The rapid and large-scale integration of photovoltaic (PV) generation into the power grid has fundamentally altered the dynamics of modern electrical power systems. A primary consequence of this shift is the marked reduction in overall system inertia and damping, as traditional rotating synchronous generators are displaced by power electronic interfaced solar inverters. This deficiency manifests as an increased susceptibility to wide-band oscillations and stability challenges, posing a significant threat to reliable grid operation. In response to these challenges, grid-forming (GFM) control technology has emerged as a pivotal solution. By emulating the intrinsic behaviors of synchronous generators—specifically inertia, damping, and voltage source characteristics—grid-forming solar inverters can actively participate in regulating grid voltage and frequency, thereby enhancing system stability and resilience. Among the various GFM strategies, the Virtual Synchronous Generator (VSG) algorithm stands out for its explicit imitation of the swing equation and excitation system of a conventional synchronous machine.
However, the transition from grid-following to grid-forming paradigms for solar inverters introduces new control complexities, particularly concerning current limitation during grid faults. Existing methodologies to address this can be broadly categorized into two approaches: the classical cascaded power-voltage-current multi-loop control architecture, and the more recent paradigm incorporating a virtual admittance loop. While effective for current limiting, the interplay between these inner control layers and the outer VSG power synchronization loop creates complex dynamic interactions that must be carefully analyzed. The impedance-based stability assessment method provides a powerful frequency-domain tool for this purpose. By deriving and analyzing the output impedance of the solar inverter and comparing it with the grid impedance, one can predict potential instability risks across different frequency ranges.
This article presents a comprehensive investigation into the impedance modeling and stability characteristics of a VSG-controlled grid-forming solar inverter that incorporates both a virtual admittance block and an inner current loop—a structure aimed at robust current limiting and control. We begin by establishing a detailed, multi-input-multi-output (MIMO) impedance model in the dq-frame, subsequently transforming it into an equivalent single-input-single-output (SISO) positive/negative sequence impedance for practical analysis. The accuracy of this theoretical model is rigorously validated against time-domain simulation results. Following this, we conduct a thorough parametric analysis to elucidate the influence of key control parameters—including those of the power outer loop, the virtual admittance, and associated low-pass filters—on the wide-band impedance characteristics of the solar inverter. Furthermore, we delve into stability analysis using the Generalized Nyquist Criterion (GNC) and the system loop impedance method to uncover the impact of these parameters and grid strength on the small-signal stability of the interconnected system. All theoretical findings are conclusively verified through detailed time-domain simulations in MATLAB/Simulink.
System Architecture and Control of the Grid-Forming Solar Inverter
The topology and control block diagram of the three-phase two-level grid-forming solar inverter under study is shown schematically below. The main power circuit consists of a DC link (representing the PV array and DC-DC converter output), a voltage source inverter (VSI), and an LCL filter for attenuating switching harmonics. The control system is hierarchically structured, featuring a VSG-based power synchronization outer loop, cascaded with a virtual admittance block and a fast inner current control loop. This architecture is specifically designed to endow the solar inverter with robust grid-forming capabilities while maintaining strict current control and limiting.
The core of the grid-forming control is the VSG algorithm, which comprises an active power-frequency loop and a reactive power-voltage loop. The active power loop replicates the rotor dynamics of a synchronous generator:
$$ \omega = \omega_0 + \frac{1}{Js + D_p} (P_{ref} – P) $$
$$ \theta = \frac{1}{s} \omega $$
where $$P_{ref}$$ and $$P$$ are the reference and measured active powers, respectively; $$\omega_0$$ is the rated angular frequency; $$J$$ is the virtual moment of inertia; $$D_p$$ is the damping coefficient; $$\omega$$ is the generated angular frequency; and $$\theta$$ is the output phase angle. This formulation provides the solar inverter with inertia and damping, crucial for frequency stability.
