We propose a global resonance suppression strategy for photovoltaic (PV) multi-inverter parallel systems operating under weak grid conditions. With the rapid expansion of renewable energy generation, the solar inverter has become a critical interface between PV arrays and the utility grid. However, the increasing number of parallel-connected solar inverters, together with the inductive grid impedance in weak grids, often leads to severe harmonic resonance, degrading power quality and threatening system stability. In this work, we first derive a Norton equivalent model of the PV multi-inverter parallel system and investigate its resonance characteristics. Based on the analysis, we develop an optimized control scheme combining capacitor current feedback and grid voltage feed-forward, and further integrate a virtual admittance branch at the point of common coupling (PCC) to achieve global resonance suppression. Extensive simulations confirm that the proposed strategy reduces the total harmonic distortion (THD) of the grid current from 17.32% to 1.71% for a four-inverter system, and maintains superior robustness when the grid impedance varies. The findings demonstrate that the proposed approach can effectively stabilize multi-inverter systems and improve the output power quality of solar inverter networks.
A typical configuration of a solar inverter used in our grid-connected PV system is illustrated below. In this setup, multiple LCL-filtered inverters are connected in parallel at the PCC, and the grid is modeled as an ideal voltage source in series with an inductive impedance, representing the weak grid condition.
In recent years, the solar inverter has been extensively studied to improve the efficiency and reliability of PV power plants. However, the interaction among multiple inverters and the grid impedance introduces complex resonance mechanisms that are not fully addressed by conventional single-inverter design methods. Therefore, a systematic global resonance suppression strategy is essential for large-scale deployment of solar inverter systems.
1. System Model and Resonance Characteristics
We consider a photovoltaic multi-inverter parallel system where each solar inverter is equipped with an LCL filter. The structure includes the inverter-side inductor \(L_1\), the filter capacitor \(C\), the grid-side inductor \(L_2\), and the grid impedance \(L_g\). The inverter number is denoted as \(n\). The control of each inverter is based on the grid-side current feedback. Using Norton’s theorem, the entire parallel system can be represented by an equivalent model where each inverter is modeled as a controlled current source in parallel with its equivalent output admittance.
For a single solar inverter, the grid current \(i_g(s)\) can be expressed as
$$ i_g(s) = G_1(s) i_{ref}(s) – Y_1(s) u_{PCC}(s) \tag{1} $$
where \(i_{ref}(s)\) is the reference current, \(u_{PCC}(s)\) is the PCC voltage disturbance, and the transfer functions are given by
$$ Y_1(s) = \frac{L_1 C s^2 + 1}{L_1 L_2 C s^3 + (L_1 + L_2)s + K_{PWM} G_{QPR}} \tag{2a} $$
$$ G_1(s) = \frac{K_{PWM} G_{QPR}}{L_1 L_2 C s^3 + (L_1 + L_2)s + K_{PWM} G_{QPR}} \tag{2b} $$
Here, \(K_{PWM}\) is the inverter gain, \(G_{QPR}\) represents the quasi-proportional-resonant controller transfer function, and \(s\) is the Laplace variable. The control structure of a single solar inverter is shown in Fig. 2 of the original study, but we omit the figure for brevity. The key point is that the output admittance \(Y_1(s)\) depends on the LCL parameters and the control parameters.
For a system with \(n\) parallel-connected solar inverters, the Norton equivalent model yields the following expression for the output current of the \(i\)-th inverter:
$$ i_{gi} = A_i(s) i_{refi} + \sum_{j=1,\, j\neq i}^n B_{ij}(s) i_{refj} + D_{gi}(s) u_g \tag{3} $$
where
$$ A_i(s) = \frac{G_i \left( \sum_{j=1,\, j\neq i}^n Y_j + Y_g \right)}{\sum_{j=1}^n Y_j + Y_g} \tag{4a} $$
$$ B_{ij}(s) = \frac{- G_j Y_i}{\sum_{k=1}^n Y_k + Y_g} \tag{4b} $$
$$ D_{gi}(s) = \frac{- Y_i Y_g}{\sum_{k=1}^n Y_k + Y_g} \tag{4c} $$
In the above equations, \(Y_g = 1/(s L_g)\) is the grid admittance, and \(G_i\) corresponds to the closed-loop gain of the \(i\)-th solar inverter. It is clear that the output current of each inverter is influenced by the reference currents of all other inverters, which gives rise to coupling resonances. The system resonance characteristics are determined by the poles of the admittance functions, which involve the LCL filter parameters, the number of inverters, and the grid impedance.
