D2AIBR_DV2 Computes 2nd derivatives of |complex current|^2 w.r.t. V. [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, 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 square of the magnitude of the branch currents. Takes sparse first derivative matrices of complex flow, complex flow vector, 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); [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = ... dIbr_dV(branch, Yf, Yt, V); Cbr = Cf; Ybr = Yf; dIbr_dVa = dIf_dVa; dIbr_dVm = dIf_dVm; Ibr = If; [Haa, Hav, Hva, Hvv] = ... d2AIbr_dV2(dIbr_dVa, dIbr_dVm, Ibr, Ybr, V, lam); Here the output matrices correspond to: Haa = (d/dVa (dAIbr_dVa.')) * lam Hav = (d/dVm (dAIbr_dVa.')) * lam Hva = (d/dVa (dAIbr_dVm.')) * lam Hvv = (d/dVm (dAIbr_dVm.')) * lam See also DIBR_DV. 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] = ... 0002 d2AIbr_dV2(dIbr_dVa, dIbr_dVm, Ibr, Ybr, V, lam) 0003 %D2AIBR_DV2 Computes 2nd derivatives of |complex current|^2 w.r.t. V. 0004 % [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, LAM) 0005 % returns 4 matrices containing the partial derivatives w.r.t. voltage 0006 % angle and magnitude of the product of a vector LAM with the 1st partial 0007 % derivatives of the square of the magnitude of the branch currents. 0008 % Takes sparse first derivative matrices of complex flow, complex flow 0009 % vector, sparse branch admittance matrix YBR, voltage vector V and 0010 % nl x 1 vector of multipliers LAM. Output matrices are sparse. 0011 % 0012 % Example: 0013 % f = branch(:, F_BUS); 0014 % Cf = sparse(1:nl, f, ones(nl, 1), nl, nb); 0015 % [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); 0016 % [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = ... 0017 % dIbr_dV(branch, Yf, Yt, V); 0018 % Cbr = Cf; 0019 % Ybr = Yf; 0020 % dIbr_dVa = dIf_dVa; 0021 % dIbr_dVm = dIf_dVm; 0022 % Ibr = If; 0023 % [Haa, Hav, Hva, Hvv] = ... 0024 % d2AIbr_dV2(dIbr_dVa, dIbr_dVm, Ibr, Ybr, V, lam); 0025 % 0026 % Here the output matrices correspond to: 0027 % Haa = (d/dVa (dAIbr_dVa.')) * lam 0028 % Hav = (d/dVm (dAIbr_dVa.')) * lam 0029 % Hva = (d/dVa (dAIbr_dVm.')) * lam 0030 % Hvv = (d/dVm (dAIbr_dVm.')) * lam 0031 % 0032 % See also DIBR_DV. 0033 % 0034 % For more details on the derivations behind the derivative code used 0035 % in MATPOWER information, see: 0036 % 0037 % [TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and 0038 % their Derivatives using Complex Matrix Notation", MATPOWER 0039 % Technical Note 2, February 2010. 0040 % http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf 0041 0042 % MATPOWER 0043 % Copyright (c) 2008-2015 by Power System Engineering Research Center (PSERC) 0044 % by Ray Zimmerman, PSERC Cornell 0045 % 0046 % $Id: d2AIbr_dV2.m 2644 2015-03-11 19:34:22Z ray $ 0047 % 0048 % This file is part of MATPOWER. 0049 % Covered by the 3-clause BSD License (see LICENSE file for details). 0050 % See http://www.pserc.cornell.edu/matpower/ for more info. 0051 0052 %% define 0053 nl = length(lam); 0054 0055 diaglam = sparse(1:nl, 1:nl, lam, nl, nl); 0056 diagIbr_conj = sparse(1:nl, 1:nl, conj(Ibr), nl, nl); 0057 0058 [Iaa, Iav, Iva, Ivv] = d2Ibr_dV2(Ybr, V, diagIbr_conj * lam); 0059 Haa = 2 * real( Iaa + dIbr_dVa.' * diaglam * conj(dIbr_dVa) ); 0060 Hva = 2 * real( Iva + dIbr_dVm.' * diaglam * conj(dIbr_dVa) ); 0061 Hav = 2 * real( Iav + dIbr_dVa.' * diaglam * conj(dIbr_dVm) ); 0062 Hvv = 2 * real( Ivv + dIbr_dVm.' * diaglam * conj(dIbr_dVm) );