em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.1799999999999997em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.8265719999999996em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1185280000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.618528em;"><span style="top:-5.3126em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.993428em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.50732em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285716em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.3674999999999997em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-2.0140719999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.993428em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.50732em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285716em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1185280000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6529999999999996em;"><span style="top:-1.6499900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.79999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3959900000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.4119800000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.653em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.15003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
+<p data-line="172" class="code-line">The function handles <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo separator="true">,</mo><mi>d</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">h, dh</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">h</span></span></span></span> are taken as they are. For <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span></span></span></span>, the entries of the copy buses are deleted, for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">dg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span></span></span></span>, the columns corresponding to the derivatives of the power flow corresponding to the copy buses are deleted</p>
+<h3 id="corresponding-functions-3" data-line="174" class="code-line">Corresponding functions</h3>
+<ul>
+<li data-line="175" class="code-line"><a href="#textttbuildlocalconstraintfunctionm" data-href="#textttbuildlocalconstraintfunctionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_constraint_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_constraint\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_constraint_function.m</span></span></span></span></span></a></li>
+<li data-line="176" class="code-line"><a href="#textttbuildlocalequalitiesm" data-href="#textttbuildlocalequalitiesm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_equalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_equalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_equalities.m</span></span></span></span></span></a></li>
+<li data-line="177" class="code-line"><a href="#textttbuildlocalinequalitiesm" data-href="#textttbuildlocalinequalitiesm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_inequalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_inequalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_inequalities.m</span></span></span></span></span></a></li>
+</ul>
+<h2 id="lagrange-function" data-line="179" class="code-line">Lagrange function</h2>
+<p data-line="180" class="code-line">The ALADIN Solver needs to be handed over a Lagrangian function and its hessian matrix with respect to x</p>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo separator="true">,</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>λ</mi><mi mathvariant="normal">.</mi><mo>∗</mo><mi mathvariant="normal">∇</mi><mi>g</mi><mo>+</mo><mi>μ</mi><mi mathvariant="normal">.</mi><mo>∗</mo><mi mathvariant="normal">∇</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">
+L(x, \lambda, \mu) = f(x) + \lambda.*\nabla g + \mu .* \nabla h
+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">λ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">μ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">λ</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">∇</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.6597200000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">μ</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">∇</span><span class="mord mathdefault">h</span></span></span></span></span></p>
+<p data-line="184" class="code-line">These functions are provided by rapidOPF.</p>
+<h3 id="corresponding-functions-4" data-line="186" class="code-line">Corresponding functions</h3>
+<p data-line="187" class="code-line"><a href="#textttbuildlocallagrangianfunctionm" data-href="#textttbuildlocallagrangianfunctionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_lagrangian_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_lagrangian\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_lagrangian_function.m</span></span></span></span></span></a></p>
+<h2 id="consensus-matrices" data-line="189" class="code-line">Consensus Matrices</h2>
+<p data-line="190" class="code-line">As consensus matrices we call the matrices that guarantee that the voltage angles and voltage magnitudes at the copy nodes correspond to the voltage angles and magnitudes at the corresponding core nodes in their core system. Therefore, the numbers of connections between all systems are needed. The local matrices are of dimension</p>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo>∗</mo><msub><mi>N</mi><mrow><mi>c</mi><mi>o</mi><mi>n</mi><mi>n</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>−</mo><mi>b</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>×</mo><mtext>dim</mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">
+4*N_{connection-buses} \times \text{dim}(x_i)
+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dim</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
+<p data-line="194" class="code-line">To fill the consensus matrix, for each system information is needed about the the indices of the local core nodes and the local copy nodes. They are given in the form of a connection table- it consists of four columns that are filled for each connection the another system for each system:</p>
+<ul>
+<li data-line="195" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">orig_system</mtext></mrow><annotation encoding="application/x-tex">\texttt{orig\_system}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">orig_system</span></span></span></span></span> global index of core system system</li>
+<li data-line="196" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_system</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_system}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_system</span></span></span></span></span> global index of copy system</li>
+<li data-line="197" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">orig_bus_loacl</mtext></mrow><annotation encoding="application/x-tex">\texttt{orig\_bus\_loacl}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">orig_bus_loacl</span></span></span></span></span> local node index of core node that is connected to another copy node</li>
+<li data-line="198" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_bus_local</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_bus\_local}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_bus_local</span></span></span></span></span> local node index of copy node connected to the core node from column 3
+Recall that</li>
+</ul>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo>∗</mo><msub><mi>N</mi><mrow><mi>c</mi><mi>o</mi><mi>n</mi><mi>n</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo>−</mo><mi>b</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>×</mo><mtext>dim</mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>∗</mo><mi>s</mi><mi>i</mi><mi>z</mi><mi>e</mi><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>n</mi><mi>n</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi mathvariant="normal">_</mi><mi>t</mi><mi>a</mi><mi>b</mi><mi>l</mi><mi>e</mi><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">
+4*N_{connection-buses} \times \text{dim}(x_i) = 2 * size(connection\_table, 1)
+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dim</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.06em;vertical-align:-0.31em;"></span><span class="mord mathdefault">s</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mord mathdefault">e</span><span class="mopen">(</span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">n</span><span class="mord mathdefault">n</span><span class="mord mathdefault">e</span><span class="mord mathdefault">c</span><span class="mord mathdefault">t</span><span class="mord mathdefault">i</span><span class="mord mathdefault">o</span><span class="mord mathdefault">n</span><span class="mord" style="margin-right:0.02778em;">_</span><span class="mord mathdefault">t</span><span class="mord mathdefault">a</span><span class="mord mathdefault">b</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p>
+<p data-line="203" class="code-line">To enforce consensus, for each row <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span> of the connection table two entries are generated:</p>
+<ul>
+<li data-line="204" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi><mi mathvariant="normal">_</mi><mi>b</mi><mi>u</mi><mi>s</mi><mi mathvariant="normal">_</mi><mi>l</mi><mi>o</mi><mi>c</mi><mi>a</mi><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A_{core}(i, core\_bus\_local) = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.06em;vertical-align:-0.31em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">e</span><span class="mord" style="margin-right:0.02778em;">_</span><span class="mord mathdefault">b</span><span class="mord mathdefault">u</span><span class="mord mathdefault">s</span><span class="mord" style="margin-right:0.02778em;">_</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault">c</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></li>
+<li data-line="205" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi><mi mathvariant="normal">_</mi><mi>b</mi><mi>u</mi><mi>s</mi><mi mathvariant="normal">_</mi><mi>l</mi><mi>o</mi><mi>c</mi><mi>a</mi><mi>l</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A_{copy}(i, copy\_bus\_local) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.06em;vertical-align:-0.31em;"></span><span class="mord"><span class="mord mathdefault">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">p</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord" style="margin-right:0.02778em;">_</span><span class="mord mathdefault">b</span><span class="mord mathdefault">u</span><span class="mord mathdefault">s</span><span class="mord" style="margin-right:0.02778em;">_</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault">c</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></li>
+</ul>
+<p data-line="207" class="code-line">We notice, that we need such a matrix not only for voltage angles but also one such matrix for voltage magnitudes, where the core_bus_entry and the copy_bus_entry need to be shifted to the column corresponding to the voltage entry of the optimization variable.</p>
+<h3 id="corresponding-functions-5" data-line="209" class="code-line">Corresponding functions</h3>
+<ul>
+<li data-line="210" class="code-line"><a href="#textttcreateconsensusmatricesm" data-href="#textttcreateconsensusmatricesm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_consensus_matrices.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_consensus\_matrices.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">create_consensus_matrices.m</span></span></span></span></span></a></li>
+</ul>
+<h2 id="dimensions-check-up" data-line="212" class="code-line">Dimensions check up</h2>
+<p data-line="213" class="code-line">For clarity, the dimensions of the outputs are summarized</p>
+<ul>
+<li data-line="215" class="code-line">The dimension of the optimization vector is given by</li>
+</ul>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>dim</mtext><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi><mi>s</mi><mi>e</mi><mi>s</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub><mo>+</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi><mi>s</mi><mi>e</mi><mi>s</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub><mo>+</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>+</mo><mn>2</mn><mo>⋅</mo><msub><mi>N</mi><mrow><mi>g</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} 
+\text{dim}(x_i) &amp; = 2\cdot N_{busses-core} + 2\cdot N_{busses-copy} + 2\cdot N_{g-core} \\ &amp; = 2\cdot N_{buses} + 2 \cdot N_{g-core}
