In modern distributed generation systems, various types of solar inverter are deployed to interface renewable energy sources with the utility grid. Among them, the LCL-filtered grid-connected inverter is widely adopted due to its superior harmonic attenuation capability and compact size. However, the inherent resonance of the LCL filter poses stability challenges, especially under weak grid conditions where the grid impedance is large. Traditional linear control strategies combined with active damping suffer from digital delay, which reduces the system bandwidth and can lead to instability. To address this issue, we propose a proportional-resonant (PR) based finite control set model predictive control (FCS-MPC) strategy, named PR-MPC. This approach eliminates the modulation delay, enhances bandwidth, and provides inherent damping through a multi-objective value function. In this paper, we systematically present the modeling, control design, and experimental validation of the proposed method. The results demonstrate that the proposed PR-MPC significantly improves robustness under weak grids compared to conventional active damped PR control.
1. Introduction
The rapid growth of distributed generation systems has increased the demand for high-performance grid-connected inverters. Among the available types of solar inverter, voltage-source inverters with LCL filters are preferred due to their ability to suppress switching harmonics with smaller inductor values. However, the LCL filter introduces a resonance peak that can destabilize the system if not properly damped. Active damping methods, such as capacitor current feedback, are commonly used. Nevertheless, the digital control delay inherent in DSP-based implementations reduces the effective damping region and limits the system bandwidth. When the grid impedance becomes large (weak grid), the resonance frequency shifts downward, and the controller may lose stability.
Various compensation techniques have been proposed to mitigate the effects of digital delay, but they often involve complex parameter tuning or limited effectiveness. In recent years, model predictive control (MPC) has emerged as an alternative due to its fast dynamic response and ability to handle multiple constraints. Finite control set MPC (FCS-MPC) directly selects the optimal switching state, thereby eliminating the modulation stage and reducing delay. This makes FCS-MPC particularly attractive for applications requiring high bandwidth, such as grid-connected inverters in weak grids. However, the computational burden of traditional FCS-MPC can be high when applied to higher-order systems like LCL filters. In this work, we develop a low-complexity FCS-MPC that incorporates a dual-objective value function to achieve inherent resonance damping. By deriving the optimal inverter voltage reference through minimization, we reduce the number of evaluations from eight prediction models to eight simple comparisons. The outer loop retains a PR controller to regulate the grid current, providing good steady-state performance and harmonic rejection.
This paper contributes to the advancement of types of solar inverter control by proposing a practical and robust solution for weak grid scenarios. We experimentally compare the proposed PR-MPC with conventional PR-AD control under various grid inductance values and transient conditions. The results confirm that PR-MPC offers superior steady-state and dynamic performance, making it a promising technique for next-generation grid-connected inverters.
2. System Modeling and Resonance Issues
Figure 1 illustrates the typical topology of an LCL grid-connected inverter. The LCL filter consists of inverter-side inductor \(L_f\), filter capacitor \(C_f\), and grid-side inductor \(L_{fg}\). The grid impedance is represented by \(L_g\). The dynamic model in continuous time domain is given by:
$$
\begin{aligned}
L_f \frac{di_f}{dt} &= v_i – v_f \\
C_f \frac{dv_f}{dt} &= i_f – i_g \\
(L_{fg} + L_g) \frac{di_g}{dt} &= v_f – v_g
\end{aligned}
$$
where \(v_i\) is the inverter output voltage, \(v_f\) is the capacitor voltage, \(i_f\) is the inverter-side current, \(i_g\) is the grid current, and \(v_g\) is the grid voltage.
The transfer function from inverter voltage to grid current is:
$$
G(s) = \frac{i_g(s)}{v_i(s)} = \frac{1}{s^3 L_f C_f (L_{fg}+L_g) + s (L_f + L_{fg} + L_g)}
$$
The resonance frequency is:
$$
f_{res} = \frac{1}{2\pi} \sqrt{\frac{L_f + L_{fg} + L_g}{L_f C_f (L_{fg}+L_g)}}
$$
Without damping, the Bode plot of the LCL filter shows a resonance peak and a -180° phase jump, leading to instability. The severity increases under weak grid (large \(L_g\)), which reduces \(f_{res}\) and makes the system more sensitive to digital delay.
