In modern distributed generation (DG) systems, the integration of renewable energy sources has become increasingly prevalent, with solar power playing a pivotal role. Among the various components, inverters are critical for converting DC power from sources like solar panels into AC power for grid or off-grid applications. When discussing the types of solar inverter, it is essential to distinguish between grid-tied, off-grid, and hybrid inverters, each serving distinct purposes in energy systems. Grid-tied inverters synchronize with the utility grid, while off-grid inverters operate independently, often in remote areas. Hybrid inverters combine features of both, allowing for battery storage integration. In this context, parallel operation of inverters is a common approach to enhance system capacity, reliability, and efficiency. However, this setup introduces challenges such as uneven power distribution and circulating currents, which can degrade performance and lead to component failure. This paper addresses these issues by proposing a control strategy that combines droop control with a voltage-current double-loop control, utilizing a quasi-Proportional-Resonant (PR) controller to minimize static errors and improve dynamic response. We will explore the theoretical foundations, design methodologies, and simulation results to demonstrate the effectiveness of this approach, with a focus on applications in off-grid systems involving various types of solar inverter configurations.
The proliferation of DG systems, particularly those based on solar energy, has underscored the importance of efficient inverter technologies. Different types of solar inverter, such as string inverters, microinverters, and central inverters, offer unique advantages in terms of scalability and maintenance. In off-grid scenarios, parallel inverter systems are employed to increase power output and provide redundancy. However, without proper control, parallel inverters can experience circulating currents—unwanted currents that flow between inverters without contributing to the load. This phenomenon arises due to discrepancies in output voltages, impedances, or control parameters among inverters. For instance, in systems using multiple types of solar inverter, variations in internal impedance or switching frequencies can exacerbate these issues. The circulating current not only increases losses and heating but also threatens the stability of the entire system. Therefore, developing robust control strategies is paramount to ensuring reliable operation.
To analyze the circulating current in a parallel inverter system, consider a simplified equivalent circuit with two inverters. Let \(E_1\) and \(E_2\) represent the output voltages of the inverters, \(V\) denote the load voltage, \(\phi\) be the phase angle difference, and \(Z\) and \(\theta\) symbolize the magnitude and phase of the output impedance, respectively. The circulating current \(I_h\) can be expressed as:
$$I_h = \frac{I_1 – I_2}{2}$$
Assuming equal output impedances for both inverters, the circulating current is given by:
$$I_h = \frac{E_1 \angle \phi_1 – E_2 \angle \phi_2}{2Z_1 \angle \theta_1}$$
This current circulates solely between the inverters, increasing system losses and reducing efficiency. In practical applications, especially when integrating different types of solar inverter, the output impedances may not be identical, leading to more complex circulating current patterns. Thus, a control mechanism that ensures power sharing and suppresses circulating currents is essential.
Droop control is a widely adopted method for power management in parallel inverter systems, inspired by the characteristics of synchronous generators. It enables decentralized control without communication links, making it suitable for systems with diverse types of solar inverter. The fundamental principle involves adjusting the frequency and amplitude of the output voltage based on the active and reactive power outputs. The droop control equations are:
$$\omega = \omega^* – mP$$
$$E = E^* – nQ$$
Here, \(\omega\) and \(E\) are the angular frequency and voltage amplitude of the inverter output, \(\omega^*\) and \(E^*\) are their reference values, and \(m\) and \(n\) are the droop coefficients for active and reactive power, respectively. The values of \(m\) and \(n\) are critical; they must be chosen to limit frequency deviations to within ±1% and voltage deviations to within ±5% to maintain power quality. For example, in systems with various types of solar inverter, the droop coefficients can be tuned to account for differences in inverter ratings or line impedances.
The relationship between power outputs and system parameters can be derived from the power flow equations. For an inverter with output voltage \(E\), load voltage \(V\), and output impedance \(Z \angle \theta\), the active power \(P\) and reactive power \(Q\) are:
$$P = \frac{EV}{Z} \cos \phi – \frac{V^2}{Z} \cos \theta + \frac{EV}{Z} \sin \phi \sin \theta$$
$$Q = \frac{EV}{Z} \cos \phi – \frac{V^2}{Z} \sin \theta – \frac{EV}{Z} \sin \phi \cos \theta$$
Assuming a small phase angle \(\phi\) (typical in stable operations), we approximate \(\sin \phi \approx \phi\) and \(\cos \phi \approx 1\). If the output impedance is predominantly inductive (common in inverter systems with large filter inductors), the equations simplify to:
$$P = \frac{EV}{X} \phi$$
$$Q = \frac{V}{X} (E – V)$$
where \(X\) is the inductive reactance. This shows that active power \(P\) is primarily influenced by the phase angle \(\phi\), while reactive power \(Q\) depends on the voltage magnitude difference. By employing droop control, the system can achieve proportional power sharing among inverters, regardless of the types of solar inverter used. For instance, if one inverter outputs excessive active power, its frequency decreases, reducing its power share and balancing the load.
To enhance the dynamic performance and disturbance rejection capability, a voltage-current double-loop control structure is implemented. The outer voltage loop regulates the output voltage, while the inner current loop controls the filter capacitor current, providing fast response to load changes. This is particularly beneficial in systems with mixed types of solar inverter, where load variations can be abrupt. The control block diagram illustrates the integration of droop control with the double-loop structure. The reference voltage \(U_{\text{ref}}\) generated by the droop controller serves as the input to the voltage loop. The voltage controller uses a quasi-PR controller to achieve zero steady-state error for sinusoidal signals, unlike traditional PI controllers that struggle with AC reference tracking.
