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function [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = dIbr_dV(branch, Yf, Yt, V) %DIBR_DV Computes partial derivatives of branch currents w.r.t. voltage. % [DIF_DVA, DIF_DVM, DIT_DVA, DIT_DVM, IF, IT] = DIBR_DV(BRANCH, YF, YT, V) % returns four matrices containing partial derivatives of the complex % branch currents 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 currents themselves. The following % explains the expressions used to form the matrices: % % If = Yf * V; % % 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)) % % Derivations for "to" bus are similar. % % Example: % [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); % [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = ... % dIbr_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 % MATPOWER % $Id: dIbr_dV.m,v 1.10 2010/11/16 16:05:47 cvs Exp $ % by Ray Zimmerman, PSERC Cornell % Copyright (c) 1996-2010 by Power System Engineering Research Center (PSERC) % % This file is part of MATPOWER. % See http://www.pserc.cornell.edu/matpower/ for more info. % % MATPOWER is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published % by the Free Software Foundation, either version 3 of the License, % or (at your option) any later version. % % MATPOWER is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with MATPOWER. If not, see <http://www.gnu.org/licenses/>. % % Additional permission under GNU GPL version 3 section 7 % % If you modify MATPOWER, or any covered work, to interface with % other modules (such as MATLAB code and MEX-files) available in a % MATLAB(R) or comparable environment containing parts covered % under other licensing terms, the licensors of MATPOWER grant % you additional permission to convey the resulting work. %% define nb = length(V); Vnorm = V ./ abs(V); if issparse(Yf) %% sparse version (if Yf is sparse) diagV = sparse(1:nb, 1:nb, V, nb, nb); diagVnorm = sparse(1:nb, 1:nb, Vnorm, nb, nb); else %% dense version diagV = diag(V); diagVnorm = diag(Vnorm); end dIf_dVa = Yf * 1j * diagV; dIf_dVm = Yf * diagVnorm; dIt_dVa = Yt * 1j * diagV; dIt_dVm = Yt * diagVnorm; %% compute currents if nargout > 4 If = Yf * V; It = Yt * V; end