Home > matpower5.0 > dSbr_dV.m

dSbr_dV

PURPOSE ^

DSBR_DV Computes partial derivatives of power flows w.r.t. voltage.

SYNOPSIS ^

function [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = dSbr_dV(branch, Yf, Yt, V)

DESCRIPTION ^

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

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

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 %   $Id: dSbr_dV.m 1720 2010-11-16 16:05:47Z cvs $
0057 %   by Ray Zimmerman, PSERC Cornell
0058 %   Copyright (c) 1996-2010 by Power System Engineering Research Center (PSERC)
0059 %
0060 %   This file is part of MATPOWER.
0061 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0062 %
0063 %   MATPOWER is free software: you can redistribute it and/or modify
0064 %   it under the terms of the GNU General Public License as published
0065 %   by the Free Software Foundation, either version 3 of the License,
0066 %   or (at your option) any later version.
0067 %
0068 %   MATPOWER is distributed in the hope that it will be useful,
0069 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0070 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
0071 %   GNU General Public License for more details.
0072 %
0073 %   You should have received a copy of the GNU General Public License
0074 %   along with MATPOWER. If not, see <http://www.gnu.org/licenses/>.
0075 %
0076 %   Additional permission under GNU GPL version 3 section 7
0077 %
0078 %   If you modify MATPOWER, or any covered work, to interface with
0079 %   other modules (such as MATLAB code and MEX-files) available in a
0080 %   MATLAB(R) or comparable environment containing parts covered
0081 %   under other licensing terms, the licensors of MATPOWER grant
0082 %   you additional permission to convey the resulting work.
0083 
0084 %% define named indices into bus, gen, branch matrices
0085 [F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, ...
0086     TAP, SHIFT, BR_STATUS, PF, QF, PT, QT, MU_SF, MU_ST, ...
0087     ANGMIN, ANGMAX, MU_ANGMIN, MU_ANGMAX] = idx_brch;
0088 
0089 %% define
0090 f = branch(:, F_BUS);       %% list of "from" buses
0091 t = branch(:, T_BUS);       %% list of "to" buses
0092 nl = length(f);
0093 nb = length(V);
0094 
0095 %% compute currents
0096 If = Yf * V;
0097 It = Yt * V;
0098 
0099 Vnorm = V ./ abs(V);
0100 if issparse(Yf)             %% sparse version (if Yf is sparse)
0101     diagVf      = sparse(1:nl, 1:nl, V(f), nl, nl);
0102     diagIf      = sparse(1:nl, 1:nl, If, nl, nl);
0103     diagVt      = sparse(1:nl, 1:nl, V(t), nl, nl);
0104     diagIt      = sparse(1:nl, 1:nl, It, nl, nl);
0105     diagV       = sparse(1:nb, 1:nb, V, nb, nb);
0106     diagVnorm   = sparse(1:nb, 1:nb, Vnorm, nb, nb);
0107     
0108     dSf_dVa = 1j * (conj(diagIf) * sparse(1:nl, f, V(f), nl, nb) - diagVf * conj(Yf * diagV));
0109     dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * sparse(1:nl, f, Vnorm(f), nl, nb);
0110     dSt_dVa = 1j * (conj(diagIt) * sparse(1:nl, t, V(t), nl, nb) - diagVt * conj(Yt * diagV));
0111     dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * sparse(1:nl, t, Vnorm(t), nl, nb);
0112 else                        %% dense version
0113     diagVf      = diag(V(f));
0114     diagIf      = diag(If);
0115     diagVt      = diag(V(t));
0116     diagIt      = diag(It);
0117     diagV       = diag(V);
0118     diagVnorm   = diag(Vnorm);
0119     temp1       = zeros(nl, nb);    temp1(sub2ind([nl,nb], (1:nl)', f)) = V(f);
0120     temp2       = zeros(nl, nb);    temp2(sub2ind([nl,nb], (1:nl)', f)) = Vnorm(f);
0121     temp3       = zeros(nl, nb);    temp3(sub2ind([nl,nb], (1:nl)', t)) = V(t);
0122     temp4       = zeros(nl, nb);    temp4(sub2ind([nl,nb], (1:nl)', t)) = Vnorm(t);
0123     
0124     dSf_dVa = 1j * (conj(diagIf) * temp1 - diagVf * conj(Yf * diagV));
0125     dSf_dVm = diagVf * conj(Yf * diagVnorm) + conj(diagIf) * temp2;
0126     dSt_dVa = 1j * (conj(diagIt) * temp3 - diagVt * conj(Yt * diagV));
0127     dSt_dVm = diagVt * conj(Yt * diagVnorm) + conj(diagIt) * temp4;
0128 end
0129 
0130 if nargout > 4
0131     Sf = V(f) .* conj(If);
0132     St = V(t) .* conj(It);
0133 end

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