As the penetration rate of photovoltaic (PV) generation continues to increase, power systems face heightened risks of wide-band oscillations, severely threatening their safe and stable operation. The fundamental cause lies in the widespread use of power electronic interfaces in PV generation systems, which significantly reduces the inertia and damping characteristics of the power system. To address this challenge, grid-forming (GFM) control techniques based on droop or virtual synchronous generator (VSG) principles have attracted considerable attention. By emulating the dynamic behavior of synchronous generators, these techniques can effectively enhance the voltage and frequency support capabilities of high-renewable-penetration power systems, such as those involving solar inverters. However, GFM PV systems encounter severe transient current control challenges during grid faults. Maintaining currents within safe thresholds has become a critical bottleneck for equipment reliability and system stability. Existing current limiting strategies can be classified into two technical approaches: one is the classical multi-loop cascaded control architecture based on power–voltage–current, and the other introduces a virtual admittance link. It is worth noting that dual-loop control is prone to instability in strong grid scenarios, while the virtual admittance method exhibits superior stability margin regulation potential through equivalent inductance enhancement.
In this work, we focus on the VSG-based grid-forming solar inverter incorporating virtual admittance and an inner current control loop. We conduct impedance modeling and grid-connected stability analysis. Firstly, an impedance model of the solar inverter under VSG control is established and validated. Secondly, the influence of key factors—such as power outer loop parameters, virtual admittance parameters and structure, and the cutoff frequency of low-pass filters—on the impedance characteristics of the solar inverter is analyzed. Thirdly, based on the generalized Nyquist stability criterion and the system loop impedance method, the influence laws of control parameters and grid strength on the small-signal stability of the VSG-based grid-forming solar inverter are investigated. Finally, MATLAB/Simulink simulations verify the correctness of the stability analysis conclusions. The results demonstrate that a smaller virtual inductance value is conducive to stable operation in the medium-low frequency range; appropriately increasing the virtual resistance in the virtual admittance link and the cutoff frequency of the low-pass filter can effectively enhance the system stability margin; under strong grid conditions, the grid-forming solar inverter system may face the risk of low-frequency oscillation instability.
System Description and Control Architecture
Figure 1 illustrates the structure and control block diagram of a grid-forming solar inverter based on VSG control. In this control system, the outer loop adopts VSG control together with a reactive power control loop that includes a primary voltage regulation link. The inner loop employs a virtual admittance and current control scheme. The virtual admittance link and the current inner loop are responsible for regulating the AC voltage and current at the point of common coupling (PCC), respectively. This control framework preserves the external characteristics of a synchronous machine while enhancing impedance tunability. Both the voltage regulation link and the virtual admittance link contain low-pass filters.
Impedance Modeling of the Grid-Forming Solar Inverter
Small-Signal Impedance Model
We adopt a frequency-domain linearization method to derive the relationship between the PCC voltage and current, thereby obtaining the AC-side sequence impedance. The detailed derivation is presented below.
Power Outer Loop
For the active power–frequency control of VSG, the rotor dynamics equation is:
$$ \theta = \frac{1}{s} \left( \omega_0 + \frac{1}{J s + D_p} (P_{\text{ref}} – P) \right) $$
The reactive power–voltage control loop adopts an improved droop characteristic. It outputs the voltage reference by detecting the difference between the reactive power reference \(Q_{\text{ref}}\) and the actual value \(Q\), as well as the difference between the voltage amplitude reference \(U_{t\text{ref}}\) and the measured value \(U_{sd}\):
$$ U_d^* = \frac{1}{K s} \left[ Q_{\text{ref}} – Q – D_q \left( H_{\text{LPF1}}(s) U_{sd} – U_{t\text{ref}} \right) \right] $$
where \(K\) is the inertia coefficient, \(D_q\) is the reactive power–voltage droop coefficient, and \(H_{\text{LPF1}}(s)\) is the low-pass filter of the primary voltage regulation loop with cutoff frequency \(f_{\text{LPF1}}\).
Virtual Admittance and Current Inner Loop
The difference between the AC voltage reference from the reactive power controller and the PCC voltage passes through the virtual admittance link to obtain the current inner loop reference \(i_{dq\text{ref}}\):
$$ i_{dq\text{ref}} = \frac{1}{R_v + s L_v} \left[ U_{dq}^* – H_{\text{LPF2}}(s) U_{sdq} \right] $$
where \(R_v\) and \(L_v\) are the virtual resistance and inductance, and \(H_{\text{LPF2}}(s)\) is the low-pass filter for voltage feedback in the virtual admittance link with cutoff frequency \(f_{\text{LPF2}}\).
