In recent years, the integration of large-scale photovoltaic (PV) systems into power grids has become a critical strategy to address energy crises and reduce environmental pollution. As a key component, the solar inverter plays a vital role in converting DC power from PV panels into AC power suitable for grid connection. However, when these solar inverters are connected to weak grids—characterized by high grid impedance—stability issues such as power oscillations and system failures often arise. This paper focuses on analyzing the stability of three-phase LCL-type solar inverters in weak grid conditions using an output impedance-based approach. I will derive the dynamic output impedance model of the inverter, incorporate the effects of phase-locked loops (PLL) and control loops, and apply impedance stability criteria to evaluate system stability. Through detailed mathematical modeling, simulation examples, and experimental validation, I aim to provide insights into the interaction between solar inverter output impedance and grid impedance, offering guidance for designing stable grid-connected PV systems.
The proliferation of solar energy has led to a significant increase in grid-connected PV systems worldwide. By the end of 2015, China’s cumulative grid-connected PV capacity reached 41.58 GW, accounting for approximately one-fifth of the global total. According to the “13th Five-Year Plan,” China aims to achieve 150 GW of installed PV capacity by 2020, with annual additions of around 20 GW. Large-scale PV plants are often connected to the ends of power grids, where the grid structure is relatively weak. In such weak grids, the point of common coupling (PCC) voltage fluctuates due to variations in PV power output, background harmonics, and three-phase voltage imbalances. These fluctuations can degrade the performance of solar inverters and even threaten system security. The weak grid effect becomes more pronounced as PV penetration increases, making stability analysis a pressing concern. The solar inverter, as the interface between PV arrays and the grid, must maintain stable operation under varying grid conditions. Traditional stability analysis methods may not fully capture the dynamics introduced by weak grids, necessitating a more comprehensive approach based on impedance modeling.
Impedance stability criteria have emerged as a powerful tool for analyzing grid-connected inverters. By modeling the solar inverter as a Norton equivalent circuit with a current source and output impedance, and the grid as a Thevenin equivalent circuit with a voltage source and grid impedance, the interaction between the inverter and grid can be studied through their impedance ratios. This method is particularly effective for systems where multiple inverters are connected in parallel or when the grid impedance is significant. In this paper, I adopt this impedance-based approach to investigate the stability of three-phase LCL-type solar inverters. The LCL filter is commonly used in solar inverters to attenuate switching harmonics and improve power quality. However, in weak grids, the LCL filter’s performance can be affected by grid impedance, potentially leading to resonance and instability. Therefore, understanding the output impedance characteristics of the solar inverter is crucial for ensuring reliable operation.
The core of this analysis lies in deriving the output impedance model of the solar inverter in the dq synchronous reference frame. Unlike single-phase systems, three-phase systems require transformation to the dq frame to obtain a linearized model around an equilibrium point. The output impedance is represented as a 2×2 matrix, including self-impedance and mutual impedance terms, which account for couplings between the d and q axes. I will consider the influences of the main circuit, PLL, and control loops on the output impedance. The PLL is essential for synchronizing the solar inverter with the grid, but in weak grids, PCC voltage disturbances can cause phase errors that propagate through the control system, affecting stability. Similarly, control loops, such as current and voltage regulators, interact with the PLL and grid impedance, creating complex dynamics. By integrating these factors, I develop a comprehensive output impedance model that accurately reflects the solar inverter’s behavior in weak grids.
To organize the paper, I first introduce the impedance stability criterion for three-phase systems. Then, I derive the output impedance model step by step, starting from the main circuit and adding PLL and control loop effects. Next, I analyze stability using the impedance ratio matrix and examine the impact of grid impedance through Bode and Nyquist plots. Simulation and experimental results are presented to validate the theoretical analysis. Finally, I conclude with key findings and implications for solar inverter design. Throughout the paper, I will use tables to summarize parameters and formulas to express mathematical relationships, ensuring a clear and detailed exposition. The goal is to provide a thorough understanding of solar inverter stability in weak grids, which can aid in the development of more robust PV integration strategies.