By solving the denominator of the system transfer function, we obtain two resonance frequencies: the inherent LCL resonance frequency \(f_1\) and the system-level coupling resonance frequency \(f_2\). Their expressions are
$$ f_1 = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} \tag{5a} $$
$$ f_2 = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2 + n L_g}{L_1 C (L_2 + n L_g)}} \tag{5b} $$
From Eq. (5b), we observe that the coupling resonance frequency \(f_2\) is inversely proportional to the number of inverters \(n\). As \(n\) increases, \(f_2\) shifts toward lower frequencies, which intensifies the low-frequency harmonic resonance. This behavior is consistent with the resonance characteristic curves reported in the literature. In Table 1, we summarize the effect of the inverter number on the resonance frequencies for the parameters used in our study.
| Number of inverters \(n\) | \(f_1\) (Hz) | \(f_2\) (Hz) |
|---|---|---|
| 2 | 1710 | 1330 |
| 4 | 1710 | 920 |
| 8 | 1710 | 610 |
The above analysis reveals that the solar inverter parallel system becomes more vulnerable to resonance as the number of units increases. Therefore, a global resonance suppression strategy is necessary to stabilize the system under weak grid conditions.
2. Proposed Resonance Suppression Strategy
To suppress the resonance in the PV multi-inverter parallel system, we first adopt an optimized control strategy that combines an inner capacitor current feedback loop and an outer grid-side current loop, with a grid voltage feed-forward compensation. The control block diagram is shown in the original study (Fig. 5). The feed-forward path includes a proportional-differential term shaped by a low-pass filter and a phase-lead compensator.
The grid voltage feed-forward function is designed as
$$ G_f(s) = \frac{1}{\alpha s + 1} \cdot C K_C s \cdot G_r(s) \tag{6} $$
where \(\alpha = 50\), \(K_C\) is the capacitor current feedback coefficient, and the phase-lead compensator is given by
$$ G_r(s) = \frac{s + 0.1}{s + 500} $$
The use of the low-pass filter attenuates high-frequency noise, while the phase-lead compensator improves the system response speed.
With the capacitor current feedback and grid voltage feed-forward, the equivalent output impedance \(Z_0(s)\) of each solar inverter becomes
$$ Z_0(s) = \frac{L_1 L_2 C s^3 + L_2 C K_C K_{PWM} s^2 + (L_1 + L_2)s + G_{QPR} K_{PWM}}{L_1 C s^2 + C K_C K_{PWM} s + 1 – G_f K_{PWM}} \tag{7} $$
For a system with \(n\) parallel inverters, the total grid impedance seen by the inverters is \(n Z_g(s)\), where \(Z_g(s) = s L_g\). The impedance ratio is defined as
$$ T_m(s) = \frac{n Z_g(s)}{Z_0(s)} \tag{8} $$
Applying the Nyquist criterion to \(T_m(s)\), we analyze the stability of the system. The Nyquist curves for different numbers of inverters are shown in the original study (Fig. 6). When \(n=2\), the curve does not encircle the critical point \((-1, j0)\), indicating that the system is stable. However, when \(n=4\), the curve encircles the critical point, which implies that the system becomes unstable. This confirms that increasing the number of inverters leads to more severe coupling resonance.
To further suppress the system-level coupling resonance, we introduce a global resonance suppression strategy based on a virtual admittance branch connected in parallel at the PCC. The concept is to extract the high-frequency harmonic component of the PCC voltage through a high-pass filter, multiply it by a virtual admittance coefficient \(Y_f\), and feed the resulting current signal back into the reference current of each inverter. This is equivalent to placing a virtual admittance across the PCC.
The high-pass filter transfer function is
$$ G_{hpf}(s) = \frac{s}{s + 50} $$
After applying the virtual admittance feedback, the modified output impedance \(Z_0^*(s)\) of the system becomes
$$ Z_0^*(s) = \frac{L_1 L_2 C s^3 + L_2 C K_C K_{PWM} s^2 + (L_1 + L_2)s + G_{QPR} K_{PWM}}{2(L_1 C s^2 + C K_C K_{PWM} + 1) + G_{QPR} K_{PWM} Y_a – G_f K_{PWM}} \tag{9} $$
with \(Y_a = G_{hpf}(s) Y_f\). The parameter \(Y_f\) is selected as 0.8 to ensure effective damping of the resonance peaks without affecting the fundamental component.
Substituting Eq. (9) into Eq. (8), we obtain the Nyquist curves for the compensated system. The results, presented in the original study (Fig. 8), show that for both \(n=2\) and \(n=4\), the impedance ratio curves do not encircle the critical point, indicating that the system remains stable even with an increased number of inverters. Thus, the proposed global resonance suppression strategy effectively mitigates both self-resonance and coupling resonance in the solar inverter parallel system.
