In recent years, the rapid development of renewable energy sources such as photovoltaics and wind power has been driven by both the fossil fuel crisis and environmental concerns. Among the various types of solar inverter, the three-phase grid-connected inverter plays a vital role in transferring DC power from solar panels to the AC grid. However, at the end of rural power grids or on islands, the grid often exhibits weak characteristics with high impedance and low short-circuit capacity due to dispersed loads and long transmission distances. As the penetration level of renewable energy continues to increase, the challenges posed by weak grid conditions become more severe. The fast fluctuation of grid impedance may induce system instability, while the voltage waveform distortion caused by high impedance — such as harmonics and unbalance — severely affects the accuracy and reliability of the phase-locked loop (PLL) of the grid-connected inverter, thereby threatening overall system stability. To improve the steady-state performance of a three-phase LCL-type grid-connected inverter under complex grid conditions — including voltage unbalance and frequency variations — and to reduce the total harmonic distortion (THD) of the grid current, two core issues must be addressed. First, the current controller must maintain high-precision and zero steady-state error tracking of the current signal even when the grid frequency fluctuates. Second, under distorted grid voltage and frequency variation, high-accuracy grid synchronization and positive-sequence component extraction must be achieved to provide the phase and frequency information of the fundamental positive-sequence component for the control loop.
Conventional proportional-integral (PI) control can achieve zero steady-state error tracking for DC signals through a voltage-and-current dual-loop structure, but it fails to track sinusoidal AC signals without error. Some researchers have proposed a combined PI and multiple proportional-resonant (PR) control strategy to improve dynamic response and reduce harmonic pollution, but the overall control module becomes computationally complex. Another composite control strategy based on quasi-PR and PI has been shown to achieve unity power factor and zero steady-state error, yet the structure is intricate and parameter tuning is difficult, potentially affecting overall system stability. A frequency-adaptive PR controller has also been proposed with the advantage of frequency adaptation, but the PR controller suffers from gain limitations that may adversely affect the system. To address these problems, I propose a frequency-adaptive quasi-proportional-resonant (FA-QPR) control strategy based on a dual second-order generalized integrator frequency-locked loop (DSOGI-FLL).
Structure of the LCL-Type Three-Phase Grid-Connected Inverter
The main circuit topology of the inverter is a three-phase full-bridge structure, with its output connected to the grid via an LCL filter. As one of the key types of solar inverter, this topology effectively suppresses high-order current harmonics generated by pulse-width modulation (PWM). However, the LCL filter introduces a sharp resonant peak at its resonant frequency, and the phase crosses -180° above that frequency, which tends to cause system oscillation and instability. Therefore, an additional damping strategy must be introduced to suppress the resonant peak and ensure stable operation. In my design, I adopt the capacitor-current-feedback active damping method. The dual closed-loop control structure consists of an outer grid current loop and an inner capacitor current loop. The difference between the actual grid current and the reference current passes through the FA-QPR controller to generate the capacitor reference current. Then the difference between the capacitor reference current and the actual capacitor current is fed into a proportional controller for the capacitor current, and the resulting signal is used for SPWM modulation to achieve closed-loop control of both the grid current and the capacitor current.
