In the context of weak grids, photovoltaic-storage power plants utilizing grid-following control are prone to instability and oscillations. The method of integrating grid-forming inverters in parallel helps to enhance the small-signal stability of power plants. However, existing research on the ratio design of grid-forming inverters only considers the capacity factor and ignores other design degrees of freedom of grid-forming inverters, resulting in conservative configuration results. Therefore, this paper proposes a parameter design method based on the partial retrofitting of inverters to grid-forming control.
First, based on impedance models and stability criteria for cascaded systems, the key mechanisms for enhancing system stability after the integration of grid-forming inverters are clarified. The influence of both the proportion and key filtering parameters of grid-forming inverters within the power plant on stability is explored. Then, a configuration process for the proportion and parameters of grid-forming inverters is proposed to obtain the parameter domain that ensures stable system operation. Finally, the effectiveness of the proposed method is verified through case analysis.
Introduction to Stability Challenges in Weak Grids
Currently, most grid-connected inverters in photovoltaic-storage power stations use a grid-following control strategy based on a phase-locked loop (PLL) to connect to the AC grid. However, in weak grids, the multi-machine system of grid-following inverters couples with the grid impedance, leading to severe oscillation instability. This makes the risk of small-signal instability relatively high.
Recent studies have found that inverters using grid-forming control achieve synchronization with the grid through a power synchronization loop, effectively reducing the risk of oscillation instability caused by the PLL. Converting or adding grid-forming control units in photovoltaic power plants enhances the stable operation capability of the system. Grid-forming inverters exhibit a “positive resistance” characteristic in the sub/super-synchronous frequency band, which can weaken the negative damping characteristics of grid-following inverters, thereby improving the system stability margin.
Building on this, research has been conducted to optimize the ratio of grid-forming inverters to ensure stable system operation. For example, from the perspective of the short-circuit ratio (SCR), adding grid-forming inverters is equivalent to increasing the generalized short-circuit ratio of the system. However, while some studies have provided guidance for capacity ratios based on state-space models and critical SCR indicators, their adaptability under multi-machine parallel operation remains unclear. Other works have analyzed the impact of the configuration proportion of grid-forming inverters on system stability using generalized SCR metrics, but the proposed design schemes are often conservative as they do not consider device-level parameter optimization.
In practice, the filter parameters of grid-forming inverters significantly affect the small-signal stability of hybrid systems. This paper addresses the gap in optimization for the configuration and design of grid-forming inverters. We propose a method for stable operation under extremely weak grid conditions by optimizing both the number of grid-forming units and their key filter parameters. The stability analysis is based on a simplified equivalent impedance model of the retrofitted power plant, and the design ensures a certain phase margin for stable operation across all operating conditions.
Small-Signal Instability in Grid-Following Power Plants
In a typical photovoltaic power station, multiple grid-following inverters are connected in parallel at the point of common coupling (PCC) via a line to the AC grid. For this analysis, we consider a station composed of N grid-following inverters. The structure and control of a single grid-following inverter include the LC filter, current loop, and PLL. Using harmonic linearization modeling, the positive-sequence and negative-sequence output impedance models for a grid-following inverter, considering frequency coupling, are shown in the following equations. This impedance acts as a variable impedance dependent on the operating point.
The equivalent output impedance of the entire grid-following power plant is the parallel combination of the output impedances of all individual inverters. Assuming all inverters in the same station have the same topology, control, and operating conditions, the plant’s positive-sequence impedance is given by:
$$ Z_{GFLp,eq}(s) = \frac{1}{N} Z_{GFL,p}(s) $$
For a balanced three-phase system, the small-signal stability is primarily determined by the positive-sequence output impedance. The system can be modeled as a voltage source \( u_s \) with grid impedance \( Z_g \) connected to the plant’s equivalent impedance \( Z_{GFL,eq} \). The Bode plot of the plant’s output impedance versus the grid impedance for different grid inductance values shows that as the grid impedance increases (weaker grid), the intersection frequency decreases. The grid-following inverter impedance has a region in the low-frequency band where its phase is less than -90 degrees. By the Nyquist criterion, the phase margin \( \theta_{PM} \) is:
$$ \theta_{PM} = 180^\circ – |\angle Z_g(j2\pi f_c) – \angle Z_{GFL}(j2\pi f_c)| $$
This analysis demonstrates that a purely grid-following power plant is prone to small-signal instability under weak grid conditions.
Analysis of Plant Characteristics after Retrofitting with Grid-Forming Inverters
We assume that of the N inverters, m are retrofitted as grid-forming inverters, leaving (N – m) as grid-following inverters. The grid-forming inverter control system uses a power synchronization loop. Its output impedance model, considering frequency coupling, is developed. The Bode plot comparing the output impedances of a grid-following and a grid-forming inverter shows that the grid-forming inverter has a significantly lower output impedance, indicating its voltage-source characteristic. The model of the retrofitted plant is shown in the figure, where multiple grid-forming inverters are connected in parallel with the grid-following inverters.
