In modern power systems, the proliferation of renewable energy sources has made the on-grid inverter a ubiquitous and critical interface. As the primary link connecting photovoltaic arrays, wind turbines, and energy storage systems to the AC grid, its dynamic behavior fundamentally shapes system stability. A pressing challenge emerges from the complex interactions between these inverters and the grid or among multiple inverters themselves. Due to the multi-time-scale dynamics inherent in digitally controlled inverters and various coupling mechanisms, the system can exhibit oscillatory instabilities across a remarkably wide frequency spectrum—from sub-synchronous ranges to several kilohertz. Furthermore, phenomena like frequency coupling can lead to simultaneous oscillations at multiple, interrelated frequencies, posing a significant threat to system security. This article delves into this complex stability problem, focusing on establishing a comprehensive theoretical framework.
The core of analyzing such wideband interactions lies in developing an accurate impedance model for the on-grid inverter. Traditional modeling approaches often focus on specific frequency ranges or neglect certain coupling effects, limiting their predictive capability for the complex, multi-frequency oscillations observed in practice. For instance, models may account for low-frequency coupling due to asymmetric dq-axis control but ignore the significant effects of digital modulation at higher frequencies, or vice-versa. A unified model capturing the inverter’s behavior across the entire frequency band of interest, while simultaneously incorporating the dominant multi-frequency coupling effects, is therefore essential. This work addresses this gap by employing a multi-harmonic linearization method to derive a holistic admittance matrix model.
The system under study is a typical three-phase, two-level voltage source on-grid inverter with an LCL filter. Its control structure includes a Phase-Locked Loop (PLL) for grid synchronization, dq-axis current controllers, and a digital Sinusoidal Pulse Width Modulation (SPWM) stage. The primary sources of frequency coupling considered are: 1) Asymmetry in the dq-axis control loops, which couples frequencies symmetric around the fundamental grid frequency $$f_1$$. If a perturbation is injected at a frequency $$f_p$$, this mechanism induces a coupled component at $$f_{ps} = 2f_1 – f_p$$. 2) The digital SPWM process, which, due to sampling and hold effects, generates significant sideband harmonics. For a positive-sequence perturbation at $$f_p$$, a dominant sideband appears at $$f_{pc} = f_c – f_p$$, where $$f_c$$ is the carrier (switching) frequency. For a negative-sequence perturbation, the coupling is at $$f_{pc} = f_c + f_p$$. To capture these interactions, the modeling approach considers three simultaneous voltage perturbations at frequencies $$f_p$$, $$f_{ps}$$, and $$f_{pc}$$ at the Point of Common Coupling (PCC).
The derivation of the unified impedance model segments the system into three parts, each modeled using linearized small-signal perturbation analysis. The relationships between voltage/current perturbations and modulation signals are established in the phase (abc) domain for clarity before being transformed for system-level analysis.
1. Control System Modeling (Low-Frequency Coupling)
The control system, including the PLL, coordinate transformations, and current controllers, is modeled first. The small-signal modulation signals in the phase domain, $$\Delta \mathbf{M}^a_{f_k}$$, are derived as functions of the PCC voltage and inverter output current perturbations. The key equations take the following form, demonstrating the coupling:
$$
\begin{cases}
\Delta M^a_{f_p} = H^{f_p}_1 \Delta U^p_{f_p} + H^{f_p}_2 \Delta I^p_{f_p} + H^{f_p}_3 \Delta U^{p*}_{f_{ps}} \\
\Delta M^a_{f_{ps}} = H^{f_{ps}}_1 \Delta U^p_{f_{ps}} + H^{f_{ps}}_2 \Delta I^p_{f_{ps}} + H^{f_{ps}}_3 \Delta U^{p*}_{f_p} \\
\Delta M^a_{f_{pc}} = H^{f_{pc}}_1 \Delta U^p_{f_{pc}} + H^{f_{pc}}_2 \Delta I^p_{f_{pc}}
\end{cases}
$$
Here, $$H_1, H_2, H_3$$ are complex transfer functions whose expressions depend on the specific frequency and system parameters (PLL bandwidth, PI controller gains, etc.). The conjugate terms (e.g., $$\Delta U^{p*}_{f_{ps}}$$) explicitly represent the coupling introduced by the dq-axis asymmetry. Note that the component at $$f_{pc}$$, being primarily a high-frequency effect from the modulator’s perspective, is assumed not to generate a further low-frequency coupled component via this mechanism in this formulation.