Proposed Control Strategy Based on DSOGI-FLL and QPR
Frequency-Adaptive Quasi-Proportional-Resonant (FA-QPR) Controller
Traditional PI control operates in the rotating dq reference frame. It uses a proportional term for fast error response and an integral term to eliminate steady-state error. However, it requires coordinate transformation and has poor tracking performance for AC signals. In contrast, proportional-resonant (PR) control works directly in the stationary αβ or abc reference frame. It provides theoretically infinite gain at a specific resonant frequency (e.g., 50 Hz), enabling zero steady-state error tracking of AC signals. This makes PR control highly suitable for applications such as grid-connected inverters, as it avoids coordinate transformation and offers high accuracy for fixed-frequency AC tracking. The ideal transfer function of a PR controller is:
$$
G_{PR}(s) = K_p + \frac{K_r s}{s^2 + \omega_0^2}
$$
where \(K_p\) is the proportional gain, \(K_r\) is the resonant gain coefficient, and \(\omega_0\) is the resonant angular frequency. This structure provides theoretically infinite gain at \(\omega_0\), achieving zero steady-state error control. Under practical grid conditions, however, frequency fluctuations significantly degrade the tracking performance of the PR controller. Since the resonator only offers high gain at a fixed frequency (e.g., the 50 Hz fundamental), any deviation of the fundamental frequency reduces the open-loop gain, resulting in steady-state current error. The quasi-PR (QPR) controller mitigates this gain limitation. Its transfer function is:
$$
G_{QPR}(s) = K_p + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2}
$$
where \(\omega_c\) is the cutoff frequency. However, the QPR controller’s resonant frequency is still fixed, and when the actual grid frequency deviates, the tracking accuracy at that frequency deteriorates significantly, causing amplitude and phase errors. To overcome the impact of grid frequency variations, maintain high-precision zero steady-state error tracking of the grid current, and preserve system stability, I propose a frequency-adaptive quasi-proportional-resonant (FA-QPR) controller. Its transfer function is:
$$
G_{FA-QPR}(s,t) = K_p + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2(t)}
$$
$$
\omega_0(t) = \omega_{grid}(t) = 2\pi f_{grid}(t)
$$
The key difference from a fixed-frequency QPR is that \(\omega_0\) is no longer constant but a time-varying variable \(\omega_0(t)\), which is updated in real time by the instantaneous grid angular frequency \(\omega_{grid}(t)\) measured by the DSOGI-FLL. The DSOGI-FLL typically updates the frequency every sampling period. To maintain dynamic performance, the FA-QPR controller also updates its discretization coefficients using the latest \(\omega_{grid}(t)\) in each control period.
To preserve the accuracy of the resonant frequency characteristics in the discrete domain, I adopt the pre-warped bilinear transformation:
$$
s = \frac{2}{T_s} \cdot \frac{z-1}{z+1} = \alpha \cdot \frac{z-1}{z+1}, \quad \alpha = \frac{2}{T_s}
$$
where \(T_s\) is the sampling period. Substituting this into the resonant part \(G_R(s,t) = \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2(t)}\) yields the discrete transfer function of the resonant part:
$$
G_R(z) = \frac{2K_r \omega_c \alpha (z^2 – 1)}{(\alpha^2 + 2\omega_c\alpha + \omega_0^2)z^2 + (-2\alpha^2 + 2\omega_0^2)z + (\alpha^2 – 2\omega_c\alpha + \omega_0^2)}
$$
Writing \(G_R(z)\) in standard form:
$$
G_R(z) = \frac{b_0 z^2 + b_1 z + b_2}{a_0 z^2 + a_1 z + a_2}
$$
By comparison, the un-normalized coefficients are:
Denominator: \(a_0 = \alpha^2 + 2\omega_c \alpha + \omega_0^2\), \(a_1 = -2\alpha^2 + 2\omega_0^2\), \(a_2 = \alpha^2 – 2\omega_c \alpha + \omega_0^2\).
Numerator: \(b_0 = 2K_r \omega_c \alpha\), \(b_1 = 0\), \(b_2 = -2K_r \omega_c \alpha = -b_0\).
