In the context of renewable energy innovation and large-scale application, the grid-tied inverter serves as a core device connecting distributed generation and the smart grid. Its stability and efficiency are crucial for the reliable operation of the energy internet. However, factors such as long-distance transmission lines and leakage inductance of isolation transformers cause the grid to exhibit characteristics of high impedance and weak grid conditions, leading to phase lag in the output impedance of the grid-tied inverter and reduced stability of the grid-connected system. This paper addresses the phase lag issue of output impedance in single-phase grid-connected inverters caused by grid voltage feedforward by proposing an output impedance phase frequency-segmented compensation method based on a single-phase LCL-type grid-tied inverter. The method improves the magnitude-frequency and phase-frequency characteristics of the grid-tied inverter’s output impedance by cascading a first-order high-pass filter (HPF) and a first-order low-pass filter (LPF) in the grid voltage feedforward path. Specifically, by reducing the delay coefficient in the low-frequency band, the gain and phase margin of the inverter’s output impedance in the mid-low frequency range are enhanced. Simultaneously, the delay coefficient in the high-frequency band is increased to strengthen the passivity of the output impedance in the high-frequency band range. Simulation and experimental results demonstrate that the proposed method effectively improves the phase margin of the output impedance in photovoltaic grid-connected systems, thereby ensuring the stable operation of the grid-tied inverter under weak grid conditions.
The stability of a grid-tied inverter is paramount in modern power systems, especially with the proliferation of distributed energy resources. The output impedance of the grid-tied inverter plays a critical role in determining system stability, particularly when interacting with grid impedance. In weak grids, where the grid impedance is significant, the phase characteristics of the output impedance can lead to instability if not properly managed. Traditional compensation methods, such as series lead compensators or model predictive control, have limitations in complexity, cost, or parameter dependency. Therefore, a frequency-segmented approach that separately addresses low-frequency and high-frequency impedance characteristics offers a robust solution for enhancing the performance of grid-tied inverters.
Output Impedance Modeling of Single-Phase Grid-Tied Inverter
To analyze the stability of a grid-tied inverter, it is essential to derive its output impedance model. For a single-phase LCL-type grid-tied inverter with weighted average current (WAC) control, the system structure can be represented using a Norton equivalent model. The grid-tied inverter is modeled as a current source in parallel with an output impedance. The output current \( i_{ele}(s) \) can be expressed as:
$$ i_{ele}(s) = \frac{1}{1 + Z_{grid}(s)/Z_{inv}(s)} \left[ i_{inv}(s) – \frac{V_{grid}(s)}{Z_{inv}(s)} \right] $$
where \( Z_{inv}(s) \) is the output impedance of the grid-tied inverter, \( Z_{grid}(s) \) is the grid impedance, \( i_{inv}(s) \) is the inverter-side equivalent current source, and \( V_{grid}(s) \) is the grid voltage. This formulation allows for stability analysis based on the impedance ratio \( Z_{grid}(s)/Z_{inv}(s) \). The grid-tied inverter’s output impedance must be designed to ensure stability under varying grid conditions.
When grid voltage feedforward is incorporated to mitigate harmonic effects, the output impedance of the grid-tied inverter is modified. The modified output impedance \( Z_{inv}(s) \) with feedforward can be derived as:
$$ Z_{inv}(s) = – \frac{V_{PCC}}{i_{ele}(s)} = \frac{1 + (1-\beta) G_{QPR}(s) G_{con}(s) G_{dis}(s)}{G_{dis}(s) [1 – G_{fee}(s) G_{con}(s)]} $$
where \( G_{QPR}(s) \) is the quasi-proportional resonant controller transfer function, \( G_{con}(s) \) is the control path transfer function, \( G_{dis}(s) is the disturbance transfer function, \( G_{fee}(s) is the feedforward path transfer function, and \( \beta \) is the weighting coefficient for WAC control. This equation highlights the impact of feedforward on the output impedance of the grid-tied inverter.
