We present a comprehensive study on a frequency-adaptive proportional resonant (PR) control strategy designed for F-type three-level grid-connected inverters. Traditional three-level inverters, such as NPC and T-type topologies, suffer from high costs and significant power losses, especially under non-ideal grid conditions including voltage imbalance and frequency fluctuations. The F-type topology offers a cost-effective solution by requiring only one switch per phase leg to withstand full DC-link voltage, while the others handle half the voltage. However, its control performance under adverse grid conditions remains a challenge. Our proposed frequency-adaptive PR controller dynamically adjusts the resonant frequency to track the actual grid frequency, thereby maintaining infinite gain at the fundamental frequency and achieving zero steady-state error. Experimental results on a 2.2 kW F-type inverter platform demonstrate that the proposed strategy reduces total harmonic distortion (THD) of the grid-connected current by 0.35% under unbalanced grid voltages and by 0.45% under frequency fluctuations, compared to conventional quasi-PR control. These findings confirm the effectiveness and superiority of the adaptive approach in enhancing grid-connected power quality for various types of solar inverter applications.
1. Introduction
Renewable energy integration, particularly photovoltaic systems, relies heavily on efficient power conversion stages. Among various topologies, three-level inverters are widely adopted due to their low harmonic content, high efficiency, and improved power density. However, conventional three-level inverters like NPC and T-type face challenges in cost and efficiency as power levels increase. The F-type three-level inverter emerges as a promising alternative, reducing component voltage stress and overall system cost. Nevertheless, controlling the grid-connected current under non-ideal grid conditions—such as voltage imbalance and frequency deviations—remains critical for maintaining power quality. This is especially relevant for different types of solar inverter deployed in weak or disturbed grids.
Proportional resonant (PR) controllers are known for their capability to track sinusoidal references without steady-state error. However, traditional PR controllers exhibit narrow bandwidth and sensitivity to frequency variations. Quasi-PR (QPR) controllers introduce a damping term to widen the bandwidth but still suffer from performance degradation when the grid frequency deviates from the preset value. To address these limitations, we propose a frequency-adaptive PR controller that updates its resonant frequency in real-time based on the measured grid frequency. This ensures that the controller maintains infinite gain at the fundamental frequency, thereby achieving zero steady-state error even under dynamic frequency changes. Our work focuses on the F-type three-level inverter platform and validates the proposed strategy through experimental tests under unbalanced grid voltages and frequency fluctuations.
2. Operating Principle of F-Type Three-Level Inverter
The main circuit of the F-type three-level inverter is shown in Figure 1 (conceptual). The DC side is connected to a DC source or a front-end DC/DC converter. The three-phase bridge outputs are connected to the grid through an LCL filter. Unlike NPC or T-type topologies, the F-type inverter uses a single switch per phase to control the connection to the positive (P), neutral (O), or negative (N) DC bus. The switching states for a single phase are summarized in Table 1, where Sx1 and Sx2 operate complementarily, as do Sx3 and Sx4. The output voltage Uxo takes three levels: +Udc/2, 0, and -Udc/2 corresponding to states P, O, and N respectively.
Table 1: Switching states and output voltage of F-type inverter (one phase)
| Sx1 | Sx2 | Sx3 | Sx4 | Uxo | State |
|---|---|---|---|---|---|
| 1 | 0 | 1 | 0 | +Udc/2 | P |
| 0 | 1 | 1 | 0 | 0 | O |
| 0 | 1 | 0 | 1 | -Udc/2 | N |
The current commutation paths are independent of the load current direction, ensuring smooth transitions. This topology significantly reduces the voltage rating requirements for most switches, making it a cost-effective solution for medium-power applications. Understanding this operation is essential for designing controllers that maintain high performance across various types of solar inverter configurations.
3. Principle of Proportional Resonant Controller
3.1 Current Proportional Resonant Control
A PR controller is derived from the internal model principle, which allows zero steady-state error tracking of sinusoidal signals. The basic resonant term can be represented in two forms:
$$ G_1(s) = \frac{s}{s^2 + \omega_0^2} $$
$$ G_2(s) = \frac{\omega_0}{s^2 + \omega_0^2} $$
where ω0 is the fundamental angular frequency. The Bode plot analysis shows that G1(s) exhibits a phase lag limited to ±90°, which provides better phase margin and stability compared to G2(s). To enhance flexibility, we introduce a proportional gain and the resonant gain Kr, leading to the standard PR controller transfer function:
$$ G_{PR}(s) = K_p + \frac{2K_r s}{s^2 + \omega_0^2} $$
where Kp is the proportional gain. This controller achieves infinite theoretical gain at ω0, ensuring zero steady-state error for sinusoidal references. However, its narrow bandwidth makes it sensitive to grid frequency variations. This sensitivity is particularly problematic for grid-connected types of solar inverter that must operate under frequency deviations.
3.2 Quasi-Proportional Resonant Control
To mitigate the bandwidth limitation, a damping term ωc is introduced, forming the quasi-PR (QPR) controller:
$$ G_{QPR}(s) = K_p + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
The QPR controller widens the resonant peak, reducing sensitivity to frequency variations. However, as shown in the Bode plots, the gain at the nominal frequency is finite, causing a small but non-zero steady-state error. Under frequency offset conditions, the gain further decreases, leading to increased THD. This motivates the need for a frequency-adaptive approach, which is critical for efficient types of solar inverter operation in weak grids.
