In this paper, I present an adaptive combined improved control strategy to enhance the stability of grid-connected inverters under weak grid conditions. Various types of solar inverter topologies, such as single-phase LCL-type inverters, are widely used in distributed generation systems. However, when the grid impedance increases, the phase margin of the system decreases significantly due to the influence of the phase-locked loop and grid voltage proportional feedforward, potentially leading to instability. I propose a combination of lead-phase compensation and a non-ideal second-order generalized integrator in the feedforward channel to improve the phase margin. Furthermore, an online impedance monitoring technique integrated with a genetic optimization algorithm is employed to adaptively tune the controller parameters, making the design more flexible and efficient. The effectiveness of the proposed method is validated through both simulation and experimental results for various types of solar inverter configurations.
1. System Modeling and Stability Analysis
I consider a single-phase LCL grid-connected inverter for a types of solar inverter system. The inverter parameters are listed in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link voltage | \(U_d\) | 400 V |
| Grid voltage | \(u_g\) | 220 V |
| Switching frequency | \(f_s\) | 10 kHz |
| Rated power | \(P\) | 6 kW |
| Inverter-side inductor | \(L_1\) | 3 mH |
| Grid-side inductor | \(L_2\) | 1 mH |
| Filter capacitor | \(C\) | 5 μF |
| Proportional gain of QPR | \(k_p\) | 0.063 |
| Resonant gain of QPR | \(k_r\) | 3.14 |
| PLL proportional gain | \(k_{pl}\) | 180 |
| PLL integral gain | \(k_{il}\) | 3143 |
| Capacitor current damping coefficient | \(k_d\) | 0.1732 |
The control block diagram of the grid-connected inverter is shown in Figure 1. The inverter uses a quasi-proportional-resonant controller \(G_i(s)\):
$$ G_i(s) = K_p + \frac{2K_r\omega_i s}{s^2 + 2\omega_i s + \omega_0^2} $$
where \(\omega_i\) is the bandwidth and \(\omega_0\) is the fundamental angular frequency. The feedforward coefficient \(H_f = 1/K_{PWM}\) and the PWM gain \(K_{PWM} = V_{in}/V_{tri}\).
The phase-locked loop transfer function is:
$$ G_{PLL}(s) = \frac{1}{2} \cdot \frac{k_{pl}(s-j\omega_0) + k_{il}}{(s-j\omega_0)^2 + u_{pcc}[k_{pl}(s-j\omega_0)+k_{il}]} $$
The equivalent output impedance of the inverter \(Z_{req}(s)\) is derived as:
$$ Z_{req}(s) = \frac{G_i(s)[1+T_o(s)]}{G_i(s)G_2(s) – H_f T_o(s) – G_i(s)I^* G_{PLL}(s)T_o(s)} $$
where \(T_o(s)\) is the open-loop transfer function and \(G_2(s)\) is the plant transfer function. To ensure stability in a weak grid with impedance \(Z_g(s)=L_g s\), the phase margin condition must be satisfied:
$$ \arg[Z_{req}(j\omega_g)] > -90^\circ $$
Figure 5 shows the Bode plots of the equivalent output impedance. Without feedforward, the system remains robust. However, with proportional feedforward and PLL, the phase margin deteriorates rapidly as \(L_g\) increases.
This instability issue affects many types of solar inverter designs, especially when the short-circuit ratio (SCR) is low. For typical types of solar inverter in weak grids, adaptive compensation is necessary.
2. Proposed Combined Control Strategy
2.1 Lead-Phase Compensation
I first introduce a lead compensator \(H_1(s)\) in the current control loop:
$$ H_1(s) = \frac{\alpha s + 1}{\beta s + 1} $$
where \(\alpha=0.0101\) and \(\beta=0.0498\) are selected to provide maximum phase boost at the crossover frequency. The phase boost is:
$$ \theta = \arctan\left(0.5\left(\sqrt{\frac{\alpha}{\beta}} – \sqrt{\frac{\beta}{\alpha}}\right)\right) $$
After adding the lead compensator, the new open-loop transfer function becomes:
$$ T’_o(s) = \frac{K_{PWM} G_i(s) H_1(s)}{L_1 L_2 C s^3 + K_d K_{PWM} L_2 C s^2 + (L_1+L_2)s} $$
The compensated equivalent impedance \(Z’_{req}(s)\) is given by:
$$ Z’_{req}(s) = Z_1(s) Z’_{ext}(s) $$
where \(Z’_{ext}(s)\) represents the additional impedance due to the compensator. As shown in Figure 10, the phase margin improves at \(L_g=2.57\) mH (SCR=10) to 56.5°, but for larger \(L_g=8.56\) mH (SCR=3), the margin drops below zero.
