A Floquet theory-based fast time-domain stability analysis for N-parallel inverters system

Received: 4 August 2021 Revised: 6 November 2021 Accepted: 2 December 2021 IET Power Electronics DOI: 10.1049/pel2.12225 ORIGINAL RESEARCH PAPER A Floquet theory-based fast time-domain stability analysis for N-parallel inverters system Hong Li1Xiaheng Jiang1Chen Liu2Zhong Li3,4Leopoldo G. Franquelo5 Sergio Vazquez5 1School of Electrical Engineering, Beijing Jiaotong University, Beijing, China 2Langfang Power Supply Company, State Grid Jibei Electric Power Co. LTD, Beijing, China 3School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, China 4Faculty of Mathematics and Computer Science, Fernuniversität in Hagen, Hagen, Germany 5Department of Electronic Engineering, University of Seville, Seville, Spain Correspondence Hong Li, School of Electrical Engineering, Beijing Jiaotong University, Beijing, China. Email: [email protected] Funding information National Natural Science Foundation of China, Grant/Award Number: U1866211; General Programs of the National Natural Science Foundation of China, Grant/Award Numbers: 51577010, 51777012; Fujian Provincial Natural Science Foundation, Grant/Award Number: 2020J01800; Excellent Youth Scholars of National Natural Science Foundation of China, Grant/Award Number: 51822701 Abstract Based on Floquet theory, a simple and fast time-domain stability analysis method for an N-parallel inverters system is proposed. Furthermore, a general time-domain model of the N-parallel inverters system in grid-connected mode and in island mode are derived, and the corresponding stability criteria are derived based on the Floquet theory. Subsequently, simulations and experiments of a two-parallel inverters system in island and grid-connected modes are conducted to verify the correctness and effectiveness of the proposed timedomain stability analysis method. To the end, the complexity comparison between the proposed time-domain method and the traditional frequency-domain method is presented to prove the advantages of the proposed method over the tedious manual derivation. 1 INTRODUCTION The voltage source inverter (VSI) enables the power transfer from a renewable energy sources (RES) to the grid [1], where the LCL filter single-phase inverters system is usually used in VSI to limit the switching current ripple [2, 3]. As the interface between renewable energy generators and an AC grid, inverters usually adopt parallel structure to inject energy into the grid [4–7]. However, due to the system bus, the dynamic interactions between complexly coupled inverters exists [8], which leads the system to have several resonance frequencies [9]. On the other hand, the required additional current-sharing control to the parallel inverters system may increase the coupling strength This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. © 2021 The Authors. IET Power Electronics published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology among the inverter modules and in turn strengthen the interaction among them, which should also be considered in the system design. Once the main circuit or control parameters are set improperly, this interaction may cause the system to oscillate or even collapse, and with the rapid development of AC–DC hybrid micro grids, these oscillations or collapse phenomenon occur more and more frequently; therefore, the modelling and stability analysis of N-parallel inverters systems have become an urgent problem for theorists and engineers [10], and the prior stability analysis for N-parallel inverters system is of practical significance. Up to now, the stability analysis methods used for parallel inverters system are mainly in frequency domain. For example, 186 wileyonlinelibrary.com/iet-pel IET Power Electron. 