In distributed power generation systems, the inherent dispersion of connection points for distributed energy resources, coupled with the presence of transmission lines and transformers, endows the public grid with a non-negligible equivalent impedance. This characteristic manifests as a “weak grid” condition. As the critical energy conversion interface between renewable energy units and the grid, the on grid inverter’s secure and stable operation is paramount for the entire distributed generation system. The stability and power quality of the on grid inverter are severely challenged by grid impedance variations.
A prevalent topology for on grid inverters is the LCL filter, prized for its superior high-frequency harmonic attenuation capability compared to simple L filters. However, the LCL filter introduces a resonant peak that can destabilize the system. Damping this resonance is essential, typically achieved through passive or active methods. While passive damping is straightforward to implement, it incurs additional power losses. Active damping, conversely, suppresses resonance by feeding back specific state variables within the control loop, offering greater flexibility and efficiency without extra hardware losses.
Traditional PI controllers, with fixed parameters, struggle to meet performance requirements under the wide range of operating conditions presented by a weak grid. Their limited bandwidth and phase margin become particularly problematic when grid impedance is high. Fuzzy control, known for its adaptability and robustness against disturbances, offers a compelling solution. Combining fuzzy logic with PI control results in a Fuzzy Adaptive PI controller, which boasts strong robustness, excellent disturbance rejection, and fast dynamic response by continuously tuning the PI parameters online based on system error.
While the Fuzzy Adaptive PI controller enhances performance under moderate grid conditions, its ability to maintain sufficient stability margin is still compromised under high-impedance weak grid scenarios. The phase lag introduced by the PI controller and the interaction with grid impedance can critically reduce the phase margin of the system’s output impedance, potentially leading to instability. This paper addresses the dual challenge of LCL resonance and insufficient stability margin for Fuzzy Adaptive PI control under high grid impedance. It employs capacitor current feedback active damping to virtually damp the resonance and introduces a phase-lead compensation strategy on top of the Fuzzy Adaptive PI control. This combined approach actively suppresses the resonant peak and significantly boosts the phase margin of the on grid inverter‘s output impedance, ensuring robust stability even in severely weak grid conditions.
1. System Modeling and Challenges of LCL-Type On Grid Inverters
The typical topology of a single-phase LCL-type on grid inverter connected to a weak grid is shown in the conceptual block diagram. The system consists of a DC voltage source, a full-bridge inverter, an LCL filter (comprising inverter-side inductor \(L_1\), filter capacitor \(C\), and grid-side inductor \(L_2\)), and the grid represented by its voltage \(U_g\) and series impedance \(Z_g\).
The open-loop transfer function from the modulating signal to the grid current, without any damping, is given by:
$$ G_{ol\_undamped}(s) = G_{PI}(s) \cdot K_{PWM} \cdot \frac{1}{s^3 L_1 L_2 C + s(L_1+L_2)} $$
Where \(G_{PI}(s)=K_p + \frac{K_i}{s}\) is the PI controller transfer function and \(K_{PWM}\) is the inverter gain. This transfer function clearly exhibits a resonant peak due to the LCL filter’s complex conjugate poles, threatening system stability.
1.1 Active Damping via Capacitor Current Feedback
To mitigate this, capacitor current feedback active damping is employed. By feeding back the capacitor current \(i_c\) through a proportional gain \(G_c(s) = K_d\), the control structure can be equivalently transformed to show a virtual resistor \(Z_s = 1/(K_{PWM} \cdot K_d)\) connected in parallel with the filter capacitor. This virtual resistor dissipates energy at the resonant frequency, effectively damping the peak.
The modified open-loop transfer function with active damping becomes:
$$ G_{ol\_damped}(s) = G_{PI}(s) \cdot \frac{K_{PWM}}{s^2 L_1 C + s C K_{PWM} K_d + 1} \cdot \frac{s^2 L_1 C + s C K_{PWM} K_d + 1}{s^3 L_1 L_2 C + s^2 L_2 C K_{PWM} K_d + s(L_1+L_2)} $$
The Bode plot comparison reveals that the active damping successfully suppresses the resonant spike, ensuring stability under nominal grid conditions.
