I have a power grid with N buses and L lines; each line is between specific nodes. o(l) for the origin bus of line l and d(l) for the destination bus of line l. I want to get a power flow and know what power is in each line and in what direction. How can I create the following constraint? this constraint shows that the power balance in each node preserves the power balance at bus n such that the inflow and outflow are equal.
note: IEEE one-area RTS-1996 system known as 'case24_ieee_rts' in Matpower
Indices and sets:
N: set of indices of buses, indexed by n
J: set of indices of generators, indexed by j
Jn: set of indices of generators connected to bus n
L: set of indices of transmission assets, indexed by l
o(l): origin bus of line l
d(l): destination bus of line l
Parameters:
Dn ; demand at bus n (in megawatts)
Decision variables:
gj : generation level of generator j (in megawatts)
pl: power flow on line l (in megawatts)
dn : load shed at node n (in megawatts)
see this link
https://www.mathworks.com/matlabcentral/fileexchange/54777-rts96-73-bus-system-ieee-benchmark
This is a constrain from paper Article Optimal power grid protection through a defender–attacker–de...
an easy and understandable way to code this constrain in Matlab with YALMIP is:
%zl: binary defense decision, 1 if line l is defended and 0 otherwise
zl = binvar(L,1,'full');
% vl: attack decision variable, 0 for attacked, 1 for otherwise
vl = binvar(L,k,'full');
% dn: load shedding at node n
dn = sdpvar(N,k,'full');
% delta_n = phase angle at node n
delta_n = sdpvar(N,k,'full');
delta_ol = sdpvar(L,k,'full');
delta_dl = sdpvar(L,k,'full');
% gj: generation level of generator j
gj = sdpvar(j,k,'full');
% pl: power flow on line l
pl = sdpvar(L,k,'full');
N=24
L=38
Jn =cell(N,1)
Jn{1} = [(1:4)']
Jn{2} = [(5:8)']
Jn{7} = [(9:11)']
Jn{13} = [(12:14)']
Jn{15} = [(15:20)']
Jn{16} = [21]
Jn{18} = [22]
Jn{21} = [23]
Jn{22} = [(24:29)']
Jn{23} = [(30:32)']
for n3 = 1:N
jcheck = numel(Jn{n3})
if n3==1
CCGMasterCon15 = [CCGMasterCon15,...
((gj(1,iter)+gj(2,iter)+gj(3,iter)+gj(4,iter)) - (pl(1,iter)+pl(2,iter) + pl(3,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==2
CCGMasterCon15 = [CCGMasterCon15,...
((gj(5,iter)+gj(6,iter)+gj(7,iter)+gj(8,iter)) - (pl(4,iter)+pl(5,iter)) + (pl(1,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==3
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(6,iter)+pl(7,iter)) + pl(2,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==4
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(8,iter)) + pl(4,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==5
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(9,iter)) + pl(3,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==6
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(10,iter)) + pl(5,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==7
CCGMasterCon15 = [CCGMasterCon15,...
((gj(9,iter)+gj(10,iter)+gj(11,iter)) -(pl(11,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==8
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(12,iter)+pl(13,iter)) + pl(11,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==9
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(14,iter)+pl(15,iter)) + pl(12,iter)+pl(8,iter)+pl(6,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==10
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(16,iter)+pl(17,iter)) + (pl(9,iter)+pl(10,iter)+pl(13,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==11
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(18,iter)+pl(19,iter)) + (pl(14,iter)+pl(16,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==12
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(20,iter)+pl(21,iter)) + (pl(15,iter)+pl(17,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==13
CCGMasterCon15 = [CCGMasterCon15,...
((gj(12,iter)+gj(13,iter)+gj(14,iter))-(pl(22,iter)) + (pl(18,iter)+pl(20,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==14
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(23,iter)) + (pl(19,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==15
CCGMasterCon15 = [CCGMasterCon15,...
((gj(15,iter)+gj(16,iter)+gj(17,iter)+gj(18,iter)+gj(19,iter)+gj(20,iter))-(pl(24,iter)+pl(25,iter)+pl(26,iter)+pl(27,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==16
CCGMasterCon15 = [CCGMasterCon15,...
((gj(21,iter)) - (pl(28,iter)+pl(29,iter)) + (pl(23,iter)+pl(24,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==17
CCGMasterCon15 = [CCGMasterCon15,...
(0 - (pl(30,iter)+pl(31,iter)) + (pl(28,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==18
CCGMasterCon15 = [CCGMasterCon15,...
((gj(22,iter))- (pl(32,iter)+pl(33,iter)) + pl(30,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==19
CCGMasterCon15 = [CCGMasterCon15,...
(0 -( pl(34,iter)+pl(35,iter)) + pl(29,iter) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==20
CCGMasterCon15 = [CCGMasterCon15,...
(0 -(pl(36,iter) + pl(37,iter)) + (pl(34,iter)+pl(35,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==21
CCGMasterCon15 = [CCGMasterCon15,...
((gj(23,iter))-(pl(38,iter)) + (pl(25,iter)+pl(26,iter)+pl(32,iter)+pl(33,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==22
CCGMasterCon15 = [CCGMasterCon15,...
((gj(24,iter)+gj(25,iter)+gj(26,iter)+gj(27,iter)+gj(28,iter)+gj(29,iter)) + (pl(31,iter)+pl(38,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==23
CCGMasterCon15 = [CCGMasterCon15,...
((gj(30,iter)+gj(31,iter)+gj(32,iter)) + (pl(21,iter)+pl(22,iter)+pl(36,iter)+pl(37,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
elseif n3==24
CCGMasterCon15 = [CCGMasterCon15,...
(0-0+(pl(7,iter)+pl(27,iter)) + dn(n3,iter) == Dn(n3)):['con15-' num2str(n3)]]
end
end
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