The global energy landscape is undergoing a profound transformation driven by the urgent need to address fossil fuel depletion and climate change. The integration of renewable energy sources, particularly wind and solar power, has seen exponential growth in both installed capacity and energy generation. A significant portion of these utility-scale renewable energy plants is situated in remote areas far from major load centers. The consequent long-distance transmission lines and multiple transformation stages impart a characteristic “weak grid” feature to the network, characterized by non-negligible grid impedance.
In such weak grid conditions, the interplay between the inherently complex control dynamics of utility interactive inverters and the grid impedance poses significant stability challenges. A common and critical issue is the excitation of wideband resonance, threatening the secure and stable operation of modern power systems. To mitigate the adverse effects of increasingly complex grid background harmonics in this “Dual High” (high penetration of renewables and power electronics) context, additional control strategies are often appended to existing schemes. The Point of Common Coupling (PCC) voltage feedforward strategy is widely adopted due to its simple structure, low hardware cost, and effective harmonic suppression capability.
However, existing research has revealed a fundamental flaw in this approach under weak grid conditions. The interaction between the inverter and the grid impedance, exacerbated by the feedforward path, can degrade the passivity of the inverter’s equivalent output impedance in the low-to-medium frequency range. This degradation brings the system phase margin perilously close to instability boundaries, making it susceptible to wideband resonance phenomena. This paper delves into this problem, analyzes the inherent defects of traditional PCC feedforward, and proposes a novel grid-side active damping strategy based on passivity theory to enhance the robustness of utility interactive inverters.
Analysis of Traditional PCC Voltage Feedforward Strategy and Passivity
To understand the limitations, we first establish the control model of a three-phase LCL-type utility interactive inverter system employing the traditional PCC voltage feedforward. The system topology and control structure in the stationary (αβ) reference frame are considered. The LCL filter parameters are denoted as inverter-side inductance \(L_1\), grid-side inductance \(L_2\), and filter capacitance \(C\). The grid impedance \(Z_g(s)\) is considered inductive (\(L_g\)) for a worst-case stability analysis. The controller includes a proportional-resonant (PR) current regulator \(G_i(s)\), a computation and PWM delay model \(G_d(s)\), and the PCC voltage feedforward gain \(G_f(s)\).
The equivalent output impedance \(Z_o(s)\), defined as the negative ratio of the PCC voltage disturbance to the injected grid current disturbance (\(-v_{pcc}(s)/i_g(s)\)), is a key metric for assessing stability and harmonic rejection. The expressions for the output impedance without (\(Z_{o2}(s)\)) and with (\(Z_{o3}(s)\)) the traditional PCC feedforward are derived as follows:
$$
Z_{o2}(s) = \frac{K(s)}{s^2 L_1 C + s C K_c G_d(s) + 1}
$$
$$
Z_{o3}(s) = \frac{K(s)}{s^2 L_1 C + s C K_c G_d(s) + 1 – G_f(s)G_d(s)}
$$
where,
$$
K(s) = s^3 L_1 L_2 C + s^2 L_2 C K_c G_d(s) + s(L_1 + L_2) + G_i(s)G_d(s)
$$
Here, \(K_c\) is the capacitor current feedback coefficient for passive damping. The delay is typically modeled as \(G_d(s) = e^{-1.5T_s s}\), where \(T_s\) is the sampling period. The feedforward gain is ideally \(G_f(s) = 1/K_{PWM}\).
Passivity Theory and Its Application
Passivity theory provides a powerful framework for analyzing the stability of interconnected systems. A network is considered passive if it cannot produce energy internally. For a utility interactive inverter, the stability of its interconnection with the grid can be assessed by examining the passivity of its equivalent output impedance \(Z_o(j\omega)\). A sufficient condition for stability is that \(Z_o(j\omega)\) is passive, meaning its real part is non-negative for all frequencies, or equivalently, its phase angle lies within [-90°, 90°].
