Circulation Suppression in Parallel Solar Inverters Based on Virtual Complex Impedance

1. Introduction

As a pivotal interface between the photovoltaic (PV) generation system and the power grid, the solar inverter carries the crucial task of injecting clean electrical energy into the grid with high efficiency and high quality. In the context of global carbon neutrality targets, the installed capacity of PV systems has grown exponentially, leading to increased demand for inverter parallel operation. Parallel connection of solar inverters enhances system flexibility and reliability by enabling capacity expansion. However, in practical engineering applications, the circulating current (also known as circulating current or loop current) issue—caused by parameter mismatches, line impedance differences, and non-ideal switching characteristics—has become a significant technical bottleneck limiting the large-scale deployment of inverter parallel systems.

Circulating currents in parallel solar inverters refer to the unbalanced current components that flow between the inverters rather than into the load. These currents reduce overall system efficiency, increase power losses, degrade power quality, and can even lead to equipment damage in severe cases. The global research community has paid extensive attention to this issue. Various control strategies have been proposed in the literature to suppress circulating currents. For example, common-mode voltage suppression methods and modified pulse-width modulation (PWM) techniques have been investigated. However, these methods often lack comprehensive multi-operation verification and have limited practical applicability. Model predictive control (MPC) approaches for multi-level inverters have also been explored; while they offer fast dynamic response, they suffer from high computational complexity and parameter sensitivity.

Droop control, which mimics the droop characteristics of synchronous generators, is among the most widely adopted methods for parallel inverter systems. However, conventional droop control in inductive environments faces challenges in power decoupling, especially in low-voltage systems where line resistance is not negligible. In this paper, we focus on the structural composition and characteristics of the parallel solar inverter system. We establish a mathematical model for parallel inverters to analyze the circulation generation mechanism. On this basis, we propose a multi-loop control strategy that integrates virtual complex impedance and adaptive regulation. Specifically, we introduce:
– A virtual negative inductance to counteract the output inductive components of the inverter;
– An adaptive virtual impedance to dynamically adjust the line impedance characteristics;
– A guaranteed resistive equivalent impedance to simplify the complex power decoupling algorithm.

The remainder of this paper is organized as follows. Section 2 presents the mathematical modeling of the parallel solar inverter system and the circulation mechanism analysis. Section 3 details the proposed virtual complex impedance-based control strategy. Section 4 validates the proposed method through comprehensive simulation tests using PSCAD/EMTDC. Section 5 provides experimental verification. Finally, Section 6 concludes the paper.

2. Mathematical Modeling and Circulation Mechanism

2.1 Equivalent Electrical Model of Parallel Solar Inverters

The topological structure of the parallel solar inverter system is illustrated in the figure below. The system consists of PV DC power sources, three-phase inverter circuits (IGBT modules S1-S6), high-frequency filter circuits (Lf, Cf), and signal acquisition circuits. Multiple solar inverters are connected in parallel through a common bus (Bus). The signal circuits collect three-phase voltage and current signals, which, after PLL synchronization and Park transformation, generate SPWM modulation waveforms to drive the switching devices.

To analyze the fundamental characteristics of this system, we consider the equivalent circuit of the parallel inverter system. For the sake of analysis, we focus on a system with two parallel solar inverters. The simplified structure is shown in Figure 2 (conceptually). In the equivalent model:
– \(E_i\angle\theta_i\) \((i=1,2)\) represents the no-load voltage of each solar inverter;
– \(Z_{\text{eqi}}\) represents the equivalent impedance of each inverter;
– \(I_i\) \((i=1,2)\) is the output current of each inverter;
– \(U_L\angle0\) is the voltage at the common bus;
– \(I_L\angle\varphi\) is the load current;
– \(Z_{\text{linei}} = R_i + jX_i\) \((i=1,2)\) is the line impedance of branch \(i\);
– \(Z_{\text{load}}\) is the load impedance.

