In the context of modern renewable energy systems, various types of solar inverters are employed to interface photovoltaic panels with the utility grid. Among these, single-phase LCL-type grid-connected inverters are widely used in distributed generation applications due to their excellent harmonic attenuation and current quality. However, when the grid impedance increases, as commonly occurs in weak grid environments, the stability of such inverters can be severely compromised. This paper presents a comprehensive study from a first-person perspective, detailing our development of an adaptive combined control strategy to enhance the robustness of grid-connected inverters under weak grid conditions, explicitly considering the influence of the phase-locked loop (PLL) and grid-voltage feedforward. Our approach integrates a lead-phase compensator with a non-ideal second-order generalized integrator (SOGI) in the feedforward path, with parameters optimally tuned via a genetic algorithm (GA) coupled with online impedance monitoring. The effectiveness is validated through both simulation and experimental results.
1. Introduction
The increasing penetration of renewable energy sources has led to a growing demand for reliable grid-connected inverters. Types of solar inverters range from string inverters to microinverters, each with distinct topologies such as L-filter, LC-filter, or LCL-filter configurations. Among these, the LCL filter offers superior high-frequency harmonic suppression but introduces resonance issues that must be actively damped. In weak grids, the grid impedance (predominantly inductive) varies significantly, interacting with the inverter’s output impedance and potentially causing instability. Previous studies have shown that both the PLL and the grid-voltage proportional feedforward deteriorate the phase margin (PM) of the system, leading to harmonic distortion or even oscillation. Various compensation techniques have been proposed, including notch filters, virtual impedance, and adaptive feedforward strategies. However, many of these methods either rely on accurate impedance estimation or involve complex parameter tuning. To address these challenges, we propose a hybrid scheme that combines a lead compensator with a SOGI-based feedforward filter, wherein the SOGI parameters are automatically adjusted using a genetic algorithm according to the estimated grid inductance. This work aims to improve the stability margin across a wide range of grid impedances while maintaining simplicity and adaptability.
2. Output Impedance Modeling of Grid-Connected Inverter
We consider a single-phase LCL grid-connected inverter as depicted in the system configuration. The inverter-side inductor \(L_1\), filter capacitor \(C\), and grid-side inductor \(L_2\) form the LCL filter. The grid impedance \(Z_g\) is assumed to be purely inductive (\(L_g\)) to represent the worst-case scenario. The control system includes a quasi-proportional-resonant (QPR) current controller \(G_i(s)\), a capacitor current active damping coefficient \(K_d\), a grid-voltage feedforward coefficient \(H_f\), and a PLL with transfer function \(G_{\text{PLL}}(s)\). The simplified control block diagram yields the inverter output impedance as seen from the point of common coupling (PCC).
The QPR controller is given by:
$$
G_i(s) = K_p + \frac{2 K_r \omega_i s}{s^2 + 2\omega_i s + \omega_0^2}
$$
where \(\omega_0\) is the fundamental angular frequency, \(\omega_i\) is the cutoff bandwidth, \(K_p\) is the proportional gain, and \(K_r\) is the resonant gain. The inverter bridge gain is \(K_{\text{PWM}} = V_{in} / V_{tri}\). The feedforward coefficient is set as \(H_f = 1/K_{\text{PWM}}\). The PLL transfer function (linearized around the fundamental frequency) is expressed as:
$$
G_{\text{PLL}}(s) = \frac{1}{2} \frac{k_{pl}(s – j\omega_0) + k_{il}}{(s – j\omega_0)^2 + u_{\text{pcc}}[k_{pl}(s – j\omega_0) + k_{il}]}
$$
The open-loop transfer function \(T_o(s)\) and the equivalent output impedance \(Z_{\text{req}}(s)\) of the inverter are derived from the block diagrams. The detailed expressions are:
$$
T_o(s) = \frac{K_{\text{PWM}} G_i(s)}{L_1 L_2 C s^3 + K_d K_{\text{PWM}} L_2 C s^2 + (L_1 + L_2) s}
$$
$$
