In this paper, we explore the application of virtual oscillator control (VOC) combined with proportional-integral (PI) current loop and quasi-proportional-resonant (QPR) voltage loop control to enhance the performance of a single phase inverter. The single phase inverter is a critical component in power conversion systems, particularly in renewable energy applications like photovoltaic systems, where it converts DC power to AC power. However, under heavy load conditions, the LC filter in a single phase inverter can exhibit resonance, leading to output waveform distortion and increased total harmonic distortion (THD). Our study aims to address this issue by integrating advanced control strategies to suppress resonance and improve power quality.
The topology of a single phase inverter typically consists of a DC input, a full-bridge inverter circuit, an LC filter, and a load. The LC filter, comprising an inductor and capacitor, is essential for smoothing the output waveform but can introduce resonance when the load impedance changes. This resonance can cause significant distortions, affecting the efficiency and reliability of the single phase inverter. To mitigate this, we propose a control scheme that leverages VOC for its fast response and simplicity, augmented with PI and QPR loops for better regulation.
The mathematical model of the single phase inverter is derived using Kirchhoff’s laws. For the LC filter, the output voltage \( V_o \) and input voltage \( V_i \) relationship can be expressed in the s-domain as:
$$ G(s) = \frac{V_o(s)}{V_i(s)} = \frac{1}{LCs^2 + RCs + 1} $$
where \( L \) is the filter inductance, \( C \) is the filter capacitance, and \( R \) is the load resistance. The natural frequency and damping ratio are given by:
$$ \omega_n = \frac{1}{\sqrt{LC}} $$
$$ \zeta = \frac{R}{2} \sqrt{\frac{C}{L}} $$
Resonance occurs when \( \zeta < 1 \), leading to peak gains that distort the output. To analyze this, we consider the Bode plot characteristics and how control loops can dampen these effects.
Virtual oscillator control emulates a nonlinear oscillator circuit, making the single phase inverter behave like an RLC resonant circuit. The VOC setup includes a current-controlled current source and a nonlinear voltage-controlled current source, described by the function:
$$ f(v) = \begin{cases}
\sigma(v – 2\phi) & \text{if } v > 2\phi \\
-\sigma v & \text{if } -2\phi \leq v \leq 2\phi \\
\sigma(v + 2\phi) & \text{if } v < -2\phi
\end{cases} $$
This function ensures that the single phase inverter oscillates at a desired frequency, with the nonlinearity helping to limit amplitude growth. The oscillation frequency is determined by the LC tank circuit, and the stability is maintained by adjusting parameters like \( K_i \) and \( K_v \), which scale the input current and output voltage, respectively.
To enhance the VOC performance, we incorporate a PI current loop and a QPR voltage loop. The PI controller for the current loop has the transfer function:
$$ G_{PI}(s) = K_{pi} + \frac{K_i}{s} $$
where \( K_{pi} \) is the proportional gain and \( K_i \) is the integral gain. This controller helps in reducing low-frequency errors and suppressing resonance peaks. The closed-loop transfer function for the current control system is:
$$ G_{cl,current}(s) = \frac{G_{PI}(s) G(s)}{1 + G_{PI}(s) G(s)} $$
The QPR controller for the voltage loop is defined as:
$$ G_{QPR}(s) = K_{pr} + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \( K_{pr} \) is the proportional gain, \( K_r \) is the resonant gain, \( \omega_c \) is the cutoff frequency, and \( \omega_0 \) is the resonant frequency. This controller provides high gain at the fundamental frequency, improving tracking accuracy and harmonic rejection.
We analyzed the impact of controller parameters on system performance. For the PI controller, varying \( K_{pi} \) and \( K_i \) affects the bandwidth and resonance damping. Similarly, for the QPR controller, parameters influence the gain and phase margin. The following table summarizes the key parameters used in our simulation:
| Parameter | Value |
|---|---|
| DC Input Voltage | 400 V |
| Filter Inductance (L) | 8 mH |
| Filter Capacitance (C) | 10 μF |
| Load Resistance | 200 Ω |
| VOC \( K_i \) | 0.00214 |
| VOC \( K_v \) | 220 |
| PI \( K_{pi} \) | 4 |
| PI \( K_i \) | 500 |
| QPR \( K_{pr} \) | 2 |
| QPR \( K_r \) | 5 |
| QPR \( \omega_c \) | 3 rad/s |
Simulations were conducted using MATLAB/Simulink to evaluate the proposed control strategy for the single phase inverter. The results showed that without the additional loops, the output voltage and current exhibited significant distortion due to LC filter resonance, with a THD of 3.73%. After implementing the PI and QPR controls, the THD reduced to 1.97%, demonstrating improved waveform quality. The output voltage and current waveforms became smoother, and resonance effects were effectively suppressed.
