In the context of weak grid environments, the inherent delay in digital control significantly impairs the resonance suppression performance of active damping strategies for LCL-type grid-connected inverters. This paper addresses the challenge by proposing a robust delay compensation control method that enhances the stability and robustness of the system under varying grid impedances. The study focuses on various types of solar inverter topologies, particularly the LCL-filtered inverter, which is widely used due to its superior harmonic attenuation. Through frequency-domain analysis, the specific impacts of control delay on resonance damping and phase margin are examined. A compensation link is introduced to eliminate the time delay in the capacitor current feedback loop, effectively restoring the active damping as a purely resistive impedance. Additionally, a phase lead correction is implemented in the forward path to counteract the phase lag caused by the delay. The unknown grid impedance, critical for accurate compensation, is estimated using a non-characteristic harmonic injection method. Parameter optimization for the current controller and active damping gain is performed under stability margin constraints. Both simulation and experimental results validate the effectiveness of the proposed strategy in maintaining system stability and power quality across a wide range of grid inductance values.
Distributed energy resources are increasingly integrated into the power grid, leading to the widespread adoption of LCL-filtered inverters for their excellent high-frequency harmonic suppression capabilities. However, the inherent resonance of the LCL filter, combined with the variable grid impedance in weak grids, poses significant stability challenges. Digital control introduces a total delay of approximately 1.5 sampling periods, which can transform the active damping from a positive resistor to a negative resistor at certain frequencies, thus rendering the resonance suppression ineffective. This paper systematically investigates the impact of control delay on different types of solar inverter configurations and presents a comprehensive compensation strategy. The proposed method not only mitigates the adverse effects of delay but also enhances the inverter’s robustness to grid impedance variations, making it suitable for modern weak grid scenarios.
System Modeling and Control Delay Analysis
The single-phase LCL-type grid-connected inverter model is considered, consisting of a DC source, a full-bridge inverter, an LCL filter, and the grid. The filter parameters are: inverter-side inductor \(L_1\), capacitor \(C\), and grid-side inductor \(L_2\). The grid impedance is represented by \(L_g\). The digital control delay is modeled as \(G_d(s) = e^{-1.5sT_s}\), where \(T_s\) is the sampling period. For small-signal analysis, a third-order Padé approximation is used to facilitate the study of zero-pole behavior.
The active damping is implemented by feeding back the capacitor current \(i_c\) through a gain \(K_C\). Considering the delay, the equivalent virtual impedance \(Z_{eq}\) seen by the filter capacitor becomes:
$$ Z_{eq} = \frac{L_1}{C K_C K_{PWM}} e^{1.5 s T_s} $$
where \(K_{PWM} = U_{dc}/V_{tri}\). The real and imaginary parts of this impedance are:
$$ R_{eq} = \frac{L_1}{C K_C K_{PWM} \cos(1.5 \omega T_s)} $$
$$ X_{eq} = \frac{L_1}{C K_C K_{PWM} \sin(1.5 \omega T_s)} $$
The real part \(R_{eq}\) is positive only for frequencies below \(f_s/6\) (where \(f_s = 1/T_s\)) and becomes negative for frequencies between \(f_s/6\) and \(f_s/2\). This negative resistance behavior eliminates the damping effect and can lead to instability. The open-loop transfer function of the system with delay is:
$$ T(s) = \frac{K_{PWM} G_i(s) G_d(s)}{s^3 L_1 (L_2 + L_g) C + s^2 (L_2 + L_g) C K_{PWM} K_C G_d(s) + s (L_1 + L_2 + L_g)} $$
Here, \(G_i(s)\) is the current controller. The resonance frequency \(f_r\) is:
$$ f_r = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2 + L_g}{L_1 (L_2 + L_g) C}} $$
Bode analysis reveals that the delay introduces a phase lag and a resonant peak, reducing the phase margin and potentially causing unstable crossings of \(-180^\circ\). Table 1 summarizes the grid strength classification based on the Short Circuit Ratio (SCR).