Passivity Analysis of the Conventional Strategy
Analyzing the derived impedance expressions and their Bode plots reveals the dual nature of the PCC feedforward strategy. The strategy effectively increases the magnitude of \(Z_o(s)\) in the low-frequency range, which improves the rejection of grid background harmonics. This is its primary advantage. However, the term \(-G_f(s)G_d(s)\) in the denominator of \(Z_{o3}(s)\) introduces significant phase lag due to the control delay \(G_d(s)\). This lag critically reduces the phase margin of the output impedance, pushing its phase closer to the -90° passivity boundary in the low-to-medium frequency spectrum, as illustrated in the comparative analysis below.
| Control Strategy | Harmonic Rejection (Low Freq.) | Passivity (Low-Medium Freq.) | Weak Grid Robustness |
|---|---|---|---|
| Without PCC Feedforward | Poor | Good (Adequate margin) | Moderate |
| With Traditional PCC Feedforward | Excellent | Poor (Near boundary) | Poor (Prone to resonance) |
The core issue is that the traditional PCC voltage feedforward strategy cannot balance the trade-off between harmonic suppression and system passivity. While it addresses power quality concerns, it inadvertently undermines stability in weak grids. This necessitates the development of a control strategy that simultaneously achieves both objectives.
Proposed Grid-Side Active Damping Strategy for Passivity Enhancement
The proposed strategy is founded on two key principles from circuit and control theory: (1) adding a virtual resistive damping in the main current path to enhance passivity, and (2) applying a disturbance rejection principle to cancel grid harmonic voltages. Direct series insertion of a physical resistor is lossy. Therefore, an active damping approach is conceived.
The initial concept involves inserting a series-compensated voltage source \(v_c\) in the grid-side inductor branch \(L_2\), where \(v_c = \alpha v_{pcc} – K_d i_g\). The term \(-K_d i_g\) acts as a virtual resistor for passivity, and \(\alpha v_{pcc}\) works as a counter-disturbance to cancel grid harmonics. To minimize power loss, this series voltage source is equivalently split using circuit transformation laws. It is realized as two separate controlled sources: one in the inverter output branch (implemented digitally in the main controller) and a second, lower-power source in the capacitor branch (implemented via an add-on active compensation device).
System Configuration and Modeling
The structure integrates the main utility interactive inverter with an active compensation device across the filter capacitor. The main inverter control incorporates a modified feedforward path and a grid-current feedback loop: \(v_{ff} = \alpha v_{pcc} G_{NF}(s) – K_d i_g\). \(G_{NF}(s)\) is a notch filter to block the fundamental component, ensuring the add-on device only processes harmonic currents, thereby minimizing its power rating and loss. The active compensation device is a low-voltage H-bridge fed by a small DC capacitor. It injects the compensating voltage \(v_{ac}\) across the filter capacitor, effectively realizing the required series voltage without handling the full fundamental current.
The equivalent output impedance \(Z_{oad}(s)\) of the system with the proposed strategy is derived as:
$$
Z_{oad}(s) = \frac{T(s)}{T_1(s)}
$$
$$
T(s) = s^3 L_1 L_2 C + s^2 L_1 C K_d G_{d1}(s)G_{NF}(s) + s(L_1 + L_2) + G_d(s)[G_i(s) + K_d G_{d1}(s)]
$$
$$
T_1(s) = 1 + s^2 L_1 C – \alpha G_d(s)G_{d1}(s) – \alpha s^2 L_1 C G_{d1}(s)G_{NF}(s)
$$
where \(G_{d1}(s)=e^{-0.5T_s s}\) represents the delay in the feedforward/feedback path. This impedance can be viewed as the parallel combination of two sub-impedances, \(Z_{K_d}(s)\) and \(Z_{\alpha}(s)\), related to the damping and feedforward terms, respectively. According to passivity theory, if both sub-impedances are passive, the overall system is passive.
Parameter Design Based on Passivity
The parameters \(K_d\) and \(\alpha\) are designed to ensure the passivity of \(Z_{K_d}(s)\) and \(Z_{\alpha}(s)\) while maintaining harmonic rejection. By analyzing the real part condition of \(Z_{K_d}(j\omega)\), a permissible range for \(K_d\) is obtained. For a given system with parameters in Table II, this range is calculated.
| Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | \(V_{dc}\) | 700 V |
| Grid Voltage (RMS) | \(V_g\) | 220 V |
| Rated Power | \(P\) | 14 kW |
| Inverter-side Inductor | \(L_1\) | 4.6 mH |
| Grid-side Inductor | \(L_2\) | 1.2 mH |
| Filter Capacitor | \(C\) | 4.5 µF |
| Switching/Sampling Frequency | \(f_s\) | 10 kHz |
| PR Controller: Proportional Gain | \(K_p\) | 15.3 |
| PR Controller: Resonant Gain | \(K_r\) | 100 |
| Grid Current Reference | \(i_{ref}\) | 40 A |
For the passivity of \(Z_{\alpha}(s)\), the parameter \(\alpha\) is tuned. A smaller \(\alpha\) improves passivity but reduces harmonic rejection capability. A value of \(\alpha=0.7\) is selected as a compromise, and \(K_d\) is chosen as 48 from within its stable range. The designed control parameters are summarized below.
