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) 1996-2015 by Power System Engineering Research Center (PSERC)
0057 %   by Ray Zimmerman, PSERC Cornell
0058 %
0059 %   $Id: dSbr_dV.m 2644 2015-03-11 19:34:22Z ray $
0060 %
0061 %   This file is part of MATPOWER.
0062 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0063 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0064 
0065 %% define named indices into bus, gen, branch matrices
0066 [F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, ...
0067     TAP, SHIFT, BR_STATUS, PF, QF, PT, QT, MU_SF, MU_ST, ...
0068     ANGMIN, ANGMAX, MU_ANGMIN, MU_ANGMAX] = idx_brch;
0069 
0070 %% define
0071 f = branch(:, F_BUS);       %% list of "from" buses
0072 t = branch(:, T_BUS);       %% list of "to" buses
0073 nl = length(f);
0074 nb = length(V);
0075 
0076 %% compute currents
0077 If = Yf * V;
0078 It = Yt * V;
0079 
0080 Vnorm = V ./ abs(V);
0081 if issparse(Yf)             %% sparse version (if Yf is sparse)
0082     diagVf      = sparse(1:nl, 1:nl, V(f), nl, nl);
0083     diagIf      = sparse(1:nl, 1:nl, If, nl, nl);
0084     diagVt      = sparse(1:nl, 1:nl, V(t), nl, nl);
0085     diagIt      = sparse(1:nl, 1:nl, It, nl, nl);
0086     diagV       = sparse(1:nb, 1:nb, V, nb, nb);
0087     diagVnorm   = sparse(1:nb, 1:nb, Vnorm, nb, nb);
0088     
0089     dSf_dVa = 1j * (conj(diagIf) * sparse(1:nl, f, V(f), nl, nb) - diagVf * conj(Yf * diagV));
0090     dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * sparse(1:nl, f, Vnorm(f), nl, nb);
0091     dSt_dVa = 1j * (conj(diagIt) * sparse(1:nl, t, V(t), nl, nb) - diagVt * conj(Yt * diagV));
0092     dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * sparse(1:nl, t, Vnorm(t), nl, nb);
0093 else                        %% dense version
0094     diagVf      = diag(V(f));
0095     diagIf      = diag(If);
0096     diagVt      = diag(V(t));
0097     diagIt      = diag(It);
0098     diagV       = diag(V);
0099     diagVnorm   = diag(Vnorm);
0100     temp1       = zeros(nl, nb);    temp1(sub2ind([nl,nb], (1:nl)', f)) = V(f);
0101     temp2       = zeros(nl, nb);    temp2(sub2ind([nl,nb], (1:nl)', f)) = Vnorm(f);
0102     temp3       = zeros(nl, nb);    temp3(sub2ind([nl,nb], (1:nl)', t)) = V(t);
0103     temp4       = zeros(nl, nb);    temp4(sub2ind([nl,nb], (1:nl)', t)) = Vnorm(t);
0104     
0105     dSf_dVa = 1j * (conj(diagIf) * temp1 - diagVf * conj(Yf * diagV));
0106     dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * temp2;
0107     dSt_dVa = 1j * (conj(diagIt) * temp3 - diagVt * conj(Yt * diagV));
0108     dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * temp4;
0109 end
0110 
0111 if nargout > 4
0112     Sf = V(f) .* conj(If);
0113     St = V(t) .* conj(It);
0114 end

dSbr_dV

PURPOSE

SYNOPSIS

DESCRIPTION

CROSS-REFERENCE INFORMATION

SOURCE CODE