opf_costfcn

OPF_COSTFCN  Evaluates objective function, gradient and Hessian for OPF.
   [F, DF, D2F] = OPF_COSTFCN(X, OM)

   Objective function evaluation routine for AC optimal power flow,
   suitable for use with MIPS or FMINCON. Computes objective function value,
   gradient and Hessian.

   Inputs:
     X : optimization vector
     OM : OPF model object

   Outputs:
     F   : value of objective function
     DF  : (optional) gradient of objective function (column vector)
     D2F : (optional) Hessian of objective function (sparse matrix)

   Examples:
       f = opf_costfcn(x, om);
       [f, df] = opf_costfcn(x, om);
       [f, df, d2f] = opf_costfcn(x, om);

   See also OPF_CONSFCN, OPF_HESSFCN.

0001 function [f, df, d2f] = opf_costfcn(x, om, varargin)
0002 %OPF_COSTFCN  Evaluates objective function, gradient and Hessian for OPF.
0003 %   [F, DF, D2F] = OPF_COSTFCN(X, OM)
0004 %
0005 %   Objective function evaluation routine for AC optimal power flow,
0006 %   suitable for use with MIPS or FMINCON. Computes objective function value,
0007 %   gradient and Hessian.
0008 %
0009 %   Inputs:
0010 %     X : optimization vector
0011 %     OM : OPF model object
0012 %
0013 %   Outputs:
0014 %     F   : value of objective function
0015 %     DF  : (optional) gradient of objective function (column vector)
0016 %     D2F : (optional) Hessian of objective function (sparse matrix)
0017 %
0018 %   Examples:
0019 %       f = opf_costfcn(x, om);
0020 %       [f, df] = opf_costfcn(x, om);
0021 %       [f, df, d2f] = opf_costfcn(x, om);
0022 %
0023 %   See also OPF_CONSFCN, OPF_HESSFCN.
0024 
0025 %   MATPOWER
0026 %   Copyright (c) 1996-2016, Power Systems Engineering Research Center (PSERC)
0027 %   by Carlos E. Murillo-Sanchez, PSERC Cornell & Universidad Nacional de Colombia
0028 %   and Ray Zimmerman, PSERC Cornell
0029 %
0030 %   This file is part of MATPOWER.
0031 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0032 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0033 
0034 %%----- initialize -----
0035 %% define named indices into data matrices
0036 [GEN_BUS, PG, QG, QMAX, QMIN, VG, MBASE, GEN_STATUS, PMAX, PMIN, ...
0037     MU_PMAX, MU_PMIN, MU_QMAX, MU_QMIN, PC1, PC2, QC1MIN, QC1MAX, ...
0038     QC2MIN, QC2MAX, RAMP_AGC, RAMP_10, RAMP_30, RAMP_Q, APF] = idx_gen;
0039 [F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, ...
0040     TAP, SHIFT, BR_STATUS, PF, QF, PT, QT, MU_SF, MU_ST, ...
0041     ANGMIN, ANGMAX, MU_ANGMIN, MU_ANGMAX] = idx_brch;
0042 [PW_LINEAR, POLYNOMIAL, MODEL, STARTUP, SHUTDOWN, NCOST, COST] = idx_cost;
0043 
0044 %% unpack data
0045 mpc = get_mpc(om);
0046 [baseMVA, gen, gencost] = deal(mpc.baseMVA, mpc.gen, mpc.gencost);
0047 cp = get_cost_params(om);
0048 [N, Cw, H, dd, rh, kk, mm] = deal(cp.N, cp.Cw, cp.H, cp.dd, ...
0049                                     cp.rh, cp.kk, cp.mm);
0050 vv = get_idx(om);
0051 
0052 %% problem dimensions
0053 ng = size(gen, 1);          %% number of dispatchable injections
0054 ny = getN(om, 'var', 'y');  %% number of piece-wise linear costs
0055 nxyz = length(x);           %% total number of control vars of all types
0056 
0057 %% grab Pg & Qg
0058 Pg = x(vv.i1.Pg:vv.iN.Pg);  %% active generation in p.u.
0059 Qg = x(vv.i1.Qg:vv.iN.Qg);  %% reactive generation in p.u.
0060 
0061 %%----- evaluate objective function -----
0062 %% polynomial cost of P and Q
0063 % use totcost only on polynomial cost; in the minimization problem
0064 % formulation, pwl cost is the sum of the y variables.
0065 ipol = find(gencost(:, MODEL) == POLYNOMIAL);   %% poly MW and MVAr costs
0066 xx = [ Pg; Qg ] * baseMVA;
0067 if ~isempty(ipol)
0068   f = sum( totcost(gencost(ipol, :), xx(ipol)) );  %% cost of poly P or Q
0069 else
0070   f = 0;
0071 end
0072 
0073 %% piecewise linear cost of P and Q
0074 if ny > 0
0075   ccost = full(sparse(ones(1,ny), vv.i1.y:vv.iN.y, ones(1,ny), 1, nxyz));
0076   f = f + ccost * x;
0077 else
