The global push towards “Carbon Peak and Carbon Neutrality” represents a fundamental strategy for addressing climate change and facilitating a green energy transition. Within the power sector, the fundamental measure to achieve these goals is the high-proportion integration of renewable energy sources to replace conventional coal-fired generation. Photovoltaic (PV) power generation stands as a pivotal technology in this endeavor, ensuring a reliable future energy supply. As these targets advance, PV generation is poised to become a dominant mode of electricity production.
Consequently, the power system is evolving into a “double-high” system characterized by a high proportion of renewable energy and a high proportion of power electronic equipment. This paradigm shift introduces new characteristics such as low system inertia, weak stabilizing capability, poor disturbance rejection, and stochastic power output. These traits diminish the adaptive and regulatory capacity of grid-connected interfaces, like the utility interactive inverter, posing significant threats to system stability. A primary manifestation of these new stability challenges is the harmonic oscillation within the medium-to-high frequency range (e.g., 100 Hz to several kHz), directly stemming from the dynamic interactions between power electronic converters and the grid.
The widespread integration of distributed generation systems, the connection of nonlinear loads, and long transmission lines cause the grid impedance at the Point of Common Coupling (PCC) to become non-negligible and variable, leading to a “weak grid” condition. In such an environment, the real-time variation of grid impedance causes the resonant frequency of the LCL filter, commonly used in utility interactive inverters, to shift dynamically. This can trigger sustained harmonic oscillations or even instability. Furthermore, when multiple utility interactive inverters are connected in parallel, forming an inverter cluster, their interactions with each other and the grid can lead to complex resonance phenomena with multiple resonant peaks, further complicating system stability. While numerous advanced resonance suppression methods have been proposed, a systematic review categorizing them for single-inverter units and inverter clusters under weak grid conditions is still needed. This article aims to provide a comprehensive overview of these hot-spot control methods, analyze their advantages and limitations, and discuss future research directions to enhance the stability of PV generation within the “double-high” power system.
Resonance Suppression for a Single Utility Interactive Inverter in Weak Grid
For a single grid-connected unit, the primary challenge in a weak grid is the dynamic shift of the LCL filter resonance frequency due to variable grid impedance ($L_g$). Fixed-parameter control strategies designed for a strong grid (where $L_g \approx 0$) often fail to maintain stability. The core objective is to enhance the adaptive capability of the utility interactive inverter to these impedance variations. We categorize and discuss the mainstream methods below.
1. Harmonic Resonant Controllers
This is a direct approach to harmonic suppression. Instead of just a fundamental Proportional-Integral (PI) or Proportional-Resonant (PR) controller, multiple quasi-Proportional-Resonant (qPR) controllers are deployed in parallel, each tuned to a specific harmonic frequency (e.g., 5th, 7th, 11th, 13th). The transfer function for a qPR controller at harmonic order $h$ is:
$$ G_{PR,h}(s) = K_{p} + \frac{2K_{r,h}\omega_{c}s}{s^{2} + 2\omega_{c}s + (\omega_{h})^{2}} $$
where $\omega_h = h \cdot \omega_0$ is the target harmonic angular frequency, $K_{p}$ is the proportional gain, $K_{r,h}$ is the resonant gain for the $h$-th harmonic, and $\omega_c$ is the cutoff frequency defining the bandwidth.
Advantage: Capable of providing high gain at specific harmonic frequencies, leading to excellent steady-state tracking and rejection performance for pre-defined harmonics.
Disadvantage: It becomes highly impractical under weak grid conditions. As the grid impedance changes, the actual resonant and problematic harmonic frequencies shift. Implementing a controller for every possible harmonic is computationally burdensome. Furthermore, accurately detecting these time-varying harmonic frequencies in real-time for controller retuning is a significant challenge, reducing the method’s adaptability.
2. Grid Voltage Feedforward
Feedforward of the PCC voltage is a common technique to improve disturbance rejection and reduce grid current distortion caused by background voltage harmonics. The typical structure feeds the measured grid voltage ($v_g$) through a feedforward function $G_{ff}(s)$ to the modulator. Ideally, $G_{ff}(s)$ is designed as the inverse of the modulator and inverter gain.
