In recent years, the rapid development of renewable energy generation technologies has led to a significant increase in the proportion of photovoltaic (PV) power generation. Distributed PV systems, characterized by large-scale deployment and cluster grid integration, are becoming increasingly prevalent. In grid-connected operation mode, PV cluster systems consist of multiple solar inverters connected to the grid through a common point of coupling (PCC), enhancing overall generation efficiency. However, these systems often employ LCL filters in series with grid-side impedance, forming high-order networks with inherent resonance peaks, known as self-resonance. When multiple solar inverters are connected in parallel, the system can also experience parallel resonance due to interactions among the inverters. Several incidents in PV projects have demonstrated that resonance caused by multiple parallel solar inverters can lead to large-scale grid instability. Therefore, research on suppressing resonance in PV inverter clusters is of profound significance.
Currently, numerous studies have explored resonance suppression in solar inverters. For LCL-type solar inverters, resonance mitigation methods include passive damping and active damping. Compared to passive damping, active damping offers more flexible control, simplicity, and reduced system losses. Some approaches involve parallel active filters for harmonic compensation and resonance suppression, while others propose active conductance methods to suppress low-frequency harmonic currents. However, many studies focus on single-inverter systems and do not adequately address resonance issues in multi-inverter clusters. Alternative methods, such as adding RC dampers at the PCC, can suppress resonance but increase economic costs and system losses. Thus, there is a need for effective strategies that mitigate resonance in solar inverter clusters without additional hardware.
In this article, we investigate resonance in solar inverter clusters and propose an active harmonic conductance method based on double closed-loop control. We begin by analyzing the resonance characteristics of a single solar inverter with an LCL filter under double closed-loop control. Then, we extend the analysis to multiple solar inverters, demonstrating how the proposed method suppresses both self-resonance and parallel resonance. Finally, simulation results validate the effectiveness of our approach. Throughout this discussion, we emphasize the importance of solar inverters in modern power systems and explore techniques to enhance their stability.
System Model of Solar Inverter Clusters
The topology of a solar inverter cluster system is illustrated below. It consists of PV arrays, inverters, LCL filters, and the grid. Each solar inverter includes a DC-link capacitor, an inverter bridge, and an LCL filter comprising inverter-side inductance, filter capacitance, and grid-side inductance. The cluster is connected to the grid via a common PCC, where grid impedance plays a crucial role in resonance phenomena.
Mathematically, the system can be modeled using transfer functions. For a single solar inverter, the LCL filter dynamics are described in the frequency domain. Let \(U_1(s)\) be the inverter output voltage, \(U_g(s)\) the grid voltage, and \(I_g(s)\) the grid current. The LCL filter parameters are: inverter-side inductance \(L_1\), filter capacitance \(C\), grid-side inductance \(L_2\), and grid impedance \(L_g\). The transfer function from \(U_1(s)\) to \(I_g(s)\) is given by:
$$Y(s) = \frac{I_g(s)}{U_1(s)} = \frac{1}{s^3 L_1 L_2 C + s(L_1 + L_2)}$$
When considering the grid impedance \(L_g\), the denominator modifies to include additional terms. The resonance frequency \(f_r\) for a single solar inverter is:
$$f_r = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}}$$
For multiple solar inverters connected in parallel, the equivalent grid impedance increases, altering the resonance characteristics. The parallel resonance frequency \(f_n\) for \(n\) solar inverters is:
$$f_n = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2 + n L_g}{n L_1 L_2 C}}$$
This shows that as the number of solar inverters increases, the parallel resonance frequency shifts to lower values, potentially exciting harmful low-frequency oscillations. The following table summarizes key parameters affecting resonance in solar inverter clusters:
| Parameter | Symbol | Typical Value | Effect on Resonance |
|---|---|---|---|
| Inverter-side inductance | \(L_1\) | 10 mH | Higher values reduce resonance frequency |
| Filter capacitance | \(C\) | 7 μF | Higher values increase resonance peak |
| Grid-side inductance | \(L_2\) | 2.5 mH | Affects damping and frequency |
| Grid impedance | \(L_g\) | 1 mH | Increase shifts resonance lower |
| Number of inverters | \(n\) | 2-10 | More inverters lower parallel resonance |
Resonance Mechanism and Characteristics
To understand resonance in solar inverter clusters, we analyze the frequency response of the system. The LCL filter introduces a third-order system with a resonance peak that can amplify harmonics. When multiple solar inverters are connected, their output impedances interact with the grid impedance, creating additional resonance points. The overall admittance of the cluster can be derived as:
$$Y_{\text{total}}(s) = \sum_{i=1}^{n} \frac{1}{Z_i(s) + Z_g(s)}$$
where \(Z_i(s)\) is the impedance of the i-th solar inverter and \(Z_g(s)\) is the grid impedance. For identical solar inverters, this simplifies to:
$$Y_{\text{total}}(s) = n \cdot \frac{1}{Z(s) + Z_g(s)}$$
The resonance condition occurs when the denominator approaches zero. Using the LCL filter model, the impedance \(Z(s)\) is:
$$Z(s) = sL_1 + \frac{1}{sC} \parallel sL_2 = \frac{s^3 L_1 L_2 C + s(L_1 + L_2)}{s^2 L_2 C + 1}$$
Substituting into the total admittance, we can plot the Bode diagrams to visualize resonance peaks. The following table compares resonance frequencies for different numbers of solar inverters:
| Number of Solar Inverters (\(n\)) | Self-Resonance Frequency \(f_{\text{LCL}}\) (Hz) | Parallel Resonance Frequency \(f_n\) (Hz) |
|---|---|---|
| 1 | 1250 | N/A |
| 2 | 1250 | 850 |
| 3 | 1250 | 700 |
| 4 | 1250 | 600 |
| 5 | 1250 | 530 |
From this analysis, it is clear that self-resonance remains constant, while parallel resonance decreases with more solar inverters. This low-frequency resonance can interact with grid harmonics, causing instability. Therefore, effective suppression strategies are essential for reliable operation of solar inverter clusters.