2.2 Fundamental Frequency Circulation Mechanism

2.2.1 Output Characteristics and Influencing Factors of Solar Inverters

From the equivalent circuit of the parallel inverter system, the complex power delivered by each solar inverter can be expressed as:

$$
S_i = P_i + Q_i = \dot{E}_i \left( \frac{\dot{E}_i – \dot{U}_L}{R_i + jX_i} \right)^* \quad (i=1,2)
\tag{1}
$$

Expanding the real and imaginary parts of the power expression yields the active and reactive power output of each inverter:

$$
P_i = \frac{U_L\left[ R_i(E_i\cos\alpha_i – U_L) + E_i X_i \sin\alpha_i \right]}{R_i^2 + X_i^2}
\tag{2}
$$

$$
Q_i = \frac{U_L\left[ X_i(E_i\cos\alpha_i – U_L) – E_i R_i \sin\alpha_i \right]}{R_i^2 + X_i^2}
\tag{3}
$$

where \(\alpha_i\) \((i=1,2)\) is the power angle of the \(i\)-th inverter.

In low-voltage PV systems, the parallel inverter system exhibits weak-grid characteristics, meaning the system resistance is much larger than the reactance. Under the condition \(R \gg X\), and considering that the power angle \(\alpha_i\) approaches zero, we have \(\sin\alpha_i \approx \alpha_i\) and \(\cos\alpha_i \approx 1\). Under these approximations, the power expressions simplify to:

$$
P_i \approx \frac{U_L (E_i – U_L)}{R_i}
\tag{4}
$$

$$
Q_i \approx -\frac{E_i U_L \alpha_i}{R_i}
\tag{5}
$$

From equations (4) and (5), we observe that in a resistive environment:
– The active power \(P_i\) is mainly determined by the voltage amplitude difference \((E_i – U_L)\);
– The reactive power \(Q_i\) is mainly determined by the power angle \(\alpha_i\).

This relationship is fundamentally different from conventional inductive-line systems, where active power depends on the power angle and reactive power depends on the voltage amplitude. This decoupling characteristic makes the resistive droop control strategy more suitable for low-voltage PV systems. Therefore, we adopt the following resistive droop control model:

$$
\begin{cases}
U = U^* – n(P – P^*) \\
f = f^* – m(Q – Q^*)
\end{cases}
\tag{6}
$$

where the quantities marked with asterisks (*) denote the reference values, and \(m\) and \(n\) are the droop coefficients. The relationship described by equation (6) reveals the voltage-active power (\(U\)-\(P\)) and frequency-reactive power (\(f\)-\(Q\)) coupling. The graphical representation of the resistive droop characteristics is shown conceptually in the following table, which summarizes the droop characteristic relationships:

2.2.2 Circulation Current Characteristics

Based on the equivalent circuit of the two parallel-connected inverters, when considering the PV power source impedance, line impedance, and load conditions, the system equations can be written as:

$$
\begin{cases}
Z_i = Z_{\text{eq}i} + Z_{\text{line}i}, \quad (i = 1,2) \\
\dot{E}_j – Z_j \dot{I}_j = \dot{U}_L, \quad (j = 1,2) \\
\dot{I}_1 + \dot{I}_2 = \dot{I}_L \\
\dot{I}_L = \dot{U}_L / Z_L
\end{cases}
\tag{7}
$$

From the above equations, the output current of each inverter can be derived as:

$$
\dot{I}_1 = \frac{\dot{U}_L}{2Z_L} + \frac{\dot{E}_1 – \dot{E}_2}{Z_1 + Z_2} = \frac{1}{2}\dot{I}_L + \frac{\Delta E}{Z_1 + Z_2}
\tag{8}
$$

$$
\dot{I}_2 = \frac{\dot{U}_L}{2Z_L} – \frac{\dot{E}_1 – \dot{E}_2}{Z_1 + Z_2} = \frac{1}{2}\dot{I}_L – \frac{\Delta E}{Z_1 + Z_2}
\tag{9}
$$