2. Digital Modulator Modeling (High-Frequency Coupling)
The digital SPWM modulator, based on a symmetrical regular-sampling scheme, is modeled next. Using a triple Fourier series approach, the relationship between the small-signal switching function $$\Delta \mathbf{S}^a_{f_k}$$ and the modulation signal $$\Delta \mathbf{M}^a_{f_k}$$ is established. This model captures the generation of sideband harmonics:
$$
\begin{cases}
\Delta S^a_{f_p} = H^{f_p, f_p}_{sf} \Delta M^a_{f_p} + H^{f_p, f_{pc}}_{cf} \Delta M^a_{f_{pc}} \\
\Delta S^a_{f_{ps}} = H^{f_{ps}, f_{ps}}_{sf} \Delta M^a_{f_{ps}} \\
\Delta S^a_{f_{pc}} = H^{f_{pc}, f_{pc}}_{sf} \Delta M^a_{f_{pc}} + H^{f_{pc}, f_p}_{cf} \Delta M^a_{f_p}
\end{cases}
$$
The transfer functions $$H_{sf}$$ and $$H_{cf}$$ describe the direct and coupling paths through the modulator. For instance, $$H^{f_p, f_{pc}}_{cf}$$ models how the modulation signal at the sideband frequency $$f_{pc}$$ influences the switching function at the primary perturbation frequency $$f_p$$. The coupling from the $$f_{ps}$$ component is considered negligible in the modulator model.
3. Main Circuit Modeling
The LCL filter and the DC-link (modeled as an ideal voltage source) form the main power circuit. Its dynamics are linear and do not introduce frequency coupling. The relationship between the switching function, PCC voltage, and output current for each frequency component can be expressed as:
$$
U_{dc} G^{f_k}_1 \Delta S^a_{f_k} + G^{f_k}_2 \Delta U^p_{f_k} = G^{f_k}_3 \Delta I^p_{f_k} \quad \text{for } f_k \in \{f_p, f_{ps}, f_{pc}\}
$$
where $$G_1, G_2, G_3$$ are transfer functions derived from the LCL filter impedances at the respective frequencies.
4. Unified Admittance Matrix Model
By combining the models from the three subsystems and solving the system of equations, the final input admittance matrix of the on-grid inverter is obtained. It relates the vector of current response perturbations to the vector of voltage perturbations at the PCC:
$$
\begin{bmatrix}
\Delta I^p_{f_p} \\
\Delta I^{p*}_{f_{ps}} \\
\Delta I^p_{f_{pc}}
\end{bmatrix} = \mathbf{Y}_{inv} \begin{bmatrix}
\Delta U^p_{f_p} \\
\Delta U^{p*}_{f_{ps}} \\
\Delta U^p_{f_{pc}}
\end{bmatrix}
$$
where the admittance matrix $$\mathbf{Y}_{inv}$$ is a 3×3 complex matrix:
$$
\mathbf{Y}_{inv} = \begin{bmatrix}
Y_{11}(f_p) & Y_{12}(f_p, f_{ps}) & Y_{13}(f_p, f_{pc}) \\
Y_{21}(f_{ps}, f_p) & Y_{22}(f_{ps}) & Y_{23}(f_{ps}, f_{pc}) \\
Y_{31}(f_{pc}, f_p) & Y_{32}(f_{pc}, f_{ps}) & Y_{33}(f_{pc})
\end{bmatrix}
$$
The diagonal elements $$Y_{kk}$$ represent the direct, same-frequency admittance. The off-diagonal elements capture the coupling: $$Y_{12}$$ and $$Y_{21}$$ model the low-frequency dq-axis coupling, while $$Y_{13}$$ and $$Y_{31}$$ model the high-frequency modulation coupling. Elements like $$Y_{23}$$ and $$Y_{32}$$ represent higher-order cross-coupling between these two mechanisms and are typically negligible. The explicit, full analytical expressions for each element are derived from the combination of the $$H$$ and $$G$$ transfer functions.
