In modern power systems, the widespread use of nonlinear and unbalanced loads poses significant challenges to the stability and power quality of three-phase inverters. Traditional control strategies often struggle to maintain balanced output voltages and suppress harmonics under such conditions. This paper addresses these issues by proposing a fractional-order quasi-proportional complex integral (FO-QPCI) control algorithm for three-phase inverters. The approach leverages fractional calculus to enhance system adaptability and robustness, ensuring effective voltage regulation and harmonic suppression in unbalanced operating scenarios. We begin by developing a fractional-order model of the three-phase inverter system, analyzing its behavior in different coordinate frames. Subsequently, the FO-QPCI controller is designed and implemented in the αβ stationary reference frame, offering improved performance over conventional methods. Experimental validation on a hardware platform confirms the efficacy of the proposed strategy, demonstrating superior voltage balance and reduced harmonic distortion.
The three-phase inverter is a critical component in various applications, including renewable energy systems, uninterruptible power supplies, and microgrids. Under ideal conditions, the output voltages of a three-phase inverter are balanced and sinusoidal. However, in practical scenarios, unbalanced loads introduce negative-sequence components and harmonics, leading to voltage distortion and potential system instability. Existing control techniques, such as proportional-integral (PI) controllers in the dq rotating frame, exhibit limitations in tracking AC signals and suppressing harmonics. Resonant controllers and repetitive control methods offer some improvements but often involve complex designs and reduced stability margins. To overcome these drawbacks, we integrate fractional-order calculus with complex domain control, resulting in a more flexible and robust solution for three-phase inverters.
The fractional-order modeling of the three-phase inverter provides a more accurate representation of system dynamics compared to integer-order models. By considering non-integer orders for inductors and capacitors, we capture additional degrees of freedom that influence current and voltage responses. The Caputo definition of fractional calculus is employed due to its suitability for engineering applications. The mathematical model is derived in the abc frame and transformed into the αβ stationary frame to simplify control design. This transformation allows for decoupled control of the α and β components, facilitating independent handling of positive and negative sequence voltages. The fractional-order model is approximated using the Oustaloup filter method, ensuring practical implementability. Key equations governing the system include the fractional Kirchhoff’s laws, expressed as:
$$ D^\lambda i_{Lx} = \frac{1}{L_f} (U_x – V_x – R_f i_{Lx}) $$
$$ D^\mu V_x = \frac{1}{C_f} (i_{Lx} – i_{llx}) $$
where \( D^\lambda \) and \( D^\mu \) denote fractional derivatives of orders \( \lambda \) and \( \mu \) for the inductor and capacitor, respectively. The values \( \lambda = 1.2 \) and \( \mu = 0.8 \) are chosen to balance dynamic performance and filtering effectiveness. The Clark transformation matrix converts the system to the αβ frame:
$$ \begin{bmatrix} x_\alpha \\ x_\beta \end{bmatrix} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} x_a \\ x_b \\ x_c \end{bmatrix} $$
In this frame, the fractional-order model becomes:
$$ D^\lambda i_{L\alpha} = \frac{1}{L_f} (U_\alpha – V_\alpha – R_f i_{L\alpha}) $$
$$ D^\lambda i_{L\beta} = \frac{1}{L_f} (U_\beta – V_\beta – R_f i_{L\beta}) $$
$$ D^\mu V_\alpha = \frac{1}{C_f} (i_{L\alpha} – i_{ll\alpha}) $$
$$ D^\mu V_\beta = \frac{1}{C_f} (i_{L\beta} – i_{ll\beta}) $$
This representation eliminates coupling between axes, enabling straightforward control design. The FO-QPCI controller is then applied to regulate the output voltages. The traditional proportional complex integral (PCI) controller offers infinite gain at the resonant frequency but suffers from stability issues. The quasi-PCI (QPCI) variant introduces a bandwidth coefficient to mitigate this problem. By incorporating fractional orders, the FO-QPCI controller achieves enhanced flexibility in adjusting the resonance bandwidth and gain characteristics. The transfer function of the FO-QPCI controller in the complex domain is:
$$ G_{v}(s) = K_p + \frac{K_i}{(s – j\omega_r)^\theta + \omega_c} + \frac{K_i}{(s + j\omega_r)^\theta + \omega_c} $$
where \( K_p \) is the proportional gain, \( K_i \) is the integral gain, \( \omega_r \) is the resonant frequency, \( \omega_c \) is the bandwidth coefficient, and \( \theta \) is the fractional order. For \( \theta = 1 \), the controller reduces to the integer-order QPCI. The fractional order allows for finer tuning of the controller’s frequency response, improving harmonic suppression and robustness to frequency variations. The implementation of the FO-QPCI controller in the αβ frame involves cross-coupling between the α and β axes to handle the complex terms. The control structure for the α-axis is depicted as follows, with similar implementation for the β-axis:
| Component | Description |
|---|---|
| Input | Voltage error \( e_\alpha \) |
| Proportional Path | Gain \( K_p \) |
| Resonant Path | Fractional integrator with cross-coupling from β-axis |
| Output | Control signal \( u_\alpha \) |
Parameter selection for the FO-QPCI controller is critical for achieving desired performance. The inner current loop uses a proportional controller with gain \( G_i = 0.42 \), set to one-tenth of the LC filter cutoff frequency to ensure stability and dynamic response. The voltage outer loop parameters are determined based on root locus and Bode plot analyses. The proportional gain \( K_p = 0.3 \) influences the low and high-frequency gains, while the integral gain \( K_i = 53.3 \) affects the resonance peak magnitude. The bandwidth coefficient \( \omega_c = 5 \, \text{rad/s} \) provides sufficient margin for frequency deviations up to \( \pm 0.5 \, \text{Hz} \). The fractional order \( \theta = 0.8 \) is chosen to optimize the resonance bandwidth without compromising stability. The open-loop transfer function of the voltage control loop is:
$$ G_{ol}(s) = G_v(s) H_2(s) $$
where \( H_2(s) \) represents the plant model in the αβ frame. The Bode plots of the FO-QPCI controller for different parameter values illustrate its adaptability. For instance, increasing \( K_p \) boosts gain at non-resonant frequencies, while increasing \( K_i \) enhances resonance tracking accuracy but may reduce stability. The root locus analysis under varying load conditions confirms that the system remains stable across the operating range when parameters are tuned for no-load conditions.