2022;15:186–202. LI ET AL.187 stability analysis methods based on impedance criterion have been widely applied for inverters systems [11–16]. However, in practical applications, as the number of inverters in the parallel system increases, or the circuit parameters of each inverter are variant, the derivation of the impedance criterion becomes quite complicated, leading to rare research on modelling and stability analysis of an N-parallel inverters system (especially for N≥3), to our best knowledge. In [17], it has been proved that the complexity of modelling the multi-stage DC–DC converters system in the frequency-domain will increase exponentially versus the number of the DC–DC converters according to the manual derivation process. Hence, the impedance modelling and prediction methods based on the black-box modelling was proposed in [18]; however, since it only uses terminal characteristics and does not pay attention to the inner parameters of the inverters, once deviations occur, it is difficult to correct it, so that the accuracy and the correctness of these methods are difficult to be ensured in practice. On the other hand, the impedance criterion only provides the sufficient condition for the stability of the parallel inverters system, which is usually very conservative for real applications [19, 20]. To avoid the conservation and complexity of impedance criterion method, the Routh criterion is also widely used for judging the stability of inverters as sufficient and necessary condition [21, 22]. To use Routh criterion, the closed-loop transfer function of the inverter system should be first derived and then it need be further simplified to obtain the closed loop characteristic equation and the effective stability criterion. However, for an N-parallel inverters system (N>2), its closed-loop transfer function is a complex high-order polynomial, and the expression of the high-order polynomial will become much more complicated with the increase of the number of paralleled inverters. Therefore, the simplified closed loop characteristic equation for N-parallel inverters system (N>2) is difficult to obtain; even if obtained, its accuracy can hardly satisfy the requirement in real applications. Besides, the state-space-based analysis method is also a tool to analyse harmonic stability for inverters, which can explain the resonance phenomenon in inverters [23–25]. However, unlike three-phase inverters, the steady-state AC variables of a singlephase inverter cannot be expressed into a DC form by the coordinate transformation [26], which leads the state-space-based analysis method not be able to be used into single-phase inverters directly. In recent years, a time-domain stability analysis method based on Floquet theory has been proposed and successfully employed into DC–DC converters system and single-phase inverter with proportional integral (PI) control and proportional resonance (PR) control [22, 27–30]. Compared with frequencydomain stability analysis method, the time-domain stability analysis method based on Floquet theory can avoid the complicated derivation of the required transfer functions in Routh criterion or the impedances expression in impedance criterion, only a state transition matrix is required, which can be obtained by the time-domain state equations for power converters system. Further, the stability analysis can be analysed by computer-aided way, which provides a possible method for the stability analysis for an N-parallel inverters system. Although Floquet theory has been employed into the stability analysis for single inverter with only PI or PR control, it cannot be directly applied into the N-parallel inverters system, since there is the interaction among the inverter modules and the double closed-loop control is used in the N-parallel inverters system due to the necessary current-sharing outer control. Furthermore, it is a challenging problem on how to bring the additional current-sharing control to the timedomain stability analysis method, as it is associated with system bus and does not appear in state variable of each inverter module. Therefore, this paper proposed a new time-domain stability analysis method for N-parallel inverters system based on Floquet theory, which can be used to reduce complex manual derivation and the high time cost in frequency-domain stability analysis methods. The rest of this paper is organised as follows: in Section 2, the general time-domain model of the N-parallel inverters system is established based on Floquet theory, both for grid-connected mode and for island mode. And the corresponding stability criterion of the N-parallel inverters system is derived. In Section 3, a two-parallel inverters system is taken as an example, its stability has been analysed, both in grid-connected mode and in island mode. In Section 4, the correctness of the stability analysis results in Section 3are verified by simulation and experiment results. In Section 5, the complexity of the frequency-domain and the proposed time-domain stability analysis methods are compared. Finally, the conclusion is drawn. 