In the context of global efforts toward carbon neutrality, renewable energy integration has become a pivotal focus. Among various technologies, solar photovoltaic (PV) systems stand out due to their environmental benefits and abundant resources. The grid-connected inverter serves as the critical interface between PV arrays and the power grid, and its performance directly impacts power quality and system stability. The Neutral-Point-Clamped (NPC) three-level grid-connected inverter is widely adopted in medium- and high-voltage applications owing to its advantages such as reduced output current harmonics, higher voltage withstand capability, and lower switching losses. However, controlling these inverters effectively remains challenging, especially when dealing with parameter uncertainties and disturbances.
Traditional control strategies, such as Proportional-Integral (PI) control and hysteresis control, have limitations in handling nonlinearities and multi-objective constraints. PI control is sensitive to parameter variations and can lead to overshoot and oscillations, while hysteresis control offers limited precision and slower response. In recent years, Finite Control Set-Model Predictive Control (FCS-MPC) has gained attention for its intuitive design and ability to manage complex system dynamics. Nonetheless, FCS-MPC relies heavily on an accurate system model, and its performance degrades under parameter mismatches caused by device aging, thermal effects, or environmental disturbances. This robustness issue is particularly critical for grid-connected inverters, where stable and high-quality current injection is essential.
To address these challenges, we propose a novel control strategy that integrates Radial Basis Function (RBF) neural networks with FCS-MPC for NPC three-level grid-connected inverters. Our approach leverages the universal approximation capability of RBF neural networks to estimate lumped disturbances in real-time, compensating for parameter deviations and enhancing system robustness. Additionally, we employ Weighted Average Current Control (WACC) to simplify the third-order LCL filter dynamics into a first-order system, reducing controller complexity and mitigating resonance risks. This article details the design, stability analysis, and experimental validation of our method, demonstrating significant improvements in current quality and parameter robustness compared to conventional FCS-MPC.
The topology of an NPC three-level grid-connected inverter with an LCL filter is depicted above. This configuration includes a DC-link with split capacitors, three-phase bridge arms capable of generating P, O, and N switching states, and filter components. The inverter connects to the grid through LCL filters, which suppress high-frequency harmonics but introduce third-order dynamics. The mathematical model of the system in the continuous-time domain is given by:
$$ L_{1j} \frac{di_{1j}}{dt} = v_j – v_{Cj} $$
$$ C_j \frac{dv_{Cj}}{dt} = i_{1j} – i_{2j} $$
$$ L_{2j} \frac{di_{2j}}{dt} = v_{Cj} – v_{gj} $$
where \( j = a, b, c \) denotes the phase, \( L_{1j} \) and \( L_{2j} \) are inverter-side and grid-side inductances, \( C_j \) is the filter capacitance, \( i_{1j} \) and \( i_{2j} \) are the respective currents, \( v_j \) is the inverter output voltage, \( v_{Cj} \) is the capacitor voltage, and \( v_{gj} \) is the grid voltage. Applying Laplace transforms, the transfer functions from inverter voltage to currents are derived. However, the LCL filter’s third-order nature can lead to resonance, complicating control design. To simplify, we adopt WACC, which defines a virtual current as a weighted average of inverter-side and grid-side currents:
$$ i_{\text{wac}j} = \upsilon i_{1j} + (1 – \upsilon) i_{2j} $$
with \( \upsilon = L_{1j} / (L_{1j} + L_{2j}) \). This reduces the system to a first-order model, as shown by the transfer function:
$$ G_{i_{\text{wac}j} – v_j}(s) = \frac{I_{\text{wac}j}(s)}{V_j(s)} = \frac{1}{s(L_{1j} + L_{2j}) + R_{1j} + R_{2j}} $$
where \( R_{1j} \) and \( R_{2j} \) represent parasitic resistances. In the dq0 synchronous reference frame, the state-space representation of the nominal system is:
$$ \dot{x}_1 = \frac{1}{L} (v_d – v_{gd} – R x_1 + \omega L x_2) $$
