Home > matpower5.1 > dAbr_dV.m

dAbr_dV

PURPOSE ^

DABR_DV Partial derivatives of squared flow magnitudes w.r.t voltage.

SYNOPSIS ^

function [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] =dAbr_dV(dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St)

DESCRIPTION ^

DABR_DV   Partial derivatives of squared flow magnitudes w.r.t voltage.
   [DAF_DVA, DAF_DVM, DAT_DVA, DAT_DVM] = ...
               DABR_DV(DFF_DVA, DFF_DVM, DFT_DVA, DFT_DVM, FF, FT)
   returns four matrices containing partial derivatives of the square of
   the branch flow magnitudes at "from" & "to" ends of each branch w.r.t
   voltage magnitude and voltage angle respectively (for all buses), given
   the flows and flow sensitivities. Flows could be complex current or
   complex or real power. Notation below is based on complex power. The
   following explains the expressions used to form the matrices:

   Let Af refer to the square of the apparent power at the "from" end of
   each branch,

       Af = abs(Sf).^2
          = Sf .* conj(Sf)
          = Pf.^2 + Qf.^2

   then ...

   Partial w.r.t real power,
       dAf/dPf = 2 * diag(Pf)

   Partial w.r.t reactive power,
       dAf/dQf = 2 * diag(Qf)

   Partial w.r.t Vm & Va
       dAf/dVm = dAf/dPf * dPf/dVm + dAf/dQf * dQf/dVm
       dAf/dVa = dAf/dPf * dPf/dVa + dAf/dQf * dQf/dVa

   Derivations for "to" bus are similar.

   Examples:
       %% squared current magnitude
       [dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft] = ...
               dIbr_dV(branch(il,:), Yf, Yt, V);
       [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] = ...
               dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft);

       %% squared apparent power flow
       [dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft] = ...
               dSbr_dV(branch(il,:), Yf, Yt, V);
       [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] = ...
               dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft);

       %% squared real power flow
       [dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft] = ...
               dSbr_dV(branch(il,:), Yf, Yt, V);
       dFf_dVa = real(dFf_dVa);
       dFf_dVm = real(dFf_dVm);
       dFt_dVa = real(dFt_dVa);
       dFt_dVm = real(dFt_dVm);
       [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] = ...
               dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft);

   See also DIBR_DV, DSBR_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

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] = ...
0002                         dAbr_dV(dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St)
0003 %DABR_DV   Partial derivatives of squared flow magnitudes w.r.t voltage.
0004 %   [DAF_DVA, DAF_DVM, DAT_DVA, DAT_DVM] = ...
0005 %               DABR_DV(DFF_DVA, DFF_DVM, DFT_DVA, DFT_DVM, FF, FT)
0006 %   returns four matrices containing partial derivatives of the square of
0007 %   the branch flow magnitudes at "from" & "to" ends of each branch w.r.t
0008 %   voltage magnitude and voltage angle respectively (for all buses), given
0009 %   the flows and flow sensitivities. Flows could be complex current or
0010 %   complex or real power. Notation below is based on complex power. The
0011 %   following explains the expressions used to form the matrices:
0012 %
0013 %   Let Af refer to the square of the apparent power at the "from" end of
0014 %   each branch,
0015 %
0016 %       Af = abs(Sf).^2
0017 %          = Sf .* conj(Sf)
0018 %          = Pf.^2 + Qf.^2
0019 %
0020 %   then ...
0021 %
0022 %   Partial w.r.t real power,
0023 %       dAf/dPf = 2 * diag(Pf)
0024 %
0025 %   Partial w.r.t reactive power,
0026 %       dAf/dQf = 2 * diag(Qf)
0027 %
0028 %   Partial w.r.t Vm & Va
0029 %       dAf/dVm = dAf/dPf * dPf/dVm + dAf/dQf * dQf/dVm
0030 %       dAf/dVa = dAf/dPf * dPf/dVa + dAf/dQf * dQf/dVa
0031 %
0032 %   Derivations for "to" bus are similar.
0033 %
0034 %   Examples:
0035 %       %% squared current magnitude
0036 %       [dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft] = ...
0037 %               dIbr_dV(branch(il,:), Yf, Yt, V);
0038 %       [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] = ...
0039 %               dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft);
0040 %
0041 %       %% squared apparent power flow
0042 %       [dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft] = ...
0043 %               dSbr_dV(branch(il,:), Yf, Yt, V);
0044 %       [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] = ...
0045 %               dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft);
0046 %
0047 %       %% squared real power flow
0048 %       [dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft] = ...
0049 %               dSbr_dV(branch(il,:), Yf, Yt, V);
0050 %       dFf_dVa = real(dFf_dVa);
0051 %       dFf_dVm = real(dFf_dVm);
0052 %       dFt_dVa = real(dFt_dVa);
0053 %       dFt_dVm = real(dFt_dVm);
0054 %       [dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm] = ...
0055 %               dAbr_dV(dFf_dVa, dFf_dVm, dFt_dVa, dFt_dVm, Ff, Ft);
0056 %
0057 %   See also DIBR_DV, DSBR_DV.
0058 %
0059 %   For more details on the derivations behind the derivative code used
0060 %   in MATPOWER information, see:
0061 %
0062 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0063 %          their Derivatives using Complex Matrix Notation", MATPOWER
0064 %          Technical Note 2, February 2010.
0065 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0066 
0067 %   MATPOWER
0068 %   Copyright (c) 1996-2015 by Power System Engineering Research Center (PSERC)
0069 %   by Ray Zimmerman, PSERC Cornell
0070 %
0071 %   $Id: dAbr_dV.m 2644 2015-03-11 19:34:22Z ray $
0072 %
0073 %   This file is part of MATPOWER.
0074 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0075 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0076 
0077 %% dimensions
0078 nl = length(Sf);
0079 
0080 %%----- partials w.r.t. real and reactive power flows -----
0081 dAf_dPf = sparse(1:nl, 1:nl, 2 * real(Sf), nl, nl);
0082 dAf_dQf = sparse(1:nl, 1:nl, 2 * imag(Sf), nl, nl);
0083 dAt_dPt = sparse(1:nl, 1:nl, 2 * real(St), nl, nl);
0084 dAt_dQt = sparse(1:nl, 1:nl, 2 * imag(St), nl, nl);
0085 
0086 %% partials w.r.t. voltage magnitudes and angles
0087 dAf_dVm = dAf_dPf * real(dSf_dVm) + dAf_dQf * imag(dSf_dVm);
0088 dAf_dVa = dAf_dPf * real(dSf_dVa) + dAf_dQf * imag(dSf_dVa);
0089 dAt_dVm = dAt_dPt * real(dSt_dVm) + dAt_dQt * imag(dSt_dVm);
0090 dAt_dVa = dAt_dPt * real(dSt_dVa) + dAt_dQt * imag(dSt_dVa);

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