2THE TIME-DOMAIN MODEL OF N-PARALLEL INVERTERS SYSTEM BASED ON FLOQUET THEORY 2.1 Floquet theory Floquet theory is used to judge the stability of periodic systems [27]. According to the zero-solution condition of the perturbation equation, the stable of system can be uniquely determined. Assume a periodic system (1), and its steady-state periodic solution is noted as X;thusX(t)=X(t+T), where, Tis the period of the steady-state periodic solution. x=G(t,x).(1) When the system suffers a small perturbation δ(t)atthe steady-state periodic solution X, the state variables at the steadystate periodic solution can be expressed as x(t)=X+𝛿(t).(2) Substituting (2)into(1), and linearising by the Taylor series expansion, the perturbation equation can be obtained, as shown 188 LI ET AL. in (3), where E(t) is the Jacobi matrix of the system. d𝛿(t) dt ≈G′(t,X)𝛿(t)=E(t)𝛿(t)=J(t,X)𝛿(t).(3) According to the Floquet theory, there is E(t)=E(t+T), since the system is a periodic system. Then, assuming that V(t)isa basic solution matrix of the system (3), there must be a nonsingular matrix Φ(t)=Φ(t+T) and a constant matrix D,sothat V(t) can be expressed as V(t)=Φ(t)etD.Further,(4)canbe deduced. V′(t+T)=E(t+T)V(t+T)=E(t)V(t+T).(4) Therefore, V(t+T) is also the basic solution matrix of the system (3). According to the uniqueness of the basic solution matrix, there is V(t+T)=CV(t). it should be noticed that the matrices V(t)andV(t+T) have the following relationship: V(t+T)=Φ(t+T)e(t+T)D=Φ(t)etDeTD =V(t)eTD . (5) Thus, Ccanbegotas(6), where Cis a reversible matrix. Specially, if E(t) is a constant matrix, it can be chosen as the matrix D. C=eTD .(6) On the other hand, assume ɛito be the disturbance factor and there is a small perturbation at t0moment, then combining the disturbance component of the system v(t0)=V(t0)ɛi,with(5) and (6) results in S=‖ ‖v(t0+nT )‖ ‖ ‖ ‖v(t0)‖ ‖=‖ ‖V(t0)Cn𝜀i‖ ‖ ‖ ‖V(t0)𝜀i‖ ‖ ≤‖ ‖V(t0)𝜀i‖ ‖‖Cn‖ ‖ ‖V(t0)𝜀i‖ ‖=‖Cn‖,(7) where Sstands for the perturbation transfer function and the necessary and sufficient conditions for the stability of the system depend on lim n→∞ ||Cn||=0.(8) According to the norm theorem, there is lim n→∞ ||Cn||=0⇔|𝜆max|<1,(9) where |𝜆max|=max{|𝜆i|,i=1,2,3…},λmax is called the spectral radius of matrix Cand λi,i=1, 2, 3… are the eigenvalues of matrix C. In summary, according to the Floquet theory, the stability of the system can be described in details: ∙If|𝜆max|<1, the system is stable; ∙Otherwise, the system is unstable. Due to the N-parallel inverters with PR control is a periodic system, its stability can be judged by the stability criterion based on Floquet theory in time domain. 2.2 The general time-domain model of N-parallel inverters system PR control was firstly proposed by Marco Liserre in 2006 to provide an optimal ac current control for inverters [30], which can offer infinite gain at a certain frequency by using generalised integrators and obtain zero steady-state error at the fundamental frequency. Therefore, the PR control is adopted to achieve current control in this paper. It should also be noticed that the required current-sharing control in the parallel inverters system might reduce the system’s stability [31], so that the additional current-sharing control also needs to be considered in modelling. Although the droop control method is usually used to achieve current and power balance among the inverter modules [18,32,33], its dynamic performance is poor because of the quite low bandwidth of droop control [18]. The average current control method is commonly used as an alternative method when inverters operate in parallel, and it can better achieve power balance and ensure the quality of the voltage or current and is useful in high-performance applications in AC distributed systems [34]. Hence, the average current control is adopted to achieve current-sharing and