Description of dSbr

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.

   Example:
       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
       [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = ...
           dSbr_dV(branch, Yf, Yt, V);

   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 [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = dSbr_dV(branch, Yf, Yt, V)
0002 %DSBR_DV   Computes partial derivatives of power flows w.r.t. voltage.
0003 %   [DSF_DVA, DSF_DVM, DST_DVA, DST_DVM, SF, ST] = DSBR_DV(BRANCH, YF, YT, V)
0004 %   returns four matrices containing partial derivatives of the complex
0005 %   branch power flows at "from" and "to" ends of each branch w.r.t voltage
0006 %   magnitude and voltage angle respectively (for all buses). If YF is a
0007 %   sparse matrix, the partial derivative matrices will be as well. Optionally
0008 %   returns vectors containing the power flows themselves. The following
0009 %   explains the expressions used to form the matrices:
0010 %
0011 %   If = Yf * V;
0012 %   Sf = diag(Vf) * conj(If) = diag(conj(If)) * Vf
0013 %
0014 %   Partials of V, Vf & If w.r.t. voltage angles
0015 %       dV/dVa  = j * diag(V)
0016 %       dVf/dVa = sparse(1:nl, f, j * V(f)) = j * sparse(1:nl, f, V(f))
0017 %       dIf/dVa = Yf * dV/dVa = Yf * j * diag(V)
0018 %
0019 %   Partials of V, Vf & If w.r.t. voltage magnitudes
0020 %       dV/dVm  = diag(V./abs(V))
0021 %       dVf/dVm = sparse(1:nl, f, V(f)./abs(V(f))
0022 %       dIf/dVm = Yf * dV/dVm = Yf * diag(V./abs(V))
0023 %
0024 %   Partials of Sf w.r.t. voltage angles
0025 %       dSf/dVa = diag(Vf) * conj(dIf/dVa)
0026 %                       + diag(conj(If)) * dVf/dVa
0027 %               = diag(Vf) * conj(Yf * j * diag(V))
0028 %                       + conj(diag(If)) * j * sparse(1:nl, f, V(f))
0029 %               = -j * diag(Vf) * conj(Yf * diag(V))
0030 %                       + j * conj(diag(If)) * sparse(1:nl, f, V(f))
0031 %               = j * (conj(diag(If)) * sparse(1:nl, f, V(f))
0032 %                       - diag(Vf) * conj(Yf * diag(V)))
0033 %
0034 %   Partials of Sf w.r.t. voltage magnitudes
0035 %       dSf/dVm = diag(Vf) * conj(dIf/dVm)
0036 %                       + diag(conj(If)) * dVf/dVm
0037 %               = diag(Vf) * conj(Yf * diag(V./abs(V)))
0038 %                       + conj(diag(If)) * sparse(1:nl, f, V(f)./abs(V(f)))
0039 %
0040 %   Derivations for "to" bus are similar.
0041 %
0042 %   Example:
0043 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0044 %       [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = ...
0045 %           dSbr_dV(branch, Yf, Yt, V);
0046 %
0047 %   For more details on the derivations behind the derivative code used
0048 %   in MATPOWER information, see:
0049 %
0050 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0051 %          their Derivatives using Complex Matrix Notation", MATPOWER
0052 %          Technical Note 2, February 2010.
0053 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0054 
0055 %   MATPOWER
0056 %   Copyright (c) 1996-2016, Power Systems Engineering Research Center (PSERC)
0057 %   by Ray Zimmerman, PSERC Cornell
0058 %
0059 %   This file is part of MATPOWER.
0060 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0061 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0062 
0063 %% define named indices into bus, gen, branch matrices
0064 [F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, ...
0065     TAP, SHIFT, BR_STATUS, PF, QF, PT, QT, MU_SF, MU_ST, ...
0066     ANGMIN, ANGMAX, MU_ANGMIN, MU_ANGMAX] = idx_brch;
0067 
0068 %% define
0069 f = branch(:, F_BUS);       %% list of "from" buses
0070 t = branch(:, T_BUS);       %% list of "to" buses
0071 nl = length(f);
0072 nb = length(V);
0073 
0074 %% compute currents
0075 If = Yf * V;
0076 It = Yt * V;
0077 
0078 Vnorm = V ./ abs(V);
0079 if issparse(Yf)             %% sparse version (if Yf is sparse)
0080     diagVf      = sparse(1:nl, 1:nl, V(f), nl, nl);
0081     diagIf      = sparse(1:nl, 1:nl, If, nl, nl);
0082     diagVt      = sparse(1:nl, 1:nl, V(t), nl, nl);
0083     diagIt      = sparse(1:nl, 1:nl, It, nl, nl);
0084     diagV       = sparse(1:nb, 1:nb, V, nb, nb);
0085     diagVnorm   = sparse(1:nb, 1:nb, Vnorm, nb, nb);
0086     
0087     dSf_dVa = 1j * (conj(diagIf) * sparse(1:nl, f, V(f), nl, nb) - diagVf * conj(Yf * diagV));
0088     dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * sparse(1:nl, f, Vnorm(f), nl, nb);
0089     dSt_dVa = 1j * (conj(diagIt) * sparse(1:nl, t, V(t), nl, nb) - diagVt * conj(Yt * diagV));
0090     dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * sparse(1:nl, t, Vnorm(t), nl, nb);
0091 else                        %% dense version
0092     diagVf      = diag(V(f));
0093     diagIf      = diag(If);
0094     diagVt      = diag(V(t));
0095     diagIt      = diag(It);
0096     diagV       = diag(V);
0097     diagVnorm   = diag(Vnorm);
0098     temp1       = zeros(nl, nb);    temp1(sub2ind([nl,nb], (1:nl)', f)) = V(f);
0099     temp2       = zeros(nl, nb);    temp2(sub2ind([nl,nb], (1:nl)', f)) = Vnorm(f);
0100     temp3       = zeros(nl, nb);    temp3(sub2ind([nl,nb], (1:nl)', t)) = V(t);
0101     temp4       = zeros(nl, nb);    temp4(sub2ind([nl,nb], (1:nl)', t)) = Vnorm(t);
0102     
0103     dSf_dVa = 1j * (conj(diagIf) * temp1 - diagVf * conj(Yf * diagV));
0104     dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * temp2;
0105     dSt_dVa = 1j * (conj(diagIt) * temp3 - diagVt * conj(Yt * diagV));
0106     dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * temp4;
0107 end
0108 
0109 if nargout > 4
0110     Sf = V(f) .* conj(If);
0111     St = V(t) .* conj(It);
0112 end

dSbr_dV

PURPOSE

SYNOPSIS

DESCRIPTION

CROSS-REFERENCE INFORMATION

SOURCE CODE