Description of makesdpmat

0001 function [Yk,Yk_,Mk,Ylineft,Ylinetf,Y_lineft,Y_linetf,YL,YL_] = makesdpmat(mpc)
0002 %MAKESDPMAT Creates the matrix functions used in the semidefinite
0003 %programming relaxation of the optimal power flow problem.
0004 %   [YK,YK_,MK,YLINEFT,YLINETF,Y_LINEFT,Y_LINETF] = MAKESDPMAT(MPC)
0005 %
0006 %   Creates functions that return the matrices used in the semidefinite
0007 %   programming relaxation of the optimal power flow problem.
0008 %
0009 %   Inputs:
0010 %       MPC : MATPOWER case variable with internal indexing.
0011 %
0012 %   Outputs:
0013 %       YK : Function to create the matrix for active power injection.
0014 %           Yk(i) is the matrix corresponding to the active power injection
0015 %           at bus i.
0016 %       YK_ : Function to create the matrix for reactive power injection.
0017 %           Yk_(i) is the matrix corresponding to the reactive power
0018 %           injection at bus i.
0019 %       MK : Function to create the matrix for the square of voltage
0020 %           magnitude. Mk(i) is the matrix corresponding to the square of
0021 %           the voltage magnitude at bus i.
0022 %       YLINEFT : Function to create the matrix for the active power flow
0023 %           on the specified line, measured from the "from" bus to the "to"
0024 %           bus. Ylineft(i) is the matrix corresponding to the active power
0025 %           flow on the line mpc.branch(i,:).
0026 %       YLINETF : Function to create the matrix for the active power flow
0027 %           on the specified line, measured from the "to" bus to the "from"
0028 %           bus. Ylinetf(i) is the matrix corresponding to the active power
0029 %           flow on the line mpc.branch(i,:).
0030 %       Y_LINEFT : Function to create the matrix for the reactive power
0031 %           flow on the specified line, measured from the "from" bus to the
0032 %           "to" bus. Ylineft(i) is the matrix corresponding to the
0033 %           reactive power flow on the line mpc.branch(i,:).
0034 %       Y_LINETF : Function to create the matrix for the reactive power
0035 %           flow on the specified line, measured from the "to" bus to the
0036 %           "from" bus. Ylineft(i) is the matrix corresponding to the
0037 %           reactive power flow on the line mpc.branch(i,:).
0038 %       YL : Function to create the matrix for the active power loss
0039 %           on the specified line (included for completeness,
0040 %           not used in the OPF formulation)
0041 %       YL_ : Function to create the matrix for the reactive power loss
0042 %           on the specified line (included for completeness,
0043 %           not used in the OPF formulation)
0044 
0045 %   MATPOWER
0046 %   Copyright (c) 2013-2019, Power Systems Engineering Research Center (PSERC)
0047 %   by Daniel Molzahn, PSERC U of Wisc, Madison
0048 %
0049 %   This file is part of MATPOWER/mx-sdp_pf.
0050 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0051 %   See https://github.com/MATPOWER/mx-sdp_pf/ for more info.
0052 
0053 %% Setup
0054 
0055 [F_BUS, T_BUS, BR_R, BR_X, BR_B, RATE_A, RATE_B, RATE_C, ...
0056     TAP, SHIFT, BR_STATUS, PF, QF, PT, QT, MU_SF, MU_ST, ...
0057     ANGMIN, ANGMAX, MU_ANGMIN, MU_ANGMAX] = idx_brch;
0058 
0059 mpc = loadcase(mpc);
0060 nbus = size(mpc.bus,1);
0061 Y = makeYbus(mpc);
0062 
0063 if have_feature('octave')
0064     emat = full(speye(nbus));
0065 else
0066     emat = speye(nbus);
0067 end
0068 e = @(k) emat(:,k); % kth standard basis vector
0069 Yk_small = @(k) e(k)*e(k).'*Y;
0070 
0071 % Set tau == 0 to 1 (tau == 0 indicates nominal voltage ratio)
0072 mpc.branch(mpc.branch(:,TAP) == 0,TAP) = 1;
0073 
0074 %% Matrices used in SDP relaxation of OPF problem
0075 
0076 Yk = @(k) (1/2)*[real(Yk_small(k) + Yk_small(k).') imag(Yk_small(k).' - Yk_small(k));
0077     imag(Yk_small(k) - Yk_small(k).') real(Yk_small(k) + Yk_small(k).')];
0078 
0079 Yk_ = @(k) -(1/2)*[imag(Yk_small(k) + Yk_small(k).') real(Yk_small(k) - Yk_small(k).');
0080     real(Yk_small(k).' - Yk_small(k)) imag(Yk_small(k) + Yk_small(k).')];
0081 
0082 Mk = @(k) blkdiag(e(k)*e(k).', e(k)*e(k).');
0083 
0084 
0085 % For the line limit matrices, specify a line index corresponding to the
0086 % entries in mpc.branch
0087 gl = @(lineidx) real( 1 / (mpc.branch(lineidx,BR_R)+1i*mpc.branch(lineidx,BR_X))); % Real part of line admittance
0088 bl = @(lineidx) imag( 1 / (mpc.branch(lineidx,BR_R)+1i*mpc.branch(lineidx,BR_X))); % Imaginary part of line admittance
