As a critical interface between photovoltaic systems and the power grid, solar inverters play a vital role in efficiently and reliably feeding clean energy into the grid. However, the parallel operation of multiple solar inverters often leads to circulating currents, which reduce equipment efficiency and increase power losses. This issue arises due to differences in inverter parameters and line impedances, posing a significant challenge for large-scale solar power integration. Existing methods, such as common-mode voltage suppression and pulse-width modulation techniques, have shown limitations in multi-scenario validation and practical applications. Similarly, model predictive control and robust voltage droop methods face complexities and sensitivity issues. To address these challenges, this paper proposes a novel control strategy based on virtual complex impedance, which dynamically adjusts system impedance to suppress circulating currents and improve power sharing among parallel solar inverters.
The proliferation of solar energy systems has intensified the need for reliable inverter technologies. Solar inverters must handle varying operational conditions while maintaining grid stability. In parallel configurations, circulating currents emerge from imbalances in voltage magnitudes, phase angles, and output impedances. These currents not only degrade performance but also pose risks to system longevity. Traditional approaches often rely on fixed impedance settings or complex decoupling algorithms, which are inadequate for dynamic environments. Our work introduces an adaptive virtual complex impedance method that simplifies control structures and enhances robustness. By leveraging real-time adjustments, this strategy ensures that the equivalent output impedance of solar inverters remains resistive, minimizing circulating currents and optimizing power distribution.
To understand the circulating current phenomenon, we first establish a mathematical model for parallel solar inverters. Consider a system with multiple solar inverters connected to a common bus, as illustrated in the following diagram. Each solar inverter comprises a DC source, a three-phase inverter circuit with IGBT switches, an LCL filter for harmonic suppression, and control units for signal processing. The control system includes phase-locked loops (PLL), Park transformations, and pulse-width modulation (PWM) to generate driving signals for the switches.
In this configuration, the equivalent electrical model for two parallel solar inverters can be represented by simplified circuits. Let \( E_i \angle \theta_i \) denote the no-load voltage of the i-th solar inverter, \( Z_{\text{eq}i} \) its equivalent output impedance, and \( Z_{\text{line}i} = R_i + jX_i \) the line impedance. The load impedance is \( Z_{\text{load}} \), and the common bus voltage is \( U_L \angle 0 \). The output currents \( I_1 \) and \( I_2 \) from the solar inverters satisfy the following power equations:
$$S_i = P_i + jQ_i = \dot{E_i} \left( \frac{\dot{E_i} – \dot{U_L}}{R + jX} \right)^* \quad (i = 1, 2)$$
Expressing the active and reactive power components:
$$\begin{cases}
P_i = \frac{U_L [R_i (E_i \cos \alpha_i – U_L) + E_i X_i \sin \alpha_i]}{R_i^2 + X_i^2} \\
Q_i = \frac{U_L [X_i (E_i \cos \alpha_i – U_L) – E_i R_i \sin \alpha_i]}{R_i^2 + X_i^2}
\end{cases}$$
where \( \alpha_i \) is the power angle. In low-voltage solar systems, the grid is typically weak, with resistance dominating reactance (\( R \gg X \)). Thus, simplifying by neglecting inductive components, \( \sin \alpha_i \approx \alpha_i \) and \( \cos \alpha_i \approx 1 \), we derive:
$$\begin{cases}
P_i \approx \frac{U_L (E_i – U_L)}{R_i} \\
Q_i \approx -\frac{E_i U_L \alpha_i}{R_i}
\end{cases}$$
This indicates that active power \( P \) depends on voltage magnitude \( U_L \), while reactive power \( Q \) relates to the power angle \( \alpha_i \). Consequently, a resistive droop control strategy is adopted, as it better aligns with low-voltage characteristics compared to inductive approaches. The droop control equations are:
$$\begin{cases}
U = U^* – n(P – P^*) \\
f = f^* – m(Q – Q^*)
\end{cases}$$
where \( U^* \) and \( f^* \) are reference values, and \( m \), \( n \) are droop coefficients. These relationships facilitate power sharing but are susceptible to impedance disparities, leading to circulating currents. The circulating current \( I_{cc} \) between two solar inverters is defined as:
$$I_{cc} = \frac{I_1 – I_2}{2} = \frac{E_1 \angle \theta_1 – E_2 \angle \theta_2}{Z_1 + Z_2}$$
where \( Z_i = Z_{\text{eq}i} + Z_{\text{line}i} \). This equation highlights that circulating currents arise from differences in voltage magnitudes, phases, and impedances, underscoring the need for impedance-based suppression strategies.
To mitigate circulating currents, we propose a multi-loop control framework incorporating virtual complex impedance. The outer loop manages power control for maximum power point tracking (MPPT), while the inner loops handle voltage and current regulation. A phase-locked loop (PLL) ensures synchronization. The core innovation lies in introducing a virtual complex impedance \( Z_{\text{vir}}(s) = R_v – sX_v \), which includes an adaptive resistive component and a negative inductive component. This virtual impedance reshapes the equivalent output impedance of the solar inverters to be predominantly resistive, thereby decoupling power control and suppressing circulating currents.
