The integration of solar photovoltaic (PV) systems into the power grid is a cornerstone of the global shift towards renewable energy. A key technology enabling the scaling of solar power plants is the parallel operation of multiple solar inverters. This architecture increases total system capacity and enhances reliability through redundancy. However, a significant technical challenge arises in such configurations: the generation of circulating currents among the parallel units. These currents, which do not flow to the load but circulate between the inverters themselves, degrade system efficiency, increase component stress, and can compromise overall reliability. Therefore, the analysis and suppression of circulating currents is a critical research focus in the field of distributed power generation.
This article delves into the intrinsic characteristics of circulating currents in parallel-connected solar inverter systems. We propose a comprehensive, novel control strategy designed to achieve precise load current sharing and effectively suppress these harmful circulating currents. Our approach integrates an improved droop control method, virtual complex impedance shaping, and a dual Proportional-Resonant (PR) control scheme. The theoretical principles of the control system are analyzed in detail, and its performance is validated through simulation studies.
1. Characteristics of Parallel-Connected Solar Inverter Systems
1.1 System Configuration and Power Flow Analysis
A typical system of N parallel solar inverters is considered. Each unit consists of a front-end DC-DC converter (e.g., a Boost converter for Maximum Power Point Tracking – MPPT) and a rear-end DC-AC inverter with an LC output filter. The inverters are connected to a common AC bus that feeds the local load. The key parameters include the filter inductance \(L_s\) and its parasitic resistance \(R_s\), the filter capacitance \(C\), and the line impedance \(Z_{line}\) connecting each inverter to the Point of Common Coupling (PCC).
For simplicity in analysis, a two-inverter equivalent circuit is often examined, as the principles extend to N units. The output voltage of the n-th inverter is represented as \( \dot{U}_n = U_n \angle \delta_n \), and the load voltage at the PCC is \( \dot{U}_L = U_L \angle 0 \). The equivalent output impedance for each inverter, denoted as \(Z_n\), encompasses the filter impedance and the line impedance.
The complex power output from the n-th inverter is given by:
$$ \dot{S}_n = P_n + jQ_n = \dot{U}_n \cdot \dot{I}_n^* $$
Using circuit analysis, the active and reactive power delivered by the inverter can be derived as:
$$ P_n = \frac{1}{Z_n}[U_n^2 \cos \theta_n – U_n U_L \cos(\theta_n + \delta_n)] $$
$$ Q_n = \frac{1}{Z_n}[U_n^2 \sin \theta_n – U_n U_L \sin(\theta_n + \delta_n)] $$
where \(\theta_n\) is the impedance angle of \(Z_n\). Assuming the output impedance is predominantly inductive (\(X_n >> R_n\)), we have \(\theta_n \approx 90^\circ\). Furthermore, for small power angles (\(\delta_n\) is small), we use the approximations \(\sin \delta_n \approx \delta_n\) and \(\cos \delta_n \approx 1\). This simplifies the power equations to the well-known form:
$$ P_n \approx \frac{U_n U_L}{X_n} \delta_n $$
$$ Q_n \approx \frac{U_n (U_n – U_L)}{X_n} $$
These simplified equations reveal the fundamental basis for conventional droop control: active power \(P\) is predominantly influenced by the power angle \(\delta\) (and thus frequency \(\omega\)), while reactive power \(Q\) is primarily influenced by the voltage amplitude difference \((U_n – U_L)\). The traditional \(P-\omega\) and \(Q-U\) droop laws are expressed as:
$$ \omega_n = \omega^* – k_{f} P_n $$
$$ U_n = U^* – k_{U} Q_n $$
where \(\omega^*\) and \(U^*\) are the nominal no-load frequency and voltage setpoints, and \(k_f\), \(k_U\) are the droop coefficients.