Normalizing the denominator polynomial gives the real-time coefficients that must be computed every sampling period:
| Coefficient | Expression |
|---|---|
| \(a’_0\) | 1 |
| \(a’_1\) | \(\frac{-2\alpha^2 + 2\omega_0^2}{\alpha^2 + 2\omega_c \alpha + \omega_0^2}\) |
| \(a’_2\) | \(\frac{\alpha^2 – 2\omega_c \alpha + \omega_0^2}{\alpha^2 + 2\omega_c \alpha + \omega_0^2}\) |
| \(b’_0\) | \(\frac{2K_r \omega_c \alpha}{\alpha^2 + 2\omega_c \alpha + \omega_0^2}\) |
| \(b’_1\) | 0 |
| \(b’_2\) | \(-b’_0\) |
The complete discrete transfer function of the FA-QPR controller is:
$$
G_{FA-QPR}(z) = K_p + G_R(z) = K_p + \frac{b’_0 z^2 + b’_1 z + b’_2}{z^2 + a’_1 z + a’_2}
$$
Dual Second-Order Generalized Integrator Frequency-Locked Loop (DSOGI-PNSC-FLL)
In an ideal three-phase balanced grid, the synchronous reference frame phase-locked loop (SRF-PLL) is widely used for efficient grid voltage phase and frequency synchronization. Under stable grid conditions, SRF-PLL can achieve accurate phase locking. However, under unbalanced grid conditions, negative-sequence components cannot be filtered out by simple filters, causing interference to SRF-PLL and preventing accurate extraction of the grid voltage phase \(\theta\). Therefore, I propose a DSOGI-PNSC-FLL, which combines dual second-order generalized integrators (DSOGI), a positive/negative sequence calculator (PNSC), and a frequency-locked loop (FLL). This combination can accurately obtain frequency information even under adverse grid conditions.
The transfer functions of a single SOGI quadrature signal generator (SOGI-QSG) are:
$$
D(s) = \frac{V’}{V(s)} = \frac{\omega’ s}{s^2 + K\omega’ s + \omega’^2}
$$
$$
Q(s) = \frac{qV’}{V(s)} = \frac{K\omega’^2}{s^2 + K\omega’ s + \omega’^2}
$$
where \(\omega’\) is the initial grid angular frequency and \(K\) is the gain coefficient. To handle unbalanced three-phase grid voltages, the PNSC module decomposes the asymmetric three-phase system into positive-sequence, negative-sequence, and zero-sequence components. The positive- and negative-sequence components in the stationary reference frame are obtained as:
$$
V^+_{\alpha\beta} = \frac{1}{2} \begin{bmatrix} 1 & -q \\ q & 1 \end{bmatrix} V_{\alpha\beta}
$$
$$
V^-_{\alpha\beta} = \frac{1}{2} \begin{bmatrix} 1 & q \\ -q & 1 \end{bmatrix} V_{\alpha\beta}
$$
where \(q = e^{-j\pi/2}\) represents a 90° phase shift. The overall DSOGI-FLL structure uses two SOGI-QSG blocks to generate two pairs of quadrature voltage signals, then the PNSC separates the positive- and negative-sequence components. The FLL accurately tracks the phase and frequency of the positive-sequence component. This method effectively separates and tracks the positive- and negative-sequence components, thereby obtaining high-precision grid phase and frequency information.
The transfer function of the DSOGI block can be expressed as:
$$
G_{DSOGI}(s) = \frac{k \omega_c s}{s^2 + k \omega_c s + \omega_0^2}
$$
Here, \(\omega_0\) is the fundamental frequency detected by the FLL, \(\omega_c\) is the cutoff frequency, and \(k\) is the damping coefficient (ideally set to 2). The bandwidth of the DSOGI is \(BW = k \cdot \omega_c\). A wider bandwidth provides faster tracking of frequency variations but reduces the rejection of noise and harmonics. The proposed DSOGI-FLL achieves a balanced trade-off between dynamic response and harmonic immunity, making it suitable for various types of solar inverter operating in weak grids.
