In recent years, the increasing integration of renewable energy sources, such as photovoltaic and wind power, into the grid has led to significant changes in grid impedance, resulting in weak grid conditions. As the interface between these sources and the grid, the on-grid inverter plays a crucial role in maintaining system stability and power quality. However, under weak grid scenarios, factors like grid voltage feedforward loops and digital control delays can couple with grid impedance, severely degrading system stability. This paper addresses these challenges by proposing an improved feedforward control strategy based on impedance reshaping for on-grid inverters, specifically focusing on LCL-filtered systems. The strategy enhances robustness against grid impedance variations and improves grid current quality, validated through detailed analysis, simulations, and experiments.
The stability of an on-grid inverter is paramount in weak grids, where the grid impedance can fluctuate, leading to potential instabilities. Traditional proportional feedforward control methods, while effective in strong grids, often introduce current positive feedback loops that reduce phase margin and compromise stability. To overcome this, we first establish the output impedance model of the on-grid inverter and analyze the limitations of conventional approaches. Subsequently, we introduce a modified feedforward control strategy that incorporates a band-pass filter in the voltage feedforward path to reshape the output impedance. Additionally, a harmonic controller is integrated to suppress background harmonics. This comprehensive method not only augments the system’s adaptability to weak grids but also ensures high-quality grid current injection. The following sections delve into the theoretical foundations, design methodology, stability analysis, and experimental verification of the proposed strategy.
The fundamental structure of a single-phase LCL-type on-grid inverter involves an inverter-side inductor \(L_1\), a grid-side inductor \(L_2\), and a filter capacitor \(C\), as shown in the control diagram. The grid impedance \(L_g\) represents the weak grid effect, and the system employs a quasi-proportional-resonant (QPR) controller for current regulation to minimize steady-state error. The transfer function of the QPR controller is given by:
$$ G_c(s) = k_p + \frac{2k_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \(k_p\) is the proportional gain, \(k_r\) is the resonant gain, \(\omega_c\) is the controller bandwidth, and \(\omega_0\) is the fundamental angular frequency. The digital control delay \(G_d(s)\) is modeled as:
$$ G_d(s) = \frac{1 – e^{-sT_s}}{sT_s} e^{-1.5sT_s} $$
with \(T_s\) as the sampling period. The system’s open-loop transfer function without feedforward can be derived, but when grid voltage feedforward is added, it introduces coupling effects that degrade performance.
In weak grids, the stability criterion relies on the phase margin at the frequency where the magnitudes of the grid impedance \(Z_g(s)\) and the inverter output impedance \(Z_o(s)\) intersect. The output impedance for an on-grid inverter with traditional proportional feedforward \(G_f(s) = 1/K_{PWM}\) is expressed as:
$$ Z_o(s) = \frac{L_1 L_2 C s^3 + K_{PWM} G_d(s) L_2 C s^2 + (L_1 + L_2)s + G_c(s) G_d(s) K_{PWM}}{L_1 C s^2 + K_c K_{PWM} G_d(s) C s + 1 – K_{PWM} G_d(s) G_f(s)} $$
where \(K_{PWM}\) is the inverter gain. Analysis of the Bode plots reveals that traditional proportional feedforward reduces the phase margin significantly in the mid-frequency range, making the on-grid inverter susceptible to instability as grid impedance increases. For instance, with \(L_g = 4 \text{ mH}\), the phase margin drops to as low as \(4.2^\circ\), compared to \(48.5^\circ\) without feedforward, highlighting the need for improvement.
