In modern power systems, the integration of renewable energy sources relies heavily on on-grid inverters to convert DC power to AC power and synchronize with the utility grid. Traditional on-grid inverters typically measure the voltage at the point of common coupling (PCC) using phase-locked loops (PLLs) to track grid phase, ensuring synchronization. However, this approach introduces harmonics from the grid into the control loop, compromising stability, especially in weak grid conditions. To address this, self-synchronization control strategies, such as virtual synchronous generators (VSGs), have emerged, mimicking synchronous generator behavior without direct grid voltage sensing. Despite advantages in robustness, these methods can lead to large inrush currents during grid connection due to phase mismatches, potentially causing on-grid inverter failure. In this paper, we propose a novel pre-synchronization control strategy for single-phase on-grid inverters based on non-grid voltage sensor control, enabling smooth and reliable grid connection by estimating grid phase from current measurements. The strategy involves applying periodic pulses with constrained duty cycles to the switching devices, operating the on-grid inverter in rectification mode at unity power factor before grid-tie. This eliminates the need for PCC voltage sensors, reducing cost and enhancing stability. We provide detailed theoretical analysis, mathematical modeling, and experimental validation, demonstrating the effectiveness of our approach for on-grid inverter applications.
The core challenge in on-grid inverter control lies in achieving synchronization without direct grid voltage feedback, which is critical for avoiding instabilities. Self-synchronization methods, derived from synchronous generator emulation, offer an alternative by using the inverter’s internal states to infer grid conditions. However, during the initial connection, the on-grid inverter may experience transient currents if its output phase deviates from the grid phase. Our pre-synchronization strategy tackles this by pre-aligning the on-grid inverter’s phase with the grid through current-based estimation. This not only mitigates inrush currents but also simplifies control complexity, making it practical for real-world on-grid inverter deployments. Throughout this work, we emphasize the importance of on-grid inverter stability and performance, particularly in weak grids where impedance interactions can exacerbate issues.
To understand our proposed strategy, we first analyze the working principle of a single-phase on-grid inverter under self-synchronization control. The system structure, as shown in the figure above, includes a DC voltage source, an H-bridge inverter, an L-type filter, and grid connection. Unlike conventional methods, the PLL input is derived from the output of the current controller’s resonant component, denoted as \( e \), rather than the PCC voltage \( V_{\text{PCC}} \). This leverages the inherent filtering properties of the current controller to achieve synchronization without grid voltage sensors. The proportional-resonant (PR) controller transfer function is given by:
$$G_c(s) = k_p \left(1 + \frac{k_r s}{s^2 + \omega_n^2}\right)$$
where \( k_p \) is the proportional gain, \( k_r \) is the resonant gain, and \( \omega_n \) is the rated fundamental angular frequency. The overall system model in the frequency domain combines the controller, delay, and plant dynamics. The delay transfer function \( G_{\text{del}}(s) \) and filter inductance transfer function \( G_p(s) \) are:
$$G_{\text{del}}(s) = e^{-1.5T_s s}, \quad G_p(s) = \frac{1}{s L_f}$$
where \( T_s \) is the sampling period and \( L_f \) is the filter inductance. Using Mason’s formula, the output current \( i(s) \) and current error \( \Delta i(s) \) can be expressed as:
$$i(s) = \frac{G_o(s)}{1 + G_o(s)} i^*(s) – \frac{G_p(s)}{1 + G_o(s)} V_{\text{PCC}}(s)$$
$$\Delta i(s) = \frac{1}{1 + G_o(s)} i^*(s) + \frac{G_p(s)}{1 + G_o(s)} V_{\text{PCC}}(s)$$
where \( G_o(s) = G_c(s) G_{\text{del}}(s) G_p(s) \) is the open-loop transfer function. The resonant component output \( e(s) \) is:
$$e(s) = k_p k_r \left( \frac{s}{s^2 + \omega_n^2} \right) \Delta i(s)$$
By analyzing the relationship between \( e(s) \) and \( V_{\text{PCC}}(s) \), we derive:
$$\frac{e(s)}{V_{\text{PCC}}(s)} = k_p k_r \left( \frac{s}{s^2 + \omega_n^2} \right) \times \frac{G_p(s)}{1 + G_o(s)}$$
A Bode plot of this transfer function shows that at the fundamental frequency, \( e \) approximates \( V_{\text{PCC}} \) in magnitude and phase, validating its use for phase estimation in an on-grid inverter. This equivalence underpins the self-synchronization capability, eliminating the need for grid voltage sensors. However, during grid connection, phase mismatch can cause large inrush currents, necessitating pre-synchronization for the on-grid inverter.
