Integrating renewable energy sources into the power grid relies heavily on the performance and stability of on-grid inverters. The LCL filter is a prevalent choice for connecting these inverters due to its superior high-frequency harmonic attenuation compared to simple L filters. However, the LCL filter introduces a resonant peak that can destabilize the system, necessitating effective damping methods. Active damping, implemented through control algorithms, is favored over passive resistor-based damping for its lack of power loss. Among various feedback schemes, inverter-side current feedback active damping (ICFAD) offers the combined benefits of resonance suppression and inherent overcurrent protection for the inverter switches, reducing sensor count. A significant challenge in digital implementation is the inherent computational and modulation delay, which distorts the intended damping effect, reduces system robustness, and can lead to instability, especially under weak grid conditions characterized by variable grid impedance. This article presents a comprehensive analysis and a novel improved control strategy for LCL-type on-grid inverters, focusing on mitigating the detrimental effects of control delay to enhance stability and robustness.
The primary circuit of a single-phase LCL-type on-grid inverter is considered. It consists of a DC source, a full-bridge inverter, an LCL filter (with inverter-side inductance \(L_1\), filter capacitance \(C\), and grid-side inductance \(L_2\)), and the grid with an equivalent impedance \(L_g\). The key variables are the inverter output voltage \(u_1\), the inverter-side current \(i_1\), the capacitor voltage \(u_c\), and the grid-injected current \(i_g\). The fundamental transfer function from \(u_1\) to \(i_1\) is given by:
$$G_{i_1u_1}(s) = \frac{(L_2 + L_g)Cs^2 + 1}{s L_1 (L_2 + L_g) C (s^2 + \omega_{res}^2)}$$
where the resonant angular frequency \(\omega_{res} = 2\pi f_{res}\) is:
$$\omega_{res} = \sqrt{\frac{L_1 + L_2 + L_g}{L_1(L_2 + L_g)C}}$$
The presence of \(L_g\) indicates that the grid impedance directly influences the resonant frequency, a critical factor in weak grid analysis for on-grid inverters.
Mathematical Modeling and Analysis of Traditional ICFAD
In a digital control system for an on-grid inverter, the control delay \(G_d(s)\) is a critical component, typically modeled as \(G_d(s) = e^{-1.5sT_s}\), where \(T_s\) is the sampling period. This delay accounts for computation and pulse-width modulation (PWM) update times. The block diagram of a traditional inverter-side current feedback active damping (ICFAD) system incorporates this delay, a current controller \(G_c(s)\) (often a PR controller for zero steady-state error at the fundamental frequency), the inverter gain \(K_{PWM}\), the plant \(G_{i_1u_1}(s)\), and an active damping coefficient \(H_{i_1}\).
The open-loop transfer function for the traditional ICFAD-controlled on-grid inverter is:
$$T_{ol2}(s) = \frac{K_{PWM} G_{i_1u_1}(s) G_d(s) G_c(s)}{1 + H_{i_1} K_{PWM} G_d(s) G_{i_1u_1}(s)}$$
Analyzing this transfer function reveals a significant drawback: while it suppresses the resonant peak, it simultaneously reduces the low-frequency gain of the system. This reduction in low-frequency gain can degrade the reference tracking performance and dynamic response of the on-grid inverter. Furthermore, stability analysis dictates that for the traditional ICFAD method to remain stable, the resonant frequency \(f_{res}\) must be less than \(f_s/6\), where \(f_s\) is the sampling frequency. This imposes a strict constraint on the filter design and limits the applicability of the on-grid inverter, especially when grid impedance variations are considered.
Proposed Improved Control Strategy with Delay Compensation
To address the limitations of the traditional approach, a two-fold improved control strategy is proposed for the LCL-type on-grid inverter.