Challenge in Weak Grid: In a weak grid, the grid impedance $L_g$ creates a positive feedback path that can destabilize the system if the feedforward path is not carefully designed. The network equation at the PCC is $v_{pcc} = v_g + sL_g i_g$. Feeding back $v_{pcc}$ directly includes a term proportional to the derivative of the output current, which can reduce phase margin.
Adaptive Approaches: Some methods propose adaptive feedforward to mitigate this. One strategy involves estimating the grid impedance online, often by injecting a perturbation or analyzing existing harmonic distortions, and then adjusting $G_{ff}(s)$ accordingly. For instance, an adaptive rule might adjust the fed-back voltage signal to be:
$$ v_{ff} = v_{pcc} – s\hat{L}_g i_g^* $$
where $\hat{L}_g$ is the estimated grid inductance and $i_g^*$ is the current reference. However, the accuracy and dynamic response of the grid impedance estimation, especially under noisy conditions with multiple harmonics, remain critical limitations.
3. Impedance Shaping (Virtual Impedance)
This method actively reshapes the output impedance of the utility interactive inverter to meet stability criteria, such as ensuring a positive real part (passivity) in a certain frequency range. It is implemented by adding a virtual impedance loop in the control software. Two primary forms exist:
- Series Virtual Impedance: A virtual impedance $Z_v(s)$ (often a virtual inductor $L_v$ or a resistor $R_v$) is added in series with the output. This is等效ally realized by subtracting $Z_v(s) \cdot i_g$ from the voltage reference. It can help to damp resonances and counteract destabilizing positive feedback loops.
- Parallel Virtual Impedance: A virtual admittance $Y_v(s)$ is added in parallel, which is等效ally similar to a grid voltage feedforward path with a specific transfer function.
The control law with series virtual impedance modifies the reference as: $v_{ref}^{‘} = v_{ref} – Z_v(s) \cdot i_g$.
Advantage: Can significantly improve system stability margins by directly influencing the inverter’s terminal characteristics.
Disadvantage: The design process becomes complex. The introduced virtual impedance alters the plant model seen by the current controller, necessitating a re-design or robust tuning of the current control loop gains to maintain performance. Finding the optimal $Z_v(s)$ that guarantees stability under all expected grid impedances is non-trivial.
4. Improved Capacitor-Current-Feedback Active Damping
Capacitor Current Feedback (CCF) is a widely adopted active damping method. Feeding back the filter capacitor current ($i_c$) through a gain $H_d$ is equivalent to placing a virtual resistor ($R_d = H_d / K_{PWM}$) in parallel with the filter capacitor, providing damping at the resonant frequency. The basic principle is shown by modifying the control signal: $d = G_c(s)(i_{ref} – i_g) – H_d \cdot i_c$.
The resonant frequency of an LCL filter is $\omega_{res} = \sqrt{(L_1 + L_2 + L_g) / (L_1 L_2 C)}$, where $L_1$ and $L_2$ are inverter-side and grid-side inductors, and $L_g$ is the grid impedance. In a weak grid, $L_g$ varies, causing $\omega_{res}$ to shift. A fixed $H_d$ optimized for one resonant frequency may be ineffective or even destabilizing at another.
Adaptive CCF Methods: Research has focused on making CCF adaptive. One approach is to online estimate the resonant frequency (e.g., by monitoring specific harmonic amplifications or using frequency scanning techniques) and then adjust $H_d$ according to a pre-designed tuning rule to maintain optimal damping. Another sophisticated approach uses Model Reference Adaptive Control (MRAC) to adjust state feedback gains, including the damping feedback, to ensure stability. However, these methods increase system complexity, require additional sensors or estimation algorithms, and their performance depends on the accuracy and speed of the resonant frequency or grid impedance identification.