3. Simulation Verification
To validate the theoretical analysis and the effectiveness of the proposed strategy, we built a detailed simulation model of the PV multi-inverter parallel system in Matlab/Simulink. The system parameters are summarized in Table 2.
| Parameter | Symbol | Value |
|---|---|---|
| DC bus voltage | \(u_{dc}\) | 700 V |
| Grid voltage (RMS) | \(u_g\) | 220 V |
| Switching frequency | \(f_{sw}\) | 20 kHz |
| Inverter gain | \(K_{PWM}\) | 25 dB |
| PCC voltage distortion | \(u_{PCC}\) | 3% |
| Inverter-side inductor | \(L_1\) | 3 mH |
| Filter capacitor | \(C\) | 5 μF |
| Grid-side inductor | \(L_2\) | 1.5 mH |
| Capacitor current feedback coefficient | \(K_C\) | 0.4 |
| Virtual admittance coefficient | \(Y_f\) | 0.8 |
| Grid admittance | \(Y_g\) | 0.32 |
First, we simulate the system with only the capacitor current feedback and grid voltage feed-forward optimization (without the PCC virtual admittance). The grid impedance is set to \(L_g = 1\) mH. Figure 9 of the original study shows the grid current waveforms for \(n=2\) and \(n=4\). When \(n=2\), the output current is nearly sinusoidal and meets the grid standards. When \(n=4\), the current waveform becomes severely distorted, and the total harmonic distortion (THD) reaches 17.32%. This confirms our theoretical prediction that the system becomes unstable as the number of inverters increases.
We then apply the proposed global resonance suppression strategy by adding the virtual admittance at the PCC. For the same operating condition (\(L_g = 1\) mH, \(n=4\)), the grid current waveform becomes much cleaner, and the THD is reduced to only 1.71%. The fast Fourier transform (FFT) analysis of the current demonstrates a significant attenuation of the resonance peaks across the frequency spectrum. These results are summarized in Table 3.
| Case condition | Grid impedance \(L_g\) | Number \(n\) | THD (%) |
|---|---|---|---|
| Without virtual admittance | 1 mH | 2 | 4.2% |
| Without virtual admittance | 1 mH | 4 | 17.32% |
| With virtual admittance | 1 mH | 4 | 1.71% |
| With virtual admittance | 3 mH | 4 | 2.86% |
To evaluate the robustness of the proposed strategy against grid impedance variations, we increase \(L_g\) to 3 mH while keeping \(n=4\). We compare the performance of the proposed global suppression strategy with two existing approaches reported in the literature: one based on active harmonic conductance (Ref. [11]) and another based on passive RC branch at PCC (Ref. [12]). The FFT analyses of the grid current under these three strategies are depicted in the original study (Fig. 12). Under the proposed strategy, the THD is maintained at 2.86%, which is well below the 5% limit of the IEEE 519 standard. In contrast, both reference methods yield THD values above 6%, indicating a higher level of harmonic distortion and less effective resonance suppression.
| Control strategy | THD (%) |
|---|---|
| Proposed global suppression strategy | 2.86 |
| Active harmonic conductance (Ref. [11]) | 6.8 |
| Passive RC branch at PCC (Ref. [12]) | 7.5 |
The simulation results clearly demonstrate the superiority of the proposed strategy in suppressing global resonance across multiple solar inverter parallel systems. The virtual admittance branch at the PCC effectively absorbs high-frequency harmonic currents, while the capacitor current feedback and grid voltage feed-forward ensure satisfactory dynamic response and low sensitivity to grid impedance changes.
4. Conclusion
In this work, we have addressed the global resonance problem in PV multi-inverter parallel systems under weak grid conditions. We constructed a Norton equivalent model that accurately captures the interactions between multiple solar inverters and the grid impedance. Through theoretical analysis, we revealed that the coupling resonance frequency decreases as the number of inverters increases, making the system more prone to instability. To overcome this issue, we proposed a combined strategy: an optimized control using capacitor current feedback and grid voltage feed-forward, plus a virtual admittance branch connected in parallel at the PCC. The Nyquist stability analysis verified that the proposed method preserves system stability even when the number of inverters is increased to four or more. Simulation results show a dramatic reduction in THD from 17.32% to 1.71% for a four-inverter system. Furthermore, the strategy exhibits excellent robustness under grid impedance variation, outperforming existing methods. The proposed global resonance suppression strategy provides a practical and effective solution for improving the reliability and power quality of large-scale solar inverter systems operating in weak grids.