The widespread integration of distributed renewable energy sources, particularly photovoltaic (PV) systems, has brought grid-connected inverters to the forefront of modern power systems. As a critical interface between the PV arrays and the utility grid, the performance and stability of the solar inverter are paramount. To enhance maximum power point tracking (MPPT) efficiency and system reliability, string-type solar inverter topologies have become increasingly prevalent in clustered PV systems. However, a significant challenge arises when these clustered inverters are connected to the grid via long transmission lines, where the line impedance is non-negligible, resulting in a weak grid condition.
In such weak grids, the interaction between the output impedance of multiple parallel solar inverter systems and the grid impedance can lead to severe resonance phenomena. This resonance destabilizes the system, causing significant harmonic distortion in the grid current, potential equipment damage, and ultimately, a failure to maintain stable grid connection. This paper delves into this critical issue and proposes an advanced active damping strategy based on an improved adaptive notch filter to effectively suppress resonance in clusters of string-type solar inverters.
System Configuration and LCL Filter Resonance Analysis
A typical structure for a clustered string-type solar inverter system is shown conceptually. Each string of PV panels is connected to a dedicated two-stage power converter (a DC-DC boost stage and a DC-AC inverter stage) and an LCL filter before being paralleled at the Point of Common Coupling (PCC). The weak grid is modeled as a Thevenin equivalent with a grid voltage source in series with a predominantly inductive impedance, $L_g$, assuming its resistive component provides natural damping. The LCL filter, comprising inverter-side inductance $L_1$, grid-side inductance $L_2$, and filter capacitance $C$, is favored for its superior high-frequency harmonic attenuation compared to simple L filters. However, it introduces a resonant peak that can excite system instability.
The open-loop transfer function from the inverter output voltage $u_{inv}(s)$ to the grid current $i_g(s)$ for a single solar inverter unit, ignoring parasitic resistances, is given by:
$$G_{LCL}(s) = \frac{i_g(s)}{u_{inv}(s)} = \frac{1}{s L_1 (L_2 + L_g) C} \cdot \frac{1}{s^2 + \omega_{res}^2}$$
where the resonant angular frequency $\omega_{res}$ is:
$$\omega_{res} = \sqrt{\frac{L_1 + L_2 + L_g}{L_1 (L_2 + L_g) C}}$$
For a system with $n$ identical solar inverters operating in parallel, the equivalent resonant frequency $f_{res_n} = \omega_{res_n} / 2\pi$ becomes:
$$f_{res_n} = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2 + n L_g}{L_1 (L_2 + n L_g) C}}$$
This relationship highlights a key challenge: the system’s resonant frequency is not fixed. It depends dynamically on both the number of active solar inverters ($n$) and the grid impedance ($L_g$). As $n$ increases or $L_g$ becomes larger (a weaker grid), $f_{res}$ decreases, moving the problematic resonance point within the controller’s bandwidth and threatening stability. A fixed-parameter damping strategy is therefore insufficient.
Limitations of Conventional Resonance Suppression Methods
Traditional active damping methods often employ a notch filter in the current control loop to attenuate the gain at the known LCL resonant frequency. A standard second-order notch filter has the transfer function:
$$G_{N}(s) = \frac{s^2 + \omega_a^2}{s^2 + \frac{\omega_a}{Q}s + \omega_a^2}$$
where $\omega_a = 2\pi f_a$ is the notch center frequency and $Q$ is the quality factor determining the bandwidth. When combined with a conventional Proportional-Integral (PI) current controller $G_c(s) = k_p + k_i/s$, the system open-loop transfer function becomes:
$$G_{ol, conv}(s) = \frac{(k_p s + k_i) G_{N}(s)}{s^4 L_1 (L_2 + L_g) C + s^2 (L_1 + L_2 + L_g)}$$
While effective for a specific, known $L_g$, this approach fails when grid conditions change. The Bode plot analysis reveals that as $L_g$ varies, the actual resonant frequency $f_{res}$ shifts away from the fixed notch frequency $f_a$. The notch filter no longer provides attenuation at the correct frequency, the resonant peak reappears in the loop gain, phase margins degrade, and the system becomes prone to oscillation. This underscores the necessity for an adaptive control strategy that can track the changing resonant frequency in real-time for a reliable solar inverter cluster operation.
