D2SBR_DV2 Computes 2nd derivatives of complex power flow w.r.t. voltage. [HAA, HAV, HVA, HVV] = D2SBR_DV2(CBR, YBR, V, LAM) returns 4 matrices containing the partial derivatives w.r.t. voltage angle and magnitude of the product of a vector LAM with the 1st partial derivatives of the complex branch power flows. Takes sparse connection matrix CBR, sparse branch admittance matrix YBR, voltage vector V and nl x 1 vector of multipliers LAM. Output matrices are sparse. Example: f = branch(:, F_BUS); Cf = sparse(1:nl, f, ones(nl, 1), nl, nb); [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); Cbr = Cf; Ybr = Yf; [Haa, Hav, Hva, Hvv] = d2Sbr_dV2(Cbr, Ybr, V, lam); Here the output matrices correspond to: Haa = (d/dVa (dSbr_dVa.')) * lam Hav = (d/dVm (dSbr_dVa.')) * lam Hva = (d/dVa (dSbr_dVm.')) * lam Hvv = (d/dVm (dSbr_dVm.')) * lam For more details on the derivations behind the derivative code used in MATPOWER information, see: [TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and their Derivatives using Complex Matrix Notation", MATPOWER Technical Note 2, February 2010. http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0001 function [Haa, Hav, Hva, Hvv] = d2Sbr_dV2(Cbr, Ybr, V, lam) 0002 %D2SBR_DV2 Computes 2nd derivatives of complex power flow w.r.t. voltage. 0003 % [HAA, HAV, HVA, HVV] = D2SBR_DV2(CBR, YBR, V, LAM) returns 4 matrices 0004 % containing the partial derivatives w.r.t. voltage angle and magnitude 0005 % of the product of a vector LAM with the 1st partial derivatives of the 0006 % complex branch power flows. Takes sparse connection matrix CBR, sparse 0007 % branch admittance matrix YBR, voltage vector V and nl x 1 vector of 0008 % multipliers LAM. Output matrices are sparse. 0009 % 0010 % Example: 0011 % f = branch(:, F_BUS); 0012 % Cf = sparse(1:nl, f, ones(nl, 1), nl, nb); 0013 % [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); 0014 % Cbr = Cf; 0015 % Ybr = Yf; 0016 % [Haa, Hav, Hva, Hvv] = d2Sbr_dV2(Cbr, Ybr, V, lam); 0017 % 0018 % Here the output matrices correspond to: 0019 % Haa = (d/dVa (dSbr_dVa.')) * lam 0020 % Hav = (d/dVm (dSbr_dVa.')) * lam 0021 % Hva = (d/dVa (dSbr_dVm.')) * lam 0022 % Hvv = (d/dVm (dSbr_dVm.')) * lam 0023 % 0024 % For more details on the derivations behind the derivative code used 0025 % in MATPOWER information, see: 0026 % 0027 % [TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and 0028 % their Derivatives using Complex Matrix Notation", MATPOWER 0029 % Technical Note 2, February 2010. 0030 % http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf 0031 0032 % MATPOWER 0033 % $Id: d2Sbr_dV2.m 1720 2010-11-16 16:05:47Z cvs $ 0034 % by Ray Zimmerman, PSERC Cornell 0035 % Copyright (c) 2008-2010 by Power System Engineering Research Center (PSERC) 0036 % 0037 % This file is part of MATPOWER. 0038 % See http://www.pserc.cornell.edu/matpower/ for more info. 0039 % 0040 % MATPOWER is free software: you can redistribute it and/or modify 0041 % it under the terms of the GNU General Public License as published 0042 % by the Free Software Foundation, either version 3 of the License, 0043 % or (at your option) any later version. 0044 % 0045 % MATPOWER is distributed in the hope that it will be useful, 0046 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0047 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0048 % GNU General Public License for more details. 0049 % 0050 % You should have received a copy of the GNU General Public License 0051 % along with MATPOWER. If not, see <http://www.gnu.org/licenses/>. 0052 % 0053 % Additional permission under GNU GPL version 3 section 7 0054 % 0055 % If you modify MATPOWER, or any covered work, to interface with 0056 % other modules (such as MATLAB code and MEX-files) available in a 0057 % MATLAB(R) or comparable environment containing parts covered 0058 % under other licensing terms, the licensors of MATPOWER grant 0059 % you additional permission to convey the resulting work. 0060 0061 %% define 0062 nl = length(lam); 0063 nb = length(V); 0064 0065 diaglam = sparse(1:nl, 1:nl, lam, nl, nl); 0066 diagV = sparse(1:nb, 1:nb, V, nb, nb); 0067 0068 A = Ybr' * diaglam * Cbr; 0069 B = conj(diagV) * A * diagV; 0070 D = sparse(1:nb, 1:nb, (A*V) .* conj(V), nb, nb); 0071 E = sparse(1:nb, 1:nb, (A.'*conj(V)) .* V, nb, nb); 0072 F = B + B.'; 0073 G = sparse(1:nb, 1:nb, ones(nb, 1)./abs(V), nb, nb); 0074 0075 Haa = F - D - E; 0076 Hva = 1j * G * (B - B.' - D + E); 0077 Hav = Hva.'; 0078 Hvv = G * F * G;