+\end{aligned}
+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000000000000004em;vertical-align:-1.2500000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">dim</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
+<ul>
+<li data-line="221" class="code-line">The number of equality constraints is equal to</li>
+</ul>
+<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>dim</mtext><mo stretchy="false">(</mo><mi>e</mi><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>N</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>s</mi></mrow></msub><mo>−</mo><msub><mi>N</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>s</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>p</mi><mi>y</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo>∗</mo><mo stretchy="false">(</mo><msub><mi>N</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi><mi>e</mi><mi>s</mi><mo>−</mo><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{dim}(eq) = 2 * (N_{buses} - N_{buses-copy}) = 2*(N_{buses-core})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dim</span></span><span class="mopen">(</span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02778em;">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
+<ul>
+<li data-line="224" class="code-line">
+<p data-line="224" class="code-line">The number of inequality constraints is equal to the number of non-zero entries of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">RATE_A</mtext></mrow><annotation encoding="application/x-tex">\texttt{RATE\_A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">RATE_A</span></span></span></span></span> in the branch specifications of the splitted opf case file</p>
+</li>
+<li data-line="226" class="code-line">
+<p data-line="226" class="code-line">The local cost functions are scalar valued, the gradients are of dimension <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">length</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\texttt{length}(x) \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord texttt">length</span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>, its Hessian is of dimension <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">length</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>×</mo><mtext mathvariant="monospace">length</mtext><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\texttt{length}(x) \times \texttt{length}(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord texttt">length</span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord texttt">length</span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span></p>
+</li>
+</ul>
+<h3 id="corresponding-functions-6" data-line="228" class="code-line">Corresponding functions</h3>
+<ul>
+<li data-line="229" class="code-line"><a href="#textttbuildlocaldimensionsm" data-href="#textttbuildlocaldimensionsm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_dimensions.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_dimensions.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_dimensions.m</span></span></span></span></span></a></li>
+</ul>
+<h2 id="initial-conditions-and-bounds" data-line="231" class="code-line">Initial conditions and bounds</h2>
+<p data-line="232" class="code-line">The initial conditions and bounds are again one by one taken by MATPOWER functions <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">params_var(om).m</mtext></mrow><annotation encoding="application/x-tex">\texttt{params\_var(om).m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">params_var(om).m</span></span></span></span></span>.</p>
+<h3 id="corresponding-functions-7" data-line="234" class="code-line">Corresponding functions</h3>
+<ul>
+<li data-line="235" class="code-line">
+<p data-line="235" class="code-line"><a href="#textttbuildlocalinitialcondistionm" data-href="#textttbuildlocalinitialcondistionm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_initial_condistion.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_initial\_condistion.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_initial_condistion.m</span></span></span></span></span></a></p>
+</li>
+<li data-line="237" class="code-line">
+<p data-line="237" class="code-line"><a href="#textttbuildlocalboundsm" data-href="#textttbuildlocalboundsm"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_bounds.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_bounds.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_bounds.m</span></span></span></span></span></a></p>
+</li>
+</ul>
+<h2 id="detailed-documentation-of-new-functions" data-line="239" class="code-line">Detailed documentation of new functions</h2>
+<h3 id="textttpreparecasefilem" data-line="241" class="code-line"><a href="#preparation-of-splitted-casefile" data-href="#preparation-of-splitted-casefile"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">prepare_case_file.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{prepare\_case\_file.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">prepare_case_file.m</span></span></span></span></span></a></h3>
+<p data-line="243" class="code-line"><code>[mpc_opf, om, copy_buses_local, mpopt] = prepare_case_file(mpc, names)</code></p>
+<p data-line="245" class="code-line"><em>Function to prepare the rapidPF splitted casefiles for OPF. Generators at the copy nodes are switched off and gen entries and gencost entries are deleted from the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> file.</em>
+Input:</p>
+<ul>
+<li data-line="247" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc</span></span></span></span></span> - splitted rapidPF case file</li>
+<li data-line="248" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">names</mtext></mrow><annotation encoding="application/x-tex">\texttt{names}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">names</span></span></span></span></span> - struct that contains the names of the mpc file</li>
+</ul>
+<p data-line="250" class="code-line">Output:</p>
+<ul>
+<li data-line="251" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> - prepared splitted rapiOPF casefile</li>
+<li data-line="252" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> - MATPOWER optimization model of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> that is needed to obtain the optimization functions</li>
+<li data-line="253" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">copy_buses_local</mtext></mrow><annotation encoding="application/x-tex">\texttt{copy\_buses\_local}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">copy_buses_local</span></span></span></span></span> - vector of local copy buses</li>
+<li data-line="254" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpopt</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpopt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpopt</span></span></span></span></span> - standard MATPOWER opf options</li>
+</ul>
+<p data-line="256" class="code-line">Tests:</p>
+<ul>
+<li data-line="257" class="code-line">added sample test to 05_UI_test</li>
+<li data-line="258" class="code-line">added assert to check whether there are still gen entries left in local <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> case file.</li>
+</ul>
+<h3 id="textttbuildlocalstatem" data-line="260" class="code-line"><a href="#local-decision-variable" data-href="#local-decision-variable"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_state.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_state.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_state.m</span></span></span></span></span></a></h3>
+<p data-line="261" class="code-line"><code>state = build_local_state(mpc, names, postfix)</code></p>
+<p data-line="263" class="code-line"><em>Function to build symbolic representation of optimization variable</em></p>
+<p data-line="265" class="code-line">Input:</p>
+<ul>
+<li data-line="266" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc</span></span></span></span></span> splitted casefile</li>
+<li data-line="267" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">names</mtext></mrow><annotation encoding="application/x-tex">\texttt{names}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">names</span></span></span></span></span> specific names of mpc struct fields</li>
+<li data-line="268" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">postfix</mtext></mrow><annotation encoding="application/x-tex">\texttt{postfix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">postfix</span></span></span></span></span> number of local system</li>
+</ul>
+<p data-line="270" class="code-line">Output</p>
+<ul>
+<li data-line="271" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">state</mtext></mrow><annotation encoding="application/x-tex">\texttt{state}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.55358em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">state</span></span></span></span></span> symbolic state</li>
+</ul>
+<p data-line="273" class="code-line">Final formate:
+<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">state</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">(Vang;</mtext><mtext> </mtext><mtext mathvariant="monospace">Vm;</mtext><mtext> </mtext><mtext mathvariant="monospace">Pg;</mtext><mtext> </mtext><mtext mathvariant="monospace">Qg)</mtext></mrow><annotation encoding="application/x-tex">\texttt{state = (Vang; Vm; Pg; Qg)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">state = (Vang; Vm; Pg; Qg)</span></span></span></span></span> (column vector) with</p>
+<ul>
+<li data-line="275" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Vang</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[Vang_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Vang_postfix_#Ncorebuses;</mtext><mtext> </mtext><mtext mathvariant="monospace">Vang_postfix_copy_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Vang_postfix_#Ncopybuses]</mtext></mrow><annotation encoding="application/x-tex">\texttt{Vang = [Vang\_postfix\_1; ... ; Vang\_postfix\_{\#Ncorebuses}; Vang\_postfix\_copy\_1; ... ; Vang\_postfix\_{\#Ncopybuses}]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Vang = [Vang_postfix_1; ... ; Vang_postfix_</span><span class="mord"><span class="mord texttt">#</span><span class="mord texttt">N</span><span class="mord texttt">c</span><span class="mord texttt">o</span><span class="mord texttt">r</span><span class="mord texttt">e</span><span class="mord texttt">b</span><span class="mord texttt">u</span><span class="mord texttt">s</span><span class="mord texttt">e</span><span class="mord texttt">s</span></span><span class="mord texttt">; Vang_postfix_copy_1; ... ; Vang_postfix_</span><span class="mord"><span class="mord texttt">#</span><span class="mord texttt">N</span><span class="mord texttt">c</span><span class="mord texttt">o</span><span class="mord texttt">p</span><span class="mord texttt">y</span><span class="mord texttt">b</span><span class="mord texttt">u</span><span class="mord texttt">s</span><span class="mord texttt">e</span><span class="mord texttt">s</span></span><span class="mord texttt">]</span></span></span></span></span></li>
+<li data-line="276" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Vm</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[Vm_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Vm_postfix_#Ncorebuses;</mtext><mtext> </mtext><mtext mathvariant="monospace">Vm_copy_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Vm_postfix_#Ncopybuses]</mtext></mrow><annotation encoding="application/x-tex">\texttt{Vm = [Vm\_postfix\_1; ... ; Vm\_postfix\_{\#Ncorebuses}; Vm\_copy\_postfix\_1; ... ; Vm\_postfix\_{\#Ncopybuses}]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Vm = [Vm_postfix_1; ... ; Vm_postfix_</span><span class="mord"><span class="mord texttt">#</span><span class="mord texttt">N</span><span class="mord texttt">c</span><span class="mord texttt">o</span><span class="mord texttt">r</span><span class="mord texttt">e</span><span class="mord texttt">b</span><span class="mord texttt">u</span><span class="mord texttt">s</span><span class="mord texttt">e</span><span class="mord texttt">s</span></span><span class="mord texttt">; Vm_copy_postfix_1; ... ; Vm_postfix_</span><span class="mord"><span class="mord texttt">#</span><span class="mord texttt">N</span><span class="mord texttt">c</span><span class="mord texttt">o</span><span class="mord texttt">p</span><span class="mord texttt">y</span><span class="mord texttt">b</span><span class="mord texttt">u</span><span class="mord texttt">s</span><span class="mord texttt">e</span><span class="mord texttt">s</span></span><span class="mord texttt">]</span></span></span></span></span></li>