The reactive power-voltage control loop incorporates a droop characteristic combined with a primary voltage regulation term, enhancing its voltage support functionality:
$$ U_d^* = \frac{1}{K s} \left[ Q_{ref} – Q – D_q \left( H_{LPF1}(s) \cdot U_{s,d} – U_{t,ref} \right) \right] $$
$$ U_q^* = 0 $$
Here, $$Q_{ref}$$ and $$Q$$ are the reference and measured reactive powers; $$K$$ is an inertial gain; $$D_q$$ is the reactive droop coefficient; $$U_{t,ref}$$ is the voltage magnitude reference; $$U_{s,d}$$ is the measured d-component of the Point of Common Coupling (PCC) voltage; and $$H_{LPF1}(s)$$ is a low-pass filter applied to the voltage feedback, with a cutoff frequency $$f_{LPF1}$$. This loop determines the reference voltage magnitude $$U_d^*$$ for the inner control layers.
The voltage reference from the outer loop is processed through a virtual admittance block to generate the current reference for the inner loop. This step is vital for shaping the output impedance of the solar inverter and managing fault currents.
$$ \mathbf{i}_{dq,ref} = \frac{1}{R_v + s L_v} \left( \mathbf{U}_{dq}^* – H_{LPF2}(s) \cdot \mathbf{U}_{s,dq} \right) $$
where $$R_v$$ and $$L_v$$ are the virtual resistance and inductance, respectively; $$\mathbf{U}_{dq}^* = [U_d^*, 0]^T$$; $$\mathbf{U}_{s,dq} = [U_{s,d}, U_{s,q}]^T$$; and $$H_{LPF2}(s)$$ is another low-pass filter (cutoff frequency $$f_{LPF2}$$) on the PCC voltage feedback. This virtual admittance actively adjusts the current reference based on the voltage error, emulating the behavior of a physical RL branch connected at the inverter terminals.
Finally, a fast proportional-integral (PI) based current controller tracks the reference current $$\mathbf{i}_{dq,ref}$$. The output of this controller yields the dq-frame modulation signals, which are then transformed back to the abc-frame using the phase angle $$\theta$$ from the VSG to generate the PWM signals for the inverter switches.
$$ \mathbf{U}_{dq, ref}^{mod} = H_i(s) (\mathbf{i}_{dq,ref} – \mathbf{i}_{s,dq}) + \mathbf{U}_{s,dq} + j \omega_0 L_f \mathbf{i}_{s,dq} $$
where $$H_i(s) = K_{p,i} + K_{i,i}/s$$ is the PI controller transfer function, and $$L_f$$ is the inverter-side filter inductance. This cascaded control structure—VSG, virtual admittance, current loop—defines the dynamic personality of the grid-forming solar inverter.
Impedance Modeling of the VSG-Based Solar Inverter
Accurate impedance modeling is the cornerstone for frequency-domain stability analysis. We derive the small-signal impedance model of the solar inverter by linearizing the system equations around a steady-state operating point and expressing the relationship between perturbations in PCC voltage and current.
Small-Signal Model Derivation
The derivation follows a systematic, multi-stage process, accounting for interactions between control loops and coordinate transformations.
1. Power Control Outer Loop: Linearizing the VSG equations yields the relationship between power perturbations, angle perturbation ($$\Delta \theta$$), and voltage reference perturbation ($$\Delta \mathbf{U}_{dq}^*$$). The power perturbations are themselves functions of steady-state operating points and perturbations in current and voltage:
$$ \Delta P = \frac{3}{2} \left( \mathbf{i}_{0}^T \Delta \mathbf{u} + \mathbf{u}_{0}^T \Delta \mathbf{i} \right) $$
$$ \Delta Q = \frac{3}{2} \left( -\mathbf{i}_{0,perp}^T \Delta \mathbf{u} + \mathbf{u}_{0,perp}^T \Delta \mathbf{i} \right) $$
where subscript $$0$$ denotes steady-state values, and the “perp” operator on a vector $$[x, y]^T$$ yields $$[-y, x]^T$$. Combining these, we obtain transfer matrices linking control system variables:
$$ s\Delta \theta = \mathbf{G}_{\omega i} \Delta \mathbf{i}^c + \mathbf{G}_{\omega u} \Delta \mathbf{u}_s^c $$
$$ \Delta \mathbf{U}^{*c} = \mathbf{G}_{ui} \Delta \mathbf{i}^c + \mathbf{G}_{uu} \Delta \mathbf{u}_s^c $$
2. Virtual Admittance and Current Inner Loop: Linearizing the virtual admittance and current controller equations provides the link between the voltage reference from the outer loop and the modulation voltage generated by the inner loop.
