In modern power systems, single-phase grid-tied inverters play a critical role in integrating renewable energy sources, such as solar and wind, into the grid. However, the performance of these single-phase inverters is often compromised by grid frequency fluctuations, which can lead to power quality issues, including harmonic distortions and unstable power control. Traditional control strategies, such as Proportional-Integral (PI) control, Proportional-Resonant (PR) control, and hysteresis control, have been widely used but exhibit limitations in handling frequency variations. For instance, PR controllers offer excellent harmonic rejection at fixed frequencies but suffer from narrow bandwidths and performance degradation under frequency shifts. Hysteresis control provides fast dynamic response but introduces significant losses and control inaccuracies. To address these challenges, we propose a novel control strategy based on a Second-Order Generalized Integrator (SOGI) combined with a Frequency-Locked Loop (FLL) and a Phase-Locked Loop (PLL). This approach enables adaptive grid synchronization and precise power decoupling control for single-phase inverters, ensuring robust performance under grid disturbances.
The core of our method lies in the SOGI-FLL/PLL structure, which generates orthogonal signals from the grid voltage and adapts to frequency variations. By transforming the grid voltage into a rotating d-q coordinate system, we achieve decoupled control of active and reactive power. The SOGI-FLL component ensures accurate frequency tracking, while the PLL locks the phase, allowing for seamless integration with current control loops. We implement a dq-decoupled PI controller to regulate the inverter output currents, facilitating independent control of real and reactive power. This strategy not only improves the stability of single-phase inverter systems but also enhances their resilience to grid anomalies. In this article, we present a comprehensive analysis of the SOGI-FLL/PLL-based control, including mathematical modeling, simulation results, and experimental validation. Our findings demonstrate that this approach effectively mitigates harmonic distortions, rapidly synchronizes with the grid, and maintains precise power control even under frequency fluctuations.
To begin, we explore the fundamental principles of the SOGI-FLL and PLL algorithms. The SOGI is a versatile filter that produces two orthogonal output signals from a single-phase input, making it ideal for single-phase inverter applications. Its transfer functions in the continuous s-domain are given by:
$$ \frac{v’}{v} = \frac{k \omega_0 s}{s^2 + k \omega_0 s + \omega_0^2} $$
$$ \frac{qv’}{v} = \frac{k \omega_0^2}{s^2 + k \omega_0 s + \omega_0^2} $$
where $v$ is the input voltage, $v’$ and $qv’$ are the orthogonal outputs, $k$ is the damping factor, and $\omega_0$ is the nominal grid frequency. For digital implementation, we apply the bilinear transform to discretize these equations into the z-domain. The transformation is defined as:
$$ s = \frac{2}{T} \frac{z-1}{z+1} $$
where $T$ is the sampling period. This allows us to implement the SOGI in a digital signal processor for real-time control of single-phase inverters. The orthogonal signals $v’$ and $qv’$ are then used in a Park transformation to convert them from the stationary α-β frame to the rotating d-q frame:
$$ \begin{bmatrix} v_d \\ v_q \end{bmatrix} = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} v’ \\ qv’ \end{bmatrix} $$
Here, $\theta$ is the phase angle obtained from the PLL. The PLL structure employs a PI controller to drive the q-component $v_q$ to zero, thereby locking the phase to the grid voltage. However, in conventional SOGI-PLL systems, fixed frequency parameters can lead to errors during grid frequency variations. To overcome this, we integrate an FLL into the SOGI, resulting in an adaptive SOGI-FLL that adjusts the frequency $\omega’$ based on the error signal. The state equations for the SOGI-FLL are derived as:
$$ \dot{x}_1 = -\omega’ x_2 + k \omega’ (v – x_1) $$
$$ \dot{x}_2 = \omega’ x_1 $$
$$ v’ = x_1 $$
$$ qv’ = x_2 $$
$$ \dot{\omega}’ = -\gamma (v – x_1) x_2 $$
where $x_1$ and $x_2$ are state variables, and $\gamma$ is a tuning parameter for the FLL. This adaptive mechanism ensures that the SOGI-FLL/PLL combination maintains accurate phase and frequency tracking even when the grid frequency deviates from its nominal value, which is crucial for reliable operation of single-phase inverters in practical environments.
