dIbr_dV

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

0001 function [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = dIbr_dV(branch, Yf, Yt, V)
0002 %DIBR_DV   Computes partial derivatives of branch currents w.r.t. voltage.
0003 %   [DIF_DVA, DIF_DVM, DIT_DVA, DIT_DVM, IF, IT] = DIBR_DV(BRANCH, YF, YT, V)
0004 %   returns four matrices containing partial derivatives of the complex
0005 %   branch currents 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 currents themselves. The following
0009 %   explains the expressions used to form the matrices:
0010 %
0011 %   If = Yf * V;
0012 %
0013 %   Partials of V, Vf & If w.r.t. voltage angles
0014 %       dV/dVa  = j * diag(V)
0015 %       dVf/dVa = sparse(1:nl, f, j * V(f)) = j * sparse(1:nl, f, V(f))
0016 %       dIf/dVa = Yf * dV/dVa = Yf * j * diag(V)
0017 %
0018 %   Partials of V, Vf & If w.r.t. voltage magnitudes
0019 %       dV/dVm  = diag(V./abs(V))
0020 %       dVf/dVm = sparse(1:nl, f, V(f)./abs(V(f))
0021 %       dIf/dVm = Yf * dV/dVm = Yf * diag(V./abs(V))
0022 %
0023 %   Derivations for "to" bus are similar.
0024 %
0025 %   Example:
0026 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0027 %       [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = ...
0028 %           dIbr_dV(branch, Yf, Yt, V);
0029 %
0030 %   For more details on the derivations behind the derivative code used
0031 %   in MATPOWER information, see:
0032 %
0033 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0034 %          their Derivatives using Complex Matrix Notation", MATPOWER
0035 %          Technical Note 2, February 2010.
0036 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0037 
0038 %   MATPOWER
0039 %   Copyright (c) 1996-2015 by Power System Engineering Research Center (PSERC)
0040 %   by Ray Zimmerman, PSERC Cornell
0041 %
0042 %   $Id: dIbr_dV.m 2644 2015-03-11 19:34:22Z ray $
0043 %
0044 %   This file is part of MATPOWER.
0045 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0046 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0047 
0048 %% define
0049 nb = length(V);
0050 
0051 Vnorm = V ./ abs(V);
0052 if issparse(Yf)             %% sparse version (if Yf is sparse)
0053     diagV       = sparse(1:nb, 1:nb, V, nb, nb);
0054     diagVnorm   = sparse(1:nb, 1:nb, Vnorm, nb, nb);
0055 else                        %% dense version
0056     diagV       = diag(V);
0057     diagVnorm   = diag(Vnorm);
0058 end
0059 dIf_dVa = Yf * 1j * diagV;
0060 dIf_dVm = Yf * diagVnorm;
0061 dIt_dVa = Yt * 1j * diagV;
0062 dIt_dVm = Yt * diagVnorm;
0063 
0064 %% compute currents
0065 if nargout > 4
0066     If = Yf * V;
0067     It = Yt * V;
0068 end