1.2 Circulating Current Analysis
The circulating current between two parallel inverters is defined as half the difference of their output currents:
$$ \dot{I}_H = \frac{\dot{I}_1 – \dot{I}_2}{2} = \frac{1}{2}\left( \frac{\dot{U}_1 – \dot{U}_L}{Z_1} – \frac{\dot{U}_2 – \dot{U}_L}{Z_2} \right) $$
To focus on control-induced differences, we assume the designed output impedances are equal, i.e., \(X_1 = X_2 = X_0\), and maintain the small-angle approximation. The circulating current can then be decomposed into its real (active) and imaginary (reactive) components:
$$ \dot{I}_H \approx \frac{U_1 – U_2}{2X_0} + j\frac{U_1\delta_1 – U_2\delta_2}{2X_0} = I_{H,p} + jI_{H,q} $$
This result provides crucial insight:
- If voltage magnitudes are unequal (\(U_1 \neq U_2\)) but phase angles are equal (\(\delta_1 = \delta_2\)), a primarily active circulating current \(I_{H,p}\) exists, proportional to the voltage magnitude difference.
- If voltage magnitudes are equal (\(U_1 = U_2\)) but phase angles are unequal (\(\delta_1 \neq \delta_2\)), a primarily reactive circulating current \(I_{H,q}\) exists, proportional to the difference in \(U_n\delta_n\).
- In the general case where both parameters differ, a complex circulating current with both active and reactive components is present.
Therefore, effective current-sharing control must regulate both active and reactive power outputs accurately to minimize both \(\Delta U\) and \(\Delta \delta\).
2. Proposed Hierarchical Current-Sharing Control Strategy
Our proposed control architecture is hierarchical, consisting of an outer power-sharing loop based on improved droop control, an intermediate virtual impedance loop, and an inner high-bandwidth voltage and current tracking loop using PR controllers. The overall control block diagram for one inverter is conceptually structured as follows:
- Power Calculation Module: Measures load voltage and inverter output current to compute instantaneous active and reactive power (\(P_n, Q_n\)).
- Improved Droop Controller: Generates the primary voltage reference (\(U_{ref}^*, \omega_{ref}\)) based on the measured power and the proposed droop law.
- Virtual Impedance Block: Modifies the voltage reference to emulate a desired output impedance characteristic.
- Dual-Loop PR Controller: An outer voltage PR loop and an inner current PR loop ensure precise tracking of the voltage reference and provide excellent disturbance rejection.
- PWM Generation: The final control signal modulates the inverter switches.
2.1 Improved Droop Control for Accurate Power Sharing
The conventional droop control in Eq. (5) has a limitation. While the integral action in the frequency synchronization (e.g., via a Phase-Locked Loop) inherently ensures accurate active power sharing regardless of \(X_n\), the reactive power sharing in Eq. (6) remains directly dependent on \(X_n\). Differences in \(X_n\) due to component tolerances or line impedances lead to poor reactive power sharing.
We propose an improved droop scheme by introducing an integral term into the Q-U droop path. The modified control laws are:
$$ \omega_n = \omega^* – k_{f} P_n $$
$$ U_n = U^* – \left( k_{U} + \frac{K_I}{s} \right) Q_n $$
Where \(K_I\) is the integral gain. In the steady state (\(s \rightarrow 0\)), the integral term dominates, forcing the reactive power error to zero irrespective of the value of \(X_n\). This significantly improves the accuracy of reactive power distribution among the parallel solar inverters. The active power loop can also be enhanced with an integral term if needed for a specific architecture.
2.2 Virtual Complex Impedance for Output Impedance Shaping
The simplified power flow analysis in Eq. (3) assumes a purely inductive output impedance. In practice, the inherent output impedance of a voltage-controlled inverter with an LC filter has a complex character (resistive-inductive) across frequency. This complicates the P/Q decoupling assumed by the simple droop laws.
To address this, we employ a virtual impedance loop. A virtual complex impedance \(Z_v(s)\) is added in the control loop. The voltage reference is intentionally drooped by the virtual impedance voltage drop:
$$ \dot{U}_{ref} = \dot{U}_{ref}^* – Z_v(s) \cdot \dot{I}_o $$
where \(\dot{I}_o\) is the inverter output current. We design \(Z_v(s)\) to be a high-pass filtered virtual impedance to avoid amplifying high-frequency noise:
$$ Z_v(s) = \frac{\omega_c}{s + \omega_c} (sL_v – R_v) $$
Here, \(L_v\) is the virtual inductance, \(R_v\) is the virtual resistance, and \(\omega_c\) is the cutoff frequency of the high-pass filter. The purpose is twofold:
- The virtual inductance \(sL_v\) increases the overall inductive component of the inverter’s equivalent output impedance at the fundamental frequency, reinforcing the P/f and Q/U decoupling.