Our pre-synchronization control strategy for the on-grid inverter operates by applying periodic pulses to the switching devices before grid-tie. We consider an L-filter-based single-phase on-grid inverter, where \( L = L_f + L_g \) represents the total inductance (filter and grid-side). In each switching period \( T_s \), modulation signals \( U_{\text{MS}} \) and \( L_{\text{MS}} \) generate pulses with a fixed duty cycle \( D \). The operation is divided into three stages: G1 and G3 for energy storage, and G2 for energy release. During G1 or G3, when the lower or upper switches are on, the inductor stores energy, and the current dynamics follow:
$$E + L \frac{di}{dt} = 0 \Rightarrow i = -\int \frac{E}{L} dt$$
where \( E = V_g \sin(\omega t) \) is the grid voltage with amplitude \( V_g \). By ensuring the initial current is zero at the start of each pulse, the current envelope becomes sinusoidal with opposite phase to the grid voltage. In stage G2, both switches are off, and the inductor releases energy through diode freewheeling paths, allowing current decay. This process enables current sampling at specific points, such as at the carrier wave troughs, to estimate grid phase without voltage sensors. For an on-grid inverter, this method provides a simple means to achieve pre-synchronization.
To demonstrate feasibility, we analyze the current characteristics over switching cycles. Let \( t_g, t_o, t_n, t_h, t_p \) denote time instants within the \( k \)-th cycle, defined as:
$$t_g = \begin{cases} 0, & k = 1 \\ (k-1)T_s – \frac{D}{2}T_s, & k > 1 \end{cases}$$
$$t_o = (k-1)T_s, \quad t_n = t_o + \frac{D}{2}T_s$$
$$t_h = (k-1)T_s + \frac{1}{2}T_s – \frac{D}{2}T_s, \quad t_p = t_h + \frac{D}{2}T_s$$
The current values at points \( n \) and \( p \) are derived as:
$$T_n = \frac{1}{L\omega} V_g [\cos(\omega t_n) – \cos(\omega t_g)]$$
$$T_p = \frac{1}{L\omega} V_g [\cos(\omega t_p) – \cos(\omega t_h)]$$
Iterating these calculations yields a sinusoidal current envelope with phase opposite to the grid voltage. Similarly, the sampled current \( i_{\text{sample}} \) at point \( o \) is:
$$i_{\text{sample}} = \frac{1}{L\omega} V_g [\cos(\omega t_o) – \cos(\omega t_g)]$$
which also exhibits a sinusoidal waveform with opposite phase. Thus, by processing \( i_{\text{sample}} \) through a phase-locked loop (PLL), we can estimate the grid phase for the on-grid inverter. We employ a second-order generalized integrator-based PLL (SOGI-PLL) for accurate phase extraction, as shown in the control block diagram. The estimated phase \( \theta_e \) is fed back to the self-synchronization controller, ensuring the on-grid inverter’s output phase \( \theta \) tracks \( \theta_e \) seamlessly during pre-synchronization.
The implementation steps for our pre-synchronization strategy in an on-grid inverter are:
- Apply periodic pulses with constrained duty cycle \( D \) to the upper and lower switches.
- Sample the output current \( i \) to obtain \( i_{\text{sample}} \).
- Enable the pre-synchronization signal and use SOGI-PLL to estimate grid phase from \( i_{\text{sample}} \).
- Feed the estimated phase into the self-synchronization control loop, activating the controller once SOGI-PLL stabilizes.
This process ensures smooth transition from rectification to inversion mode for the on-grid inverter, minimizing transients.
