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power_gen_optimization.m
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power_gen_optimization.m
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%% CE 295 - Energy Systems and Control
% Optimal Economic Dispatch in Distribution Feeders with Renewables
% Prof. Moura
% Last updated: February 20, 2018
% TSANG_NATHAN_HW3.m
clear; close all;
fs = 15; % Font Size for plots
%% 13 Node IEEE Test Feeder Parameters
%%% Node (aka Bus) Data
% l_j^P: Active power consumption [MW]
l_P = [0; 0.2; 0; 0.4; 0.17; 0.23; 1.155; 0; 0.17; 0.843; 0; 0.17; 0.128];
% l_j^Q: Reactive power consumption [MVAr]
l_Q = [0; 0.116; 0; 0.29; 0.125; 0.132; 0.66; 0; 0.151; 0.462; 0; 0.08; 0.086];
% l_j^S: Apparent power consumption [MVA]
l_S = sqrt(l_P.^2 + l_Q.^2);
% s_j,max: Maximal generating power [MW]
s_max = [5; 0; 0; 3; 0; 0; 0; 0; 0; 3; 0; 0; 0];
% c_j: Marginal generation cost [USD/MW]
c = [100; 0; 0; 150; 0; 0; 0; 0; 0; 50; 0; 0; 0];
% V_min, V_max: Minimum and maximum nodal voltages [V]
v_min = 0.95;
v_max = 1.05;
%%% Edge (aka Line) Data
% r_ij: Resistance [p.u.]
r = xlsread('HW3_Data.xlsx', 'Line-Data', 'B6:N18');
% x_ij: Reactance [p.u.]
x = xlsread('HW3_Data.xlsx', 'Line-Data', 'B24:N36');
% I_max_ij: Maximal line current [p.u.]
I_max = xlsread('HW3_Data.xlsx', 'Line-Data', 'B42:N54');
% A_ij: Adjacency matrix; A_ij = 1 if i is parent of j
A = xlsread('HW3_Data.xlsx', 'Line-Data', 'B60:N72');
%%% Set Data (add +1 everywhere for Matlab indexing)
% \rho(j): Parent node of node j
rho = [0; 0; 1; 2; 1; 4; 1; 6; 6; 8; 6; 10; 10]+1;
% Node index set - CREATED BY NATE
j = [0 1 2 3 4 5 6 7 8 9 10 11 12];
%% Problem 1
% Plot active and reactive power consumption
figure(1);
bar(j, [l_P,l_Q]);
legend('Real power (MW)','Reactive power (MVAR)')
xlabel('Node index number')
ylabel('Power consumption')
set(gca,'FontSize',fs)
%% Problem 2
% Assumptions:
% - Disregard the entire network diagram
% - Balance supply and demand, without any network considerations
% - Goal is to minimize generation costs, given by c^T s
% Solve with CVX
cvx_begin
variables s(13,1) p(13,1) q(13,1) % declare your optimization variables here
minimize(c'*s) % objective function here
subject to % constraints
% Balance power generation with power consumption
sum(p) == sum(l_P);
sum(q) == sum(l_Q);
% Loop over each node
for jj = 1:13
% Non-negative power generation
q(jj) >= 0;
p(jj) >= 0;
% Compute apparent power from active & reactive power
norm([p(jj) q(jj)],2) <= s(jj);
end
% Apparent Power Limits
s <= s_max;
cvx_end
% Output Results
fprintf(1,'------------------- PROBLEM 2 --------------------\n');
fprintf(1,'--------------------------------------------------\n');
fprintf(1,'Minimum Generating Cost : %4.2f USD\n',cvx_optval);
fprintf(1,'\n');
fprintf(1,'Node 0 [Grid] Gen Power : p_0 = %1.3f MW | q_0 = %1.3f MW | s_0 = %1.3f MW\n',p(1),q(1),s(1));
