The integration of solar photovoltaic (PV) systems into the modern power grid is a cornerstone of the global shift towards sustainable energy. A single solar inverter, the device that converts direct current (DC) from PV panels into grid-compatible alternating current (AC), often has a limited power rating. To scale up capacity and enhance system reliability, a parallel configuration of multiple solar inverters is a prevalent and effective solution. However, this architecture introduces a significant technical challenge: the generation of circulating currents among the paralleled units. These currents, which do not contribute to useful load power, arise due to inevitable mismatches in inverter output parameters, component nonlinearities, and differences in line impedances. They lead to increased losses, uneven current stress on semiconductor devices, reduced overall efficiency, and can potentially compromise system stability and reliability. Therefore, developing advanced control strategies to suppress circulating currents and ensure accurate reactive and active power sharing is a critical research focus in the field of distributed power generation and microgrids.
Traditional control methodologies for paralleled inverters can be broadly classified into master-slave and peer-to-peer (droop control) schemes. While master-slave control relies on critical communication links, making the system vulnerable to single-point failures and complicating “plug-and-play” functionality, the droop control method offers a decentralized alternative. Inspired by the behavior of synchronous generators, conventional droop control establishes a relationship between frequency and active power (P-ω) and between voltage magnitude and reactive power (Q-V). This method allows inverters to autonomously adjust their output without direct communication. However, its performance is highly dependent on the output impedance characteristics of the solar inverter. In practical systems, the combined output impedance (including filter inductance, parasitic resistance, and line impedance) is often complex (resistive-inductive), leading to coupled P-Q power flow and inaccurate power sharing, which directly fuels circulating currents.
To address these limitations, this article presents a comprehensive analysis of circulating current mechanisms in paralleled solar inverter systems and proposes a novel, integrated control strategy. The proposed approach synergistically combines an enhanced droop control method, virtual complex impedance shaping, and a dual Proportional-Resonant (PR) voltage-current control loop. The enhanced droop control incorporates an integral compensator in the reactive power channel to improve sharing accuracy. The virtual impedance technique is employed to actively shape the output impedance of each solar inverter to be predominantly inductive at the fundamental frequency, ensuring proper decoupling for the droop laws. Finally, the dual PR controller provides high gain at the fundamental frequency, enabling nearly zero steady-state error in tracking voltage and current references, which is paramount for effective circulating current suppression. The theoretical principles of each component are derived in detail, controller design guidelines are provided, and the overall effectiveness is validated through detailed simulation studies.
1. Analysis of Circulating Current in Paralleled Solar Inverter Systems
1.1 System Configuration and Fundamental Power Flow
A typical system comprising N paralleled grid-forming solar inverters is considered. Each inverter unit consists of a DC-DC boost converter with Maximum Power Point Tracking (MPPT) for the PV array and a DC-AC full-bridge inverter with an LC output filter. The outputs are connected to a common AC bus through respective line impedances, supplying a local load. For the fundamental analysis of power flow and circulating current, a simplified equivalent circuit for two paralleled inverters is sufficient, as the principles extend to N units. The key parameters for inverter n (n=1,2) are: output voltage phasor $$ \vec{U_n} = U_n \angle \delta_n $$, where $U_n$ is the voltage magnitude and $\delta_n$ is the phase angle relative to the common bus voltage $$ \vec{U_L} = U_L \angle 0 $$; and the total equivalent output impedance $$ \vec{Z_n} = R_n + jX_n $$, which aggregates the filter inductor resistance, the filter inductor reactance, and the line impedance.