To simplify the analysis, we first analyze the characteristics of the grid-forming inverter’s output impedance. The Bode plot for different active and reactive power outputs shows that the impedance varies mostly near the fundamental frequency. At medium and high frequencies, the influence of the operating point is small. Further analysis reveals that the actual output impedance of the grid-forming inverter closely matches its LC filter output impedance, especially at non-fundamental frequencies. This suggests that for stability analysis, the grid-forming inverter’s output impedance can be approximated by its LC filter impedance.
The LC filter output impedance is dominated by the filter inductance \( L_f \) in the low-frequency range and by the filter capacitance \( C_f \) in the high-frequency range. A study of the impedance variation with \( L_f \) and \( C_f \) shows that \( L_f \) has a much more significant impact on the overall impedance characteristic across a wide frequency range. Therefore, the filter inductance is identified as the key parameter for grid-forming inverter design in this context.
Since the grid-forming inverter output impedance is relatively constant and close to its LC filter impedance, it can be approximated as an ideal voltage source in series with a fixed impedance. Applying Thevenin’s theorem, the grid side and the grid-forming inverters can be combined into an equivalent voltage source \( u_{s,eq} \) and an equivalent grid impedance \( Z_{g,eq} \).
The expression for the equivalent grid impedance, assuming all m grid-forming inverters are identical, is:
$$ Z_{g,eq}(s) = \frac{Z_g(s) Z_{GFM}(s)}{m Z_g(s) + Z_{GFM}(s)} $$
By substituting the grid impedance (with negligible resistance) and the LC filter impedance, and assuming the grid-forming inverter impedance \( Z_{GFM}(s) \) equals the LC impedance \( Z_{LC}(s) \), we obtain:
$$ Z_{g,eq}(s) = \frac{sL_g (R_C L_f C_f s^2 + L_f s + R_L)}{m s L_g [L_f C_f s^2 + (R_L + R_C) C_f s + 1] + R_C L_f C_f s^2 + L_f s + R_L} $$
Mechanism of Stability Enhancement by Grid-Forming Inverters
The fundamental reason for the instability of grid-following units under weak grids is the large grid impedance, which causes the intersection frequency of the grid-following plant impedance and the grid impedance to fall within the negative damping region. By retrofitting some inverters to grid-forming operation, the equivalent grid impedance magnitude is reduced. This increases the intersection frequency, moving it away from the negative damping region and reducing the risk of instability.
The key parameters affecting the equivalent grid impedance are the number of grid-forming inverters (m) and their filter inductance (\( L_f \)). We analyze the influence of these two parameters on system stability using a specific example where the grid impedance \( L_g = 0.1 \) mH (SCR = 1.4).
Effect of Filter Inductance on Stability
When \( m = 1 \) and \( L_f = 0.1 \) mH, the intersection frequency \( f_c \) is 214.8 Hz, and the system has a positive phase margin. The system meets the Nyquist criterion and is stable. When \( L_f \) is increased to 0.5 mH, the intersection frequency decreases to 169 Hz, which is within the negative damping region. The phase margin becomes negative (-18.22 degrees), and the system becomes unstable. This shows that a smaller filter inductance leads to a lower equivalent grid impedance magnitude, which helps stabilize the system.
Effect of Number of Grid-Forming Inverters on Stability
For the case where \( L_f = 0.5 \) mH and \( m = 1 \), the system is unstable. If we increase the number of grid-forming inverters to \( m = 2 \) (ratio 14:2), the equivalent grid impedance magnitude decreases. The intersection frequency increases to 208.9 Hz, and the phase margin becomes 8.14 degrees. Increasing \( m \) to 3 further increases the phase margin to 41.18 degrees. This demonstrates that increasing the number of grid-forming inverters enhances stability by reducing the equivalent grid impedance.
Design Process for Retrofitting Power Plants
The optimization of a photovoltaic-storage power plant involves selecting the number of grid-forming inverters (m) and their filter inductance (\( L_f \)). The goal is to ensure a phase margin greater than 30 degrees for stable operation. The design process follows these steps:
- Start with all inverters as grid-following and set \( m = 1 \).
- Calculate the equivalent grid-following plant impedance \( Z_{GFL,eq} \) and the equivalent grid impedance \( Z_{g,eq} \) using the LC filter model for the grid-forming inverters.
- Find the intersection frequency \( f_c \) of these two impedance curves and calculate the phase margin \( \theta_{PM} \).
- If \( \theta_{PM} < 30^\circ \), decrease the filter inductance \( L_f \) of the grid-forming inverters and recalculate.