To validate the accuracy of this unified model, frequency-scanning simulations were performed on a detailed time-domain model in MATLAB/Simulink. The results confirmed that the theoretical model accurately predicts both the magnitude and phase of the major diagonal and off-diagonal admittance terms across a wide frequency band from tens of Hz to near the switching frequency.
5. Wideband Stability Analysis Using the Generalized Nyquist Criterion
With the inverter modeled by the 3×3 admittance matrix $$\mathbf{Y}_{inv}(s)$$ and the grid represented by a corresponding diagonal impedance matrix $$\mathbf{Z}_g(s) = \text{diag}[Z_g(s_{f_p}), Z_g(s_{f_{ps}}), Z_g(s_{f_{pc}})]$$, the system stability is assessed using the Generalized Nyquist Stability Criterion (GNSC). The loop gain matrix is $$\mathbf{L}(s) = \mathbf{Y}_{inv}(s) \mathbf{Z}_g(s)$$. The system is stable if and only if the net number of counterclockwise encirclements of the critical point (-1, j0) by the set of eigenvalues of $$\mathbf{L}(j\omega)$$, as $$\omega$$ sweeps from 0 to ∞, equals the number of right-half-plane poles of $$\mathbf{L}(s)$$.
This analysis was applied to two key scenarios:
5.1 Single On-Grid Inverter System
Under weak grid conditions (high grid inductance) and with substantial digital control delay, the Nyquist plots revealed eigenvalue loci that encircled the (-1, j0) point. Crucially, these encirclements originated from trajectories associated with both low-frequency and mid/high-frequency dynamics. The analysis predicted simultaneous oscillation risks near 112/212 Hz (low-frequency coupling) and 2450/3550 Hz (delay and modulator coupling). Time-domain simulations confirmed this prediction, showing a grid current that, upon a small disturbance, developed sustained oscillations containing spectral components at both frequency clusters. This validates the model’s ability to identify complex, multi-frequency instability that simpler models would miss.
5.2 Two Parallel On-Grid Inverter System
The analysis was extended to a system with two parallel inverters connected to a weak grid. An additional practical factor was introduced: asynchronous carrier waves (one inverter’s carrier at 0° initial phase, the other at 180°). The GNSC analysis of the combined system model indicated instability with predicted oscillatory modes around 95/195 Hz, 2630 Hz, and 2490/3510 Hz. The time-domain simulation results showed a total grid current rich in oscillatory content, including low-frequency components (~75/175 Hz), a mid-frequency component (~2450 Hz), and a high-frequency circulating current between the inverters (~3550 Hz) due to the asynchronous carriers. The correlation between prediction and simulation confirms the model’s applicability to multi-inverter scenarios.
6. Impact of System Parameters on Wideband Oscillations
A significant finding from this unified modeling approach is that key system parameters can have opposing effects on stability in different frequency bands. The grid voltage feedforward coefficient $$K_f$$ serves as a prime example. The table below summarizes its distinct impact:
| Parameter | Effect on Low-Fband (e.g., ~100-500 Hz) Stability | Effect on Mid/High-Fband (e.g., ~1-4 kHz) Stability | Design Implication |
|---|---|---|---|
| Grid Voltage Feedforward Gain ($$K_f$$) | Increasing $$K_f$$ improves stability. A lower $$K_f$$ increases oscillation risk. | Increasing $$K_f$$ reduces stability. A higher $$K_f$$ increases oscillation risk. | Trade-off required. For very weak grids, prioritize high $$K_f$$. For systems with low switching frequency/long delay, consider lower $$K_f$$. |
| PLL Bandwidth | Higher bandwidth generally reduces stability in weak grids. | Minimal direct effect, but can influence coupling terms. | Should be tuned conservatively for weak-grid operation. |
| Current Controller Bandwidth | Affects phase margin in lower frequency range. | Directly impacts the phase lag at higher frequencies, influencing resonance risk. | Optimal design must consider the entire frequency range of interaction. |
| Grid Strength (Short-Circuit Ratio) | Weaker grid (lower SCR) drastically increases oscillation risk. | Weaker grid can also trigger higher-frequency resonances depending on grid impedance profile. | A primary factor; system must be stable across the expected range of grid impedance. |
This dichotomy underscores the necessity of a wideband perspective in parameter design. Optimizing a parameter to suppress one type of oscillation might inadvertently exacerbate another. The unified model enables a co-optimization process that considers the system’s stability margin across the entire relevant frequency spectrum.