0089 bsl= @(lineidx) mpc.branch(lineidx,BR_B); % Line shunt susceptance
0090 Rl = @(lineidx) mpc.branch(lineidx,BR_R);
0091 Xl = @(lineidx) mpc.branch(lineidx,BR_X);
0092 
0093 % tau (col TAP)
0094 % theta (col SHIFT)
0095 % gbcosft = g*cos(theta)+b*cos(theta+pi/2)
0096 % gbsinft = g*sin(theta)+b*sin(theta+pi/2)
0097 % gbcostf = g*cos(-theta)+b*cos(-theta+pi/2)
0098 % gbsintf = g*sin(-theta)+b*sin(-theta+pi/2)
0099 
0100 tau = @(lineidx) mpc.branch(lineidx,TAP); 
0101 theta = @(lineidx) mpc.branch(lineidx,SHIFT)*pi/180;
0102 gbcosft = @(lineidx) gl(lineidx)*cos(theta(lineidx)) + bl(lineidx)*cos(theta(lineidx)+pi/2);
0103 gbsinft = @(lineidx) gl(lineidx)*sin(theta(lineidx)) + bl(lineidx)*sin(theta(lineidx)+pi/2);
0104 gbcostf = @(lineidx) gl(lineidx)*cos(-theta(lineidx)) + bl(lineidx)*cos(-theta(lineidx)+pi/2);
0105 gbsintf = @(lineidx) gl(lineidx)*sin(-theta(lineidx)) + bl(lineidx)*sin(-theta(lineidx)+pi/2);
0106 
0107 Ylineft = @(lidx) 0.5*(sparse(    [mpc.branch(lidx,F_BUS)     mpc.branch(lidx,F_BUS)     mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus  mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus ], ...
0108                                   [mpc.branch(lidx,F_BUS)     mpc.branch(lidx,T_BUS)     mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus  mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus ], ...
0109                                   [gl(lidx)/(tau(lidx)^2)     -gbcosft(lidx)/tau(lidx)   gbsinft(lidx)/tau(lidx)     gl(lidx)/(tau(lidx)^2)       -gbsinft(lidx)/tau(lidx)    -gbcosft(lidx)/tau(lidx)    ] ...
0110                                   ,2*nbus,2*nbus) + ...
0111                        sparse(    [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)     mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus  mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus ], ...
0112                                   [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,T_BUS)     mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus  mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus ], ...
0113                                   [gl(lidx)/(tau(lidx)^2)    -gbcosft(lidx)/tau(lidx)   gbsinft(lidx)/tau(lidx)     gl(lidx)/(tau(lidx)^2)       -gbsinft(lidx)/tau(lidx)    -gbcosft(lidx)/tau(lidx)    ] ...
0114                                   ,2*nbus,2*nbus).');
0115                                                            
0116 Y_lineft = @(lidx) 0.5*(sparse(     [mpc.branch(lidx,F_BUS)                mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus            mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus ], ...
0117                                     [mpc.branch(lidx,F_BUS)                mpc.branch(lidx,T_BUS)    mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus            mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus ], ...
0118                                     [-(bl(lidx)+bsl(lidx)/2)/(tau(lidx)^2) gbsinft(lidx)/tau(lidx)   gbcosft(lidx)/tau(lidx)     -(bl(lidx)+bsl(lidx)/2)/(tau(lidx)^2)  -gbcosft(lidx)/tau(lidx)    gbsinft(lidx)/tau(lidx)     ] ...
0119                                   ,2*nbus,2*nbus) + ...
0120                         sparse(     [mpc.branch(lidx,F_BUS)                mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus           mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus ], ...
0121                                     [mpc.branch(lidx,F_BUS)                mpc.branch(lidx,T_BUS)    mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus           mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus ], ...
0122                                     [-(bl(lidx)+bsl(lidx)/2)/(tau(lidx)^2) gbsinft(lidx)/tau(lidx)   gbcosft(lidx)/tau(lidx)     -(bl(lidx)+bsl(lidx)/2)/(tau(lidx)^2) -gbcosft(lidx)/tau(lidx)    gbsinft(lidx)/tau(lidx)     ] ...
0123                                   ,2*nbus,2*nbus).');
0124 
0125 Ylinetf = @(lidx) 0.5*(sparse(    [mpc.branch(lidx,F_BUS)     mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,T_BUS) mpc.branch(lidx,T_BUS)+nbus ], ...
0126                                   [mpc.branch(lidx,T_BUS)     mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS) mpc.branch(lidx,T_BUS)+nbus ], ...
0127                                   [-gbcostf(lidx)/tau(lidx)   -gbsintf(lidx)/tau(lidx)    gbsintf(lidx)/tau(lidx)     -gbcostf(lidx)/tau(lidx)    gl(lidx)               gl(lidx)                    ] ...
