d2Sbus_dV2

D2SBUS_DV2   Computes 2nd derivatives of power injection w.r.t. voltage.
   [GAA, GAV, GVA, GVV] = D2SBUS_DV2(YBUS, V, LAM) returns 4 matrices
   containing the partial derivatives w.r.t. voltage angle and magnitude
   of the product of a vector LAM with the 1st partial derivatives of the
   complex bus power injections. Takes sparse bus admittance matrix YBUS,
   voltage vector V and nb x 1 vector of multipliers LAM. Output matrices
   are sparse.

   Example:
       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
       [Gaa, Gav, Gva, Gvv] = d2Sbus_dV2(Ybus, V, lam);

   Here the output matrices correspond to:
       Gaa = (d/dVa (dSbus_dVa.')) * lam
       Gav = (d/dVm (dSbus_dVa.')) * lam
       Gva = (d/dVa (dSbus_dVm.')) * lam
       Gvv = (d/dVm (dSbus_dVm.')) * lam

   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 [Gaa, Gav, Gva, Gvv] = d2Sbus_dV2(Ybus, V, lam)
0002 %D2SBUS_DV2   Computes 2nd derivatives of power injection w.r.t. voltage.
0003 %   [GAA, GAV, GVA, GVV] = D2SBUS_DV2(YBUS, V, LAM) returns 4 matrices
0004 %   containing the partial derivatives w.r.t. voltage angle and magnitude
0005 %   of the product of a vector LAM with the 1st partial derivatives of the
0006 %   complex bus power injections. Takes sparse bus admittance matrix YBUS,
0007 %   voltage vector V and nb x 1 vector of multipliers LAM. Output matrices
0008 %   are sparse.
0009 %
0010 %   Example:
0011 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0012 %       [Gaa, Gav, Gva, Gvv] = d2Sbus_dV2(Ybus, V, lam);
0013 %
0014 %   Here the output matrices correspond to:
0015 %       Gaa = (d/dVa (dSbus_dVa.')) * lam
0016 %       Gav = (d/dVm (dSbus_dVa.')) * lam
0017 %       Gva = (d/dVa (dSbus_dVm.')) * lam
0018 %       Gvv = (d/dVm (dSbus_dVm.')) * lam
0019 %
0020 %   For more details on the derivations behind the derivative code used
0021 %   in MATPOWER information, see:
0022 %
0023 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0024 %          their Derivatives using Complex Matrix Notation", MATPOWER
0025 %          Technical Note 2, February 2010.
0026 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0027 
0028 %   MATPOWER
0029 %   $Id: d2Sbus_dV2.m 1720 2010-11-16 16:05:47Z cvs $
0030 %   by Ray Zimmerman, PSERC Cornell
0031 %   Copyright (c) 2008-2010 by Power System Engineering Research Center (PSERC)
0032 %
0033 %   This file is part of MATPOWER.
0034 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0035 %
0036 %   MATPOWER is free software: you can redistribute it and/or modify
0037 %   it under the terms of the GNU General Public License as published
0038 %   by the Free Software Foundation, either version 3 of the License,
0039 %   or (at your option) any later version.
0040 %
0041 %   MATPOWER is distributed in the hope that it will be useful,
0042 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0043 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
0044 %   GNU General Public License for more details.
0045 %
0046 %   You should have received a copy of the GNU General Public License
0047 %   along with MATPOWER. If not, see <http://www.gnu.org/licenses/>.
0048 %
0049 %   Additional permission under GNU GPL version 3 section 7
0050 %
0051 %   If you modify MATPOWER, or any covered work, to interface with
0052 %   other modules (such as MATLAB code and MEX-files) available in a
0053 %   MATLAB(R) or comparable environment containing parts covered
0054 %   under other licensing terms, the licensors of MATPOWER grant
0055 %   you additional permission to convey the resulting work.
0056 
0057 n = length(V);
0058 Ibus    = Ybus * V;
0059 diaglam = sparse(1:n, 1:n, lam, n, n);
0060 diagV   = sparse(1:n, 1:n, V, n, n);
0061 
0062 A = sparse(1:n, 1:n, lam .* V, n, n);
0063 B = Ybus * diagV;
0064 C = A * conj(B);
0065 D = Ybus' * diagV;
0066 E = conj(diagV) * (D * diaglam - sparse(1:n, 1:n, D*lam, n, n));
0067 F = C - A * sparse(1:n, 1:n, conj(Ibus), n, n);
0068 G = sparse(1:n, 1:n, ones(n, 1)./abs(V), n, n);
0069 
0070 Gaa = E + F;
0071 Gva = 1j * G * (E - F);
0072 Gav = Gva.';
0073 Gvv = G * (C + C.') * G;