d2ASbr_dV2

D2ASBR_DV2   Computes 2nd derivatives of |complex power flow|^2 w.r.t. V.
   [HAA, HAV, HVA, HVV] = D2ASBR_DV2(DSBR_DVA, DSBR_DVM, SBR, 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 square of the magnitude of branch complex power flows.
   Takes sparse first derivative matrices of complex flow, complex flow
   vector, sparse connection matrix CBR, 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);
       [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = ...
               dSbr_dV(branch, Yf, Yt, V);
       Cbr = Cf;
       Ybr = Yf;
       dSbr_dVa = dSf_dVa;
       dSbr_dVm = dSf_dVm;
       Sbr = Sf;
       [Haa, Hav, Hva, Hvv] = ...
             d2ASbr_dV2(dSbr_dVa, dSbr_dVm, Sbr, Cbr, Ybr, V, lam);

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

   See also DSBR_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     d2ASbr_dV2(dSbr_dVa, dSbr_dVm, Sbr, Cbr, Ybr, V, lam)
0003 %D2ASBR_DV2   Computes 2nd derivatives of |complex power flow|^2 w.r.t. V.
0004 %   [HAA, HAV, HVA, HVV] = D2ASBR_DV2(DSBR_DVA, DSBR_DVM, SBR, CBR, 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 branch complex power flows.
0008 %   Takes sparse first derivative matrices of complex flow, complex flow
0009 %   vector, sparse connection matrix CBR, sparse branch admittance matrix YBR,
0010 %   voltage vector V and nl x 1 vector of multipliers LAM. Output matrices
0011 %   are sparse.
0012 %
0013 %   Example:
0014 %       f = branch(:, F_BUS);
0015 %       Cf =  sparse(1:nl, f, ones(nl, 1), nl, nb);
0016 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0017 %       [dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St] = ...
0018 %               dSbr_dV(branch, Yf, Yt, V);
0019 %       Cbr = Cf;
0020 %       Ybr = Yf;
0021 %       dSbr_dVa = dSf_dVa;
0022 %       dSbr_dVm = dSf_dVm;
0023 %       Sbr = Sf;
0024 %       [Haa, Hav, Hva, Hvv] = ...
0025 %             d2ASbr_dV2(dSbr_dVa, dSbr_dVm, Sbr, Cbr, Ybr, V, lam);
0026 %
0027 %   Here the output matrices correspond to:
0028 %     Haa = (d/dVa (dASbr_dVa.')) * lam
0029 %     Hav = (d/dVm (dASbr_dVa.')) * lam
0030 %     Hva = (d/dVa (dASbr_dVm.')) * lam
0031 %     Hvv = (d/dVm (dASbr_dVm.')) * lam
0032 %
0033 %   See also DSBR_DV.
0034 %
0035 %   For more details on the derivations behind the derivative code used
0036 %   in MATPOWER information, see:
0037 %
0038 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0039 %          their Derivatives using Complex Matrix Notation", MATPOWER
0040 %          Technical Note 2, February 2010.
0041 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0042 
0043 %   MATPOWER
0044 %   Copyright (c) 2008-2016, Power Systems Engineering Research Center (PSERC)
0045 %   by Ray Zimmerman, PSERC Cornell
0046 %
0047 %   This file is part of MATPOWER.
0048 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0049 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0050 
0051 %% define
0052 nl = length(lam);
0053 
0054 diaglam = sparse(1:nl, 1:nl, lam, nl, nl);
0055 diagSbr_conj = sparse(1:nl, 1:nl, conj(Sbr), nl, nl);
0056 
0057 [Saa, Sav, Sva, Svv] = d2Sbr_dV2(Cbr, Ybr, V, diagSbr_conj * lam);
0058 Haa = 2 * real( Saa + dSbr_dVa.' * diaglam * conj(dSbr_dVa) );
0059 Hva = 2 * real( Sva + dSbr_dVm.' * diaglam * conj(dSbr_dVa) );
0060 Hav = 2 * real( Sav + dSbr_dVa.' * diaglam * conj(dSbr_dVm) );
0061 Hvv = 2 * real( Svv + dSbr_dVm.' * diaglam * conj(dSbr_dVm) );