the corresponding time-domain model is established in this paper. Although the grid-connected inverter with reactive power capability are done some research [35–37], the grid-connected inverters with unity power factor are more common in real applications, therefore, the grid-connected inverters system with unit power factor is taken as the example in this paper. According to the above mentioned, the circuit model of the N-parallel single-phase inverters system in grid-connected mode is firstly established, as shown in Figure 1. The current sharing link adopts instantaneous current average control and the current loops of each inverter adopt PR control and use the same current reference signal. In addition, Nis the number of inverters in the system and the meanings of the other symbols are shown in Table 1. Subsequently, for the jth inverter (j=1, 2, ..., N)in Figure 1, the state Equation (10) of the main circuit can be obtained according to Kirchhoff current laws (KCL) and Kirchhoff voltage laws (KVL). ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ du Cj dt =1 Cj i1_ j−1 Cj i2_ j di1_ j dt =Udj L1_ j imj −R1_ j+Rdj L1_ j i1_ j+Rdj L1_ j i2_ j−uCj L1_ j di2_ j dt =1 L2_ j uCj +Rdj L2_ j i1_ j−R2_ j+Rdj L2_ j i2_ j−us L2_ j is= N ∑ j=1(i2_ j) , (10) LI ET AL.189 FIGURE 1 The circuit model of the N-parallel inverters system in grid-connected mode 190 LI ET AL. TABLE 1 IndexofsymbolsinFigure1 Symbol Index Udj DCvoltagesourceofthejth inverter uij output voltage of H-bridge of the jth inverter L1_jinverter-side inductance of the jth inverter R1_jparasitic resistance of L1_j L2_jgrid-side inductance of the jth inverter R2_jparasitic resistance of L2_j i1_jinverter-side inductor current of the jth inverter i2_jgrid-side inductor current of the jth inverter u1_jvoltage of L1_jand R1_jof the jth inverter u2_jvoltage of L2_jand R2_jof the jth inverter Cjfilter capacitor of the jth inverter Rdj the additional damping resistor uCj voltage of capacitor of the jth inverter usgrid voltage isgrid current iref the closed-loop reference current ejthe difference between the output current of the jth inverter and the closed-loop reference current im_j the output signal of the PR controller d1_j-d4_jduty signals of the four switches of the jth inverter In the control circuit, since the PR controller is a secondorder system, in addition to selecting imj as the state variable for the jth inverter, one more state variable should be selected. The state variable can be arbitrarily selected in the PR control block diagram. For convenience of calculation, ic1_jis selected as the system state variable. Also, the average current control that strengthens the coupling interaction with other modules is considered, as this current loop collects the output current of all inverters and feeds the sum of current back to each module. According to Figure 1, the expression of the state variable in the control circuit of the jth inverter can be obtained as (11). ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ imj =𝜔 0∫(ic1_ j)dt +kpj (iref −is N) ic1_ j=2krj 𝜔0(iref −is N)−𝜔 0∫[imj −kpj (iref −is N)]dt . (11) Therefore, one has ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dimj dt =𝜔 0⋅ic1_ j+kpj ⋅diref dt −kpj N⋅dis dt dic1_ j dt =2krj 𝜔0 ⋅diref dt −2krj 𝜔0N⋅dis dt −𝜔 0[imj −kpj (iref −is N)] ,(12) Substituting (10)into(12) and using state variables i2_1, …, i2_j,…,i2_Nto decouple the sum of current is, result in (13). ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dimj dt =− kpj N N ∑ k=1 ⎛⎜⎜⎜⎜⎝ 1 L2_k uCk +Rdk L2_k i1_k− Rdk +R2_k L2_k i2_k−us L2_k ⎞⎟⎟⎟⎟⎠ +𝜔0⋅ic1_k +kpj ⋅diref dt dic1_ j dt =−2krj 𝜔0N N ∑ k=1 ⎛⎜⎜⎜⎜⎝ 1 L2_k uCk +Rdk L2_k i1_k− Rdk +R2_k L2_k i2_k−us L2_k ⎞⎟⎟⎟⎟⎠ −𝜔 0imj +𝜔0kpj N N ∑ k=1(i2_k)+𝜔 0kpjiref +2krj 𝜔0 ⋅diref dt , (13) According to (10)and(13), selecting uC1,i1_1,i2_1,im1, ic1_1,…,uCj,i1_j,i2_j,imj,ic1_j,…,uCN,i1_N,i2_N,imN and ic1_N,j=1, 2, … , N, as the state variables of the system, the state expression of the N-parallel inverters system in grid-connected mode can be obtained. Furthermore, the Jacobi matrix EsN(t) of the inverters system can be written as EsN (t)= ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ B11 B12 B13 ⋯B1N B21 B22 B23 ⋯B2N B31 B32 B33 ⋯B3N ⋮⋮⋮⋱⋮ BN1BN2BN3⋯BNN ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ,(14) EsN(t)isa5Nth order matrix and consists of submatricesBkj(k,j=1,2,…,N). In (14), Bjj represents the self-action matrix of each inverter, as shown in (15), which is caused by the main circuit of each module;Bkj(k,j=1,2,…,N;k≠j)represents the interaction matrix between each module, as shown in (16), which is caused by the interaction among modules and the currentsharing control strategy. Since the obtained Jacobi matrix EsN(t) does not contain time variables, based on Floquet theory, there is DsN =EsN(t). Then, the state transition matrix of the N-parallel inverters system in the grid-connect mode can be expressed as CsN =eTDsN, where the time constant Tis set as T=1/50s as same as the period of inverter system. According to Floquet theory, the stability of the system is analyzed by calculating the modulus of the maximum eigenvalue of CsN =eTDsN, namely, |λmax|: ∙If|𝜆max|<1, the system is stable; ∙Otherwise, the system is unstable. LI ET AL.191 Furthermore, the imaginary part of the eigenvalue can represent the oscillation frequency. Bjj (j=1,2,…,N)= ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 01 Cj −1 Cj 00 −1 L1_ j −R1_ j+Rdj L1_ j Rdj L1_ j Udj L1_ j 0 1 L2_ j Rdj L2_ j −R2_ j+Rdj L2_ j 00 −kpj NL2_ j −kpjRdj NL2_ j kpj (R2_ j+Rdj ) NL2_ j 0𝜔0 −2krj 𝜔0NL2_ j −2krjRdj 𝜔0NL2_ j 2krj (R2_ j+Rdj ) 𝜔0NL2_ j −𝜔0kpj N −𝜔00 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , (15) Bkj (k≠j;k,j=1,2,…,N)= ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ 000 00 000 00 000 00 −kpk NL2_ j −kpkRdj NL2_ j kpk (R2_ j+Rdj) NL2_ j 00 −2krk 𝜔0NL2_ j −2krkRdj 𝜔0NL2_ j 2krk (R2_ j+Rdj) 𝜔0NL2_ j −𝜔0kpk N 00 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (16) Similarly, this proposed method can be deduced and applied to an inverters system with similar structures, not only for the N-parallel inverters system in island mode, but also for the variation parameters PWM filters. Further, the proposed method can easy extend to N≥3 paralleled inverters through (14). And the theoretical stability analysis results, simulation and experimental results both in grid-connected mode and in island mode are given in Section 3. TABLE 2 The parameters in simulation and experiment of two parallel inverters system in grid-connected mode Main circuit parameters CONTROL PARAMETERS Ud1=350 V;L 1_1 =5.8 mH; L2_1 =1mH;C 1=1.5 μF; R1_1 =0.01 Ω;R 2_1 =0.01 Ω;R d1=4.5 Ω; Ud2=350 V;L 1_2 =5.8 mH; L2_2 =1mH;C 2=1.5 μF; R1_2 =0.01 Ω;R 2_2 =0.01 Ω;R d2=4.5 Ω; us=220 V Case I: kp1=kp2=0.08–0.12; kr1=kr2=500; Case II: kp1=kp2=0.09; kr1=kr2=500–600. FIGURE 2 Eigenvalues trend of Cs2when increasing kpj and fixed krj =500 FIGURE 3 Eigenvalues trend of Cs2when increasing krj and fixed kpj =0.09 3STABILITY ANALYSIS BASED ON THE FLOQUET THEORY 3.1 The two-parallel inverters system in grid-connected mode Here, a two-parallel inverters system in grid-connected mode is taken as an example. According to Section 2, when N=2, we can be obtained the system state transition matrix Cs2=eTDs2. The parameters of the parallel inverters system in gridconnected mode is given in Table 2. The time-domain stability analysis results are presented with the change of the control parameters kpj and krj. Figures 2and 3show the trend of eigenvalue changes with increasing the kpj and krj parameters, 192 LI ET AL. FIGURE 4 Case I: Eigenvalue distribution of Cs2when kpj changing (a) kpj =0.09, krj =500. (b) kpj =0.10, krj =500 FIGURE 5 Case II: Eigenvalue distribution of Cs2when krj changing. (a) krj =570, kpj =0.09. (b) krj =580, kpj =0.09 respectively. It can be seen from Figures 2and 3that as the control parameters increase, the eigenvalues of the state transition matrix gradually pass out of the unit circle in the form of conjugates. Case I: Stepwise increase kpj(j=1, 2), when kpj(j=1, 2) =0.09, as shown in Figure 4(a), the modulus of the maximum eigenvalue of the system state transition matrix Cs2is 0.9883 <1, which is within the unit circle. According to Floquet theory, system is stable. When kpj(j=1, 2) =0.10, as shown in Figure 4(b), the eigenvalue of the system state transition matrix Cs2partially exceeds the unit circle, and the modulus of the maximum eigenvalue is 60.0592 >1. According to Floquet theory, system is unstable. Case II: Stepwise increase krj(j=1, 2), when krj(j=1, 2) =570 as shown in Figure 5(a), the modulus of the maximum eigenvalue of the system state transition matrix Cs2is 0.9883 <1, which is within the unit circle. According to Floquet theory, system is stable. When krj(j=1, 2) =580, as shown in Figure 5(b), the eigenvalue of the system state transition matrix Cs2partially exceeds the unit circle, and the modulus of the maximum eigenvalue is 1.1179 >1. According to Floquet theory, system is unstable. 