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Modelling and stability analysis of virtual synchronous machine using harmonic state‐space modelling method

2018; Institution of Engineering and Technology; Volume: 2019; Issue: 16 Linguagem: Inglês

10.1049/joe.2018.8601

2051-3305

Xudong Chen, Siru Yu, Xinglai Ge,

Power Systems and Renewable Energy

The Journal of EngineeringVolume 2019, Issue 16 p. 2597-2603 Session – Poster EFOpen Access Modelling and stability analysis of virtual synchronous machine using harmonic state-space modelling method Xu-Dong Chen, Xu-Dong Chen Key Laboratory of Magnetic Suspension Technology and Maglev Vehicle, Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of China Department of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of ChinaSearch for more papers by this authorSi-Ru Yu, Si-Ru Yu Department of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of ChinaSearch for more papers by this authorXing-Lai Ge, Corresponding Author Xing-Lai Ge xlgee@163.com Department of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of ChinaSearch for more papers by this author Xu-Dong Chen, Xu-Dong Chen Key Laboratory of Magnetic Suspension Technology and Maglev Vehicle, Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of China Department of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of ChinaSearch for more papers by this authorSi-Ru Yu, Si-Ru Yu Department of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of ChinaSearch for more papers by this authorXing-Lai Ge, Corresponding Author Xing-Lai Ge xlgee@163.com Department of Electrical Engineering, Southwest Jiaotong University, Chengdu, Sichuan, People's Republic of ChinaSearch for more papers by this author First published: 19 December 2018 https://doi.org/10.1049/joe.2018.8601Citations: 5AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract A modelling and stability analysis method of a virtual synchronous machine (VSM) is proposed in this study. VSM technology is used to solve the problems caused by the high permeability of distributed energy. Though VSM has inertia and damping characteristics, it is more prone to oscillate than a synchronous generator because of the characteristic and unexpected harmonics generated by power electronic elements. The traditional linear time invariant based model is difficult to analyse a time-variant system and cannot analyse harmonic coupling mechanism accurately. This study proposes a new model of a grid-connected VSM by using the harmonic state-space modelling approach, which is a typical linear time-varying periodic model and considers different harmonics of the system. The stability of the system is analysed by eigenvalues of system matrix, which is based on the harmonic state-space model. The theoretical modelling method and stability analysis are verified by direct time-domain simulation results. 1 Introduction Nowadays, energy and environmental problems have been drawing considerable attention. The distributed energy such as wind and solar energy has received much attention, and the grid-connected inverter plays an important role in the distributed generation field for its flexible and controllable characteristics [1]. With the high permeability of distribution energy sources, the inertia and damping characteristics of the grid are reduced, which may cause some problem. To solve this problem, a virtual synchronous machine (VSM) method is proposed, which mimics the external characteristics of the synchronous machine [2-5]. However, studies have shown that VSM has the power oscillation problem that is similar to the traditional synchronous generator [6]. Although the control strategy of the VSM is superior to that of the traditional droop method in frequency stability, it is easier to cause low frequency oscillation [7]. So, the modelling and stability analysis of VSM are essential. Some scholars have studied parameter design scheme of a VSM based on a small-signal modelling method [8], but the small-signal model method is a linear-time invariant (LTI) model and can only linearise around a constant value, which is not suitable for a grid inverter because of the time-varying characteristics of the system. Thus, linear time-varying periodic (LTP), which is based on the assumption that all signals are periodically time varying, is used to model the grid-connected inverter [9, 10]. The non-linear time-varying periodic (NLTP) system