With the continuous increase in the penetration rate of photovoltaic generation, solar inverter systems are required to possess grid-forming capability to enhance system stability. Existing methods to meet current limiting requirements generally adopt control architectures with cascaded power outer loop, voltage loop, and current loop, or introduce virtual admittance links in the control loop. This article focuses on impedance modeling and grid-connected stability of virtual synchronous generator (VSG)-based grid-forming solar inverters incorporating virtual admittance and current inner control loops. The impedance model of the solar inverter under VSG control is established and validated, followed by an analysis of key factors influencing impedance characteristics, such as power outer loop parameters, virtual admittance parameters and structures, and low-pass filter cutoff frequencies. Subsequently, based on the generalized Nyquist stability criterion and the system loop impedance method, the influence of control parameters and grid strength on the small-signal stability of VSG-based grid-forming solar inverters is investigated. Finally, the accuracy of stability conclusions is verified through simulations, demonstrating that smaller virtual inductance values facilitate stable operation in medium-low frequency bands, appropriately increasing virtual resistance in the virtual admittance link and the cutoff frequency of the low-pass filter can effectively enhance system stability margin, and under strong grid conditions, grid-forming solar inverter systems may face low-frequency oscillation instability risks.
The structure and control diagram of a VSG-based grid-forming solar inverter are illustrated in the following figure, which includes an outer loop with VSG control and reactive power control with a primary voltage regulation link, and an inner loop with virtual admittance and current control. The virtual admittance link and current inner loop are responsible for controlling the AC voltage and current at the point of common coupling (PCC), respectively. This control architecture retains the external characteristics of a synchronous machine while enhancing impedance adjustability. The voltage regulation link and virtual admittance link both include low-pass filters.
The active-frequency control of the VSG follows the rotor dynamics equation of a synchronous machine:
$$ \theta = \frac{1}{s} \left[ \omega_0 + \frac{1}{Js + D_p} (P_{\text{ref}} – P) \right] $$
where \( P_{\text{ref}} \) and \( P \) are the reference and measured active power, \( \omega_0 \) is the rated angular frequency, \( J \) is the virtual inertia, \( D_p \) is the damping coefficient, \( \omega \) is the system angular frequency, and \( \theta \) is the phase angle. The reactive-voltage control link uses an improved droop characteristic:
$$ U_d^* = \frac{1}{Ks} \left[ Q_{\text{ref}} – Q – D_q (H_{\text{LPF1}}(s) \cdot U_{sd} – U_{t\text{ref}}) \right], \quad U_q^* = 0 $$
where \( K \) is the inertia coefficient, \( D_q \) is the reactive-voltage droop coefficient, \( U_{t\text{ref}} \) and \( U_{sd} \) are the voltage magnitude reference and measured values, and \( H_{\text{LPF1}}(s) \) is the low-pass filter for the primary voltage regulation link with cutoff frequency \( f_{\text{LPF1}} \). The virtual admittance link outputs the current inner loop reference \( i_{dq\text{ref}} \):
$$ i_{dq\text{ref}} = \frac{1}{R_v + s L_v} \left[ U_{dq}^* – H_{\text{LPF2}}(s) \cdot U_{sdq} \right] $$
where \( R_v \) and \( L_v \) are the virtual resistance and inductance, and \( H_{\text{LPF2}}(s) \) is the voltage feedback low-pass filter with cutoff frequency \( f_{\text{LPF2}} \). The current inner loop uses a PI controller:
$$ U_{d\text{ref}} = H_i(s) (i_{d\text{ref}} – i_d) + \omega_0 L_f i_q + U_{sd}, \quad U_{q\text{ref}} = H_i(s) (i_{q\text{ref}} – i_q) – \omega_0 L_f i_d + U_{sq} $$
where \( H_i(s) \) is the PI controller transfer function.
