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My state space model of VSI inverter with SVPWM won't produce output voltage and current in abc!

Sunday Olayanju on 4 Jul 2023
Latest activity Edit by John on 22 Aug 2023

Kindly help me correct this code to function properly. I am just learning MATLAB. i cannot get the output in abc frame. This is the code:
%----------- Define input and state parameters-----------------------------
clc
v_dc = 350; % DC input voltage in V
m = 0.841; % modulation index
C = 4000e-6; % DC buss capacitance in uf
L_1 = 2.5e-3; % Inverter side inductance in mH
L_2 = 2.5e-3; % Load side inductance in mH
L = 0; % load inductance
C_f = 10e-6; % filter capacitance in uf
R_f = 0.7; % damping resistance in ohms
R_L = 20; % load resistance in ohms
f_s = 10e3; % switching frequency
f = 60; % System frequency
R_s = 0.01; % Capacitance of the DC circuit
I_d = 8.594; % steady state current
w = 2*pi*f; % System angular Frequency
% Define initial steady state values
v_c = 349.4; i_d = 8.594; i_q = -0.213; v_df = 285; v_qf = -120; i_Ld = 8.594; i_Lq = 0.85;
%------------------S V P W M Generator-------------------------------------
% Define reference vector Uref
U_mag = m*v_dc/2; % Magnitude of Uref
% Define switching vectors
U1 = [v_dc/2;0]; % Vector Q1
U2 = [v_dc/4;sqrt(3)*v_dc/4]; % Vector Q2
U3 = [-v_dc/4;sqrt(3)*v_dc/4]; % Vector Q3
U4 = [-v_dc/2;0]; % Vector Q4
U5 = [-v_dc/4;-sqrt(3)*v_dc/4]; % Vector Q5
U6 = [v_dc/4;-sqrt(3)*v_dc/4]; % Vector Q6
% Define sector angles
theta1 = pi/6;
theta2 = pi/2;
theta3 = 5*pi/6;
theta4 = 7*pi/6;
theta5 = 3*pi/2;
theta6 = 11*pi/6;
% Define duty cycles for each switch using a for loop
for t=0:1/f_s:1/f % Time variable from 0 to one cycle of system frequency with steps of switching frequency
U_phase = w*t; % Phase of Uref (t is time variable)
U_alpha = U_mag*cos(U_phase); % Alpha component of Uref
U_beta = U_mag*sin(U_phase); % Beta component of Uref
if (0 <= U_phase) && (U_phase < theta1) % Sector 1
T1 = (sqrt(3)*U_beta + U_alpha)/(2*v_dc);
T2 = (-sqrt(3)*U_beta + U_alpha)/(2*v_dc);
T0 = 1 - T1 - T2;
d_a(round(t)+1) = T1 + T0/2;
d_b(round(t)+1) = T2 + T0/2;
d_c(round(t)+1) = T0/2;
elseif (theta1 <= U_phase) && (U_phase < theta2) % Sector 2
T3 = (sqrt(3)*U_beta - U_alpha)/(2*v_dc);
T2 = (sqrt(3)*U_beta + U_alpha)/(2*v_dc);
T0 = 1 - T3 - T2;
d_a(round(t)+1) = T0/2;
d_b(round(t)+1) = T2 + T0/2;
d_c(round(t)+1) = T3 + T0/2;
elseif (theta2 <= U_phase) && (U_phase < theta3) % Sector 3
T3 = (sqrt(3)*U_beta - U_alpha)/(2*v_dc);
T4 = (-sqrt(3)*U_beta - U_alpha)/(2*v_dc);
T0 = 1 - T3 - T4;
d_a(round(t)+1) = T0/2;
d_b(round(t)+1) = T0/2;
d_c(round(t)+1) = T3 + T0/2;
elseif (theta3 <= U_phase) && (U_phase < theta4) % Sector 4
T5 = (-sqrt(3)*U_beta + U_alpha)/(2*v_dc);
T4 = (-sqrt(3)*U_beta - U_alpha)/(2*v_dc);
T0 = 1 - T5 - T4;
d_a(round(t)+1) = T5 + T0/2;
d_b(round(t)+1) = T0/2;
d_c(round(t)+1) = T4 + T0/2;
elseif (theta4 <= U_phase) && (U_phase < theta5) % Sector 5
T5 = (-sqrt(3)*U_beta + U_alpha)/(2*v_dc);
T6 = (sqrt(3)*U_beta + U_alpha)/(2*v_dc);
T0 = 1 - T5 - T6;
d_a(round(t)+1) = T5 + T0/2;