The circulating current \(I_{\text{cc}}\) between the two parallel inverters is defined as:

$$
I_{\text{cc}} = \frac{I_1 – I_2}{2} = \frac{\dot{E}_1 – \dot{E}_2}{Z_1 + Z_2}
\tag{10}
$$

Equation (10) reveals the fundamental mechanism of circulating current generation. The circulating current \(I_{\text{cc}}\) depends on:
1. The voltage magnitude difference between the two inverters \((\dot{E}_1 – \dot{E}_2)\);
2. The voltage phase difference between the two inverters;
3. The total equivalent impedance \(Z_1 + Z_2\), which includes both the inverter output impedance and the line impedance.

This analysis indicates that to suppress the circulating current, one must either minimize the voltage differences between the parallel inverters or increase the equivalent impedance between them. Since increasing impedance negatively affects voltage regulation, the preferred approach in this paper is to reshape the inverter output impedance characteristics through virtual impedance control, thereby reducing the voltage differences and achieving circulation suppression.

3. Circulation Suppression Strategy Design

3.1 Control Architecture Overview

To address the circulating current challenges, we design a multi-loop control architecture for the parallel solar inverter system:
– Outer power loop: Achieves maximum power point tracking (MPPT) for PV generation and generates current/voltage references;
– Inner voltage/current loops: Combined with PLL technology to ensure phase and amplitude synchronization of the inverters;
– Virtual impedance loop: Introduces virtual complex impedance, including negative inductance and adaptive resistance, to reshape the output impedance of the inverter.

A key innovation in our control design is the introduction of a virtual negative reactance to cancel the inductive component of the inverter output impedance. However, our analysis shows that relying solely on virtual negative reactance is insufficient to solve voltage drop, circulation suppression, and power sharing problems simultaneously. Therefore, we incorporate adaptive dynamic adjustment. By feeding back real-time power variations to adjust the virtual impedance, the system output impedance exhibits predominantly resistive characteristics. This ultimately improves circulation suppression accuracy and achieves power sharing among parallel solar inverters.

3.2 Adaptive Virtual Complex Impedance

Unlike high-voltage transmission systems, low-voltage systems predominantly exhibit resistive line impedance characteristics. However, the inverter output impedance consists of both resistive and inductive components. The proportion of these components varies depending on filter design and control methods, leading to power coupling phenomena. To solve this problem, we incorporate virtual impedance into the control loop to reshape the equivalent output impedance so that it becomes resistive.

The voltage control structure with a virtual impedance feedback loop is illustrated in Figure 5 of the original paper. The composite virtual impedance \(Z_{\text{vir}}(s)\) consists of two parts:
– An adaptive virtual resistance component \(R_v\);
– A negative virtual inductance component \(-X_v\).

When the virtual impedance is introduced into the feedback loop, the output voltage \(U_0(s)\) can be expressed as:

$$
U_0(s) = G(s)U_n(s) – Z(s)i_0(s)
\tag{11}
$$

where \(G(s)\) is the closed-loop voltage gain and \(Z(s)\) is the equivalent output impedance. More explicitly:

$$
U_0(s) = \frac{K_{\text{PWM}} T_i(s) T_u(s)}{LCs^2 + T_i(s)K_{\text{PWM}}Cs + T_i(s)K_{\text{PWM}}T_u(s) + 1} U_n(s) – \frac{\left[ L_s + Z_{\text{vir}}(s) K_{\text{PWM}} T_i(s) T_u(s) \right]}{LCs^2 + T_i(s)K_{\text{PWM}}Cs + T_i(s)K_{\text{PWM}}T_u(s) + 1} i_0(s)
\tag{12}
$$