2.2 Feedforward with Non-Ideal SOGI
To further enhance the phase margin for all types of solar inverter in weak grids, I propose replacing the proportional feedforward with a non-ideal second-order generalized integrator (SOGI) \(G_I(s)\):
$$ G_I(s) = \frac{\omega_1^2 s}{s^2 + \omega_c s + \omega_1^2} $$
where \(\omega_c\) is the cutoff frequency and \(\omega_1\) is the maximum gain frequency. The overall output impedance after both modifications is:
$$ Z”_{req}(s) = Z_1(s) Z”_{ext}(s) $$
The new extra impedance \(Z”_{ext}(s)\) provides additional phase lift in the low-frequency range. The phase margins for different grid inductances after adding SOGI are summarized in Table 2.
| Grid inductance \(L_g\) | SCR | Phase margin (deg) |
|---|---|---|
| 2.57 mH | 10 | 76.7 |
| 4.23 mH | 6 | 60.3 |
| 8.56 mH | 3 | 29.7 |
All values exceed the required 0° margin, confirming stability. This demonstrates that the combined strategy is effective for all types of solar inverter operating in weak grids.
3. Adaptive Parameter Tuning Using Genetic Algorithm
The SOGI parameters \(\omega_c\) and \(\omega_1\) are traditionally difficult to design manually. I employ an online impedance monitoring technique to estimate the grid inductance \(L_{g,est}\), and then use a genetic algorithm to optimize the SOGI parameters. The optimization objective is to minimize the phase error:
$$ \min f = |PM – 30^\circ| $$
subject to constraints on \(\omega_c\) in [10^4, 1.5×10^4] rad/s and \(\omega_1\) in [50, 100] rad/s. The genetic algorithm evolves over 100 generations to find the optimal values. For \(L_g=8.56\) mH, the optimized parameters are \(\omega_c=13177\) rad/s and \(\omega_1=86.85\) rad/s. The convergence process is illustrated in Figure 16, showing that the fitness function rapidly decreases to an acceptable level.
Figure 18 verifies that even with a ±20% estimation error in grid inductance, the system remains stable, confirming the robustness of the adaptive tuning for various types of solar inverter installations.
4. Simulation and Experimental Results
I built a 6 kW single-phase LCL grid-connected inverter simulation model in MATLAB/Simulink and conducted experiments using an RTU-BOX204 platform. The results are compared for three control schemes: conventional proportional feedforward, lead compensation only, and the proposed combined method.
4.1 Simulation Results
Figure 19 shows the output current waveforms and total harmonic distortion for different grid inductances. At \(L_g=2.57\) mH, the lead compensation gives THD=2.69% and the combined method gives THD=2.12%. At \(L_g=8.56\) mH, the lead compensation fails (THD=14.9%) while the combined method maintains THD=2.93%. Figure 20 demonstrates that during a sudden grid impedance change, the combined strategy maintains stable operation without severe distortion.
4.2 Experimental Results
Experimental waveforms are shown in Figures 22 to 28. With conventional control at \(L_g=2.57\) mH, the current is severely distorted (Figure 22). Lead compensation improves the waveform (Figure 23), but at \(L_g=8.56\) mH it still distorts (Figure 24). The proposed combined strategy remains stable for both small and large grid inductances (Figures 25 and 26). Load transient tests (full-load to half-load and vice versa) in Figures 27 and 28 confirm that the proposed method maintains high-quality output under varying operating conditions, which is critical for all types of solar inverter deployed in weak grids.
5. Conclusion
In this work, I have developed an adaptive combined improved control strategy for grid-connected inverters in weak grids, addressing the detrimental effects of the phase-locked loop and grid voltage feedforward. The key contributions are:
- Lead-phase compensation provides initial phase margin improvement but is insufficient for high grid impedance.
- Adding a non-ideal SOGI in the feedforward path further boosts the phase margin, enabling stable operation for SCR as low as 3.
- Genetic algorithm-based adaptive tuning simplifies parameter design and ensures robustness against grid impedance estimation errors.
The proposed method is applicable to various types of solar inverter topologies and significantly extends the impedance adaptation range. Simulation and experimental results consistently verify the effectiveness and practicality of the strategy.