In our pursuit of accelerating the green and low‑carbon transition, we have set new targets for “carbon peak” and “carbon neutrality”. As the core of energy transformation, the power industry carries the critical mission of implementing these goals. Grid‑connected inverters are the essential equipment for renewable energy generation, and their diverse applications, circuit topologies, and control strategies have significantly increased the complexity of resonance analysis in multi‑inverter clusters. In recent years, several resonance events in inverter clusters have been reported worldwide. For instance, a wind farm in Guyuan, Hebei Province, experienced subsynchronous oscillations at 3–10 Hz in 2011, while a wind farm in Hami, Xinjiang, suffered frequent subsynchronous oscillations at 20–40 Hz in 2015. High‑frequency harmonic resonances have also emerged, such as a 1271 Hz harmonic resonance in the Luxi asynchronous interconnection project in Yunnan in 2017, and a 2500 Hz oscillation in a photovoltaic station in Huyanghe in 2021. These cases demonstrate that oscillation problems under inverter‑cluster grid‑connection conditions span a wide frequency band, involve complex mechanisms, and have significant engineering impacts. Once such oscillations become uncontrolled, they can cause equipment overheating, shortened lifespan, cascading faults, or even large‑scale blackouts—as highlighted by the Spanish blackout. Therefore, systematically analyzing the resonance mechanisms and developing suppression strategies based on multiple device types are crucial for ensuring reliable operation of modern power systems.
The resonance phenomenon originates from the interaction between the inherent characteristics of inverters and the equivalent grid impedance at the point of common coupling (PCC). Passive damping methods were initially used, but later active monitoring and suppression strategies have become dominant. For example, a combined feedforward and virtual admittance method enhances robustness against line and grid impedance variations. Active damping strategies based on auto‑disturbance rejection control (ADRC) and quasi‑PR control have also been proposed. However, as the power‑electronic nature of the grid becomes more prominent, the interaction among inverter clusters changes the resonance mechanism, requiring more sophisticated models and suppression strategies. In this article, we systematically review recent progress in resonance analysis and suppression for multi‑inverter grid‑connected systems, covering influencing factors, modeling methods, and suppression strategies. Our aim is to provide a reference for future research.
1. Modeling of Multi‑Inverter Grid‑Connected Systems
A typical multi‑inverter system is shown conceptually in Figure 1 (not reproduced here). The filter topologies can be L‑type, LC‑type, or LCL‑type, each with different characteristics. The resonance mechanism in multi‑inverter systems is more complex than that of a single inverter, because we must consider not only the inverter’s own resonance but also the couplings among parallel inverters. It has been shown that when the total output impedance’s inductive part cancels its capacitive part at a certain frequency, the system forms a low‑damped oscillation at that frequency. Figures 2 and 3 (not reproduced) illustrate the current and voltage waveforms of a single inverter during resonance and the FFT analysis of the grid‑connected current, respectively.
We divide the factors influencing resonance into three categories: grid impedance, nonlinear characteristics of inverters, and parameter mismatches among inverters. The number of parallel inverters also affects resonance frequencies, but we focus on the qualitative factors.
2. Factors Influencing Resonance in Multi‑Inverter Systems
2.1 Grid Impedance
In a multi‑inverter system, each inverter is connected through transmission lines and a common bus. The distributed parameters of line impedance and bus impedance create a complex coupled impedance network. This network modifies the potential resonance frequencies and mechanisms. Additionally, grid impedance affects the bandwidth of phase‑locked loops (PLLs) and the equivalent output admittance, thus impacting resonance stability.
2.2 Nonlinear Characteristics
The nonlinearities of inverters mainly arise from three aspects: the PWM modulation process of power switches (inherent and unavoidable), the time delay of digital control systems, and non‑ideal device characteristics such as dead‑time effects. Although dead‑time compensation can suppress low‑order harmonics, the harmonic distortion becomes more severe when multiple inverters operate in parallel, increasing the complexity of resonance analysis.
2.3 Parameter Mismatches Among Inverters
In modeling, assuming identical parameters simplifies the system. However, differences in capacity, filter parameters, and control parameters can cause diverse resonance phenomena. When capacities differ, coupling effects generate complex resonance modes and reduce stability. This significantly increases modeling complexity.
