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.pdf0001 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 % $Id: dAbr_dV.m,v 1.11 2010/11/16 16:05:47 cvs Exp $ 0069 % by Ray Zimmerman, PSERC Cornell 0070 % Copyright (c) 1996-2010 by Power System Engineering Research Center (PSERC) 0071 % 0072 % This file is part of MATPOWER. 0073 % See http://www.pserc.cornell.edu/matpower/ for more info. 0074 % 0075 % MATPOWER is free software: you can redistribute it and/or modify 0076 % it under the terms of the GNU General Public License as published 0077 % by the Free Software Foundation, either version 3 of the License, 0078 % or (at your option) any later version. 0079 % 0080 % MATPOWER is distributed in the hope that it will be useful, 0081 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0082 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0083 % GNU General Public License for more details. 0084 % 0085 % You should have received a copy of the GNU General Public License 0086 % along with MATPOWER. If not, see <http://www.gnu.org/licenses/>. 0087 % 0088 % Additional permission under GNU GPL version 3 section 7 0089 % 0090 % If you modify MATPOWER, or any covered work, to interface with 0091 % other modules (such as MATLAB code and MEX-files) available in a 0092 % MATLAB(R) or comparable environment containing parts covered 0093 % under other licensing terms, the licensors of MATPOWER grant 0094 % you additional permission to convey the resulting work. 0095 0096 %% dimensions 0097 nl = length(Sf); 0098 0099 %%----- partials w.r.t. real and reactive power flows ----- 0100 dAf_dPf = sparse(1:nl, 1:nl, 2 * real(Sf), nl, nl); 0101 dAf_dQf = sparse(1:nl, 1:nl, 2 * imag(Sf), nl, nl); 0102 dAt_dPt = sparse(1:nl, 1:nl, 2 * real(St), nl, nl); 0103 dAt_dQt = sparse(1:nl, 1:nl, 2 * imag(St), nl, nl); 0104 0105 %% partials w.r.t. voltage magnitudes and angles 0106 dAf_dVm = dAf_dPf * real(dSf_dVm) + dAf_dQf * imag(dSf_dVm); 0107 dAf_dVa = dAf_dPf * real(dSf_dVa) + dAf_dQf * imag(dSf_dVa); 0108 dAt_dVm = dAt_dPt * real(dSt_dVm) + dAt_dQt * imag(dSt_dVm); 0109 dAt_dVa = dAt_dPt * real(dSt_dVa) + dAt_dQt * imag(dSt_dVa);