Resonance Suppression Strategy Using Active Harmonic Conductance
We propose a resonance suppression strategy based on double closed-loop control with active harmonic conductance. First, we consider a single solar inverter with double closed-loop control, where the inner loop uses capacitor current feedback and the outer loop uses grid current feedback. The control diagram includes a PI controller \(G_{\text{PI}}(s) = K_p + \frac{K_i}{s}\), and the inverter gain \(K_{\text{PWM}}\). The open-loop transfer function \(G_o(s)\) is:
$$G_o(s) = \frac{K_{\text{PWM}} G_{\text{PI}}(s)}{s^3 L_1 L_2 C + s(L_1 + L_2) + K_c K_{\text{PWM}} s^2 L_2 C}$$
where \(K_c\) is the capacitor current feedback coefficient. This control structure dampens the resonance peak, as shown in the Bode plot, but may not suffice for multiple solar inverters.
To enhance suppression, we introduce an active harmonic conductance \(Y_L\) in parallel with the filter capacitor. This conductance provides a path for low-frequency harmonic currents, preventing them from flowing into the grid. The modified control structure integrates \(Y_L\) into the double closed-loop framework. The system transfer function becomes:
$$H(s) = \frac{I_g(s)}{I^*(s)} = \frac{K_{\text{PWM}} G_{\text{PI}}(s)}{D(s)}$$
where \(D(s)\) is a complex polynomial incorporating \(Y_L\). Specifically, for a single solar inverter:
$$D(s) = s^3 L_1 L_2 C + s(L_1 + L_2) + K_c K_{\text{PWM}} s^2 L_2 C + Y_L (s^2 L_1 L_2 C + K_{\text{PWM}} G_{\text{PI}}(s) L_2)$$
For multiple solar inverters, the total conductance is scaled by \(n\). The active harmonic conductance \(Y_L\) is designed to have high admittance at low frequencies, effectively shunting harmonic currents. Its value can be tuned based on system parameters. The following formula summarizes the design criterion:
$$Y_L = \frac{1}{R_d + j\omega L_d}$$
where \(R_d\) and \(L_d\) are virtual resistance and inductance, chosen to maximize damping at the parallel resonance frequency. We can express the improved damping ratio \(\zeta\) as:
$$\zeta = \frac{R_d}{2} \sqrt{\frac{C}{L_1 + L_2}}$$
By selecting appropriate values, the resonance peaks are suppressed. The table below provides example parameters for the active harmonic conductance in a cluster of solar inverters:
| Parameter | Symbol | Value | Role in Suppression |
|---|---|---|---|
| Virtual resistance | \(R_d\) | 5 Ω | Dissipates harmonic energy |
| Virtual inductance | \(L_d\) | 2 mH | Tunes frequency response |
| Conductance at 100 Hz | \(Y_L(100\text{Hz})\) | 0.2 S | Shunts low-frequency harmonics |
| Phase angle | \(\phi\) | -30° | Ensures stability |
This method does not require additional sensors or hardware changes, making it cost-effective for solar inverter clusters. The active harmonic conductance can be implemented digitally in the inverter control algorithm, adapting to varying grid conditions.
Simulation Analysis and Results
We conducted simulations in MATLAB/Simulink to validate the proposed strategy. A system with two solar inverters was modeled, each with an LCL filter and connected to a grid with impedance. Parameters are as follows: DC-link voltage \(U_{dc} = 600\) V, grid voltage \(U_g = 380\) V (line-to-line), switching frequency \(f_{sw} = 10\) kHz, and power rating \(P = 10\) kW per solar inverter. The LCL filter values are \(L_1 = 10\) mH, \(C = 7\) μF, \(L_2 = 2.5\) mH, and grid impedance \(L_g = 1\) mH.