$$ \Delta \mathbf{i}_{ref}^c = H_v(s) ( \Delta \mathbf{U}^{*c} – H_{LPF2}(s) \Delta \mathbf{u}_s^c ) $$
$$ \Delta \mathbf{U}_{ref}^c = H_i(s)(\Delta \mathbf{i}_{ref}^c – \Delta \mathbf{i}^c) + \Delta \mathbf{u}_s^c + \mathbf{H}_L \Delta \mathbf{i}^c $$
where $$H_v(s)=1/(R_v + s L_v)$$ and $$\mathbf{H}_L$$ is a matrix representing the cross-coupling terms ($$\omega_0 L_f$$). Combining these eliminates the intermediate current reference variable:
$$ \Delta \mathbf{U}_{ref}^c = \mathbf{G}_{iv} \Delta \mathbf{U}^{*c} + \mathbf{G}_{ivc} \Delta \mathbf{u}_s^c + \mathbf{G}_{Li} \Delta \mathbf{i}^c $$
3. Coordinate Transformation: A critical aspect of modeling grid-forming inverters is the distinction between the controller’s rotating reference frame (denoted by superscript $$c$$) and the electrical system’s frame (denoted by superscript $$s$$), which are separated by the power angle $$\delta_0$$. Small-signal perturbations in one frame relate to those in the other as follows:
$$ \Delta \mathbf{x}^s = \mathbf{T}_s \Delta \mathbf{x}^c + \mathbf{T}_{v,x} \Delta \delta $$
$$ \Delta \mathbf{x}^c = \mathbf{T}_s^{-1} \Delta \mathbf{x}^s + \mathbf{T}_{c,x} \Delta \delta $$
where $$\mathbf{T}_s$$ is the rotation matrix by angle $$\delta_0$$, and the $$\mathbf{T}_{v,x}, \mathbf{T}_{c,x}$$ matrices arise from the linearization of products involving steady-state values.
4. LCL Filter Circuit: The dynamics of the main power circuit are described by the LCL filter equations. Their linearized form gives the relationship between inverter output voltage $$\Delta \mathbf{U}^s$$, PCC voltage $$\Delta \mathbf{U}_s^s$$, and PCC current $$\Delta \mathbf{i}_s^s$$:
$$ \Delta \mathbf{U}_s^s = \mathbf{G}_{uvv} \Delta \mathbf{U}^s + \mathbf{G}_{uiv} \Delta \mathbf{i}_c^s $$
$$ \Delta \mathbf{i}_c^s = \mathbf{G}_i \Delta \mathbf{i}_s^s + \mathbf{G}_{vi} \Delta \mathbf{U}^s $$
where $$\Delta \mathbf{i}_c^s$$ is the inverter-side current perturbation.
5. Assembly of Full Impedance Model: By systematically combining all the linearized relationships from the control system (including coordinate transformations) and the power circuit, we can eliminate all internal state variables (like $$\Delta \mathbf{U}^s$$, $$\Delta \theta$$, $$\Delta \mathbf{i}_c^s$$) to arrive at the fundamental relationship between PCC voltage and current perturbations in the electrical system’s dq-frame:
$$ \Delta \mathbf{U}_s^s = \mathbf{Z}_{PV}^{dq}(s) \Delta \mathbf{i}_s^s $$
Here, $$\mathbf{Z}_{PV}^{dq}(s)$$ is the 2×2 MIMO dq-impedance matrix of the solar inverter. Its final analytical form, derived from the sequential substitution of all intermediate equations, is:
$$ \mathbf{Z}_{PV}^{dq}(s) = \left[ \mathbf{I} – \mathbf{D} – \frac{\mathbf{E}\mathbf{G}}{a} \right]^{-1} \left[ \mathbf{C} + \frac{\mathbf{E}\mathbf{F}}{a} \right] $$
where matrices $$\mathbf{C}, \mathbf{D}, \mathbf{E}$$ and scalars $$a, \mathbf{F}, \mathbf{G}$$ are complex aggregates of all the previously defined transfer matrices ($$\mathbf{G}_{ui}, \mathbf{G}_{iv}, \mathbf{G}_{\omega i}$$, etc.), steady-state operating points, and filter parameters.