Step 1: Enhanced Active Damping with Gain Function
First, to eliminate the problem of reduced low-frequency gain, the constant damping coefficient \(H_{i_1}\) is replaced by a dynamic gain function \(G_{AD}(s)\). The modified inner active damping loop is designed as:
$$G_{AD}(s) = \frac{1}{K_{PWM}} \cdot \frac{\mu_0 s}{s + \gamma}$$
where \(\mu_0\) is a gain coefficient and \(\gamma\) controls the bandwidth of this damping path. This modification effectively decouples the damping action from the low-frequency control performance. The open-loop transfer function of the on-grid inverter with this improved ICFAD becomes:
$$T_{olAD}(s) = \frac{K_{PWM} G_d(s) G_{i_1u_1}(s) G_c(s)}{1 + G_{AD}(s) K_{PWM} G_d(s) G_{i_1u_1}(s)}$$
This active damping method is equivalent to virtually inserting an impedance \(Z_{eq1}\) in series with the inverter-side inductor \(L_1\). The equivalent virtual impedance is derived as:
$$Z_{eq1} = G_{AD}(s) K_{PWM} G_d(s) = \frac{s \mu_0}{s + \gamma} e^{-1.5sT_s}$$
This virtual impedance can be decomposed into a frequency-dependent virtual resistance \(R_{eq1}(f)\) and virtual reactance \(X_{eq1}(f)\). The sign of \(R_{eq1}(f)\) determines whether it provides positive damping (stabilizing) or negative damping (destabilizing). Analysis shows that with this improved structure, the frequency range over which \(R_{eq1}(f) > 0\) (the positive damping region) can be extended up to \((0, f_s/3)\), which is a significant improvement over the traditional method’s \((0, f_s/6)\) limit. This expansion enhances the stability margin of the on-grid inverter.
Step 2: Delay Compensation via a Cascaded Lead Compensator
Despite the improvement, the control delay \(e^{-1.5sT_s}\) within \(Z_{eq1}\) still causes a phase lag. This lag makes the virtual impedance highly frequency-sensitive and can cause the effective resonant frequency of the system to shift into a region where \(R_{eq1}(f) < 0\) (negative resistance) when grid impedance \(L_g\) increases, threatening the robustness of the on-grid inverter in weak grids.
To combat this, a cascaded lead compensator \(G_p(s)\) is introduced into the active damping loop. The proposed compensator is designed as:
$$G_p(s) = \left( \frac{p+1}{1 + p e^{-sT_s}} \right)^2, \quad 0 < p \le 1$$
The parameter \(p\) controls the amount of phase lead. When \(p\) is close to 1, it provides significant phase advance around the critical frequencies, effectively compensating for the phase lag introduced by the control delay. The final block diagram of the proposed robust control strategy for the on-grid inverter is shown below, integrating the gain function and the delay compensator.
The equivalent virtual impedance with the complete proposed strategy is now:
$$Z_{eq2} = G_{AD}(s) G_p(s) K_{PWM} G_d(s) = \frac{s \mu_0}{s + \gamma} \left( \frac{p+1}{1 + p e^{-sT_s}} \right)^2 e^{-1.5sT_s}$$
The corresponding expressions for \(R_{eq2}(f)\) and \(X_{eq2}(f)\) are more complex but reveal the key benefit. By properly choosing \(p\) (e.g., \(p=0.92\)), the positive damping region where \(R_{eq2}(f) > 0\) can be dramatically expanded to approximately \((0, 0.48f_s)\). This ensures that even under wide variations of grid impedance \(L_g\) in a weak grid, the system’s resonant frequency is highly likely to remain within this positive damping zone, guaranteeing stability and robustness for the on-grid inverter.
Parameter Design and Stability Analysis
The performance of the proposed on-grid inverter control strategy depends on the careful selection of parameters \(\mu_0\), \(\gamma\), and \(p\). A systematic design method is proposed to achieve a balance between resonance suppression, stability margin, and dynamic performance.
The current controller \(G_c(s)\) is a Proportional-Resonant (PR) type:
$$G_c(s) = k_p + \frac{2k_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2}$$
where \(k_p\) and \(k_r\) are the proportional and resonant gains, \(\omega_c\) is the cutoff bandwidth, and \(\omega_0\) is the fundamental grid angular frequency.
The parameter \(\gamma\) is designed based on the desired damping characteristic. We define a damping ratio \(\lambda\) representing the proportion of virtual resistance to virtual reactance at the nominal resonant frequency \(\lambda = |R_{eq} / X_{eq}|_{f=f_{res}}\). A larger \(\lambda\) indicates stronger and more effective damping. By setting a target (e.g., \(\lambda > 10\)), a suitable range for \(\gamma\) can be determined from the derived equations. Typically, a larger \(\gamma\) within the stable range is preferred.
With \(\gamma\) fixed, \(\mu_0\) is tuned. While a larger \(\mu_0\) increases the virtual resistance and improves damping, it can also push the system poles towards instability. Root locus analysis in the discrete domain is used to select a value for \(\mu_0\) that provides strong damping while maintaining an adequate stability margin for the on-grid inverter.