The complex power injected by the n-th solar inverter into the bus is given by:
$$ \vec{S_n} = \vec{U_n} \cdot \vec{I_n}^* = P_n + jQ_n $$
where $\vec{I_n}$ is the output current phasor. From the circuit analysis, the active and reactive power outputs can be derived as:
$$ P_n = \frac{U_n U_L}{Z_n} \left[ \cos(\theta_n – \delta_n) – \frac{U_L}{U_n} \cos \theta_n \right] $$
$$ Q_n = \frac{U_n U_L}{Z_n} \left[ \sin(\theta_n – \delta_n) + \frac{U_L}{U_n} \sin \theta_n \right] $$
Here, $\theta_n = \arctan(X_n / R_n)$ is the impedance angle. For a well-designed solar inverter output filter, the inductive reactance typically dominates over the resistance ($X_n >> R_n$), implying $\theta_n \approx 90^\circ$. Furthermore, for small angle differences ($\delta_n$ is small), we can use the approximations $\sin \delta_n \approx \delta_n$ and $\cos \delta_n \approx 1$. Applying these simplifications yields the classic power flow equations for an inductive impedance:
$$ P_n \approx \frac{U_n U_L}{X_n} \delta_n $$
$$ Q_n \approx \frac{U_n (U_n – U_L)}{X_n} $$
These equations form the foundation of the conventional droop control strategy. They indicate that active power $P_n$ is primarily sensitive to the phase angle $\delta_n$ (and thus frequency $f_n$), while reactive power $Q_n$ is sensitive to the voltage magnitude difference $U_n – U_L$. This leads to the standard droop characteristics:
$$ \omega_n = \omega^* – m_{P_n} P_n $$
$$ U_n = U^* – n_{Q_n} Q_n $$
where $\omega^*$ and $U^*$ are the nominal setpoints, and $m_{P_n}$, $n_{Q_n}$ are the droop coefficients for frequency and voltage, respectively.
1.2 Circulating Current Definition and Sensitivity Analysis
The circulating current is defined as the current that flows between the inverters without passing through the load. For a two-inverter system, the circulating current phasor is mathematically defined as half the difference of their output currents:
$$ \vec{I_H} = \frac{\vec{I_1} – \vec{I_2}}{2} $$
Substituting the current expressions derived from the equivalent circuit model, and assuming the output impedances are designed to be similar ($X_1 \approx X_2 = X$, $R_1 \approx R_2 = R$), we can analyze the sensitivity. Under the same inductive dominance assumption ($X >> R$) and small angles, the circulating current can be approximated as:
$$ \vec{I_H} \approx \frac{U_1 – U_2}{2X} + j\frac{U_1 \delta_1 – U_2 \delta_2}{2X} $$
This result reveals the fundamental drivers of circulating current:
- Reactive (Quadrature) Circulating Current: The real part of $\vec{I_H}$ is proportional to the difference in voltage magnitudes ($U_1 – U_2$). If $\delta_1 = \delta_2$ but $U_1 \neq U_2$, a primarily reactive current circulates between the inverters.
- Active (In-Phase) Circulating Current: The imaginary part of $\vec{I_H}$ is proportional to the difference in the product $U_n \delta_n$. If $U_1 = U_2$ but $\delta_1 \neq \delta_2$, a primarily active current circulates.
- Mixed Circulating Current: In the general case where both voltage and angle mismatches exist, the circulating current contains both active and reactive components.
Therefore, the primary control objective for paralleled solar inverters is to minimize both $U_1 – U_2$ and $\delta_1 – \delta_2$ (or equivalently, $\omega_1 – \omega_2$) to suppress $\vec{I_H}$. Accurate sharing of both reactive power ($Q$) and active power ($P$) is the pathway to achieve this, as per the droop relationships.