Next, we develop the mathematical model of the single-phase grid-tied inverter with an LCL filter. The LCL filter is preferred over simpler L or LC filters due to its superior harmonic attenuation and smaller inductor sizes, making it cost-effective for single-phase inverter systems. The circuit topology consists of an H-bridge formed by MOSFET switches (Q1 to Q4), with an LCL filter comprising inductors L1 and L2, and a capacitor C. The grid voltage and current are denoted as $v_g$ and $i_g$, respectively. The dynamic equations of the system are:
$$ \frac{di_1}{dt} = \frac{v_{dc} – v_c}{L_1} $$
$$ \frac{di_2}{dt} = \frac{v_c – v_g}{L_2} – \frac{R i_2}{L_2} $$
$$ \frac{dv_c}{dt} = \frac{i_1 – i_2}{C} $$
where $i_1$ is the inverter-side current, $i_2$ is the grid-side current, $v_c$ is the capacitor voltage, $v_{dc}$ is the DC-link voltage, and $R$ represents parasitic resistance. By applying Laplace transforms, we derive the transfer function $G(s)$ between the grid current $i_2$ and the inverter output voltage $v_{dc}$:
$$ G(s) = \frac{i_2(s)}{v_{dc}(s)} = \frac{1}{s^3 L_1 L_2 C + s^2 L_1 L_2 + s (L_1 + L_2)} $$
This transfer function exhibits a first-order characteristic at low frequencies and a third-order behavior at high frequencies, which is typical for LCL filters in single-phase inverters. For digital control, we discretize $G(s)$ using the bilinear transform to obtain $G(z)$, enabling implementation in a microcontroller. The overall system structure incorporates the SOGI-FLL/PLL for grid synchronization and a dq-based current controller. The control loop involves transforming the grid current into the d-q frame, where $i_d$ and $i_q$ represent the active and reactive current components, respectively. PI controllers are used to regulate these currents by comparing them with reference values $i_d^*$ and $i_q^*$. The output of the PI controllers generates modulation signals for pulse-width modulation (PWM), which drives the H-bridge switches. This setup allows for decoupled control of active and reactive power in the single-phase inverter, enhancing its grid-support capabilities.
To validate the proposed control strategy, we conducted extensive simulations using MATLAB/Simulink. The parameters for the single-phase inverter system are summarized in Table 1. We compared the performance of the SOGI-FLL/PLL-based control with traditional SOGI-PLL under various scenarios, including step changes in active power and grid frequency variations. The simulation results demonstrate that our approach achieves faster synchronization, lower total harmonic distortion (THD), and improved stability for single-phase inverters.
| Parameter | Value |
|---|---|
| DC-link voltage | 70 V |
| Grid voltage (AC) | 22 V |
| Inverter-side inductance L1 | 1 mH |
| Grid-side inductance L2 | 1 mH |
| Filter capacitance C | 47 μF |
| Switching frequency | 40 kHz |
| Nominal grid frequency | 50 Hz |
In one simulation, we set the reference active current to 3 A and reactive current to 0 A, with a grid frequency of 50 Hz. At t = 0.4 s, we introduced a step change in active current to 1.5 A. The results show that the grid current quickly stabilized within one cycle, maintaining synchronization with the grid voltage. The THD of the grid current was measured at 2.55%, indicating minimal harmonic distortion. The FFT analysis revealed a dominant fundamental component at 50 Hz with negligible higher-order harmonics, underscoring the effectiveness of the SOGI-FLL/PLL in enhancing the power quality of single-phase inverters.