7. Experimental Validation and Model Comparison
The superiority of the proposed unified model was demonstrated through comparative analysis and hardware-in-the-loop (HIL) experimentation. Four models were compared:
- Proposed Unified Model: Includes both low-frequency (dq) and high-frequency (modulator) coupling.
- Low-F Coupling Only Model: Includes dq-axis coupling but neglects modulator dynamics and detailed delay.
- High-F Coupling Only Model: Includes modulator coupling and delay but neglects dq-axis coupling.
- Basic dq Model: SISO impedance per dq-axis, neglecting all frequency coupling.
Under a specific set of weak-grid parameters (grid inductance = 5 mH), the stability predictions differed markedly:
| Model | Predicted Stability | Predicted Oscillation Frequencies |
|---|---|---|
| Unified Model (Proposed) | Unstable | ~430/530 Hz, ~2480 Hz, ~2670/3330 Hz |
| Low-F Coupling Only | Stable | None |
| High-F Coupling Only | Unstable | Only ~3350 Hz (misses low-frequency risk) |
| Basic dq Model | Stable | None |
A time-domain simulation with the detailed switch model confirmed the prediction of the unified model, showing growing oscillations with components near 400/500 Hz and 2500 Hz. Finally, a hardware experiment was conducted on a lab-scale on-grid inverter prototype with a programmable grid emulator. When the grid inductance was increased to the critical value, a small perturbation triggered an oscillation. The measured grid current waveform and its FFT spectrum clearly revealed the simultaneous presence of low-frequency oscillatory components (around 350/450 Hz) and a high-frequency component (around 2300 Hz), aligning well with the predictions of the unified model.
8. Conclusion
This work establishes a comprehensive framework for analyzing wideband stability in systems dominated by on-grid inverters. The key contributions are:
- Unified Impedance Modeling: A multi-harmonic linearization method was successfully employed to develop a 3×3 admittance matrix model for a three-phase on-grid inverter. This model uniquely integrates the effects of both low-frequency (dq-axis control asymmetry) and high-frequency (digital modulator) coupling mechanisms, providing an accurate representation of the inverter’s terminal behavior across a wide frequency band up to the switching frequency.
- Analysis of Complex Oscillations: Using the Generalized Nyquist Stability Criterion, the analysis demonstrated that inverter-based systems are susceptible to complex instability scenarios where oscillations occur simultaneously at multiple, coupled frequencies. This was validated for both single-inverter and multi-inverter grid connections through detailed time-domain simulations and hardware experiments.
- Parameter Impact Insight: The analysis revealed that common control parameters, such as the grid voltage feedforward coefficient, can have opposing effects on stability in different frequency bands. This highlights the critical need for a wideband perspective during the controller design and tuning process for on-grid inverters, moving beyond traditional single-frequency-range optimization.
- Validation and Superiority: The proposed model was shown to be significantly more accurate than conventional simplified impedance models. Its predictions of multi-frequency instability were confirmed both in simulation and on a physical experimental platform, proving its practical utility for diagnosing and preventing wideband oscillations in modern power systems.
The presented modeling approach and stability analysis methodology provide a essential tool for engineers and researchers tasked with ensuring the reliable integration of high penetrations of inverter-based resources into the power grid. Future work can focus on extending this framework to other inverter topologies (e.g., modular multilevel converters), incorporating nonlinear saturation effects, and developing wideband damping control strategies based on the insights gained from this unified model.