fprintf(1,'Node 3 [Gas] Gen Power : p_3 = %1.3f MW | q_3 = %1.3f MW | s_3 = %1.3f MW\n',p(4),q(4),s(4));
fprintf(1,'Node 9 [Solar] Gen Power : p_9 = %1.3f MW | q_9 = %1.3f MW | s_9 = %1.3f MW\n',p(10),q(10),s(10));
fprintf(1,'\n');
fprintf(1,'Total active power : %1.3f MW consumed | %1.3f MW generated\n',sum(l_P),sum(p));
fprintf(1,'Total reactive power : %1.3f MVAr consumed | %1.3f MVAr generated\n',sum(l_Q),sum(q));
fprintf(1,'Total apparent power : %1.3f MVA consumed | %1.3f MVA generated\n',sum(l_S),sum(s));
%% Problem 3
% Assumptions:
% - Disregard L_ij, the squared magnitude of complex line current
% - Disregard nodal voltage equation
% - Disregard nodal voltage limits
% - Disregard maximum line current
% - Goal is to minimize generation costs, given by c^T s
% Solve with CVX
cvx_begin
variables p(13,1) q(13,1) s(13,1) P(13,13) Q(13,13)
dual variable mu_s
minimize(c' * s)
subject to
% Boundary condition for power line flows
P( 1 , 1 ) == 0;
Q( 1 , 1 ) == 0;
% Loop over each node
for jj = 1:13
% Parent node, i = \rho(j)
ii = rho(jj);
% Line Power Flows
P(ii,jj) == l_P(jj) - p(jj) + sum(A(jj,:).* P(jj,:));
Q(ii,jj) == l_Q(jj) - q(jj) + sum(A(jj,:).* Q(jj,:));
% Compute apparent power from active & reactive power
norm([p(jj) q(jj)],2) <= s(jj);
q(jj) >= 0;
p(jj) >= 0;
end
% Apparent Power Limits
s <= s_max : mu_s;
cvx_end
% Output Results
fprintf(1,'------------------- PROBLEM 3 --------------------\n');
fprintf(1,'--------------------------------------------------\n');
fprintf(1,'Minimum Generating Cost : %4.2f USD\n',cvx_optval);
fprintf(1,'\n');
fprintf(1,'Node 0 [Grid] Gen Power : p_0 = %1.3f MW | q_0 = %1.3f MW | s_0 = %1.3f MW || mu_s0 = %3.0f USD/MW\n',p(1),q(1),s(1),mu_s(1));
fprintf(1,'Node 3 [Gas] Gen Power : p_3 = %1.3f MW | q_3 = %1.3f MW | s_3 = %1.3f MW || mu_s3 = %3.0f USD/MW\n',p(4),q(4),s(4),mu_s(4));
fprintf(1,'Node 9 [Solar] Gen Power : p_9 = %1.3f MW | q_9 = %1.3f MW | s_9 = %1.3f MW || mu_s9 = %3.0f USD/MW\n',p(10),q(10),s(10),mu_s(10));
fprintf(1,'\n');
fprintf(1,'Total active power : %1.3f MW consumed | %1.3f MW generated\n',sum(l_P),sum(p));
fprintf(1,'Total reactive power : %1.3f MVAr consumed | %1.3f MVAr generated\n',sum(l_Q),sum(q));
fprintf(1,'Total apparent power : %1.3f MVA consumed | %1.3f MVA generated\n',sum(l_S),sum(s));
%% Problem 4
% Assumptions:
% - Add back all previously disregarded terms and constraints
% - Relax squared line current equation into inequality
% - Goal is to minimize generation costs, given by c^T s
% Solve with CVX
cvx_begin
variables p(13,1) q(13,1) s(13,1) P(13,13) Q(13,13) L(13,13) V(13,1)
dual variables mu_s mu_L mu_vmin mu_vmax
minimize(c'*s)
subject to
% Boundary condition for power line flows
P( 1 , 1 ) == 0;
Q( 1 , 1 ) == 0;
% Boundary condition for squared line current
L( 1 , 1 ) == 0;
% Fix node 0 voltage to be 1 "per unit" (p.u.)
