Description of fmincopf

0001 function [results, success, raw] = fmincopf_solver(om, mpopt)
0002 %FMINCOPF_SOLVER  Solves AC optimal power flow using FMINCON.
0003 %
0004 %   [RESULTS, SUCCESS, RAW] = FMINCOPF_SOLVER(OM, MPOPT)
0005 %
0006 %   Inputs are an OPF model object and a MATPOWER options struct.
0007 %
0008 %   Outputs are a RESULTS struct, SUCCESS flag and RAW output struct.
0009 %
0010 %   RESULTS is a MATPOWER case struct (mpc) with the usual baseMVA, bus
0011 %   branch, gen, gencost fields, along with the following additional
0012 %   fields:
0013 %       .order      see 'help ext2int' for details of this field
0014 %       .x          final value of optimization variables (internal order)
0015 %       .f          final objective function value
0016 %       .mu         shadow prices on ...
0017 %           .var
0018 %               .l  lower bounds on variables
0019 %               .u  upper bounds on variables
0020 %           .nln    (deprecated) 2*nb+2*nl - Pmis, Qmis, Sf, St
0021 %               .l  lower bounds on nonlinear constraints
0022 %               .u  upper bounds on nonlinear constraints
0023 %           .nle    nonlinear equality constraints
0024 %           .nli    nonlinear inequality constraints
0025 %           .lin
0026 %               .l  lower bounds on linear constraints
0027 %               .u  upper bounds on linear constraints
0028 %
0029 %   SUCCESS     1 if solver converged successfully, 0 otherwise
0030 %
0031 %   RAW         raw output in form returned by MINOS
0032 %       .xr     final value of optimization variables
0033 %       .pimul  constraint multipliers
0034 %       .info   solver specific termination code
0035 %       .output solver specific output information
0036 %
0037 %   See also OPF, FMINCON.
0038 
0039 %   MATPOWER
0040 %   Copyright (c) 2000-2017, Power Systems Engineering Research Center (PSERC)
0041 %   by Ray Zimmerman, PSERC Cornell
0042 %   and Carlos E. Murillo-Sanchez, PSERC Cornell & Universidad Nacional de Colombia
0043 %
0044 %   This file is part of MATPOWER.
0045 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0046 %   See https://matpower.org for more info.
0047 
0048 %%----- initialization -----
0049 %% define named indices into data matrices
0050 [PQ, PV, REF, NONE, BUS_I, BUS_TYPE, PD, QD, GS, BS, BUS_AREA, VM, ...
0051     VA, BASE_KV, ZONE, VMAX, VMIN, LAM_P, LAM_Q, MU_VMAX, MU_VMIN] = idx_bus;
0052 [GEN_BUS, PG, QG, QMAX, QMIN, VG, MBASE, GEN_STATUS, PMAX, PMIN, ...
0053     MU_PMAX, MU_PMIN, MU_QMAX, MU_QMIN, PC1, PC2, QC1MIN, QC1MAX, ...
0054     QC2MIN, QC2MAX, RAMP_AGC, RAMP_10, RAMP_30, RAMP_Q, APF] = idx_gen;
0055 [F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, ...
0056     TAP, SHIFT, BR_STATUS, PF, QF, PT, QT, MU_SF, MU_ST, ...
0057     ANGMIN, ANGMAX, MU_ANGMIN, MU_ANGMAX] = idx_brch;
0058 [PW_LINEAR, POLYNOMIAL, MODEL, STARTUP, SHUTDOWN, NCOST, COST] = idx_cost;
0059 
0060 %% unpack data
0061 mpc = om.get_mpc();
0062 [baseMVA, bus, gen, branch, gencost] = ...