$$ \dot{x}_2 = \frac{1}{L} (v_q – v_{gq} – R x_2 – \omega L x_1) $$
with \( x = [x_1, x_2]^T = [i_{\text{wac}d}, i_{\text{wac}q}]^T \), \( L = L_{1j} + L_{2j} \), \( R = R_{1j} + R_{2j} \), \( \omega \) as the grid angular frequency, and \( v_d, v_q, v_{gd}, v_{gq} \) as dq-axis voltages. For practical grid-connected inverters, parameters often deviate due to aging or thermal effects, leading to a mismatched model. Let \( L_t = L + \Delta L \) and \( R_t = R + \Delta R \) be the actual values. The disturbed system dynamics become:
$$ \dot{x}_1 = \frac{1}{L_t} (v_d – v_{gd} – R_t x_1 + \omega L_t x_2) $$
$$ \dot{x}_2 = \frac{1}{L_t} (v_q – v_{gq} – R_t x_2 – \omega L_t x_1) $$
We define lumped disturbances \( f_d \) and \( f_q \) encompassing parameter variations and coupling terms:
$$ f_d = (-\Delta R x_1 + \Delta L \omega x_2 – \Delta L \dot{x}_1) / L $$
$$ f_q = (-\Delta R x_2 – \Delta L \omega x_1 – \Delta L \dot{x}_2) / L $$
This allows rewriting the state equation as:
$$ \dot{x} = A x + B v + D v_g + Z f $$
where \( A = \begin{bmatrix} -R/L & \omega \\ -\omega & -R/L \end{bmatrix} \), \( B = D = Z = \begin{bmatrix} 1/L & 0 \\ 0 & 1/L \end{bmatrix} \), \( x = [i_{\text{wac}d}, i_{\text{wac}q}]^T \), \( v = [v_d, v_q]^T \), \( v_g = [v_{gd}, v_{gq}]^T \), and \( f = [f_d, f_q]^T \). Discretizing with a sampling time \( T_s \) using forward Euler method yields:
$$ x(k+1) = A_\delta x(k) + B_\delta v(k) + D_\delta v_g(k) + Z_\delta f(k) $$
with \( A_\delta = I + T_s A \), \( B_\delta = D_\delta = Z_\delta = T_s I \), and \( I \) as the identity matrix. In conventional FCS-MPC, the cost function minimizes errors between predicted and reference currents, but disturbances \( f(k) \) degrade performance. Our solution incorporates an RBF neural network to estimate \( f(k) \) online, compensating the prediction model. The RBF network has three layers: input, hidden, and output. For input \( z = [z_1, z_2, \dots, z_l]^T \), the hidden layer outputs Gaussian functions:
$$ h_p(z) = \exp\left(-\frac{\| z – c_p \|^2}{2\sigma_p^2}\right) $$
where \( c_p \) and \( \sigma_p \) are center and width vectors. The output layer produces the disturbance estimate:
$$ \hat{f} = \Phi^T h(x) $$
with \( \Phi \) as the weight matrix. We design a state estimator:
$$ \dot{\hat{x}} = A \hat{x} + B v + D v_g + \Phi^T h(x) – \gamma_1 \tilde{x} $$
where \( \tilde{x} = \hat{x} – x \) is the estimation error, and \( \gamma_1 \) is a feedback gain. The weight update law is:
$$ \dot{\Phi} = -\eta [h(x) \tilde{x}^T + \gamma_2 \Phi] $$
with learning rate \( \eta \) and regularization factor \( \gamma_2 \). Stability is proven via Lyapunov theory. Consider the Lyapunov function:
$$ V = \frac{1}{2} \tilde{x}^T \tilde{x} + \frac{1}{2\eta} \text{tr}(\tilde{\Phi}^T \tilde{\Phi}) $$
where \( \tilde{\Phi} = \Phi – \Phi^* \) and \( \Phi^* \) is the optimal weight. Its derivative leads to:
$$ \dot{V} \leq -\tau \|\tilde{x}\|^2 – \frac{\gamma_2}{2} \|\tilde{\Phi}\|_F^2 + \frac{\gamma_2}{2} \|\Phi^*\|_F^2 + \frac{1}{2} \|\epsilon\|^2 $$
with \( \tau = \gamma_1 + 1/2 \) and \( \epsilon \) as approximation error. This ensures input-to-state stability, guaranteeing bounded errors under disturbances. The discretized compensated predictor is:
$$ \hat{x}(k+1) = A_\delta \hat{x}(k) + B_\delta v(k) + D_\delta v_g(k) + Z_\delta \hat{f}(k) – T_s \gamma_1 \tilde{x}(k) $$
For the grid-connected inverter, we also address DC-link midpoint potential balance. The capacitor voltages dynamics are:
$$ C \dot{v}_{C1} = i_{C1}, \quad C \dot{v}_{C2} = i_{C2} $$
Discretized, the predicted voltages are:
$$ v_{C1}^r(k+1) = v_{C1}(k) + \frac{T_s}{C} i_{C1}(k) $$
$$ v_{C2}^r(k+1) = v_{C2}(k) + \frac{T_s}{C} i_{C2}(k) $$
where \( i_{C1}(k) \) and \( i_{C2}(k) \) depend on switching states and phase currents. The overall cost function combines current tracking and midpoint balance:
$$ g_\Sigma = \| x_{\text{ref}}(k+1) – \hat{x}(k+1) \|^2 + \psi \| v_{C1}^r(k+1) – v_{C2}^r(k+1) \|^2 $$
with weighting factor \( \psi \). The controller evaluates all 27 switching states per sampling period, selecting the one that minimizes \( g_\Sigma \). This integrated approach enhances robustness for grid-connected inverter operations.