| SCR Range | Grid Strength | \(L_g\) Range (mH) |
|---|---|---|
| SCR > 10 | Strong | < 2.56 |
| 2 < SCR < 3 | Weak | 8.6 – 12.8 |
| SCR < 2 | Very Weak | > 12.8 |
Control Delay Compensation
Active Damping Loop Compensation
To eliminate the delay from the capacitor current feedback path, a compensation link is derived by moving the delay outside the active damping loop. Starting from the original block diagram, the goal is to make the delay appear only in the forward path. Through equivalent transformations, the required compensator is found to be:
$$ G(s) = \left[ 1 – G_d(s) \right] K_C G_{iC}(s) $$
where \(G_{iC}(s) = s K_{PWM} / [L_1 (s^2 + \omega_r^2)]\). This compensator can be implemented digitally using known transfer functions. The unknown grid impedance \(L_g\) needed for the compensator is estimated online by injecting a non-characteristic harmonic (e.g., 75 Hz) current and measuring the voltage and current at the Point of Common Coupling (PCC). The grid impedance is calculated as:
$$ Z_g(h) = \frac{u_{pcc(h)}}{i_{pcc(h)}} $$
$$ R_g = \text{Re}[Z_g(h)], \quad \omega_h L_g = \text{Im}[Z_g(h)] $$
After adding this compensation, the open-loop transfer function becomes:
$$ T'(s) = \frac{K_{PWM} G_i(s) G_d(s)}{s^3 L_1 (L_2 + L_g) C + s^2 (L_2 + L_g) C K_{PWM} K_C + s (L_1 + L_2 + L_g)} $$
The \(s^2\) term no longer contains the delay, so the active damping behaves as a constant positive resistance, effectively suppressing the resonance peak. Bode plots confirm that the resonant peak is eliminated.
Phase Lag Compensation
Despite the removal of delay from the damping loop, the forward path still suffers from phase lag due to \(G_d(s)\), which reduces the phase margin (PM). To compensate, a phase lead corrector is introduced in the forward path. The lead compensator is designed as:
$$ G_{lead}(s) = \frac{s + \omega_m \sqrt{\frac{1 – \sin\theta_m}{1 + \sin\theta_m}}}{s + \omega_m \sqrt{\frac{1 + \sin\theta_m}{1 – \sin\theta_m}}} $$
where \(\omega_m\) is the frequency at which maximum phase lead occurs, and \(\theta_m\) is the maximum phase lead angle. Selecting \(\omega_m = 306 \times 2\pi\) rad/s (the crossover frequency of \(T'(s)\)) and \(\theta_m = 45^\circ\) yields the compensator parameters. The overall open-loop transfer function with the lead compensator is:
$$ T_A(s) = \frac{K_{PWM} G_i(s) G_d(s) G_{lead}(s)}{s^3 L_1 (L_2 + L_g) C + s^2 (L_2 + L_g) C K_{PWM} K_C + s (L_1 + L_2 + L_g)} $$
Bode analysis shows that the phase margin improves from 5.82° to 41.6°, while the gain margin decreases from 20 dB to 10.4 dB. The remaining margin is still acceptable, but further parameter optimization can improve robustness.
Parameter Optimization
Current Controller Parameters
A quasi-proportional-resonant (PR) controller is used for the grid current loop:
$$ G_i(s) = k_p + \frac{2 \omega_i k_r s}{s^2 + 2 \omega_i s + \omega_0^2} $$
where \(\omega_0 = 100\pi\) rad/s and \(\omega_i = \pi\) rad/s. The proportional gain \(k_p\) is determined by the crossover frequency \(f_c = 1\) kHz (one-tenth of the switching frequency):
$$ k_p \approx \frac{2\pi f_c (L_1 + L_2 + L_g)}{K_{PWM}} = 0.115 $$
The resonant gain \(k_r\) is constrained by the required base-frequency gain \(T_{f0} > 73\) dB and phase margin \(PM > 45^\circ\). Solving yields:
$$ k_r \in (27.23, 61.09) $$
A value of \(k_r = 30\) is selected.