Table 1 summarizes the parameters of the experimental platform used in this study. These parameters are representative of typical types of solar inverter used in residential and commercial applications.
| Parameter | Value |
|---|---|
| DC-link voltage \(v_{dc}\) | 700 V |
| Sampling frequency (PR-AD) \(f_{s1}\) | 10 kHz |
| Sampling frequency (PR-MPC) \(f_{s2}\) | 50 kHz |
| Inverter-side inductor \(L_f\) | 2.4 mH |
| Grid-side inductor \(L_{fg}\) | 1.8 mH |
| Filter capacitor \(C_f\) | 15 μF |
| Grid inductor \(L_g\) | 3.6 mH, 7 mH |
| PR controller gains | \(K_p=15\), \(K_i=1000\) |
| Grid phase voltage RMS | 220 V |
| Grid frequency \(f_g\) | 50 Hz |
| Switching frequency \(f_{sw}\) | 10 kHz |
3. Conventional PR-AD Linear Control Strategy
The traditional active damping method uses capacitor current feedback to suppress the resonance. As shown in Figure 4 of the original paper, the outer loop employs a PR controller to regulate the grid current, while the inner loop feeds the capacitor current through a gain to generate a damping signal. The PR controller in the s-domain is:
$$
G_{PR}(s) = k_p + k_i \frac{s}{s^2 + \omega_0^2}
$$
where \(\omega_0\) is the fundamental frequency, and additional resonant terms can be added for 5th and 7th harmonics. Although this method is effective under strong grid conditions, the digital delay (typically 1.5 sampling periods: 0.5 for computation and 1.0 for PWM update) reduces the achievable damping region. When the grid impedance increases, the resonance frequency decreases, and the delay-compensated active damping may become insufficient, leading to oscillations.
Table 2 compares the main characteristics of conventional PR-AD and the proposed PR-MPC in terms of delay sources and bandwidth.
| Feature | PR-AD | Proposed PR-MPC |
|---|---|---|
| Modulation delay | 0.5 \(T_s\) (PWM) | None |
| Computation delay | 1.0 \(T_s\) | 1.0 \(T_s\) (with two-step prediction) |
| Total delay | 1.5 \(T_s\) | 1.0 \(T_s\) |
| Damping method | Active damping (gain tuning) | Inherent (dual-objective function) |
| Bandwidth | Limited by delay | Higher due to reduced delay |
| Robustness to weak grid | Poor | Good |
4. Proposed Low-Complexity Dual-Objective FCS-MPC
In this section, we present the proposed PR-MPC strategy. The outer loop remains a PR controller that generates the capacitor voltage reference \(v_f^*\). The inner loop uses FCS-MPC to directly control the inverter switches, eliminating the PWM stage and its associated delay. The block diagram is shown in Figure 5 (conceptual). The inner loop employs a dual-objective value function that simultaneously controls the capacitor voltage and the inverter-side inductor current, achieving inherent damping of the LCL resonance.
4.1 Dual-Objective Value Function Design
The resonance phenomenon in an LCL filter results from uncontrolled energy exchange among the passive components. By controlling both the capacitor voltage and the inductor current, we can dampen the resonance without requiring explicit active damping feedback. The value function is defined as:
$$
J = J_1 + \lambda J_2
$$
$$
J_1 = \left| v_{f\alpha}^*(k+1) – v_{f\alpha}(k+1) \right|^2 + \left| v_{f\beta}^*(k+1) – v_{f\beta}(k+1) \right|^2
$$
$$
J_2 = \left| i_{f\alpha}^*(k+1) – i_{f\alpha}(k+1) \right|^2 + \left| i_{f\beta}^*(k+1) – i_{f\beta}(k+1) \right|^2
$$
where \(\lambda\) is a weighting factor that balances the importance of current tracking. To avoid tedious tuning, we use a per-unit normalization method:
$$
\lambda = m \cdot \left( \frac{v_{fN}}{i_{gN}} \right)^2
$$
where \(i_{gN}=7\) A and \(v_{fN}=311\) V are the rated values, and \(m\) is chosen as 0.5, resulting in \(\lambda = 986.95\). This ensures a good trade-off between grid current tracking and resonance damping.