A critical aspect is determining the maximum duty cycle \( D \) to guarantee current decay to zero before each pulse. Assuming constant grid voltage over a switching period, the current at point \( p \), \( T_p \), is approximated as:
$$T_p \approx -\frac{D T_s}{L} V_g \sin[(k-1)\omega T_s]$$
During energy release in stage G2, the circuit equation is:
$$E + L \frac{di}{dt} – V_{\text{dc}} = 0 \Rightarrow i = \int \frac{V_{\text{dc}} – E}{L} dt + T_p$$
where \( V_{\text{dc}} \) is the DC-link voltage. For current to decay to zero within time \( t – t_p \leq \frac{T_s}{2} – D T_s \), we derive the constraint:
$$D \leq \frac{V_{\text{dc}} – V_g \sin[(k-1)\omega T_s]}{2V_{\text{dc}}}$$
The worst-case occurs at peak grid voltage, i.e., \( \sin[(k-1)\omega T_s] = 1 \), yielding the critical duty cycle for the on-grid inverter. Using typical parameters, we compute \( D_{\text{critical}} \approx 0.2933 \), ensuring proper operation.
To validate our strategy, we conducted semi-physical experiments with a single-phase on-grid inverter. The parameters are summarized in Table 1, which illustrates key values for the on-grid inverter setup.
| Parameter | Symbol | Value |
|---|---|---|
| Filter Inductance | \( L_f \) | 3 mH |
| Grid-Side Inductance | \( L_g \) | 10 mH |
| Grid Voltage Amplitude | \( V_g \) | 310 V |
| Switching Period | \( T_s \) | 5 × 10⁻⁴ s |
| Angular Frequency | \( \omega \) | 314 rad/s |
| Duty Cycle | \( D \) | 0.25 |
| DC-Link Voltage | \( V_{\text{dc}} \) | 750 V |
We tested current characteristics under periodic pulses with \( D = 0.25 \). The results, shown in graphical plots, confirm that the actual current \( i \) and sampled current \( i_{\text{sample}} \) envelopes are sinusoidal with phase opposite to grid voltage, aligning with theoretical predictions for the on-grid inverter. Additionally, we verified that using an LCL filter (with \( C_f = 100 \mu F \)) yields similar behavior, proving the strategy’s robustness to filter type for on-grid inverters.
We further examined duty cycle effects by comparing \( D = 0.29 \) and \( D = 0.30 \). For \( D = 0.29 \) (below critical), current envelopes maintain opposite phase, enabling accurate phase estimation. However, for \( D = 0.30 \) (above critical), currents fail to decay fully, distorting envelopes and compromising phase estimation. This validates the duty cycle constraint for reliable on-grid inverter operation.
The efficacy of our pre-synchronization strategy was assessed by comparing on-grid inverter performance with and without it. Under self-synchronization control alone, the on-grid inverter experienced inrush currents exceeding 60 A during connection, triggering protection shutdowns. In contrast, with pre-synchronization, the on-grid inverter smoothly transitioned to grid-tie mode, with currents staying within safe limits. Table 2 summarizes the comparative results, highlighting the benefits for on-grid inverter stability.
| Aspect | Without Pre-Synchronization | With Pre-Synchronization |
|---|---|---|
| Inrush Current | > 60 A (unsafe) | < 30 A (safe) |
| Phase Alignment | Poor, leading to transients | Accurate, minimizing mismatches |
| Grid Connection | Failed due to protection | Successful and smooth |
| Control Complexity | High, requires voltage sensors | Low, sensorless operation |
These experiments demonstrate that our pre-synchronization control strategy effectively reduces transients, ensuring reliable grid integration for single-phase on-grid inverters. The approach is simple to implement, cost-effective due to eliminated voltage sensors, and compatible with various filter configurations, making it suitable for real-world on-grid inverter applications.