Simulation Analysis
I built a simulation model of the LCL-type three-phase grid-connected inverter in MATLAB/Simulink to verify the effectiveness of the proposed control method. The key parameters of the entire system are listed in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| DC bus voltage | \(U_{dc}\) | 750 V |
| Inverter-side inductor | \(L_1\) | 4.5 mH |
| Filter capacitor | \(C_f\) | 5 µF |
| Grid-side inductor | \(L_2\) | 0.9 mH |
| Grid frequency (nominal) | \(f_{grid}\) | 50 Hz |
| Proportional gain | \(K_p\) | 0.4 |
| Resonant gain | \(K_r\) | 20 |
| Cutoff frequency | \(\omega_c\) | 5 rad/s |
| Sampling frequency | \(f_s\) | 10 kHz |
First, I analyzed the performance of the DSOGI-FLL under non-ideal grid conditions. A frequency disturbance of +3 Hz was injected into the three-phase grid from 0.2 s to 0.3 s. The results show that compared with the conventional SRF-PLL, the DSOGI-FLL exhibited much smoother frequency transitions during the disturbance. The conventional PLL took about 0.26 s to recover and its output frequency peaked near 54 Hz, while the DSOGI-FLL maintained a stable output at about 53 Hz with much less overshoot. When a voltage dip of 50% was injected from 0.2 s to 0.3 s, the conventional PLL suffered severe frequency fluctuations with a deviation of about 5 Hz from the nominal value, whereas the DSOGI-FLL’s output deviated by only about 0.2 Hz. Moreover, the conventional PLL needed about 0.37 s to fully stabilize, while the DSOGI-FLL stabilized by 0.35 s. These results demonstrate that the DSOGI-FLL is significantly less affected by disturbances such as voltage sags and frequency jumps, providing a more accurate and robust frequency signal. Since the performance of the FA-QPR controller depends heavily on the accuracy of the frequency reference, the DSOGI-FLL’s superior anti-interference capability and precision offer a more reliable frequency reference.
Next, I evaluated the overall system performance under the proposed FA-QPR control strategy with the DSOGI-FLL. The measured A-phase grid voltage and grid current waveforms show that they are in phase and have the same frequency, confirming that the DSOGI-FLL accurately captures the grid voltage phase and the FA-QPR controller tracks the current reference precisely. The system maintains stable operation. Both active power and reactive power reach their setpoints within a very short response time, indicating fast power control capability that meets grid standards and practical requirements.
To verify the frequency adaptability of the FA-QPR controller, I tested the system at three different grid frequencies: 45 Hz, 50 Hz, and 55 Hz. Without any modification to the controller parameters, the resonant frequency coefficient in the controller automatically follows the actual grid frequency detected by the DSOGI-FLL in real time. In all three cases, the three-phase grid currents reach steady state after a very short transient, demonstrating excellent frequency adaptation. This feature is crucial for types of solar inverter operating in weak grids where frequency variations are common.
I also compared the total harmonic distortion (THD) of the grid current under the proposed FA-QPR control and under conventional PI control under the same operating conditions. The results are summarized in the table below.
| Control Strategy | THD (%) |
|---|---|
| FA-QPR (Proposed) | 1.82 |
| Conventional PI | 4.67 |
The FA-QPR control reduces the THD by more than 60% compared to the conventional PI control, confirming its superior harmonic suppression capability. This improvement is especially important for meeting grid codes and ensuring high power quality in different types of solar inverter applications.
Conclusion
In this paper, I have proposed and validated a frequency-adaptive quasi-proportional-resonant (FA-QPR) control strategy based on a dual second-order generalized integrator frequency-locked loop (DSOGI-FLL) for LCL-type three-phase grid-connected inverters. The following conclusions are drawn from the simulation analysis:
1. Compared with the conventional synchronous reference frame phase-locked loop (SRF-PLL), the proposed DSOGI-FLL can extract the phase and frequency of the grid voltage fundamental positive-sequence component more quickly and accurately under complex conditions such as voltage dips and sudden frequency changes. The DSOGI-FLL exhibits stronger anti-interference ability and better dynamic stability, providing a real-time and reliable frequency reference for the FA-QPR controller. This is a prerequisite for maintaining high-performance operation of the inverter.
2. Compared with conventional PI control, the FA-QPR controller dynamically adjusts its resonant frequency parameters according to the real-time grid frequency detected by the DSOGI-FLL. This adaptive mechanism ensures that the system can achieve precise zero steady-state error tracking of the reference current signal at different grid frequencies, thereby consistently maintaining unity power factor between the grid voltage and current. This strategy fundamentally improves current tracking quality and significantly reduces the total harmonic distortion (THD) of the grid current.
The proposed control method offers an effective solution for enhancing the robustness and power quality of three-phase grid-connected inverters, particularly in weak grid environments where frequency variations and voltage distortions are common. The approach can be readily adapted to various types of solar inverter configurations, including string inverters and central inverters, by tuning the controller parameters accordingly.