+<li data-line="277" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Pg</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[Pg_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Pg_postfix_#switched-on-generators]</mtext></mrow><annotation encoding="application/x-tex">\texttt{Pg = [Pg\_postfix\_1; ... ; Pg\_postfix\_{\#switched-on-generators}]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Pg = [Pg_postfix_1; ... ; Pg_postfix_</span><span class="mord"><span class="mord texttt">#</span><span class="mord texttt">s</span><span class="mord texttt">w</span><span class="mord texttt">i</span><span class="mord texttt">t</span><span class="mord texttt">c</span><span class="mord texttt">h</span><span class="mord texttt">e</span><span class="mord texttt">d</span><span class="mord texttt">-</span><span class="mord texttt">o</span><span class="mord texttt">n</span><span class="mord texttt">-</span><span class="mord texttt">g</span><span class="mord texttt">e</span><span class="mord texttt">n</span><span class="mord texttt">e</span><span class="mord texttt">r</span><span class="mord texttt">a</span><span class="mord texttt">t</span><span class="mord texttt">o</span><span class="mord texttt">r</span><span class="mord texttt">s</span></span><span class="mord texttt">]</span></span></span></span></span></li>
+<li data-line="278" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Qg</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[Qg_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Qg_postfix_#switched-on-generators]</mtext></mrow><annotation encoding="application/x-tex">\texttt{Qg = [Qg\_postfix\_1; ... ; Qg\_postfix\_{\#switched-on-generators}]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Qg = [Qg_postfix_1; ... ; Qg_postfix_</span><span class="mord"><span class="mord texttt">#</span><span class="mord texttt">s</span><span class="mord texttt">w</span><span class="mord texttt">i</span><span class="mord texttt">t</span><span class="mord texttt">c</span><span class="mord texttt">h</span><span class="mord texttt">e</span><span class="mord texttt">d</span><span class="mord texttt">-</span><span class="mord texttt">o</span><span class="mord texttt">n</span><span class="mord texttt">-</span><span class="mord texttt">g</span><span class="mord texttt">e</span><span class="mord texttt">n</span><span class="mord texttt">e</span><span class="mord texttt">r</span><span class="mord texttt">a</span><span class="mord texttt">t</span><span class="mord texttt">o</span><span class="mord texttt">r</span><span class="mord texttt">s</span></span><span class="mord texttt">]</span></span></span></span></span></li>
+</ul>
+<h3 id="textttcreatestatempm" data-line="280" class="code-line"><a href="#local-decision-variable" data-href="#local-decision-variable"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_state_mp.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_state\_mp.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">create_state_mp.m</span></span></span></span></span></a></h3>
+<p data-line="281" class="code-line"><code>[vang, vmag, pg, qg] = create\_state\_mp(postfix, Nbus, Ngen)</code></p>
+<p data-line="283" class="code-line"><em>Subfunction of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_state.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_state.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_state.m</span></span></span></span></span></em></p>
+<p data-line="285" class="code-line">Input:</p>
+<ul>
+<li data-line="286" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">postfix</mtext></mrow><annotation encoding="application/x-tex">\texttt{postfix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">postfix</span></span></span></span></span> number of local system the</li>
+<li data-line="287" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Nbus</mtext></mrow><annotation encoding="application/x-tex">\texttt{Nbus}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Nbus</span></span></span></span></span> number of buses</li>
+<li data-line="288" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Ngen</mtext></mrow><annotation encoding="application/x-tex">\texttt{Ngen}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Ngen</span></span></span></span></span> umber of generators</li>
+</ul>
+<p data-line="290" class="code-line">Output:</p>
+<ul>
+<li data-line="291" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">[vang,</mtext><mtext> </mtext><mtext mathvariant="monospace">vmag,</mtext><mtext> </mtext><mtext mathvariant="monospace">pg,</mtext><mtext> </mtext><mtext mathvariant="monospace">qg]</mtext></mrow><annotation encoding="application/x-tex">\texttt{[vang, vmag, pg, qg]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">[vang, vmag, pg, qg]</span></span></span></span></span> with
+<ul>
+<li data-line="292" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">vang</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[vang_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">vang_postfix_Nbus]</mtext></mrow><annotation encoding="application/x-tex">\texttt{vang = [vang\_postfix\_1; ... ; vang\_postfix\_Nbus]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">vang = [vang_postfix_1; ... ; vang_postfix_Nbus]</span></span></span></span></span></li>
+<li data-line="293" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">vm</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[vm_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">vm_postfix_Nbus]</mtext></mrow><annotation encoding="application/x-tex">\texttt{vm = [vm\_postfix\_1; ... ; vm\_postfix\_Nbus]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">vm = [vm_postfix_1; ... ; vm_postfix_Nbus]</span></span></span></span></span></li>
+<li data-line="294" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Pg</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[Pg_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Pg_postfix_Ngen]</mtext></mrow><annotation encoding="application/x-tex">\texttt{Pg = [Pg\_postfix\_1; ... ; Pg\_postfix\_Ngen]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Pg = [Pg_postfix_1; ... ; Pg_postfix_Ngen]</span></span></span></span></span></li>
+<li data-line="295" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Qg</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[Qg_postfix_1;</mtext><mtext> </mtext><mtext mathvariant="monospace">...</mtext><mtext> </mtext><mtext mathvariant="monospace">;</mtext><mtext> </mtext><mtext mathvariant="monospace">Qg_postfix_Ngen]</mtext></mrow><annotation encoding="application/x-tex">\texttt{Qg = [Qg\_postfix\_1; ... ; Qg\_postfix\_Ngen]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Qg = [Qg_postfix_1; ... ; Qg_postfix_Ngen]</span></span></span></span></span></li>
+</ul>
+</li>
+</ul>
+<h3 id="textttbuildlocalcostfunctionm" data-line="298" class="code-line"><a href="#local-decision-variable" data-href="#local-decision-variable"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_cost_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_cost\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_cost_function.m</span></span></span></span></span></a></h3>
+<p data-line="299" class="code-line"><code>[cost, grad, hess] = build_local_cost_function(om)</code></p>
+<p data-line="301" class="code-line"><em>Function to extract opf cost function from MATPOWER to be usable in rapidOPF</em></p>
+<p data-line="303" class="code-line">Input:</p>
+<ul>
+<li data-line="304" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> MATPOWER optimization model</li>
+</ul>
+<p data-line="306" class="code-line">Output:</p>
+<ul>
+<li data-line="307" class="code-line">function handles for
+<ul>
+<li data-line="308" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">cost</mtext></mrow><annotation encoding="application/x-tex">\texttt{cost}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.55358em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">cost</span></span></span></span></span> optimization cost function</li>
+<li data-line="309" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">grad</mtext></mrow><annotation encoding="application/x-tex">\texttt{grad}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">grad</span></span></span></span></span> gradient of cost function</li>
+<li data-line="310" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">hess</mtext></mrow><annotation encoding="application/x-tex">\texttt{hess}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">hess</span></span></span></span></span> hessian of cost function</li>
+</ul>
+</li>
+</ul>
+<h3 id="textttbuildlocalconstraintfunctionm" data-line="312" class="code-line"><a href="#local-equality-and-inequality-constraints" data-href="#local-equality-and-inequality-constraints"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_constraint_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_constraint\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_constraint_function.m</span></span></span></span></span></a></h3>
+<p data-line="313" class="code-line"><code>[constraint_function, Lxx] = build_local_constraint_function(mpc_opf, om, mpopt)</code></p>
+<p data-line="315" class="code-line"><em>returns the local constraint functions and the Lagrangian of the original problem</em></p>
+<p data-line="317" class="code-line">Input:</p>
+<ul>
+<li data-line="318" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> splitted and cleaned opf file</li>
+<li data-line="319" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> MATPOWER optimization model of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span></li>
+<li data-line="320" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpopt</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpopt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7757999999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpopt</span></span></span></span></span> MATPOWER standard options
+Output:</li>
+<li data-line="322" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">constraint_functions</mtext></mrow><annotation encoding="application/x-tex">\texttt{constraint\_functions}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">constraint_functions</span></span></span></span></span> function handle for constraints and there derivatives</li>
+<li data-line="323" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">Lxx</mtext></mrow><annotation encoding="application/x-tex">\texttt{Lxx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Lxx</span></span></span></span></span> Lagrangian function of MATPOWER for entire system including deleted equality constraints</li>
+</ul>
+<h3 id="textttbuildlocalequalitiesm" data-line="325" class="code-line"><a href="#local-equality-and-inequality-constraints" data-href="#local-equality-and-inequality-constraints"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_equalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_equalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_equalities.m</span></span></span></span></span></a></h3>
+<p data-line="326" class="code-line"><code>[eq, eq_jac] = build_local_equalities(constraint_function, local_buses_to_remove)</code></p>
+<p data-line="328" class="code-line"><em>Extracts the relevant power flow equation for the core buses. Power flow equations of the copy buses are removed</em></p>
+<p data-line="330" class="code-line">Input:</p>
+<ul>
+<li data-line="331" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">constraint_function</mtext></mrow><annotation encoding="application/x-tex">\texttt{constraint\_function}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">constraint_function</span></span></span></span></span> vector with function handles for constraints</li>
+<li data-line="332" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">local_buses_to_remove</mtext></mrow><annotation encoding="application/x-tex">\texttt{local\_buses\_to\_remove}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">local_buses_to_remove</span></span></span></span></span> indices of buses that were copied from neighbour system</li>
+</ul>
+<p data-line="334" class="code-line">Output:</p>
+<ul>
+<li data-line="335" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">eq</mtext></mrow><annotation encoding="application/x-tex">\texttt{eq}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">eq</span></span></span></span></span> relevant equality constraints for</li>