To derive the impedance model of the grid-forming solar inverter, a small-signal linearization approach is applied in the frequency domain. The relationship between the PCC voltage \( U_{sabc} \) and current \( i_{abc} \) is established to obtain the AC-side sequence impedance. The derivation involves the power control outer loop, virtual admittance and current inner loop, coordinate transformation, and filter环节. For the power control outer loop, small-signal linearization of the active-frequency and reactive-voltage controls yields:
$$ -\Delta P = (Js + D_p) s \Delta \theta, \quad -\Delta Q – D_q \cdot H_{\text{LPF1}}(s) \Delta U_{csd} = K s \Delta U_d^{*c} $$
Using power conservation, \( \Delta P \) and \( \Delta Q \) can be expressed as:
$$ \Delta P = \frac{3}{2} \left[ i_{d0}^c \quad i_{q0}^c \right] \begin{bmatrix} \Delta U_{sd}^c \\ \Delta U_{sq}^c \end{bmatrix} + \frac{3}{2} \left[ U_{sd0}^c \quad U_{sq0}^c \right] \begin{bmatrix} \Delta i_d^c \\ \Delta i_q^c \end{bmatrix} $$
$$ \Delta Q = \frac{3}{2} \left[ -i_{q0}^c \quad i_{d0}^c \right] \begin{bmatrix} \Delta U_{sd}^c \\ \Delta U_{sq}^c \end{bmatrix} + \frac{3}{2} \left[ U_{sq0}^c \quad -U_{sd0}^c \right] \begin{bmatrix} \Delta i_d^c \\ \Delta i_q^c \end{bmatrix} $$
Substituting these into the linearized equations gives:
$$ s \Delta \theta = G_{\omega i} \Delta i_{dq}^c + G_{\omega u} \Delta U_{sdq}^c, \quad \Delta U_{dq}^{*c} = G_{ui} \Delta i_{dq}^c + G_{uu} \Delta U_{sdq}^c $$
where \( G_{\omega i} \), \( G_{\omega u} \), \( G_{ui} \), and \( G_{uu} \) are transfer matrices derived from the steady-state operating points.
For the virtual admittance and current inner loop, small-signal linearization results in:
$$ \Delta i_{dq\text{ref}}^c = H_v \cdot (\Delta U_{dq}^{*c} – H_{\text{LPF2}} \cdot \Delta U_{sdq}^c), \quad H_v = \frac{1}{R_v + s L_v} $$
$$ \Delta U_{d\text{ref}}^c = H_i (\Delta i_{d\text{ref}}^c – \Delta i_d^c) + \omega_0 L_f \Delta i_q^c + \Delta U_{sd}^c, \quad \Delta U_{q\text{ref}}^c = H_i (\Delta i_{q\text{ref}}^c – \Delta i_q^c) – \omega_0 L_f \Delta i_d^c + \Delta U_{sq}^c $$
Combining these, the relationship between outer loop voltage and inner loop electrical quantities is:
$$ \Delta U_{dq\text{ref}}^c = G_{ivc} \Delta U_{sdq}^c + G_{Li} \Delta i_{dq}^c + G_{iv} \Delta U_{dq}^{*c} $$
where \( G_{ivc} \), \( G_{Li} \), and \( G_{iv} \) are matrices involving \( H_i \), \( H_v \), and \( H_L \).
Coordinate transformation between the control system (superscript c) and electrical system (superscript s) coordinates accounts for the steady-state phase difference \( \delta_0 \). The transformation matrices are:
$$ T_s = \begin{bmatrix} \cos \delta_0 & -\sin \delta_0 \\ \sin \delta_0 & \cos \delta_0 \end{bmatrix}, \quad T_s^{-1} = \begin{bmatrix} \cos \delta_0 & \sin \delta_0 \\ -\sin \delta_0 & \cos \delta_0 \end{bmatrix} $$
Small-signal transformations yield:
$$ \Delta U_{sdq} = T_s \Delta U_{dq}^c + T_v \Delta \delta, \quad \Delta U_{sdq}^c = T_s^{-1} \Delta U_{sdq}^s + T_{cs} \Delta \delta, \quad \Delta i_{dq}^c = T_s^{-1} \Delta i_{dq}^s + T_c \Delta \delta $$
where \( T_v \), \( T_{cs} \), and \( T_c \) are matrices involving steady-state values.
The LCL filter环节 relates the main circuit electrical quantities:
$$ \Delta U_{sdq}^i = \Delta U_{sdq} + G_{ui} \Delta i_{cdq}^s, \quad \Delta i_{cdq}^s = \Delta i_{dq}^s + Y_f \Delta U_{sdq}^i, \quad \Delta U_{sdq}^s = \Delta U_{sdq}^i + G_{Lg} \Delta i_{dq}^s $$
where \( G_{ui} \), \( Y_f \), and \( G_{Lg} \) are impedance and admittance matrices. Combining these, the PCC voltage and current relationship is:
$$ \Delta U_{sdq}^s = A \Delta i_{dq}^s + B \Delta U_{sdq}, \quad A = G_{uiv} G_i, \quad B = G_{uiv} G_{vi} + G_{uvv} $$
Further combining with previous equations gives the solar inverter output impedance \( Z_{PV} \):
$$ Z_{PV} = \left( I – D – \frac{E G}{a} \right)^{-1} \left( C + \frac{E F}{a} \right) $$
where \( C \), \( D \), \( E \), \( F \), \( G \), and \( a \) are derived matrices and scalars.
