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, IANG, MPOPT)

   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
     IANG : index vector of branches corresponding to rows of AANG, LANG, UANG
     MPOPT : MATPOWER options struct

   Outputs:
     D2VADIF : Hessian of branch angle difference constraints.

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

   See also OPF_BRANCH_ANG_FCN.

0001 function d2VaDif = opf_branch_ang_hess(x, lambda, Aang, lang, uang, iang, mpopt)
0002 %OPF_BRANCH_ANG_HESS  Evaluates Hessian of branch angle difference constraints.
0003 %   D2VADIF = OPF_BRANCH_ANG_HESS(X, LAMBDA, AANG, LANG, UANG, IANG, MPOPT)
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 %     IANG : index vector of branches corresponding to rows of AANG, LANG, UANG
0016 %     MPOPT : MATPOWER options struct
0017 %
0018 %   Outputs:
0019 %     D2VADIF : Hessian of branch angle difference constraints.
0020 %
0021 %   Example:
0022 %       d2VaDif = opf_branch_ang_hess(x, lambda, Aang, lang, uang, iang, mpopt);
0023 %
0024 %   See also OPF_BRANCH_ANG_FCN.
0025 
0026 %   MATPOWER
0027 %   Copyright (c) 2018, Power Systems Engineering Research Center (PSERC)
0028 %   by Ray Zimmerman, PSERC Cornell
0029 %   and Baljinnyam Sereeter, Delft University of Technology
0030 %
0031 %   This file is part of MATPOWER.
0032 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0033 %   See https://matpower.org for more info.
0034 
0035 
0036 %% unpack data
0037 [Vr, Vi] = deal(x{:});
0038 nb = length(Vr);
0039 
0040 %%----- evaluate Hessian of branch angle difference constraints -----
0041 nlam = length(lambda) / 2;
0042 if nlam
0043     lamL = lambda(1:nlam);
0044     lamU = lambda((1:nlam)+nlam);
0045 else
0046     lamL = zeros(0,1);
0047     lamU = zeros(0,1);
0048 end
0049 
0050 Vr2 = Vr.^2;
0051 Vi2 = Vi.^2;
0052 
0053 lamL_Vm4 = (Aang' * lamL) ./ (Vr2 + Vi2).^2;
0054 lamU_Vm4 = (Aang' * lamU) ./ (Vr2 + Vi2).^2;
0055 
0056 VaDifL_rr = sparse(1:nb, 1:nb, 2 * lamL_Vm4 .*  Vr .* Vi,   nb, nb);
0057 VaDifL_ri = sparse(1:nb, 1:nb,     lamL_Vm4 .* (Vi2 - Vr2), nb, nb);
0058 VaDifL_ir =  VaDifL_ri;
0059 VaDifL_ii = -VaDifL_rr;
0060 
0061 VaDifU_rr = sparse(1:nb, 1:nb, 2 * lamU_Vm4 .*  Vr .* Vi,   nb, nb);
0062 VaDifU_ri = sparse(1:nb, 1:nb,     lamU_Vm4 .* (Vi2 - Vr2), nb, nb);
0063 VaDifU_ir =  VaDifU_ri;
0064 VaDifU_ii = -VaDifU_rr;
0065 
0066 %% construct Hessian
0067 d2VaDif = -[ VaDifL_rr VaDifL_ri;
0068              VaDifL_ir VaDifL_ii ] + ...
0069            [ VaDifU_rr VaDifU_ri;
0070              VaDifU_ir VaDifU_ii ];