+<li data-line="336" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">eq_jac</mtext></mrow><annotation encoding="application/x-tex">\texttt{eq\_jac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">eq_jac</span></span></span></span></span> relevant entries of jacobian matrix</li>
+</ul>
+<h3 id="textttbuildlocalinequalitiesm" data-line="338" class="code-line"><a href="#local-equality-and-inequality-constraints" data-href="#local-equality-and-inequality-constraints"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_inequalities.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_inequalities.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_inequalities.m</span></span></span></span></span></a></h3>
+<p data-line="340" class="code-line"><code>[ineq, ineq_jac] = build_local_equalities(constraint_function, local_buses_to_remove)</code></p>
+<p data-line="342" class="code-line"><em>Extracts the relevant power flow equation for the core buses. Power flow equations of the copy buses are removed</em></p>
+<p data-line="344" class="code-line">Input:</p>
+<ul>
+<li data-line="345" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">constraint_function</mtext></mrow><annotation encoding="application/x-tex">\texttt{constraint\_function}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">constraint_function</span></span></span></span></span> vector with function handles for constraints</li>
+<li data-line="346" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">local_buses_to_remove</mtext></mrow><annotation encoding="application/x-tex">\texttt{local\_buses\_to\_remove}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">local_buses_to_remove</span></span></span></span></span> indices of buses that were copied from neighbour system</li>
+</ul>
+<p data-line="348" class="code-line">Output:</p>
+<ul>
+<li data-line="349" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">ineq</mtext></mrow><annotation encoding="application/x-tex">\texttt{ineq}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">ineq</span></span></span></span></span> relevant inequality constraints for</li>
+<li data-line="350" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">ineq_jac</mtext></mrow><annotation encoding="application/x-tex">\texttt{ineq\_jac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">ineq_jac</span></span></span></span></span> relevant entries of jacobian matrix</li>
+</ul>
+<h3 id="textttbuildlocallagrangianfunctionm" data-line="352" class="code-line"><a href="#lagrange-function" data-href="#lagrange-function"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_lagrangian_function.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_lagrangian\_function.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_lagrangian_function.m</span></span></span></span></span></a></h3>
+<p data-line="354" class="code-line"><code>Builds the Lagrangian of the local mpc file</code></p>
+<p data-line="356" class="code-line">Input:</p>
+<ul>
+<li data-line="357" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">f</mtext></mrow><annotation encoding="application/x-tex">\texttt{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">f</span></span></span></span></span> scalar valued cost function</li>
+<li data-line="358" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">df</mtext></mrow><annotation encoding="application/x-tex">\texttt{df}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">df</span></span></span></span></span> gradient of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">f</mtext></mrow><annotation encoding="application/x-tex">\texttt{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">f</span></span></span></span></span></li>
+<li data-line="359" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">g</mtext></mrow><annotation encoding="application/x-tex">\texttt{g}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">g</span></span></span></span></span> equality constraints</li>
+<li data-line="360" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dg</mtext></mrow><annotation encoding="application/x-tex">\texttt{dg}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">dg</span></span></span></span></span> transposed jacobian of the 'reduced' equality constraints</li>
+<li data-line="361" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">h</mtext></mrow><annotation encoding="application/x-tex">\texttt{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">h</span></span></span></span></span> inequality constraints</li>
+<li data-line="362" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dh</mtext></mrow><annotation encoding="application/x-tex">\texttt{dh}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">dh</span></span></span></span></span> transposed jacobian of the inequality constraints</li>
+</ul>
+<p data-line="364" class="code-line">Output:</p>
+<ul>
+<li data-line="365" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">L</mtext></mrow><annotation encoding="application/x-tex">\texttt{L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">L</span></span></span></span></span> 'reduced' Lagrangefunction</li>
+<li data-line="366" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dLdx</mtext></mrow><annotation encoding="application/x-tex">\texttt{dLdx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">dLdx</span></span></span></span></span> grandient of the Lagrangian</li>
+<li data-line="367" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">d2Ldx</mtext></mrow><annotation encoding="application/x-tex">\texttt{d2Ldx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">d2Ldx</span></span></span></span></span> hessian of the Lagrangian</li>
+</ul>
+<p data-line="369" class="code-line">Remark:</p>
+<ul>
+<li data-line="370" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">lambda</mtext></mrow><annotation encoding="application/x-tex">\texttt{lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">lambda</span></span></span></span></span> is the Lagrange multiplier for the ??? constraints (as column vector)</li>
+<li data-line="371" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mu</mtext></mrow><annotation encoding="application/x-tex">\texttt{mu}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">mu</span></span></span></span></span> is the Lagrange multiplier for the ??? constraint (as column vector)</li>
+<li data-line="372" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">kappa</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">[lambda;mu]</mtext></mrow><annotation encoding="application/x-tex">\texttt{kappa = [lambda;mu]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">kappa = [lambda;mu]</span></span></span></span></span> concatenated multiplier</li>
+<li data-line="373" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">L</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">f</mtext><mtext> </mtext><mtext mathvariant="monospace">+</mtext><mtext> </mtext><mtext mathvariant="monospace">lambda’g</mtext><mtext> </mtext><mtext mathvariant="monospace">+</mtext><mtext> </mtext><mtext mathvariant="monospace">mu’h</mtext><mtext> </mtext></mrow><annotation encoding="application/x-tex">\texttt{L = f + lambda&#x27;g + mu&#x27;h }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">L = f + lambda’g + mu’h </span></span></span></span></span></li>
+<li data-line="374" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dL</mtext><mtext> </mtext><mtext mathvariant="monospace">=</mtext><mtext> </mtext><mtext mathvariant="monospace">grad_f</mtext><mtext> </mtext><mtext mathvariant="monospace">+</mtext><mtext> </mtext><mtext mathvariant="monospace">sum</mtext><mtext> </mtext><mtext mathvariant="monospace">lambda_i</mtext><mtext> </mtext><mtext mathvariant="monospace">(Jac_g)’(:,</mtext><mtext> </mtext><mtext mathvariant="monospace">i)</mtext><mtext> </mtext><mtext mathvariant="monospace">+</mtext><mtext> </mtext><mtext mathvariant="monospace">sum</mtext><mtext> </mtext><mtext mathvariant="monospace">mu_i(Jac_h)’(:,</mtext><mtext> </mtext><mtext mathvariant="monospace">i)</mtext></mrow><annotation encoding="application/x-tex">\texttt{dL = grad\_f + sum lambda\_i (Jac\_g)&#x27;(:, i) + sum mu\_i(Jac\_h)&#x27;(:, i)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9166599999999999em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">dL = grad_f + sum lambda_i (Jac_g)’(:, i) + sum mu_i(Jac_h)’(:, i)</span></span></span></span></span></li>
+</ul>
+<h3 id="textttbuildlocalinitialcondistionm" data-line="377" class="code-line"><a href="#initial-conditions-and-bounds" data-href="#initial-conditions-and-bounds"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_initial_condistion.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_initial\_condistion.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_initial_condistion.m</span></span></span></span></span></a></h3>
+<p data-line="379" class="code-line"><code>x0 = build_local_initial_conditions(om)</code></p>
+<p data-line="381" class="code-line"><em>extracts initial condition for x from MATPOWER</em></p>
+<p data-line="383" class="code-line">INPUT:</p>
+<ul>
+<li data-line="384" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> optimization model of reduced splitted case file</li>
+</ul>
+<p data-line="386" class="code-line">OUTPUT:</p>
+<ul>
+<li data-line="387" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">x0</mtext></mrow><annotation encoding="application/x-tex">\texttt{x0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">x0</span></span></span></span></span> initial condition for objective variable <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> of model <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> of subsystem <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span></li>
+</ul>
+<h3 id="textttbuildlocalboundsm" data-line="390" class="code-line"><a href="#initial-conditions-and-bounds" data-href="#initial-conditions-and-bounds"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_bounds.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_bounds.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_bounds.m</span></span></span></span></span></a></h3>
+<p data-line="392" class="code-line"><code>[lb, ub] = build_local_bounds(om)</code></p>
+<p data-line="394" class="code-line"><em>extracts local bounds for x from MATPOWER</em></p>
+<p data-line="396" class="code-line">INPUT:</p>
+<ul>
+<li data-line="397" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> optimization model of reduced splitted case file</li>
+</ul>
+<p data-line="399" class="code-line">OUTPUT:</p>
+<ul>
+<li data-line="400" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">[lb,</mtext><mtext> </mtext><mtext mathvariant="monospace">ub]</mtext></mrow><annotation encoding="application/x-tex">\texttt{[lb, ub]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333299999999999em;vertical-align:-0.13889em;"></span><span class="mord text"><span class="mord texttt">[lb, ub]</span></span></span></span></span> lower and upper bond for objective variable <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span> of model <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">om</mtext></mrow><annotation encoding="application/x-tex">\texttt{om}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">om</span></span></span></span></span> of subsystem <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span></li>
+</ul>
+<h3 id="textttbuildlocaldimensionsm" data-line="403" class="code-line"><a href="#dimensions-check-up" data-href="#dimensions-check-up"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_dimensions.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_dimensions.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">build_local_dimensions.m</span></span></span></span></span></a></h3>
+<p data-line="405" class="code-line"><code>dims = build_local_dimensions(mpc_opf, eq, ineq, local_buses_to_remove)</code></p>
+<p data-line="407" class="code-line"><em>creates a field of dimensions as how they should look like and uses it for testing</em></p>
+<p data-line="409" class="code-line">INPUT:</p>
+<ul>
+<li data-line="410" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc_opf</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc\_opf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc_opf</span></span></span></span></span> splitted case files</li>