0063     deal(mpc.baseMVA, mpc.bus, mpc.gen, mpc.branch, mpc.gencost);
0064 [vv, ll, nne, nni] = om.get_idx();
0065 
0066 %% problem dimensions
0067 nb = size(bus, 1);          %% number of buses
0068 nl = size(branch, 1);       %% number of branches
0069 ny = om.getN('var', 'y');   %% number of piece-wise linear costs
0070 
0071 %% bounds on optimization vars
0072 [x0, xmin, xmax] = om.params_var();
0073 
0074 %% linear constraints
0075 [A, l, u] = om.params_lin_constraint();
0076 
0077 %% split l <= A*x <= u into less than, equal to, greater than, and
0078 %% doubly-bounded sets
0079 ieq = find( abs(u-l) <= eps );          %% equality
0080 igt = find( u >=  1e10 & l > -1e10 );   %% greater than, unbounded above
0081 ilt = find( l <= -1e10 & u <  1e10 );   %% less than, unbounded below
0082 ibx = find( (abs(u-l) > eps) & (u < 1e10) & (l > -1e10) );
0083 Af  = [ A(ilt, :); -A(igt, :); A(ibx, :); -A(ibx, :) ];
0084 bf  = [ u(ilt);   -l(igt);     u(ibx);    -l(ibx)];
0085 Afeq = A(ieq, :);
0086 bfeq = u(ieq);
0087 
0088 %% build admittance matrices
0089 [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0090 
0091 %% try to select an interior initial point, unless requested not to
0092 if mpopt.opf.start < 2
0093     s = 1;                      %% set init point inside bounds by s
0094     lb = xmin; ub = xmax;
0095     lb(xmin == -Inf) = -1e10;   %% replace Inf with numerical proxies
0096     ub(xmax ==  Inf) =  1e10;
0097     x0 = (lb + ub) / 2;         %% set x0 mid-way between bounds
0098     k = find(xmin == -Inf & xmax < Inf);    %% if only bounded above
0099     x0(k) = xmax(k) - s;                    %% set just below upper bound
0100     k = find(xmin > -Inf & xmax == Inf);    %% if only bounded below
0101     x0(k) = xmin(k) + s;                    %% set just above lower bound
0102     Varefs = bus(bus(:, BUS_TYPE) == REF, VA) * (pi/180);
0103     Vmax = min(bus(:, VMAX), 1.5);
0104     Vmin = max(bus(:, VMIN), 0.5);
0105     Vm = (Vmax + Vmin) / 2;
0106     if mpopt.opf.v_cartesian
0107         V = Vm * exp(1j*Varefs(1));
0108         x0(vv.i1.Vr:vv.iN.Vr) = real(V);
0109         x0(vv.i1.Vi:vv.iN.Vi) = imag(V);
0110     else
0111         x0(vv.i1.Va:vv.iN.Va) = Varefs(1);  %% angles set to first reference angle
0112         x0(vv.i1.Vm:vv.iN.Vm) = Vm;         %% voltage magnitudes
0113         if ny > 0
0114             ipwl = find(gencost(:, MODEL) == PW_LINEAR);
0115             c = gencost(sub2ind(size(gencost), ipwl, NCOST+2*gencost(ipwl, NCOST)));    %% largest y-value in CCV data
0116             x0(vv.i1.y:vv.iN.y) = max(c) + 0.1 * abs(max(c));
0117         end
0118     end
0119 end
0120 
0121 %% find branches with flow limits
0122 il = find(branch(:, RATE_A) ~= 0 & branch(:, RATE_A) < 1e10);
0123 nl2 = length(il);           %% number of constrained lines
0124 
0125 %% basic optimset options needed for fmincon
0126 fmoptions = optimset('GradObj', 'on', 'GradConstr', 'on', ...
0127             'TolCon', mpopt.opf.violation, 'TolX', mpopt.fmincon.tol_x, ...
0128             'TolFun', mpopt.fmincon.tol_f );
0129 if mpopt.fmincon.max_it ~= 0
0130     fmoptions = optimset(fmoptions, 'MaxIter', mpopt.fmincon.max_it, ...