d_b(round(t)+1) = T6 + T0/2;
d_c(round(t)+1) = T0/2;
elseif (theta5 <= U_phase) && (U_phase < theta6) % Sector 6
T1 = (sqrt(3)*U_beta + U_alpha)/(2*v_dc);
T6 = (sqrt(3)*U_beta - U_alpha)/(2*v_dc);
T0 = 1 - T1 - T6;
d_a(round(t)+1) = T1 + T0/2;
d_b(round(t)+1) = T0/2;
d_c(round(t)+1) = T6 + T0/2;
end
end
%-------------------------Define system matrices---------------------------
% Create Three-phase SVPWM VSI Inverter
% System matrix Nx-by-Nx matrix
A = [-1/(C*R_s),-sqrt(3)*m/(2*C),0,0,0,0,0;
sqrt(3)*m/(3*L_1),-R_f/(3*L_1),w,-1/(2*L_1),-sqrt(3)/(6*L_1),-R_f/(3*L_1),0;
0,-w,-R_f/(3*L_1),-sqrt(3)/(6*L_1),-1/(2*L_1),0,R_f/(3*L_1);
0,1/(2*C_f),-sqrt(3)/(6*C_f),0,w,-1/(2*C_f),sqrt(3)/(6*C_f);
0,sqrt(3)/(6*C_f),1/(2*C_f),-w,0,-sqrt(3)/(6*C_f),-1/(2*C_f);
0,R_f/(3*(L_2+L)),0,1/(2*(L_2+L)),sqrt(3)/(6*(L_2+L)),((-3*R_L-R_f)/(3*(L_2+L))),w;
0, 0, R_f/(3*(L_2+L)), -sqrt(3)/(6*(L_2+L)), 1/(2*(L_2+L)), -w, ((-3*R_L-R_f)/(3*(L_2+L)))];
% Define input matrix
B = [1/(C*R_s),-sqrt(3)*i_d/(2*C);d_a*v_dc,(sqrt(3)*v_c)/L_1;d_b*v_dc,0;d_c*v_dc,0;0,0;0,0;0,0]; % Nx-by-Nu input matrix
% Define output matrix
C = [0 1 0 0 0 0 0; % Ny-by-Nx matrix
0 0 1 0 0 0 0;
0 0 0 1 0 0 0;
0 0 0 0 1 0 0;
0 0 0 0 0 1 0;
0 0 0 0 0 0 1];
% Feedthrough matrix
D = zeros(6, 2); % Ny-by-Nu matrix
% create state-space model object
sys = ss(A,B,C,D);
% Define initial conditions and input
x0 = [v_c; i_d; i_q; v_df; v_qf; i_Ld; i_Lq]; % Initial state vector
t = 0:1e-6:0.5; % Time vector for simulation
u = repmat([v_dc;m],1,length(t)); % repeat u for each time step
% Simulate the system
[y, ~, x] = lsim(sys, u, t, x0);
% Extract the states
v_c_sim = x(:, 1);
i_d_sim = x(:, 2);
i_q_sim = x(:, 3);
v_df_sim = x(:, 4);
v_qf_sim = x(:, 5);
i_Ld_sim = x(:, 6);
i_Lq_sim = x(:, 7);
% Extract the outputs
v_abc_sim = y(:, 1:3);
i_abc_sim = y(:, 4:6);
v_dq_sim = y(:, 4:5);
i_dq_sim = y(:, 2:3);
% Plot the variables
figure;
subplot(4, 2, 1);
plot(t, v_c_sim);
xlabel('Time');
ylabel('v_c');
title('Capacitor Voltage');
subplot(4, 2, 2);
plot(t, i_d_sim);
xlabel('Time');
ylabel('i_d');
title('d-Axis Current');
subplot(4, 2, 3);
plot(t, i_q_sim);
xlabel('Time');
ylabel('i_q');
title('q-Axis Current');
subplot(4, 2, 4);
plot(t, v_df_sim);
xlabel('Time');
ylabel('v_df');
title('d-Component Filter Voltage');
subplot(4, 2, 5);
plot(t, v_qf_sim);
xlabel('Time');
ylabel('v_qf');
title('q-Component Filter Voltage');
subplot(4, 2, 6);
plot(t, i_Ld_sim);
xlabel('Time');
ylabel('i_Ld');
title('d-Axis Load Current');
subplot(4, 2, 7);
plot(t, i_Lq_sim);
xlabel('Time');
ylabel('i_Lq');
title('q-Axis Load Current');
% Perform coordinate transformation from dq frame to abc frame for currents
i_a_sim = cos(w*t)*i_d_sim - sin(w*t)*i_q_sim;
i_b_sim = cos(w*t - 2*pi/3)*i_d_sim - sin(w*t - 2*pi/3)*i_q_sim;
i_c_sim = cos(w*t + 2*pi/3)*i_d_sim - sin(w*t + 2*pi/3)*i_q_sim;
% Perform coordinate transformation from dq frame to abc frame for voltages
v_a_sim = cos(w*t)*v_df_sim - sin(w*t)*v_qf_sim;
v_b_sim = cos(w*t - 2*pi/3)*v_df_sim - sin(w*t - 2*pi/3)*v_qf_sim;
v_c_sim = cos(w*t + 2*pi/3)*v_df_sim - sin(w*t + 2*pi/3)*v_qf_sim;
Many thanks