- If \( L_f \) becomes smaller than its minimum feasible value (determined by hardware constraints like current ripple), increase the number of grid-forming inverters by 1 (i.e., \( m = m + 1 \)) and reset \( L_f \) to its maximum value.
- Repeat steps 4 and 5 until \( \theta_{PM} \geq 30^\circ \).
From an economic perspective, it is preferable to first reduce the filter inductance of the existing grid-forming inverters before increasing their number, as this minimizes the amount of hardware modification required.
Case Study and Validation
We consider the same 16-machine parallel system, now operating with a grid impedance of 0.14 mH (SCR = 1.1, an extremely weak grid). A system with only grid-following inverters is unstable. Based on the design method, we establish the feasible range for the grid-forming filter inductance as 0.1 mH ≤ \( L_f \) ≤ 0.5 mH, where the lower limit is defined by current ripple constraints and the upper limit by voltage drop considerations. Within this range, the current THD of a single grid-forming inverter is acceptable, as shown in the table below.
The optimization process yields a relationship between the number of grid-forming inverters, filter inductance, and phase margin. To achieve a phase margin greater than 30 degrees, the parameters should be within the yellow region of the design space. For this case, we select point A, which corresponds to \( m = 2 \) grid-forming inverters (12.5% of total capacity) and a filter inductance \( L_f = 0.2 \) mH. Simulation results show that this configuration is stable under the given extremely weak grid conditions.
| Filter Inductance (Lf) | Current THD |
|---|---|
| 0.1 mH | 0.37% |
| 0.2 mH | 0.30% |
| 0.3 mH | 0.29% |
| 0.4 mH | 0.31% |
| 0.5 mH | 0.40% |
For comparison, we test two other points from the design map. Point B (\( L_f = 0.4 \) mH, \( m = 2 \)) is stable but shows larger harmonic components. Point C (\( L_f = 0.5 \) mH, \( m = 1 \)) leads to significant oscillations and instability, confirming that both the number of grid-forming inverters and their filter parameters are crucial for stability.
To further validate the approach, we test the designed system in a 3-machine, 9-bus system. The 16-machine plant is connected to a bus. The table below summarizes the stability results for different configurations.
| Configuration | Stable? | Current THD | Voltage THD | Matches Theory? |
|---|---|---|---|---|
| 1 GFM, Lf=0.1 mH | Stable | 0.21% | 1.93% | Yes |
| 1 GFM, Lf=0.2 mH | Unstable | – | – | Yes |
| 2 GFM, Lf=0.2 mH | Stable | 0.17% | 1.80% | Yes |
| 2 GFM, Lf=0.5 mH | Unstable | – | – | Yes (at low margin) |
| 3 GFM, Lf=0.1 mH | Stable | 0.11% | 2.17% | Yes |
The simulation results are consistent with the theoretical stability analysis. It is noted that at parameter boundaries with low stability margins (e.g., 2 GFM units with \( L_f = 0.5 \) mH), the system is predicted to be unstable, which confirms the need to maintain a sufficient phase margin of 30 degrees.
Conclusion
This paper presented a methodology for enhancing the small-signal stability of photovoltaic-storage power plants through the optimal retrofitting of a portion of their grid-following inverters to grid-forming control. The key findings and contributions are summarized as follows:
- Mechanism of Stability Enhancement: Retrofitting inverters to grid-forming operation significantly reduces the equivalent grid impedance as seen by the remaining grid-following inverters. This effectively increases the electrical strength of the grid (increase in equivalent SCR), thereby improving system stability margins. The primary parameters influencing stability are the number of grid-forming inverters and their filter inductance.
- Key Parameter Identification: The output impedance of a grid-forming inverter is largely dominated by its LC filter parameters, with the filter inductance (\( L_f \)) having the most pronounced effect across a wide frequency range. This parameter, along with the number of grid-forming inverters, forms the basis for optimization.
- Optimized Design Process: A systematic design process was proposed to determine the minimum number of grid-forming inverters and their optimal filter inductance required to guarantee a phase margin of at least 30 degrees. The process prioritizes reducing the filter inductance of individual units over increasing their number, balancing economic cost with technical performance.
- Validation: The method was validated through detailed simulations of a 16-machine system under extremely weak grid conditions (SCR = 1.1). The results confirmed that a retrofitting plan of just 12.5% of the inverters with an optimal filter inductance can ensure stable operation. The analysis also demonstrated that employing a single grid-forming unit or an inappropriately designed filter inductance is insufficient, highlighting the importance of the joint optimization of both variables.
This research provides a more flexible and efficient guiding principle for the engineering configuration of different types of solar inverter technologies in modern power systems. Future work will focus on the impact of dynamic interactions and mutual coupling between inverters in hybrid systems to further refine the modeling and optimization methods.