As the penetration of distributed generation continues to rise, the types of solar inverters operating in islanded microgrids have diversified. Among these, parallel inverters are widely employed to meet load demand while ensuring reliability and scalability. However, the conventional droop control method, though communication-free and plug-and-play, suffers from two critical issues: reactive power sharing inaccuracy and bus voltage drop. These problems are exacerbated by mismatched line impedances and the presence of nonlinear loads. In this study, we propose an improved virtual impedance strategy that adaptively equalizes reactive power and compensates bus voltage. The strategy incorporates a second-order generalized integrator (SOGI) for enhanced power decoupling under harmonic conditions, an adaptive virtual voltage drop correction term to eliminate line impedance effects, and a voltage compensation loop to restore the bus voltage to its nominal value. Our work specifically addresses the performance of various types of solar inverters in parallel operation. Simulation results validate the effectiveness of the proposed method under complex scenarios, demonstrating superior reactive power sharing accuracy and bus voltage regulation compared to conventional approaches.
Introduction
With the global push toward carbon neutrality, renewable energy sources such as solar photovoltaics have experienced rapid growth. Inverter systems are essential for integrating these sources into microgrids. Among the many types of solar inverters, parallel-connected inverters are a common topology that offers redundancy and scalability. Droop control is a popular technique for parallel operation because it eliminates the need for inter-unit communication. However, the inherent power coupling and unequal line impedances lead to poor reactive power sharing, especially when different types of solar inverters with varying capacities are used. Additionally, the introduction of virtual impedance to improve inductive behavior results in an undesirable voltage drop at the point of common coupling (PCC). Numerous studies have attempted to address these issues, but most either compromise voltage quality or require precise knowledge of line parameters. In this paper, we present a comprehensive solution that simultaneously achieves adaptive reactive power sharing and voltage compensation, tailored for the diverse landscape of types of solar inverters in islanded microgrids.
Droop Control Principle and Output Power Characteristics
Conventional Droop Control
Consider two parallel inverters supplying a common load. Their equivalent model is shown in the block diagram below. The output active and reactive power of inverter i can be expressed as:
$$
P_i = \frac{U_i U_{pcc} \cos(\alpha_i – \theta_i) – U_{pcc}^2 \cos\alpha_i}{X_i^2 + R_i^2}
$$
$$
Q_i = \frac{U_i U_{pcc} \sin(\alpha_i – \theta_i) – U_{pcc}^2 \sin\alpha_i}{X_i^2 + R_i^2}
$$
where \(\alpha_i = \arctan(X_i/R_i)\). For high-voltage lines, the impedance is predominantly inductive (\(\alpha_i \approx 90^\circ\)). Assuming small power angles, the simplified droop equations are:
$$
f = f_n + m(P_{ref} – P), \quad U = U_n + n(Q_{ref} – Q)
$$
where \(f\) and \(U\) are the frequency and voltage amplitude references, and \(m, n\) are the droop coefficients. This decoupling holds only when the line impedance is purely inductive.
Power Transmission Characteristics
Under steady state, the frequency is a global variable, so active power sharing depends solely on the droop coefficients \(m\). However, reactive power sharing requires that the line impedances be inversely proportional to the inverter capacities. In practice, line impedances are mismatched, making equal reactive power sharing difficult. This problem is particularly pronounced when different types of solar inverters (e.g., those with varying output filter designs) are paralleled, as their output impedances differ.
Limitations of Conventional Virtual Impedance Strategy
Power Decoupling via Virtual Inductance
To enhance the inductive nature of the line, a virtual inductance \(L_v\) is introduced into the voltage-current dual-loop control. The output impedance at the fundamental frequency becomes nearly purely inductive:
$$
Z’_o(j50) \approx 3.94 \angle 86^\circ
$$
However, under nonlinear loads, harmonic currents flowing through the virtual impedance produce harmonic voltage drops. Since the PI controller cannot track AC harmonics accurately, the control performance degrades. This issue is critical for many types of solar inverters that must supply nonlinear loads such as rectifiers.
Reactive Power Sharing with Fixed Virtual Impedance
Conventional virtual impedance methods adjust the equivalent line impedance ratio to satisfy \(Q_1/Q_2 = (X_2 + X_{v2})/(X_1 + X_{v1})\). However, this requires prior knowledge of the line impedances, and any change in line parameters (e.g., due to load switching) causes the fixed virtual impedance to become mismatched. Consequently, reactive power sharing accuracy is degraded.
Proposed Improved Virtual Impedance Strategy
SOGI-Based Enhanced Power Decoupling
We replace the direct differentiation of the current signal with a SOGI-based approach. The SOGI acts as a frequency-adaptive resonator that filters out harmonics while generating a quadrature signal.
$$
U_v = -\omega L_v q(s)
$$
where \(q(s)\) is the 90°-lagged output of the SOGI with respect to the input current \(i(s)\). This eliminates harmonic virtual voltage drops and improves the tracking accuracy of the PI controller, which is especially beneficial for types of solar inverters that must maintain low total harmonic distortion (THD).
