As a researcher specializing in power electronics and renewable energy integration, I have focused on improving the performance of grid-connected inverters, particularly those employing LCL filters. The increasing penetration of renewable energy sources, such as wind and solar photovoltaic systems, demands high-quality power injection into the grid. Among various types of solar inverters, the LCL-filtered inverter is widely adopted due to its superior harmonic attenuation capability. However, the inherent resonance of LCL filters poses stability challenges, especially under weak grid conditions with background harmonics. In this study, I propose an active damping (AD) control strategy based on inverter-side inductor current feedback combined with proportional-resonant (PR) and harmonic compensation (HC) controllers. This approach not only suppresses resonance effectively but also reduces the number of required sensors, making it cost-effective and robust for diverse grid conditions.
1. Introduction
The global shift towards low-carbon energy has accelerated the deployment of grid-connected inverters, which serve as the interface between renewable energy sources and the AC grid. Among the various types of solar inverters, three-phase inverters with LCL filters are preferred for their ability to attenuate high-frequency current harmonics generated by pulse-width modulation (PWM). However, the LCL filter introduces a resonance peak that can cause instability when the grid impedance varies. Passive damping (PD) methods, such as adding resistors in series with the filter capacitor, are simple but increase power losses and reduce efficiency. On the other hand, active damping (AD) techniques modify the control algorithm to emulate damping without additional hardware. Many existing AD strategies rely on capacitor current or voltage feedback, which requires extra sensors and increases system complexity. Therefore, there is a need for a sensor-reduced AD strategy that maintains high harmonic suppression and robust stability under both stiff and weak grids.
In this paper, I present a novel AD control scheme that uses only inverter-side inductor current feedback combined with a PR+HC controller. The proposed method, termed Proportional Harmonic-Compensation Inverter Current Feedback Active Damping (PHICFAD), eliminates the need for capacitor current or grid current sensors. I derive the equivalent circuit model and optimal damping gain analytically. Simulation and experimental results validate that the proposed strategy achieves excellent resonance damping and harmonic rejection for both stiff and weak grid conditions, while reducing sensor count and thus system cost.
2. System Model and Problem Formulation
Figure 1 (conceptually) shows the topology of a voltage source inverter (VSI) connected to the grid through an LCL filter, with control architecture. The system parameters are listed in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link voltage | \(U_{dc}\) | 700 V |
| Grid phase voltage (rms) | \(u_x\) | 220 V |
| Inverter gain | \(K_{PWM}\) | 200 |
| Switching frequency | \(f_{sw}\) | 5 kHz |
| Sampling frequency | \(f_s\) | 10 kHz |
| Inverter-side inductor | \(L_1\) | 3 mH |
| Grid-side inductor | \(L_2\) | 1.5 mH |
| Filter capacitor | \(C\) | 29 μF |
| Grid impedance | \(L_g\) | 0 / 2.5 / 5 mH |
The LCL filter resonance angular frequency is given by:
\[
\omega_r = \sqrt{\frac{L_1 + L_T}{L_1 L_T C}}
\]
where \(L_T = L_2 + L_g\). The current regulator is a PR+HC controller:
\[
H_{AC}(s) = G_{pr}(s) + G_{hc}(s)
\]
with:
\[
G_{pr}(s) = K_p + \frac{2 K_i s}{s^2 + \omega_0^2}
\]
\[
G_{hc}(s) = \sum_{h=3,5,7,9} \frac{K_{ih} s}{s^2 + ( \omega_0 h )^2}
\]
The digital control delay is modeled as:
\[
G_d(s) = e^{-1.5 s T_s}
\]
where \(T_s\) is the sampling period. The inverter-side current feedback is used for active damping.
3. Proposed PHICFAD Control Strategy
3.1 Equivalent Circuit Derivation
The proposed PHICFAD method uses inverter-side current feedback with a proportional gain \(K_p\) and an additional damping gain \(K\) (including both inherent and active damping). The equivalent circuit model is derived by matching the denominator of the closed-loop transfer function to that of a passive damping resistor \(R_d\) placed in parallel with the filter capacitor. After simplification, the equivalent damping resistance becomes:
\[
R_d = \frac{\omega_r}{K_{PWM} K}
\]
Considering the digital delay, the equivalent impedance presented to the capacitor is frequency-dependent:
\[
Z_{eq}(s) = R_d e^{-1.5 s T_s}
\]
And its real and imaginary parts at frequency \(\omega\) are:
\[
R_{eq}(\omega) = R_d \cos(1.5 \omega T_s), \quad X_{eq}(\omega) = R_d \sin(1.5 \omega T_s)
\]
The effective capacitance becomes:
\[
C_{eq}(\omega) = \frac{C}{1 – \omega C X_{eq}(\omega)}
\]
The damping ratio of the system is given by:
\[
\zeta(\omega’_r) = \frac{K_{PWM} K \sin(1.5 \omega’_r T_s) – \omega_r C \omega’_r}{2 \omega_r C \omega’_r \cos(1.5 \omega’_r T_s)}
\]
where \(\omega’_r\) is the resonance frequency of the equivalent circuit. By solving for the optimal damping ratio \(\zeta = 0.5009\) under various grid impedances, I obtain the optimal total damping gain \(K_{opt}\).
3.2 Optimal Damping Gain Design
Using the derived expression, I plot the damping ratio versus \(K\) for different grid inductances (Lg = 0, 2.5, 5 mH). The results are summarized in Table 2.
| \(L_g\) (mH) | Optimal \(K\) | Maximum \(\zeta\) |
|---|---|---|
| 0 | 0.5666 | 0.5009 |
| 2.5 | 0.6553 | 0.5009 |
| 5 | 0.7363 | 0.5009 |
It is evident that the optimal damping ratio remains constant, while the required gain \(K\) increases with grid inductance. This provides a systematic design guideline for the proposed PHICFAD method.
3.3 Bode Plot Analysis
Figure 2 (not shown here) illustrates the open-loop Bode plots of the system with Lg = 0 mH for different \(K\) values. When \(K = 0.5666\) (optimal), the resonance peak is effectively suppressed, and the phase margin is increased, ensuring stability. For underdamped (\(K = 0.48\)) or overdamped (\(K = 0.92\)) cases, the resonance suppression degrades. Similar behavior is observed for Lg = 5 mH.
4. Stability Analysis Under Weak Grid Conditions
To evaluate the robustness of the proposed method, I analyze the system with varying grid impedance. Figure 3 (conceptual) shows the open-loop Bode plots for Lg = 0, 2.5, 5, and 9 mH using the optimal damping gain. As the grid inductance increases, the resonance peak flattens, and the low-frequency gain decreases, but the gain margin remains positive. This confirms that the PHICFAD strategy maintains stability even in extremely weak grids. This robustness is crucial for the reliable operation of various types of solar inverters in remote areas where the grid strength is low.
5. Simulation Verification
I built a simulation model in Matlab/Simulink based on Table 1. The grid voltage contains 5th, 7th, and 9th harmonic components (11 V, 6.6 V, 4.4 V, respectively). The inverter current reference steps from 20 A to 40 A at t = 0.1 s.
5.1 Stiff Grid (Lg = 0 mH)
Figure 4 (not shown) presents the simulation waveforms for three gain values: \(K = 0.48\) (underdamped), \(K = 0.5666\) (optimal), and \(K = 0.92\) (overdamped). The total harmonic distortion (THD) of the grid current under steady-state (20 A) and transient (40 A) conditions is listed in Table 3.
| \(K\) | THD (20 A) | THD (40 A) |
|---|---|---|
| 0.48 | 1.17% | 0.73% |
| 0.5666 | 1.12% | 0.54% |
| 0.92 | 1.52% | 1.25% |
The optimal gain yields the lowest THD, confirming the effectiveness of the designed damping ratio.
5.2 Weak Grid (Lg = 5 mH)
Similarly, I simulate the system with Lg = 5 mH. The THD results are shown in Table 4.
| \(K\) | THD (20 A) | THD (40 A) |
|---|---|---|
| 0.62 | 2.91% | 1.25% |
| 0.7363 | 2.65% | 1.23% |
| 1.25 | 3.34% | 1.41% |
The optimal gain again provides the best performance, demonstrating that the proposed PHICFAD strategy adapts well to varying grid strengths without requiring retuning.
6. Experimental Validation
I constructed a 5 kW three-phase two-level inverter prototype with LCL filter, using STM32G474 controller. The experimental parameters are given in Table 5. I tested both stiff (Lg = 0 mH) and weak (Lg = 5 mH) grid conditions with background harmonics.
| Parameter | Value |
|---|---|
| DC voltage | 130 V |
| Grid line voltage (rms) | 315 V |
| Grid current reference | 3 A / 6 A |
| \(L_1\) | 4.5 mH |
| \(L_2\) | 1.8 mH |
| \(C\) | 29 μF |
The experimental waveforms for Lg = 0 mH are shown in Figure 5 (conceptual). The results confirm that with optimal gain \(K = 0.5666\), the grid current is sinusoidal with low harmonic content. For Lg = 5 mH, Figure 6 (conceptual) shows similar trends, with the optimal gain \(K = 0.7363\) yielding the best quality. Table 6 summarizes the THD measurements.
| \(L_g\) (mH) | \(K\) | THD (20A) | THD (40A) |
|---|---|---|---|
| 0 | 0.5666 | 1.45% | 0.74% |
| 5 | 0.7363 | 2.98% | 1.52% |
These experimental results corroborate the simulation findings and demonstrate the practical feasibility of the proposed PHICFAD strategy for various types of solar inverters used in modern renewable energy systems.
7. Conclusion
In this work, I have proposed an active damping control strategy for LCL-type grid-connected inverters that combines inverter-side inductor current feedback with a proportional-resonant plus harmonic compensation controller. The key contributions are:
- A systematic derivation of the equivalent circuit model, allowing optimal damping gain design.
- Demonstration that the optimal damping ratio remains constant (\(\zeta = 0.5009\)) regardless of grid impedance, simplifying controller tuning.
- Reduction of sensor requirements (only two current sensors for inverter-side currents and PCC voltage for synchronization), lowering system cost.
- Comprehensive simulation and experimental validation under both stiff and weak grid conditions, confirming excellent harmonic suppression and dynamic response.
The proposed PHICFAD method is well-suited for various types of solar inverters, particularly in large-scale photovoltaic and offshore wind power plants where grid impedance can vary widely. Future work will focus on extending the strategy to multi-parallel inverters and adaptive tuning under uncertain grid conditions.