Model Transformation and Validation
The derived MIMO dq-impedance is comprehensive but can be challenging to interpret directly for stability studies. A more convenient approach is to transform it into an equivalent SISO positive-sequence impedance $$Z_{PV}^p(s)$$. This is achieved through a two-step process: first, a linear transformation converts the dq-impedance into a modified sequence-domain impedance matrix; second, a model reduction step accounts for the interaction with the grid impedance to obtain the final SISO equivalent. The transformation is lossless in terms of stability information.
To validate the accuracy of the derived analytical impedance model, a frequency sweep is performed on a detailed time-domain simulation model of the solar inverter in MATLAB/Simulink. The system parameters used for both modeling and simulation are listed in the table below.
| Parameter Category | Symbol | Value |
|---|---|---|
| Power Circuit | DC Link Voltage | 1200 V |
| Inverter-side Inductor (L_f) | 54 μH | |
| Grid-side Inductor (L_g) | 8 μH | |
| Filter Capacitor (C_f) | 28 μF | |
| Damping Resistor (R_d) | 0.1 Ω | |
| Controller | Virtual Inertia (J) | 8.0 pu |
| Damping Coefficient (D_p) | 80 pu | |
| Inertial Gain (K) | 2.3 pu | |
| Reactive Droop (D_q) | 5 pu | |
| Virtual Resistance (R_v) | 0.1 pu | |
| Virtual Inductance (L_v) | 1e-6 pu | |
| Current PI: K_{p,i} / K_{i,i} | 0.1606 / 208.01 | |
| LPF1 Cutoff (f_{LPF1}) | 44 Hz | |
| LPF2 Cutoff (f_{LPF2}) | 89 Hz | |
| Operating Point | Active Power (P) | 330 kW |
| Reactive Power (Q) | 0 Var | |
| Grid Voltage (U_g) | 800 V (L-L) | |
| Short-Circuit Ratio (SCR) | 2.5 |
The comparison between the impedance magnitude and phase obtained from the analytical model and the simulation sweep shows excellent agreement across a wide frequency range (from sub-synchronous to several hundred Hz). This close match validates the correctness and fidelity of the derived small-signal impedance model for the VSG-based grid-forming solar inverter.
Impedance Characteristics Analysis: Influence of Control Parameters and Structures
With a validated impedance model in hand, we can systematically investigate how various control parameters and structural choices shape the output characteristics of the grid-forming solar inverter. This analysis is crucial for understanding its interactive behavior with the grid.
Impact of Power Outer Loop Parameters
The parameters of the VSG outer loop primarily govern the low-frequency behavior of the solar inverter, near the fundamental frequency (50/60 Hz). Analysis reveals that:
- Damping Coefficient ($$D_p$$): Increasing $$D_p$$ reduces the impedance magnitude slightly around the fundamental frequency and marginally increases the phase. Its effect is concentrated in a narrow band near the line frequency.
- Virtual Inertia ($$J$$): A larger $$J$$ increases the impedance magnitude in the sub-synchronous region (below fundamental frequency) and slightly reduces the capacitive (negative resistor) characteristic in that band. It has minimal impact on higher frequencies.
- Reactive Droop ($$D_q$$): This parameter exhibits an almost negligible influence on the impedance profile of the solar inverter across the studied frequency range.
- Inertial Gain ($$K$$): Increasing $$K$$ raises the impedance magnitude below the fundamental frequency and lowers it above, while the phase characteristic remains largely unaffected.
In summary, the VSG power loop parameters are significant for stability involving power synchronization modes (very low frequency) but have limited direct influence on mid-to-high frequency impedance characteristics, which are dominated by the inner loops.
Impact of Virtual Admittance Parameters
The virtual admittance block ($$R_v$$, $$L_v$$) is a powerful tool for shaping the impedance of the solar inverter, particularly in the mid-to-high frequency range. The real part of the impedance (Re{Z}) reflects the damping characteristic, which is critical for stability.