+<li data-line="411" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">eq</mtext></mrow><annotation encoding="application/x-tex">\texttt{eq}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">eq</span></span></span></span></span> power flow equations for core nodes plus their jacbian'</li>
+<li data-line="412" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">ineq</mtext></mrow><annotation encoding="application/x-tex">\texttt{ineq}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">ineq</span></span></span></span></span> flow limits inequality constraints plus their jacobian'</li>
+<li data-line="413" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">local_buses_to_remove</mtext></mrow><annotation encoding="application/x-tex">\texttt{local\_buses\_to\_remove}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">local_buses_to_remove</span></span></span></span></span> copy nodes</li>
+</ul>
+<p data-line="415" class="code-line">OUTPUT:</p>
+<ul>
+<li data-line="416" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dims</mtext></mrow><annotation encoding="application/x-tex">\texttt{dims}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">dims</span></span></span></span></span> struct containing dimensions</li>
+</ul>
+<h3 id="textttcreateconsensusmatricesm" data-line="419" class="code-line"><a href="#consensus-matrices" data-href="#consensus-matrices"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">create_consensus_matrices.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{create\_consensus\_matrices.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.70625em;vertical-align:-0.09514em;"></span><span class="mord text"><span class="mord texttt">create_consensus_matrices.m</span></span></span></span></span></a></h3>
+<p data-line="421" class="code-line"><code>A = create_consensus_matrices_opf(tab, number_of_buses_in_region, number_of_generators_in_region)</code></p>
+<p data-line="423" class="code-line"><em>creates optimal power flow consensus matrix for distributed optization</em></p>
+<p data-line="425" class="code-line">INPUT:</p>
+<ul>
+<li data-line="426" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">tab</mtext></mrow><annotation encoding="application/x-tex">\texttt{tab}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">tab</span></span></span></span></span> connection table</li>
+<li data-line="427" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">number_of_buses_in_region</mtext></mrow><annotation encoding="application/x-tex">\texttt{number\_of\_buses\_in\_region}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">number_of_buses_in_region</span></span></span></span></span></li>
+<li data-line="428" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">number_of_generators_in_region</mtext></mrow><annotation encoding="application/x-tex">\texttt{number\_of\_generators\_in\_region}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">number_of_generators_in_region</span></span></span></span></span></li>
+</ul>
+<p data-line="430" class="code-line">OUTPUT:</p>
+<ul>
+<li data-line="431" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">A</mtext></mrow><annotation encoding="application/x-tex">\texttt{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">A</span></span></span></span></span> cell with consensus matrices</li>
+</ul>
+<h3 id="textttgeneratedistributedopfm" data-line="433" class="code-line"><a href="#generation-of-distributed-opf" data-href="#generation-of-distributed-opf"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">generate_distributed_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{generate\_distributed\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">generate_distributed_opf.m</span></span></span></span></span></a></h3>
+<p data-line="435" class="code-line"><code>problem = generate_distributed_opf(mpc, names, ~)</code></p>
+<p data-line="437" class="code-line"><em>stores the information from the splitted case files to run a distributed optimization with ALADIN</em></p>
+<p data-line="439" class="code-line">INPUT:</p>
+<ul>
+<li data-line="440" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc</span></span></span></span></span> struct with splitted case files</li>
+<li data-line="441" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">names</mtext></mrow><annotation encoding="application/x-tex">\texttt{names}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">names</span></span></span></span></span> struct containing the names of the fields of
+$\texttt{mpc}
+Output:</li>
+<li data-line="444" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">problem</mtext></mrow><annotation encoding="application/x-tex">\texttt{problem}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">problem</span></span></span></span></span> struct with the following fields</li>
+<li data-line="445" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">locFuns</mtext></mrow><annotation encoding="application/x-tex">\texttt{locFuns}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">locFuns</span></span></span></span></span> struct with fields of cells for local costs,
+equality constraints, inequality constraints and imensions</li>
+<li data-line="447" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">sens</mtext></mrow><annotation encoding="application/x-tex">\texttt{sens}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">sens</span></span></span></span></span> struct with fields of cells for radient of costs,
+jacobian of equality and inequality and the Hessian f the
+Lagrangian</li>
+<li data-line="450" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">zz0</mtext></mrow><annotation encoding="application/x-tex">\texttt{zz0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">zz0</span></span></span></span></span> cell of initial conditions</li>
+<li data-line="451" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">AA</mtext></mrow><annotation encoding="application/x-tex">\texttt{AA}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">AA</span></span></span></span></span> cell of consensus matrices</li>
+<li data-line="452" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">state</mtext></mrow><annotation encoding="application/x-tex">\texttt{state}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.55358em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">state</span></span></span></span></span> cell that contans the local states</li>
+<li data-line="453" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">llbx</mtext></mrow><annotation encoding="application/x-tex">\texttt{llbx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">llbx</span></span></span></span></span> cell that contains the local lower ounds</li>
+<li data-line="454" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">uubx</mtext></mrow><annotation encoding="application/x-tex">\texttt{uubx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">uubx</span></span></span></span></span> cell that cotains the local upper ounds</li>
+</ul>
+<h3 id="textttbuildlocalopfm" data-line="456" class="code-line"><a href="#generation-of-distributed-opf" data-href="#generation-of-distributed-opf"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">build_local_opf.m</mtext></mrow><annotation encoding="application/x-tex">\texttt{build\_local\_opf.m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">build_local_opf.m</span></span></span></span></span></a></h3>
+<p data-line="457" class="code-line"><code>[cost, ineq, eq, x0, grad_cost, eq_jac, ineq_jac, lagrangian_hessian, state, dims, lb, ub] = build_local_opf(mpc, names, postfix)</code></p>
+<p data-line="459" class="code-line"><em>extracts the local functions and information needed for opf</em></p>
+<p data-line="461" class="code-line">INPUT:</p>
+<ul>
+<li data-line="462" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">mpc</mtext></mrow><annotation encoding="application/x-tex">\texttt{mpc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">mpc</span></span></span></span></span> splitted full casefile</li>
+<li data-line="463" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">names</mtext></mrow><annotation encoding="application/x-tex">\texttt{names}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">names</span></span></span></span></span> struct of names corresponding to the fields
+of mpc</li>
+<li data-line="465" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">postfix</mtext></mrow><annotation encoding="application/x-tex">\texttt{postfix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">postfix</span></span></span></span></span> index of subsystem
+OUTPUT:</li>
+<li data-line="467" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">cost</mtext></mrow><annotation encoding="application/x-tex">\texttt{cost}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.55358em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">cost</span></span></span></span></span> scalar local cost function</li>
+<li data-line="468" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">ineq</mtext></mrow><annotation encoding="application/x-tex">\texttt{ineq}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">ineq</span></span></span></span></span> vector of equality constraint functions</li>
+<li data-line="469" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">eq</mtext></mrow><annotation encoding="application/x-tex">\texttt{eq}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65278em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">eq</span></span></span></span></span> vector of inequality constraint functions</li>
+<li data-line="470" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">x0</mtext></mrow><annotation encoding="application/x-tex">\texttt{x0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">x0</span></span></span></span></span> intial condition of x</li>
+<li data-line="471" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">grad_cost</mtext></mrow><annotation encoding="application/x-tex">\texttt{grad\_cost}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">grad_cost</span></span></span></span></span> gradient of cost function</li>
+<li data-line="472" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">eq_jac</mtext></mrow><annotation encoding="application/x-tex">\texttt{eq\_jac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">eq_jac</span></span></span></span></span> jacobian of eqality contraints</li>
+<li data-line="473" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">ineq_jac</mtext></mrow><annotation encoding="application/x-tex">\texttt{ineq\_jac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">ineq_jac</span></span></span></span></span> jacobian of inequality constraints</li>
+<li data-line="474" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">lagrangian_hessian</mtext></mrow><annotation encoding="application/x-tex">\texttt{lagrangian\_hessian}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">lagrangian_hessian</span></span></span></span></span> hesisan of reduced agrange
+function</li>
+<li data-line="476" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">state</mtext></mrow><annotation encoding="application/x-tex">\texttt{state}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.55358em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">state</span></span></span></span></span> representation of objective ariable</li>
+<li data-line="477" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">dims</mtext></mrow><annotation encoding="application/x-tex">\texttt{dims}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">dims</span></span></span></span></span> struct of local dimensions</li>
+<li data-line="478" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">lb</mtext></mrow><annotation encoding="application/x-tex">\texttt{lb}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">lb</span></span></span></span></span> lower bound of x</li>
+<li data-line="479" class="code-line"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext mathvariant="monospace">ub</mtext></mrow><annotation encoding="application/x-tex">\texttt{ub}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">ub</span></span></span></span></span> upper bond of x</li>
+</ul>
 
 </body></html>
\ No newline at end of file
diff --git a/docs/extension_to_opf.md b/docs/extension_to_opf.md
index 99fe4e0..8720b0a 100644
--- a/docs/extension_to_opf.md
+++ b/docs/extension_to_opf.md
@@ -1,4 +1,4 @@
-# RapidOPF - Extension to Optimal Power Flow
+# RapidOPF - Extension of RapidPF to Optimal Power Flow
 
 <!--!!! warning "Extension to optimal power flow = work in progress"
