d2Ibr_dV2

D2IBR_DV2   Computes 2nd derivatives of complex branch current w.r.t. voltage.
   [HAA, HAV, HVA, HVV] = D2IBR_DV2(CBR, YBR, 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 branch currents. Takes sparse branch admittance matrix YBR,
   voltage vector V and nl x 1 vector of multipliers LAM. Output matrices
   are sparse.

   Example:
       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
       Ybr = Yf;
       [Haa, Hav, Hva, Hvv] = d2Ibr_dV2(Ybr, V, lam);

   Here the output matrices correspond to:
       Haa = (d/dVa (dIbr_dVa.')) * lam
       Hav = (d/dVm (dIbr_dVa.')) * lam
       Hva = (d/dVa (dIbr_dVm.')) * lam
       Hvv = (d/dVm (dIbr_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 [Haa, Hav, Hva, Hvv] = d2Ibr_dV2(Ybr, V, lam)
0002 %D2IBR_DV2   Computes 2nd derivatives of complex branch current w.r.t. voltage.
0003 %   [HAA, HAV, HVA, HVV] = D2IBR_DV2(CBR, YBR, 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 branch currents. Takes sparse branch admittance matrix YBR,
0007 %   voltage vector V and nl x 1 vector of multipliers LAM. Output matrices
0008 %   are sparse.
0009 %
0010 %   Example:
0011 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0012 %       Ybr = Yf;
0013 %       [Haa, Hav, Hva, Hvv] = d2Ibr_dV2(Ybr, V, lam);
0014 %
0015 %   Here the output matrices correspond to:
0016 %       Haa = (d/dVa (dIbr_dVa.')) * lam
0017 %       Hav = (d/dVm (dIbr_dVa.')) * lam
0018 %       Hva = (d/dVa (dIbr_dVm.')) * lam
0019 %       Hvv = (d/dVm (dIbr_dVm.')) * lam
0020 %
0021 %   For more details on the derivations behind the derivative code used
0022 %   in MATPOWER information, see:
0023 %
0024 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0025 %          their Derivatives using Complex Matrix Notation", MATPOWER
0026 %          Technical Note 2, February 2010.
0027 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0028 
0029 %   MATPOWER
0030 %   $Id: d2Ibr_dV2.m 1720 2010-11-16 16:05:47Z cvs $
0031 %   by Ray Zimmerman, PSERC Cornell
0032 %   Copyright (c) 2008-2010 by Power System Engineering Research Center (PSERC)
0033 %
0034 %   This file is part of MATPOWER.
0035 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0036 %
0037 %   MATPOWER is free software: you can redistribute it and/or modify
0038 %   it under the terms of the GNU General Public License as published
0039 %   by the Free Software Foundation, either version 3 of the License,
0040 %   or (at your option) any later version.
0041 %
0042 %   MATPOWER is distributed in the hope that it will be useful,
0043 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0044 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
0045 %   GNU General Public License for more details.
0046 %
0047 %   You should have received a copy of the GNU General Public License
0048 %   along with MATPOWER. If not, see <http://www.gnu.org/licenses/>.
0049 %
0050 %   Additional permission under GNU GPL version 3 section 7
0051 %
0052 %   If you modify MATPOWER, or any covered work, to interface with
0053 %   other modules (such as MATLAB code and MEX-files) available in a
0054 %   MATLAB(R) or comparable environment containing parts covered
0055 %   under other licensing terms, the licensors of MATPOWER grant
0056 %   you additional permission to convey the resulting work.
0057 
0058 %% define
0059 nb = length(V);
0060 
0061 diaginvVm = sparse(1:nb, 1:nb, ones(nb, 1)./abs(V), nb, nb);
0062 
0063 Haa = sparse(1:nb, 1:nb, -(Ybr.' * lam) .* V, nb, nb);
0064 Hva = -1j * Haa * diaginvVm;
0065 Hav = Hva;
0066 Hvv = sparse(nb, nb);