The table below summarizes the key characteristics of these single-inverter resonance suppression methods.
| Method | Core Principle | Key Advantages | Major Challenges in Weak Grid |
|---|---|---|---|
| Harmonic Resonant Control | High-gain control at specific harmonic frequencies. | Excellent steady-state harmonic suppression for known frequencies. | Requires detection of time-varying harmonics; computationally heavy; poor adaptability. |
| Grid Voltage Feedforward | Pre-cancels grid voltage disturbance. | Simple, effective, easy parameter selection in strong grid. | Can cause instability; requires accurate, adaptive grid impedance estimation for robustness. |
| Impedance Shaping | Actively modifies inverter output impedance. | Can significantly improve system stability margins. | Complicates overall controller design; parameters are hard to optimize for wide impedance range. |
| Adaptive CCF Active Damping | Provides virtual damping at the resonant frequency. | Good dynamic performance; no physical power loss. | Requires real-time identification of resonant frequency or grid impedance; performance hinges on estimation accuracy. |
A critical observation is that most adaptive methods rely on some form of online grid impedance estimation. However, a frequently overlooked aspect is the tolerance and potential variation of the filter inductors ($L_1$, $L_2$) themselves due to manufacturing, temperature, and aging. Future research should investigate joint parameter estimation techniques that can identify both the grid impedance and the actual filter parameters concurrently, leading to truly robust and self-adaptive control for the utility interactive inverter.
Resonance Analysis and Suppression for Clusters of Utility Interactive Inverters
Even if individual inverters are stabilized, connecting multiple utility interactive inverters in parallel to a common PCC introduces complex interactions. The equivalent grid impedance seen by any single inverter is no longer just $Z_g(s)$ but a function of $Z_g(s)$ and the equivalent impedances of all other parallel inverters. This can lead to multiple resonant peaks and complex stability issues not present in single-inverter systems. Accurate modeling is the first step to understanding these phenomena.
1. Modeling and Stability Analysis for Inverter Clusters
Several modeling approaches have been developed to analyze multi-inverter systems.
a) Multiple-Input-Multiple-Output (MIMO) State-Space Model:
This is a general approach where each inverter is represented by its state-space model. The entire system is described by a global state matrix $\mathbf{A}_{sys}$. The stability is assessed by examining the eigenvalues of $\mathbf{A}_{sys}$. A resonance mode is excited if its corresponding eigenvalue has a small or negative damping ratio.
$$ \dot{\mathbf{x}}_{sys} = \mathbf{A}_{sys} \mathbf{x}_{sys} + \mathbf{B}_{sys} \mathbf{u} $$
$$ \mathbf{y}_{sys} = \mathbf{C}_{sys} \mathbf{x}_{sys} $$
Advantage: Very accurate and can handle inverters with different parameters and topologies.
Disadvantage: The model order increases dramatically with the number of inverters, making analysis complex. It provides a global stability view but less direct insight into individual inverter resonances or the frequency-domain interaction.
b) Norton Equivalent Circuit with Admittance Model:
This is a widely used frequency-domain approach. Each closed-loop controlled utility interactive inverter is represented by its Norton equivalent: an ideal current source $I_{o,k}(s)$ in parallel with its output admittance $Y_{o,k}(s)$. The grid is a voltage source $V_g(s)$ in series with grid impedance $Z_g(s)$. The cluster of $N$ inverters and the grid are connected at the PCC.
For a system with $N$ identical inverters ($Y_o = Y_{o,1}=…=Y_{o,N}$), the total admittance at the PCC is $Y_{total}(s) = N Y_o(s) + 1/Z_g(s)$. The grid current of a single inverter can be derived using current division:
$$ I_g(s) = \frac{I_o(s) – Y_o(s) V_g(s)}{1 + N Y_o(s) Z_g(s)} $$
The term $1 + N Y_o(s) Z_g(s) = 0$ is the characteristic equation whose roots indicate potential instability. Crucially, the term $N Y_o(s) Z_g(s)$ shows that the effective loop gain is scaled by $N$, meaning an inverter cluster is more prone to instability than a single unit. For non-identical inverters, the analysis involves the sum of individual admittances $\sum_{k=1}^{N} Y_{o,k}(s)$.
Advantage: Intuitive frequency-domain interpretation; clearly shows the impedance interaction mechanism.
