In my research, I have focused on mitigating high-frequency oscillations in LCL-type grid-connected inverters that arise from digital control delays under weak grid conditions. The increasing penetration of distributed generation systems has made LCL-type inverters a core topology for interfacing renewable energy sources with the utility grid, owing to their superior attenuation of high-frequency harmonics. However, the inherent 1.5 sampling period delay in digital control introduces phase lag that can destabilize the system, particularly when the grid impedance is high. I have developed a comprehensive compensation strategy that addresses this issue by combining active damping loop reconfiguration, phase-lead compensation, and parameter optimization. This work is essential for ensuring the reliable operation of various types of solar inverter used in modern photovoltaic systems, as they often face weak grid scenarios. By rigorously analyzing the loop gain and applying stability margin constraints, I have achieved effective suppression of oscillations across a wide range of grid strengths.
To establish a foundation, I first review the fundamental architecture of a grid-connected inverter. The single-phase LCL topology consists of a DC input, a full-bridge inverter, and an LCL filter composed of inverter-side inductor \(L_1\), filter capacitor \(C\), and grid-side inductor \(L_2\). The grid impedance \(L_g\) represents the inductive nature of the actual grid. The digital controller employs a current loop with a proportional-resonant (PR) regulator, and the control delay is modeled as \(G_d(s)=e^{-1.5T_s s}\), where \(T_s\) is the sampling period. To enable linear analysis, I use a third-order Padé approximation to represent this delay. The inverter bridge gain is \(K_{pwm}=U_{dc}/V_{tri}\), where \(U_{dc}\) is the DC-link voltage and \(V_{tri}\) is the triangular carrier amplitude. The system’s stability heavily depends on the interaction between the LCL filter resonance and the digital delay. In weak grids (short-circuit ratio SCR < 3), the grid inductance \(L_g\) can range from 8.6 mH to 12.8 mH, causing the resonance peak to shift and amplify, which often leads to high-frequency oscillations. Understanding these dynamics is critical for all types of solar inverter, as they must operate reliably under varying grid conditions.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link voltage | \(U_{dc}\) | 400 V |
| Grid voltage (RMS) | \(U_g\) | 220 V |
| Nominal power | \(P_n\) | 5 kW |
| Inverter-side inductor | \(L_1\) | 2.5 mH |
| Filter capacitor | \(C\) | 10 µF |
| Grid-side inductor | \(L_2\) | 1.5 mH |
| Sampling frequency | \(f_s\) | 10 kHz |
| Switching frequency | \(f_{sw}\) | 10 kHz |
| Resonant frequency (LCL) | \(f_r\) | ~1.8 kHz |
The core contribution of my study is the time-delay compensation strategy, which consists of three interconnected parts: active damping loop compensation, phase-lag compensation, and parameter optimization. First, I address the issue that the digital control delay degrades the effectiveness of the conventional capacitor-current-feedback active damping. By shifting the delay element out of the damping loop through equivalent transformation, I eliminate its adverse effect on the virtual damping characteristic. The original capacitor-current feedback path includes the delay \(G_d(s)\), which can cause the equivalent virtual resistance to become negative at the resonance frequency. My compensation approach extracts the delay from the loop and inserts a compensating block that restores the ideal damping behavior. The compensated active damping transfer function becomes \(G_{ic}(s) = \frac{1}{L_1 L_2 C s^2 + (L_1+L_2) s / K_{pwm} \omega_r^2}\), where \(\omega_r\) is the LCL resonant angular frequency. This ensures that the damping remains positive even when the grid impedance varies.
Table 2 summarizes the comparison between conventional active damping and my proposed compensation method for different types of solar inverter operating under varying grid conditions. The results show that the compensated approach maintains a stable phase margin above 30° across the weak grid region, whereas the conventional method exhibits negative phase margins and oscillations.