Stability Analysis for Grid-Tied Inverters
The stability of a grid-connected system involving a grid-tied inverter depends on the phase margin at the frequency where the magnitudes of \( Z_{inv}(s) \) and \( Z_{grid}(s) \) intersect, denoted as \( f_{cut} \). According to the Nyquist stability criterion, the system remains stable if the phase margin \( \gamma \) is positive:
$$ \gamma = 180^\circ – \left[ \angle Z_{grid}(j2\pi f_{cut}) – \angle Z_{inv}(j2\pi f_{cut}) \right] = 90^\circ + \angle Z_{inv}(j2\pi f_{cut}) > 0 $$
This implies that for stability, the phase of \( Z_{inv}(s) \) at \( f_{cut} \) must be greater than \( -90^\circ \). In practice, the region where \( \angle Z_{inv}(s) > -90^\circ \) is considered passive and stable, while \( \angle Z_{inv}(s) < -90^\circ \) is active and potentially unstable. Therefore, enhancing the phase characteristics of the output impedance is crucial for the reliable operation of grid-tied inverters in weak grids.
Proposed Phase Frequency-Segmented Compensation Method
The proposed method aims to compensate for the phase lag in the output impedance of a grid-tied inverter by segmenting the frequency domain into low-frequency and high-frequency bands. This is achieved by cascading a first-order high-pass filter and a first-order low-pass filter in the grid voltage feedforward path. The compensated output impedance \( Z’_{inv}(s) \) is given by:
$$ Z’_{inv}(s) = \frac{1 + (1-\beta) G_{QPR}(s) G_{con}(s) G_{dis}(s)}{G_{dis}(s) \left[1 – G_{fee}(s) \left( G_{MLPC}(s) + G_{HPC}(s) \right) G_{con}(s) \right]} $$
where \( G_{MLPC}(s) \) is the mid-low frequency phase compensator and \( G_{HPC}(s) \) is the high-frequency phase compensator, defined as:
$$ G_{MLPC}(s) = k_1 \frac{\omega_{cor1}}{s + \omega_{cor1}} e^{-\lambda_1 T_s} $$
$$ G_{HPC}(s) = k_2 \frac{s}{s + \omega_{cor2}} e^{-\lambda_2 T_s} $$
Here, \( k_1 \) and \( k_2 \) are gains, \( \omega_{cor1} \) and \( \omega_{cor2} \) are corner frequencies, \( \lambda_1 \) and \( \lambda_2 \) are delay coefficients for low-frequency and high-frequency bands, respectively, and \( T_s \) is the sampling period. This structure allows independent tuning of impedance characteristics across frequency bands, improving the overall performance of the grid-tied inverter.
Parameter Design for the Compensators
The design of parameters for the phase compensators is critical to ensure that the output impedance of the grid-tied inverter meets stability requirements. The goal is to enhance phase margin in the mid-low frequency range and strengthen passivity in the high-frequency range. Below, a summary of parameter design considerations is provided in tabular form.
| Parameter | Description | Design Criteria | Typical Value |
|---|---|---|---|
| \( k_1 \) | Gain for low-frequency compensator | Ensure phase margin > 40° at \( f_{cut} \) | 0.6 |
| \( \omega_{cor1} \) | Corner frequency for HPF (rad/s) | Avoid interaction with LCL resonance | 1000π |
| \( \lambda_1 \) | Delay coefficient for low-frequency band | Minimize digital control delay | 1.5 |
| \( k_2 \) | Gain for high-frequency compensator | Maintain positive real part of impedance | 0.4 |
| \( \omega_{cor2} \) | Corner frequency for LPF (rad/s) | Attenuate high-frequency components | 4000π |
| \( \lambda_2 \) | Delay coefficient for high-frequency band | Provide phase lag near resonance | 3.5 |
The selection of \( k_1 \) is based on ensuring a sufficient phase margin at the crossover frequency. For the grid-tied inverter, a phase margin of 40° is targeted, leading to \( k_1 = 0.6 \). The delay coefficient \( \lambda_1 \) is set to 1.5 to account for digital control delays, including sampling and computation. For the high-frequency compensator, \( k_2 \) is chosen as 0.4 to keep the real part of the output impedance positive, thereby enhancing passivity. The delay coefficient \( \lambda_2 \) is set to 3.5 to introduce a phase lag of approximately -180° at the LCL filter resonance frequency, forming a negative feedback loop that suppresses high-frequency oscillations. These parameters collectively improve the impedance characteristics of the grid-tied inverter across all frequency bands.