3.2 A two-parallel inverters system in island mode Similarly, a two-parallel inverters system in island mode is taken as an example in Figure 6, to prove this stability analysis method is also suitable for system in island mode. TABLE 3 The parameters in simulation and experiment of two parallel inverters system in island mode Main circuit parameters CONTROL PARAMETERS Ud1=350 V; L1_1 =5.8 mH; L2_1 =1mH;C1=1.5 μF; R1_1 =0.01 Ω;R2_1 =0.01 Ω; Rd1=4.3 Ω; Ud2=350 V; L1_2 =5.8 mH; L2_2 =1mH;C2=1.5 μF; R1_2 =0.01 Ω;R2_2 =0.01 Ω; Rd2=4.3 Ω; Ro=100 Ω,uo=220 V kp1=kp2=0.08; kr1=kr2=500; Case I: kcp =0.010 -0.015; kci =10; Case II: kcp =0.010; kci =15–25. The controls of system adopt the voltage and current doubleloop control strategy, the outer voltage loop uses PI control, the inner current loop applies PR control, and each inverter takes the same current reference signal. Rois the load resistance, uois the output voltage, uref is the closed-loop reference voltage and iref is the closed-loop reference current of PI controller output, the rest of the symbols are the same as Table 1. According to Section 2, when N=2, the system state transition matrix can be obtained as C2=eTD 2. The time-domain stability analysis results are presented with the change of the control parameters kcp and kci. Figures 7and 8show the trend of eigenvalue changes with increasing the kcp and kci parameters, respectively. The parameters of the parallel inverters system in island mode are given in Table 3. Case I: Stepwise increase kcp, as can be obtained from Figure 9(a), when kcp =0.011, the modulus of the maximum eigenvalue of the system state transition matrix C2is 0.8576 <1, which is located in the unit circle. According to Floquet theory, the system is stable. As shown in Figure 9(b), when kcp increases to 0.012, the eigenvalue of the 1qstate transition matrix C2partially exceeds the unit circle, and the modulus of the maximum eigenvalue is 3.1282 >1. According to Floquet theory, system is unstable. Case II: Stepwise increase kci, as can be obtained from Figure 10(a), when kci =17, the modulus of the maximum eigenvalue of the system state transition matrix C2is 0.9867 <1, which is located in the unit circle. According to Floquet theory, the system is stable. As shown in Figure 10(b), when kci increases to 18, the eigenvalue of the system state transition matrix C2partially exceeds the unit circle, and the modulus of the maximum eigenvalue is 1.2680 >1. According to Floquet theory, system is unstable. 4SIMULATION AND EXPERIMENT 4.1 Simulation results 4.1.1 The two-parallel inverters system in grid-connected mode Case I:AsshowninFigure11, when kpj(j=1, 2) =0.09, the grid-connected current of the two inverters tends to be LI ET AL.193 FIGURE 6 The circuit model of the N-parallel inverters system in island mode FIGURE 7 Eigenvalues trend of C2when increasing kcp and fixed kci =10 stable after 0.03s, the amplitude of current can track the reference value without difference. System is stable. When kpj(j=1, FIGURE 8 Eigenvalues trend of C2when increasing kci and fixed kcp =0.01 2) =0.10, as shown in Figure 12, the grid-connected currents of the two inverters are resonant, the resonance peak value is 194 LI ET AL. FIGURE 9 Case I: Eigenvalue distribution of C2when kcp changing. (a) kcp =0.011, kci =10. (b) kcp =0.012, kci =10 FIGURE 10 Case II: Eigenvalue distribution of C2when kci changing. (a) kci =17, kcp =0.010. (b) kci =18, kcp =0.010 FIGURE 11 Output current waveform of two inverters in grid-connected mode when kpj =0.09, krj =500 FIGURE 12 Output current waveform of two inverters in grid-connected mode when kpj =0.10, krj =500 FIGURE 13 Output current waveform of two inverters in grid-connected mode when krj =530, kpj =0.09 FIGURE 14 Output current waveform of two inverters in grid-connected mode when krj =540, kpj =0.09 greater than the reference value, the output currents of the two inverters are greatly distorted, and system is unstable. Case II: It can be seen from Figure 13 that the gridconnected current of the system is stable when krj(j=1, 2) =530. Figure 14 depicts that the system is unstable when krj(j=1, 2) =540. The simulation results are almost the same with the theoretical analysis results. 