| Grid Inductance \(L_g\) (mH) | Conventional Damping – Phase Margin (°) | Proposed Compensated Damping – Phase Margin (°) |
|---|---|---|
| 2.57 (Strong) | 5.82 | 41.6 |
| 8.6 (Weak) | -3.2 | 38.9 |
| 12.8 (Very Weak) | -15.6 | 42.1 |
Second, I implement a phase-lead compensation stage to counteract the phase lag introduced by the 1.5-sample delay and the LCL filter. The phase-lead compensator is designed as a first-order rational function:
$$ G_{\text{lead}}(s) = \frac{s + w_m \frac{1+\sin\theta_m}{1-\sin\theta_m}}{s + w_m \frac{1-\sin\theta_m}{1+\sin\theta_m}} $$
Here, \(w_m\) is the frequency at which the maximum phase lead occurs, and \(\theta_m\) is the maximum phase lead angle. I set \(w_m\) to approximately 1.2–1.5 times the system cutoff frequency to avoid interaction with the LCL resonance peak, and \(\theta_m\) is chosen as 45° to compensate the typical 45°–60° phase loss caused by the digital delay and filter. This correction ensures that the phase margin of the loop gain remains within the stable range of 30°–60°. The phase-lead compensator works synergistically with the active damping loop: while the damping loop maintains positive virtual resistance, the phase-lead block restores the phase reserve, thereby preventing high-frequency oscillations. Such a combined approach is applicable to many types of solar inverter, especially those using LCL filters in weak grids.
Third, I optimize the controller parameters using stability margin constraints and impedance-based criteria. The current controller is a quasi-PR regulator with proportional gain \(k_p\) and resonant gain \(k_r\). The proportional gain is derived from the cutoff frequency requirement:
$$ k_p = \frac{2\pi f_c (L_1 + L_2)}{K_{pwm}} $$
where \(f_c\) is the desired cutoff frequency (typically 300–500 Hz). The resonant gain \(k_r\) is selected between 27.23 and 61.09 based on the base-frequency gain requirement and phase margin, while the resonant bandwidth \(\omega_i\) is widened to improve grid-frequency adaptability. The active damping coefficient \(k_C\) is co-optimized with virtual admittance reconstruction. I inject a 75 Hz non-characteristic harmonic to measure the grid impedance in real time, using discrete Fourier transform to extract the real and imaginary parts. The virtual admittance coefficient \(H_i\) is then adjusted dynamically within the range [0.0206, 0.4550] to balance oscillation suppression and stability. The optimization constraints are: phase margin between 30° and 60°, gain margin ≥ 10 dB, and all closed-loop poles in the left half-plane for \(L_g\) from 0 to 12.8 mH. I also verify the impedance-based stability criterion: at the intersection frequency of the inverter output impedance and the grid impedance, the phase angle of the inverter output impedance must be greater than -90°.
Table 3 lists the optimized parameters for the quasi-PR controller and active damping under different grid strengths. These parameters ensure robust performance for various types of solar inverter.
| Grid Condition | \(k_p\) | \(k_r\) | \(\omega_i\) (rad/s) | \(k_C\) | \(H_i\) |
|---|---|---|---|---|---|
| Strong (\(L_g=2.57\) mH) | 0.115 | 30 | 10 | 0.25 | 0.0206 |
| Weak (\(L_g=8.6\) mH) | 0.115 | 30 | 10 | 0.30 | 0.150 |
| Very Weak (\(L_g=12.8\) mH) | 0.115 | 30 | 10 | 0.35 | 0.455 |
The effectiveness of the proposed compensation strategy is validated through comprehensive simulation tests under three representative grid scenarios: strong grid (SCR=10, \(L_g=2.57\) mH), weak grid (SCR=3, \(L_g=8.6\) mH), and very weak grid (SCR=2, \(L_g=12.8\) mH). Table 4 summarizes the performance metrics before and after applying the time-delay compensation. In the strong grid case, the uncompensated system already exhibits a high-frequency oscillation at around 1.8 kHz with a resonance peak of 12 dB and a phase margin of only 5.82°. After compensation, the oscillation amplitude is attenuated by 92%, the total harmonic distortion (THD) of the grid current drops from 2.63% to 1.17%, and the phase margin improves to 41.6°. The point of common coupling (PCC) voltage distortion is reduced to 1.2%.
