With the rapid advancement of renewable energy generation, photovoltaic (PV) power systems have gained widespread adoption due to their safety, reliability, flexibility, and environmental benefits. In large-scale PV power plants, multiple solar inverters are often connected in parallel to the grid to enhance power generation efficiency. However, as the scale of grid-connected systems expands, the inductive impedance inherent in weak grids poses significant challenges to system stability and reliability. These challenges manifest as inadequate global resonance suppression and potential instability risks. This paper addresses these issues by developing a comprehensive strategy to mitigate resonance in parallel solar inverter systems, ensuring robust operation under weak grid conditions.
The integration of multiple solar inverters in parallel configurations introduces complex interactions between the inverters and the grid impedance. Solar inverters, which serve as the core interface between PV arrays and the grid, typically employ LCL filters to attenuate switching harmonics and improve current quality. However, the high-order dynamics of LCL filters, combined with grid impedance, can lead to resonant peaks that compromise system stability. In multi-inverter systems, these resonances can be categorized into inherent resonances, dependent solely on LCL parameters, and coupling resonances, influenced by the number of solar inverters and grid impedance. This paper proposes a novel global resonance suppression strategy that combines optimized control techniques with virtual admittance at the point of common coupling (PCC) to effectively dampen these resonances.
System Modeling of Parallel Solar Inverters
To analyze the resonance characteristics, a Norton equivalent model of the parallel solar inverter system is established. Consider a system with n solar inverters connected in parallel at the PCC, each utilizing an LCL filter. The LCL filter parameters include the inverter-side inductance \( L_{1n} \), grid-side inductance \( L_{2n} \), and filter capacitance \( C_n \). The grid impedance is represented by \( L_g \), and the grid voltage is denoted as \( u_g \). The control structure for each solar inverter employs grid-side current feedback, as illustrated in the following block diagram.
The output current \( i_g(s) \) of a single solar inverter can be expressed as:
$$ i_g(s) = G_1(s) i_{\text{ref}}(s) – Y_1(s) u_{\text{PCC}}(s) $$
where \( G_1(s) \) and \( Y_1(s) \) are derived from the LCL filter dynamics and control parameters. For multiple solar inverters, the Norton equivalent model simplifies the analysis by representing each inverter as a controlled current source with output admittance. The equivalent circuit accounts for the parallel connection of n solar inverters and the grid impedance \( L_g \). The output current of the i-th solar inverter is given by:
$$ i_{gi} = A_i(s) i_{\text{ref}i} + \sum_{j=1, j \neq i}^{n} B_{ij}(s) i_{\text{ref}j} + D_{gi}(s) u_g $$
where \( A_i(s) \), \( B_{ij}(s) \), and \( D_{gi}(s) \) are transfer functions that depend on the output admittances of the solar inverters and the grid admittance \( Y_g \). These coefficients capture the interactions between solar inverters and the grid, highlighting the potential for resonant behavior.
The parameters of the solar inverters and system components are summarized in Table 1.
| Parameter | Value |
|---|---|
| DC-link voltage \( u_{dc} \) | 700 V |
| Grid voltage \( u_g \) | 220 V |
| Switching frequency \( f_{sw} \) | 20 kHz |
| Inverter gain \( K_{\text{PWM}} \) | 25 dB |
| PCC voltage disturbance \( u_{\text{PCC}} \) | 3% |
| Inverter-side inductance \( L_1 \) | 3 mH |
| Grid-side inductance \( L_2 \) | 1.5 mH |
| Filter capacitance \( C \) | 5 μF |
| Capacitor current feedback coefficient \( K_C \) | 0.4 |
| Number of solar inverters \( n \) | 2 to 4 |
Resonance Characteristics Analysis
The resonance behavior in parallel solar inverter systems is characterized by two distinct frequencies: the inherent resonance frequency \( f_1 \) and the coupling resonance frequency \( f_2 \). These frequencies are derived from the system’s impedance model and are critical for stability analysis. The inherent resonance frequency depends solely on the LCL filter parameters:
$$ f_1 = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} $$
In contrast, the coupling resonance frequency is influenced by the number of solar inverters \( n \) and the grid impedance \( L_g \):
$$ f_2 = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2 + n L_g}{L_1 C (L_2 + n L_g)}} $$
As the number of solar inverters increases, \( f_2 \) shifts to lower frequencies, exacerbating resonance issues. This relationship is illustrated in the resonance characteristics curve, which shows that for \( n = 2 \), the system remains stable, but for \( n = 4 \), significant resonance peaks appear, leading to potential instability. The impedance ratio \( T_m(s) \) between the grid impedance and the output impedance of the solar inverters is used to assess stability via the Nyquist criterion:
$$ T_m(s) = \frac{n Z_g(s)}{Z_0(s)} $$
where \( Z_0(s) \) is the output impedance of a single solar inverter. The Nyquist plots for different values of \( n \) demonstrate that as \( n \) increases, the system becomes more prone to encirclement of the critical point (-1, j0), indicating instability.