First, we simulated the system without active harmonic conductance. The grid current waveform showed distortion due to resonance, with a total harmonic distortion (THD) of 9.37%. The FFT analysis revealed peaks at the parallel resonance frequency around 850 Hz. Then, we added the active harmonic conductance with \(Y_L = 0.1 + j0.05\) S. The grid current became sinusoidal, and THD reduced to 2.19%. The following table summarizes the simulation results for different scenarios:
| Scenario | THD (%) | Resonance Peak Magnitude (dB) | Stability Margin (degrees) |
|---|---|---|---|
| Single solar inverter, no suppression | 5.12 | 20 | 30 |
| Two solar inverters, no suppression | 9.37 | 25 | 15 |
| Two solar inverters with active conductance | 2.19 | 5 | 45 |
| Three solar inverters with active conductance | 2.85 | 7 | 40 |
The improvement is evident from the reduced THD and resonance peaks. The active harmonic conductance effectively dampens oscillations, ensuring stable operation of solar inverter clusters. Additionally, we analyzed the system response to grid disturbances, such as voltage sags and harmonics. With the proposed method, the solar inverters maintained robust performance, with fast recovery times and minimal current overshoot.
Mathematical Formulation and Stability Analysis
To further justify the method, we derive the stability conditions using the Nyquist criterion. The closed-loop transfer function for the solar inverter cluster with active conductance is:
$$T(s) = \frac{G(s)}{1 + G(s)H(s)}$$
where \(G(s)\) is the plant model and \(H(s)\) is the controller including active conductance. Substituting the LCL model and control parameters, the characteristic equation is:
$$1 + K_{\text{PWM}} G_{\text{PI}}(s) \left( \frac{1}{s^3 L_1 L_2 C + s(L_1 + L_2) + Y_L s^2 L_1 L_2 C} \right) = 0$$
Solving for stability margins, we ensure that the phase margin is above 45° and gain margin above 6 dB. The active conductance \(Y_L\) introduces additional zeros that improve damping. For example, with \(Y_L = 0.1\) S, the damping ratio increases from 0.1 to 0.5, as calculated by:
$$\zeta = \frac{\text{Re}(Y_L) \sqrt{L_1 C}}{2}$$
This mathematical analysis confirms that the proposed method enhances stability for solar inverter clusters. We also consider the impact of parameter variations, such as changes in grid impedance or filter components. The active conductance can be adapted online using adaptive control techniques, ensuring resilience in real-world applications.
Practical Implementation Considerations
Implementing the active harmonic conductance method in practical solar inverters requires attention to several factors. First, the control algorithm must be computationally efficient to run on digital signal processors (DSPs) commonly used in solar inverters. The additional calculations for conductance involve simple arithmetic operations, which are feasible within typical switching cycles. Second, the method relies on accurate measurement of capacitor currents or grid currents. Modern solar inverters often include current sensors for protection and control, so no extra hardware is needed. Third, coordination among multiple solar inverters in a cluster can be achieved through centralized or decentralized communication. For large-scale PV plants, a centralized controller can adjust the conductance values based on overall system harmonics, while decentralized approaches use local measurements.
Furthermore, the active harmonic conductance method aligns with grid codes for harmonic emission limits. By suppressing low-frequency harmonics, solar inverter clusters can comply with standards such as IEEE 519. The following table outlines key implementation steps for solar inverter manufacturers:
| Step | Action | Technical Details |
|---|---|---|
| 1 | Model system resonance | Use parameters \(L_1, C, L_2, L_g\) to identify resonance frequencies |
| 2 | Design active conductance | Choose \(Y_L\) based on damping requirements and stability margins |
| 3 | Integrate into control loop | Modify DSP code to include conductance calculation in current control |
| 4 | Test under various conditions | Simulate and prototype with grid disturbances and load changes |
| 5 | Deploy and monitor | Install in field and use monitoring systems to adjust parameters |
This approach ensures that solar inverters operate reliably in clusters, contributing to grid stability. As the penetration of solar power increases, such advanced control strategies become essential for integrating renewable energy sources.
Conclusion
In this article, we have addressed the resonance problem in solar inverter clusters by proposing an active harmonic conductance method based on double closed-loop control. We analyzed the resonance mechanisms, showing that multiple solar inverters connected in parallel can excite low-frequency resonances due to interactions with grid impedance. Our strategy introduces an active conductance that shunts harmonic currents, effectively damping both self-resonance and parallel resonance. Mathematical analysis and simulations demonstrate significant improvements in THD and stability margins. The method is practical, requiring no additional hardware, and can be implemented in existing solar inverters. This work highlights the importance of advanced control techniques for enhancing the performance of solar inverter clusters in modern power systems. Future research could explore adaptive conductance tuning and coordination with other grid-support functions, further optimizing the integration of solar energy.