| Test Condition (\(L_g\) / SCR) | Metric | Without Compensation | With Compensation |
|---|---|---|---|
| \(L_g=2.57\) mH (SCR=10) | Oscillation frequency band | 1.2–1.9 kHz | No significant oscillation |
| Grid current THD | 2.63% | 1.17% | |
| PCC voltage distortion | 2.1% | 1.2% | |
| Phase margin | 5.82° | 41.6° | |
| \(L_g=8.6\) mH (SCR=3) | Oscillation frequency band | 1.1–2.0 kHz | No significant oscillation |
| Grid current THD | 4.13% | 1.56% | |
| PCC voltage distortion | 3.5% | 1.8% | |
| Phase margin | -3.2° | 38.9° | |
| \(L_g=12.8\) mH (SCR=2) | Oscillation frequency band | 0.9–2.2 kHz (unstable) | No significant oscillation |
| Grid current THD | 12.6% | 2.16% | |
| PCC voltage distortion | 8.7% | 2.0% | |
| Phase margin | -15.6° | 42.1° |
For the weak grid case (\(L_g=8.6\) mH, SCR=3), the uncompensated system suffers from a phase margin of -3.2° and a grid current THD of 4.13%. After applying the proposed compensation, the 75 Hz harmonic injection method accurately estimates the grid impedance with an error below 5%. The optimized controller parameters (\(k_p=0.115\), \(k_r=30\), \(k_C=0.3\)) together with the phase-lead compensator completely eliminate the oscillations. The THD reduces to 1.56%, and the phase margin increases to 38.9°. The dynamic response time to a grid current step change is 0.05 s with no overshoot. In the very weak grid scenario (\(L_g=12.8\) mH, SCR=2), the uncompensated system shows severe oscillations leading to instability, with a THD of 12.6% and a phase margin of -15.6°. The time-delay compensation, by moving the delay out of the active damping loop, ensures that the equivalent virtual resistance remains positive. The phase-lead compensator provides up to 90° of phase boost at critical frequencies, effectively restoring stability. The resulting THD is only 2.16%, and the PCC voltage distortion is kept below 2.0%. These results confirm that the proposed solution is robust across all tested grid conditions, making it suitable for a wide range of types of solar inverter deployed in weak or very weak grids.
A practical example of such an inverter is a hybrid solar inverter designed for off-grid and grid-tied applications, such as the one shown in the figure below. The hardware implementation of the proposed compensation algorithm on a digital signal processor or FPGA is straightforward, requiring only minor modifications to the existing control code. The overall energy efficiency is also improved because the compensation avoids excessive switching or power losses.
In addition to the steady-state performance, I have also evaluated the transient behavior under abrupt grid impedance changes. For instance, when the grid inductance suddenly jumps from 2.57 mH to 12.8 mH (simulating a grid disconnection event), the compensated system maintains stable operation with only a brief transient of 0.08 s, whereas the uncompensated system loses synchronism. This robustness is critical for types of solar inverter that must support islanding detection and ride-through capabilities. The impedance-based stability analysis further shows that the inverter output impedance phase angle remains above -90° at all intersecting frequencies, satisfying the Nyquist criterion. The gain margin is consistently above 12 dB, providing ample tolerance against parameter variations.
Table 5 summarizes the transient performance comparison under a grid impedance step change from 2.57 mH to 12.8 mH.
| Metric | Without Compensation | With Compensation |
|---|---|---|
| Settling time (99% of steady-state) | 0.35 s (oscillatory, unstable) | 0.08 s (no overshoot) |
| Peak grid current overshoot | 45% | 8% |
| Recovery of phase margin | Lost synchronism | 38.5° within 0.1 s |
I have also investigated the scalability of the proposed method to three-phase LCL-type inverters and other types of solar inverter, such as those with multi-level topologies or transformerless designs. The underlying principle of isolating the control delay from the active damping loop is topology-independent. However, the parameter tuning for the phase-lead compensator and the virtual admittance coefficients must be adjusted according to the specific LCL filter resonant frequency and the digital control sampling rate. For single-stage inverters, the DC-link dynamics may introduce additional low-frequency interactions, which I plan to address in future work. Nonetheless, for the common two-stage types of solar inverter, the proposed compensation strategy yields a universal solution.
In conclusion, my research provides a systematic methodology for suppressing high-frequency oscillations in grid-connected LCL-type inverters caused by digital control delays. By combining active damping loop compensation, phase-lead correction, and parameter optimization under stability margin constraints, I have achieved significant improvements in THD, phase margin, and dynamic response across strong, weak, and very weak grid scenarios. The results demonstrate that the proposed approach effectively addresses the oscillation problem that plagues many types of solar inverter in weak grids, thereby enhancing the reliability and power quality of distributed generation systems. Future work will focus on extending the method to multi-parallel inverters and considering the impact of communication delays in large-scale photovoltaic plants.