The closed-loop transfer function \(G_v(s)\) and the equivalent output impedance \(Z_q(s)\) are defined as:

$$
G_v(s) = \frac{K_{\text{PWM}} T_i(s) T_u(s)}{LCs^2 + T_i(s)K_{\text{PWM}}Cs + T_i(s)K_{\text{PWM}}T_u(s) + 1}
\tag{13}
$$

$$
Z_q(s) = \frac{L_s + Z_{\text{vir}}(s) K_{\text{PWM}} T_i(s) T_u(s)}{LCs^2 + T_i(s)K_{\text{PWM}}Cs + T_i(s)K_{\text{PWM}}T_u(s) + 1}
\tag{14}
$$

We adopt a quasi-proportional-resonant (PR) controller for the voltage outer loop to achieve zero steady-state error tracking. The PR controller can avoid output impedance amplitude deviations caused by parameter variations and attenuate frequency disturbance. The current inner loop adopts a P controller. The transfer functions of these controllers are given by:

$$
T_u(s) = k_{\text{PR}} + \frac{2k_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2}
\tag{15}
$$

$$
T_i(s) = k_e
\tag{16}
$$

where:
– \(k_{\text{PR}}\) is the voltage loop proportional gain;
– \(k_r\) is the resonant gain;
– \(\omega_0\) is the resonant frequency;
– \(\omega_c\) is the resonant bandwidth factor;
– \(k_e\) is the current loop proportional gain.

From equation (6) in the previous analysis, we know that the voltage amplitude differences between parallel inverters in steady state are a primary cause of circulating currents. Merely introducing a fixed-value virtual impedance cannot achieve satisfactory circulation suppression. Therefore, we design the resistive part of the virtual complex impedance as a voltage-dependent dynamic virtual resistance. By incorporating a proportional-derivative (PD) link, the response speed is accelerated, and the voltage amplitude difference between the parallel inverters is minimized. The adaptive virtual resistance is expressed as:

$$
R_v = R_x – \alpha \Delta U – \beta \frac{d(\Delta U)}{dt}
\tag{17}
$$

where:
– \(R_x\) is the pre-set fixed virtual resistance;
– \(\alpha\) is the proportional regulation coefficient;
– \(\beta\) is the derivative regulation coefficient;
– \(\Delta U\) is the difference between the inverter output voltage and the pre-set voltage reference \(U_{\text{ref}}\).

The pre-set voltage reference is given by:

$$
U_{\text{ref}} = U_n – Z_{\text{vir}} i_0
\tag{18}
$$

where:
– \(U_n\) is the set rated voltage value;
– \(Z_{\text{vir}}\) is the virtual complex impedance;
– \(i_0\) is the measured output current of the inverter.

To optimize power decoupling accuracy, the equivalent output impedance of the inverter should be designed to be resistive. One approach is to increase the virtual resistance value to reduce the proportion of inductive components. However, increasing the resistance value also affects voltage quality. Therefore, we instead incorporate a virtual negative reactance feedback \(-X_v\) in the control loop to attenuate the inductive component of the equivalent output impedance.

The equivalent output impedance vector diagram with virtual negative reactance is shown in Figure 6 of the original. The new equivalent output impedance \(Z_{vq}\) can be seen as the vector sum of the original impedance \(Z_q(s)\) and the negative reactance component \(-X_v G_v(s)\). At the fundamental frequency, the voltage gain is approximately 1, so \(-X_v\) can be treated as a negative vector on the imaginary axis. Consequently:
– The negative inductance primarily affects the inductive part of the output impedance;
– The resistive part remains largely unaffected;
– The output impedance characteristics are optimized to predominantly exhibit resistive behavior, resulting in better power decoupling effects.

3.3 Adaptive Droop Coefficient Regulation Algorithm

By introducing the negative reactance component and the adaptive virtual impedance algorithm described in Section 3.2, the inductive components in the system are maximally attenuated. However, residual differences in the resistive parts of the parallel inverters still exist. This leads to imbalance in the active power output among parallel inverters, which in turn produces active power circulating current and affects the normal system operation.