| Parameter | Symbol | Designed Value |
|---|---|---|
| Active Damping Coefficient | \(K_d\) | 48 |
| Feedforward Coefficient | \(\alpha\) | 0.7 |
| Notch Filter Center Frequency | \(\omega_1\) | 3141 rad/s (50 Hz) |
| Notch Filter Bandwidth | \(f_1\) | 250 Hz |
Passivity and Performance Verification of the Proposed Design
The Bode plot of the resulting \(Z_{oad}(s)\) demonstrates the success of the proposed grid-side active damping strategy. Compared to the traditional PCC feedforward, the proposed method maintains a high impedance magnitude in the low-frequency region, ensuring excellent rejection of grid background harmonics. Crucially, the phase of \(Z_{oad}(s)\) remains well within the passive region (above -50°) across a wide frequency spectrum (from 100 Hz to 10 kHz), even under varying weak grid impedances. This indicates significantly enhanced robustness and stability margins for the utility interactive inverter when connected to weak grids.
Simulation Verification
A detailed simulation model is built in MATLAB/Simulink to validate the theoretical analysis and the performance of the proposed control strategy for the utility interactive inverter under various weak grid scenarios.
Scenario 1: Traditional PCC Feedforward Strategy
First, the system with traditional PCC feedforward is tested. Under a moderately weak grid (SCR=8, \(L_g=4.12\) mH) with injected background harmonics (5th, 7th, 9th, 13th, 15th), the strategy effectively suppresses current distortion, confirming its harmonic rejection merit. However, when the grid strength weakens further (SCR decreases to 3, \(L_g=11\) mH), the system exhibits sustained oscillation and resonance, demonstrating its inherent instability under very weak grid conditions.
Scenario 2: Proposed Grid-Side Active Damping Strategy
In the same severe weak grid condition (SCR=3), the system initially with traditional feedforward goes unstable. At t=0.3s, the proposed grid-side active damping strategy is activated. The results show a rapid suppression of the resonance. The grid current recovers to a stable, sinusoidal waveform within a short transient period. Furthermore, the utility interactive inverter maintains a high-quality current output even in the presence of grid background harmonics, proving that the proposed strategy successfully decouples harmonic rejection from stability.
Scenario 3: Dynamic Performance
To verify that the proposed strategy does not compromise dynamic response, a step change in the current reference from 40 A to 80 A is applied. The grid current tracks the new reference quickly and smoothly, confirming that the system retains good dynamic performance.
| Scenario & Condition | Grid Strength (SCR) | Control Strategy | Key Observation |
|---|---|---|---|
| 1a: Harmonics Present | 8 (Moderately Weak) | Traditional PCC Feedforward | Good harmonic suppression. |
| 1b: Grid Weakens | 3 (Very Weak) | Traditional PCC Feedforward | System instability and resonance occur. |
| 2: Recovery & Stability | 3 (Very Weak) | Proposed Active Damping (Activated at 0.3s) | Resonance is damped rapidly; stable, low-THD current is maintained. |
| 3: Reference Step | 3 (Very Weak) | Proposed Active Damping | Fast and accurate current tracking, good dynamic response. |
Conclusion
This paper has addressed the critical challenge of balancing harmonic rejection and stability for utility interactive inverters in weak grids. The analysis confirmed that the widely used traditional PCC voltage feedforward strategy, while effective against harmonics, degrades the passivity of the inverter’s output impedance, making the system prone to wideband resonance. To solve this problem, a novel grid-side active damping strategy was proposed and developed based on passivity theory.
The proposed strategy ingeniously combines an add-on active compensation device across the filter capacitor with modified feedforward and feedback control in the main inverter. This approach introduces necessary damping and controlled disturbance rejection without significant additional loss. The parameter design ensures the passivity of the overall system. The main conclusions are as follows:
- The traditional PCC voltage feedforward strategy inherently compromises the passivity of a utility interactive inverter in weak grids, creating a risk of resonance instability.
- The proposed grid-side active damping strategy successfully enhances the passivity of the equivalent output impedance across a wide frequency range while preserving strong rejection capability against grid background harmonics. This significantly improves the stability and adaptability of the utility interactive inverter in weak and variable grid conditions.
- The strategy is practical, with a straightforward implementation structure. It does not adversely affect the dynamic response of the system, making it a viable solution for enhancing the robustness of utility interactive inverters in future power systems with high renewable penetration.