opf_vref_hess

OPF_VREF_HESS  Evaluates Hessian of voltage angle reference.
   D2VREF = OPF_VREF_HESS(X, LAMBDA, MPC, REFS, MPOPT)

   Hessian evaluation function for voltage angle reference.

   Inputs:
     X : optimization vector
     LAMBDA : column vector of Lagrange multipliers on active and reactive
              power balance constraints
     MPC : MATPOWER case struct
     REFS : reference vector
     MPOPT : MATPOWER options struct

   Outputs:
     D2VREF : Hessian of voltage angle reference.

   Example:
       d2Vref = opf_vref_hess(x, lambda, mpc, refs, mpopt);

   See also OPF_VREF_FCN.

0001 function d2Vref = opf_vref_hess(x, lam, mpc, refs, mpopt)
0002 %OPF_VREF_HESS  Evaluates Hessian of voltage angle reference.
0003 %   D2VREF = OPF_VREF_HESS(X, LAMBDA, MPC, REFS, MPOPT)
0004 %
0005 %   Hessian evaluation function for voltage angle reference.
0006 %
0007 %   Inputs:
0008 %     X : optimization vector
0009 %     LAMBDA : column vector of Lagrange multipliers on active and reactive
0010 %              power balance constraints
0011 %     MPC : MATPOWER case struct
0012 %     REFS : reference vector
0013 %     MPOPT : MATPOWER options struct
0014 %
0015 %   Outputs:
0016 %     D2VREF : Hessian of voltage angle reference.
0017 %
0018 %   Example:
0019 %       d2Vref = opf_vref_hess(x, lambda, mpc, refs, mpopt);
0020 %
0021 %   See also OPF_VREF_FCN.
0022 
0023 %   MATPOWER
0024 %   Copyright (c) 2018, Power Systems Engineering Research Center (PSERC)
0025 %   by Baljinnyam Sereeter, Delft University of Technology
0026 %   and Ray Zimmerman, PSERC Cornell
0027 %
0028 %   This file is part of MATPOWER.
0029 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0030 %   See https://matpower.org for more info.
0031 
0032 %% unpack data
0033 [Vr, Vi] = deal(x{:});
0034 nb = length(Vr);
0035 
0036 %%----- evaluate Hessian of voltage angle constraints -----
0037 VVr = Vr(refs);
0038 VVi = Vi(refs);
0039 VVr2 = VVr.^2;
0040 VVi2 = VVi.^2;
0041 lamVm4 = lam ./ (VVr2 + VVi2).^2;
0042 
0043 d2Vref_rr = sparse(refs, refs, 2 * lamVm4 .*  VVr .* VVi,   nb, nb);
0044 d2Vref_ri = sparse(refs, refs,     lamVm4 .* (VVi2 - VVr2), nb, nb);
0045 d2Vref_ir =  d2Vref_ri;
0046 d2Vref_ii = -d2Vref_rr;
0047 
0048 %% construct Hessian
0049 d2Vref = [  d2Vref_rr d2Vref_ri;
0050             d2Vref_ir d2Vref_ii   ];