Disadvantage: Deriving an accurate $Y_o(s)$ for complex controllers can be involved. The analysis for non-identical inverters is more cumbersome.
c) Modal Analysis based on Network Admittance Matrix:
This method treats the entire system as an electrical network. All inverters and the grid are represented by their admittances connected at various nodes (PCC being the key node). The network equation is $\mathbf{I}(s) = \mathbf{Y}_{bus}(s) \mathbf{V}(s)$, where $\mathbf{Y}_{bus}$ is the bus admittance matrix. Resonance modes are associated with eigenvalues of the admittance matrix. A resonance near frequency $\omega$ occurs when an eigenvalue $\lambda_i(j\omega) \approx 0$, implying a high impedance mode.
Advantage: Well-established from power systems; good for identifying critical resonance frequencies and participating inverters.
Disadvantage: Primarily a system-level analysis; it does not easily reveal the stability margin or resonance characteristics of a specific utility interactive inverter within the cluster.
d) Quantitative Interaction Indices:
Some researchers propose indices to quantify the interaction strength or the contribution of each inverter to system instability. For example, a “participation factor” or “interaction admittance” can be defined. These metrics help identify the most critical units in a cluster that should be retuned or damped.
Advantage: Provides a metric for system planning and vulnerability assessment.
Disadvantage: Often based on simplified models and may not capture all dynamic interactions under weak grid conditions with harmonic distortions.
The table below compares these modeling approaches.
| Modeling Method | Basis | Primary Advantage | Primary Limitation |
|---|---|---|---|
| MIMO State-Space | State-space equations of interconnected systems. | Accurate for heterogeneous systems; detailed transient analysis. | High model order; complex eigenvalue analysis for large N. |
| Norton Equivalent & Admittance Model | Frequency-domain equivalent circuit. | Intuitive; clearly shows the $N$-scaling effect and impedance-based stability criterion. | Deriving $Y_o(s)$ can be complex; analysis for non-identical units is less straightforward. |
| Modal/Admittance Matrix Analysis | Network theory and matrix eigenvalues. | Excellent for identifying system-wide resonance modes and participation. | Does not directly describe single-inverter closed-loop behavior. |
| Quantitative Indices | Derived metrics from models. | Useful for ranking and planning. | May lack comprehensiveness for dynamic stability assessment. |
2. Resonance Suppression Methods for Inverter Clusters
Suppressing resonances in a cluster requires strategies that address the collective behavior. The methods can be implemented locally on each inverter or via a centralized device.
a) Active Damping Applied to All Inverters:
This involves implementing damping strategies (like the adaptive CCF or virtual impedance discussed earlier) on every utility interactive inverter in the cluster. The goal is to ensure each unit’s output admittance $Y_o(s)$ is well-damped, which in turn makes the total parallel admittance $N Y_o(s)$ stable. Advanced techniques combine active damping with phase-lead compensation or notch filters tuned to the problematic resonant frequencies. The notch filter, with a transfer function like $G_{notch}(s) = (s^2 + \omega_z^2)/(s^2 + 2\zeta\omega_n s + \omega_n^2)$, can be placed in the current feedback or reference path to attenuate signals at the resonance frequency $\omega_n$.
Challenge: Requires precise detection of the cluster’s resonant frequencies, which may be multiple and time-varying. Coordinating the tuning of parameters across many inverters is difficult.
b) Global Impedance Shaping:
Rather than just stabilizing individual units, this approach aims to shape the aggregate impedance of the cluster or the impedance at the PCC. One concept is the “Active Harmonic Conductance” method, where inverters are controlled to exhibit a small positive conductance at harmonic frequencies, effectively absorbing harmonic energy and preventing it from circulating. Another method proposes adding a “virtual negative admittance” on a designated master inverter to cancel out a destabilizing positive admittance from the rest of the cluster. The control law for such a unit might include: $I_{ref,harmonic} = G_h \cdot V_{pcc,harmonic}$, where $G_h$ is a positive conductance gain at harmonic frequencies.