is linearised around a steady-state periodic operating trajectory. Furthermore, the LTP model is analysed in the extended harmonic domain (EHD), and the harmonic state-space (HSS) modelling approach is introduced to analyse the power system [11]. Then, the stability state of the system can be analysed by the pole zero diagram, the generalised Nyquist curve or eigenvalues of system matrix [9, 12]. In this paper, the HSS model of VSM is derived, and the eigenvalue analysis method is used to judge whether the system is stable or not. 2 Review of HSS modelling Before discussing the HSS modelling, the differences between the LTI and the LTP models are explained in this section. Most of power electronics devices are typically non-linear time-varying (NLTV) systems caused by the non-linear characteristic of a switch and controller. Some NLTI systems, such as DC systems, can be linearised at a constant state operating point and transferred to the LTI model (1); the system parameters A, B, C and D are all time-invariant. On the contrary, the state operating point of AC system is time-varying, and the LTI model is no longer applicable. Usually, we assume that the system parameters, state variables, input and output signals change periodically with the same periodic T 0, so the NLTP systems can be linearised around a steady-state operating trajectory and transferred to LTP modelling, which can be represented by (2). The HSS model is a typical LTP model that is based on the harmonic domain (HD) theory: (1) (2) The LTP model assumes that all the state transition matrices and input and output signals have the same minimum periodic T 0; the periodic angular frequency is ω 0, and the time-varying system matrix A (t) can be expressed as Fourier series: (3) where n is the order of considered harmonic, and An is the Fourier coefficient, the same as B (t), C (t) and D (t). Considering the dynamic performance and maintaining the same space of input and output signals, exponentially modulated periodic (EMP) function is used to describe the state variables and input and output signals, as shown in the following equation: (4) where , and Xn, Un and Yn are Fourier coefficients: (5) Substituting (4) and (5) in (2), we obtain (6) The HSS model can be written as (7) or the matrix format (8): (7) (8) with N = diag[… − 2jω 0, − jω 0, 0, jω 0, 2jω 0 …], and N is a modulation frequency, which is derived from (7); the dimension of N depends on the maximum harmonic order that is considered. A is a Toeplitz matrix, which can be defined as follows: (9) A is a double infinite matrix ideally; however, exorbitant orders increase the burden of compute, so an appropriate truncation order needs to be chosen. Suppose there are d state variables and the truncation frequency order is q, the size of matrix A will be d (2q + 1) × d (2q + 1) and X will be a d (2q + 1)-dimensional column matrix. Taking TP [cos(t)] and TP [sin(t)] for example, (10) (11) Similar to LTI modelling, the harmonic transfer function can be described as follows: (12) Then, the stability of the system can be evaluated through the eigenvalues of the matrix (A − N). If the real part of all of the eigenvalues is distributed in the left half-plane (LHP), it means Re(λi) ≤ 0; when Re(λi) = 0, the algebraic multiplicity is 1 and the system is stable; otherwise, the whole system is unstable. Suppose that there are d state variables x and the truncation frequency order is q, then the system has d (2q + 1) eigenvalues, but only d of them are original eigenvalues, as shown in Fig. 1 by ‘◦’, and the others are translated copies, labelled ‘x’ in Fig. 1, because of harmonics. The trajectory of eigenvalues of the HSS model is a series of vertical lines, and the eigenvalues near the truncation frequency may be distorted, deviating from a straight line; when analysing the stability of a system, the edge effect can be ignored. Fig. 1Open in figure viewerPowerPoint Eigenvalues loci of different systems (a) Eigenvalues loci of a stable system, (b) Eigenvalues loci of an unstable system According to the stability theory that is introduced, case (a) in Fig. 1 is stable for all of the eigenvalues in LHP and case (b) is unstable. 