0131             'MaxFunEvals', 4 * mpopt.fmincon.max_it);
0132 end
0133 
0134 if mpopt.verbose == 0,
0135   fmoptions.Display = 'off';
0136 elseif mpopt.verbose == 1
0137   fmoptions.Display = 'iter';
0138 else
0139   fmoptions.Display = 'testing';
0140 end
0141 
0142 %% select algorithm
0143 if have_fcn('fmincon_ipm')
0144   switch mpopt.fmincon.alg
0145     case 1              %% active-set (does not use sparse matrices, not suitable for large problems)
0146       fmoptions = optimset(fmoptions, 'Algorithm', 'active-set');
0147       Af = full(Af);
0148       Afeq = full(Afeq);
0149     case 2              %% interior-point, w/ default 'bfgs' Hessian approx
0150       fmoptions = optimset(fmoptions, 'Algorithm', 'interior-point');
0151     case 3              %% interior-point, w/ 'lbfgs' Hessian approx
0152       fmoptions = optimset(fmoptions, 'Algorithm', 'interior-point', 'Hessian','lbfgs');
0153     case 4              %% interior-point, w/ exact user-supplied Hessian
0154       fmc_hessian = @(x, lambda)opf_hessfcn(x, lambda, 1, om, Ybus, Yf(il,:), Yt(il,:), mpopt, il);
0155       fmoptions = optimset(fmoptions, 'Algorithm', 'interior-point', ...
0156           'Hessian', 'user-supplied', 'HessFcn', fmc_hessian);
0157     case 5              %% interior-point, w/ finite-diff Hessian
0158       fmoptions = optimset(fmoptions, 'Algorithm', 'interior-point', 'Hessian','fin-diff-grads', 'SubProblem', 'cg');
0159     case 6              %% sqp (does not use sparse matrices, not suitable for large problems)
0160       fmoptions = optimset(fmoptions, 'Algorithm', 'sqp');
0161       Af = full(Af);
0162       Afeq = full(Afeq);
0163     otherwise
0164       error('fmincopf_solver: unknown algorithm specified in ''fmincon.alg'' option');
0165   end
0166 else
0167   fmoptions = optimset(fmoptions, 'LargeScale', 'off');
0168   Af = full(Af);
0169   Afeq = full(Afeq);
0170 end
0171 % fmoptions = optimset(fmoptions, 'DerivativeCheck', 'on', 'FinDiffType', 'central', 'FunValCheck', 'on');
0172 % fmoptions = optimset(fmoptions, 'Diagnostics', 'on');
0173 
0174 %%-----  run opf  -----
0175 f_fcn = @(x)opf_costfcn(x, om);
0176 gh_fcn = @(x)opf_consfcn(x, om, Ybus, Yf(il,:), Yt(il,:), mpopt, il);
0177 [x, f, info, Output, Lambda] = ...
0178   fmincon(f_fcn, x0, Af, bf, Afeq, bfeq, xmin, xmax, gh_fcn, fmoptions);
0179 success = (info > 0);
0180 
0181 %% update solution data
0182 if mpopt.opf.v_cartesian
0183     Vi = x(vv.i1.Vi:vv.iN.Vi);
0184     Vr = x(vv.i1.Vr:vv.iN.Vr);
0185     V = Vr + 1j*Vi;
0186     Va = angle(V);
0187     Vm = abs(V);
0188 else
0189     Va = x(vv.i1.Va:vv.iN.Va);
0190     Vm = x(vv.i1.Vm:vv.iN.Vm);
0191     V = Vm .* exp(1j*Va);
0192 end
0193 Pg = x(vv.i1.Pg:vv.iN.Pg);
0194 Qg = x(vv.i1.Qg:vv.iN.Qg);
0195 
0196 %%-----  calculate return values  -----
0197 %% update voltages & generator outputs
0198 bus(:, VA) = Va * 180/pi;
0199 bus(:, VM) = Vm;
0200 gen(:, PG) = Pg * baseMVA;
0201 gen(:, QG) = Qg * baseMVA;
0202 gen(:, VG) = Vm(gen(:, GEN_BUS));