The transfer function of the quasi-PR controller is:
$$G_{\text{PR}}(s) = k_P + \frac{2k_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2}$$
where \(k_P\) is the proportional gain, \(k_r\) is the resonant gain, \(\omega_c\) is the cutoff frequency, and \(\omega_0\) is the fundamental angular frequency (e.g., 314 rad/s for 50 Hz systems). This controller provides high gain at the fundamental frequency, ensuring accurate tracking of the voltage reference and suppression of low-order harmonics. The current inner loop employs a proportional controller with gain \(K_P\) to improve response speed. The overall output voltage \(U_o(s)\) in terms of the reference voltage \(U_{\text{ref}}(s)\) and output current \(I_o(s)\) is derived as:
$$U_o(s) = \frac{G_{\text{PR}}(s) K_P K_{\text{PWM}}}{(R_s + sL_f + K_P K_{\text{PWM}}) s C_f + G_{\text{PR}}(s) K_P K_{\text{PWM}} + 1} U_{\text{ref}}(s) – \frac{R_s + sL_f + K_P K_{\text{PWM}}}{(R_s + sL_f + K_P K_{\text{PWM}}) s C_f + G_{\text{PR}}(s) K_P K_{\text{PWM}} + 1} I_o(s)$$
where \(K_{\text{PWM}}\) is the inverter gain, \(L_f\) is the filter inductance, \(C_f\) is the filter capacitance, and \(R_s\) is the equivalent series resistance. The voltage gain \(G(s)\) is:
$$G(s) = \frac{G_{\text{PR}}(s) K_P K_{\text{PWM}}}{(R_s + sL_f + K_P K_{\text{PWM}}) s C_f + G_{\text{PR}}(s) K_P K_{\text{PWM}} + 1}$$
The characteristic equation of the closed-loop system is:
$$D(s) = L_f C_f s^4 + (R_s C_f + 2\omega_c L_f C_f + K_P K_{\text{PWM}} C_f) s^3 + (2 R_s \omega_c C_f + L_f C_f \omega_0^2 + 2\omega_c K_P K_{\text{PWM}} C_f + k_P K_P K_{\text{PWM}} + 1) s^2 + (R_s C_f \omega_0^2 + K_P K_{\text{PWM}} \omega_0^2 C_f + 2 k_P K_P K_{\text{PWM}} \omega_c + 2 k_r \omega_c K_P K_{\text{PWM}} + 2\omega_c) s + (k_P K_P K_{\text{PWM}} + 1) \omega_0^2$$
To ensure stability, the poles are placed with a damping ratio of 0.707. This design approach is applicable to various types of solar inverter, as it accommodates differences in filter parameters and switching frequencies.
In practical implementations, the selection of control parameters is crucial. The following table summarizes key parameters used in a typical off-grid parallel inverter system, which can be adapted for different types of solar inverter:
| Parameter | Symbol | Value |
|---|---|---|
| DC Input Voltage | \(V_{\text{dc}}\) | 400 V |
| Rated Power | \(P_{\text{rated}}\) | 10 kW |
| Filter Capacitance | \(C_f\) | 20 μF |
| Filter Inductance | \(L_f\) | 5 mH |
| Active Power Droop Coefficient | \(m\) | 3 × 10-4 |
| Reactive Power Droop Coefficient | \(n\) | 1.5 × 10-2 |
| Switching Frequency | \(f_{\text{sw}}\) | 10 kHz |
| Inverter Gain | \(K_{\text{PWM}}\) | 200 |
| Proportional Gain (Current Loop) | \(K_P\) | 0.5 |
| Proportional Gain (Voltage Loop) | \(k_P\) | 1.2 |
| Resonant Gain | \(k_r\) | 25 |
| Cutoff Frequency | \(\omega_c\) | 5 rad/s |
| Fundamental Frequency | \(\omega_0\) | 314 rad/s |
Simulation studies using MATLAB/Simulink were conducted to validate the proposed control strategy for a system with two parallel inverters. The system was tested under load variations to assess power sharing, circulating current suppression, and dynamic response. The results demonstrate that the combination of droop control and double-loop control with quasi-PR controller effectively maintains system stability and adapts to load changes. For example, during a load step change from 1 s to 2 s, the output powers of both inverters remain balanced, with minimal circulating current. The output frequency deviates by only 0.17 Hz under steady state and 0.24 Hz during transients, within acceptable limits. Similarly, the voltage total harmonic distortion (THD) is 1.63%, indicating high power quality. These findings are relevant for various types of solar inverter, as the control strategy can be scaled for larger systems.
The output voltage and current waveforms from the simulation show smooth transitions with low distortion. The FFT analysis of the output voltage reveals a dominant fundamental component and minimal harmonics, affirming the effectiveness of the quasi-PR controller in tracking the reference signal. This is particularly important in off-grid systems with nonlinear loads, where harmonic currents can cause additional circulating currents. By implementing the proposed control, systems incorporating different types of solar inverter can achieve improved reliability and efficiency.
In conclusion, the integration of droop control with a voltage-current double-loop control using a quasi-PR controller offers a robust solution for parallel inverter systems in off-grid DG applications. This approach addresses key challenges such as power sharing inaccuracies and circulating currents, which are common when combining various types of solar inverter. The droop control enables decentralized power management, while the double-loop structure enhances dynamic response and disturbance rejection. Simulation results confirm that the strategy maintains stability under load variations, with minimal frequency and voltage deviations. Future work could explore adaptive droop coefficients for heterogeneous inverter systems or real-time optimization techniques to further improve performance. Overall, this control methodology contributes to the advancement of sustainable energy systems by ensuring efficient and reliable operation of parallel inverters.