The theoretical impedance model is validated by converting the dq-domain multi-input multi-output impedance to an equivalent single-input single-output positive/negative sequence impedance using linear transformation and model order reduction. The positive sequence impedance \( Z_{PVp}(s) \) and negative sequence impedance \( Z_{PVn}(s) \) are:
$$ Z_{PVp}(s) = Z_{PVpp}(s_p) – Z_{PVpn}(s_p) \frac{Z_{PVnp}(s_p) + Z_{gnp}(s_p)}{Z_{PVnn}(s_p) + Z_{gnn}(s_p)} $$
$$ Z_{PVn}(s) = Z_{PVnn}(s_n) – Z_{PVnp}(s_n) \frac{Z_{PVpn}(s_n) + Z_{gpn}(s_n)}{Z_{PVpp}(s_n) + Z_{gpp}(s_n)} $$
where \( s_p = s – j\omega_0 \), \( s_n = s + j\omega_0 \). Simulation results confirm the accuracy of the impedance model across a wide frequency range.
The impedance characteristics of the grid-forming solar inverter are influenced by various control parameters. The power outer loop parameters primarily affect the impedance characteristics near the fundamental frequency, with minimal impact on other frequency bands. For instance, increasing the damping coefficient \( D_p \) reduces the impedance magnitude near the fundamental frequency and increases the phase, while increasing the virtual inertia \( J \) raises the impedance magnitude below the fundamental frequency and weakens the capacitive negative resistance特性. The reactive-voltage droop coefficient \( D_q \) has negligible effect, and increasing the inertia coefficient \( K \) increases the impedance magnitude below the fundamental frequency and decreases it above, with little change in phase characteristics.
Virtual admittance parameters significantly impact the wide-band impedance magnitude-phase characteristics and real/imaginary parts. The impedance real part reflects damping特性. Increasing the virtual resistance \( R_v \) elevates the overall impedance magnitude and real part, significantly reducing or eliminating the inductive negative resistance特性 at high frequencies and slightly weakening the capacitive negative resistance特性 below the fundamental frequency. Increasing the virtual inductance \( L_v \) raises the impedance magnitude, with little change in resistance characteristics at medium-low frequencies but prominent negative resistance特性 at high frequencies, which shifts to lower frequencies as \( L_v \) increases.
Low-pass filter parameters also affect impedance特性. The cutoff frequency \( f_{\text{LPF1}} \) of the primary voltage regulation link has almost no effect, while increasing the cutoff frequency \( f_{\text{LPF2}} \) of the virtual admittance link voltage feedback reduces the impedance magnitude in most frequency bands except near the fundamental and above 500 Hz, shifts the negative resistance interval at medium frequencies to higher frequencies, and weakens the capacitive特性 at sub-synchronous frequencies.
Comparing control structures, constant-term admittance versus transfer function admittance forms shows differences mainly in medium-high frequencies. The transfer function form results in higher impedance resonance peaks and more prone inductive negative resistance特性.
Stability analysis using the generalized Nyquist criterion and system loop impedance method reveals that virtual resistance \( R_v \), virtual inductance \( L_v \), and the cutoff frequency \( f_{\text{LPF2}} \) are sensitive parameters affecting the stability of VSG-based grid-forming solar inverters. The system loop impedance \( Z_{P\text{loop}}(s) \) is defined as the sum of the solar inverter and grid equivalent single-input single-output positive sequence impedances: \( Z_{P\text{loop}}(s) = Z_{PVp}(s) + Z_{gp}(s) \). Its real part reflects the overall damping特性, and the imaginary part’s zero-crossing frequency indicates potential resonance points. Stability is assessed based on the real part magnitude at these frequencies.
For a grid with short-circuit ratio (SCR) of 2.5, increasing \( R_v \) shifts the Nyquist curve away from (-1,0) and increases the real part magnitude at the imaginary part’s zero-crossing frequency, enhancing stability. Increasing \( L_v \) brings the Nyquist curve closer to (-1,0) and decreases the real part magnitude, degrading stability. Decreasing \( f_{\text{LPF2}} \) also brings the Nyquist curve closer to (-1,0), reducing stability. Under strong grid conditions (e.g., SCR=10.8), the Nyquist curve critically encircles (-1,0) at around 1.18 Hz, indicating low-frequency oscillation risks.