To validate our strategy, we conducted experiments on a dSPACE platform. The system parameters are summarized in Table 1. We compared our neural network-based FCS-MPC with traditional FCS-MPC under steady-state, parameter perturbation, and dynamic scenarios. The RBF network had 2 input nodes (dq-axis currents), 8 hidden nodes, and 2 output nodes (disturbance estimates), with \( \gamma_1 = 100 \), \( \eta = 10 \), and \( \gamma_2 = 0.01 \).
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC-link voltage \( V_{dc} \) | 400 V | Filter capacitance \( C_j \) | 5 μF |
| Grid phase voltage \( v_g \) | 110 V | DC-link capacitance \( C \) | 2200 μF |
| Inverter-side inductance \( L_{1j} \) | 2 mH | Sampling time \( T_s \) | 50 μs |
| Grid-side inductance \( L_{2j} \) | 1 mH | Grid frequency \( f \) | 50 Hz |
| Inverter-side resistance \( R_{1j} \) | 0.2 Ω | Weighting factor \( \psi \) | 0.1 |
| Grid-side resistance \( R_{2j} \) | 0.1 Ω | – | – |
In steady-state with reference currents \( i_{\text{ref}d} = 6 \, \text{A} \) and \( i_{\text{ref}q} = 0 \, \text{A} \), both methods produced sinusoidal grid currents. However, FFT analysis revealed that traditional FCS-MPC had a Total Harmonic Distortion (THD) of 2.48%, while our approach reduced THD to 1.35%, demonstrating superior harmonic suppression. This improvement is crucial for grid-connected inverters to meet power quality standards.
Parameter robustness was tested by varying inductances and resistances. In Scenario 1, actual values were set to 1.5 times nominal values (\( L_t = 1.5L \), \( R_t = 1.5R \)). Traditional FCS-MPC resulted in a THD of 2.36%, whereas our method achieved 1.13%. In Scenario 2, with values at 0.5 times nominal (\( L_t = 0.5L \), \( R_t = 0.5R \)), THDs were 3.41% and 1.59%, respectively. The increase in THD for traditional FCS-MPC was 0.93%, compared to only 0.24% for our method, highlighting enhanced robustness against parameter perturbations. These results underscore the effectiveness of RBF neural networks in estimating and compensating disturbances, ensuring reliable performance for grid-connected inverters under uncertainties.
Dynamic performance was evaluated by stepping the d-axis reference current from 6 A to 8 A. Both strategies responded quickly without overshoot, maintaining stable grid connection. The midpoint potential fluctuations remained within acceptable bounds, as regulated by the cost function. Table 2 summarizes the comparative THD results under different conditions, emphasizing the consistency of our approach.
| Condition | Traditional FCS-MPC THD | Neural Network-Based FCS-MPC THD |
|---|---|---|
| Steady-state (nominal parameters) | 2.48% | 1.35% |
| Scenario 1 (1.5× parameters) | 2.36% | 1.13% |
| Scenario 2 (0.5× parameters) | 3.41% | 1.59% |
Our control strategy offers several advantages for grid-connected inverters. First, the WACC simplifies the LCL filter dynamics, easing implementation. Second, the RBF neural network provides adaptive disturbance estimation without requiring precise system models, enhancing parameter robustness. Third, the Lyapunov-based stability assurance ensures reliable operation under varying grid conditions. These features make the method suitable for real-world applications where parameter deviations are common, such as in solar PV systems with aging components or temperature fluctuations.
Further analysis considers the computational aspect. The RBF network’s forward pass and weight update involve simple operations, feasible for modern digital signal processors. The prediction horizon in FCS-MPC is one step, keeping computational load manageable. For a grid-connected inverter, this enables high switching frequencies and precise current control. Additionally, the method can be extended to other multilevel inverter topologies or different filter configurations, showcasing its versatility.
In conclusion, we have developed a neural network-based model predictive control strategy for NPC three-level grid-connected inverters. By integrating RBF neural networks with FCS-MPC and WACC, we achieve robust performance against parameter uncertainties and reduced current harmonics. Experimental validations confirm significant improvements in THD and dynamic response compared to traditional methods. Future work may explore online tuning of neural network parameters or integration with other adaptive techniques to further optimize grid-connected inverter performance. This contribution advances the control of renewable energy systems, supporting the transition toward sustainable power grids.
The proposed approach underscores the importance of intelligent control in modern power electronics. As grid-connected inverters become more prevalent, robustness and efficiency will remain key research areas. Our method demonstrates a practical solution, leveraging neural networks to enhance model predictive control, ultimately ensuring stable and high-quality power injection from renewable sources into the grid.