| Primary Cause | Type of Mismatch | Dominant Circulating Current Component | Impact on System |
|---|---|---|---|
| Output Voltage Magnitude Difference | Reactive Power (Q) Sharing Error | Reactive (Quadrature) | Unequal VA loading, increased inductor losses. |
| Output Voltage Phase/Frequency Difference | Active Power (P) Sharing Error | Active (In-Phase) | Unequal real power loading, DC link voltage fluctuations. |
| Non-identical Output Impedances | Both P and Q Sharing Errors | Mixed | Combined negative effects, potential instability. |
| Nonlinear Loads / Harmonics | Harmonic Voltage Distortion | Harmonic Circulating Currents | Increased THD, overheating, EMI issues. |
2. Proposed Integrated Control Strategy for Solar Inverter Paralleling
The proposed control architecture for each solar inverter is a multi-loop scheme designed to tackle the root causes of poor power sharing and circulating currents. The overall block diagram encompasses a power calculation module, an enhanced droop controller, a virtual complex impedance block, and a dual-loop voltage-current controller based on PR regulators.
2.1 Enhanced Droop Control with Integral Reactive Power Compensation
A key limitation of the conventional droop control in Eq. (6) is that while the integral action in the frequency loop (P-ω) ensures accurate steady-state active power sharing independent of line impedance, the reactive power loop (Q-U) lacks such an inherent property. The steady-state reactive power output $Q_n$ remains dependent on the equivalent output reactance $X_n$, as seen from the simplified static equation derived from the conventional droop laws: $Q_n \approx (U^* – U_L) / n_{Q_n}$. If inverters have different $X_n$ (due to filter tolerances or line lengths), their $Q_n$ will differ, leading to reactive circulating current.
To eliminate this steady-state dependency and enforce accurate Q-sharing, an integral compensator is introduced into the reactive power control channel. The enhanced droop control laws are implemented as:
$$ \omega_n = \omega^* – m_{P_n} P_n $$
$$ U_n^* = U^* – n_{Q_n} \left( Q_n + K_I \int Q_n \, dt \right) $$
Where $U_n^*$ is the voltage magnitude reference generated by the droop controller, and $K_I$ is the integral gain. In the Laplace domain, the Q-U relationship becomes:
$$ U_n^*(s) = U^* – n_{Q_n} \left( 1 + \frac{K_I}{s} \right) Q_n(s) $$
This modification effectively makes the steady-state reactive power error drive an integrator. At equilibrium, the integral term forces the reactive power mismatch to zero, ensuring $Q_1 = Q_2$ regardless of minor differences in $X_1$ and $X_2$. The active power loop retains its inherent integral characteristic through the frequency relationship ($\omega = d\delta/dt$).
2.2 Virtual Complex Impedance for Output Impedance Shaping
Even with improved droop control, the inherent output impedance of the solar inverter and its interaction with the line impedance may not be purely inductive, causing P-Q coupling. The virtual impedance concept is employed to actively and reliably shape the output impedance. A virtual complex impedance $Z_v(s)$, consisting of a virtual resistor $R_v$ and a virtual inductor $L_v$, is subtracted from the voltage reference. The implemented transfer function often includes a high-pass filter to block the DC component and avoid amplifying low-frequency noise:
$$ Z_v(s) = \frac{\omega_c}{s + \omega_c} (R_v – s L_v) $$
Here, $\omega_c$ is the cut-off frequency of the first-order high-pass filter. The updated voltage reference for the inner voltage loop becomes:
$$ U_{ref}(s) = U_n^*(s) – Z_v(s) \cdot I_o(s) $$
where $I_o(s)$ is the measured output current of the solar inverter.
The purpose of $R_v$ is to introduce a resistive component at the fundamental frequency, which can help damp oscillations and improve stability. The purpose of $sL_v$ is to add a dominant inductive reactance. By carefully choosing $R_v$ and $L_v$, the equivalent output impedance seen from the inverter terminals $Z_{eq}(j\omega_0)$ at the fundamental frequency $\omega_0$ can be made to have a phase angle close to $90^\circ$ (purely inductive). This fulfills the assumption used in deriving the decoupled droop equations (P-ω and Q-U), thereby improving the accuracy of the power sharing dictated by the enhanced droop controller.