- Virtual Resistance ($$R_v$$): Increasing $$R_v$$ significantly enhances the real part of the impedance across a broad spectrum. This action effectively dampens or even eliminates inductive negative-resistance regions that often appear at high frequencies. It also slightly mitigates the capacitive negative-resistance behavior in the sub-synchronous range. A larger $$R_v$$ makes the solar inverter appear more resistive and damped to the grid.
- Virtual Inductance ($$L_v$$): Increasing $$L_v$$ increases the overall impedance magnitude. More importantly, it induces a prominent inductive negative-resistance characteristic at high frequencies. As $$L_v$$ grows, this detrimental negative-resistance region expands and shifts towards lower frequencies. In the mid-to-low frequency band, however, the resistive characteristic remains relatively unchanged.
The design of the virtual admittance is therefore a critical trade-off: $$R_v$$ enhances damping, while $$L_v$$, though useful for current limiting and voltage control, can introduce destabilizing negative damping at specific frequencies.
Impact of Low-Pass Filter Parameters
Low-pass filters are essential for attenuating measurement noise and preventing aliasing. Their cutoff frequencies subtly but importantly influence the solar inverter’s dynamics.
- Voltage Regulation LPF ($$f_{LPF1}$$): The cutoff frequency of the low-pass filter in the primary voltage regulation loop has a negligible impact on the inverter’s impedance characteristics.
- Virtual Admittance LPF ($$f_{LPF2}$$): This filter’s cutoff frequency has a pronounced effect. Decreasing $$f_{LPF2}$$ (making the filter slower) increases the impedance magnitude in the mid-frequency range, causes the negative-resistance region to shift to lower frequencies, and strengthens the capacitive characteristic in the sub-synchronous band. A faster filter (higher $$f_{LPF2}$$) generally yields a more benign impedance profile.
Impact of Control Structure
The structure of the virtual admittance itself can be modified. A comparison between the described dynamic admittance ($$1/(R_v + sL_v)$$) and a simple constant conductance ($$1/R_v$$) reveals significant differences. The constant-conductance structure lacks the inductive frequency dependence. Consequently, the solar inverter with a dynamic virtual admittance exhibits a higher resonant peak and is more prone to developing the aforementioned inductive negative-resistance特性 at high frequencies, compared to its constant-conductance counterpart. This highlights the importance of the virtual inductance term in the dynamic interaction.
Grid-Connected Stability Analysis
The ultimate goal of impedance modeling is to assess the stability of the interconnected system where the grid-forming solar inverter is connected to the grid via a network impedance. The stability can be evaluated using the Generalized Nyquist Criterion (GNC) applied to the minor loop gain or, equivalently, by examining the system loop impedance $$Z_{loop}(s) = Z_{PV}^p(s) + Z_g^p(s)$$, where $$Z_g^p(s)$$ is the positive-sequence grid impedance. Instability is predicted if the Nyquist plot of $$Z_{PV}^p(s)/Z_g^p(s)$$ encircles the (-1, j0) point, or if the real part of $$Z_{loop}(s)$$ is negative at frequencies where its imaginary part crosses zero (indicating negative damping at a resonant frequency).
Influence of Key Control Parameters on Stability
Parametric stability analysis focuses on the most influential parameters identified earlier: $$R_v$$, $$L_v$$, and $$f_{LPF2}$$, under a weak grid condition (SCR = 2.5).
- Virtual Resistance ($$R_v$$): Increasing $$R_v$$ moves the Nyquist plot away from the critical (-1, j0) point. Concurrently, it increases the real part of $$Z_{loop}(s)$$ at the critical imaginary-axis crossing frequencies. This confirms that a larger virtual resistance unambiguously enhances the stability margin of the grid-connected solar inverter system.
- Virtual Inductance ($$L_v$$): Increasing $$L_v$$ has the opposite effect. The Nyquist plot moves closer to (-1, j0), and the damping (real part of $$Z_{loop}$$) at resonance decreases. Therefore, while some $$L_v$$ is necessary for control, an excessively large value is detrimental to the small-signal stability of the solar inverter.