To address this residual circulation issue, we propose an adaptive droop coefficient regulation algorithm. The principle is to dynamically associate the droop coefficients with the inverter output active power, replacing fixed coefficients to improve system active power sharing accuracy. The specific implementation steps are as follows:
– (1) Collect the active power of the parallel modules: \(P_1\) and \(P_2\);
– (2) Calculate the average active power \(P_{\text{ref}} = (P_1 + P_2)/2\);
– (3) Calculate the power differences \(\Delta P_1 = P_{\text{ref}} – P_1\) and \(\Delta P_2 = P_{\text{ref}} – P_2\);
– (4) Ship \(\Delta P_i\) into a PI controller and subtract from the given droop coefficient to obtain the dynamic droop coefficient \(n_{si}\);
– (5) Apply the dynamic droop coefficient to the improved voltage-active power control branch.

The improved active power control structure is shown in Figure 8 of the original. The corresponding control equation is:

$$
\begin{cases}
U_n = U^* + n_s(P_i – P^*) \\
n_{si} = n_i – \left(k_p + \frac{k_i}{s}\right)(P_{\text{ref}} – P_i) \\
P_{\text{ref}} = \frac{P_1 + P_2}{2}
\end{cases}
\tag{19}
$$

In equation (19):
– \(n_i\) is the traditional active power droop coefficient;
– \(k_p\) and \(k_i\) are the proportional and integral coefficients of the adaptive controller;
– \(P_{\text{ref}}\) is the average active power of the parallel inverters;
– \(P_i\) is the active power output of the \(i\)-th inverter;
– \(n_{si}\) is the dynamic droop coefficient obtained by the adaptive regulation.

The range of the droop coefficient \(n_i\) is determined as:

$$
0 < n_i \leq \frac{f^* – f_{\min}}{P_{\max} – P_{\text{ref}}}
\tag{20}
$$

where:
– \(f^*\) is the reference voltage frequency of the inverter;
– \(f_{\min}\) is the minimum operating frequency of the system;
– \(P_{\max}\) is the maximum allowable active power output of the inverter.

3.4 Adaptive Virtual Complex Impedance Strategy Summary

The complete adaptive virtual complex impedance control strategy is designed to:
– Set negative reactance components to cancel the inductive components in the equivalent impedance;
– Eliminate the influence of connection line impedance as much as possible;
– Make the output equivalent impedance appear resistive over a wide frequency range;
– Avoid complex power decoupling algorithms;
– Enhance system operation stability.

The output impedance characteristics and control performance improvements of various control strategies are compared in Table 3.

4. Simulation Validation

4.1 Simulation Setup

To verify the effectiveness of the proposed circulation suppression strategy, we performed comprehensive simulation tests using the large-scale digital simulation software PSCAD/EMTDC. The simulation model consists of two same-capacity solar inverters operating in parallel. The PV sources are represented by adjustable DC power sources to simplify the simulation while retaining the key dynamic characteristics. The simulation parameters are presented in Table 1 in Section 2.1.

We compared the performance under two scenarios:

– Scenario A: Traditional droop control strategy without virtual impedance and adaptive regulation.

– Scenario B: The proposed adaptive virtual complex impedance control strategy.

The simulation duration was set to 3 seconds. Key monitored variables include:
– Output current of each inverter (\(I_1\), \(I_2\));
– Circulating current (\(I_{\text{cc}}\));
– Active power output of each inverter (\(P_1\), \(P_2\));
– Reactive power output of each inverter (\(Q_1\), \(Q_2\));
– Bus voltage in per-unit value (\(U_p\));
– Bus frequency (\(f\)).

4.2 Simulation Results of Traditional Control Strategy

The simulation waveforms under traditional droop control (Scenario A) are shown in Figure 10 of the original paper. The key observations are summarized as follows:

These results confirm that in the traditional control strategy:
– The circulating current between parallel inverters is large (about 2.3 A);
– The active power sharing accuracy is degraded by line impedance differences;
– Voltage and frequency deviate from their rated values;
– System stability is threatened.