Table 1 summarizes the influence of these factors on resonance frequency and amplitude.
| Factor | Effect on Resonance Frequency | Effect on Resonance Amplitude |
|---|---|---|
| Grid impedance | Frequency shifts with impedance variations | Maximum when grid impedance conjugately matches inverter output impedance |
| Nonlinear characteristics | Harmonic frequencies introduced by nonlinearities couple with natural frequencies | Increased by harmonic amplification; stronger nonlinearities yield larger amplitudes |
| Parameter mismatches | Multiple potential resonance frequencies; new cross‑resonance frequencies appear | Larger mismatches cause impedance superposition at certain frequencies, leading to amplitude surge |
3. Modeling Methods and Analysis
3.1 Norton Equivalent Model
The Norton equivalent model is effective for characterizing the dynamics of multi‑inverter systems. Multiple Norton current sources and admittances are connected in parallel, and the superimposed impedance network can describe resonance mode coupling caused by parameter dispersion (e.g., filter parameter differences, control bandwidth variations). The Norton equivalent model for the whole system is an extension of the single‑inverter model. Figure 4 (not reproduced) shows the typical topology. However, when inverter parameters differ, the modeling complexity increases dramatically.
3.2 Eigenvalue‑Based Modal Analysis
Eigenvalues are key indicators for linear system stability. In multi‑inverter scenarios, traditional transfer‑function methods become computationally intensive as the number of inverters grows. Modal analysis through eigenvalue‑eigenvector decomposition can systematically incorporate control strategies, grid impedance, and parameter differences, revealing modal coupling mechanisms. The node voltage equation at frequency \(f\) is:
\[
\mathbf{U}_f = \mathbf{Y}_f^{-1} \mathbf{I}_f
\]
where \(\mathbf{U}_f\) is the node voltage vector, \(\mathbf{Y}_f\) is the node admittance matrix, and \(\mathbf{I}_f\) is the current injection vector. Decomposing \(\mathbf{Y}_f\) gives:
\[
\mathbf{Y}_f = \mathbf{L} \mathbf{\Lambda} \mathbf{T}
\]
with \(\mathbf{\Lambda}\) being the diagonal eigenvalue matrix, and \(\mathbf{L}\), \(\mathbf{T}\) the left and right eigenvectors satisfying \(\mathbf{L} = \mathbf{T}^{-1}\). Then we define modal voltage and current vectors:
\[
\mathbf{V}_f = \mathbf{T} \mathbf{U}_f, \quad \mathbf{J}_f = \mathbf{T} \mathbf{I}_f
\]
leading to:
\[
\begin{bmatrix}
V_1 \\ V_2 \\ \vdots \\ V_n
\end{bmatrix}
=
\begin{bmatrix}
\lambda_1^{-1} & 0 & \cdots & 0 \\
0 & \lambda_2^{-1} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \lambda_n^{-1}
\end{bmatrix}
\begin{bmatrix}
J_1 \\ J_2 \\ \vdots \\ J_n
\end{bmatrix}
\]
Each diagonal element \(\lambda_m^{-1}\) corresponds to a modal impedance. Modal analysis can accurately locate resonance points via eigenvalues and quantify node participation. However, traditional modal analysis only uses the magnitude of modal impedance, which may cause misjudgment. Improved methods, such as the half‑power bandwidth method, address this limitation.
3.3 Harmonic State Space (HSS) Method
The HSS method overcomes the limitations of time‑domain simulation and frequency‑domain modeling for time‑varying systems. It converts periodic time‑varying variables into DC quantities in the complex frequency domain, avoiding complex iterative calculations. For a periodic time‑varying system described by:
\[
\dot{x}(t) = A(t)x(t) + B(t)u(t), \quad y(t) = C(t)x(t) + D(t)u(t)
\]
with period \(T\), we expand each variable in Fourier series, e.g., \(x(t) = \sum_{k=-\infty}^{\infty} \hat{x}_k(t) e^{(\alpha + jk\omega_0)t}\). Using harmonic balance, we obtain the HSS model:
\[
(\alpha + jk\omega_0) \hat{x}_k = \sum_{m=-\infty}^{\infty} \hat{A}_m \hat{x}_{k-m} + \sum_{m=-\infty}^{\infty} \hat{B}_m \hat{u}_{k-m}
\]
\[
\hat{y}_k = \sum_{m=-\infty}^{\infty} \hat{C}_m \hat{x}_{k-m} + \sum_{m=-\infty}^{\infty} \hat{D}_m \hat{u}_{k-m}
\]
In practice, we truncate to a finite number of harmonics (e.g., \(k = -N\) to \(N\)). The input‑output relationship becomes:
\[
\hat{\mathbf{Y}} = \mathbf{H}(\omega) \hat{\mathbf{U}}
\]
where \(\hat{\mathbf{Y}} = [\hat{y}_{-N}, \ldots, \hat{y}_0, \ldots, \hat{y}_N]^T\) and \(\hat{\mathbf{U}}\) is similarly defined. The HSS method explicitly describes multi‑frequency coupling but is only applicable to strictly periodic systems.
