Solar energy is a pollution-free and renewable new energy source, and photovoltaic power generation is a current research hotspot. Currently, the multi-inverter parallel structure can effectively increase the installed capacity of solar energy and realize large-scale utilization of solar power. However, due to differences in inverter parameters, device nonlinearities, and line impedances, parallel solar inverters are prone to circulating currents, which affect system efficiency and reliability. To suppress circulating currents, the parallel control strategy of inverters has attracted the attention of many scholars and is a key research area in the inverter field.
The parallel control methods for solar inverters mainly include master-slave control and peer-to-peer control. With the master-slave strategy, inverters need interconnected communication, which makes the control system complex and less stable. With peer-to-peer control, i.e., droop control, all inverters participate in regulating the AC bus voltage and frequency, and there is no interconnection between inverters, enabling plug-and-play with good stability. To solve the circulating current problem in parallel inverter systems using droop control, virtual impedance technology has been proposed and widely applied. Virtual impedance technology can effectively address the poor current sharing effect caused by differences in output impedance and line impedance. To further reduce the influence of inverter external characteristics on current sharing, multi-loop voltage and current control methods have been proposed and applied, improving the system transient response capability. Currently, the parallel system of solar inverters still has deficiencies in voltage and current tracking error, anti-interference capability, and circulating current suppression, which need further improvement.
To address the poor current sharing effect of parallel solar inverter systems, this paper analyzes the circulating current characteristics of solar inverters. By improving the droop control and adopting virtual complex impedance technology, the load power is effectively matched among parallel inverters. By adopting voltage and current dual PR loop control technology, the voltage and current are tracked without steady-state error, and the circulating current is effectively suppressed. The principle of circulating current suppression is analyzed, and the parallel controller is designed. Finally, modeling and simulation are carried out on the Matlab/Simulink platform for verification.
Parallel Characteristics of Solar Inverters
Power Characteristics Analysis of Parallel Solar Inverter System
Figure 1 (a) shows the schematic of a multi-solar inverter parallel system, where each solar inverter consists of a front-end Boost circuit and a back-end inverter circuit. The front-end circuit achieves boosting and maximum power point tracking to obtain a stable DC voltage, while the back-end circuit realizes DC-AC inversion and grid connection of the solar power source. In Figure 1 (a), \(Z_{line}\) is the line impedance; \(L_s\) and \(R_s\) are the filter inductor and its parasitic resistance; \(C\) is the filter capacitor; \(Z_L\) is the load impedance. Figure 1 (b) shows the equivalent circuit of two inverters in parallel, where \(\dot{U}_n\) is the inverter output voltage; \(\delta_n\) is the phase difference between the inverter output voltage and the load voltage; \(Z_n\) is the transmission impedance (including \(L_s\), \(R_s\), \(C\), and \(Z_{line}\)).
According to Figure 1 (b), the active and reactive power outputted by the \(n\)-th (\(n = 1,2\)) solar inverter are:
$$ P_n = \frac{1}{Z_n} \left[ U_n^2 \cos \theta_n – U_n U_L \cos (\delta_n + \theta_n) \right] $$
$$ Q_n = \frac{1}{Z_n} \left[ U_n^2 \sin \theta_n – U_n U_L \sin (\delta_n + \theta_n) \right] $$
where \(\theta_n\) is the impedance angle of \(Z_n\). When \(X_n \gg R_n\) and \(R_n\) is neglected, \(\theta_n = 90^\circ\); and since \(\delta_n\) is small, \(\sin \delta_n \approx \delta_n\), \(\cos \delta_n \approx 1\). Then the equations become:
$$ P_n \approx \frac{U_n U_L}{X_n} \delta_n $$
$$ Q_n \approx \frac{U_n (U_n – U_L)}{X_n} $$
It can be seen that active power is related to the angle (frequency), so active power \(P\) can be adjusted by regulating the frequency \(f\); reactive power is related to the voltage difference, so reactive power \(Q\) can be adjusted by regulating the voltage \(U\). Therefore, the droop control equations for \(P/f\) and \(Q/U\) are:
$$ \omega_n = \omega^* – k_f P_n $$
$$ U_n = U^* – k_U Q_n $$
where \(\omega^*\) is the reference angular frequency at no-load; \(\omega_n\) is the actual output voltage frequency; \(U^*\) is the reference output voltage amplitude; \(U_n\) is the actual output voltage amplitude; \(P_n\) and \(Q_n\) are the actual active and reactive power output of the \(n\)-th inverter; \(k_f\) and \(k_U\) are the frequency and voltage droop gains, respectively.