The voltage control structure with virtual impedance feedback is depicted in a block diagram, where \( U_n \) is the reference voltage, \( U_0 \) the output voltage, \( K_{\text{PWM}} \) the inverter gain, and \( i_0 \), \( i_L \) the output and inductor currents, respectively. The controllers include a quasi-proportional-resonant (PR) controller for voltage and a proportional (P) controller for current, with transfer functions:
$$\begin{cases}
T_u(s) = k_{\text{PR}} + \frac{2k_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} \\
T_i(s) = k_e
\end{cases}$$
The output voltage transfer function is:
$$U_0(s) = G(s)U_n(s) – Z(s)i_0(s)$$
where:
$$G(s) = \frac{K_{\text{PWM}} T_i(s) T_u(s)}{L C s^2 + T_i(s) K_{\text{PWM}} C s + T_i(s) K_{\text{PWM}} T_u(s) + 1}$$
and the equivalent output impedance \( Z_q(s) \) is:
$$Z_q(s) = \frac{L s + Z_{\text{vir}}(s) K_{\text{PWM}} T_i(s) T_u(s)}{L C s^2 + T_i(s) K_{\text{PWM}} C s + T_i(s) K_{\text{PWM}} T_u(s) + 1}$$
The adaptive virtual resistance \( R_v \) is dynamically adjusted based on voltage deviations:
$$R_v = R_x – \alpha \Delta U – \beta \frac{d(\Delta U)}{dt}$$
where \( R_x \) is a preset resistance, \( \alpha \) and \( \beta \) are tuning coefficients, and \( \Delta U = U_{\text{ref}} – U_0 \) with \( U_{\text{ref}} = U_n – Z_{\text{vir}} i_0 \). The negative virtual inductance \( -X_v \) counteracts the inherent inductive components, as shown in the impedance vector diagram. By introducing \( -X_v \), the equivalent impedance shifts toward resistive behavior without significantly altering the resistive part, thus enhancing power decoupling.
Furthermore, to address residual errors in active power sharing, we implement an adaptive droop coefficient algorithm. The droop coefficients \( n_{si} \) are dynamically adjusted based on the average active power \( P_{\text{ref}} \) and individual inverter outputs \( P_i \):
$$\begin{cases}
U_n = U^* – n_s (P_i – P^*) \\
n_{si} = n_i – \left(k_p + \frac{k_i}{s}\right) (P_{\text{ref}} – P_i) \\
P_{\text{ref}} = \frac{P_1 + P_2}{2}
\end{cases}$$
where \( n_i \) is the conventional droop coefficient, bounded by:
$$0 < n_i \leq \frac{f^* – f_{\text{min}}}{P_{\text{max}} – P_{\text{ref}}}$$
This adaptive approach ensures precise active power distribution and reduces circulating currents by continuously optimizing control parameters.
The overall control strategy integrates these elements into a cohesive framework. Measured currents are processed through virtual negative inductance and adaptive virtual impedance before entering the dual-loop control. This setup simplifies traditional algorithms and improves circulating current suppression accuracy. The control diagram illustrates the interconnection of power loops, droop control, voltage-current loops, and PWM generation, emphasizing the role of virtual impedance in achieving desired performance.
To validate the proposed strategy, we conducted simulations using PSCAD/EMTDC for a system of two parallel solar inverters with identical capacities. The solar sources were modeled as adjustable DC sources. Key parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| DC Link Voltage \( U_{dc} \) | 800 V |
| Carrier Frequency | 10 kHz |
| Rated Frequency | 50 Hz |
| DG1 Line Impedance \( Z_{\text{line1}} \) | 0.15 + j0.02 Ω |
| DG2 Line Impedance \( Z_{\text{line2}} \) | 0.1 + j0.01 Ω |
| Load 1 \( Z_{\text{load1}} \) | 1.8 kW + 0.7 kvar |
| Load 2 \( Z_{\text{load2}} \) | 1.8 kW + 0.7 kvar |
| Active Droop Coefficient \( m_1 = m_2 \) | 1 × 10-3 |
| Reactive Droop Coefficient \( n_1 = n_2 \) | 1.5 × 10-4 |
The simulation duration was set to 3 seconds. Under traditional control methods, the output currents of the solar inverters exhibited significant deviations, with a circulating current amplitude of approximately 2.3 A. This led to uneven power sharing and voltage fluctuations. In contrast, with the proposed virtual complex impedance strategy, the circulating current amplitude reduced to about 0.8 A, a decrease of over 60%. The output currents became balanced, and active power sharing improved markedly. Additionally, the bus voltage stabilized near the rated value, and frequency maintained at 50 Hz, demonstrating enhanced system stability.
Further experimental validation on a renewable energy test platform confirmed these findings. The experimental parameters matched the simulation settings. Waveforms under traditional control showed current imbalances and substantial circulating currents, whereas the improved strategy resulted in synchronized currents with equal magnitudes and phases, effectively suppressing circulating currents and ensuring accurate power distribution. These results align with simulation outcomes, affirming the practicality of the proposed approach for real-world solar inverter applications.
In conclusion, this paper presents a comprehensive solution for circulating current suppression in parallel solar inverters using virtual complex impedance. By dynamically adjusting system impedance to resistive characteristics, the strategy mitigates the effects of parameter variations and line impedance disparities. The integration of adaptive virtual resistance and negative virtual inductance simplifies control algorithms and enhances performance. Simulation and experimental results validate the method’s efficacy in reducing circulating currents and improving power sharing. This work provides valuable insights for the design and optimization of large-scale solar power systems, contributing to the advancement of renewable energy integration.
The implications of this research extend to various applications in solar energy systems. Future work could explore scalability to larger arrays of solar inverters and integration with energy storage systems. Additionally, the adaptability of this strategy to different grid conditions and fault scenarios warrants further investigation. Overall, the virtual complex impedance approach offers a robust framework for addressing circulating current challenges, promoting the reliable and efficient operation of solar inverters in modern power networks.