Simulation and Experimental Verification
To validate the effectiveness of the proposed phase frequency-segmented compensation method, simulations and experiments were conducted on a 3 kW single-phase LCL-type grid-tied inverter. The system parameters are summarized in the table below.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Grid Voltage \( V_{grid} \) | 110 V | DC Link Voltage \( V_{dc} \) | 220 V |
| Inverter-side Inductance \( L_{inv} \) | 1 mH | Grid-side Inductance \( L_{ele} \) | 0.8 mH |
| Filter Capacitance \( C \) | 10 μF | Fundamental Frequency \( f_0 \) | 50 Hz |
| Switching Frequency \( f_{sw} \) | 15 kHz | Sampling Frequency \( f_{sam} \) | 30 kHz |
| Weighting Coefficient \( h \) | 0.02 | Low-frequency Gain \( k_1 \) | 0.6 |
| High-frequency Gain \( k_2 \) | 0.4 | Corner Frequency \( \omega_{cor1} \) | 3140 rad/s |
| Corner Frequency \( \omega_{cor2} \) | 13000 rad/s | Delay Coefficient \( \lambda_1 \) | 1.5 |
| Delay Coefficient \( \lambda_2 \) | 3.5 |
Simulation results under weak grid conditions with short-circuit ratio (SCR) values of 6 and 8 demonstrate that the proposed compensation method significantly improves stability. Without compensation, the grid-tied inverter exhibits oscillatory behavior and high total harmonic distortion (THD). With the phase frequency-segmented compensation, the grid current and voltage waveforms become smooth, with THD reduced to below 2.5%. Moreover, the method maintains stability even when the grid impedance is capacitive, showcasing its robustness for grid-tied inverters in diverse weak grid scenarios.
Experimental verification was performed using an RT-LAB hardware-in-the-loop platform. The results align with simulations, confirming that the compensated grid-tied inverter operates stably under low SCR conditions. The dynamic response of the grid current during step changes from 10 A to 20 A shows fast tracking and minimal oscillation, indicating enhanced performance of the grid-tied inverter. These findings underscore the practicality of the proposed method for real-world applications involving grid-tied inverters.
Conclusion
This paper has presented a phase frequency-segmented compensation method to address the phase lag issue in the output impedance of single-phase LCL-type grid-tied inverters caused by grid voltage feedforward and digital delays. By cascading a first-order high-pass filter and a first-order low-pass filter in the feedforward path, the output impedance characteristics of the grid-tied inverter are reshaped across frequency bands. The mid-low frequency compensator enhances phase margin, while the high-frequency compensator strengthens passivity, collectively improving stability under weak grid conditions. Simulation and experimental results confirm that the method ensures stable operation of the grid-tied inverter at low short-circuit ratios and with various grid impedance types. The proposed approach offers a effective solution for enhancing the robustness and performance of grid-tied inverters in renewable energy systems, contributing to the reliable integration of distributed generation into the power grid.
The grid-tied inverter is a pivotal component in modern power electronics, and its impedance characteristics directly impact system stability. The frequency-segmented compensation method provides a flexible framework for tuning these characteristics, making it suitable for adaptive control in dynamically changing grid environments. Future work could explore the integration of this method with advanced grid-support functions for grid-tied inverters, such as virtual inertia or harmonic filtering, to further enhance grid resilience. Overall, the development of such compensation strategies is essential for the continued expansion of grid-tied inverter applications in smart grids and renewable energy systems.