4. Frequency-Adaptive Proportional Resonant Control
We propose a frequency-adaptive PR controller that continuously adjusts its resonant frequency based on the real-time grid frequency measurement. The modified transfer function is:
$$ G_{APR}(s) = K_p + \frac{2K_r s}{s^2 + (2\pi f_r)^2} $$
where fr is the instantaneous grid frequency (default 50 Hz). The block diagram of the adaptive controller is shown conceptually in Figure 6. The grid frequency fr is estimated using a phase-locked loop (PLL) and fed into the resonant term. This ensures that the controller always maintains infinite theoretical gain at the actual fundamental frequency.
Under steady-state grid conditions (fr = 50 Hz), the gain expression is:
$$ A_{dB} = |K_p + \frac{2K_r \cdot j\omega}{(j\omega)^2 + (2\pi f_r)^2}| $$
When ω = 2πfr, the denominator becomes zero, and the gain tends to infinity. Under frequency deviation, the adaptive controller updates fr dynamically, thus maintaining infinite gain at the new fundamental frequency. Consequently:
$$ I_g = I_{ref} \quad \text{and} \quad B = \frac{G_2(s)}{1+K_{PWM}G_1(s)G_2(s)} \approx 0 $$
This leads to zero steady-state error and high robustness against grid voltage disturbances. The frequency-adaptive PR control is especially beneficial for various types of solar inverter that must comply with stringent grid codes and operate under dynamic conditions.
Table 2: Comparison of control strategies characteristics
| Feature | PR | QPR | APR (Proposed) |
|---|---|---|---|
| Theoretical gain at ω0 | Infinite | Finite | Infinite (adaptive) |
| Bandwidth | Narrow | Wide | Narrow but adaptive |
| Sensitivity to frequency variation | High | Low | None (real-time tracking) |
| Steady-state error under frequency offset | Non-zero | Small | Zero |
| THD under grid imbalance | High | Moderate | Low |
5. Experimental Validation
We built a 2.2 kW experimental platform consisting of a DC adjustable power supply, an F-type three-level inverter, a three-phase variac, and the grid. Key parameters are listed in Table 3.
Table 3: Experimental system parameters
| Parameter | Value |
|---|---|
| Rated output power (kW) | 2.2 |
| Input voltage (V) | 600 |
| Switching frequency (kHz) | 100 |
| Grid voltage (V) | 380 |
| Grid-side inductor (μH) | 5 |
| Filter capacitor (μF) | 4.7 |
| Inverter-side inductor (mH) | 1 |
We tested two scenarios: unbalanced grid voltages and frequency fluctuations. In the unbalanced test, phase A and B voltages were set to 198 V (10% drop) while phase C remained at 220 V. Figure 9 and Figure 10 (not explicitly referenced) show the grid-connected current waveforms and FFT analysis under quasi-PR and frequency-adaptive PR control respectively.
5.1 Performance Under Unbalanced Grid Voltages
Under unbalanced voltages, the quasi-PR controller produced a grid current THD of 1.20%. In contrast, our frequency-adaptive PR controller reduced THD to 0.85%, a 0.35% improvement. The adaptive scheme effectively suppressed the negative-sequence harmonics and maintained sinusoidal current waveforms. This demonstrates the superiority of the adaptive method for types of solar inverter operating in unbalanced grids.
5.2 Performance Under Grid Frequency Fluctuation
We set the grid frequency to 51 Hz (within permissible limits). Quasi-PR control resulted in a THD of 1.65%, while frequency-adaptive PR control achieved 1.20% THD, a reduction of 0.45%. The adaptive controller maintained high gain at the off-nominal frequency, significantly reducing low-order harmonics. The results confirm that the adaptive PR controller is robust against frequency deviations, a common requirement for modern types of solar inverter connected to weak grids.
Table 4: THD comparison under different operating conditions
| Operating condition | QPR (%) | APR (Proposed) (%) | Improvement |
|---|---|---|---|
| AB two-phase voltage drop 10% | 1.20 | 0.85 | -0.35% |
| Grid frequency 51 Hz | 1.65 | 1.20 | -0.45% |
The experimental results clearly demonstrate that the frequency-adaptive PR controller outperforms the conventional quasi-PR controller in both unbalanced and frequency-varying conditions. This makes it an ideal solution for various types of solar inverter that require high power quality and grid compliance.
6. Conclusion
We have proposed and experimentally validated a frequency-adaptive proportional resonant control strategy for F-type three-level grid-connected inverters. The controller adaptively tracks the actual grid frequency, ensuring infinite gain at the fundamental frequency and zero steady-state error. Compared to traditional quasi-PR control, our method reduces total harmonic distortion by 0.35% under unbalanced grid voltages and by 0.45% under frequency fluctuations. The F-type inverter, combined with the adaptive control, offers a cost-effective and high-performance solution for grid-connected applications. This research contributes to the enhancement of power quality for different types of solar inverter operating in complex grid environments. Future work will extend the adaptive principle to multi-resonant controllers for selective harmonic elimination.
Acknowledgments
This work was supported by the State Grid Shaanxi Electric Power Co., Ltd. and the Xi’an University of Technology. We thank all team members for their contributions.