1.2 Impedance-Based Stability Analysis in Weak Grids
Under weak grid conditions, the grid impedance \(Z_g(s)\) is no longer negligible. The system stability can be analyzed using the impedance-based criterion. The on grid inverter can be modeled as a Norton equivalent circuit: an ideal current source \(i_s(s)\) in parallel with its output impedance \(Z_{inv}(s)\). The grid is modeled as a voltage source \(U_g(s)\) in series with its impedance \(Z_g(s)\).
The grid current is:
$$ i_g(s) = \frac{Z_{inv}(s)}{Z_{inv}(s)+Z_g(s)} i_s(s) – \frac{U_g(s)}{Z_{inv}(s)+Z_g(s)} $$
The stability of the interconnected system depends on the minor loop gain:
$$ M_1(s) = \frac{Z_g(s)}{Z_{inv}(s)} $$
According to the Nyquist stability criterion, the system remains stable if the phase margin PM at the frequency where \(|Z_{inv}(j\omega)| = |Z_g(j\omega)|\) is positive:
$$ PM = 180^\circ – \angle Z_g(j\omega_c) + \angle Z_{inv}(j\omega_c) > 0 $$
where \(\omega_c\) is the crossover frequency. A high grid impedance (large \(|Z_g|\)) and the inherent phase lag of the PI-controlled inverter (low \(\angle Z_{inv}\)) can drastically reduce this phase margin, leading to potential instability.
2. Fuzzy Adaptive PI Control Design
The core of the adaptive controller is a fuzzy inference system that dynamically adjusts the PI parameters (\(K_p\) and \(K_i\)) based on the grid current error \(e = i_{ref} – i_g\) and its derivative \(\dot{e}\).
The final PI parameters are calculated as:
$$ K_p = K_{p0} + \Delta K_p $$
$$ K_i = K_{i0} + \Delta K_i $$
where \(K_{p0}\) and \(K_{i0}\) are initial base values, and \(\Delta K_p\), \(\Delta K_i\) are the adjustments from the fuzzy controller.
2.1 Fuzzy Rule Base and Surfaces
The fuzzy sets for inputs and outputs are defined as: Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (Z), Positive Small (PS), Positive Medium (PM), Positive Big (PB). The membership functions are triangular. The rule base is formulated from operational experience to achieve fast response with minimal overshoot and good steady-state accuracy.
The fuzzy control rules for \(\Delta K_p\) and \(\Delta K_i\) are summarized in the following table:
| \(\Delta K_p\) / \(\Delta K_i\) | Rate of Error Change (\(\dot{e}\)) | |||||||
|---|---|---|---|---|---|---|---|---|
| NB | NM | NS | Z | PS | PM | PB | ||
| Error (e) | NB | PB/NB | PB/NB | PM/NB | PM/NM | PS/NM | PS/Z | Z/Z |
| NM | PB/NB | PB/NB | PM/NM | PM/NM | PB/NS | Z/Z | Z/Z | |
| NS | PM/NM | PM/NM | PM/NS | PS/NS | Z/Z | NS/PS | NM/PS | |
| Z | PM/NM | PS/NS | PS/NS | Z/Z | NS/PS | NM/PS | NM/PM | |
| PS | PS/NS | PS/NS | Z/Z | NS/PS | NS/PS | NM/PM | NM/PM | |
| PM | Z/Z | Z/Z | NS/PS | NM/PM | NM/PM | NM/PB | NB/PB | |
| PB | Z/Z | NS/Z | NS/PS | NM/PM | NM/PB | NB/PB | NB/PB | |
The corresponding three-dimensional rule surfaces for \(\Delta K_p\) and \(\Delta K_i\) show a smooth, non-linear mapping from the error inputs to the parameter adjustments. This adaptive mechanism allows the on grid inverter to maintain good performance during transients and under varying load conditions. However, analysis shows that as the grid inductance \(L_g\) increases, the phase margin provided solely by the Fuzzy Adaptive PI controller diminishes. For very high grid impedance, the margin may become inadequate, necessitating an additional stabilization measure.
3. Phase-Lead Compensation Control Strategy
To address the stability margin shortfall in high-impedance weak grids, a phase-lead compensator is introduced in series with the Fuzzy Adaptive PI controller. The compensator is placed after the PI block in the forward path. Its transfer function is:
$$ G_{lead}(s) = \frac{s + z}{s + p}, \quad \text{with} \quad p > z > 0 $$
The purpose of \(G_{lead}(s)\) is to add positive phase shift near the crossover frequency of the system’s loop gain or output impedance.