3.4 Comparison of Modeling Methods
Table 2 summarizes the advantages, disadvantages, and typical applications of the three main modeling methods.
| Method | Advantages | Disadvantages | Typical Applications |
|---|---|---|---|
| Norton equivalent model | Simplifies complex networks to single‑port; computationally efficient; intuitive for multi‑harmonic source interaction | Limited to steady‑state; high‑frequency errors (e.g., LCL filter switching noise); requires known parameters | Steady‑state harmonic evaluation; compatibility analysis of multiple filters; preliminary resonance screening |
| Modal analysis | Accurate identification of resonance frequencies and dominant modes; sensitivity analysis for parameter optimization; supports global analysis of multi‑node systems | Requires complete system parameters, low robustness; cannot handle time‑varying or nonlinear systems directly; ignores control algorithms | Filter resonance frequency design; parameter optimization (L/C values); modal coupling analysis in multi‑filter systems |
| HSS method | Captures dynamic characteristics (transient harmonics, control loop effects); supports nonlinear and time‑varying systems; verifies damping stability | Complex modeling, high computation; requires precise dynamic equations (e.g., LCL damping loops); sensitive to parameters | Dynamic harmonic suppression verification; control strategy (e.g., active damping) stability analysis; interaction between nonlinear loads and filters |
4. Progress in Resonance Suppression Techniques
4.1 Passive Damping
The simplest approach is to add a series or parallel resistor in the LCL filter (six basic passive damping configurations, not shown). The characteristics differ:
- Inductor‑series resistor reduces low‑frequency gain significantly as resistance increases, with little effect on high‑frequency gain.
- Inductor‑parallel resistor weakens high‑frequency harmonic attenuation as resistance decreases; low‑frequency gain almost unchanged.
- Capacitor‑series resistor weakens high‑frequency attenuation as resistance increases; low‑frequency gain almost unchanged.
- Capacitor‑parallel resistor does not affect low‑ or high‑frequency magnitude response.
Capacitor‑parallel resistor provides the best damping effect with no impact on magnitude characteristics. Passive damping is simple and robust but causes additional power loss, especially at high frequencies.
4.2 Active Suppression Strategies
4.2.1 Active Damping Methods
Active damping avoids energy loss by constructing virtual impedance through control algorithms. For example, injecting harmonic conductance only at the resonance frequency reduces the active damper’s capacity. A virtual harmonic resistor design can suppress resonance peaks from about 35 dB to 5 dB when a 6.5 Ω virtual resistor is injected. Methods using feedback of filter capacitor current to create virtual damping have also been proposed. Under weak grid conditions, background harmonics can worsen resonance when grid voltage feedforward introduces a positive feedback loop. Combining full feedforward with an active damper can mitigate both effects. An improved weighted current control method, adding virtual admittance in parallel at the PCC, further optimizes suppression.
Notch‑filter‑based active damping is another approach with frequency selectivity. A typical notch filter transfer function is:
\[
H(s) = \frac{s^2 + \omega_n^2}{s^2 + \frac{\omega_n}{Q} s + \omega_n^2}
\]
where \(\omega_n\) is the notch angular frequency and \(Q\) is the quality factor. Using a second‑order generalized integrator (SOGI), a multistage notch filter can attenuate specific frequencies while preserving signal integrity at other frequencies. This strategy effectively suppresses resonance in weak‑grid multi‑inverter systems.
4.2.2 Impedance Reshaping
Impedance reshaping techniques can be narrowband or broadband. Notch filters and SOGI methods are narrowband and sensitive to frequency deviations. Broadband methods, such as introducing additional resonance suppression devices at the PCC, using low‑pass filter networks, or applying impedance‑model‑based active damping, can suppress resonances over a wide frequency range, improving robustness against frequency variations.