4.3 Simulation Results of the Proposed Strategy

In contrast, the simulation waveforms under the proposed control strategy (Scenario B) are shown in Figure 11 of the original. The comparison between the two scenarios is presented below:

From the comparison of Figures 10 and 11 in the original paper, we draw the following conclusions:
– Circulating current suppression: The circulating current amplitude is reduced by more than 60%, from around 2.3 A to around 0.8 A;
– Active power sharing: With the adaptive droop coefficient regulation, the active power is evenly shared between the two solar inverters, ensuring the capacity-based load sharing;
– Voltage and frequency regulation: The bus voltage is maintained at the rated value, and the frequency is stabilized at 50 Hz;
– Reactive power sharing: The reactive power distribution is maintained at the predetermined ratio according to the droop coefficients;
– System stability: The overall system stability and power quality are enhanced.

These results clearly demonstrate the superiority of the proposed virtual complex impedance control strategy for parallel solar inverters.

5. Experimental Verification

To further validate the correctness and engineering applicability of the proposed control strategy, we designed and constructed an experimental platform with two parallel solar inverters on a renewable energy power generation platform. The experimental parameters are kept consistent with the simulation parameters.

5.1 Experimental Results and Analysis

The experimental waveforms are shown in Figure 12 of the original paper. Key observations from the experiments are as follows:

Under traditional control strategy:
– The output currents of the two solar inverters exhibit amplitude deviations;
– This current imbalance prevents even power sharing;
– Consequently, a large circulating current flows between the inverters.

Under the proposed control strategy:
– The output currents of the two solar inverters have equal amplitudes and are in phase;
– Equal amplitude and phase enable even active power sharing;
– The circulating current is effectively suppressed to a low level.

The conclusion drawn from the experimental results is consistent with the simulation results. The proposed control strategy demonstrates the following improvements over the traditional strategy:
– Better power sharing accuracy;
– Stronger circulation suppression capability;
– Improved system stability;
– Practical engineering applicability.

These findings collectively support the utility of our adaptive virtual complex impedance approach for parallel solar inverter systems.

6. Conclusion

In this paper, we proposed and validated a circulation suppression strategy for parallel-connected solar inverters in photovoltaic power generation systems. The main contributions and findings of this work are summarized as follows:

1) Circulation mechanism analysis: We established a mathematical model for parallel solar inverters and systematically analyzed the circulation generation mechanism. The analysis shows that the circulating current depends on the voltage magnitude differences, phase angle differences, and the equivalent impedance between the parallel-connected inverters.

2) Virtual complex impedance design: We presented a multi-loop control strategy that integrates virtual complex impedance with adaptive regulation. The key idea is to:
– Introduce a virtual negative inductance to cancel the inductive component of the inverter output impedance;
– Use an adaptive virtual resistance to dynamically balance the system equivalent impedance characteristics;
– Ensure the system’s equivalent impedance exhibits predominantly resistive behavior.
This simplifies the complex power decoupling algorithm while improving the circulation suppression accuracy.

3) Adaptive droop coefficient regulation: To address the residual active power imbalance caused by resistive component mismatch, we proposed an adaptive droop coefficient regulation algorithm. This approach dynamically adjusts the droop coefficients based on the real-time power difference between the parallel inverters, achieving accurate active power sharing and thus further reducing active power circulating current.

4) Comprehensive validation: We performed extensive simulations using PSCAD/EMTDC software and validated the results with experimental tests. The results of the comparison between the traditional control method and the proposed method confirm:
– The circulating current is reduced by more than 60% (from approximately 2.3 A to approximately 0.8 A);
– Active power is evenly shared between the solar inverters;
– The bus voltage is maintained at the rated value and the frequency remains stable at 50 Hz;
– System stability and power quality are enhanced.

In summary, the proposed virtual complex impedance-based circulation suppression strategy exhibits excellent power sharing performance, strong circulation suppression capability, and practical feasibility. This research provides a solid theoretical foundation and engineering guidance for the control of large-scale photovoltaic power generation systems with multiple parallel-connected solar inverters.

References and Nomenclature