     __The documentation and code is currently being written.
@@ -39,21 +39,49 @@ The main changes happen during step 3. and 4. as the cost functions, power balan
 ## Extensions to existing Code
 New files:
 
-* $\texttt{generate\_distributed\_opf.m}$
-* $\texttt{create\_consensus\_matrices\_opf.m}$
-* $\texttt{build\_local\_opf.m}$
-* [$\texttt{prepare\_case\_file.m}$](#textttprepare_case_filem)
-* [$\texttt{build\_local\_state.m}$](#local-decision-variable)
-* $\texttt{create\_state\_mp}$
-* $\texttt{build\_local\_dims.m}$
-* $\texttt{build\_local\_cost\_functon.m}$
-* $\texttt{build\_local\_equalities.m}$
-* $\texttt{build\_local\_inequalities.m}$
-* $\texttt{build\_local\_lagrangian\_function.m}$ 
-* $\texttt{build\_local\_initial\_condition.m}$
-* $\texttt{build\_local\_bounds.m}$
+- [RapidOPF - Extension of RapidPF to Optimal Power Flow](#rapidopf---extension-of-rapidpf-to-optimal-power-flow)
+  - [OPF Problem Formulation](#opf-problem-formulation)
+  - [Subsumption in RapidPf](#subsumption-in-rapidpf)
+  - [Extensions to existing Code](#extensions-to-existing-code)
+  - [Generation of distributed opf](#generation-of-distributed-opf)
+  - [Preparation of splitted Casefile](#preparation-of-splitted-casefile)
+    - [Corresponding functions:](#corresponding-functions)
+  - [Local Decision Variable](#local-decision-variable)
+    - [Corresponding functions](#corresponding-functions-1)
+  - [Local cost function](#local-cost-function)
+    - [Corresponding functions](#corresponding-functions-2)
+  - [Local equality and inequality constraints](#local-equality-and-inequality-constraints)
+    - [Corresponding functions](#corresponding-functions-3)
+  - [Lagrange function](#lagrange-function)
+    - [Corresponding functions](#corresponding-functions-4)
+  - [Consensus Matrices](#consensus-matrices)
+    - [Corresponding functions](#corresponding-functions-5)
+  - [Dimensions check up](#dimensions-check-up)
+    - [Corresponding functions](#corresponding-functions-6)
+  - [Initial conditions and bounds](#initial-conditions-and-bounds)
+    - [Corresponding functions](#corresponding-functions-7)
+  - [Detailed documentation of new functions](#detailed-documentation-of-new-functions)
+    - [$\texttt{prepare\_case\_file.m}$](#textttprepare_case_filem)
+    - [$\texttt{build\_local\_state.m}$](#textttbuild_local_statem)
+    - [$\texttt{create\_state\_mp.m}$](#textttcreate_state_mpm)
+    - [$\texttt{build\_local\_cost\_function.m}$](#textttbuild_local_cost_functionm)
+    - [$\texttt{build\_local\_constraint\_function.m}$](#textttbuild_local_constraint_functionm)
+    - [$\texttt{build\_local\_equalities.m}$](#textttbuild_local_equalitiesm)
+    - [$\texttt{build\_local\_inequalities.m}$](#textttbuild_local_inequalitiesm)
+    - [$\texttt{build\_local\_lagrangian\_function.m}$](#textttbuild_local_lagrangian_functionm)
+    - [$\texttt{build\_local\_initial\_condistion.m}$](#textttbuild_local_initial_condistionm)
+    - [$\texttt{build\_local\_bounds.m}$](#textttbuild_local_boundsm)
+    - [$\texttt{build\_local\_dimensions.m}$](#textttbuild_local_dimensionsm)
+    - [$\texttt{create\_consensus\_matrices.m}$](#textttcreate_consensus_matricesm)
+    - [$\texttt{generate\_distributed\_opf.m}$](#textttgenerate_distributed_opfm)
+    - [$\texttt{build\_local\_opf.m}$](#textttbuild_local_opfm)
 
 
+
+## Generation of distributed opf
+From the casefiles, all information needed for the distributed solver need to be extracted from the casefile of splitted casefiles. To do so, several functions are called within the fuctions 
+- [$\texttt{generate\_distributed\_opf.m}$](#textttgenerate_distributed_opfm)
+- [$\texttt{build\_local\_opf.m}$]()
 ## Preparation of splitted Casefile
 The RapidPF splitted casefiles consist of the following fields:
 *  $\texttt{baseMVA}$
@@ -78,22 +106,9 @@ Here, $n_b$ corresponds to the number of buses in the system and $n_g$ correspon
 
 At this point we need to be careful as the fields $\texttt{gen}$ and $\texttt{gencost}$ also do contain the generators at the copy nodes that shall not have any impact on the local objectives. Switching them of in the casefile as it is done in the function $\texttt{prepare\_case\_file}$ leads them in MATPOWER to be treated as non existing, thus giving us the wanted effect. 
 
-### $\texttt{prepare\_case\_file.m}$
-Function to prepare the rapidPF splitted case for OPF. Generators at the copy nodes are switched off and gen entries and gencost entries are deleted from the $\texttt{mpc\_opf}$ file.
-Input:
-* $\texttt{mpc}$ - splitted rapidPF case file
-* $\texttt{names}$ - struct that contains the names of the mpc file
-  
-Output:
-* $\texttt{mpc\_opf}$ - splitted rapidOPF case file
-* $\texttt{om}$ - MATPOWER optimization model of $\texttt{mpc\_opf}$ that is needed to obtain the optimization functions
-* $\texttt{mpc\_opf}$ - prepared rapiOPF casefile
-* $\texttt{copy\_buses\_local}$ - vector of local copy buses
-* $\texttt{mpopt}$ - standard MATPOWER opf options
+### Corresponding functions:
+- [$\texttt{prepare\_case\_file.m}$](#textttprepare_case_filem)
 
-Tests: 
-* added sample test to 05_UI_test
-* added assert to check whether there are still gen entries left in local $\texttt{mpc\_opf}$ case file.
 
 ## Local Decision Variable
 Each system has a local decision variable. We need the following values:
@@ -116,4 +131,350 @@ Thus the total dimension of $x_i$ is
 $$\begin{aligned} 
 \text{dim}(x_i) & = 2\cdot N_{b-core} + 2\cdot N_{b-copy} + 2\cdot N_{g-core} \\ & = 2\cdot N_b + 2 \cdot N_{g-core}
 \end{aligned}
-$$   
+$$
+
+### Corresponding functions
+- [$\texttt{build\_local\_state.m}$](#textttbuild_local_statem)
+- [$\texttt{create\_state\_mp.m}$](#textttcreate_state_mpm)
+
+## Local cost function
+To calculate the cost function for rapidOPF, the MATPOWER function [$\texttt{opf\_costfcn.m}$](https://matpower.org/docs/ref/matpower5.0/opf_costfcn.html). The call of $\texttt{opf\_costfcn(x,om)}$ returns
+- the evaluation of the cost function $f$ at $x$ ( dimension $1 \times 1$)
+- the evaluation of the gradient $\nabla f$ of $f$ at $x$ (dimension $\text{length}(x)\times 1$) and 
+- the evaluation of the hessian matrix $\nabla^2 f$ of $f$ at $x$ (dimension $\text{length}(x) \times \text{length}(x)$)
+ 
+with respect to the structure of $x$ illustrated above.
+
+The results are one to one used by rapidOPF
+
+### Corresponding functions
+- [$\texttt{build\_local\_cost\_function.m}$](#textttbuild_local_cost_functionm)
+
+
+## Local equality and inequality constraints
+To calculate the local equality and inequality constraints for rapidOPF, the MATPOWER function [$\texttt{opf\_consfcn.m}$](https://matpower.org/docs/ref/matpower5.0/opf_consfcn.html) is used. We recall the output formate of $\texttt{opf\_consfcn(x, om, Ybus, Yf(il,:), Yt(il,:), mpopt, il)}$: 
+- $\texttt{h(x)}$ inequality constraints at $x$ of dim Nineq $\times 1$
+- $\texttt{g(x)}$ equality constraints at $x$ of dim Neq $\times 1$
+- $\texttt{dh(x)}$ transposed of the jacobian of $h$ at $x$ of dimension dim Nx $\times$ dim Nineq, i.e. 
+$$
+\left[ \begin{array}{ccc}
+\frac{\partial h_1}{\partial x_1} & \cdots & \frac{\partial h_{ng}}{\partial x_{1}} \\ \vdots & \ddots & \vdots \\ 
+\frac{\partial h_{1}}{\partial x_{nx}} & \cdots & \frac{\partial h_{ng}}{\partial x_{nx}}
+\end{array}\right] 
+$$
+- $\texttt{dg(x)}$ transposed of jacobian of $g$ at $x$ of dimension dim Nx $\times$ dim Neq, i.e.
+$$
+\left[ \begin{array}{ccc}
+\frac{\partial g_1}{\partial x_1} & \cdots & \frac{\partial g_{ng}}{\partial x_{1}} \\ \vdots & \ddots & \vdots \\ 
+\frac{\partial g_{1}}{\partial x_{nx}} & \cdots & \frac{\partial g_{ng}}{\partial x_{nx}}
+\end{array}\right] 
+$$
+
+The function handles $h, dh$ are taken as they are. For $g$, the entries of the copy buses are deleted, for $dg$, the columns corresponding to the derivatives of the power flow corresponding to the copy buses are deleted
+
+### Corresponding functions
+- [$\texttt{build\_local\_constraint\_function.m}$](#textttbuild_local_constraint_functionm)
+- [$\texttt{build\_local\_equalities.m}$](#textttbuild_local_equalitiesm)
+- [$\texttt{build\_local\_inequalities.m}$](#textttbuild_local_inequalitiesm)
+
+## Lagrange function
+The ALADIN Solver needs to be handed over a Lagrangian function and its hessian matrix with respect to x
+$$
+L(x, \lambda, \mu) = f(x) + \lambda.*\nabla g + \mu .* \nabla h
+$$.