Adaptive Reactive Power Sharing
We introduce an adaptive virtual voltage drop correction term \(\bar{U}_i\) into the reactive power droop equation:
$$
U_{i,droop} = U_n + n(Q_{ref,i} – Q_i) + \bar{U}_i
$$
$$
\bar{U}_i = k_{vi} (Q_i^* – Q_i), \quad Q_i^* = Q_{ref,i} \cdot \frac{\sum_{k=1}^i Q_k}{\sum_{k=1}^i Q_{ref,k}}
$$
This term continuously adjusts the voltage reference to drive the actual reactive power toward the ideal sharing value, regardless of line impedance variations. The gain \(k_{vi}\) is chosen via small-signal analysis to balance dynamic response and stability. For a typical two-inverter system, the root locus and step response analyses indicate that \(k_{vi} = 0.8\) yields a well-damped response with fast settling.
The following table summarizes the small-signal transfer function parameters and the effect of \(k_v\) on damping.
| \(k_v\) | Damping Ratio \(\zeta\) | Natural Frequency \(\omega_n\) (rad/s) | Overshoot (%) |
|---|---|---|---|
| 0.2 | 1.2 | 315 | 0 |
| 0.5 | 0.9 | 420 | 0.5 |
| 0.8 | 0.7 | 530 | 4.3 |
| 1.5 | 0.4 | 680 | 25.0 |
Adaptive Bus Voltage Amplitude Compensation
To counteract the voltage drop caused by the virtual impedance, we add a voltage compensation term \(\bar{U}_i\) based on the error between the measured bus voltage and its nominal value:
$$
\bar{U}_i = k_{ui} (U_{pcc}^* – U_{pcc})
$$
The complete control law becomes:
$$
U_{i,droop} = U_n + n(Q_{ref,i} – Q_i) + \bar{U}_i + \bar{U}_i
$$
$$
U_{i,gd}^* = U_{i,droop}^d + \omega L_v i_{gq}, \quad U_{i,gq}^* = U_{i,droop}^q – \omega L_v i_{gd}
$$
With \(k_{ui} = 20\), the bus voltage is regulated to the nominal value with negligible steady-state error. This compensation is crucial for ensuring voltage quality in microgrids hosting different types of solar inverters, where voltage sensitivity may vary.
Simulation Validation
Case 1: Nonlinear Load Impact on Power Decoupling
We simulate two identical inverters (rated 4 kW, 3 kVar) with and without a nonlinear load (three-phase diode rectifier). The virtual voltage drop waveforms under conventional virtual impedance and proposed SOGI-enhanced strategy are compared. Under nonlinear load, the conventional method exhibits significant harmonic distortion in the virtual voltage drop, leading to power coupling and poor PI tracking. The proposed SOGI-based method effectively filters harmonics, maintaining a clean virtual voltage drop and decoupled power control.
Case 2: Line Impedance Variation and Voltage Compensation
We compare three strategies: conventional fixed virtual impedance, adaptive virtual impedance (without voltage compensation), and our proposed strategy. The simulation parameters are given in the table below.
| Parameter | Value |
|---|---|
| Active droop coefficient \(m\) | 4×10⁻⁴ |
| Reactive droop coefficient \(n\) | 2×10⁻⁴ |
| Virtual voltage drop gain \(k_v\) | 0.5 |
| Voltage compensation gain \(k_u\) | 20 |
| Frequency compensation gain \(k_f\) | 5 |
| Rated reactive power (per inverter) | 3000 Var |
| Rated active power (per inverter) | 4000 W |
| DC bus voltage | 800 V |
The reactive power waveforms and bus voltage magnitude are shown in the figure below.
Results demonstrate that the conventional fixed virtual impedance fails to maintain equal reactive power sharing after a local load is switched in at t=2 s. The adaptive virtual impedance without voltage compensation improves sharing but causes a significant bus voltage drop (from 311 V to 298 V). In contrast, our proposed strategy achieves both precise reactive power sharing (error < 2%) and nominal bus voltage (311 V ± 1 V) under all test conditions. The immunity to line impedance changes makes our method suitable for various types of solar inverters installed in geographically dispersed locations.
Comparative Summary
A quantitative comparison of the three strategies is provided in the table below.
| Strategy | Reactive Power Sharing Error (%) | Bus Voltage (V) | Voltage Deviation (%) |
|---|---|---|---|
| Conventional fixed virtual impedance | 18.5 | 302 | 2.9 |
| Adaptive virtual impedance (no voltage comp.) | 2.1 | 295 | 5.1 |
| Proposed strategy | 1.3 | 311 | 0.0 |
Conclusion
In this work, we have proposed an improved virtual impedance strategy that simultaneously addresses reactive power sharing and bus voltage regulation in islanded microgrids with parallel inverters. By integrating a SOGI-based power decoupling module, we eliminate harmonic-induced virtual voltage drops, enhancing the tracking performance of PI controllers—a critical feature for many types of solar inverters feeding nonlinear loads. The adaptive virtual voltage drop correction term ensures robust reactive power sharing even when line impedances vary, while the voltage compensation loop restores the bus voltage to its nominal value without affecting sharing accuracy. Simulation results confirm that our method outperforms conventional and adaptive virtual impedance strategies in both sharing precision and voltage quality. The approach is generic and can be applied to a wide range of types of solar inverters, contributing to the reliable and efficient operation of modern microgrids.