I. INTRODUCTION

Distributed power generation systems have attracted more and more research due to their feasibility and efficiency [1]. As a flexible and efficient grid interface, power electronic converters are widely accepted for connecting renewable energy systems [2]. In the grid-connected mode, an inverter is typically controlled as a current source that injects a certain amount of current into the grid. Most grid-connected inverters are connected in parallel due to the characteristics of current sources.

Due to favorable low-pass filter characteristics, the LCL filter is more widely used to suppress the harmonics around the switching frequency produced by inverter power devices [3], [4]. However, the resonance problem caused by LCL filter making makes it easier for the grid connected inverter to be unstable. As a result, damping solutions are needed. Multiple paralleled inverters connected to a low voltage grid via the point of common connection (PCC) are inter-coupling due to the grid impendence [8]-[10]. Therefore, passive and active solutions for a single LCL inverter model to damp the resonance, which have been studied extensively in [5]-[7] are not variable. Previous studies [11]-[13] show that the resonant behavior of multiple inverters is different from that of a single grid-connected inverter. The authors of [11] established an open-loop transfer function matrix to describe the interaction between parallel inverters. However, the control strategies of inverters are usually MPPT (maximum power point tracking) schemes. Therefore, the transient behavior of resonances cannot be described by this model. An inverter is considered to be an ideal voltage source when analyzing the resonant characteristics of a parallel LCL filter [12]. Without considering the effect of the inverters, a model cannot describe the resonant characteristics between inverters. The discrete domain closed-loop transfer function matrix was established and the resonant characteristics of multiple paralleled inverters were analyzed in [13]. However, the parameters, control strategies and control parameters that directly affect the degree and amount of the resonance frequency are considered to be same. The resonant characteristics of multiple paralleled grid-connected inverters with an LCL Filter are analyzed, but the resonant mechanism and suppression strategy are not studied [14].

Complex resonant characteristics increase the difficulty of harmonic suppression, and seriously affect both the power quality and the system stability. Therefore, this paper develops a multiple input multiple output model of a grid-connected inverter system using a closed-loop transfer function to analyze the system resonance mechanism. In addition, it explores whether the resonance suppression strategy can be successfully applied to applications (such as compressors) where a simple starting control is required [15].

The resonance problem caused by multiple paralleled grid-connected inverters with an LCL filter brings a great challenge to the stability of power systems [16], [17]. In order to improve the capacity of adapting to a weak grid, many passive damping [18] and active damping [19]-[21] methods have been proposed to suppress the resonance of inverters. Active damping methods are widely used in grid-connected inverters, because they can increase structural damping without reducing system efficiency. However, it is difficult to design a damping controller when it is used in multiple paralleled grid-connected inverters which have very complex resonant characteristics and need a high bandwidth [22], [23]. This paper proposes a scheme for changing grid harmonic damping by controlling the harmonic voltage based on an analysis of the resonant characteristics.

In Section II, a transfer function model with multiple-input and multiple-output describing multiple paralleled grid-connected inverters is established based on a Norton model. In Section III, the relationships between the resonant characteristics and the control parameters of the inverters as well as the power grid circumstances are invested. In Section IV, a scheme based on controlling the harmonic voltage to damp the resonant of multiple paralleled grid-connected inverters is proposed. In Section V, an experiment is carried out to verify the analysis of the resonant characteristics and the proposed scheme to damp resonance. Some conclusions are presented in Section VI.

II. MODEL OF MULTIPLE PARALLELED GRID-CONNECTED INVERTERS WITH LCL FILTER

Fig. 1 shows the system studied in this paper. Several inverters connected into a grid via the PCC composing a multiple inverter paralleled grid. In this figure, ZL1i and ZL2i are the inverter side and grid side filter impedances, respectively; ZCi is the filter capacitor; Udc,i is the DC capacitance voltage; and i=1,2,⋯,n is the serial number of the parallel inverters. Lg is the equivalent inductance of the low voltage grid, whose corresponding impedance is Zg, and ug is the grid voltage.

Fig. 1.Block diagram of grid-connected system with multiple LCL inverters.

