opf_branch_ang_hess

OPF_BRANCH_ANG_HESS  Evaluates Hessian of branch angle difference constraints.
   D2VADIF = OPF_BRANCH_ANG_HESS(X, LAMBDA, AANG, LANG, UANG)

   Hessian evaluation function for branch angle difference constraints
   for voltages in cartesian coordinates.

   Inputs:
     X : optimization vector
     LAMBDA : column vector of Lagrange multipliers on branch angle
           difference constraints, lower, then upper
     AANG : constraint matrix, see MAKEAANG
     LANG : lower bound vector, see MAKEAANG
     UANG : upper bound vector, see MAKEAANG

   Outputs:
     D2VADIF : Hessian of branch angle difference constraints.

   Example:
       d2VaDif = opf_branch_ang_hess(x, lambda, Aang, lang, uang);

   See also OPF_BRANCH_ANG_FCN.

0001 function d2VaDif = opf_branch_ang_hess(x, lambda, Aang, lang, uang)
0002 %OPF_BRANCH_ANG_HESS  Evaluates Hessian of branch angle difference constraints.
0003 %   D2VADIF = OPF_BRANCH_ANG_HESS(X, LAMBDA, AANG, LANG, UANG)
0004 %
0005 %   Hessian evaluation function for branch angle difference constraints
0006 %   for voltages in cartesian coordinates.
0007 %
0008 %   Inputs:
0009 %     X : optimization vector
0010 %     LAMBDA : column vector of Lagrange multipliers on branch angle
0011 %           difference constraints, lower, then upper
0012 %     AANG : constraint matrix, see MAKEAANG
0013 %     LANG : lower bound vector, see MAKEAANG
0014 %     UANG : upper bound vector, see MAKEAANG
0015 %
0016 %   Outputs:
0017 %     D2VADIF : Hessian of branch angle difference constraints.
0018 %
0019 %   Example:
0020 %       d2VaDif = opf_branch_ang_hess(x, lambda, Aang, lang, uang);
0021 %
0022 %   See also OPF_BRANCH_ANG_FCN.
0023 
0024 %   MATPOWER
0025 %   Copyright (c) 2018-2020, Power Systems Engineering Research Center (PSERC)
0026 %   by Ray Zimmerman, PSERC Cornell
0027 %   and Baljinnyam Sereeter, Delft University of Technology
0028 %
0029 %   This file is part of MATPOWER.
0030 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0031 %   See https://matpower.org for more info.
0032 
0033 
0034 %% unpack data
0035 [Vr, Vi] = deal(x{:});
0036 nb = length(Vr);
0037 
0038 %%----- evaluate Hessian of branch angle difference constraints -----
0039 nlam = length(lambda) / 2;
0040 if nlam
0041     lam = lambda(nlam+1:2*nlam) - lambda(1:nlam);   %% lamU - lamL
0042 else
0043     lam = zeros(0,1);
0044 end
0045 
0046 Vr2 = Vr.^2;
0047 Vi2 = Vi.^2;
0048 
0049 lam_Vm4 = (Aang' * lam) ./ (Vr2 + Vi2).^2;
0050 
0051 VaDif_rr = sparse(1:nb, 1:nb, 2 * lam_Vm4 .*  Vr .* Vi,   nb, nb);
0052 VaDif_ri = sparse(1:nb, 1:nb,     lam_Vm4 .* (Vi2 - Vr2), nb, nb);
0053 VaDif_ir =  VaDif_ri;
0054 VaDif_ii = -VaDif_rr;
0055 
0056 %% construct Hessian
0057 d2VaDif = [ VaDif_rr VaDif_ri;
0058             VaDif_ir VaDif_ii ];