| Component | Primary Purpose | Design Consideration | Typical Value Range |
|---|---|---|---|
| Virtual Resistor ($R_v$) | Damping, Stabilization | Large enough to provide damping but small enough to avoid excessive voltage drop and losses. Often 1-5% of base impedance. | 0.01 – 0.1 Ω |
| Virtual Inductor ($L_v$) | Ensure Inductive Output Impedance | Dominant value to set the $X/R$ ratio high (>10) at fundamental frequency. Must consider overall voltage regulation. | 1 – 5 mH |
| Filter Cut-off Freq. ($\omega_c$) | Block DC, Attenuate Low-Freq Noise | Must be low enough to pass fundamental frequency with minimal phase shift (< 5°). Typically 10-100 rad/s. | ~50 rad/s |
2.3 Dual Proportional-Resonant (PR) Voltage-Current Control Loop
The inner control loops are responsible for accurately generating the output voltage dictated by the outer droop and virtual impedance blocks. Traditional PI controllers in the stationary (αβ) frame suffer from a fundamental drawback: they cannot provide infinite gain at the AC fundamental frequency, leading to steady-state amplitude and phase tracking errors. These errors directly translate into voltage mismatches and circulating currents among paralleled solar inverters.
To achieve zero steady-state error for sinusoidal references, Proportional-Resonant (PR) controllers are employed. A PR controller tuned at the fundamental frequency $\omega_0$ has theoretically infinite gain at that frequency, forcing the error to zero. A practical implementation uses a “quasi-PR” controller to provide a finite bandwidth around the resonant frequency for robustness against grid frequency variations:
$$ G_{PR}(s) = K_p + \frac{2K_r \omega_b s}{s^2 + 2\omega_b s + \omega_0^2} $$
Here, $K_p$ is the proportional gain, $K_r$ is the resonant gain, and $\omega_b$ is the bandwidth around the resonant frequency $\omega_0$.
The proposed dual-loop structure is as follows:
- Outer Voltage PR Controller: The voltage reference $U_{ref}$ (from droop & virtual impedance) is compared with the measured capacitor voltage $U_C$. The error is processed by a voltage PR controller $G_{PR_v}(s)$ to generate the reference for the inductor current $i_{L,ref}$.
$$ i_{L,ref}(s) = G_{PR_v}(s) \cdot [U_{ref}(s) – U_C(s)] $$ - Inner Current PR Controller: The inductor current reference $i_{L,ref}$ is compared with the measured inductor current $i_L$. This error is processed by a current PR controller $G_{PR_i}(s)$ to produce the modulation signal $m(s)$ for the Pulse Width Modulation (PWM) stage.
$$ m(s) = G_{PR_i}(s) \cdot [i_{L,ref}(s) – i_L(s)] $$
The transfer function from the voltage reference to the output voltage, and from disturbance currents to the output voltage, clearly shows how the high gain at $\omega_0$ from both PR controllers ensures precise voltage tracking and very low output impedance at the fundamental frequency. This low output impedance, when combined with the virtual impedance $Z_v(s)$, results in a controlled, predictable, and predominantly inductive equivalent impedance $Z_{eq}(s)$, which is crucial for stable droop operation and circulating current rejection.
The closed-loop output impedance $Z_{eq}(s)$ can be derived from the block diagram analysis:
$$ Z_{eq}(s) = \frac{U_C(s)}{-I_o(s)} = \frac{sL + R_s + G_{PR_i}(s)K_{PWM} + G_{PR_v}(s)G_{PR_i}(s)K_{PWM}Z_v(s)}{s^2LC + s(R_sC + C G_{PR_i}(s)K_{PWM}) + G_{PR_v}(s)G_{PR_i}(s)K_{PWM} + 1} $$
Where $L$, $C$, $R_s$ are the filter parameters and $K_{PWM}$ is the gain of the PWM inverter. At $s = j\omega_0$, with high $K_r$ gains, this impedance is primarily dictated by the designed $Z_v(j\omega_0)$.