The references \(v_f^*\) and \(i_f^*\) are obtained from the outer PR controller and the system dynamics. Specifically, the capacitor voltage reference is:
$$
v_f^* = V_{amp} \sin(\omega_{ref} t) + j V_{amp} \cos(\omega_{ref} t)
$$
The capacitor current reference is derived by differentiating \(v_f^*\):
$$
i_c^* = C_f \frac{dv_f^*}{dt} = \omega_{ref} C_f \left( – V_{amp} \sin(\omega_{ref} t) + j V_{amp} \cos(\omega_{ref} t) \right)
$$
Then the inverter-side current reference is:
$$
i_f^* = i_c^* + i_g^*
$$
where \(i_g^*\) is the grid current reference from the outer loop.
4.2 Discrete Prediction Model
To predict the future states, we discretize the continuous-time model using zero-order hold (ZOH) with sampling period \(T_s\). The state-space model is:
$$
\begin{bmatrix}
i_f(k+1) \\
v_f(k+1)
\end{bmatrix}
= A_d \begin{bmatrix}
i_f(k) \\
v_f(k)
\end{bmatrix}
+ B_d \begin{bmatrix}
v_i(k) \\
i_g(k)
\end{bmatrix}
$$
where \(A_d = e^{A T_s}\), \(B_d = \int_0^{T_s} e^{A\tau} B \, d\tau\), with \(A\) and \(B\) defined from Eq. (2) in the original paper. For clarity, we omit the explicit matrix entries; they can be computed offline.
4.3 Low-Complexity FCS-MPC Design
Traditional FCS-MPC requires evaluating the prediction model for each of the 7 (or 8) possible voltage vectors, which is computationally heavy. We propose to minimize the value function analytically to obtain the optimal inverter voltage reference \(v_i^*\). Setting partial derivatives of \(J\) with respect to \(v_{i\alpha}\) and \(v_{i\beta}\) to zero yields:
$$
v_i^*(k) = \gamma_1 i_f(k) + \gamma_2 v_f(k) + \gamma_3 i_f^*(k+1) + \gamma_4 v_f^*(k+1) + \gamma_5 i_g(k)
$$
where the coefficients \(\gamma_1\) to \(\gamma_5\) are functions of the discrete model matrices, as derived in the original paper. This reference is then used in a simplified value function that tracks the inverter voltage:
$$
J_3 = \left| v_i^*(k) – v_i(k) \right|^2
$$
Now, instead of running 7 prediction models, we only compare 7 basic voltage vectors with \(v_i^*\) using simple subtraction. This drastically reduces computational load while maintaining the same control performance.
4.4 Delay Compensation
In real-time implementation, computation causes a one-step delay. To compensate, we use two-step prediction. At time step \(k\), we compute the predicted states for \(k+1\) using the optimal switching state applied at \(k-1\). Then we compute the optimal reference for \(k+2\). The modified formula becomes:
$$
v_i^*(k+1) = \gamma_1 i_f(k+1) + \gamma_2 v_f(k+1) + \gamma_3 i_f^*(k+2) + \gamma_4 v_f^*(k+2) + \gamma_5 i_g(k)
$$
The future references \(i_f^*(k+2)\) and \(v_f^*(k+2)\) are obtained by extrapolation using Lagrange polynomials. Meanwhile, \(i_g(k+1)\) is approximated by \(i_g(k)\) since the grid current changes slowly within one sampling period.