Impedance Stability Criterion for Grid-Connected Solar Inverters
The impedance stability criterion is based on modeling the grid-connected solar inverter as a current source with output impedance and the grid as a voltage source with grid impedance. For small-signal analysis, the system can be represented by the following equivalent circuit in the frequency domain:
The solar inverter is modeled as a Norton equivalent: an ideal current source \(I_{\text{inv}}(s)\) in parallel with the inverter’s dynamic output impedance \(Z_{\text{inv}}(s)\). The grid is modeled as a Thevenin equivalent: an ideal voltage source \(U_s(s)\) in series with the grid impedance \(Z_g(s)\). The PCC voltage \(U_g(s)\) and grid current \(I_g(s)\) are then derived from this circuit. The stability of the interconnected system can be assessed by examining the impedance ratio \(Z_g(s)/Z_{\text{inv}}(s)\). According to the criterion, if the impedance ratio satisfies the Nyquist stability criterion, the system remains stable. This approach assumes that the grid is stable when the solar inverter is disconnected and that the inverter is stable when connected to an ideal grid (zero grid impedance).
For three-phase systems, the analysis is conducted in the dq reference frame. The output impedance of the solar inverter becomes a matrix:
$$ Z_{\text{inv}}(s) = \begin{bmatrix} Z_{dd}(s) & Z_{dq}(s) \\ Z_{qd}(s) & Z_{qq}(s) \end{bmatrix} $$
where \(Z_{dd}(s)\) and \(Z_{qq}(s)\) are the self-impedances in the d and q axes, respectively, and \(Z_{dq}(s)\) and \(Z_{qd}(s)\) are the mutual impedances representing cross-coupling between axes. Similarly, the grid impedance matrix is:
$$ Z_g(s) = \begin{bmatrix} Z_{g,dd}(s) & Z_{g,dq}(s) \\ Z_{g,qd}(s) & Z_{g,qq}(s) \end{bmatrix} $$
In many cases, the mutual impedances are small compared to self-impedances, especially when decoupling control strategies are employed. Thus, the impedance ratio matrix can be approximated as diagonal:
$$ Z_g(s) Z_{\text{inv}}^{-1}(s) \approx \begin{bmatrix} \frac{Z_{g,dd}(s)}{Z_{dd}(s)} & 0 \\ 0 & \frac{Z_{g,qq}(s)}{Z_{qq}(s)} \end{bmatrix} $$
Stability is then determined by applying the Nyquist criterion to each diagonal element. This simplification allows for easier analysis while capturing the essential dynamics of the solar inverter in weak grids.
Output Impedance Model of Three-Phase LCL-Type Solar Inverter
The output impedance model of the solar inverter is derived by considering the main circuit, PLL, and control loops. I start with the main circuit of a three-phase LCL-type solar inverter connected to a weak grid, as shown in the system diagram. The LCL filter consists of inverter-side inductance \(L_1\), grid-side inductance \(L_2\), capacitance \(C_f\), and damping resistance \(R_d\). The grid impedance includes resistance \(R_g\) and inductance \(L_g\). The solar inverter operates with a DC-link voltage \(u_{dc}\) and generates PWM signals to control the switches.
In the dq frame, the small-signal equations of the main circuit are derived by linearizing around a steady-state operating point. The resulting equations describe the relationships between voltage and current perturbations. From these, the output impedance of the main circuit \(Z_{\text{inv,main}}\) can be expressed as a function of circuit parameters and frequency. The derivation involves solving the network equations to find the transfer functions from grid current perturbations to PCC voltage perturbations.