4.1.2 A two-par allel inverters system in island mode Case I: It can be seen from Figure 15 that the output current of the system is stable when kcp =0.011. Figure 16 depicts that the system is unstable when kcp =0.012. The simulation results are the same with the theoretical analysis results. Case II: As can be obtained from Figure 17, when kci =17, the two inverters output sinusoidal currents without oscillation, and the system is in a stable state. However, when kci increases to 18, the system has a current resonance begin from 0.08 s, as is shown in Figure 18, and according to the waveform, the system is already in an unstable state. LI ET AL.201 TABLE 5 Complexity comparisons of N-parallel inverters stability analysis based on Floquet criterion and Routh criterion Criterion name Number of steps Number of formulas When N≥3 Floquet 4 5N+1 Feasible Routh 6 N2+14N+3 Infeasible transfer function of the system needs to be obtained, namely, it is necessary to standardise closed-loop transfer function. However, since the simplification of the frequency-domain transfer function has no rules to follow, the standardisation process is extremely complicated. Hence, for the parallel system with N≥3, the stability analysis based on the Routh criterion is completely infeasible. Based on the Flowchart in Figure 30, complexity comparisons of N-parallel inverters stability analysis based on Floquet criterion and Routh criterion can be summarised, as show in Table 5and Figure 31. It is clear that Routh criterion requires more steps and formulas, and as the number Nof parallel inverters in the system increasing, the number of formulas increases at quadratic growth coefficients. Therefore, Floquet criterion is more suitable for analysing the stability of complex systems than Routh criterion. 6CONCLUSIONS This paper provides a simple and fast stability analysis method for N-parallel inverters systems. Aiming at N-parallel inverters system with the PR control and the average current control, a general time-domain model is established. Taking a 2- parallel inverters system as an example, the correctness of the proposed method is verified based on simulations and experiments, both for grid-connected mode and for island mode; further, the result of complexity comparisons show that the proposed time-domain method has advantages over traditional frequency-domain method in terms of time consumption, both in number of steps and number of formulas. These mean that the derived general time-domain model with the corresponding stability criterion based on the Floquet theory is effective and is more suitable for a complex N-parallel inverters system. Hence, this paper provides a new fast stability analysis method for N- parallel inverters system. ACKNOWLEDGEMENTS This work was supported in part by the Excellent Youth Scholars of National Natural Science Foundation of China under Grant 51822701, in part by the key project of National Natural Science Foundation of China under Grant U1866211, in part by the General Programs of the National Natural Science Foundation of China under Grants 51577010, 51777012 and in part by the Fujian Provincial Natural Science Foundation under the grant Nr. 2020J01800. CONFLICT OF INTEREST The authors declare no conflict of interest. ORCID Hong Li https://orcid.org/0000-0002-8117-8980 REFERENCES 1. Komurcugil, H., Bayhan, S., Abu-Rub, H.: Lyapunov-function based control approach with cascaded PR controllers for single-phase grid-tied LCL- filtered quasi-Z-source inverters. 2017 11th IEEE International Conference on Compatibility, Power Electronics and Power Engineering (CPEPOWERENG), Cadiz, pp. 510–515 (2017) 2. 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