To test the adaptability to frequency changes, we simulated a grid frequency step from 50 Hz to 45 Hz at t = 0.4 s, with the active current reference set to 3 A. Under the SOGI-FLL/PLL control, the single-phase inverter regained stability within two cycles, and the steady-state current THD was 2.94%. In contrast, the conventional SOGI-PLL strategy resulted in a distorted current waveform with a THD of 16.52%, primarily due to significant third-harmonic components. This comparison highlights the superiority of the SOGI-FLL/PLL in mitigating the impact of grid frequency fluctuations on single-phase inverter performance.
We further verified the control strategy through experimental tests using a prototype single-phase inverter. The setup included an STM32F407ZET6 microcontroller, IRF540NPBF MOSFETs, and an LCL filter with parameters listed in Table 2. Voltage and current sensors, along with an ADS7606 ADC, were used for feedback. The experimental results corroborate the simulation findings, showing stable grid connection, accurate current tracking, and robust performance during transients.
| Parameter | Value |
|---|---|
| Grid voltage (AC) | 22 V |
| Inverter-side inductance L1 | 1 mH |
| Grid-side inductance L2 | 1 mH |
| Filter capacitance C | 40 μF |
| Switching frequency | 20 kHz |
In a steady-state test with an active current of 1.5 A and unity power factor, the grid current and voltage were perfectly synchronized, exhibiting a sinusoidal waveform. When the active current was stepped from 1.5 A to 3 A, the system responded rapidly, achieving stability within one cycle. Additionally, tests with simultaneous changes in active and reactive currents (e.g., from 1.5 A to 3 A active and 0 A to 1.5 A reactive) demonstrated that the single-phase inverter maintained decoupled control, with quick transient recovery and minimal overshoot. These experiments confirm that the SOGI-FLL/PLL-based control strategy is practical and effective for real-world single-phase inverter applications.
In conclusion, our proposed SOGI-FLL/PLL-based power decoupling control for single-phase grid-tied inverters offers significant advantages in terms of adaptability, precision, and robustness. By integrating frequency and phase locking with dq-current decoupling, this approach addresses the challenges posed by grid frequency variations, ensuring high-quality power injection and stable operation. The mathematical analysis, simulations, and experimental results collectively validate the efficacy of this method. Future work will focus on optimizing the control parameters for wider operating ranges, reducing computational complexity, and extending the strategy to three-phase systems and microgrid applications. Overall, this research contributes to the advancement of single-phase inverter technologies, supporting the integration of renewable energy into modern power grids.
The key equations and transfer functions derived in this work are essential for designing and implementing such control systems. For instance, the discrete-time implementation of the SOGI-FLL can be summarized using the following difference equations after bilinear transformation:
$$ x_1[n] = a_1 x_1[n-1] + a_2 x_2[n-1] + b_1 v[n] $$
$$ x_2[n] = a_3 x_1[n-1] + a_4 x_2[n-1] + b_2 v[n] $$
$$ \omega'[n] = \omega'[n-1] – \gamma T (v[n] – x_1[n]) x_2[n] $$
where $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, and $b_2$ are coefficients derived from the discretization process. Similarly, the dq-current PI controller outputs can be expressed as:
$$ u_d = k_p (i_d^* – i_d) + k_i \int (i_d^* – i_d) dt $$
$$ u_q = k_p (i_q^* – i_q) + k_i \int (i_q^* – i_q) dt $$
These equations form the basis for the real-time control algorithms in single-phase inverters. Additionally, the power decoupling principle ensures that active power $P$ and reactive power $Q$ are independently controlled through the d and q axes, respectively:
$$ P = \frac{1}{2} v_d i_d $$
$$ Q = -\frac{1}{2} v_d i_q $$
This decoupling is vital for maintaining grid stability and power factor correction in single-phase inverter systems. In summary, the SOGI-FLL/PLL strategy represents a robust solution for enhancing the performance of single-phase inverters in fluctuating grid conditions, paving the way for more reliable and efficient renewable energy integration.