To address these issues, we propose an improved feedforward control strategy that reshapes the output impedance of the on-grid inverter. The key modification involves adding a band-pass filter \(G_{bpf}(s)\) in the voltage feedforward path, defined as:
$$ G_{bpf}(s) = \frac{K \omega_i s}{s^2 + \omega_i s + \omega_L \omega_H} $$
where \(K\) is the filter gain, \(\omega_i\) is the resonance width coefficient, and \(\omega_L\) and \(\omega_H\) are the lower and upper cutoff frequencies, respectively. This filter attenuates the feedforward effect in the mid-to-high frequency range, thereby enhancing the phase margin. The modified output impedance becomes:
$$ Z_o'(s) = \frac{L_1 L_2 C s^3 + K_{PWM} G_d(s) L_2 C s^2 + (L_1 + L_2)s + G_c(s) G_d(s) K_{PWM}}{L_1 C s^2 + K_c K_{PWM} G_d(s) C s + 1 – K_{PWM} G_d(s) G_f(s)[1 – G_{bpf}(s)]} $$
By carefully selecting parameters such as \(K=150\), \(\omega_L = 314 \text{ rad/s}\), and \(\omega_H = 13700 \text{ rad/s}\), the output impedance phase is boosted, improving stability margins. Furthermore, a harmonic controller \(G_h(s)\) is串联 in the feedforward channel to suppress low-order grid voltage harmonics:
$$ G_h(s) = \sum_{h=1,3,5} \frac{\omega_b s}{s^2 + \omega_b s + (h\omega_0)^2} $$
where \(\omega_b\) is the bandwidth and \(h\) is the harmonic order. This combination ensures robust performance of the on-grid inverter under distorted grid conditions.
The stability analysis through Bode plots demonstrates the effectiveness of the improved strategy. For example, with \(L_g = 4 \text{ mH}\), the phase margin increases to \(35.7^\circ\) compared to the traditional method, and the crossover frequency shifts higher, broadening the adaptability range. The open-loop transfer function of the combined system is:
$$ G_l(s) = \frac{K_c K_{PWM} G_c(s) G_d(s)}{L_1 (L_2 + L_g) C s^3 + K_c K_{PWM} G_d(s)(L_2 C s^2 + (L_2 + L_g)s) – L_g K_{PWM} G_d(s) G_{bpf}(s) s} $$
This formulation highlights how the impedance reshaping mitigates the detrimental effects of feedforward loops. The following table summarizes the parameter values used in the analysis:
| Parameter | Symbol | Value |
|---|---|---|
| DC bus voltage | \(V_{dc}\) | 400 V |
| Grid voltage | \(U_g\) | 220 V |
| Inverter-side inductor | \(L_1\) | 0.6 mH |
| Grid-side inductor | \(L_2\) | 0.1 mH |
| Filter capacitor | \(C\) | 10 μF |
| Proportional gain | \(k_p\) | 2.8 |
| Resonant gain | \(k_r\) | 114 |
| Band-pass filter gain | \(K\) | 150 |
To validate the proposed strategy, simulations and experiments were conducted under varying grid conditions. The performance of the on-grid inverter was assessed in terms of stability and current quality. Simulation results for traditional proportional feedforward show severe current oscillations with \(L_g = 4 \text{ mH}\), whereas the improved strategy maintains smooth grid current waveforms. The total harmonic distortion (THD) of the grid current is a critical metric, and the improved method reduces THD to \(1.27\%\), well below the \(5\%\) limit, compared to \(33.84\%\) for the traditional approach. This underscores the efficacy of impedance reshaping in enhancing the on-grid inverter’s robustness.
Experimental verification using a hardware-in-the-loop platform confirms the simulation findings. The on-grid inverter with the improved feedforward control strategy stable under weak grid conditions, with minimal harmonic distortion. The following table compares key performance indicators:
| Control Strategy | Grid Impedance \(L_g\) | Phase Margin | Current THD | Stability |
|---|---|---|---|---|
| No Feedforward | 4 mH | 48.5° | Low | Stable |
| Traditional Proportional Feedforward | 4 mH | 4.2° | 33.84% | Unstable |
| Improved Feedforward | 4 mH | 35.7° | 1.27% | Stable |
The impedance reshaping achieved through the band-pass filter effectively decouples the feedforward path from grid impedance variations, making the on-grid inverter more resilient. Additionally, the harmonic controller suppresses background harmonics, ensuring compliance with grid codes. The Bode plots of the output impedance illustrate the phase boost in the mid-frequency range, which is crucial for maintaining stability in weak grids. For instance, the magnitude and phase curves of \(Z_o'(s)\) show improved characteristics, with the crossover frequency moved to a higher value, reducing the risk of instability.