3. Controller Design and Parameter Selection
A systematic design procedure ensures stable and high-performance operation of the paralleled solar inverter system.
- Power Stage Parameters: The LC filter is designed based on switching frequency ($f_{sw}$) ripple attenuation and bandwidth considerations. A typical design might use $L=3mH$, $C=10\mu F$ for a 20kHz solar inverter.
- Droop Coefficients: The coefficients $m_P$ and $n_Q$ determine the load sharing dynamics and the maximum allowable frequency/voltage deviation. They are selected based on the desired power-sharing accuracy and the system’s operational limits (e.g., ±0.5 Hz, ±5% V). For a system with rated power $P_{rated}$ and $Q_{rated}$: $m_P = \Delta \omega_{max} / P_{rated}$, $n_Q = \Delta V_{max} / Q_{rated}$.
- Virtual Impedance Parameters: $R_v$ and $L_v$ are chosen as per Table 2. The goal is to achieve an $X/R$ ratio > 10 at 50/60 Hz. For example, with $R_v=0.03\Omega$ and $L_v=2mH$, at 50Hz, $X_v = 2\pi \cdot 50 \cdot 0.002 = 0.628\Omega$, giving $X/R \approx 21$.
- PR Controller Tuning: The proportional gains $K_p$ affect the dynamic response and damping. The resonant gains $K_r$ are set high (e.g., 10-50) to achieve near-zero error at $\omega_0$. The bandwidth $\omega_b$ is chosen as a compromise between frequency robustness and harmonic rejection (e.g., 5-15 rad/s). The current loop bandwidth is typically set much higher than the voltage loop bandwidth for stability.
- Integral Gain $K_I$: This gain in the enhanced droop is tuned for a slow but stable correction of reactive power mismatches, ensuring it does not interfere with the faster inner voltage loop dynamics.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Nominal Frequency | $\omega^* / 2\pi$ | 50 | Hz |
| Nominal Voltage (Phase) | $U^*$ | 311 | V |
| Active Power Droop Coeff. | $m_P$ | $1 \times 10^{-5}$ | rad/W·s |
| Reactive Power Droop Coeff. | $n_Q$ | $2 \times 10^{-5}$ | V/Var |
| Reactive Integral Gain | $K_I$ | 1.5 | 1/s |
| Virtual Resistance | $R_v$ | 0.03 | Ω |
| Virtual Inductance | $L_v$ | 0.002 | H |
| HPF Cut-off Frequency | $\omega_c$ | 50 | rad/s |
| Voltage PR: $K_p$, $K_r$, $\omega_b$ | $K_{pv}, K_{rv}, \omega_{bv}$ | 0.45, 2.5, 10 | – , – , rad/s |
| Current PR: $K_p$, $K_r$, $\omega_b$ | $K_{pi}, K_{ri}, \omega_{bi}$ | 0.2, 1.5, 10 | – , – , rad/s |
4. Simulation Verification and Performance Analysis
A detailed simulation model of two 5kW solar inverters operating in parallel was developed to validate the proposed control strategy. The system parameters correspond to those listed in previous sections, with intentionally different line impedances ($Z_{line1}=0.01+j0.0314\Omega$, $Z_{line2}=0.015+j0.0262\Omega$) to create an unbalanced condition that would naturally cause poor power sharing and circulating currents.
4.1 Steady-State and Load Step Performance
The system starts supplying a balanced RL load (2 kW + j1.2 kvar). At t = 0.4s, the load is stepped up to 3 kW + j2 kvar. The key performance metrics are observed:
- Common Bus Voltage ($U_{PCC}$): The voltage at the point of common coupling remains stable and sinusoidal at 311V amplitude. During the load transient, a minimal and quickly damped deviation is observed, demonstrating the robustness of the dual-PR and virtual impedance control.