Challenge: These methods often require communication or sophisticated decentralized algorithms to estimate the global harmonic voltage or cluster impedance. Their dynamic performance under rapidly changing conditions needs careful evaluation.
c) Dedicated Active Damper:
This is a hardware-based solution. A separate power electronic device, the Active Damper, is connected at the PCC. Its sole purpose is to stabilize the network by injecting damping currents or modulating its own impedance. It acts as a “harmonic sink.” For instance, it can be controlled to present a resistive impedance ($R_d$) at high frequencies: $I_{damper} = V_{pcc} / R_d$ in the frequency range of concern. A related concept is using a high-bandwidth grid-forming inverter in parallel to provide active damping.
Advantage: Independent solution; does not require modification of existing utility interactive inverters; can be optimized purely for stability.
Disadvantage: Adds cost, hardware, and control complexity to the overall system.
| Method Category | Implementation | Key Advantage | Key Challenge |
|---|---|---|---|
| Distributed Active Damping | Damping control (e.g., adaptive CCF, notch filter) on each inverter. | Good resonance suppression if well-tuned; utilizes existing hardware. | Requires accurate, possibly coordinated, frequency detection and parameter tuning. |
| Global Impedance Shaping | Control algorithms to shape collective cluster admittance (e.g., harmonic conductance). | Addresses the root cause of cluster interaction; can be very effective. | Often needs system-wide information; control design is complex. |
| Dedicated Active Damper | Additional stabilizing power converter at PCC. | Plug-and-play solution; non-intrusive to existing inverters. | Increases system cost and hardware footprint. |
A persistent challenge in cluster resonance suppression is dealing with multiple, time-varying resonant peaks, especially when inverters are not synchronized or have different control parameters. A unified damping strategy must be robust across these frequencies, which often leads to conservative and potentially performance-limiting controller designs. Future work needs to develop cooperative or hierarchical control strategies where inverters can share minimal information to collectively and adaptively damp multiple resonance modes.
Conclusions and Future Directions
The advancement towards “Carbon Peak and Carbon Neutrality” will undoubtedly accelerate the deployment of PV generation systems. The utility interactive inverter, as the core grid interface, plays a critical role in determining the performance and stability of these systems integrated into the evolving “double-high” power grid. This review has summarized and analyzed the primary resonance suppression methods for both single inverters and inverter clusters operating under weak grid conditions. Key challenges remain in achieving high efficiency, modularity, low grid current distortion, and strong adaptive capability. Based on our analysis, we identify the following crucial directions for future research:
- Joint Online Estimation of Grid and Plant Parameters: Most adaptive methods depend on accurate grid impedance ($L_g$) estimation. However, the inherent tolerance and variation of filter components ($L_1$, $L_2$, $C$) are often neglected. Future control schemes should integrate robust online parameter estimation techniques (e.g., extended state observers, recursive least squares, or AI-based estimators) that can simultaneously identify grid impedance and actual filter parameters. This will form the foundation for truly self-adaptive and resilient control of the utility interactive inverter.
- Stability in the Framework of New-Type Power Systems: As renewable penetration reaches very high levels, the classic stability definitions and analysis tools need expansion. Research must delve into the new stability problems defined for “double-high” systems, particularly focusing on how a large population of utility interactive inverters affects and can be controlled to enhance system-wide stability in the medium/high-frequency range. Furthermore, with the proliferation of power electronic converters, new protection principles and logics, potentially leveraging the controllability of the utility interactive inverter itself, must be developed for the new-type power system.
- Advanced Analysis and Design for Inverter Clusters: The resonant characteristics of clusters, such as the effective impedance scaling by factor $N$ for identical units and the emergence of multiple resonant peaks for non-identical or non-synchronized units, are not yet fully incorporated into standard inverter design procedures. Future work should focus on:
- Developing clear analytical frameworks to predict multi-resonant peak phenomena.
- Formulating simplified, yet robust, parameter design guidelines for inverter controllers that inherently account for their operation within a cluster under weak grid conditions.
- Designing distributed or decentralized cooperative control strategies that enable a cluster of utility interactive inverters to autonomously and collectively damp resonances without extensive communication, enhancing overall system stability and scalability.
Addressing these challenges will be essential to ensure the reliable, stable, and high-quality integration of massive PV generation, ultimately supporting the successful realization of the “dual-carbon” goals.