Further analysis involves examining the sensitivity of the on-grid inverter to parameter variations. The improved strategy demonstrates reduced sensitivity to changes in grid impedance, thanks to the impedance reshaping. This is quantified by the stability margin over a range of \(L_g\) values, as shown in the following formula for phase margin calculation:
$$ PM = \min \left[ \angle Z_g(j\omega) – \angle Z_o(j\omega) \right] \text{ at } |Z_g(j\omega)| = |Z_o(j\omega)| $$
With the improved feedforward, the phase margin remains positive even for \(L_g\) up to 10 mH, indicating a wider stable operating region. This is essential for on-grid inverters deployed in areas with fluctuating grid strength.
The design of the band-pass filter involves trade-offs between harmonic suppression and phase enhancement. A higher gain \(K\) improves harmonic rejection but may reduce phase margin, while a lower gain does the opposite. Through iterative analysis, \(K=150\) was selected as optimal. The harmonic controller targets specific low-order harmonics (e.g., 3rd, 5th, 7th), which are common in grid voltages. The combined transfer function of the feedforward path becomes:
$$ G_{ff}(s) = G_{bpf}(s) \cdot G_h(s) $$
This ensures that the on-grid inverter not only stabilizes the system but also improves power quality. The effectiveness of this approach is evident in the spectral analysis of grid current, where harmonic peaks are significantly reduced.
In practical implementations, digital control delays pose additional challenges. The proposed strategy accounts for these delays by incorporating them into the impedance model. The modified output impedance formula includes \(G_d(s)\), ensuring accurate stability prediction. Simulation waveforms for grid voltage and current under both strong and weak grid conditions validate the theoretical models. The on-grid inverter with improved feedforward exhibits negligible oscillation even under severe grid distortions, highlighting its superior performance.
To further elaborate on the impedance reshaping concept, consider the equivalent circuit of the on-grid inverter. The output impedance can be viewed as a series combination of passive components and controlled sources. The band-pass filter introduces a frequency-dependent adjustment that counteracts the negative phase contribution of the feedforward loop. This is analogous to adding virtual impedance in the control loop, but without additional hardware. The mathematical derivation shows how the filter parameters influence the Nyquist plot, ensuring no encirclements of the critical point.
The robustness of the on-grid inverter is also tested against grid voltage sags and swells. The improved strategy maintains stable operation, with fast transient response due to the QPR controller. The current tracking error is minimized, as demonstrated by the error dynamics equation:
$$ e(s) = i_{ref}(s) – i_g(s) = \frac{1 – G_{ff}(s) G_{x1}(s)}{1 + G_c(s) G_{x1}(s) G_{x2}(s)} u_{pcc}(s) $$
where \(G_{x1}(s)\) and \(G_{x2}(s)\) are plant transfer functions. With the improved feedforward, the numerator term \(1 – G_{ff}(s) G_{x1}(s)\) is minimized at harmonic frequencies, reducing error.
Experimental results include waveforms of grid current and voltage under various conditions. The on-grid inverter with the proposed control shows clean sinusoidal current even with \(L_g = 4 \text{ mH}\), while the traditional method fails. The efficiency of the system was measured, reaching up to 96.8% at rated power, indicating that the improved strategy does not compromise efficiency. This is crucial for commercial on-grid inverter applications where energy conversion efficiency is paramount.
In conclusion, the improved feedforward control strategy based on impedance reshaping offers a viable solution for enhancing the stability and power quality of on-grid inverters in weak grids. By integrating a band-pass filter and harmonic controller, the output impedance is reshaped to boost phase margin and suppress harmonics. This method effectively addresses the limitations of traditional proportional feedforward, providing robust performance across varying grid conditions. Future work could explore adaptive tuning of filter parameters for further optimization. Overall, this strategy advances the reliability of on-grid inverter systems in renewable energy integration, contributing to grid stability and efficiency.
The theoretical insights presented here are supported by comprehensive simulations and experiments, confirming the practicality of the approach. For engineers designing on-grid inverter systems, this method provides a systematic way to mitigate weak grid challenges. The use of impedance reshaping as a design paradigm can be extended to other power converter topologies, broadening its impact. As grid codes evolve towards stricter harmonic standards, such advanced control strategies will become increasingly essential for on-grid inverter deployments worldwide.