d2Sbr_dV2

D2SBR_DV2   Computes 2nd derivatives of complex power flow w.r.t. voltage.
   [HAA, HAV, HVA, HVV] = D2SBR_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 power flows. Takes 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);
       Cbr = Cf;
       Ybr = Yf;
       [Haa, Hav, Hva, Hvv] = d2Sbr_dV2(Cbr, Ybr, V, lam);

   Here the output matrices correspond to:
       Haa = (d/dVa (dSbr_dVa.')) * lam
       Hav = (d/dVm (dSbr_dVa.')) * lam
       Hva = (d/dVa (dSbr_dVm.')) * lam
       Hvv = (d/dVm (dSbr_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] = d2Sbr_dV2(Cbr, Ybr, V, lam)
0002 %D2SBR_DV2   Computes 2nd derivatives of complex power flow w.r.t. voltage.
0003 %   [HAA, HAV, HVA, HVV] = D2SBR_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 power flows. Takes sparse connection matrix CBR, sparse
0007 %   branch admittance matrix YBR, voltage vector V and nl x 1 vector of
0008 %   multipliers LAM. Output matrices are sparse.
0009 %
0010 %   Example:
0011 %       f = branch(:, F_BUS);
0012 %       Cf =  sparse(1:nl, f, ones(nl, 1), nl, nb);
0013 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0014 %       Cbr = Cf;
0015 %       Ybr = Yf;
0016 %       [Haa, Hav, Hva, Hvv] = d2Sbr_dV2(Cbr, Ybr, V, lam);
0017 %
0018 %   Here the output matrices correspond to:
0019 %       Haa = (d/dVa (dSbr_dVa.')) * lam
0020 %       Hav = (d/dVm (dSbr_dVa.')) * lam
0021 %       Hva = (d/dVa (dSbr_dVm.')) * lam
0022 %       Hvv = (d/dVm (dSbr_dVm.')) * lam
0023 %
0024 %   For more details on the derivations behind the derivative code used
0025 %   in MATPOWER information, see:
0026 %
0027 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0028 %          their Derivatives using Complex Matrix Notation", MATPOWER
0029 %          Technical Note 2, February 2010.
0030 %             http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf
0031 
0032 %   MATPOWER
0033 %   Copyright (c) 2008-2016, Power Systems Engineering Research Center (PSERC)
0034 %   by Ray Zimmerman, PSERC Cornell
0035 %
0036 %   This file is part of MATPOWER.
0037 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0038 %   See http://www.pserc.cornell.edu/matpower/ for more info.
0039 
0040 %% define
0041 nl = length(lam);
0042 nb = length(V);
0043 
0044 diaglam = sparse(1:nl, 1:nl, lam, nl, nl);
0045 diagV   = sparse(1:nb, 1:nb, V, nb, nb);
0046 
0047 A = Ybr' * diaglam * Cbr;
0048 B = conj(diagV) * A * diagV;
0049 D = sparse(1:nb, 1:nb, (A*V) .* conj(V), nb, nb);
0050 E = sparse(1:nb, 1:nb, (A.'*conj(V)) .* V, nb, nb);
0051 F = B + B.';
0052 G = sparse(1:nb, 1:nb, ones(nb, 1)./abs(V), nb, nb);
0053 
0054 Haa = F - D - E;
0055 Hva = 1j * G * (B - B.' - D + E);
0056 Hav = Hva.';
0057 Hvv = G * F * G;