Active Damping Coefficient
The active damping gain \(K_C\) affects both the damping of the resonant peak and the phase margin. To maintain stability margins \(GM > 10\) dB and \(PM > 30^\circ\), the allowable range for \(K_C\) is found from Bode analysis:
$$ K_C \in (0.0206, 0.455) $$
A value of \(K_C = 0.3\) is chosen, resulting in \(GM = 18.5\) dB and \(PM = 42.1^\circ\). The zero-pole map for varying \(L_g\) from 0 to 12.8 mH shows that all poles remain in the left half-plane, confirming robust stability.
Robustness Analysis via Impedance Criterion
Using the impedance-based stability criterion, the inverter output impedance must have a phase angle greater than \(-90^\circ\) at the intersection with the grid impedance. The output impedance before compensation is:
$$ Z_{out0} = \frac{s^3 L_1 (L_2 + L_g) C + s^2 (L_2 + L_g) C K_C K_{PWM} G_d + s (L_1 + L_2 + L_g) + G_d G_i K_{PWM}}{s^2 L_1 C + s C K_C K_{PWM} G_d + 1} $$
After compensation, it becomes:
$$ Z_{out1} = \frac{s^3 L_1 (L_2 + L_g) C + s^2 (L_2 + L_g) C K_C K_{PWM} + s (L_1 + L_2 + L_g) + G_d G_i K_{PWM} G_{lead}}{s^2 L_1 C + s C K_C K_{PWM} + 1} $$
Bode comparison shows that \(Z_{out1}\) has a significantly higher phase margin (minimum 16.4°) compared to \(Z_{out0}\), which can become negative. This confirms that the proposed strategy greatly enhances robustness against grid impedance variation.
Simulation and Experimental Results
Simulations were conducted in MATLAB/Simulink with parameters listed in Table 2.
| Parameter | Value |
|---|---|
| DC voltage \(U_{dc}\) | 700 V |
| Grid line voltage \(V_{grms}\) | 380 V |
| Rated power \(P_{out}\) | 18 kW |
| Grid fundamental frequency \(f_0\) | 50 Hz |
| Switching frequency \(f_{sw}\) | 10 kHz |
| Inverter-side inductor \(L_1\) | 2 mH |
| Filter capacitor \(C\) | 1.5 μF |
| Grid-side inductor \(L_2\) | 0.5 mH |
First, a system using only the phase lead compensator (without removing delay from the damping loop) was tested. At \(L_g = 12.8\) mH (SCR=2), the grid current THD reached 12.6%, violating the IEEE 519 standard (THD < 5%). When the full proposed compensation was applied, the current THD reduced to 2.16% even at \(L_g = 12.8\) mH, and remained below 2% for stronger grid conditions. The waveforms show clean sinusoidal current and stable PCC voltage.
Experimental validation was performed on a hardware platform. For various types of solar inverter configurations, the proposed strategy was tested with grid inductances of 0.4 mH, 2.57 mH, 8.6 mH, and 12.8 mH. Before applying the compensation, weak grid conditions caused severe current distortion and resonance. After activating the compensation, the current quality improved dramatically, confirming the effectiveness of the method across different types of solar inverter designs.
Conclusion
This paper presents a robust delay compensation strategy for LCL-type grid-connected inverters operating in weak grids. By introducing a compensation link that removes the control delay from the active damping loop, the virtual impedance becomes purely resistive, eliminating resonance. A phase lead corrector in the forward path restores the phase margin. The unknown grid impedance is estimated via harmonic injection. Parameter optimization ensures stability margins under varying grid conditions. Both simulation and experiments demonstrate that the proposed method maintains low THD and high stability for a wide range of grid inductances, making it suitable for various types of solar inverter applications in modern distributed generation systems.