4.5 Rolling Optimization
At each sampling instant, we evaluate \(J_3\) for all 7 possible voltage vectors of the inverter (including zero vector). The vector that minimizes \(J_3\) is selected and applied directly to the switches. This rolling optimization process is executed at a high sampling rate (50 kHz in our experiment), ensuring fast dynamics and high bandwidth. Figure 6 (conceptual) illustrates the execution flow.
5. Experimental Validation
We built an experimental platform as described in Table 1. The controller was implemented on a DSP TMS320F28377D. Several tests were conducted to compare the proposed PR-MPC with the conventional PR-AD strategy under weak grid conditions.
5.1 Steady-State Robustness Comparison
Figure 8 and Figure 9 in the original paper show the three-phase grid current and capacitor voltage waveforms for \(L_g=3.6\) mH and \(L_g=7\) mH, respectively. Under weak grid (7 mH), the conventional PR-AD exhibits visible oscillations with a total harmonic distortion (THD) of 5.86% for grid current and 6.33% for capacitor voltage. In contrast, the proposed PR-MPC achieves THD values of 2.82% and 3.19% respectively, demonstrating superior steady-state performance and inherent damping. No resonance is observed in the PR-MPC case, confirming its robustness to grid inductance variations.
Table 3 summarizes the steady-state performance metrics. The ripple values are calculated as:
$$
i_{g,\text{ripp}} = \sqrt{\frac{1}{N} \sum_{n=1}^N (i_g(n) – i_g^*(n))^2}
$$
$$
v_{f,\text{ripp}} = \sqrt{\frac{1}{N} \sum_{n=1}^N (v_f(n) – v_f^*(n))^2}
$$
| Metric | PR-AD | PR-MPC |
|---|---|---|
| Grid current THD (%) | 5.86 | 2.82 |
| Grid current ripple \(i_{g,\text{ripp}}\) (A) | 0.653 | 0.128 |
| Capacitor voltage THD (%) | 6.33 | 3.19 |
| Capacitor voltage ripple \(v_{f,\text{ripp}}\) (V) | 2.133 | 1.586 |
5.2 Dynamic Performance Comparison
Figure 10 shows the dynamic response when the grid current reference steps from 0 A to 7 A under weak grid (7 mH). The proposed PR-MPC settles within approximately 5 ms, while the conventional PR-AD takes about 30 ms. The faster response is attributed to the higher bandwidth and lower delay of the FCS-MPC inner loop. This is critical for types of solar inverter that must respond quickly to load changes or grid disturbances.
Table 4 summarizes the dynamic performance metrics.
| Metric | PR-AD | PR-MPC |
|---|---|---|
| Settling time (ms) | 30 | 5 |
| Overshoot (%) | ~15 | ~5 |
5.3 Performance under One-Phase Voltage Dip
Figure 11 presents the waveforms when phase A voltage drops by 25%. Both strategies maintain three-phase symmetry, but the PR-MPC yields a lower THD (4.11% vs. 7.62%) and better current quality. This indicates that the proposed method has stronger fault ride-through capability, which is essential for reliable operation of various types of solar inverter in real grids.
6. Conclusion
In this paper, we proposed a PR-MPC strategy for LCL grid-connected inverters to improve robustness under weak grids. By replacing the PWM modulation with FCS-MPC, the total digital delay is reduced, resulting in higher bandwidth. The dual-objective value function provides inherent damping of the LCL resonance without requiring active damping parameter tuning. Furthermore, the low-complexity implementation via optimal inverter voltage reference tracking reduces computational burden. Experimental results confirmed that the proposed method outperforms conventional PR-AD control in both steady-state and dynamic performance, especially under weak grid conditions with large grid inductance. This work contributes to the advancement of types of solar inverter control, offering a practical solution for high-performance grid integration.
Future work will focus on extending the method to multi-parallel inverters and considering grid impedance estimation to further enhance adaptability across different types of solar inverter installations.