Proposed Adaptive Notch-Based Control Strategy
To overcome the limitations of fixed-parameter methods, a comprehensive adaptive control strategy is proposed, featuring three core innovations: an enhanced tunable notch filter, a Quasi-Proportional-Resonant (QPR) current controller, and a real-time grid impedance observation module.
1. Enhanced Tunable Notch Filter
The conventional notch filter is redesigned into a more flexible biquad form with independent adjustment coefficients $m_1$ and $m_2$:
$$G_{n}(s) = \frac{ \left( \frac{s}{2\pi f_a} \right)^2 + 2m_1 \left( \frac{s}{2\pi f_a} \right) + 1 }{ \left( \frac{s}{2\pi f_a} \right)^2 + 2m_2 \left( \frac{s}{2\pi f_a} \right) + 1 }$$
This structure provides independent control over notch depth and bandwidth. The notch depth is primarily governed by the ratio $m_1/m_2$, while the bandwidth is adjusted by the absolute values of $m_1$ and $m_2$. This flexibility is crucial for precisely shaping the loop gain to ensure stability under varying conditions without excessive phase loss at other frequencies.
2. QPR Current Controller
The PI controller is replaced by a QPR controller to achieve zero steady-state error at the fundamental grid frequency (e.g., 50/60 Hz) while maintaining good tracking performance. The QPR controller transfer function is:
$$G_{QPR}(s) = k_p + \frac{2\omega_i k_r s}{s^2 + 2\omega_i s + \omega_0^2}$$
where $k_p$ is the proportional gain, $k_r$ is the resonant gain, $\omega_0$ is the fundamental angular frequency, and $\omega_i$ is the cutoff angular frequency that introduces a bandwidth around the resonant frequency. This bandwidth makes the controller robust against slight variations in the grid fundamental frequency, a common issue in weak grids that a pure resonant controller would struggle with. The integration of this QPR controller significantly enhances the reference tracking capability of the solar inverter.
3. Grid Impedance Adaptive System
The cornerstone of the adaptive strategy is the real-time identification of the grid impedance $L_g$. A quasi-passive measurement scheme is employed. This method continuously monitors grid-side currents and calculates a residual signal $R_{est}$. When $R_{est}$ exceeds a predefined threshold, indicating a significant change in grid impedance, an active measurement routine is triggered to update the estimate of $L_g$. This updated value is fed into two critical adaptation mechanisms:
- Notch Frequency Adaptation: The center frequency $f_a$ of the enhanced notch filter $G_n(s)$ is continuously set equal to the calculated resonant frequency $f_{res}$ from Eq. (3), ensuring the notch always aligns with the active resonance point.
- QPR Controller Gain Adaptation: To maintain optimal phase margin and bandwidth as seen from each solar inverter, the QPR controller gains are scaled by an adaptive factor $F(\alpha)$. This factor is derived from the ratio $\alpha = (f_a / f_b)^2$, where $f_b$ is a base frequency, and the filter parameters, ensuring consistent dynamic performance.
The block diagram of the complete proposed control system for each solar inverter in the cluster incorporates these elements, forming a cohesive and adaptive resonance suppression unit.
Stability Analysis and Parameter Design
The open-loop transfer function of the system with the proposed controller is complex but can be represented in the form:
$$G_{ol, prop}(s) = \frac{N(s)}{D(s)}$$
where $N(s)$ and $D(s)$ are polynomials whose coefficients are functions of $k_p, k_r, \omega_i, m_1, m_2, f_a, L_1, L_2, C,$ and $L_g$. A comparative Bode plot analysis between the conventional fixed-notch method and the proposed adaptive method under increasing $L_g$ clearly demonstrates the superiority of the latter. While the conventional method shows a growing resonant peak and degrading phase margin, the proposed method maintains a smooth gain curve with adequate phase margin, as the notch and controller parameters adapt to cancel the shifting resonance.