Fig. 2 shows the configuration of a grid-connected inverter with an LCL filter. The upper part shows the power circuit while the lower part shows the digital control diagram. The grid currents ig,i and iC,i are current feedbacks to increase damping of the system. A block diagram of the dual-loop control strategy is shown in Fig. 3, where iref ,i(s) represents the injected current reference of the outer loop, Hi1(s) represent sensor gains of the filter capacitor current, Hi2 (s) represent the injected current, Gt,i(s) represents the controller of the outer loop, and Ginv(s) represents the gain of the PWM inverter [24]. Applying the equivalent conversion principle to Fig. 3, the equivalent simplified diagram can be obtained as Fig. 4.

Fig. 2.System structure of single-phase grid-connected inverter with an LCL filter.

Fig. 3.Block diagram of the dual-loop control strategy of grid-connected inverter.

Fig. 4.The equivalent block diagram of the dual-loop control strategy.

The results from the equivalent block diagram are shown in Fig. 4, where:

The relationship between the input and the output can be expressed as follows:

Where T represents the loop gain of the system, which can be shown as follows:

According to equation (2), the Norton equivalent circuit shown in Fig. 5 can be established to indicates the external characteristics of grid-connected inverters with LCL filter, were Gi represents the control coefficient of the controlled current source, and Yi represents the equivalent internal impedance.

Fig. 5.Norton model of LCL grid-connected inverter control.

Replacing the inverters in Fig. 1 by the Norton equivalent model shown in Fig. 5, the Norton model of multiple paralleled grid-connected inverters with a LCL filter is shown in Fig. 6.

Fig. 6.Norton model of multiple paralleled grid-connected inverters.

Applying Kirchhoff’s law to the circuit, the voltage of the PCC is shown in equation (4).

From (3) and (4), the current of inverter ig,i can be expressed as:

Where Pi,t is the parallel resonance of the inverters t and i; and Si is the series resonance of the inverter i and the grid. Ri, Pi,t and Si are shown as follows:

Where Gi=Gi1/Gi2. Equation (7) describes the current of multiple paralleled grid-connected inverters.

As can be seen from equation (7), the nonzero impedance of a low voltage grid causes the coupling between parallel inverters. Moreover, the resonant characteristics are more complicated because of this interaction.

III. RESONANT CHARACTERISTICS ANALYSIS

To determine how the numbers, types and composition ratios of parallel inverters affect the resonant characteristics, this paper studied three inverters A, B and C. Inverters using the same dual-loop control strategy but with different parameters are shown in Table I. The inverters are connected to a weak grid through the PCC. The grid voltage is 220V and the frequency is 50Hz. The equivalent inductance and resistance of the power grid are set to 1.5mH and 0.05Ω, respectively, based on the relationship between the SCR and the equivalent impedance of the power grid [25]. The zeros and poles of Ri, Pi,t and Si are the same. As a result, analyzing only Ri can fully describe the resonant characteristics.

TABLE IPARAMETERS OF INVERTER A, B AND C

A. The Number of Paralleled Grid-Connected Inverters

Fig. 7(a), (b) and (c) indicate how the number of grid-connected inverters with an LCL filter affects the resonant characteristics. The lines with different colors represent the number of the paralleled inverters. Fig. 7(a) shows that all of the paralleled inverters are A; Fig. 7(b) shows that the composition ratio of inverters A and B is 1:1; Fig. 7(c) shows that the composition ratio of inverters A, B and C is 1:1:1. The total number of paralleled inverters is shown on the top right corner.

Fig. 7.The resonant characteristics of different grid-connected inverter number.

Multiple positive resonant peaks are shown in Fig. 7 when multiple inverters are paralleled into a grid. The lower resonant frequency increases with a decreasing number of parallel inverters, and the type of inverters affects the number of resonant peaks. When the composition ratio of the inverters is fixed, the resonant peak is stable without the effect of parallel inverter number.

