DSBUS_DV Computes partial derivatives of power injection w.r.t. voltage.
[DSBUS_DVM, DSBUS_DVA] = DSBUS_DV(YBUS, V) returns two matrices containing
partial derivatives of the complex bus power injections w.r.t voltage
magnitude and voltage angle respectively (for all buses). If YBUS is a
sparse matrix, the return values will be also. The following explains
the expressions used to form the matrices:
S = diag(V) * conj(Ibus) = diag(conj(Ibus)) * V
Partials of V & Ibus w.r.t. voltage magnitudes
dV/dVm = diag(V./abs(V))
dI/dVm = Ybus * dV/dVm = Ybus * diag(V./abs(V))
Partials of V & Ibus w.r.t. voltage angles
dV/dVa = j * diag(V)
dI/dVa = Ybus * dV/dVa = Ybus * j * diag(V)
Partials of S w.r.t. voltage magnitudes
dS/dVm = diag(V) * conj(dI/dVm) + diag(conj(Ibus)) * dV/dVm
= diag(V) * conj(Ybus * diag(V./abs(V)))
+ conj(diag(Ibus)) * diag(V./abs(V))
Partials of S w.r.t. voltage angles
dS/dVa = diag(V) * conj(dI/dVa) + diag(conj(Ibus)) * dV/dVa
= diag(V) * conj(Ybus * j * diag(V))
+ conj(diag(Ibus)) * j * diag(V)
= -j * diag(V) * conj(Ybus * diag(V))
+ conj(diag(Ibus)) * j * diag(V)
= j * diag(V) * conj(diag(Ibus) - Ybus * diag(V))
Example:
[Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
[dSbus_dVm, dSbus_dVa] = dSbus_dV(Ybus, 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.pdf0001 function [dSbus_dVm, dSbus_dVa] = dSbus_dV(Ybus, V) 0002 %DSBUS_DV Computes partial derivatives of power injection w.r.t. voltage. 0003 % [DSBUS_DVM, DSBUS_DVA] = DSBUS_DV(YBUS, V) returns two matrices containing 0004 % partial derivatives of the complex bus power injections w.r.t voltage 0005 % magnitude and voltage angle respectively (for all buses). If YBUS is a 0006 % sparse matrix, the return values will be also. The following explains 0007 % the expressions used to form the matrices: 0008 % 0009 % S = diag(V) * conj(Ibus) = diag(conj(Ibus)) * V 0010 % 0011 % Partials of V & Ibus w.r.t. voltage magnitudes 0012 % dV/dVm = diag(V./abs(V)) 0013 % dI/dVm = Ybus * dV/dVm = Ybus * diag(V./abs(V)) 0014 % 0015 % Partials of V & Ibus w.r.t. voltage angles 0016 % dV/dVa = j * diag(V) 0017 % dI/dVa = Ybus * dV/dVa = Ybus * j * diag(V) 0018 % 0019 % Partials of S w.r.t. voltage magnitudes 0020 % dS/dVm = diag(V) * conj(dI/dVm) + diag(conj(Ibus)) * dV/dVm 0021 % = diag(V) * conj(Ybus * diag(V./abs(V))) 0022 % + conj(diag(Ibus)) * diag(V./abs(V)) 0023 % 0024 % Partials of S w.r.t. voltage angles 0025 % dS/dVa = diag(V) * conj(dI/dVa) + diag(conj(Ibus)) * dV/dVa 0026 % = diag(V) * conj(Ybus * j * diag(V)) 0027 % + conj(diag(Ibus)) * j * diag(V) 0028 % = -j * diag(V) * conj(Ybus * diag(V)) 0029 % + conj(diag(Ibus)) * j * diag(V) 0030 % = j * diag(V) * conj(diag(Ibus) - Ybus * diag(V)) 0031 % 0032 % Example: 0033 % [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); 0034 % [dSbus_dVm, dSbus_dVa] = dSbus_dV(Ybus, V); 0035 % 0036 % For more details on the derivations behind the derivative code used 0037 % in MATPOWER information, see: 0038 % 0039 % [TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and 0040 % their Derivatives using Complex Matrix Notation", MATPOWER 0041 % Technical Note 2, February 2010. 0042 % http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf 0043 0044 % MATPOWER 0045 % $Id: dSbus_dV.m 1720 2010-11-16 16:05:47Z cvs $ 0046 % by Ray Zimmerman, PSERC Cornell 0047 % Copyright (c) 1996-2010 by Power System Engineering Research Center (PSERC) 0048 % 0049 % This file is part of MATPOWER. 0050 % See http://www.pserc.cornell.edu/matpower/ for more info. 0051 % 0052 % MATPOWER is free software: you can redistribute it and/or modify 0053 % it under the terms of the GNU General Public License as published 0054 % by the Free Software Foundation, either version 3 of the License, 0055 % or (at your option) any later version. 0056 % 0057 % MATPOWER is distributed in the hope that it will be useful, 0058 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0059 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0060 % GNU General Public License for more details. 0061 % 0062 % You should have received a copy of the GNU General Public License 0063 % along with MATPOWER. If not, see <http://www.gnu.org/licenses/>. 0064 % 0065 % Additional permission under GNU GPL version 3 section 7 0066 % 0067 % If you modify MATPOWER, or any covered work, to interface with 0068 % other modules (such as MATLAB code and MEX-files) available in a 0069 % MATLAB(R) or comparable environment containing parts covered 0070 % under other licensing terms, the licensors of MATPOWER grant 0071 % you additional permission to convey the resulting work. 0072 0073 n = length(V); 0074 Ibus = Ybus * V; 0075 0076 if issparse(Ybus) %% sparse version (if Ybus is sparse) 0077 diagV = sparse(1:n, 1:n, V, n, n); 0078 diagIbus = sparse(1:n, 1:n, Ibus, n, n); 0079 diagVnorm = sparse(1:n, 1:n, V./abs(V), n, n); 0080 else %% dense version 0081 diagV = diag(V); 0082 diagIbus = diag(Ibus); 0083 diagVnorm = diag(V./abs(V)); 0084 end 0085 0086 dSbus_dVm = diagV * conj(Ybus * diagVnorm) + conj(diagIbus) * diagVnorm; 0087 dSbus_dVa = 1j * diagV * conj(diagIbus - Ybus * diagV);