The parameters for the notch filter ($m_1$, $m_2$) and the QPR controller ($k_p$, $k_r$, $\omega_i$) are designed based on frequency-domain stability criteria, ensuring sufficient gain margin (GM) and phase margin (PM) across the expected range of $L_g$ and $n$. A key design table is summarized below.
| System Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | $U_{dc}$ | 660 V |
| Grid Frequency | $f_0$ | 50 Hz |
| Inverter-side Inductor | $L_1$ | 2.0 mH |
| Filter Capacitor | $C$ | 7.0 μF |
| Grid-side Inductor | $L_2$ | 0.9 mH |
| Notch Coefficient | $m_1$ | $5 \times 10^{-4}$ |
| Notch Coefficient | $m_2$ | $5 \times 10^{-1}$ |
| Grid Impedance Range | $L_g$ | 0 – 5 mH |
Simulation Verification
The effectiveness of the proposed adaptive resonance suppression method is verified using a detailed MATLAB/Simulink model of a cluster with two parallel string-type solar inverters. The system parameters are listed in the table above.
Case 1: Steady-State Performance under Weak Grid ($L_g = 1$ mH). First, the performance is compared with the traditional fixed-notch method. With the conventional method, the grid current waveform is severely distorted due to uncontrolled resonance, exhibiting a high Total Harmonic Distortion (THD) of 8.42%. In contrast, with the proposed adaptive method activated, the grid current is sinusoidal and stable. The FFT analysis confirms a drastic reduction in THD to 3.10%, with the remaining harmonics primarily being low-order background harmonics from the grid voltage, not resonance-related. This demonstrates the fundamental ability of the adaptive solar inverter control to suppress resonance.
Case 2: Dynamic Performance during Grid Impedance Change. To test the adaptability, a scenario is simulated where the grid impedance $L_g$ changes instantaneously from 4 mH to 1 mH at t = 0.15 s. Without the adaptive mechanism, the system loses stability post-change, resulting in oscillatory and divergent grid currents. With the proposed adaptive scheme, the impedance observation module detects the change, recalculates the new resonant frequency, and updates the notch filter and QPR parameters accordingly. The grid current experiences a brief, minor transient and quickly re-stabilizes to a clean sinusoidal waveform within approximately 20 ms. This validates the dynamic robustness of the control strategy for a solar inverter cluster in a realistically varying weak grid environment.
Conclusion
This paper has addressed the critical challenge of resonance instability in clusters of string-type photovoltaic inverters connected to weak grids. The analysis confirms that the resonant frequency of such a system is not a fixed parameter but varies inversely with both the number of parallel solar inverters and the grid impedance, directly undermining system stability. To solve this, an advanced active damping strategy based on an adaptive notch filter has been proposed and developed in detail.
The core of the method involves three synergistic components: 1) an enhanced, tunable biquad notch filter for precise resonance peak cancellation, 2) a QPR current controller for accurate fundamental frequency tracking, and 3) a quasi-passive grid impedance observation module that enables real-time adaptation. This module continuously estimates the grid condition and feeds the updated resonant frequency back to the notch filter center frequency while adaptively tuning the controller gains to preserve stability margins.
Comprehensive frequency-domain analysis and time-domain simulations for a two-inverter cluster demonstrate the method’s superiority. It effectively suppresses resonance, maintains high-quality grid currents with low THD under steady-state weak grid conditions, and exhibits excellent dynamic robustness by quickly stabilizing the system following abrupt changes in grid impedance. This adaptive approach provides a reliable and effective solution for ensuring the stable and high-performance operation of large-scale, clustered solar inverter systems in modern power networks characterized by increasing renewable penetration and weak grid conditions.