function [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = dSbr_dV(branch, Yf, Yt, V)%DSBR_DV   Computes partial derivatives of power flows w.r.t. voltage.%   [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = dSbr_dV(branch, Yf, Yt, V)%   returns four matrices containing partial derivatives of the complex%   branch power flows at "from" and "to" ends of each branch w.r.t voltage%   magnitude and voltage angle respectively (for all buses). If Yf is a%   sparse matrix, the partial derivative matrices will be as well. Optionally%   returns vectors containing the power flows themselves. The following%   explains the expressions used to form the matrices:%%   If = Yf * V;%   Sf = diag(Vf) * conj(If) = diag(conj(If)) * Vf%%   Partials of V, Vf & If w.r.t. voltage angles%       dV/dVa  = j * diag(V)%       dVf/dVa = sparse(1:nl, f, j * V(f)) = j * sparse(1:nl, f, V(f))%       dIf/dVa = Yf * dV/dVa = Yf * j * diag(V)%%   Partials of V, Vf & If w.r.t. voltage magnitudes%       dV/dVm  = diag(V./abs(V))%       dVf/dVm = sparse(1:nl, f, V(f)./abs(V(f))%       dIf/dVm = Yf * dV/dVm = Yf * diag(V./abs(V))%%   Partials of Sf w.r.t. voltage angles%       dSf/dVa = diag(Vf) * conj(dIf/dVa)              + diag(conj(If)) * dVf/dVa%               = diag(Vf) * conj(Yf * j * diag(V))     + conj(diag(If)) * j * sparse(1:nl, f, V(f))%               = -j * diag(Vf) * conj(Yf * diag(V))    + j * conj(diag(If)) * sparse(1:nl, f, V(f))%               = j * (conj(diag(If)) * sparse(1:nl, f, V(f)) - diag(Vf) * conj(Yf * diag(V)))%%   Partials of Sf w.r.t. voltage magnitudes%       dSf/dVm = diag(Vf) * conj(dIf/dVm)              + diag(conj(If)) * dVf/dVm%               = diag(Vf) * conj(Yf * diag(V./abs(V))) + conj(diag(If)) * sparse(1:nl, f, V(f)./abs(V(f)))%%   Derivations for "to" bus are similar.%%   MATPOWER%   $Id: dSbr_dV.m,v 1.5 2004/08/23 20:56:13 ray Exp $%   by Ray Zimmerman, PSERC Cornell%   Copyright (c) 1996-2004 by Power System Engineering Research Center (PSERC)%   See http://www.pserc.cornell.edu/matpower/ for more info.%% constantj = sqrt(-1);%% define named indices into bus, gen, branch matrices[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] = idx_brch;%% definef = branch(:, F_BUS);       %% list of "from" busest = branch(:, T_BUS);       %% list of "to" busesnl = length(f);nb = length(V);%% compute currentsIf = Yf * V;It = Yt * V;Vnorm = V ./ abs(V);if issparse(Yf)             %% sparse version (if Yf is sparse)    diagVf      = spdiags(V(f), 0, nl, nl);    diagIf      = spdiags(If, 0, nl, nl);    diagVt      = spdiags(V(t), 0, nl, nl);    diagIt      = spdiags(It, 0, nl, nl);    diagV       = spdiags(V, 0, nb, nb);    diagVnorm   = spdiags(Vnorm, 0, nb, nb);        dSf_dVa = j * (conj(diagIf) * sparse(1:nl, f, V(f), nl, nb) - diagVf * conj(Yf * diagV));    dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * sparse(1:nl, f, Vnorm(f), nl, nb);    dSt_dVa = j * (conj(diagIt) * sparse(1:nl, t, V(t), nl, nb) - diagVt * conj(Yt * diagV));    dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * sparse(1:nl, t, Vnorm(t), nl, nb);else                        %% dense version    diagVf      = diag(V(f));    diagIf      = diag(If);    diagVt      = diag(V(t));    diagIt      = diag(It);    diagV       = diag(V);    diagVnorm   = diag(Vnorm);    temp1       = zeros(nl, nb);    temp1(sub2ind([nl,nb],[1:nl]', f)) = V(f);    temp2       = zeros(nl, nb);    temp2(sub2ind([nl,nb],[1:nl]', f)) = Vnorm(f);    temp3       = zeros(nl, nb);    temp3(sub2ind([nl,nb],[1:nl]', t)) = V(t);    temp4       = zeros(nl, nb);    temp4(sub2ind([nl,nb],[1:nl]', t)) = Vnorm(t);        dSf_dVa = j * (conj(diagIf) * temp1 - diagVf * conj(Yf * diagV));    dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * temp2;    dSt_dVa = j * (conj(diagIt) * temp3 - diagVt * conj(Yt * diagV));    dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * temp4;endif nargout > 4    Sf = V(f) .* conj(If);    St = V(t) .* conj(It);endreturn;