This paper analyzed a grid connecting six inverters in two cases. Case I: inverters A and C compose the parallel inverter system. Case II: inverters A, B and C compose the parallel inverter system. Table II shows the composition ratio of these six inverters. Fig. 8(a) and (b) show the resonant characteristics analysis results, in which the colored lines represent the ratios of the inverters as defined in Table II. Fig. 8 indicates that the composition ratios and types of the inverters have great effects on the resonance frequency and quantity.

TABLE IITHE COMPOSITION RATIO OF INVERTERS

Fig. 8.Resonance characteristics of six paralleled grid-connected inverters. (a) case I. (b) case II.

B. The Effect of Parameters on the Resonant Characteristics

The control parameters and the strength of the grid are factors that greatly affect the resonant characteristics of multiple paralleled grid-connected inverters. The parameters for the sensor gain (Hi1), the outer current loop control (Kp) and (Ki), and the grid strength (Lg) are considered in this paper. Fig. 9 show the relationship between the resonant characteristics and the parameters, in the case of a system composed of inverter A and B. Fig. 9(a), (b), (c) and (d) show the effects of the sensor gain (Hi1), (Kp), (Ki), and the grid strength (Lg), respectively.

Fig. 9.Influence of the parameters to the resonant characteristics. (a) Hi1. (b) Kp. (c)Ki.(d) grid impedance.

As can be seen from Fig. 9(a), the resonant frequency increases as the sensor gain Hi1 decreases. Fig. 9(b) shows that the resonant frequency decreases as the control parameter of Kp decreases. Fig. 9(c) indicates that the control parameter Ki affects the resonant peak, but has no effect on the resonant frequency. Fig. 9(d) demonstrates that with the enhancement of the power grid strength, the resonant frequency increases.

IV. SCHEME FOR RESONANT SUPPRESSION

The above research shows that the resonant characteristics are influenced by the number of inverters and their control parameters. To suppress resonance, the damping of a system should be enhanced. Active damping can be realized by impendence reshaping using the voltage of the PCC. However, there are several resonant frequencies in a multiple inverter system, and the resonant frequency changes as the working conditions shift. Therefore, a high bandwidth damping controller is needed in order to cover a wide range of resonant frequencies. This introduces difficulty to the design of the damping controller.

It can be shown from the above analysis that resonance is caused by grid impendence. Therefore, this resonance can be suppressed by reshaping the impedance of the grid. A global suppression strategy based on virtual admittance is proposed to solve the resonance problem. The equivalent circuit of a system with virtual admittance is shown in Fig. 10.

Fig. 10.Equivalent circuit of system with virtual admittance.

With virtual admittance, the grid current of the inverter i can be expressed as follows:

where:

To suppress the resonance of a system and to guarantee that the fundamental wave control is not affected, the virtual admittance Yf near the resonance frequency should be much larger than L to minimize Mi* and Ni*, while Yf should be much smaller than L at other frequencies, which does not affect the control under this frequency. Considering that a band-pass filter has a very large gain at the pass-band frequency, while it has a very small gain at other frequencies, a second order band-pass should be chosen as the virtual admittance. The transfer function of Yf is as follows:

ω 0 and Q denote the resonance frequency and the bandwidth of the filter, respectively. A control block can be embed in a inverter if its power rating is high enough or if another converter without active power control can be used to achieve the control.

Therefore, the impedance of the grid can be reshaped by controlling the resonant voltage of the PCC to suppress resonance. The resonant voltage compensation scheme is shown in Fig. 11. Since only one controller is needed to control the resonance voltage of the PCC, the burden of the other controllers is reduced. As a result, their performances are improved. Simulation and experimental results of the proposed scheme will be presented in section V.

Fig. 11.Control Block Diagram of the resonant compensation scheme.

V. SIMULATION AND EXPERIMENTAL RESULTS

A. Simulation Results of the Resonant Characteristics

A MATLAB/Simulink electromagnetic transient model of a micro-grid composing two paralleled grid-connected inverters is established to verify the analysis results. As can be seen from Fig. 7(a) and Fig. 7(b), the resonance frequency of a two grid-connected inverter system consisting of A is 420Hz and one consisting of A and B is 490Hz. Therefore, a 5% harmonic of 400Hz was injected into inverter A when the system consisted of A, while a 5% harmonic of 500Hz was injected into inverter A when the system consisted of A and B, considering that the harmonic frequency is an integral multiple of the fundamental frequency (50Hz). The equivalent inductance and resistance of the power grid are stetted to 1.5mH and 0.05Ω, respectively. The simulation parameters of inverters A and B are shown in Table I.

