In modern renewable energy systems, the three phase inverter plays a critical role in converting DC power from sources like solar panels into AC power for grid integration or standalone applications. Traditional control strategies, such as standard model predictive control (MPC), often suffer from issues like poor prediction accuracy and variable switching frequency, leading to high current harmonic distortion. This paper addresses these challenges by proposing a novel dual-vector modulation-based MPC strategy for the three phase inverter, incorporating an improved Euler method for discretization. The approach significantly enhances steady-state and dynamic performance while reducing total harmonic distortion (THD) in output currents.
The three phase inverter topology considered here is a standard three-wire system, where each phase consists of switching devices that generate AC output from a DC source. The mathematical model in the abc stationary frame is derived using Kirchhoff’s laws, resulting in the following state equation:
$$ L \frac{d}{dt} \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} = \begin{bmatrix} u_a \\ u_b \\ u_c \end{bmatrix} – R \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} $$
where \( i_a, i_b, i_c \) represent the line currents, and \( u_a, u_b, u_c \) denote the phase voltages. To simplify analysis, the Clark transformation is applied to convert the system into the αβ stationary coordinate frame:
$$ \begin{bmatrix} x_\alpha \\ x_\beta \\ x_0 \end{bmatrix} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} x_a \\ x_b \\ x_c \end{bmatrix} $$
This yields the reduced state equation:
$$ L \frac{d}{dt} \begin{bmatrix} i_\alpha \\ i_\beta \end{bmatrix} = \begin{bmatrix} u_\alpha \\ u_\beta \end{bmatrix} – R \begin{bmatrix} i_\alpha \\ i_\beta \end{bmatrix} $$
Solving this differential equation accurately is crucial for predictive control. Traditional MPC uses forward Euler discretization, which has first-order accuracy and can lead to errors. Instead, the improved Euler method, a second-order numerical technique, is employed for higher precision. The discretization process involves predicting the next state using:
$$ y_{i+1}’ = y_i + T_s f(x_i, y_i) $$
$$ y_{i+1} = y_i + \frac{T_s}{2} \left[ f(x_i, y_i) + f(x_{i+1}, y_{i+1}’) \right] $$
where \( T_s \) is the sampling period. For the three phase inverter current prediction, this becomes:
$$ i_{\alpha\beta}(k+1) = i_{\alpha\beta}(k) + \frac{T_s}{2L} \left[ u_{\alpha\beta}(k) – R i_{\alpha\beta}(k) + u_{\alpha\beta}(k+1) – R i_{\alpha\beta}(k+1)’ \right] $$
with \( i_{\alpha\beta}(k+1)’ \) as the intermediate prediction. This method reduces discretization errors, improving the overall control performance of the three phase inverter.
The control strategy combines PI control with dual-vector MPC in a cascaded structure. The outer loop uses PI control to regulate the output voltage, generating a current reference for the inner MPC loop. This ensures high power factor and robust performance in non-grid-connected scenarios. The reference current is derived as:
$$ I_{\text{ref}} = K_p (u_{\text{ref}} – u_0) + K_i \int (u_{\text{ref}} – u_0) \, dt $$
where \( u_{\text{ref}} \) is the sinusoidal voltage reference, and \( u_0 \) is the load voltage. The phase factor adjusts the current phase for optimal power factor. The inner MPC loop then minimizes the cost function:
$$ g = |i_{\alpha,\text{ref}} – i_\alpha| + |i_{\beta,\text{ref}} – i_\beta| $$
by evaluating all possible voltage vectors. In conventional finite control set MPC (FCS-MPC), the three phase inverter has eight switching states, corresponding to six active and two zero vectors. The voltage vectors in αβ coordinates are listed in Table 1.
| Vector | Switching State | \( u_\alpha \) | \( u_\beta \) |
|---|---|---|---|
| \( u_0 \) | 000 | 0 | 0 |
| \( u_1 \) | 100 | \( \frac{2U_{dc}}{3} \) | 0 |
| \( u_2 \) | 110 | \( \frac{U_{dc}}{3} \) | \( \frac{\sqrt{3}U_{dc}}{3} \) |
| \( u_3 \) | 010 | \( -\frac{U_{dc}}{3} \) | \( \frac{\sqrt{3}U_{dc}}{3} \) |
| \( u_4 \) | 011 | \( -\frac{2U_{dc}}{3} \) | 0 |
| \( u_5 \) | 001 | \( -\frac{U_{dc}}{3} \) | \( -\frac{\sqrt{3}U_{dc}}{3} \) |
| \( u_6 \) | 101 | \( \frac{U_{dc}}{3} \) | \( -\frac{\sqrt{3}U_{dc}}{3} \) |
| \( u_7 \) | 111 | 0 | 0 |
To enhance precision, dual-vector modulation is introduced, where each control period combines two voltage vectors. This expands the finite set to 12 vectors, such as \( u_1′ = u_0 + u_1 \), \( u_2′ = u_0 + u_2 \), etc., improving resolution and fixed switching frequency. The combined vector is synthesized as:
$$ u_i’ = \frac{t_j}{T} u_j + \frac{t_k}{T} u_k $$
with \( t_j + t_k = T \), where \( T \) is the control period. The dwell times are proportional to the inverse of individual cost functions:
$$ t_j = \frac{g_j}{g_j + g_k} T, \quad t_k = \frac{g_k}{g_j + g_k} T $$
This ensures optimal vector selection and reduces current ripple in the three phase inverter. For example, combining vectors \( u_1 \) and \( u_2 \) results in a modulated output where the switch states vary within the period, achieving smoother transitions.
Simulation studies validate the proposed strategy using a three phase inverter model with a 500 V DC input and 120 V AC output reference. Steady-state performance shows a THD of 0.98% for the improved Euler-based dual-vector MPC, compared to 1.09% for traditional MPC. The output current waveform is sinusoidal with minimal distortion, and the response settles within three cycles. Table 2 compares THD under different discretization methods.
| Method | THD (%) |
|---|---|
| Euler Method | 1.09 |
| Improved Euler Method | 0.98 |
| Runge-Kutta Method | 1.32 |
The improved Euler method offers better stability and accuracy than higher-order methods like Runge-Kutta, which may suffer from numerical instability in power electronics applications. Dynamic performance is tested under input voltage changes from 400 V to 500 V and reference voltage changes from 120 V to 150 V. The three phase inverter maintains low THD variations (0.2% and 0.01%, respectively), demonstrating robustness against disturbances common in solar power systems.
In conclusion, the integration of improved Euler discretization and dual-vector MPC in a cascaded PI-MPC structure significantly enhances the performance of the three phase inverter. This approach reduces harmonic distortion, ensures fixed switching frequency, and provides excellent dynamic response. Future work could explore adaptive control and AI-based optimization for further improvements in three phase inverter applications.