The current inner loop uses a PI controller to regulate the dq-axis currents:
$$ U_{d\text{ref}} = H_i(s) (i_{d\text{ref}} – i_d) + \omega_0 L_f i_q + U_{sd} $$
$$ U_{q\text{ref}} = H_i(s) (i_{q\text{ref}} – i_q) – \omega_0 L_f i_d + U_{sq} $$
where \(H_i(s)\) is the PI controller transfer function of the current loop.
Linearization and Coordinate Transformation
Linearizing the power equations and combining them with the coordinate transformation between the control system frame (superscript c) and the electrical system frame (superscript s) yields the small-signal relationship. The steady-state phase difference \(\delta_0\) exists between the two frames. The transformation matrices are:
$$ T_s = \begin{bmatrix} \cos \delta_0 & -\sin \delta_0 \\ \sin \delta_0 & \cos \delta_0 \end{bmatrix}, \quad T_s^{-1} = \begin{bmatrix} \cos \delta_0 & \sin \delta_0 \\ -\sin \delta_0 & \cos \delta_0 \end{bmatrix} $$
Filter Stage and Output Impedance
The main circuit uses an LCL filter. The small-signal relationships of the filter components are derived, and after combining all the above equations, we obtain the output impedance \(Z_{\text{PV}}\) of the solar inverter in the dq domain:
$$ Z_{\text{PV}} = \left( I – D – \frac{E G}{a} \right)^{-1} \left( C + \frac{E F}{a} \right) $$
where matrices \(C, D, E, F, G\) and scalar \(a\) are given by the linearized system equations (detailed expressions are omitted here for brevity).
Impedance Model Validation
The theoretical dq-domain impedance model is transformed into equivalent single-input single-output positive/negative sequence impedances using a linear transformation and model reduction method. The simulation parameters used for validation are listed in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC voltage / V | 1200 | Virtual inertia J (p.u.) | 8 |
| Inverter-side filter inductance \(L_f\) / μH | 54 | Damping coefficient \(D_p\) (p.u.) | 80 |
| Grid-side filter inductance \(L_g\) / μH | 8 | Inertia coefficient \(K\) (p.u.) | 2.3 |
| Filter capacitance \(C_f\) / μF | 28 | Reactive droop coefficient \(D_q\) (p.u.) | 5 |
| Damping resistor \(R_d\) / Ω | 0.1 | Virtual resistance \(R_v\) (p.u.) | 0.1 |
| Active power \(P\) / kW | 330 | Virtual inductance \(L_v\) (p.u.) | 10⁻⁶ |
| Reactive power \(Q\) / Var | 0 | Current loop proportional gain \(K_{gp\_i}\) | 0.1606 |
| Grid short-circuit ratio (SCR) | 2.5 | Current loop integral gain \(K_{gi\_i}\) | 208.01 |
| Grid voltage \(U_g\) / V | 800 | LPF1 cutoff frequency \(f_{\text{LPF1}}\) / Hz | 44 |
| Control delay \(T_d\) / μs | 80 | LPF2 cutoff frequency \(f_{\text{LPF2}}\) / Hz | 89 |
Frequency-domain scanning results from a time-domain simulation model in MATLAB/Simulink confirm that the theoretical impedance model closely matches the measured impedance amplitude and phase over a wide frequency range, thereby validating the accuracy of the established model.
Impedance Characteristic Analysis
Influence of Power Outer Loop Parameters
The power outer loop parameters primarily affect the impedance characteristics near the fundamental frequency. As the damping coefficient \(D_p\) increases, the impedance magnitude near the fundamental frequency decreases while the phase increases. Increasing the virtual inertia \(J\) raises the impedance magnitude below the fundamental frequency and weakens the capacitive negative-resistance characteristic. The reactive droop coefficient \(D_q\) has negligible impact, while increasing the inertia coefficient \(K\) increases the impedance magnitude below the fundamental frequency and decreases it above.
Influence of Virtual Admittance Parameters
We analyze the impact of virtual resistance \(R_v\) and virtual inductance \(L_v\) on the wideband impedance amplitude/phase and real/imaginary parts. The real part of the impedance reflects the damping characteristic. When \(R_v\) increases, both the impedance magnitude and the real part increase, the inductively negative-resistance characteristic in the high-frequency range is significantly weakened or eliminated, and the capacitive negative-resistance characteristic below the fundamental frequency is also reduced. As \(L_v\) increases, the impedance magnitude increases, the resistive characteristic in the medium–low frequency band remains almost unchanged, but a prominent negative-resistance characteristic appears in the high-frequency band, and its frequency range extends toward lower frequencies as the inductance grows.