The key steps are as follows. First, write the voltage-current equations for the LCL filter and grid impedance in the dq frame. Then, apply Laplace transforms to obtain algebraic equations. By manipulating these equations, I derive the output impedance matrix for the main circuit. The elements of this matrix are given by:
$$ Z_{dd,\text{main}} = Z_{qq,\text{main}} = \frac{s^2 L_1 L_2 G + s R_d G (L_1 + L_2) – s L_1 H – \omega^2 L_1 L_2 (G – 1)}{G R_d} $$
$$ Z_{dq,\text{main}} = -Z_{qd,\text{main}} = -\frac{s \omega L_1 L_2 (2G – 1) + \omega R_d G (L_1 + L_2) – \omega L_1 H}{G R_d} $$
where \(G = (1 + s R_d C_f)^2 + (\omega R_d C_f)^2\), \(H = s L_2 (1 + s R_d C_f) + \omega^2 R_d C_f L_2\), and \(\omega\) is the grid angular frequency. The parameters are defined in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | \(P_n\) | 500 kW |
| Switching Frequency | \(f_s\) | 3.3 kHz |
| DC-Link Capacitance | \(C\) | 4700 µF |
| Inverter-Side Inductance | \(L_1\) | 0.108 mH |
| Grid-Side Inductance | \(L_2\) | 0.117 mH |
| Filter Capacitance | \(C_f\) | 218 µF |
| Damping Resistance | \(R_d\) | 0.17 Ω |
| DC-Link Voltage | \(U_{dc}\) | 600 V |
| Grid Voltage (Line-to-Line) | \(U_s\) | 270 V |
| Grid Frequency | \(f\) | 50 Hz |
Next, I incorporate the effects of the PLL. The PLL tracks the PCC voltage phase to synchronize the solar inverter with the grid. In weak grids, PCC voltage disturbances cause phase errors \(\Delta \theta\) that affect the control signals. The PLL transfer function \(G_{\text{PLL}}(s)\) relates the q-axis voltage perturbation to the phase error. This error propagates through the control system, modifying the output impedance. The influence of the PLL on the output impedance is modeled by adding terms that account for the coupling between voltage perturbations and control actions.
The PLL-adjusted output impedance \(Z_{\text{inv,PLL}}\) is derived by integrating the PLL transfer functions into the main circuit model. The resulting matrix elements become more complex, involving products of \(G_{\text{PLL}}(s)\) with steady-state operating points. For example, the d-axis self-impedance with PLL effects includes additional terms proportional to \(U_{gq} G_{\text{PLL}}(s)\), where \(U_{gq}\) is the steady-state q-axis PCC voltage.
Finally, I include the control loops. The solar inverter typically uses dual-loop control with inner current control and outer voltage or power control. The current controller employs PI regulators with decoupling terms to independently control d and q axes currents. The control loops introduce dynamics that further modify the output impedance. By combining the main circuit, PLL, and control loops, the complete output impedance model \(Z_{\text{inv,PLL,Ci}}\) is obtained.
The derivation yields the following expression for the complete output impedance matrix:
$$ Z_{\text{inv,PLL,Ci}} = -\left\{ -Z_{\text{inv}}^{-1} + G_{\text{im}} G_{\text{del}} \left[ G_{\text{PLL}}^m + U_{dc}^{-1} G_{\text{PLL}}^u + U_{dc}^{-1} (G_{\text{dec}}^i – G_{\text{Cpi}}^i) G_{\text{PLL}}^i \right] F \right\}^{-1} \left[ G_{\text{im}} G_{\text{del}} U_{dc}^{-1} (G_{\text{Cpi}}^i – G_{\text{dec}}^i) F + E \right] $$
where \(G_{\text{im}}\), \(G_{\text{del}}\), \(F\) are transfer functions for current measurement, PWM delay, and filtering, respectively; \(G_{\text{PLL}}^m\), \(G_{\text{PLL}}^u\), \(G_{\text{PLL}}^i\) are PLL-related transfer functions for modulation, voltage, and current; \(G_{\text{dec}}^i\) and \(G_{\text{Cpi}}^i\) are decoupling and PI control matrices; and \(E\) is the identity matrix. This comprehensive model captures the intricate dynamics of the solar inverter in weak grids.