+These functions are provided by rapidOPF.
+
+### Corresponding functions
+[$\texttt{build\_local\_lagrangian\_function.m}$](#textttbuild_local_lagrangian_functionm)
+
+## Consensus Matrices
+As consensus matrices we call the matrices that guarantee that the voltage angles and voltage magnitudes at the copy nodes correspond to the voltage angles and magnitudes at the corresponding core nodes in their core system. Therefore, the numbers of connections between all systems are needed. The local matrices are of dimension 
+$$
+4*N_{connection-buses} \times \text{dim}(x_i)
+$$
+To fill the consensus matrix, for each system information is needed about the the indices of the local core nodes and the local copy nodes. They are given in the form of a connection table- it consists of four columns that are filled for each connection the another system for each system:
+ - $\texttt{orig\_system}$ global index of core system system
+ - $\texttt{copy\_system}$ global index of copy system 
+ - $\texttt{orig\_bus\_loacl}$ local node index of core node that is connected to another copy node
+ - $\texttt{copy\_bus\_local}$ local node index of copy node connected to the core node from column 3
+Recall that 
+$$
+4*N_{connection-buses} \times \text{dim}(x_i) = 2 * size(connection\_table, 1)
+$$
+To enforce consensus, for each row $i$ of the connection table two entries are generated: 
+- $A_{core}(i, core\_bus\_local) = 1$
+- $A_{copy}(i, copy\_bus\_local) = -1$
+
+We notice, that we need such a matrix not only for voltage angles but also one such matrix for voltage magnitudes, where the core_bus_entry and the copy_bus_entry need to be shifted to the column corresponding to the voltage entry of the optimization variable.
+
+### Corresponding functions
+- [$\texttt{create\_consensus\_matrices.m}$](#textttcreate_consensus_matricesm)
+
+## Dimensions check up
+For clarity, the dimensions of the outputs are summarized
+
+- The dimension of the optimization vector is given by
+$$\begin{aligned} 
+\text{dim}(x_i) & = 2\cdot N_{busses-core} + 2\cdot N_{busses-copy} + 2\cdot N_{g-core} \\ & = 2\cdot N_{buses} + 2 \cdot N_{g-core}
+\end{aligned}
+$$
+
+- The number of equality constraints is equal to 
+$$\text{dim}(eq) = 2 * (N_{buses} - N_{buses-copy}) = 2*(N_{buses-core})$$, i.e. we have two equations for each core node
+
+- The number of inequality constraints is equal to the number of non-zero entries of $\texttt{RATE\_A}$ in the branch specifications of the splitted opf case file
+
+- The local cost functions are scalar valued, the gradients are of dimension $\texttt{length}(x) \times 1$, its Hessian is of dimension $\texttt{length}(x) \times \texttt{length}(x)$ 
+
+### Corresponding functions
+- [$\texttt{build\_local\_dimensions.m}$](#textttbuild_local_dimensionsm)
+
+## Initial conditions and bounds
+The initial conditions and bounds are again one by one taken by MATPOWER functions $\texttt{params\_var(om).m}$.
+
+### Corresponding functions
+- [$\texttt{build\_local\_initial\_condistion.m}$](#textttbuild_local_initial_condistionm)
+
+- [$\texttt{build\_local\_bounds.m}$](#textttbuild_local_boundsm)
+
+## Detailed documentation of new functions
+
+### [$\texttt{prepare\_case\_file.m}$](#preparation-of-splitted-casefile)
+
+`[mpc_opf, om, copy_buses_local, mpopt] = prepare_case_file(mpc, names)`
+
+_Function to prepare the rapidPF splitted casefiles for OPF. Generators at the copy nodes are switched off and gen entries and gencost entries are deleted from the $\texttt{mpc\_opf}$ file._
+Input:
+* $\texttt{mpc}$ - splitted rapidPF case file
+* $\texttt{names}$ - struct that contains the names of the mpc file
+  
+Output:
+* $\texttt{mpc\_opf}$ - prepared splitted rapiOPF casefile
+* $\texttt{om}$ - MATPOWER optimization model of $\texttt{mpc\_opf}$ that is needed to obtain the optimization functions
+* $\texttt{copy\_buses\_local}$ - vector of local copy buses
+* $\texttt{mpopt}$ - standard MATPOWER opf options
+
+Tests: 
+* added sample test to 05_UI_test
+* added assert to check whether there are still gen entries left in local $\texttt{mpc\_opf}$ case file.
+
+### [$\texttt{build\_local\_state.m}$](#local-decision-variable)
+`state = build_local_state(mpc, names, postfix)`
+
+_Function to build symbolic representation of optimization variable_
+
+Input:
+ - $\texttt{mpc}$ splitted casefile
+ - $\texttt{names}$ specific names of mpc struct fields
+ - $\texttt{postfix}$ number of local system
+  
+ Output
+   - $\texttt{state}$ symbolic state
+
+ Final formate:
+ $\texttt{state = (Vang; Vm; Pg; Qg)}$ (column vector) with 
+ - $\texttt{Vang = [Vang\_postfix\_1; ... ; Vang\_postfix\_{\#Ncorebuses}; Vang\_postfix\_copy\_1; ... ; Vang\_postfix\_{\#Ncopybuses}]}$
+ - $\texttt{Vm = [Vm\_postfix\_1; ... ; Vm\_postfix\_{\#Ncorebuses}; Vm\_copy\_postfix\_1; ... ; Vm\_postfix\_{\#Ncopybuses}]}$
+ - $\texttt{Pg = [Pg\_postfix\_1; ... ; Pg\_postfix\_{\#switched-on-generators}]}$
+ - $\texttt{Qg = [Qg\_postfix\_1; ... ; Qg\_postfix\_{\#switched-on-generators}]}$
+
+### [$\texttt{create\_state\_mp.m}$](#local-decision-variable)
+`[vang, vmag, pg, qg] = create\_state\_mp(postfix, Nbus, Ngen)`
+ 
+_Subfunction of $\texttt{build\_local\_state.m}$_
+
+Input:
+ - $\texttt{postfix}$ number of local system the 
+ - $\texttt{Nbus}$ number of buses
+ - $\texttt{Ngen}$ umber of generators
+
+Output:
+ - $\texttt{[vang, vmag, pg, qg]}$ with 
+   - $\texttt{vang = [vang\_postfix\_1; ... ; vang\_postfix\_Nbus]}$
+   - $\texttt{vm = [vm\_postfix\_1; ... ; vm\_postfix\_Nbus]}$
+   - $\texttt{Pg = [Pg\_postfix\_1; ... ; Pg\_postfix\_Ngen]}$
+   - $\texttt{Qg = [Qg\_postfix\_1; ... ; Qg\_postfix\_Ngen]}$
+
+
+### [$\texttt{build\_local\_cost\_function.m}$](#local-decision-variable)
+`[cost, grad, hess] = build_local_cost_function(om)`
+
+_Function to extract opf cost function from MATPOWER to be usable in rapidOPF_
+
+Input: 
+- $\texttt{om}$ MATPOWER optimization model
+
+Output:
+ - function handles for 
+   - $\texttt{cost}$ optimization cost function 
+   - $\texttt{grad}$ gradient of cost function
+   - $\texttt{hess}$ hessian of cost function
+
+### [$\texttt{build\_local\_constraint\_function.m}$](#local-equality-and-inequality-constraints)
+`[constraint_function, Lxx] = build_local_constraint_function(mpc_opf, om, mpopt)`
+
+  _returns the local constraint functions and the Lagrangian of the original problem_ 
+
+ Input:
+  - $\texttt{mpc\_opf}$ splitted and cleaned opf file
+  - $\texttt{om}$ MATPOWER optimization model of $\texttt{mpc\_opf}$
+  - $\texttt{mpopt}$ MATPOWER standard options
+ Output: 
+  - $\texttt{constraint\_functions}$ function handle for constraints and there derivatives
+  - $\texttt{Lxx}$ Lagrangian function of MATPOWER for entire system including deleted equality constraints
+  
+### [$\texttt{build\_local\_equalities.m}$](#local-equality-and-inequality-constraints)
+`[eq, eq_jac] = build_local_equalities(constraint_function, local_buses_to_remove)`
+
+   _Extracts the relevant power flow equation for the core buses. Power flow equations of the copy buses are removed_
+
+Input:
+  - $\texttt{constraint\_function}$ vector with function handles for constraints
+  - $\texttt{local\_buses\_to\_remove}$ indices of buses that were copied from neighbour system
+
+Output:
+  - $\texttt{eq}$ relevant equality constraints for 
+  - $\texttt{eq\_jac}$ relevant entries of jacobian matrix
+
+### [$\texttt{build\_local\_inequalities.m}$](#local-equality-and-inequality-constraints)
+
+`[ineq, ineq_jac] = build_local_equalities(constraint_function, local_buses_to_remove)`
+
+   _Extracts the relevant power flow equation for the core buses. Power flow equations of the copy buses are removed_
+
+Input:
+  - $\texttt{constraint\_function}$ vector with function handles for constraints