Circulating Current Characteristic Analysis
The circulating current of parallel solar inverters is defined as:
$$ \Delta \dot{I}_H = \frac{\dot{I}_1 – \dot{I}_2}{2} = \frac{1}{2} \left( \frac{U_1 \angle \delta_1 – U_L \angle 0}{Z_1} – \frac{U_2 \angle \delta_2 – U_L \angle 0}{Z_2} \right) $$
Assuming the equivalent output impedances are designed to be the same, i.e., \(X_1 = X_2 = X_0\), and \(\delta\) is small, then:
$$ \Delta \dot{I}_H = \frac{U_1 – U_2}{2X_0} + j \frac{U_1 \delta_1 – U_2 \delta_2}{2X_0} $$
According to the above equation, if \(U_1 \neq U_2\) and \(\delta_1 = \delta_2\), there is mainly active circulating current, which is proportional to the voltage amplitude difference and can be reduced by regulating the active power difference. If \(U_1 = U_2\) and \(\delta_1 \neq \delta_2\), there is mainly reactive circulating current, which increases with the phase difference and can be reduced by regulating the reactive power difference. If both \(U_1 \neq U_2\) and \(\delta_1 \neq \delta_2\), both active and reactive circulating currents exist, and both active and reactive power differences need to be adjusted simultaneously.
Current Sharing Control
Control System Structure and Principle
The structure of the parallel solar inverter control system is shown in Figure 2. It mainly consists of a power calculation module, a droop control module, a voltage and current controller, and an SPWM driving module. The functions of each module are as follows: the power calculation module collects the load voltage and inductor current and calculates the instantaneous active and reactive power; the droop control module generates the outer-loop voltage reference; the voltage and current controller takes the difference between the voltage reference and the actual output voltage through a PR regulator to generate the inner-loop current reference, then the difference between the current reference and the feedback value is sent to another PR regulator to obtain the modulation signal; the SPWM driving module generates the driving signals.
Improved Droop Controller
Combining the power expressions and droop equations, we can obtain:
$$ P_n(s) = \left[ \omega^*(s) – \omega_0(s) \right] \frac{1}{s} \frac{U_n U_L}{X_n} \left( 1 + \frac{1}{s} k_f \frac{U_n U_L}{X_n} \right)^{-1} $$
$$ Q_n(s) = (U_L – U^*) \frac{U_0}{X_n} \left( 1 + k_U \frac{U_0}{X_n} \right)^{-1} $$
From the above equations, under steady-state conditions, due to the integral term, the active power output of the inverter is not affected by the equivalent output impedance \(X_n\), so the active power can be evenly shared among parallel inverters. However, the reactive power output is related to \(X_n\), so it varies with \(X_n\), affecting the accuracy of reactive power distribution. Therefore, the conventional droop control has a poor current sharing effect and low power distribution accuracy.
To improve power distribution accuracy and current sharing, an integral term \(K/s\) (with compensation coefficient \(K\)) is introduced in the reactive power loop so that the steady-state reactive power is not affected by the equivalent output impedance. The improved \(P/f\) and \(Q/U\) control loops are illustrated in Figure 3.