To illustrate the parameters involved, Table 2 summarizes the control and PLL parameters used in the analysis.
| Parameter | Symbol | Value |
|---|---|---|
| Current Loop Proportional Gain | \(k_{pi}\) | 0.65 |
| Current Loop Integral Gain | \(k_{ii}\) | 6.5 s⁻¹ |
| Voltage Loop Proportional Gain | \(k_{pv}\) | 0.14 |
| Voltage Loop Integral Gain | \(k_{iv}\) | 61.67 s⁻¹ |
| PLL Proportional Gain | \(k_{p,\text{PLL}}\) | 10.07 |
| PLL Integral Gain | \(k_{i,\text{PLL}}\) | 7987 s⁻¹ |
| PWM Delay Time | \(T_{\text{del}}\) | 1.5/\(f_s\) s |
Stability Analysis Based on Impedance Ratios
With the output impedance model of the solar inverter established, I now analyze stability using the impedance ratio matrix. The focus is on how grid impedance affects the stability of the solar inverter system. I consider varying levels of grid impedance, expressed in per unit (pu) values relative to the base impedance of the system. The base impedance is calculated from the rated power and voltage.
The impedance ratio matrix is evaluated for different grid impedances. For simplicity, I assume the mutual impedances are negligible due to decoupling control, so the matrix is diagonal. Thus, stability depends on the Nyquist plots of \(Z_{g,dd}/Z_{dd}\) and \(Z_{g,qq}/Z_{qq}\). I compute these ratios using the derived output impedance model and typical grid impedance values.
For a grid impedance of \(Z_g^* = 0.10\) pu (where * denotes per unit), the Nyquist plots of both diagonal elements do not encircle the point (-1, j0), indicating system stability. However, for \(Z_g^* = 0.20\) pu, the Nyquist plot for the q-axis ratio \(Z_{g,qq}/Z_{qq}\) encircles (-1, j0) twice, indicating instability. This shows that the q-axis self-impedance interaction is critical for solar inverter stability in weak grids.
To further investigate, I plot Bode diagrams of the output impedance and grid impedance for different grid impedances. The Bode plots reveal the frequency-dependent behavior and phase margins at crossover frequencies. For the d-axis self-impedance, the phase margin remains positive even at high grid impedances, suggesting robustness. In contrast, for the q-axis self-impedance, the phase margin becomes negative when grid impedance exceeds 0.20 pu, leading to instability. This underscores the importance of the q-axis dynamics in solar inverter control.
The analysis can be summarized with the following key points:
- The solar inverter’s output impedance is highly influenced by PLL and control loops, especially in the q-axis.
- Grid impedance exacerbates interactions, particularly when it exceeds 0.20 pu.
- The q-axis self-impedance is the dominant factor in stability degradation.
To quantify the stability margins, I calculate the phase margins for different grid impedances. Table 3 shows the phase margins at the crossover frequencies for the d and q axes.
| Grid Impedance (pu) | d-Axis Phase Margin | q-Axis Phase Margin | Stability Conclusion |
|---|---|---|---|
| 0.01 | 117° | 101° | Stable |
| 0.10 | 79° | 75° | Stable |
| 0.20 | 63° | -55° | Unstable |
The negative phase margin for the q-axis at \(Z_g^* = 0.20\) pu confirms instability. This occurs because the grid impedance introduces additional phase lag that, when combined with the solar inverter’s dynamics, leads to a positive feedback loop. The PLL amplifies this effect by introducing phase errors that distort current control, further reducing stability margins.
Simulation and Experimental Validation
To validate the theoretical analysis, I conduct simulations and experiments on a three-phase LCL-type solar inverter system. The simulation model is built in PSCAD/EMTDC, replicating the parameters from Table 1. The solar inverter is connected to a weak grid with adjustable grid impedance. I test two scenarios: \(Z_g^* = 0.10\) pu and \(Z_g^* = 0.20\) pu, under varying solar irradiance conditions.