Z_{\text{req}}(s) = \frac{G_i(s)[1 + T_o(s)]}{G_i(s) G_2(s) – H_f T_o(s) – G_i(s) I^* G_{\text{PLL}}(s) T_o(s)}
$$
where \(G_2(s) = \frac{L_1 C s^2 + K_d K_{\text{PWM}} C s + 1}{L_1 L_2 C s^3 + K_d K_{\text{PWM}} L_2 C s^2 + (L_1 + L_2) s}\). The system parameters used in this study are listed in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(U_d\) (V) | 400 | \(C\) (\(\mu\)F) | 5 |
| \(u_g\) (V) | 220 | \(K_p\) | 0.063 |
| \(f_s\) (kHz) | 10 | \(K_r\) | 3.14 |
| \(P\) (kW) | 6 | \(k_{pl}\) | 180 |
| \(L_1\) (mH) | 3 | \(k_{il}\) | 3143 |
| \(L_2\) (mH) | 1 | \(K_d\) | 0.1732 |
3. Stability Analysis via Impedance Criterion
To assess system stability under weak grid conditions, we employ the impedance-based stability criterion. The grid impedance \(Z_g = L_g s\) intersects with the inverter output impedance \(Z_{\text{req}}(s)\) at a certain frequency \(\omega_{\text{int}}\). The phase margin (PM) is defined as:
$$
\text{PM} = 180^\circ – \arg[Z_g(2\pi f_{\text{int}})] + \arg[Z_{\text{req}}(2\pi f_{\text{int}})]
$$
For a purely inductive grid, the condition reduces to \(\arg[Z_{\text{req}}(2\pi f_{\text{int}})] > -90^\circ\). When the grid is strong (\(L_g = 0\)), the inverter is stable if the current loop itself is stable. As \(L_g\) increases, the phase of \(Z_{\text{req}}\) at the intersection frequency decreases. Our analysis reveals that the proportional feedforward and the PLL significantly reduce the phase of the equivalent output impedance in the low-frequency range, thereby shrinking the phase margin. Figure 1 illustrates the Bode plots of the inverter output impedance under three conditions: without feedforward, with proportional feedforward, and with both feedforward and PLL. It is evident that the inclusion of PLL and feedforward causes the phase to drop below -90° at a lower frequency, making the system prone to instability as the grid inductance increases. This observation motivates the need for phase compensation.
4. Proposed Combined Improved Control Strategy
4.1 Lead-Phase Compensation
To restore the phase margin, we first introduce a lead compensator in series with the current loop. The transfer function of the lead compensator is:
$$
H_1(s) = \frac{\alpha s + 1}{\beta s + 1}
$$
where \(\alpha\) and \(\beta\) are positive constants. The maximum phase boost occurs at frequency \(\omega_m = 1/\sqrt{\alpha\beta}\), with the maximum compensation angle \(\theta_{\max} = \arctan\left(0.5\left(\sqrt{\alpha/\beta} – \sqrt{\beta/\alpha}\right)\right)\). By selecting \(\alpha = 0.0101\) and \(\beta = 0.0498\), we achieve a phase boost around the expected crossover frequency for a grid inductance of \(L_g = 2.57\) mH (SCR=10). After adding the lead compensator, the modified open-loop transfer function becomes:
$$
T’_o(s) = T_o(s) H_1(s)
$$
The new equivalent output impedance \(Z’_{\text{req}}(s)\) is obtained by substituting \(T’_o(s)\) into the impedance expression. The lead compensator effectively raises the phase at the intersection point, as shown in the Bode plot (Figure 9 in the original text). However, as the grid inductance further increases (e.g., to 8.56 mH), the PM again becomes insufficient, indicating the limitation of a single lead compensator.
4.2 Feedforward Path Modification with Non-Ideal SOGI
To further enhance the phase margin under higher grid impedance, we propose to replace the original proportional feedforward with a non-ideal second-order generalized integrator (SOGI) in the feedforward path. The SOGI transfer function is:
$$
G_I(s) = \frac{\omega_1^2 s}{s^2 + \omega_c s + \omega_1^2}
$$
where \(\omega_c\) is the cutoff frequency and \(\omega_1\) is the resonant frequency. By properly tuning \(\omega_c\) and \(\omega_1\), the SOGI can provide a phase lead in the frequency range of interest, compensating for the phase lag introduced by the PLL and the original feedforward. The block diagram after incorporating the lead compensator and the SOGI is shown conceptually. The resulting equivalent output impedance becomes:
$$
Z”_{\text{req}}(s) = \frac{1 + T’_o(s)}{G_2(s) – \frac{H_f T’_o(s) G_I(s)}{G_i(s) H_1(s)} – I^* G_{\text{PLL}}(s) T’_o(s)}
$$
The extra impedance term \(Z”_{\text{ext}}(s)\) now includes the effect of the SOGI, which improves the phase response.