After adding the integral term, the equivalent output impedance becomes \(sX_n/K\), and the power expressions become:
$$ P_n(s) = \left[ \omega^*(s) – \omega_0(s) \right] \frac{K}{s} \frac{U_L^2}{X_n} \left( 1 + \frac{K}{s} k_f \frac{U_L^2}{X_n} \right)^{-1} $$
$$ Q_n(s) = (U_L – U^*) \frac{K}{s} \frac{U_0}{X_n} \left( 1 + \frac{K}{s} k_U \frac{U_0}{X_n} \right)^{-1} $$
Voltage and Current Loop Design
Equivalent Output Impedance Design Based on Virtual Complex Impedance
In practice, the equivalent output impedance of a solar inverter is usually resistive-inductive, so the active power-frequency and reactive power-voltage are not fully decoupled. To solve this problem, virtual impedance technology is used. In this paper, the equivalent output impedance is designed to be purely inductive by introducing a virtual complex impedance \(Z_v(s)\) that contains both resistive and inductive components:
$$ Z_v(s) = \frac{\omega}{s + \omega} (s L_v – R_v) $$
where \(L_v\) is the virtual inductance; \(R_v\) is the virtual resistance; \(\omega\) is the cutoff frequency of the high-pass filter. The derivative term produced by the inductive component of the virtual impedance can cause high-frequency noise in the control process, so a high-pass filter is used to reduce its impact.
According to the above equation, the virtual inductance component can increase the equivalent output inductive reactance of the solar inverter, and the virtual resistance component can reduce the equivalent output resistance. Therefore, the output impedance of the parallel solar inverters can be designed to be purely inductive at the power frequency, thereby improving power distribution and suppressing circulating currents. After adding the virtual impedance, the voltage reference for the outer loop is related to the voltage generated by the droop module as:
$$ U_{ref} = U_{ref}^* – Z_v(s) i_o $$
where \(U_{ref}^*\) is the voltage generated by the droop module, and \(U_{ref}\) is the reference for the voltage outer loop.
Voltage and Current Dual PR Loop Control
The block diagram of dual-loop control with virtual impedance is shown in Figure 4. Traditional PI regulators do not have infinite gain at the fundamental frequency, making it difficult to achieve zero steady-state error. Therefore, quasi-PR controllers are adopted. The voltage loop uses a quasi-PR controller to achieve zero steady-state error of the output voltage, improving the current sharing effect of the parallel system. The current loop also uses a quasi-PR controller to suppress current fluctuations and improve the system’s anti-interference capability. The transfer function of the quasi-PR controller is:
$$ G(s) = k_p + \frac{2 k_r \omega_b s}{s^2 + 2 \omega_b s + \omega_1^2} $$
where \(k_p\) is the proportional coefficient; \(k_r\) is the resonant coefficient; \(\omega_b\) is the bandwidth frequency; \(\omega_1\) is the fundamental angular frequency. At the fundamental frequency, the gain of the quasi-PR controller reaches its maximum:
$$ G(s) = k_p + k_r $$
Taking the inner current loop as an example, based on the current control loop in Figure 4:
$$ i_L = \frac{G_i(s) K_{PWM}}{s L_s + R_s + G_i(s) K_{PWM}} i_{ref} – \frac{1}{s L_s + R_s + G_i(s) K_{PWM}} u_o $$
By selecting appropriate proportional and resonant coefficients, the inverter output current \(i_L\) can closely follow the current reference \(i_{ref}\), achieving zero steady-state error and suppressing circulating currents.
Analysis of Equivalent Output Impedance
Based on the voltage and current closed-loop control block diagram in Figure 4, the equivalent output impedance after introducing virtual impedance is:
$$ Z'(s) = \frac{s L_s + G_u(s) G_i(s) K_{PWM} Z_v(s) + R_s + G_i(s) K_{PWM}}{s^2 L_s C + s \left[ R_s C + C G_i(s) K_{PWM} \right] + G_i(s) G_u(s) K_{PWM} + 1} $$
where
$$ G_i(s) = K_{pi} + \frac{2 K_{ri} \omega_{bi} s}{s^2 + 2 \omega_{bi} s + \omega_{1i}^2} $$
$$ G_u(s) = K_{pu} + \frac{2 K_{ru} \omega_{bu} s}{s^2 + 2 \omega_{bu} s + \omega_{1u}^2} $$
Table 1 lists the parameters of the control loop and virtual impedance.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| \(k_{ri}\) | 1.5 | \(k_{ru}\) | 2.5 |
| \(k_{pi}\) | 0.2 | \(k_{pu}\) | 0.45 |
| \(\omega_{bi}\) (rad/s) | 10 | \(\omega_{bu}\) (rad/s) | 10 |
| \(\omega_{1i}\) (rad/s) | 314 | \(\omega_{1u}\) (rad/s) | 314 |
| \(R_{v1}\) (mΩ) | 3 | \(L_{v1}\) (mH) | 2 |
| \(R_{v2}\) (mΩ) | 4 | \(L_{v2}\) (mH) | 2.5 |
| \(k_f\) | 1e-5 | \(k_U\) | 2e-5 |
| \(K\) | 1.5 | \(\omega\) (rad/s) | 50 |
The Bode plot of the equivalent output impedance before and after introducing virtual complex impedance is shown in Figure 5. After introducing the virtual impedance, the resonance peak at high frequency is significantly attenuated. From the phase characteristic, before introducing the virtual impedance, the output equivalent impedance is resistive-inductive near the power frequency; after introducing the virtual complex impedance, the equivalent output impedance at the power frequency becomes purely inductive, which is beneficial for realizing the \(P/f\) and \(Q/U\) droop control, and at high frequencies it becomes capacitive, which can largely suppress harmonic injection.