1 Comment

John
John on 22 Aug 2023 (Edited on 22 Aug 2023)
Hello,
The issue is that you are not correctly transforming the signals from dq to abc frame. Here is the corrected code:
% Perform coordinate transformation from dq frame to abc frame for currents
i_a_sim = cos(wt)i_d_sim - sin(wt)i_q_sim;
i_b_sim = cos(w
t - 2
pi/3)i_d_sim - sin(wt - 2pi/3)i_q_sim;
i_c_sim = cos(w
t + 2
pi/3)i_d_sim - sin(wt + 2*pi/3)*i_q_sim;
% Perform coordinate transformation from dq frame to abc frame for voltages
v_a_sim = cos(wt)v_df_sim - sin(wt)vqf_sim;
v_b_sim = cos(w
t - 2
pi/3)v_df_sim - sin(wt - 2pi/3)vqf_sim;
v_c_sim = cos(w
t + 2
pi/3)v_df_sim - sin(wt + 2*pi/3)*vqf_sim;
% Plot the abc frame currents and voltages
figure;
subplot(2,1,1)
plot(t, [i_a_sim i_b_sim i_c_sim])
title('ABC frame currents')
subplot(2,1,2)
plot(t, [v_a_sim v_b_sim v_c_sim])
title('ABC frame voltages')
The key changes:
  1. Transform voltages from dq to abc using v_df and vqf instead of v_dq
  2. Plot i_a_sim, i_b_sim, i_c_sim together to see the abc currents
  3. Plot v_a_sim, v_b_sim, v_c_sim together to see the abc voltages
This should now correctly transform and plot the signals in the abc frame. Let me know if any part is still unclear!

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