4.3 Adaptive Parameter Tuning Using Genetic Algorithm
The two parameters \(\omega_c\) and \(\omega_1\) of the SOGI are not trivial to determine analytically, especially considering that the grid impedance varies with operating conditions. Therefore, we employ an adaptive optimization framework that combines online impedance estimation with a genetic algorithm (GA). The grid inductance estimate \(L_{g,\text{est}}\) is obtained via a real-time impedance monitoring technique (with a typical accuracy of ±20%). The GA is then used to minimize the following objective function:
$$
\min f = |\text{PM} – 30^\circ|
$$
subject to constraints:
$$
\omega \in (\omega_{\min}, \omega_{\max}), \quad \omega_c \in (\omega_{c,\min}, \omega_{c,\max})
$$
The PM is computed using Eq. (1) with \(Z”_{\text{req}}(s)\) and the estimated grid inductance. The GA searches over \(\omega_c\) and \(\omega_1\) to bring the PM as close as possible to a target of 30°, ensuring sufficient stability margin. Figure 15 in the original text shows the GA flow chart, and Figure 16 displays the optimization process for \(L_g = 8.56\) mH, where the optimal values converge to \(\omega_c = 13177\) rad/s and \(\omega_1 = 86.85\) rad/s after about 100 iterations. The final PM obtained is 29.7°, which meets the design requirement. Table 2 summarizes the phase margins under different grid conditions for the proposed combined strategy.
| Grid Inductance \(L_g\) (mH) | SCR | Phase Margin (deg) |
|---|---|---|
| 2.57 | 10 | 76.7 |
| 4.23 | 6 | 60.3 |
| 8.56 | 3 | 29.7 |
5. Simulation and Experimental Verification
To validate the proposed strategy, we built a 6 kW single-phase LCL grid-connected inverter model in MATLAB/Simulink and implemented the control algorithm on an RTU-BOX204 platform. The system parameters are listed in Table 1. We compared three scenarios: (i) no compensation, (ii) lead compensation only, and (iii) the proposed adaptive combined control (lead + SOGI with GA tuning). The total harmonic distortion (THD) of the grid current under different grid impedances is summarized in Table 3.
| Grid Inductance \(L_g\) | No Compensation | Lead Compensation | Proposed Combined Control |
|---|---|---|---|
| 2.57 mH | Severe distortion | 2.69% | 2.12% |
| 8.56 mH | Severe distortion | 14.9% | 2.93% |
As shown, without any compensation, the inverter fails to operate stably under both weak grid conditions. The lead compensator alone can only maintain stability at \(L_g = 2.57\) mH (THD = 2.69%), but at the higher inductance of 8.56 mH, the THD rises to 14.9%, indicating instability. In contrast, our proposed combined control strategy achieves THD values of 2.12% and 2.93%, respectively, well within the IEEE 519 standard for grid connection. Experimental waveforms captured from the RTU-BOX204 platform confirm these results. Figure 22 corresponds to the traditional control at \(L_g = 2.57\) mH showing distorted current; Figure 23 shows improvement with lead compensation; Figure 24 shows that lead compensation fails at \(L_g = 8.56\) mH; Figures 25 and 26 demonstrate clean sinusoidal current with the proposed strategy at both inductance levels. Additionally, load transient tests (Figure 27 and 28) verify that the controller maintains stability during step changes from full load to half load and vice versa.
6. Conclusion
This paper presented a comprehensive adaptive combined control strategy for grid-connected inverters operating in weak grids, taking into account the adverse effects of PLL and grid-voltage feedforward. By combining a lead-phase compensator with a non-ideal SOGI in the feedforward path and utilizing a genetic algorithm for online parameter optimization, we significantly improved the phase margin of the inverter output impedance. The proposed method enhances the robustness of various types of solar inverters under varying grid impedance conditions, ensuring stable operation even with large grid inductance. Simulation and experimental results confirmed that the THD of the grid current remains below 3% for SCR values down to 3, demonstrating the effectiveness and practicality of the approach. Future work will focus on accelerating the GA optimization to enable real-time parameter updates for highly dynamic grid conditions.