0128                                   ,2*nbus,2*nbus) + ...
0129                        sparse(    [mpc.branch(lidx,F_BUS)     mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,T_BUS) mpc.branch(lidx,T_BUS)+nbus ], ...
0130                                   [mpc.branch(lidx,T_BUS)     mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS) mpc.branch(lidx,T_BUS)+nbus ], ...
0131                                   [-gbcostf(lidx)/tau(lidx)   -gbsintf(lidx)/tau(lidx)    gbsintf(lidx)/tau(lidx)     -gbcostf(lidx)/tau(lidx)    gl(lidx)               gl(lidx)                ] ...
0132                                   ,2*nbus,2*nbus).');
0133                               
0134 Y_linetf = @(lidx) 0.5*(sparse(   [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,T_BUS)  mpc.branch(lidx,T_BUS)+nbus ], ...
0135                                   [mpc.branch(lidx,T_BUS)    mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS)  mpc.branch(lidx,T_BUS)+nbus ], ...
0136                                   [gbsintf(lidx)/tau(lidx)   -gbcostf(lidx)/tau(lidx)    gbcostf(lidx)/tau(lidx)     gbsintf(lidx)/tau(lidx)     -(bl(lidx)+bsl(lidx)/2) -(bl(lidx)+bsl(lidx)/2)     ] ...
0137                                   ,2*nbus,2*nbus) + ...
0138                        sparse(    [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,F_BUS)+nbus mpc.branch(lidx,T_BUS)   mpc.branch(lidx,T_BUS)+nbus ], ...
0139                                   [mpc.branch(lidx,T_BUS)    mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS)      mpc.branch(lidx,T_BUS)+nbus mpc.branch(lidx,T_BUS)   mpc.branch(lidx,T_BUS)+nbus ], ...
0140                                   [gbsintf(lidx)/tau(lidx)   -gbcostf(lidx)/tau(lidx)    gbcostf(lidx)/tau(lidx)     gbsintf(lidx)/tau(lidx)     -(bl(lidx)+bsl(lidx)/2)  -(bl(lidx)+bsl(lidx)/2)     ] ...
0141                                   ,2*nbus,2*nbus).');
0142                          
0143                               
0144 % Matrices to calculate active and reactive power line losses
0145 YL = @(lidx)          sparse(    [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus      mpc.branch(lidx,F_BUS)+nbus   mpc.branch(lidx,T_BUS)  mpc.branch(lidx,T_BUS)  mpc.branch(lidx,T_BUS)+nbus  mpc.branch(lidx,T_BUS)+nbus ], ...
0146                                   [mpc.branch(lidx,F_BUS)   mpc.branch(lidx,T_BUS)      mpc.branch(lidx,F_BUS)+nbus      mpc.branch(lidx,T_BUS)+nbus   mpc.branch(lidx,F_BUS)  mpc.branch(lidx,T_BUS)  mpc.branch(lidx,F_BUS)+nbus  mpc.branch(lidx,T_BUS)+nbus ], ...
0147                                   [1                        -1                          1                                -1                            -1                      1                       -1                           1                           ] ...
0148                                   ,2*nbus,2*nbus) * Rl(lidx)*(gl(lidx)^2+bl(lidx)^2);
0149                               
0150 YL_ = @(lidx)          sparse(    [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)      mpc.branch(lidx,F_BUS)+nbus      mpc.branch(lidx,F_BUS)+nbus   mpc.branch(lidx,T_BUS)  mpc.branch(lidx,T_BUS)  mpc.branch(lidx,T_BUS)+nbus  mpc.branch(lidx,T_BUS)+nbus ], ...
0151                                   [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,T_BUS)      mpc.branch(lidx,F_BUS)+nbus      mpc.branch(lidx,T_BUS)+nbus   mpc.branch(lidx,F_BUS)  mpc.branch(lidx,T_BUS)  mpc.branch(lidx,F_BUS)+nbus  mpc.branch(lidx,T_BUS)+nbus ], ...
0152                                   [1                         -1                          1                                -1                            -1                      1                       -1                           1                           ] ...
0153                                   ,2*nbus,2*nbus) * Xl(lidx) * (gl(lidx)^2+bl(lidx)^2) + ...
0154                        -sparse(   [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)+nbus   mpc.branch(lidx,T_BUS)   mpc.branch(lidx,T_BUS)+nbus   ], ...
0155                                   [mpc.branch(lidx,F_BUS)    mpc.branch(lidx,F_BUS)+nbus   mpc.branch(lidx,T_BUS)   mpc.branch(lidx,T_BUS)+nbus   ], ...
0156                                   [1                         1                             1                        1                             ] ...
0157                                   ,2*nbus,2*nbus) * bsl(lidx)/2;
makesdpmat

PURPOSE

SYNOPSIS

DESCRIPTION

CROSS-REFERENCE INFORMATION

SOURCE CODE