Simulations in MATLAB/Simulink validate these findings. For example, when \( L_v \) is increased from 0.0001 pu to 0.0005 pu, the PCC current exhibits growing oscillations, indicating instability. Similarly, reducing \( f_{\text{LPF2}} \) from 89 Hz to 4.5 Hz causes divergent oscillations. Under SCR=10.8, the PCC current oscillates at 48.8 Hz and 51.2 Hz in the three-phase stationary frame (mirror frequency coupling components), corresponding to 1.2 Hz in the dq-frame, consistent with theoretical analysis.
In conclusion, the impedance modeling and stability analysis of VSG-based grid-forming solar inverters provide insights into parameter design for enhanced system robustness. Key recommendations include using smaller virtual inductance values for medium-low frequency stability, appropriately increasing virtual resistance and low-pass filter cutoff frequencies to improve stability margins, and addressing low-frequency oscillation risks in strong grids. These findings contribute to the optimal control of solar inverter systems in modern power networks.
| Parameter Name | Value |
|---|---|
| DC-side voltage (V) | 1200 |
| Inverter-side filter inductance \( L_f \) (μH) | 54 |
| Grid-side filter inductance \( L_g \) (μH) | 8 |
| Filter capacitance \( C_f \) (μF) | 28 |
| Damping resistance \( R_d \) (Ω) | 0.1 |
| Active power \( P \) (kW) | 330 |
| Reactive power \( Q \) (Var) | 0 |
| Grid short-circuit ratio (SCR) | 2.5 |
| Grid voltage \( U_g \) (V) | 800 |
| Control delay \( T_d \) (μs) | 80 |
| Virtual inertia \( J \) (pu) | 8 |
| Damping coefficient \( D_p \) (pu) | 80 |
| Inertia coefficient \( K \) (pu) | 2.3 |
| Reactive-voltage droop coefficient \( D_q \) (pu) | 5 |
| Virtual resistance \( R_v \) (pu) | 0.1 |
| Virtual inductance \( L_v \) (pu) | 10^{-6} |
| Current loop proportional coefficient \( K_{gp_i} \) | 0.1606 |
| Current loop integral coefficient \( K_{gi_i} \) | 208.01 |
| LPF1 cutoff frequency \( f_{\text{LPF1}} \) (Hz) | 44 |
| LPF2 cutoff frequency \( f_{\text{LPF2}} \) (Hz) | 89 |
The impedance characteristics can be summarized using the following equations for key transfer functions. The power control outer loop matrices are:
$$ G_{\omega i} = \frac{3}{2(Js + D_p)} \begin{bmatrix} -U_{sd0}^c & -U_{sq0}^c \end{bmatrix}, \quad G_{\omega u} = \frac{3}{2(Js + D_p)} \begin{bmatrix} -i_{d0}^c & -i_{q0}^c \end{bmatrix} $$
$$ G_{ui} = \frac{3}{2Ks} \begin{bmatrix} -U_{sq0}^c & U_{sd0}^c \\ 0 & 0 \end{bmatrix}, \quad G_{uu} = \frac{3}{2Ks} \begin{bmatrix} i_{q0}^c & -i_{d0}^c \\ 0 & 0 \end{bmatrix} – \frac{D_q}{Ks} \begin{bmatrix} H_{\text{LPF1}}(s) & 0 \\ 0 & 0 \end{bmatrix} $$
The virtual admittance and current inner loop matrices are:
$$ G_{ivc} = I – H_{\text{LPF2}}(s) H_i H_v, \quad G_{Li} = H_L – H_i, \quad G_{iv} = H_i H_v $$
where \( H_i = \begin{bmatrix} H_i & 0 \\ 0 & H_i \end{bmatrix} \), \( H_v = \begin{bmatrix} H_v & 0 \\ 0 & H_v \end{bmatrix} \), \( H_L = \begin{bmatrix} 0 & \omega_0 L_f \\ -\omega_0 L_f & 0 \end{bmatrix} \). The filter matrices are:
$$ G_{ui} = – \begin{bmatrix} s L_f + R_f & -\omega_0 L_f \\ \omega_0 L_f & s L_f + R_f \end{bmatrix}, \quad Y_f = \left( \begin{bmatrix} s C_f & -\omega_0 C_f \\ \omega_0 C_f & s C_f \end{bmatrix}^{-1} + \begin{bmatrix} R_d & 0 \\ 0 & R_d \end{bmatrix}^{-1} \right), \quad G_{Lg} = \begin{bmatrix} -s L_g & \omega_0 L_g \\ -\omega_0 L_g & -s L_g \end{bmatrix} $$
These models and analyses ensure that solar inverters can be designed for stable operation in various grid conditions, highlighting the importance of impedance-based approaches for modern solar inverter systems.