- LPF Cutoff Frequency ($$f_{LPF2}$$): Reducing $$f_{LPF2}$$ (slowing down the filter) draws the Nyquist contour closer to the instability point. A sufficiently low $$f_{LPF2}$$ can indeed trigger instability. Thus, a higher cutoff frequency for this filter is generally preferable for stability.
The conclusion is clear: for stable operation of this VSG-based solar inverter in weak grids, one should favor larger $$R_v$$, smaller $$L_v$$, and higher $$f_{LPF2}$$.
Influence of Grid Strength
Grid strength, characterized by the Short-Circuit Ratio (SCR), is a major external factor affecting stability. Analysis shows a non-intuitive trend: as the grid becomes stronger (higher SCR, lower grid impedance), the stability of the grid-forming solar inverter system can actually deteriorate in the low-frequency range. In strong grid scenarios, the Nyquist plot can critically approach or even encircle the (-1, j0) point at very low frequencies (e.g., around 1-2 Hz). This indicates a risk of low-frequency oscillation instability, a phenomenon distinct from the high-frequency resonances typical of weak grids. This finding underscores that grid-forming solar inverters require careful tuning not just for weak-grid operation, but also for very strong grid connections.
Time-Domain Simulation Verification
To conclusively validate the stability predictions from the impedance-based analysis, time-domain simulations are conducted in MATLAB/Simulink. The system is initialized in a stable operating state, and at a specific time instant (t=3.7s), a critical parameter is abruptly changed to a value predicted to cause instability.
- Case 1: Increasing Virtual Inductance ($$L_v$$). With SCR=2.5, increasing $$L_v$$ from 0.0001 pu to 0.0005 pu (as per the stability plot indicating instability for the latter) causes the PCC currents to develop growing oscillations, confirming system instability.
- Case 2: Decreasing LPF Cutoff Frequency ($$f_{LPF2}$$). With SCR=2.5, reducing $$f_{LPF2}$$ from 89 Hz to 4.5 Hz triggers oscillatory and divergent current waveforms, validating the destabilizing effect of a very slow voltage feedback filter in the virtual admittance path of the solar inverter.
- Case 3: Increasing Grid Strength (SCR). Changing the SCR from 2 to 10.8 while keeping control parameters constant induces low-frequency oscillations in the PCC currents. A Fast Fourier Transform (FFT) analysis reveals dominant frequency components at 48.8 Hz and 51.2 Hz in the abc-frame, which correspond to a 1.2 Hz oscillation in the dq-frame. This matches perfectly with the ~1.18 Hz instability frequency predicted by the Nyquist analysis for SCR=10.8.
These simulation results provide definitive proof for the correctness of the impedance modeling and the stability boundaries established through the frequency-domain analysis.
Conclusion
This work has presented a detailed framework for the impedance modeling and stability analysis of a practical grid-forming solar inverter based on Virtual Synchronous Generator control, incorporating virtual admittance and inner current loops for enhanced performance. The derivation of a detailed MIMO dq-impedance model and its validation against simulations establishes a reliable foundation for analysis. The parametric investigation reveals that the impedance characteristics, and hence the stability properties, of the solar inverter are predominantly shaped by the inner control layers—the virtual admittance and the associated filters—in the mid-to-high frequency range, while the VSG outer loop governs very low-frequency behavior.
The stability analysis yields critical design insights: (1) The virtual admittance is a double-edged sword; its resistive component ($$R_v$$) is a key stability enhancer, while its inductive component ($$L_v$$) must be kept relatively small to avoid inducing negative damping, especially in the mid-frequency range. (2) The low-pass filter on the voltage feedback within the virtual admittance loop should have a sufficiently high cutoff frequency to avoid degrading stability margins. (3) Contrary to traditional understanding, grid-forming solar inverters may face low-frequency oscillation risks when connected to very strong grids, necessitating tailored design and tuning for all network conditions.
The comprehensive methodology—combining rigorous impedance modeling, frequency-domain stability criteria, and time-domain validation—provides a powerful toolkit for the design, parameter tuning, and stability assessment of modern grid-forming solar inverters, which are essential components for the future resilient power system.