The compensator parameter \(p\) is chosen as a trade-off. \(p=1\) offers maximum phase lead and the widest positive damping region but may introduce excessive gain at high frequencies. A value like \(p=0.92\) provides a very wide positive damping region up to \(0.48f_s\) while maintaining reasonable high-frequency gain.
The stability and robustness are rigorously analyzed by examining the closed-loop pole locations in the z-domain as the grid impedance \(L_g\) varies from 0 to a value representing a very weak grid. The proposed strategy demonstrates that all poles remain inside the unit circle for this entire range, confirming robust stability. In contrast, the poles for the system without delay compensation may approach or even cross the unit circle boundary for certain \(L_g\) values.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (RMS) | \(U_g\) | 220 V |
| Sampling/Switching Frequency | \(f_s\) | 10 kHz |
| Grid Fundamental Frequency | \(f_0\) | 50 Hz |
| Inverter-side Inductor | \(L_1\) | 3.6 mH |
| Grid-side Inductor | \(L_2\) | 1.6 mH |
| Filter Capacitor | \(C\) | 4.7 μF |
| Damping Gain Coefficient | \(\mu_0\) | 4 |
| Damping Bandwidth | \(\gamma\) | \(1.6 \times 10^4\) rad/s |
| Compensator Parameter | \(p\) | 0.92 |
| PR Controller Proportional Gain | \(k_p\) | 3 |
| PR Controller Resonant Gain | \(k_r\) | 800 |
Simulation and Experimental Verification
The proposed control strategy for the on-grid inverter was validated through detailed simulations and experiments on a 2 kW single-phase prototype. The parameters used are listed in Table 1.
Simulation Results
Simulations were conducted under both strong grid (\(L_g = 0\) mH) and weak grid (\(L_g = 4\) mH) conditions. The performance of the traditional ICFAD, the improved ICFAD (with gain function only), and the fully proposed strategy (with gain function and delay compensator) were compared.
First, the improved ICFAD successfully restored the low-frequency gain and stabilized the system where the traditional method showed deteriorating waveforms. Second, and more critically, under conditions where the resonant frequency was close to the stability boundary of the improved ICFAD, the system exhibited significant oscillation. Upon activating the proposed delay compensator, the grid current (\(i_g\)) and point of common coupling voltage (\(u_{pcc}\)) became stable and smooth.
Table 2 summarizes the harmonic performance under a weak grid scenario (\(L_g=4\) mH) with the proposed strategy active. The Total Harmonic Distortion (THD) of the grid current is well below the standard 5% limit, demonstrating the excellent steady-state performance of the controlled on-grid inverter.
| Variable | Fundamental Magnitude | Total Harmonic Distortion (THD) |
|---|---|---|
| Grid Current (\(i_g\)) | 19.9 A | 1.34% |
| PCC Voltage (\(u_{pcc}\)) | 154.5 V | 3.39% |
Experimental Results
The experimental platform was built using a TMS320F28335 DSP. The results consistently corroborated the simulations. With \(L_g=0\) mH and a high resonant frequency, the traditional ICFAD failed, while the improved ICFAD worked. However, pushing the system to its limit showed instability with the improved ICFAD, which was immediately corrected by enabling the delay compensator.
Under a weak grid condition (\(L_g=4\) mH), the superiority of the proposed method was even more pronounced. The waveforms for the traditional and improved ICFAD methods showed visible oscillations, whereas the proposed strategy maintained clean, stable grid current and voltage. Furthermore, a load step-change test (from half to full load) confirmed the excellent dynamic response and stability of the on-grid inverter with the proposed controller, as the current tracked its reference rapidly and smoothly without any instability.
Conclusion
This article has presented a robust control strategy for LCL-type on-grid inverters, specifically designed to overcome the limitations imposed by digital control delays. The strategy involves two key innovations: first, replacing the constant damping coefficient with a dynamic gain function to preserve low-frequency performance; and second, introducing a novel cascaded lead compensator in the active damping loop to directly counteract the phase lag from the delay. This compensation dramatically expands the frequency range over which the equivalent virtual impedance provides positive damping, thereby greatly enhancing the system’s stability margin and its robustness against grid impedance variations—a critical requirement for reliable operation in weak grids. Comprehensive stability analysis, parameter design guidelines, simulation results, and experimental verification on a prototype on-grid inverter have conclusively demonstrated the effectiveness and practical viability of the proposed method. This strategy provides a valuable solution for ensuring the high-performance and reliable integration of renewable energy sources through advanced on-grid inverter controls.