In the first scenario (\(Z_g^* = 0.10\) pu), the solar inverter operates stably as irradiance increases from 400 W/m² to 1000 W/m². The output power ramps up smoothly and settles at the rated value. The system frequency remains within ±0.5 Hz of the nominal 50 Hz, complying with grid standards. This aligns with the impedance analysis, which predicted stability for this grid impedance.
In the second scenario (\(Z_g^* = 0.20\) pu), when irradiance reaches 1000 W/m², the output power exhibits sustained oscillations with a period of approximately 0.3 seconds. The frequency also fluctuates beyond acceptable limits. This instability matches the prediction from the impedance ratio analysis, where the q-axis Nyquist plot encircled (-1, j0). The oscillations are caused by the interaction between the solar inverter’s output impedance and grid impedance, particularly in the q-axis.
For experimental validation, I set up a 5 kW T-type three-level solar inverter testbed. The parameters are listed in Table 4. The inverter is connected to a weak grid emulated with series inductors to represent grid impedance. I test the system under a step change in PV power from 3 kW to 4 kW with a grid impedance of \(Z_g^* = 0.23\) pu (corresponding to 4.7 mH inductance).
| Parameter | Value |
|---|---|
| Rated Power | 5 kW |
| Switching Frequency | 20 kHz |
| DC-Link Capacitance | 300 µF |
| Inverter-Side Inductance | 2 mH |
| Grid-Side Inductance | 3 mH |
| Filter Capacitance | 3.3 µF |
| Current Loop Gains | \(k_{pi} = 3.9\), \(k_{ii} = 0.37\) s⁻¹ |
| Grid Voltage | 180 V (line-to-line) |
| Grid Frequency | 50 Hz |
The experimental results show that after the power step, the grid current and PCC voltage begin to oscillate, and the DC-link capacitor voltages become unbalanced. Eventually, the inverter triggers protection and disconnects from the grid due to overvoltage or instability. This behavior corroborates the simulation and theoretical findings, demonstrating that grid impedance above 0.20 pu can destabilize the solar inverter.
The consistency between simulation, experiment, and theory reinforces the validity of the output impedance-based stability analysis. It highlights the need to consider grid impedance in the design and operation of solar inverters, especially for large-scale PV plants connected to weak grids.
Conclusion
In this paper, I have analyzed the stability of three-phase LCL-type solar inverters connected to weak grids using an output impedance approach. The key contributions include deriving a comprehensive output impedance model that incorporates the main circuit, PLL, and control loops of the solar inverter, and applying impedance stability criteria to evaluate system stability. The analysis reveals that the solar inverter’s output impedance is a 2×2 matrix in the dq frame, with self-impedance and mutual impedance terms. Among these, the q-axis self-impedance plays a decisive role in stability when grid impedance increases.
The findings indicate that for the solar inverter system studied, stability is maintained when grid impedance is below 0.20 pu. Beyond this threshold, the interaction between the solar inverter’s output impedance and grid impedance leads to negative phase margins in the q-axis, causing power oscillations and potential system failure. Simulation and experimental results validate these conclusions, showing instability at grid impedances of 0.20 pu and above.
This research underscores the importance of impedance-based analysis for solar inverter design in weak grid environments. Future work could explore adaptive control strategies to mitigate instability, such as adjusting PLL bandwidth or control gains based on grid impedance estimation. Additionally, the methodology can be extended to other types of solar inverters or renewable energy systems. By enhancing the stability of solar inverters, we can facilitate the secure integration of photovoltaic power into modern grids, supporting the global transition to sustainable energy.
In summary, the output impedance model provides a powerful tool for understanding and improving the stability of grid-connected solar inverters. As PV penetration continues to grow, such analyses will be crucial for ensuring reliable and efficient power systems.