The digital implementation of the FO-QPCI controller involves approximating the fractional operators \( s^\lambda \) and \( (j\omega_0)^\lambda \). The Oustaloup filter method is used to fit \( s^\lambda \) over a frequency range \( (\omega_b, \omega_h) = (10^{-6}, 10^6) \, \text{rad/s} \). The approximation is given by:
$$ s^\lambda \approx K \prod_{k=-N}^{N} \frac{s + \omega_k’}{s + \omega_k} $$
where \( \omega_k’ \) and \( \omega_k \) are recursively calculated frequencies, and \( K \) is a constant. For the term \( (j\omega_0)^\lambda \), Euler’s formula is applied to decompose it into real and imaginary parts. The resulting expressions are:
$$ (j\omega_0)^\lambda = \omega_0^\lambda \left( \cos\left(\frac{\lambda \pi}{2}\right) + j \sin\left(\frac{\lambda \pi}{2}\right) \right) $$
This allows the resonant part of the controller to be implemented using real-valued filters and cross-coupling between axes. The digital realization ensures compatibility with modern digital signal processors and microcontrollers, making the FO-QPCI controller practical for real-time applications in three-phase inverters.
Experimental validation of the proposed control strategy is conducted on a hardware platform featuring a three-level neutral-point-clamped (NPC) inverter. The system parameters are summarized in the table below:
| Parameter | Value |
|---|---|
| DC Link Voltage | 200 V |
| Filter Inductance \( L_f \) | 2 mH |
| Filter Capacitance \( C_f \) | 20 μF |
| Switching Frequency | 10 kHz |
| Output Voltage | 77 V (phase) |
Under balanced load conditions, the three-phase inverter with FO-QPCI control maintains sinusoidal output voltages with minimal distortion. The voltage and current waveforms are stable, and the neutral point potential remains balanced within \( \pm 0.3 \, \text{V} \). In unbalanced load scenarios, the performance of the FO-QPCI controller is compared with traditional PI and QPCI methods. The FO-QPCI controller achieves faster dynamic response, reaching steady state in 0.01 seconds compared to 0.02 seconds for QPCI. The output voltages exhibit lower unbalancedness and harmonic distortion, as quantified in the following table:
| Control Method | Voltage Unbalancedness (%) | THD (%) |
|---|---|---|
| PI | 5.2 | 8.1 |
| QPCI | 0.2 | 4.0 |
| FO-QPCI | 0.1 | 2.0 |
The superior performance of the FO-QPCI controller is attributed to its ability to independently regulate positive and negative sequence components while suppressing harmonics through adjustable fractional orders. The controller’s robustness to load variations and frequency shifts makes it suitable for diverse applications involving three-phase inverters. Further analysis of the harmonic spectrum shows that the FO-QPCI controller significantly reduces low-order harmonics, contributing to improved power quality.
In conclusion, the fractional-order quasi-proportional complex integral control strategy offers a compelling solution for managing unbalanced conditions in three-phase inverters. The integration of fractional calculus enhances the controller’s degrees of freedom, enabling precise tuning of resonance characteristics and harmonic suppression. The mathematical modeling, parameter design, and digital implementation provide a comprehensive framework for practical applications. Experimental results validate the effectiveness of the FO-QPCI controller in maintaining voltage balance and reducing distortion, underscoring its potential for advancing the performance of three-phase inverters in modern power systems. Future work could explore adaptive fractional-order control for varying operating conditions and integration with other renewable energy sources.
The development of advanced control strategies for three-phase inverters remains a critical area of research, especially with the increasing penetration of distributed generation. The FO-QPCI algorithm represents a step forward in addressing the challenges posed by unbalanced and nonlinear loads. By leveraging fractional-order theory, we can achieve more resilient and efficient power conversion systems. The insights gained from this study may inspire further innovations in the control of three-phase inverters, contributing to the stability and reliability of future energy networks.
In summary, the key contributions of this work include the fractional-order modeling of the three-phase inverter, the design of the FO-QPCI controller, and its experimental validation. The controller’s ability to handle unbalanced conditions while suppressing harmonics demonstrates its superiority over conventional methods. As power systems evolve, the adoption of such advanced control techniques will be essential for ensuring high-quality power delivery and system stability. The three-phase inverter, as a fundamental power electronic device, stands to benefit significantly from these advancements, enabling more robust and adaptable energy solutions.