0078   ccost = zeros(1, nxyz);
0079 end
0080 
0081 %% generalized cost term
0082 if ~isempty(N)
0083     nw = size(N, 1);
0084     r = N * x - rh;                 %% Nx - rhat
0085     iLT = find(r < -kk);            %% below dead zone
0086     iEQ = find(r == 0 & kk == 0);   %% dead zone doesn't exist
0087     iGT = find(r > kk);             %% above dead zone
0088     iND = [iLT; iEQ; iGT];          %% rows that are Not in the Dead region
0089     iL = find(dd == 1);             %% rows using linear function
0090     iQ = find(dd == 2);             %% rows using quadratic function
0091     LL = sparse(iL, iL, 1, nw, nw);
0092     QQ = sparse(iQ, iQ, 1, nw, nw);
0093     kbar = sparse(iND, iND, [   ones(length(iLT), 1);
0094                                 zeros(length(iEQ), 1);
0095                                 -ones(length(iGT), 1)], nw, nw) * kk;
0096     rr = r + kbar;                  %% apply non-dead zone shift
0097     M = sparse(iND, iND, mm(iND), nw, nw);  %% dead zone or scale
0098     diagrr = sparse(1:nw, 1:nw, rr, nw, nw);
0099     
0100     %% linear rows multiplied by rr(i), quadratic rows by rr(i)^2
0101     w = M * (LL + QQ * diagrr) * rr;
0102 
0103     f = f + (w' * H * w) / 2 + Cw' * w;
0104 end
0105 
0106 %%----- evaluate cost gradient -----
0107 if nargout > 1
0108   %% index ranges
0109   iPg = vv.i1.Pg:vv.iN.Pg;
0110   iQg = vv.i1.Qg:vv.iN.Qg;
0111 
0112   %% polynomial cost of P and Q
0113   df_dPgQg = zeros(2*ng, 1);        %% w.r.t p.u. Pg and Qg
0114   df_dPgQg(ipol) = baseMVA * polycost(gencost(ipol, :), xx(ipol), 1);
0115   df = zeros(nxyz, 1);
0116   df(iPg) = df_dPgQg(1:ng);
0117   df(iQg) = df_dPgQg((1:ng) + ng);
0118 
0119   %% piecewise linear cost of P and Q
0120   df = df + ccost';  % As in MINOS, the linear cost row is additive wrt
0121                      % any nonlinear cost.
0122 
0123   %% generalized cost term
0124   if ~isempty(N)
0125     HwC = H * w + Cw;
0126     AA = N' * M * (LL + 2 * QQ * diagrr);
0127     df = df + AA * HwC;
0128     
0129     %% numerical check
0130     if 0    %% 1 to check, 0 to skip check
0131       ddff = zeros(size(df));
0132       step = 1e-7;
0133       tol  = 1e-3;
0134       for k = 1:length(x)
0135         xx = x;
0136         xx(k) = xx(k) + step;
0137         ddff(k) = (opf_costfcn(xx, om) - f) / step;
0138       end
0139       if max(abs(ddff - df)) > tol
0140         idx = find(abs(ddff - df) == max(abs(ddff - df)));
0141         fprintf('\nMismatch in gradient\n');
0142         fprintf('idx             df(num)         df              diff\n');
0143         fprintf('%4d%16g%16g%16g\n', [ 1:length(df); ddff'; df'; abs(ddff - df)' ]);
0144         fprintf('MAX\n');
0145         fprintf('%4d%16g%16g%16g\n', [ idx'; ddff(idx)'; df(idx)'; abs(ddff(idx) - df(idx))' ]);
0146         fprintf('\n');
0147       end
0148     end     %% numeric check
0149   end
0150 
0151   %% ---- evaluate cost Hessian -----
0152   if nargout > 2
0153     pcost = gencost(1:ng, :);
0154     if size(gencost, 1) > ng
0155         qcost = gencost(ng+1:2*ng, :);
0156     else
0157         qcost = [];
0158     end
0159     
0160     %% polynomial generator costs
0161     d2f_dPg2 = sparse(ng, 1);               %% w.r.t. p.u. Pg
0162     d2f_dQg2 = sparse(ng, 1);               %% w.r.t. p.u. Qg
0163     ipolp = find(pcost(:, MODEL) == POLYNOMIAL);
0164     d2f_dPg2(ipolp) = baseMVA^2 * polycost(pcost(ipolp, :), Pg(ipolp)*baseMVA, 2);
0165     if ~isempty(qcost)          %% Qg is not free
0166         ipolq = find(qcost(:, MODEL) == POLYNOMIAL);
0167         d2f_dQg2(ipolq) = baseMVA^2 * polycost(qcost(ipolq, :), Qg(ipolq)*baseMVA, 2);
0168     end
0169     i = [iPg iQg]';
0170     d2f = sparse(i, i, [d2f_dPg2; d2f_dQg2], nxyz, nxyz);
0171 
0172     %% generalized cost
0173     if ~isempty(N)
0174         d2f = d2f + AA * H * AA' + 2 * N' * M * QQ * sparse(1:nw, 1:nw, HwC, nw, nw) * N;
0175     end
0176   end
0177 end