0203 
0204 %% compute branch flows
0205 Sf = V(branch(:, F_BUS)) .* conj(Yf * V);  %% cplx pwr at "from" bus, p.u.
0206 St = V(branch(:, T_BUS)) .* conj(Yt * V);  %% cplx pwr at "to" bus, p.u.
0207 branch(:, PF) = real(Sf) * baseMVA;
0208 branch(:, QF) = imag(Sf) * baseMVA;
0209 branch(:, PT) = real(St) * baseMVA;
0210 branch(:, QT) = imag(St) * baseMVA;
0211 
0212 %% line constraint is typically on square of limit
0213 %% so we must fix multipliers
0214 muSf = zeros(nl, 1);
0215 muSt = zeros(nl, 1);
0216 if ~isempty(il)
0217     if upper(mpopt.opf.flow_lim(1)) == 'P'
0218         muSf(il) = Lambda.ineqnonlin(nni.i1.Sf:nni.iN.Sf);
0219         muSt(il) = Lambda.ineqnonlin(nni.i1.St:nni.iN.St);
0220     else
0221         muSf(il) = 2 * Lambda.ineqnonlin(nni.i1.Sf:nni.iN.Sf) .* branch(il, RATE_A) / baseMVA;
0222         muSt(il) = 2 * Lambda.ineqnonlin(nni.i1.St:nni.iN.St) .* branch(il, RATE_A) / baseMVA;
0223     end
0224 end
0225 
0226 %% fix Lambdas
0227 %% (shadow prices on equality variable bounds can come back on wrong limit)
0228 kl = find(Lambda.lower < 0 & Lambda.upper == 0);
0229 Lambda.upper(kl) = -Lambda.lower(kl);
0230 Lambda.lower(kl) = 0;
0231 ku = find(Lambda.upper < 0 & Lambda.lower == 0);
0232 Lambda.lower(ku) = -Lambda.upper(ku);
0233 Lambda.upper(ku) = 0;
0234 
0235 %% update Lagrange multipliers
0236 if mpopt.opf.v_cartesian
0237     if om.userdata.veq
0238         lam = Lambda.eqnonlin(nne.i1.Veq:nne.iN.Veq);
0239         mu_Vmax = zeros(size(lam));
0240         mu_Vmin = zeros(size(lam));
0241         mu_Vmax(lam > 0) =  lam(lam > 0);
0242         mu_Vmin(lam < 0) = -lam(lam < 0);
0243         bus(om.userdata.veq, MU_VMAX) = mu_Vmax;
0244         bus(om.userdata.veq, MU_VMIN) = mu_Vmin;
0245     end
0246     bus(om.userdata.viq, MU_VMAX) = Lambda.ineqnonlin(nni.i1.Vmax:nni.iN.Vmax);
0247     bus(om.userdata.viq, MU_VMIN) = Lambda.ineqnonlin(nni.i1.Vmin:nni.iN.Vmin);
0248 else
0249     bus(:, MU_VMAX)  = Lambda.upper(vv.i1.Vm:vv.iN.Vm);
0250     bus(:, MU_VMIN)  = Lambda.lower(vv.i1.Vm:vv.iN.Vm);
0251 end
0252 gen(:, MU_PMAX)  = Lambda.upper(vv.i1.Pg:vv.iN.Pg) / baseMVA;
0253 gen(:, MU_PMIN)  = Lambda.lower(vv.i1.Pg:vv.iN.Pg) / baseMVA;
0254 gen(:, MU_QMAX)  = Lambda.upper(vv.i1.Qg:vv.iN.Qg) / baseMVA;
0255 gen(:, MU_QMIN)  = Lambda.lower(vv.i1.Qg:vv.iN.Qg) / baseMVA;
0256 if mpopt.opf.current_balance
0257     %% convert current balance shadow prices to equivalent lamP and lamQ
0258     %% P + jQ = (Vr + jVi) * (M - jN)
0259     %% M = (Vr P + Vi Q) / (Vr^2 + Vi^2)