V(1) == 1;
% Loop over each node
for jj = 1:13
% Parent node, i = \rho(j)
ii = rho(jj);
% Line Power Flows
P(ii,jj) == l_P(jj) - p(jj) + sum(A(jj,:).* P(jj,:)) + r(ii,jj)*L(ii,jj);
Q(ii,jj) == l_Q(jj) - q(jj) + sum(A(jj,:).* Q(jj,:)) + x(ii,jj)*L(ii,jj);
% Nodal voltage
V(jj) == V(ii)+(r(ii,jj)^2 + x(ii,jj)^2) * L(ii,jj) - 2*(r(ii,jj) * P(ii,jj) + x(ii,jj) * Q(ii,jj));
% Squared current magnitude on lines
L(ii,jj) >= quad_over_lin(P(ii,jj),V(jj))+ quad_over_lin(Q(ii,jj),V(jj));
% Compute apparent power from active & reactive power
norm([p(jj) q(jj)],2) <= s(jj);
q(jj) >= 0;
p(jj) >= 0;
end
% Squared line current limits
L <= I_max.^2 : mu_L;
% Nodal voltage limits
V <= v_max.^2 : mu_vmax;
V >= v_min.^2 : mu_vmin;
% Apparent Power Limits
s <= s_max : mu_s;
cvx_end
% Output Results
fprintf(1,'------------------- PROBLEM 4 --------------------\n');
fprintf(1,'--------------------------------------------------\n');
fprintf(1,'Minimum Generating Cost : %4.2f USD\n',cvx_optval);
fprintf(1,'\n');
fprintf(1,'Node 0 [Grid] Gen Power : p_0 = %1.3f MW | q_0 = %1.3f MW | s_0 = %1.3f MW || mu_s0 = %3.0f USD/MW | mu_vmax0 = %3.2f p.u.| mu_vmin0 = %3.2f p.u./MW\n',p(1),q(1),s(1),mu_s(1),mu_vmax(1),mu_vmin(1));
fprintf(1,'Node 3 [Gas] Gen Power : p_3 = %1.3f MW | q_3 = %1.3f MW | s_3 = %1.3f MW || mu_s3 = %3.0f USD/MW | mu_vmax3 = %3.2f p.u.| mu_vmin3 = %3.2f p.u./MW\n',p(4),q(4),s(4),mu_s(4),mu_vmax(4),mu_vmin(4));
fprintf(1,'Node 9 [Solar] Gen Power : p_9 = %1.3f MW | q_9 = %1.3f MW | s_9 = %1.3f MW || mu_s9 = %3.0f USD/MW | mu_vmax9 = %3.2f p.u.| mu_vmin9 = %3.2f p.u./MW\n',p(10),q(10),s(10),mu_s(10),mu_vmax(10),mu_vmin(10));
fprintf(1,'\n');
fprintf(1,'Total active power : %1.3f MW consumed | %1.3f MW generated\n',sum(l_P),sum(p));
fprintf(1,'Total reactive power : %1.3f MVAr consumed | %1.3f MVAr generated\n',sum(l_Q),sum(q));
fprintf(1,'Total apparent power : %1.3f MVA consumed | %1.3f MVA generated\n',sum(l_S),sum(s));
fprintf(1,'\n');
for jj = 1:13
fprintf(1,'Node %2.0f Voltage : %1.3f p.u.\n',jj,sqrt(V(jj)));
end
%% Problem 5
% Assumptions:
% - Assume solar generator at node 9 has uncertain power capacity
% - Goal is to minimize generation costs, given by c^T s, in face of uncertainty
% Solve with CVX
% define new variables
a_bar = [-1.25; -1.25; 1];
E = diag([0.25 0.25 0]);
b = 0;
cvx_begin
variables p(13,1) q(13,1) s(13,1) P(13,13) Q(13,13) L(13,13) V(13,1) y(3,1)
dual variables mu_s mu_L mu_vmin mu_vmax
minimize(c'*s)
subject to
% Boundary condition for power line flows
P( 1 , 1 ) == 0;
Q( 1 , 1 ) == 0;
% Boundary condition for squared line current
L( 1 , 1 ) == 0;
% Fix node 0 voltage to be 1 "per unit" (p.u.)
