d2AIbr_dV2

D2AIBR_DV2   Computes 2nd derivatives of |complex current|^2 w.r.t. V.
   [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, 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 square of the magnitude of the branch currents.
   Takes sparse first derivative matrices of complex flow, complex flow
   vector, sparse branch admittance matrix YBR, voltage vector V and
   nl x 1 vector of multipliers LAM. Output matrices are sparse.

   Example:
       f = branch(:, F_BUS);
       Cf =  sparse(1:nl, f, ones(nl, 1), nl, nb);
       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
       [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = ...
               dIbr_dV(branch, Yf, Yt, V);
       Cbr = Cf;
       Ybr = Yf;
       dIbr_dVa = dIf_dVa;
       dIbr_dVm = dIf_dVm;
       Ibr = If;
       [Haa, Hav, Hva, Hvv] = ...
             d2AIbr_dV2(dIbr_dVa, dIbr_dVm, Ibr, Ybr, V, lam);

   Here the output matrices correspond to:
     Haa = (d/dVa (dAIbr_dVa.')) * lam
     Hav = (d/dVm (dAIbr_dVa.')) * lam
     Hva = (d/dVa (dAIbr_dVm.')) * lam
     Hvv = (d/dVm (dAIbr_dVm.')) * lam

   See also DIBR_DV.

   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] = ...
0002     d2AIbr_dV2(dIbr_dVa, dIbr_dVm, Ibr, Ybr, V, lam)
0003 %D2AIBR_DV2   Computes 2nd derivatives of |complex current|^2 w.r.t. V.
0004 %   [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, LAM)
0005 %   returns 4 matrices containing the partial derivatives w.r.t. voltage
0006 %   angle and magnitude of the product of a vector LAM with the 1st partial
0007 %   derivatives of the square of the magnitude of the branch currents.
0008 %   Takes sparse first derivative matrices of complex flow, complex flow
0009 %   vector, sparse branch admittance matrix YBR, voltage vector V and
0010 %   nl x 1 vector of multipliers LAM. Output matrices are sparse.
0011 %
0012 %   Example:
0013 %       f = branch(:, F_BUS);
0014 %       Cf =  sparse(1:nl, f, ones(nl, 1), nl, nb);
0015 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0016 %       [dIf_dVa, dIf_dVm, dIt_dVa, dIt_dVm, If, It] = ...
0017 %               dIbr_dV(branch, Yf, Yt, V);
0018 %       Cbr = Cf;
0019 %       Ybr = Yf;
0020 %       dIbr_dVa = dIf_dVa;
0021 %       dIbr_dVm = dIf_dVm;
0022 %       Ibr = If;
0023 %       [Haa, Hav, Hva, Hvv] = ...
0024 %             d2AIbr_dV2(dIbr_dVa, dIbr_dVm, Ibr, Ybr, V, lam);
0025 %
0026 %   Here the output matrices correspond to:
0027 %     Haa = (d/dVa (dAIbr_dVa.')) * lam
0028 %     Hav = (d/dVm (dAIbr_dVa.')) * lam
0029 %     Hva = (d/dVa (dAIbr_dVm.')) * lam
0030 %     Hvv = (d/dVm (dAIbr_dVm.')) * lam
0031 %
0032 %   See also DIBR_DV.
0033 %
0034 %   For more details on the derivations behind the derivative code used
0035 %   in MATPOWER information, see:
0036 %
0037 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0038 %          their Derivatives using Complex Matrix Notation", MATPOWER
0039 %          Technical Note 2, February 2010.
0040 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0041 
0042 %   MATPOWER
0043 %   $Id: d2AIbr_dV2.m 1720 2010-11-16 16:05:47Z cvs $
0044 %   by Ray Zimmerman, PSERC Cornell
0045 %   Copyright (c) 2008-2010 by Power System Engineering Research Center (PSERC)
0046 %
0047 %   This file is part of MATPOWER.
0048 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0049 %
0050 %   MATPOWER is free software: you can redistribute it and/or modify
0051 %   it under the terms of the GNU General Public License as published
0052 %   by the Free Software Foundation, either version 3 of the License,
0053 %   or (at your option) any later version.
0054 %
0055 %   MATPOWER is distributed in the hope that it will be useful,
0056 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0057 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
0058 %   GNU General Public License for more details.
0059 %
0060 %   You should have received a copy of the GNU General Public License
0061 %   along with MATPOWER. If not, see <http://www.gnu.org/licenses/>.
0062 %
0063 %   Additional permission under GNU GPL version 3 section 7
0064 %
0065 %   If you modify MATPOWER, or any covered work, to interface with
0066 %   other modules (such as MATLAB code and MEX-files) available in a
0067 %   MATLAB(R) or comparable environment containing parts covered
0068 %   under other licensing terms, the licensors of MATPOWER grant
0069 %   you additional permission to convey the resulting work.
0070 
0071 %% define
0072 nl = length(lam);
0073 
0074 diaglam = sparse(1:nl, 1:nl, lam, nl, nl);
0075 diagIbr_conj = sparse(1:nl, 1:nl, conj(Ibr), nl, nl);
0076 
0077 [Iaa, Iav, Iva, Ivv] = d2Ibr_dV2(Ybr, V, diagIbr_conj * lam);
0078 Haa = 2 * real( Iaa + dIbr_dVa.' * diaglam * conj(dIbr_dVa) );
0079 Hva = 2 * real( Iva + dIbr_dVm.' * diaglam * conj(dIbr_dVa) );
0080 Hav = 2 * real( Iav + dIbr_dVa.' * diaglam * conj(dIbr_dVm) );
0081 Hvv = 2 * real( Ivv + dIbr_dVm.' * diaglam * conj(dIbr_dVm) );