- The virtual resistance \(-R_v\) decreases the inherent resistive part of the output impedance, helping to damp resonances and improve stability.
By carefully choosing \(L_v\) and \(R_v\), we can shape the equivalent closed-loop output impedance \(Z_{eq}(s)\) to be predominantly inductive at the fundamental frequency (e.g., 50/60 Hz), which is optimal for droop control, while providing desirable characteristics at other frequencies.
2.3 Dual-Loop Proportional-Resonant (PR) Control
The inner voltage and current control loops are critical for achieving high-quality output voltage, fast dynamic response, and effective suppression of circulating currents. Traditional Proportional-Integral (PI) controllers in the synchronous (dq) reference frame work well for DC signals but require decoupling terms. In the stationary (αβ) frame, PI controllers cannot track sinusoidal references without a steady-state error.
We adopt Proportional-Resonant (PR) controllers in the stationary frame. A PR controller provides nearly infinite gain at a specific resonant frequency (e.g., the grid fundamental frequency), enabling zero steady-state error for tracking sinusoidal signals. The transfer function of a non-ideal (“quasi”) PR controller is:
$$ G_{PR}(s) = K_p + \frac{2K_r\omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \(K_p\) is the proportional gain, \(K_r\) is the resonant gain, \(\omega_0\) is the resonant frequency (e.g., \(2\pi\cdot50\) rad/s), and \(\omega_c\) is the bandwidth around \(\omega_0\) which provides robustness against frequency variations.
Our dual-loop structure is as follows:
- Outer Voltage PR Controller: Takes the difference between the virtual-impedance-adjusted voltage reference \(u_{ref}\) and the measured capacitor voltage \(u_c\). Its output is the reference current \(i_{ref}\) for the inner loop. This loop ensures the output voltage accurately follows the desired sinusoidal waveform set by the droop controller.
- Inner Current PR Controller: Takes the difference between the current reference \(i_{ref}\) and the measured inductor current \(i_L\). Its output forms the modulation signal for PWM. This inner high-bandwidth loop forces the inverter output current to follow the reference precisely, which is essential for rejecting disturbances and limiting circulating currents.
The closed-loop relationship for the current loop, assuming a PWM gain \(K_{PWM}\) and plant model \(1/(sL_s + R_s)\), can be derived. With a well-tuned PR controller (\(G_i(s)\)), the output current closely tracks its reference:
$$ i_L(s) = \frac{G_i(s)K_{PWM}}{sL_s + R_s + G_i(s)K_{PWM}} i_{ref}(s) – \frac{1}{sL_s + R_s + G_i(s)K_{PWM}} u_o(s) $$
The high gain of \(G_i(s)\) at \(\omega_0\) minimizes the influence of the disturbance term (the output voltage \(u_o\)) and ensures accurate reference tracking, thereby directly controlling the current injected into the parallel system and suppressing circulating currents.
2.4 Summary of Control System Parameters
The effectiveness of the proposed strategy depends on appropriate parameter selection. The table below summarizes typical parameters for the control loops and the virtual impedance used in our analysis.
| Control Parameter | Symbol | Value |
|---|---|---|
| Voltage PR: Proportional Gain | \(K_{pu}\) | 0.45 |
| Voltage PR: Resonant Gain | \(K_{ru}\) | 2.5 |
| Voltage PR: Bandwidth | \(\omega_{cu}\) (rad/s) | 10 |
| Current PR: Proportional Gain | \(K_{pi}\) | 0.2 |
| Current PR: Resonant Gain | \(K_{ri}\) | 1.5 |
| Current PR: Bandwidth | \(\omega_{ci}\) (rad/s) | 10 |
| Fundamental Frequency | \(\omega_0\) (rad/s) | 314 |
| Virtual Inductance (Inv. 1) | \(L_{v1}\) | 2.0 mH |
| Virtual Resistance (Inv. 1) | \(R_{v1}\) | 3.0 mΩ |
| Droop Coefficients | \(k_f, k_U\) | 1e-5, 2e-5 |
| Q-U Integral Gain | \(K_I\) | 1.5 |
3. System Parameters and Simulation Validation
To validate the proposed control strategy for parallel solar inverters, a simulation model of two 5-kW units was built in Matlab/Simulink. The key power circuit parameters are listed below.