In conclusion, we have proposed and validated a pre-synchronization control strategy for single-phase on-grid inverters based on non-grid voltage sensor control. By applying constrained periodic pulses, the on-grid inverter operates in rectification mode to estimate grid phase from current samples, enabling smooth grid connection. Theoretical analysis derived critical duty cycle constraints, and experimental tests confirmed the strategy’s ability to mitigate inrush currents and enhance stability. Compared to existing methods, our approach reduces complexity and hardware costs, offering a practical solution for on-grid inverter deployments in weak grids. Future work could extend this to three-phase on-grid inverters or integrate adaptive duty cycle control for varying grid conditions. Overall, this research contributes to advancing on-grid inverter technology, promoting safer and more efficient renewable energy integration.
The mathematical foundations of our strategy involve extensive modeling of the on-grid inverter dynamics. For instance, the state-space representation of the system during pre-synchronization can be expressed as:
$$\frac{d}{dt} \begin{bmatrix} i \\ v_c \end{bmatrix} = \begin{bmatrix} 0 & -\frac{1}{L} \\ \frac{1}{C} & 0 \end{bmatrix} \begin{bmatrix} i \\ v_c \end{bmatrix} + \begin{bmatrix} \frac{1}{L} \\ 0 \end{bmatrix} E – \begin{bmatrix} 0 \\ \frac{1}{C} \end{bmatrix} i_{\text{load}}$$
where \( v_c \) is the capacitor voltage in LCL filters, and \( i_{\text{load}} \) represents load current. This model helps in analyzing transient responses and optimizing control parameters for the on-grid inverter. Additionally, the frequency domain analysis using impedance models reveals stability margins. The output impedance of the on-grid inverter under self-synchronization control is given by:
$$Z_{\text{out}}(s) = \frac{V_{\text{PCC}}(s)}{i(s)} = \frac{G_p(s)}{1 + G_o(s)}$$
which shows reduced sensitivity to grid harmonics compared to PLL-based methods, enhancing the on-grid inverter’s robustness.
To further illustrate key concepts, we provide additional formulas and tables. The SOGI-PLL transfer functions for phase estimation are:
$$H_{\text{SOGI}}(s) = \frac{k \omega s}{s^2 + k \omega s + \omega^2}$$
where \( k \) is a damping factor. This ensures accurate phase tracking from the sampled current in the on-grid inverter. Moreover, the power flow equations during pre-synchronization emphasize unity power factor operation:
$$P = V_g I \cos(\phi), \quad Q = V_g I \sin(\phi)$$
with \( \phi = 0 \) for rectification mode, aligning current and voltage phases inversely. Table 3 summarizes the control parameters used in our experiments for the on-grid inverter, ensuring reproducibility.
| Parameter | Symbol | Value |
|---|---|---|
| Proportional Gain | \( k_p \) | 5 to 30 (adjustable) |
| Resonant Gain | \( k_r \) | 100 |
| SOGI-PLL Gain | \( k \) | 1.414 |
| PI Controller Gains | \( K_P, K_I \) | 0.5, 10 |
| Switching Frequency | \( f_s \) | 10 kHz |
In practice, the on-grid inverter’s performance depends on these parameters, and our strategy allows tuning for different grid strengths. For weak grids with high impedance, the on-grid inverter can maintain stability by adjusting \( k_p \) and \( k_r \) to dampen resonances. We also analyzed harmonic distortion using total harmonic distortion (THD) calculations:
$$\text{THD} = \frac{\sqrt{\sum_{n=2}^{\infty} I_n^2}}{I_1} \times 100\%$$
where \( I_n \) are harmonic current components. Our pre-synchronization strategy keeps THD below 5%, meeting grid standards for on-grid inverters.
Finally, we discuss scalability aspects. The proposed method can be extended to multi-level on-grid inverters or microgrid applications, where multiple on-grid inverters operate in parallel. Synchronization among inverters can be achieved through communication-less techniques using frequency droop control, enhancing reliability. The core principle remains: using current-based phase estimation to pre-synchronize each on-grid inverter before grid connection. This underscores the versatility of our approach for future smart grid integrations.
In summary, this paper presents a comprehensive study on pre-synchronization control for single-phase on-grid inverters, addressing critical challenges in sensorless operation. Through rigorous analysis and validation, we demonstrate that our strategy ensures safe, efficient, and cost-effective grid integration, paving the way for advanced on-grid inverter technologies in renewable energy systems.