The current of inverter A and its THD are shown in Fig. 11. Where (a) represents two inverters composed of A connecting to the grid, (b) represents two inverters composed of A and B connecting to the grid, and (c) represents only one inverter A connecting to the grid.

It can be seen that the 400Hz harmonic is enlarged by comparing Fig. 12(a) and Fig. 12(c). In other words, the inherent resonance frequency of an LCL filter is added by a lower resonant frequency when two inverters are connected to the PCC. The resonant frequency and peaks are closely related to the type of inverter, as given in a comparison of Fig. 12(a) and Fig. 12(b). The effectiveness of the proposed analysis method is demonstrated by the consistency of the simulation results and theoretical analysis.

Fig. 12The Current and its THD of grid-connected inverter A. (a) two grid-connected inverters consisted A. (b) grid-connected inverters consisted A and B. (c) only inverter A connected to grid.

As can be seen in Fig. 10(a) and (d), the resonant characteristics are mainly influenced by Hi1 and Lg. Therefore, only the influence of these two parameters is analyzed. A simulation is executed with a two grid-connected inverter system. The first converter has parameters that are similar to those of inverter A, as shown in Table 1, except that Hi1 and Lg are different. In addition, the second inverter has the same parameters as those of inverter B, as can be seen in Table 1. When a 5% harmonic of 400Hz, 500Hz and 750Hz was injected into the first inverter, the current of inverter B and its THD are shown in Fig. 13(a), (b) and (c). Fig. 13(a) represents the situation where Hi1=0.2 and Lg=1.5mH, (b) represents the situation where Hi1=0.3 and Lg=1.5mH, and (c) represents the situation where Hi1=0.2 and Lg=0.5mH.

Fig. 13.The current and its THD of grid-connected inverter A. (a) Hi1=0.2 and Lg=1.5mH. (b) Hi1=0.3 and Lg=1.5mH. (c) Hi1=0.2 and Lg=0.5mH.

It can be seen that the resonant frequency decreases as the sensor gain Hi1 increases by comparing Fig. 13(a) and Fig. 13(b), which verifies the correctness of the analysis results in Fig. 9(a). A comparison of the results of Fig. 13(a) and Fig. 13(c) show that the resonant frequency increases as the grid impedance decreases, which validate the correctness of the analysis results in Fig. 10(d).

B. Experiment Results of the Resonant Characteristics

Experiments have been carried out to verify the theoretical analysis. Fig.14 shows the converters and experimental platform used in the experiments. The experiments have been carried on two eight-cell cascaded converters, where one cell was reconstructed for this experiment. A digital signal processor (TMS320F2812) and a field programmable gate array (EP2C50F484C8) compose the central controller. Three-phase diode rectifiers are used to provide the dc link voltages for these inverters. A reactor is used to enlarge the equivalent impedance of the grid.

Fig. 14.The inverter and experiment platform used in the experiment.

The experiments were conducted under the same conditions as the simulation. A 5% harmonic of 400Hz was injected into inverter A when the system consisted of A, while a 5% harmonic of 500Hz was injected into inverter A when the system consisted of A and B. The equivalent inductance 15mH and resistance 0.05Ω were simulated through a series reactor to decrease the strength of the grid. The filter parameters of the inverter used in the experiment are same as those used in the simulation, while the parameters for the system damping are different from those used in the simulation. Therefore, in order to obtain the same resonant frequency, the control parameters need to be different from the simulation. The experiment parameters of inverter A and B are shown in Table III.

TABLE IIIPARAMETERS OF INVERTER D AND E USED IN EXPERIMENT

The current of grid-connected inverter A is shown in Fig. 15(a), (b) and (c). Fig. 15(a) represents two inverter A connecting to the grid. Fig. 15(b) represents two inverter A and B connecting to the grid. Fig. 15(c) represents only one inverter A connecting to the grid.