3.1 Compensator Design and Impact on Output Impedance
The phase contributed by the lead compensator at a frequency \(\omega\) is:
$$ \phi_{lead}(\omega) = \arctan\left(\frac{\omega}{z}\right) – \arctan\left(\frac{\omega}{p}\right) $$
The maximum phase boost \(\phi_{max}\) occurs at the geometric mean frequency \(\omega_m = \sqrt{z \cdot p}\), and is given by:
$$ \phi_{max} = \arcsin\left(\frac{p-z}{p+z}\right) $$
The amount of phase boost required is determined by the deficiency in the phase margin of the on grid inverter‘s output impedance \(Z_{inv}(s)\) when interacting with the anticipated maximum grid impedance. By carefully selecting the zero \(z\) and pole \(p\), the compensator increases the phase of the modified inverter output impedance \(Z_{inv\_comp}(s)\) in the mid-frequency range.
The modified output impedance with the compensator is:
$$ Z_{inv\_comp}(s) = \frac{1 + G_{PI}(s)G_{lead}(s)G_x(s)}{G_{x2}(s)} $$
where \(G_x(s)\) and \(G_{x2}(s)\) are transfer functions related to the actively-damped LCL filter and PWM. A comparative Bode plot of \(Z_{inv}(s)\) and \(Z_{inv\_comp}(s)\) demonstrates the significant increase in phase angle around the critical frequency where \(|Z_{inv}| \approx |Z_g|\). For instance, the phase margin can be increased from a precarious few degrees to over 45 degrees, providing a robust stability buffer for the on grid inverter operating in a weak grid.
4. Simulation Verification and Analysis
To validate the proposed integrated control strategy combining capacitor current feedback active damping, Fuzzy Adaptive PI, and phase-lead compensation, simulations are conducted under weak grid conditions. The key parameters for the on grid inverter system are listed below:
| Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | \(U_{dc}\) | 400 V |
| Grid Voltage (RMS) | \(U_g\) | 220 V |
| Grid Frequency | \(f\) | 50 Hz |
| Inverter-side Inductor | \(L_1\) | 3 mH |
| Grid-side Inductor | \(L_2\) | 1 mH |
| Filter Capacitor | \(C\) | 20 μF |
| Switching/Sampling Frequency | \(f_s\) | 10 kHz |
| Lead Compensator Zero | \(z\) | 187 rad/s |
| Lead Compensator Pole | \(p\) | 1087 rad/s |
Scenario 1: System without Phase-Lead Compensation. The system operates stably under a normal grid condition (\(L_g = 0.5 \text{mH}\)). At time t = 0.4s, the grid impedance is stepped to a high value (\(L_g = 5.0 \text{mH}\)). The grid current waveform exhibits severe oscillation and distortion, indicating that the Fuzzy Adaptive PI controller alone cannot maintain stability under such a weak grid condition.
Scenario 2: System with the Proposed Phase-Lead Compensation. Under the same test sequence, the system with the phase-lead compensator remains stable after the grid impedance step. The grid current experiences a minor transient disturbance but recovers to a stable, low-THD sinusoidal waveform within one grid cycle. This clearly demonstrates that the added phase margin from the compensator successfully stabilizes the on grid inverter in the high-impedance weak grid.
5. Conclusion
This paper addresses the critical stability challenges for LCL-type on grid inverters in weak grid environments. The resonant peak of the LCL filter is effectively suppressed using capacitor current feedback active damping, which implements a virtual resistor in parallel with the filter capacitor. While a Fuzzy Adaptive PI controller provides excellent dynamic and steady-state performance under normal conditions, its ability to ensure stability is compromised when the grid impedance becomes very high.
The proposed solution integrates a phase-lead compensation strategy with the Fuzzy Adaptive PI control. The lead compensator is strategically designed to increase the phase angle of the inverter’s equivalent output impedance in the frequency range critical for impedance-based stability interaction with the grid. This added phase boost directly translates to a larger stability phase margin according to the impedance ratio criterion. Comprehensive simulation results confirm that the combined strategy not only damps the LCL resonance but also guarantees stable and high-quality grid current injection for the on grid inverter even under severely weak grid conditions characterized by high network impedance. The method is thus a robust and effective control solution for enhancing the reliability of distributed generation systems interfaced with the grid through power electronic inverters.