Proposed Global Resonance Suppression Strategy
To address the resonance issues in parallel solar inverter systems, a global resonance suppression strategy is proposed. This strategy combines capacitor current feedback and grid voltage feed-forward optimization control with virtual admittance at the PCC. The capacitor current feedback introduces active damping to suppress the inherent resonance peaks, while the grid voltage feed-forward compensates for grid voltage distortions. The control law for the solar inverter is enhanced as follows:
The output impedance with capacitor current feedback is modified to:
$$ Z_0(s) = \frac{L_1 L_2 C s^3 + L_2 C K_C K_{\text{PWM}} s^2 + (L_1 + L_2) s + G_{\text{QPR}} K_{\text{PWM}}}{L_1 C s^2 + C K_C K_{\text{PWM}} s + 1 – G_f K_{\text{PWM}}} $$
where \( G_f(s) \) is the grid voltage feed-forward function, designed as:
$$ G_f(s) = \frac{1}{\alpha s + 1} C K_C s G_r(s) $$
Here, \( G_r(s) = \frac{s + 0.1}{s + 500} \) is a phase lead compensator, and \( \alpha = 50 \) is the filter coefficient. This feed-forward strategy mitigates low-frequency harmonics from the grid voltage.
Furthermore, to suppress global coupling resonance, a virtual admittance \( Y_f \) is connected in parallel at the PCC. This virtual admittance absorbs high-frequency harmonic currents and is implemented using a high-pass filter \( G_{\text{hpf}}(s) = \frac{s}{s + 50} \). The compensated output impedance becomes:
$$ Z_0^*(s) = \frac{L_1 L_2 C s^3 + L_2 C K_C K_{\text{PWM}} s^2 + (L_1 + L_2) s + G_{\text{QPR}} K_{\text{PWM}}}{2(L_1 C s^2 + C K_C K_{\text{PWM}} s + 1) + G_{\text{QPR}} K_{\text{PWM}} Y_a – G_f K_{\text{PWM}}} $$
where \( Y_a = G_{\text{hpf}}(s) Y_f \). The virtual admittance effectively dampens the resonance without affecting the fundamental frequency component, ensuring improved stability.
Simulation Results and Discussion
The proposed strategy is validated through simulations in MATLAB/Simulink. The system parameters are as listed in Table 1, and the grid impedance is set to \( L_g = 1 \) mH. The performance is evaluated for different numbers of solar inverters, and the total harmonic distortion (THD) of the output current is used as a key metric.
First, the system with only capacitor current feedback and grid voltage feed-forward is tested. For \( n = 2 \) solar inverters, the output current THD is within acceptable limits. However, for \( n = 4 \) solar inverters, the THD increases to 17.32%, indicating severe resonance. The FFT analysis reveals prominent harmonic peaks around the resonance frequencies.
After applying the virtual admittance at the PCC, the resonance is significantly suppressed. For \( n = 4 \) solar inverters, the THD is reduced to 1.71%, demonstrating the effectiveness of the global suppression strategy. The current waveforms become sinusoidal, and the harmonic content is minimized.
To assess robustness, the grid impedance is increased to \( L_g = 3 \) mH, and the proposed strategy is compared with existing methods. The results show that the proposed strategy maintains a low THD of 2.86%, while other methods exhibit THD values above 6%. This confirms the superior performance and robustness of the proposed approach under varying grid conditions.
| Strategy | THD for \( n = 4 \), \( L_g = 3 \) mH |
|---|---|
| Proposed Global Suppression | 2.86% |
| Method from Reference [11] | 6.45% |
| Method from Reference [12] | 6.87% |
The Nyquist plots further validate the stability improvements. With the virtual admittance, the impedance ratio curves no longer encircle the critical point (-1, j0) for \( n = 4 \), ensuring system stability. The optimized control parameters for the solar inverters are summarized in Table 3.
| Parameter | Value |
|---|---|
| Capacitor current feedback coefficient \( K_C \) | 0.4 |
| Grid voltage feed-forward coefficient \( G_f(s) \) | As defined |
| Virtual admittance \( Y_f \) | 0.8 |
| High-pass filter \( G_{\text{hpf}}(s) \) | \( \frac{s}{s + 50} \) |
| Phase lead compensator \( G_r(s) \) | \( \frac{s + 0.1}{s + 500} \) |
Conclusion
This paper has presented a comprehensive global resonance suppression strategy for parallel solar inverter systems in photovoltaic applications. By developing a Norton equivalent model, the resonance characteristics were analyzed, revealing the negative correlation between coupling resonance frequency and the number of solar inverters. The proposed strategy, which combines capacitor current feedback, grid voltage feed-forward optimization, and virtual admittance at the PCC, effectively suppresses both inherent and coupling resonances. Simulation results demonstrate significant improvements in system stability and output current quality, with THD values reduced to below 3% even under weak grid conditions. The strategy exhibits strong robustness against grid impedance variations, making it a reliable solution for large-scale PV systems with multiple solar inverters. Future work will focus on experimental validation and adaptation to more complex grid scenarios.