Influence of Low-Pass Filter Cutoff Frequency
Increasing the cutoff frequency \(f_{\text{LPF1}}\) of the primary voltage regulation loop has almost no effect on the impedance. However, increasing the cutoff frequency \(f_{\text{LPF2}}\) of the voltage feedback low-pass filter in the virtual admittance link reduces the impedance magnitude in most frequency ranges except near the fundamental and above 500 Hz, shifts the negative-resistance region toward higher frequencies, and weakens the capacitive characteristic in the subsynchronous range.
Influence of Control Structure
We compare the impedance characteristics of two control structures: one using a constant-term admittance form and the other using a transfer-function admittance form (the one adopted in this work). The differences are mainly in the medium–high frequency range. The transfer-function form yields a higher resonant peak and is more prone to exhibiting an inductively negative-resistance characteristic.
Grid-Connected Stability Analysis of the Grid-Forming Solar Inverter
Based on the impedance characteristic analysis, virtual resistance \(R_v\), virtual inductance \(L_v\), and the cutoff frequency \(f_{\text{LPF2}}\) of the voltage feedback low-pass filter are identified as the most influential parameters. Therefore, we utilize the generalized Nyquist criterion (GNC) and the system loop impedance method to investigate their effects on stability. The system loop impedance \(Z_{\text{Ploop}}(s)\) is defined as the sum of the equivalent single-input single-output positive-sequence impedance of the solar inverter and the grid impedance. The real part of \(Z_{\text{Ploop}}(s)\) reflects the overall system damping, and the zero crossings of its imaginary part indicate potential resonant frequencies.
Influence of Virtual Resistance \(R_v\)
When \(R_v\) increases, the Nyquist curve shifts away from the point \((-1,0)\), and the real part of the loop impedance at the frequency where the imaginary part crosses zero increases. This indicates that in a weak grid (SCR = 2.5), increasing the virtual resistance appropriately improves the stability of the grid-connected solar inverter system.
Influence of Virtual Inductance \(L_v\)
As \(L_v\) increases, the Nyquist curve moves closer to \((-1,0)\), and the real part of the loop impedance at the zero-crossing frequency decreases. Therefore, within the studied range, increasing the virtual inductance deteriorates the system stability.
Influence of Low-Pass Filter Cutoff Frequency \(f_{\text{LPF2}}\)
A decrease in \(f_{\text{LPF2}}\) shifts the Nyquist curve toward \((-1,0)\), indicating that reducing the cutoff frequency of the voltage feedback low-pass filter destabilizes the system. Thus, a larger \(f_{\text{LPF2}}\) is beneficial for stability.
Influence of Grid Strength
When the grid strength increases (higher SCR), the Nyquist curve in the low-frequency range gradually approaches and critically encircles \((-1,0)\). For example, at SCR = 10.8, the curve critically encircles \((-1,0)\) at a frequency of 1.18 Hz, implying a risk of low-frequency oscillation near 1.18 Hz. This suggests that under strong grid conditions, the grid-forming solar inverter system may face low-frequency instability.
Simulation Verification
Time-domain simulations using MATLAB/Simulink are performed to validate the stability analysis results. The parameters are consistent with those used in the theoretical analysis.
When the virtual inductance \(L_v\) (p.u.) is increased from 0.0001 to 0.0005 at 3.7 s, the PCC current waveform gradually exhibits oscillations, confirming instability. Similarly, when \(f_{\text{LPF2}}\) is reduced from 89 Hz to 4.5 Hz at 3.7 s, the PCC current diverges, validating the destabilizing effect. When the grid SCR is increased from 2 to 10.8 at 3.7 s, the PCC current oscillates with dominant frequencies at 48.8 Hz and 51.2 Hz in the three-phase stationary frame, corresponding to a 1.2 Hz component in the dq frame, which matches the theoretical prediction of low-frequency oscillation.
Conclusion
In this work, we have established an impedance model for a VSG-controlled grid-forming solar inverter with virtual admittance and current inner loop. Based on impedance characteristic analysis and small-signal stability study, the following conclusions are drawn:
- The virtual inductance value significantly affects system stability; a smaller virtual inductance is more favorable for stable operation in the medium–low frequency range.
- Appropriately increasing the virtual resistance in the virtual admittance link and the cutoff frequency of the voltage feedback low-pass filter can effectively enhance the stability margin of the system.
- Under strong grid conditions (high short-circuit ratio), the grid-forming solar inverter system may face the risk of low-frequency oscillation instability, with the oscillatory mode appearing near 1.2 Hz in the dq frame.
These findings provide theoretical guidance for the control parameter optimization and impedance reshaping of grid-forming solar inverters in practical applications.