0260     %% N = (Vi P - Vr Q) / (Vr^2 + Vi^2)
0261     %% lamP = df/dP = df/dM * dM/dP + df/dN + dN/dP
0262     %% lamQ = df/dQ = df/dM * dM/dQ + df/dN + dN/dQ
0263     VV = V ./ (V .* conj(V));   %% V / Vm^2
0264     VVr = real(VV);
0265     VVi = imag(VV);
0266     lamM = Lambda.eqnonlin(nne.i1.rImis:nne.iN.rImis);
0267     lamN = Lambda.eqnonlin(nne.i1.iImis:nne.iN.iImis);
0268     bus(:, LAM_P) = (VVr.*lamM + VVi.*lamN) / baseMVA;
0269     bus(:, LAM_Q) = (VVi.*lamM - VVr.*lamN) / baseMVA;
0270 else
0271     bus(:, LAM_P) = Lambda.eqnonlin(nne.i1.Pmis:nne.iN.Pmis) / baseMVA;
0272     bus(:, LAM_Q) = Lambda.eqnonlin(nne.i1.Qmis:nne.iN.Qmis) / baseMVA;
0273 end
0274 branch(:, MU_SF) = muSf / baseMVA;
0275 branch(:, MU_ST) = muSt / baseMVA;
0276 
0277 %% package up results
0278 nlnN = 2*nb + 2*nl;     %% because muSf and muSt are nl x 1, not nl2 x 1
0279 nlt = length(ilt);
0280 ngt = length(igt);
0281 nbx = length(ibx);
0282 
0283 %% extract multipliers for nonlinear constraints
0284 kl = find(Lambda.eqnonlin < 0);
0285 ku = find(Lambda.eqnonlin > 0);
0286 nl_mu_l = zeros(nlnN, 1);
0287 nl_mu_u = [zeros(2*nb, 1); muSf; muSt];
0288 nl_mu_l(kl) = -Lambda.eqnonlin(kl);
0289 nl_mu_u(ku) =  Lambda.eqnonlin(ku);
0290 
0291 %% extract multipliers for linear constraints
0292 kl = find(Lambda.eqlin < 0);
0293 ku = find(Lambda.eqlin > 0);
0294 
0295 mu_l = zeros(size(u));
0296 mu_l(ieq(kl)) = -Lambda.eqlin(kl);
0297 mu_l(igt) = Lambda.ineqlin(nlt+(1:ngt));
0298 mu_l(ibx) = Lambda.ineqlin(nlt+ngt+nbx+(1:nbx));
0299 
0300 mu_u = zeros(size(u));
0301 mu_u(ieq(ku)) = Lambda.eqlin(ku);
0302 mu_u(ilt) = Lambda.ineqlin(1:nlt);
0303 mu_u(ibx) = Lambda.ineqlin(nlt+ngt+(1:nbx));
0304 
0305 mu = struct( ...
0306   'var', struct('l', Lambda.lower, 'u', Lambda.upper), ...
0307   'nln', struct('l', nl_mu_l, 'u', nl_mu_u), ...
0308   'nle', Lambda.eqnonlin, ...
0309   'nli', Lambda.ineqnonlin, ...
0310   'lin', struct('l', mu_l, 'u', mu_u) );
0311 
0312 results = mpc;
0313 [results.bus, results.branch, results.gen, ...
0314     results.om, results.x, results.mu, results.f] = ...
0315         deal(bus, branch, gen, om, x, mu, f);
0316 
0317 pimul = [ ...
0318   results.mu.nln.l - results.mu.nln.u;
0319   results.mu.lin.l - results.mu.lin.u;
0320   -ones(ny>0, 1);
0321   results.mu.var.l - results.mu.var.u;
0322 ];
0323 raw = struct('xr', x, 'pimul', pimul, 'info', info, 'output', Output);
fmincopf_solver

PURPOSE

SYNOPSIS

DESCRIPTION

CROSS-REFERENCE INFORMATION

SOURCE CODE