3 Modelling of VSM 3.1 Topology modelling The topology of VSM is shown in Fig. 2 a, and the parameters of the system are shown in Table 1. According to the circuit relation, the average model of this part can be derived as (13), where sw _abc represents the switch state of phases A, B and C, respectively, the same as i Ls_abc, i Lg_abc, v c_abc and v g_abc; the other symbols are shown in Fig. 2 a : (13) Fig. 2Open in figure viewerPowerPoint Virtual synchronous machine (a) Topology of VSM, (b) Controller of VSM Table 1. System parameters Parameters Values Parameters Values Parameters Values V gm V dc 800 V f pwm 12.5 kHz L s 5 mH L g 0.1 mH C 50 μF R s 0.035 Ω R g 0 Ω D q 10,000 f g 50 Hz J 0.13 kg m2 D p 30 Using ‘Δ’ to represent the disturbance around the steady-state periodic trajectory and ignoring the second-order terms, the system can linearised as (14), which is the LTP model of topology: (14) where sw_ abc0 and v dc0 represent the steady-state values of switching and DC side voltage, respectively, both are periodic time-varying if taking harmonics into consideration and can be obtained by simulation or mathematical calculation. Equation (14) can be transformed to the HSS model in frequency domain given by the following equation: (15) where the state variable matrix X T = [ΔI Lsa, ΔI Lsb, ΔI Lsc, ΔI Lga, ΔI Lgb, ΔI Lgc, ΔV dc, ΔV capa, ΔV capb, ΔV capc]T, the input signal U T = [ΔV ga, ΔV gc, ΔV gc, Δsw a, Δsw b, Δsw c, ΔV dc, ΔV capa, ΔV capb, ΔV capc]T, then A T can be represented by a partitioned matrix with (16) where A T1 = diag( − R s /L s I − N), A T2 = diag( − 1/L s I), A T3 = diag( − R g /L g I -N), A T4 = diag(1/L g I), A T5 = diag(1/CI), A T6 = diag( − 1/CI), Z is zero matrix, I is the identity matrix, N = diag [… − 2jω 0, − jω 0, 0, jω 0 , 2jω 0 …], and I and N have the same matrix size with the harmonics that considered: (17) where B T1 = diag(TP [v dc0]/L s), B T2 = diag( − I /L g), B T3 = diag(I /L g), B T4 = diag( − I /L g), B T5 = column(TP [sw a0], TP [sw b0], TP [sw c0]). ‘TP []’ means the Toeplitz matrix, and v dc0, sw a0, sw b0 and sw c0 are steady-state solutions of the DC side voltage and switching, which change periodically. In this paper, v dc0 is considered as a constant value with no harmonics. 3.2 Control loop modelling The control loop is shown in Fig. 2 b; chosing x c = [θ, ω, M f i f]T as the state variables and input signal u c = [T m, Q set, ω n, v m, v r, i Lsa, i Lsb, i Lsc]T, the non-LTP model of the controller is shown in (18) with all signals being non-linear and time-varying: (18) where T e, T m, J and Q are electromagnetic torque, mechanical torque, inertia and output reactive power, respectively, ω and v m are the angular speed and amplitude of output voltage, θ is the phase angle of the output voltage of inverter of phase A v inva and M f i f is the virtual mutual inductance. According to the theory of VSM [6], v inv, T e and Q can be calculated by (19) and (20): (19) (20) where 〈, 〉 means the conventional inner product: (21) As a result, (18) and (20) should be turned into the LTP format based on the assumption that all signals are varying periodically and linear around a periodic trajectories. Then, the linear model can be represented by (22) (23) Equation (23) can be simplified to (24) (24) In the steady state, ω is ω 0, and the output reactive power can be represented by (25) Similarly, the reactive power can be linearised as (26) According to (22), (24) and (26), choosing the state variable matrix X C = [ΔI Lsa, ΔI Lsb, ΔI Lsc, ΔI Lga, ΔI Lgb, ΔI Lgc, ΔV dc, ΔV capa, ΔV capb, ΔV capc, Δθ, Δω, ΔM f i f]T, the input signal U C = [ΔV ga, ΔV gb, ΔV gc, ΔV dc, ΔV capa, ΔV capb, ΔV capc, ΔT m, ΔQ set, Δω 0, Δv m, Δv r, Δi Lsa, Δi Lsb, Δi Lsc]T, the HSS model of the controller can also be represented by the matrix format in the frequency domain: (27) (28) with (29) where M f i f0 is the steady-state solution of virtual mutual inductance. (30) where (31) 3.3 Closed-loop system modelling In the first and second subsections of this section, the HSS model of topology and controller has been derived; however, the stability analysis is based on the closed-loop system matrix (A −N); thus, this subsection derives the closed-loop model of the VSM: During this procedure, we give some assumptions: (i) the DC side voltage fluctuation is considered to be 0, (ii) the system is three-phase balanced and (iii) the on–off action of the switch is ignored; then, the switch state can be represented by (32) and linearised by (33): (32) (33) According to (15)–(17), (27)–(31) and (33), choosing the state variable matrix X = [ΔI Lsa, ΔI Lsb, ΔI Lsc, ΔI Lga, ΔI Lgb, ΔI Lgc, ΔV dc, ΔV capa, ