In the context of offshore wind power systems, grid-connected inverters equipped with LCL-type filters play a crucial role in attenuating high-frequency current harmonics while ensuring compliance with grid-connection standards. However, the presence of background harmonics introduced by complex power grids often leads to current distortion, necessitating effective suppression strategies. This paper proposes an active damping (AD) method that replaces the conventional PI controller with a proportional resonant (PR) controller combined with harmonic compensation (HC). The proposed approach, termed Proportional Harmonic-Compensation Inverter Current Feedback Active Damping (PHICFAD), enhances harmonic mitigation performance while reducing the number of required sensors. This study addresses the challenges associated with different types of solar inverter configurations, particularly in weak grid scenarios, where traditional damping methods may fall short.
The increasing integration of renewable energy sources, such as wind and solar, has driven the demand for high-performance power converters. Among various types of solar inverter topologies, those employing LCL filters are widely adopted due to their superior harmonic attenuation capabilities. Nevertheless, the inherent resonance of the LCL filter poses stability risks, especially when the grid impedance varies. Passive damping (PD) techniques, although simple, introduce additional power losses and reduce system efficiency. In contrast, active damping (AD) methods, such as capacitor current feedback or inverter-side current feedback, offer lossless resonance suppression. The proposed PHICFAD strategy builds upon inverter-side current feedback and incorporates a PR+HC controller to mitigate grid background harmonics effectively. This method is particularly suited for different types of solar inverter applications, ranging from residential to utility-scale installations.
System Modeling and Problem Formulation
The three-phase voltage source inverter (VSI) connected to the grid via an LCL filter is depicted in Figure 1 (refer to the conceptual block diagram). The system parameters used in this study are summarized in Table 1. The LCL filter consists of inverter-side inductance \(L_1\), grid-side inductance \(L_2\), and filter capacitance \(C\). The grid impedance is represented by \(L_g\), and the total grid-side inductance is \(L_T = L_2 + L_g\). The resonant angular frequency of the LCL filter is given by:
$$
\omega_r = \sqrt{\frac{L_1 + L_T}{L_1 L_T C}}.
$$
The PR+HC controller \(H_{AC}(s)\) is expressed as:
$$
H_{AC}(s) = G_{pr}(s) + G_{hc}(s),
$$
where the PR term is:
$$
G_{pr}(s) = K_p + \frac{2K_i s}{s^2 + \omega_0^2},
$$
and the harmonic compensation term for the 3rd, 5th, 7th, and 9th harmonics is:
$$
G_{hc}(s) = \sum_{h=3,5,7,9} \frac{K_{ih} s}{s^2 + (\omega_0 \cdot h)^2}.
$$
The digital control delay is modeled as:
$$
G_d(s) = e^{-1.5 s T_s},
$$
where \(T_s\) is the sampling period. The PWM gain is denoted as \(K_{PWM}\). The key system parameters for simulation and experiment are provided in Table 1.
| Parameter | Symbol | Value (Simulation) | Value (Experiment) |
|---|---|---|---|
| DC-link voltage | \(U_{dc}\) | 700 V | 130 V |
| Grid phase voltage (RMS) | \(u_x\) | 220 V | 315 V (line-to-line) |
| Inverter gain | \(K_{PWM}\) | 200 | 200 |
| Switching frequency | \(f_{sw}\) | 5 kHz | 5 kHz |
| Sampling frequency | \(f_s\) | 10 kHz | 10 kHz |
| Inverter-side inductance | \(L_1\) | 3 mH | 4.5 mH |
| Grid-side inductance | \(L_2\) | 1.5 mH | 1.8 mH |
| Filter capacitance | \(C\) | 29 μF | 29 μF |
| Grid impedance | \(L_g\) | 0 / 2.5 / 5 mH | 0 / 5 mH |
Proposed PHICFAD Control Strategy
The proposed PHICFAD method employs inverter-side current feedback with an additional PR+HC controller to suppress grid background harmonics. Unlike conventional approaches that require extra sensors for capacitor current or voltage, this strategy reduces the sensor count while maintaining effective damping. The open-loop gain of the system is derived as:
$$
T(s) = \frac{H_{AC} K_{PWM} G_d G_{iL2}}{1 + K_{PWM} G_d (H_{i1} G_{iL1} + K_p G_{ic})},
$$
where \(G_{iL2}\) is the transfer function from inverter voltage to grid current, and \(G_{ic}\) is the transfer function from inverter voltage to capacitor current. The equivalent circuit of the proposed AD method is shown in Figure 2 (conceptual equivalent model). By equating the denominator of the closed-loop transfer function with that of a passive damping system, the damping coefficient \(\zeta\) can be expressed as:
$$
\zeta = \frac{1}{2 C_{eq} R_{eq} \omega_r’},
$$
where \(C_{eq}\) and \(R_{eq}\) are the equivalent capacitance and resistance after considering the control delay. The total active damping gain \(K\) is related to the damping resistor \(R_d\) by:
$$
R_d = \frac{\omega_r}{K_{PWM} K}.
$$
Further analysis yields the optimal damping coefficient \(\zeta_{opt} = 0.5009\) for various grid conditions. The relationship between \(\zeta\) and \(K\) under different grid impedances is listed in Table 2. It is evident that the optimal total damping gain \(K_{opt}\) increases with higher grid inductance, but the optimal damping coefficient remains constant.
| Grid impedance \(L_g\) (mH) | Optimal gain \(K_{opt}\) | Optimal damping \(\zeta_{opt}\) |
|---|---|---|
| 0 | 0.5666 | 0.5009 |
| 2.5 | 0.6553 | 0.5009 |
| 5.0 | 0.7363 | 0.5009 |
| 9.0 | 0.8500 | 0.5009 |
The Bode analysis indicates that as \(K\) approaches the optimal value, the phase margin increases and the system becomes more stable. The proposed PHICFAD strategy is directly applicable to various types of solar inverter designs, including those for single-phase and three-phase applications, as it only requires inverter-side current sensing.
Stability Analysis Under Varying Grid Conditions
To evaluate the robustness of the proposed method, the open-loop Bode diagrams are analyzed for grid inductances \(L_g = 0, 2.5, 5.0,\) and \(9.0\) mH. With the optimal gain \(K_{opt}\) selected for each case, the gain margin (GM) remains positive, confirming stable operation under both stiff and weak grid conditions. The results are summarized in Table 3. As the grid inductance increases, the resonant peak becomes smoother, indicating better damping. However, the low-frequency gain decreases slightly, which is acceptable for grid-connected inverters.
| Grid impedance \(L_g\) (mH) | Gain margin (dB) | Phase margin (°) |
|---|---|---|
| 0 | 6.8 | 48.2 |
| 2.5 | 5.5 | 42.1 |
| 5.0 | 4.2 | 36.5 |
| 9.0 | 2.9 | 30.8 |
This analysis demonstrates that the PHICFAD method maintains stability across a wide range of grid impedances, making it suitable for different types of solar inverter installations, especially in remote or weak grid areas.
Simulation Validation
Simulations were conducted using Matlab/Simulink with the parameters listed in Table 1. The grid voltage contained background harmonics of 5th, 7th, and 9th orders. The system was tested under stiff grid (\(L_g = 0\) mH) and weak grid (\(L_g = 5\) mH) conditions. Three values of total gain \(K\) were selected: below optimal, optimal, and above optimal. The grid current reference was stepped from 20 A to 40 A at 0.1 s. The total harmonic distortion (THD) of the grid current was measured over one cycle starting at 0.06 s (before the step) and 0.16 s (after the step). The results are presented in Table 4.
| Grid condition | \(K\) | THD at 20 A (%) | THD at 40 A (%) |
|---|---|---|---|
| \(L_g = 0\) mH | 0.48 | 1.17 | 0.73 |
| 0.5666 | 1.12 | 0.54 | |
| 0.92 | 1.52 | 1.25 | |
| \(L_g = 5\) mH | 0.62 | 2.91 | 1.25 |
| 0.7363 | 2.65 | 1.23 | |
| 1.25 | 3.34 | 1.41 |
As shown in Table 4, the optimal gain yields the lowest THD under both stiff and weak grid conditions. The THD increases when \(K\) deviates from the optimal value, confirming the analysis. The proposed method effectively suppresses background harmonics, meeting grid standards for various types of solar inverter output.
Experimental Validation
An experimental prototype based on a two-level three-phase inverter with an LCL filter was built. The control platform used an STM32G474 microcontroller with a sampling frequency twice the switching frequency. The grid background harmonics were emulated by injecting 5th (11 V), 7th (6.6 V), and 9th (4.4 V) voltage harmonics. The experiment was conducted for \(L_g = 0\) mH and \(L_g = 5\) mH, with the grid current reference stepped from 3 A to 6 A. The THD results are summarized in Table 5.
| Grid condition | \(K\) | THD at 3 A (%) | THD at 6 A (%) |
|---|---|---|---|
| \(L_g = 0\) mH | 0.48 | 1.69 | 0.83 |
| 0.5666 | 1.45 | 0.74 | |
| 0.92 | 2.67 | 1.82 | |
| \(L_g = 5\) mH | 0.62 | 3.31 | 1.78 |
| 0.7363 | 2.98 | 1.52 | |
| 1.25 | 3.74 | 1.92 |
The experimental results align well with the simulations. The optimal gain provides the best harmonic rejection, while underdamped or overdamped gains lead to higher THD. The proposed PHICFAD method maintains good dynamic response and steady-state performance, demonstrating its applicability to different types of solar inverter systems, including those in weak grid environments.
To further illustrate the practical implementation, the experimental setup is depicted in the following image:
Conclusion
This paper presents an active damping control strategy (PHICFAD) for LCL-type grid-connected inverters, which integrates a proportional resonant controller with harmonic compensation. The method addresses the resonance issue inherent in LCL filters while effectively suppressing grid background harmonics. By deriving the optimal damping gain through an equivalent circuit analysis, the proposed approach achieves a damping coefficient of 0.5009, which remains constant across varying grid impedances. Simulation and experimental results confirm that the PHICFAD strategy outperforms conventional underdamped and overdamped configurations, yielding lower total harmonic distortion under both stiff and weak grid conditions. Furthermore, the reduction in sensor count makes it cost-effective for various types of solar inverter applications, from residential to utility-scale implementations. The proposed control scheme enhances system stability and power quality, contributing to the reliable integration of renewable energy into modern power grids.