+  - $\texttt{local\_buses\_to\_remove}$ indices of buses that were copied from neighbour system
+
+Output:
+  - $\texttt{ineq}$ relevant inequality constraints for 
+  - $\texttt{ineq\_jac}$ relevant entries of jacobian matrix
+
+### [$\texttt{build\_local\_lagrangian\_function.m}$](#lagrange-function)
+
+`Builds the Lagrangian of the local mpc file`
+   
+  Input: 
+  - $\texttt{f}$ scalar valued cost function
+  - $\texttt{df}$ gradient of $\texttt{f}$ 
+  - $\texttt{g}$ equality constraints
+  - $\texttt{dg}$ transposed jacobian of the 'reduced' equality constraints
+  - $\texttt{h}$ inequality constraints
+  - $\texttt{dh}$ transposed jacobian of the inequality constraints
+ 
+  Output:
+  - $\texttt{L}$ 'reduced' Lagrangefunction
+  - $\texttt{dLdx}$ grandient of the Lagrangian
+  - $\texttt{d2Ldx}$ hessian of the Lagrangian
+
+  Remark:
+   - $\texttt{lambda}$ is the Lagrange multiplier for the ??? constraints (as column vector)
+   - $\texttt{mu}$ is the Lagrange multiplier for the ??? constraint (as column vector)
+   - $\texttt{kappa = [lambda;mu]}$ concatenated multiplier
+   - $\texttt{L = f + lambda'g + mu'h }$
+   - $\texttt{dL = grad\_f + sum lambda\_i (Jac\_g)'(:, i) + sum mu\_i(Jac\_h)'(:, i)}$
+
+
+### [$\texttt{build\_local\_initial\_condistion.m}$](#initial-conditions-and-bounds)
+
+   `x0 = build_local_initial_conditions(om)`
+
+   _extracts initial condition for x from MATPOWER_
+
+   INPUT:
+   - $\texttt{om}$ optimization model of reduced splitted case file
+
+  OUTPUT:
+  - $\texttt{x0}$ initial condition for objective variable $x$ of model $\texttt{om}$ of subsystem $i$
+
+
+### [$\texttt{build\_local\_bounds.m}$](#initial-conditions-and-bounds)
+
+   `[lb, ub] = build_local_bounds(om)`
+
+   _extracts local bounds for x from MATPOWER_
+
+   INPUT:
+   - $\texttt{om}$ optimization model of reduced splitted case file
+
+  OUTPUT:
+  - $\texttt{[lb, ub]}$ lower and upper bond for objective variable $x$ of model $\texttt{om}$ of subsystem $i$    
+
+
+### [$\texttt{build\_local\_dimensions.m}$](#dimensions-check-up)
+
+   `dims = build_local_dimensions(mpc_opf, eq, ineq, local_buses_to_remove)`
+
+   _creates a field of dimensions as how they should look like and uses it for testing_
+
+   INPUT:  
+   - $\texttt{mpc\_opf}$ splitted case files
+   - $\texttt{eq}$ power flow equations for core nodes plus their jacbian'
+   - $\texttt{ineq}$ flow limits inequality constraints plus their jacobian'
+   - $\texttt{local\_buses\_to\_remove}$ copy nodes
+  
+   OUTPUT: 
+   - $\texttt{dims}$ struct containing dimensions
+
+
+### [$\texttt{create\_consensus\_matrices.m}$](#consensus-matrices)
+
+   `A = create_consensus_matrices_opf(tab, number_of_buses_in_region, number_of_generators_in_region)`
+
+   _creates optimal power flow consensus matrix for distributed optization_
+
+   INPUT:
+   - $\texttt{tab}$ connection table
+   - $\texttt{number\_of\_buses\_in\_region}$
+   - $\texttt{number\_of\_generators\_in\_region}$
+
+  OUTPUT:
+  - $\texttt{A}$ cell with consensus matrices
+
+### [$\texttt{generate\_distributed\_opf.m}$](#generation-of-distributed-opf)
+
+   `problem = generate_distributed_opf(mpc, names, ~)`
+
+   _stores the information from the splitted case files to run a distributed optimization with ALADIN_
+
+   INPUT:
+   - $\texttt{mpc}$ struct with splitted case files
+   - $\texttt{names}$ struct containing the names of the fields of
+     $\texttt{mpc}
+  Output:
+   - $\texttt{problem}$ struct with the following fields
+   - $\texttt{locFuns}$ struct with fields of cells for local costs,
+  equality constraints, inequality constraints and imensions
+  - $\texttt{sens}$ struct with fields of cells for radient of costs,
+  jacobian of equality and inequality and the Hessian f the
+  Lagrangian
+  - $\texttt{zz0}$ cell of initial conditions
+  - $\texttt{AA}$ cell of consensus matrices
+  - $\texttt{state}$ cell that contans the local states
+  - $\texttt{llbx}$ cell that contains the local lower ounds
+  - $\texttt{uubx}$ cell that cotains the local upper ounds
+
+### [$\texttt{build\_local\_opf.m}$](#generation-of-distributed-opf)
+   `[cost, ineq, eq, x0, grad_cost, eq_jac, ineq_jac, lagrangian_hessian, state, dims, lb, ub] = build_local_opf(mpc, names, postfix)`
+
+   _extracts the local functions and information needed for opf_
+
+   INPUT: 
+  - $\texttt{mpc}$ splitted full casefile
+  - $\texttt{names}$ struct of names corresponding to the fields
+    of mpc
+  - $\texttt{postfix}$ index of subsystem
+   OUTPUT:
+  - $\texttt{cost}$ scalar local cost function
+  - $\texttt{ineq}$ vector of equality constraint functions
+  - $\texttt{eq}$ vector of inequality constraint functions
+  - $\texttt{x0}$ intial condition of x
+  - $\texttt{grad\_cost}$ gradient of cost function
+  - $\texttt{eq\_jac}$ jacobian of eqality contraints
+  - $\texttt{ineq\_jac}$ jacobian of inequality constraints
+  - $\texttt{lagrangian\_hessian}$ hesisan of reduced agrange
+  function
+  - $\texttt{state}$ representation of objective ariable
+  - $\texttt{dims}$ struct of local dimensions
+  - $\texttt{lb}$ lower bound of x
+  - $\texttt{ub}$ upper bond of x
\ No newline at end of file
diff --git a/mpc_merge.m b/mpc_merge.m
index 9fb853c..7fa34d0 100644
--- a/mpc_merge.m
+++ b/mpc_merge.m
@@ -20,12 +20,12 @@
 	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	10	1	0	0	0	0	1	1.1	0	230	1	1.1	0.9;
-	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,14 +35,15 @@
 	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	5	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	15	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
@@ -67,7 +68,7 @@
 	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -84,7 +85,8 @@
 	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
-	2	0	0	2	10	0;
 ];
diff --git a/mpc_merge_split.m b/mpc_merge_split.m
index d911eeb..209b807 100644
--- a/mpc_merge_split.m
+++ b/mpc_merge_split.m
@@ -20,12 +20,12 @@
 	7	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	8	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	9	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	10	1	0	0	0	0	1	1.1	0	230	1	1.1	0.9;
-	11	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	10	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	11	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
 	12	1	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	13	2	300	98.61	0	0	1	1	0	230	1	1.1	0.9;
 	14	2	400	131.47	0	0	1	1	0	230	1	1.1	0.9;
-	15	2	0	0	0	0	1	1	0	230	1	1.1	0.9;
+	15	1	0	0	0	0	1	1	0	230	1	1.1	0.9;
 ];
 
 %% generator data
@@ -35,14 +35,15 @@
 	1	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	3	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	4	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	5	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
+	5	466.51	0	450	-450	1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
 	6	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	8	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	9	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	40	0	30	-30	1	100	1	40	0	0	0	0	0	0	0	0	0	0	0	0;
+	11	170	0	127.5	-127.5	1	100	1	170	0	0	0	0	0	0	0	0	0	0	0	0;
 	13	323.49	0	390	-390	1	100	1	520	0	0	0	0	0	0	0	0	0	0	0	0;
 	14	0	0	150	-150	1	100	1	200	0	0	0	0	0	0	0	0	0	0	0	0;
-	15	466.51	0	450	-450	1.1	100	1	600	0	0	0	0	0	0	0	0	0	0	0	0;
 ];
 
 %% branch data
@@ -67,7 +68,7 @@
 	12	13	0.00108	0.0108	0.01852	0	0	0	0	0	1	-360	360;
 	13	14	0.00297	0.0297	0.00674	0	0	0	0	0	1	-360	360;
 	14	15	0.00297	0.0297	0.00674	240	240	240	0	0	1	-360	360;
-	9	11	0	0.00623	0	0	0	0	0.985	0	1	0	0;
+	6	15	0	0.00623	0	0	0	0	0.985	0	1	0	0;
 ];
 
 %%-----  OPF Data  -----%%
@@ -84,7 +85,8 @@
 	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
+	2	0	0	2	14	0;
+	2	0	0	2	15	0;
 	2	0	0	2	30	0;
 	2	0	0	2	40	0;
-	2	0	0	2	10	0;
 ];