The Bode plots for the system under different PI parameter values illustrate how \( K_{pi} \) influences the high-frequency response and resonance peak, while \( K_i \) affects the low-frequency gain. For the QPR controller, adjustments in \( K_{pr} \), \( K_r \), and \( \omega_c \) alter the bandwidth and noise rejection capabilities. These insights are crucial for optimizing the single phase inverter design.
In conclusion, integrating PI current loop and QPR voltage loop control with virtual oscillator control significantly enhances the performance of a single phase inverter. This approach mitigates LC filter resonance, reduces waveform distortion, and lowers THD, making it suitable for high-quality power applications. Future work could explore adaptive tuning of controller parameters for varying load conditions in single phase inverter systems.
The dynamics of the single phase inverter can be further described by state-space equations. Let \( x_1 \) represent the inductor current and \( x_2 \) represent the capacitor voltage. The state equations are:
$$ \dot{x_1} = \frac{1}{L} (V_i – x_2 – R x_1) $$
$$ \dot{x_2} = \frac{1}{C} (x_1 – \frac{x_2}{R}) $$
These equations help in designing the control loops for better stability. The transfer function from the input to output for the single phase inverter with the LC filter can be rewritten as:
$$ G(s) = \frac{1}{s^2 LC + s RC + 1} $$
To incorporate the VOC, the nonlinear function \( f(v) \) is linearized around the operating point, leading to a describing function analysis. The equivalent impedance of the VOC circuit is given by:
$$ Z_{eq} = \frac{1}{Y_{eq}} = \frac{1}{G(s) + jB(s)} $$
where \( G(s) \) and \( B(s) \) are the conductance and susceptance, respectively. This facilitates the integration with the PI and QPR controllers.
For the current loop, the open-loop transfer function is:
$$ G_{ol,current}(s) = G_{PI}(s) \cdot G(s) $$
The closed-loop transfer function then becomes:
$$ G_{cl,current}(s) = \frac{G_{ol,current}(s)}{1 + G_{ol,current}(s)} $$
Similarly, for the voltage loop with QPR control, the overall system transfer function is:
$$ G_{system}(s) = \frac{G_{QPR}(s) G_{cl,current}(s)}{1 + G_{QPR}(s) G_{cl,current}(s) H(s)} $$
where \( H(s) \) is the feedback transfer function. In our case, \( H(s) = 1 \) for voltage feedback.
The performance metrics for the single phase inverter include rise time, settling time, and overshoot, which are optimized through controller tuning. The following table compares the key performance indicators before and after adding the control loops:
| Performance Indicator | Without Control Loops | With PI and QPR Control |
|---|---|---|
| Rise Time (ms) | 5.2 | 3.8 |
| Settling Time (ms) | 15.1 | 10.5 |
| Overshoot (%) | 12.3 | 5.7 |
| THD (%) | 3.73 | 1.97 |
These results highlight the effectiveness of the proposed control strategy in improving the transient and steady-state response of the single phase inverter.
In summary, the combination of virtual oscillator control with PI and QPR loops offers a robust solution for managing resonance and enhancing power quality in single phase inverter applications. This approach is particularly beneficial in renewable energy systems, where stable and efficient power conversion is paramount. Further research could involve real-time implementation and testing under dynamic load conditions to validate the simulation findings.
The design of the single phase inverter also considers the impact of parameter variations on system stability. Sensitivity analysis shows that changes in load resistance or filter components can affect the resonance frequency. The damping ratio \( \zeta \) plays a crucial role, and it can be expressed as:
$$ \zeta = \frac{R}{2} \sqrt{\frac{C}{L}} $$
To maintain stability, \( \zeta \) should be kept above a critical value, which is achieved through proper controller design. The VOC parameters \( K_i \) and \( K_v \) are tuned to ensure oscillation at the desired frequency, while the PI and QPR gains are optimized for damping and tracking.
In practice, the single phase inverter must operate efficiently across a range of conditions. The use of digital signal processors (DSPs) for implementing VOC and control loops allows for flexibility and adaptability. The simulation models developed in this work can be extended to include nonlinear loads and grid-connected scenarios for comprehensive analysis.
Overall, this research demonstrates that advanced control techniques can significantly improve the performance of single phase inverters, making them more reliable and suitable for modern power systems. The integration of VOC with voltage-current loops represents a promising direction for future innovations in power electronics.