4.2.3 Active Dampers
An active damper is a separate hardware device connected in parallel at the PCC. It reshapes the system impedance without interfering with the inverter’s own control. A typical port equivalent circuit is shown in Figure 6 (not reproduced). The damper consists of a fundamental current source and an impedance composed of the original port impedance and a virtual impedance. An adaptive virtual resistance control strategy can adjust the virtual resistor in real time to suppress resonance while reducing energy loss. However, at low switching frequencies, digital delay and limited control bandwidth degrade the accuracy of virtual resistance. A compensation method has been proposed to restore ideal resistive characteristics in the target damping frequency band.
Table 3 compares typical resonance suppression methods.
| Method | Advantages | Disadvantages |
|---|---|---|
| Passive damping | Simple, high reliability, low cost | Additional power loss; fixed damping; limited for wideband |
| Active damping (state‑variable feedback) | No power loss; dynamically adjustable; fast response for low‑frequency | High sensitivity to measurement noise; complex controller design; limited high‑frequency suppression |
| Notch‑filter‑based active damping | Simple implementation; effective for fixed frequency; low computation | Sensitive to frequency drift; multiple notch filters may cause phase distortion |
| Impedance reshaping | Fundamental stability improvement; wideband applicability; robust | Requires accurate impedance model; high controller bandwidth; difficult impedance matching |
| Hybrid damping | Combines low loss and high damping; robust; reduces design pressure | More complex; coordination between components |
| Active damper | Independent from main inverter; high compatibility; robust to parameter changes | Additional hardware cost; complex coordination; high power demand for large systems |
5. Future Trends and Challenges
As renewable energy integration continues to expand, resonance issues in multi‑inverter systems will become increasingly critical. We envision the following developments:
- Multi‑factor coupling: Current research addresses single‑factor resonance; future work must tackle the dynamic coupling of time‑varying impedance, multiple nonlinearities, and heterogeneous parameters. Understanding the evolution of resonance under varying renewable generation and load conditions is essential.
- Modeling complexity: Methods like Norton, modal analysis, and HSS each have trade‑offs. Multi‑factor models become highly complex. Distributed modeling frameworks and model order reduction techniques will be needed to reduce computational burden while maintaining accuracy for real‑time applications.
- Wideband and adaptive suppression: For heterogeneous multi‑inverter systems with multiple dynamic resonance frequencies, existing strategies struggle to track and suppress simultaneously. Future solutions include wideband impedance reshaping, multistage notch filter optimization, and adaptive coordination of hybrid damping parameters.
- Artificial intelligence (AI) integration: AI is a natural direction. AI can process massive real‑time data from inverters (voltages, currents, grid frequency fluctuations, line impedances) to detect and classify resonance modes, and adaptively tune damping parameters. Some work has already applied AI to single‑inverter resonance analysis; extending this to multi‑inverter systems is a promising frontier.
The image above illustrates a modern solar inverter system. In our studies, we consider various types of solar inverter topologies, such as L‑filter, LC‑filter, and LCL‑filter types. The types of solar inverter strongly influence the resonance characteristics and the choice of suppression strategy. For example, two‑level and three‑level types of solar inverter have different output impedance profiles and control bandwidths, affecting the multi‑inverter system’s resonance behavior. Understanding the interaction among different types of solar inverter is a key step in designing robust resonance mitigation solutions. Future work should also explore hybrid configurations combining multiple types of solar inverter to achieve better overall stability.
6. Conclusion
Driven by the “dual‑carbon” goals, grid‑connected inverters are vital for renewable energy integration, but their scaling introduces significant resonance risks. In this article, we have systematically reviewed the resonance mechanisms, influencing factors, modeling methods, and suppression strategies for multi‑inverter systems.
Resonance in multi‑inverter systems arises from the interplay between inherent inverter characteristics and grid impedance, as well as couplings among parallel inverters. Key factors include grid impedance, nonlinearities (PWM modulation, digital delay, dead‑time), and parameter mismatches among inverters.
Modeling methods we analyzed include the Norton equivalent model (simple but steady‑state only), eigenvalue‑based modal analysis (precise but requires complete parameters and linearity), and the harmonic state space method (accurate for periodic time‑varying systems but computationally heavy). Each has its place depending on the analysis objectives.
For suppression, passive damping provides simplicity with energy loss; active damping, impedance reshaping, hybrid damping, and active dampers offer various trade‑offs between efficiency, complexity, and robustness. The choice must align with system requirements.
Future challenges include multi‑factor coupling, computational complexity in modeling, wideband adaptive suppression, and AI integration. By addressing these, we can ensure stable operation of future power systems with high penetration of renewable energy and various types of solar inverter.