V(1) == 1;
% Loop over each node
for jj = 1:13
% Parent node, i = \rho(j)
ii = rho(jj);
if jj == 10
a_bar'*y + norm([E'*y],2) <= b;
y(1:2,1) >= 0;
y(1:2,1) <= 1;
s(jj) == y(3);
% Line Power Flows
P(ii,jj) == l_P(jj) - p(jj) + sum(A(jj,:).* P(jj,:)) + r(ii,jj)*L(ii,jj);
Q(ii,jj) == l_Q(jj) - q(jj) + sum(A(jj,:).* Q(jj,:)) + x(ii,jj)*L(ii,jj);
% Nodal voltage
V(jj) == V(ii)+(r(ii,jj)^2 + x(ii,jj)^2) * L(ii,jj) - 2*(r(ii,jj) * P(ii,jj) + x(ii,jj) * Q(ii,jj));
% Squared current magnitude on lines
L(ii,jj) >= quad_over_lin(P(ii,jj),V(jj))+ quad_over_lin(Q(ii,jj),V(jj));
% Compute apparent power from active & reactive power
norm([p(jj) q(jj)],2) <= s(jj);
q(jj) >= 0;
p(jj) >= 0;
else
% Line Power Flows
P(ii,jj) == l_P(jj) - p(jj) + sum(A(jj,:).* P(jj,:)) + r(ii,jj)*L(ii,jj);
Q(ii,jj) == l_Q(jj) - q(jj) + sum(A(jj,:).* Q(jj,:)) + x(ii,jj)*L(ii,jj);
% Nodal voltage
V(jj) == V(ii)+(r(ii,jj)^2 + x(ii,jj)^2) * L(ii,jj) - 2*(r(ii,jj) * P(ii,jj) + x(ii,jj) * Q(ii,jj));
% Squared current magnitude on lines
L(ii,jj) >= quad_over_lin(P(ii,jj),V(jj))+ quad_over_lin(Q(ii,jj),V(jj));
% Compute apparent power from active & reactive power
norm([p(jj) q(jj)],2) <= s(jj);
q(jj) >= 0;
p(jj) >= 0;
end
end
% Squared line current limits
L <= I_max.^2 : mu_L;
% Nodal voltage limits
V <= v_max.^2 : mu_vmax;
V >= v_min.^2 : mu_vmin;
% Apparent Power Limits
s <= s_max : mu_s;
cvx_end
% Output Results
fprintf(1,'------------------- PROBLEM 5 --------------------\n');
fprintf(1,'--------------------------------------------------\n');
fprintf(1,'Minimum Generating Cost : %4.2f USD\n',cvx_optval);
fprintf(1,'\n');
fprintf(1,'Node 0 [Grid] Gen Power : p_0 = %1.3f MW | q_0 = %1.3f MW | s_0 = %1.3f MW || mu_s0 = %3.0f USD/MW | mu_vmax0 = %3.2f p.u.| mu_vmin0 = %3.2f p.u./MW\n',p(1),q(1),s(1),mu_s(1),mu_vmax(1),mu_vmin(1));
fprintf(1,'Node 3 [Gas] Gen Power : p_3 = %1.3f MW | q_3 = %1.3f MW | s_3 = %1.3f MW || mu_s3 = %3.0f USD/MW | mu_vmax3 = %3.2f p.u.| mu_vmin3 = %3.2f p.u./MW\n',p(4),q(4),s(4),mu_s(4),mu_vmax(4),mu_vmin(4));
fprintf(1,'Node 9 [Solar] Gen Power : p_9 = %1.3f MW | q_9 = %1.3f MW | s_9 = %1.3f MW || mu_s9 = %3.0f USD/MW | mu_vmax9 = %3.2f p.u.| mu_vmin9 = %3.2f p.u./MW\n',p(10),q(10),s(10),mu_s(10),mu_vmax(10),mu_vmin(10));
fprintf(1,'\n');
fprintf(1,'Total active power : %1.3f MW consumed | %1.3f MW generated\n',sum(l_P),sum(p));
fprintf(1,'Total reactive power : %1.3f MVAr consumed | %1.3f MVAr generated\n',sum(l_Q),sum(q));
fprintf(1,'Total apparent power : %1.3f MVA consumed | %1.3f MVA generated\n',sum(l_S),sum(s));
fprintf(1,'\n');
for jj = 1:13
fprintf(1,'Node %2.0f Voltage : %1.3f p.u.\n',jj,sqrt(V(jj)));
end