Fig. 15.The current of grid-connected inverter A. (a) two grid-connected inverters consisted A. (b) grid-connected inverters consisted A and B. (c) only inverter A connected to grid.

A comparison of Fig.12 and Fig.15 shows that the two parallel grid-connected converter system add a lower resonant frequency into the inherent resonance frequency of the LCL filter. It can also be seen that the type of the inverters has great effect on the resonant frequencies, which matches with the frequency domain analysis in Fig. 7(a) and (b).

An experiment is carried out with the two grid-connected inverter systems. The experimental conditions are the same as those for the simulation executed above. A 5% harmonic of 400Hz, 500Hz and 750Hz was injected into the first inverter. The current and its THD are shown separately in Fig. 17(a), (b) and (c). Fig. 17(a) represents the situation where Hi1=0.2 and Lg=1.5mH, (b) represents the situation where Hi1=0.3 and Lg =1.5mH, and (c) represents the situation where Hi1=0.2 and Lg =0.5mH.

A comparison of Fig. 13 and Fig. 16 illustrates that the resonance frequency of the multiple parallel inverter system is related to the control parameters and grid strength, which is in accordance with the analysis results of Fig. 10.

Fig. 16The Current of grid-connected inverter A. (a) Hi1=0.2 and Lg=1.5mH. (b) Hi1=0.3 and Lg=1.5mH. (c) Hi1=0.2 and Lg=0.5mH.

Fig. 17.Simulated currents waveform of inverter B before and after applying the harmonic suppression strategy.

C. Simulation Results of the Resonant Suppression

Corresponding to the analysis currents shown in Fig. 7(b), the two grid-connected inverters consist of A and B, and a 5% harmonic 500Hz is injected, where the setting value of inverter A at 0.1s. Meanwhile, inverter B uses the harmonic suppression strategy shown in Fig.10 at 0.2s. The equivalent inductance and resistance of the power grid are preferable to 1.5mH and 0.05Ω, respectively.

Fig. 17 shows simulated current waveforms of the inverters before and after applying the harmonic suppression strategy at 0.2s. The simulations shown in Fig. 17 indicate that the resonance caused by the dynamic interactions between the control loops of the paralleled inverters is stabilized by the harmonic suppression strategy. Fig. 18 shows the change of the PCC voltage once the harmonic suppression strategy is enabled at the instant of 0.2s. An effective resonance damping of the PCC voltage can be observed.

Fig. 18.Simulated voltage waveform of PCC without and with applying the harmonic suppression strategy.

D. Experiment Results of the Resonant Suppression

The experimental platform is shown in Fig. 14. The experimental and simulation conditions are the same, while the control parameters used in experiment are different from those used in the simulation to obtain the same resonant frequency.

Fig. 19 shows the measured output currents of the inverters before and after applying the scheme for resonant suppression. In addition, the measured PCC voltage waveform in this case is shown in Fig. 20. The experimental results show that the resonance can be suppressed by controlling the harmonic voltage of the PCC completed by one inverter. This simplifies the design of the other damping controllers.

Fig. 19.Simulated currents waveform of inverter B before and after applying the harmonic suppression strategy.

Fig. 20.Simulated voltage waveform of PCC without and with applying the harmonic suppression strategy.

VI. CONCLUSIONS

A model of multiple paralleled grid-connected inverters with a LCL filter has been established in this paper. The resonant characteristics have been theoretically analyzed based on this model, which indicates that there is more than one resonant frequency when multiple inverters are paralleled into grid, and that the number, type and composition ratio of the parallel inverters greatly affect the resonant frequency. In addition, the lower resonance frequency increases with a decreasing number of parallel inverters, and the control parameters and the grid strength are the main effect factors when the number of inverters is fixed. The method for implementing the direct resonant voltage compensation on only one inverter simplifies the design of the other damping controllers. The capability of the model and the accuracy of the analytical results are verified by time-domain simulation and experimental results and the effectiveness of the resonant suppressing scheme.