Simulation Verification
To verify the proposed method, a parallel system of two solar inverters is simulated on the Matlab/Simulink platform. Each solar inverter has a capacity of 5 kW. The circuit parameters are shown in Table 2.
| Parameter | Symbol | Value |
|---|---|---|
| DC side voltage (V) | \(U_{dc}\) | 400 |
| Bus voltage (V) | \(u_g\) | 220 |
| Filter inductance (mH) | \(L_S\) | 3 |
| Filter capacitor (\(\mu\)F) | \(C\) | 10 |
| Switching frequency (kHz) | \(f_s\) | 20 |
| Line impedance 1 (Ω) | \(Z_{line1}\) | 0.01 + j0.0314 |
| Line impedance 2 (Ω) | \(Z_{line2}\) | 0.015 + j0.0262 |
The simulation process is set as follows: the load is resistive-inductive; initially, the load active and reactive powers are 2 kW and 1.2 kvar, respectively; at \(t = 0.4\) s, the load changes to 3 kW and 2 kvar. Figure 6 shows the PCC voltage waveform. It can be seen that the PCC voltage amplitude is 311 V at steady state, and the voltage fluctuation is small after the load change at 0.4 s. This indicates that the controller has a good control effect.
Figure 7 (a) and (b) show the reactive power outputs of inverter 1 and inverter 2 using the conventional droop controller and the improved droop controller, respectively. After the load change, the reactive power distribution difference is larger with the conventional droop control, reaching about 50 var at 0.8 s; with the improved droop control, the difference is only about 15 var at 0.8 s. This demonstrates that the improved controller, by constructing the integral term in the Q/U loop, makes the steady-state reactive power distribution less affected by the equivalent output impedance, thus achieving better power sharing.
Figure 7 (c) shows the active power outputs of the two inverters, which are well shared. Figure 7 (d) shows the output currents; the amplitudes and phases of the currents are also very close, indicating effective current sharing.
Figure 8 compares the circulating current between the two inverters using traditional PI control and dual PR control. Compared with traditional PI control, the dual PR control loop effectively reduces the circulating current, and the current fluctuation is also smaller, indicating that the combination of dual PR control and quasi-PR control can both reduce the circulating current and its fluctuations.
To further verify the circulating current suppression effect, a nonlinear rectifier load is added in the simulation, as shown in Figure 9. It can be observed that the output current fluctuation of the solar inverters is small, indicating that the proposed method is also suitable for nonlinear loads.
Conclusion
To address the poor current sharing effect in parallel solar inverter control systems, a novel current sharing control technique is proposed based on the analysis of circulating current characteristics. The following conclusions are obtained:
- By adopting virtual complex impedance technology, the equivalent output impedance of the solar inverter is designed to be purely inductive, reducing the influence of impedance differences.
- By using an improved droop control method with an integral term in the Q/U loop, the accuracy of reactive power distribution is improved.
- By adopting dual PR loop control, zero steady-state error tracking of voltage and current is achieved, and the circulating current and its fluctuations are effectively reduced.