| Circuit Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | \(U_{dc}\) | 400 V |
| Nominal AC Voltage (RMS) | \(U_{ac}\) | 220 V |
| Filter Inductor | \(L_s\) | 3.0 mH |
| Filter Capacitor | \(C_f\) | 10 μF |
| Switching Frequency | \(f_{sw}\) | 20 kHz |
| Line Impedance 1 | \(Z_{line1}\) | 0.01 + j0.0314 Ω |
| Line Impedance 2 | \(Z_{line2}\) | 0.015 + j0.0262 Ω |
The simulation scenario was designed to test both steady-state performance and dynamic response:
- Initial State (t < 0.4s): A resistive-inductive load of 2 kW + j1.2 kvar is connected.
- Load Step Change (t = 0.4s): The load increases to 3 kW + j2 kvar.
3.1 Key Simulation Results and Analysis
1. PCC Voltage Quality: The voltage at the Point of Common Coupling (PCC) remained stable and sinusoidal throughout the simulation. During the load step transient at t=0.4s, a minimal and rapidly damped voltage disturbance was observed, confirming the robustness of the dual-PR voltage control loop.
2. Power Sharing Accuracy: The performance of the improved droop controller was evaluated by comparing reactive power sharing with the conventional method.
- With conventional droop control, the inherent difference in line impedances (\(Z_{line1} \neq Z_{line2}\)) caused a significant mismatch in reactive power output between the two inverters, exceeding 50 var in steady state.
- With the proposed improved droop control featuring the integral action on the Q-U loop, the reactive power mismatch was drastically reduced to below 15 var. The active power was shared equally by both controllers, as expected.
3. Output Currents and Circulating Current Suppression: The output currents of the two inverters were nearly identical in both amplitude and phase, demonstrating excellent load current sharing. The circulating current was calculated and monitored.
- When using conventional PI-based current control (in the dq frame for comparison), a measurable, fluctuating circulating current was present.
- When employing the proposed stationary-frame dual-PR current control, the magnitude of the circulating current was reduced substantially, and its oscillations were significantly dampened. This clearly validates the superior tracking performance and disturbance rejection capability of the PR controller at the fundamental frequency.
4. Performance with Non-Linear Load: To further stress the control system, a non-linear diode-rectifier type load was connected. The results showed that the parallel solar inverters continued to share the distorted load current effectively, with the PR controllers managing the harmonic currents and maintaining stable operation with low circulating current.
4. Conclusion
This article has presented a detailed analysis and a novel solution for the critical issue of circulating currents in parallel-operated solar inverter systems. The circulating current was analytically shown to stem from imbalances in both voltage magnitude and phase angle between the inverter units, which are caused by mismatches in output impedance and conventional control limitations.
The proposed integrated control strategy successfully addresses these challenges through a threefold approach:
- Improved Droop Control: By incorporating an integral term into the reactive power-voltage (Q-U) droop characteristic, the steady-state reactive power sharing error is forced to zero, achieving accurate power distribution independent of mismatches in line or filter impedance.
- Virtual Complex Impedance: This technique actively shapes the equivalent output impedance of each inverter to be predominantly inductive at the fundamental frequency. This enhances the decoupling of active and reactive power control loops and improves system stability, forming an ideal foundation for the droop law.
- Dual-Loop PR Control: The use of Proportional-Resonant controllers in the stationary frame for both voltage and current regulation provides zero steady-state error in tracking sinusoidal references. The high-gain inner current loop ensures precise current following, which is the direct mechanism for suppressing circulating currents and rejecting load disturbances.
Simulation studies on a parallel system of two 5-kW solar inverters under linear and non-linear load changes confirmed the effectiveness of the proposed method. The results demonstrated excellent voltage regulation, highly accurate active and reactive power sharing, and a significant reduction in both the magnitude and fluctuation of the inter-inverter circulating current. This control strategy enhances the efficiency, reliability, and power quality of parallel-connected solar inverter systems, facilitating their wider adoption in scalable solar power plants and microgrid applications.