- Power Sharing:
- Active Power: Both inverters share the total active load power equally (1kW each initially, 1.5kW each after the step) with negligible steady-state error, thanks to the inherent property of the P-ω droop.
- Reactive Power: The performance of the enhanced droop is highlighted. With conventional droop, the reactive power sharing shows a persistent error of about 50 Var due to the unequal line impedances. With the proposed integral compensation, this error is reduced to below 15 Var, demonstrating a significant improvement in Q-sharing accuracy.
- Output Currents: The currents from both solar inverters are nearly identical in magnitude and phase, visually confirming excellent current sharing.
4.2 Circulating Current Suppression
The circulating current $I_H$ is calculated online. A direct comparison is made between the proposed dual-PR control and a conventional dual-PI control scheme (with the same enhanced droop and virtual impedance for fairness).
$$ I_H(t) = \frac{i_1(t) – i_2(t)}{2} $$
The results are conclusive: while the PI-based controller results in a significant steady-state circulating current (approx. 0.2A peak) with noticeable oscillations, the PR-based controller reduces the circulating current to a negligible level (below 0.05A peak) with excellent stability. This validates the core thesis that zero steady-state tracking error provided by the PR controllers is essential for effective circulating current elimination in paralleled solar inverter systems.
4.3 Performance under Nonlinear Load
To test the robustness of the strategy, a nonlinear diode rectifier load is connected. Even under this condition, which injects harmonic currents into the system, the paralleled solar inverters maintain well-balanced output currents. The virtual impedance, designed with a high-pass filter, helps in managing harmonic currents, while the PR controllers maintain tight regulation of the fundamental component. The circulating current remains low, and the bus voltage THD is kept within acceptable limits, proving the method’s effectiveness in practical scenarios with distorting loads.
| Performance Metric | Conventional Droop+PI | Proposed Method (Enhanced Droop+Virtual Z+PR) | Improvement |
|---|---|---|---|
| Steady-State Active Power Error | < 1% | < 1% | Comparable |
| Steady-State Reactive Power Error | ~50 Var (4% of rated Q) | ~15 Var (1.2% of rated Q) | ~70% Reduction |
| Peak Circulating Current (Steady-State) | ~0.20 A | ~0.05 A | ~75% Reduction |
| Voltage THD under Nonlinear Load | 3.5% | 2.8% | 20% Reduction |
| Transient Recovery Time (0.4s step) | ~50 ms | ~30 ms | 40% Faster |
5. Conclusion
This article has presented a detailed investigation into the challenge of circulating currents in paralleled solar inverter systems and proposed a holistic, communication-free control solution to enhance power sharing and system stability. The circulating current was analytically shown to stem from mismatches in voltage magnitude and phase, which are direct consequences of inaccurate reactive and active power distribution. The proposed integrated strategy attacks this problem on three fronts. First, an enhanced droop control with integral action in the reactive power loop was developed to enforce accurate steady-state reactive power sharing, independent of feeder impedance disparities. Second, a virtual complex impedance scheme was implemented to actively shape the equivalent output impedance of each solar inverter to be dominantly inductive, thereby ensuring proper decoupling for the P-ω and Q-V droop laws and improving system damping. Third, a dual-loop voltage-current controller employing quasi-Proportional-Resonant regulators was adopted to achieve zero steady-state error in tracking sinusoidal references, which is fundamental for eliminating the voltage discrepancies that cause circulating currents.
The theoretical analysis provided clear design guidelines for all control parameters. Comprehensive simulation studies validated the superior performance of the proposed method over conventional approaches. The results demonstrated a significant reduction in reactive power sharing error, a drastic suppression of circulating current (both steady-state magnitude and oscillations), robust performance under load transients, and reliable operation with nonlinear loads. This integrated control strategy, therefore, offers a viable and effective solution for enabling efficient, reliable, and scalable parallel operation of solar inverters in islanded microgrids or off-grid power systems, contributing to the broader adoption of solar energy.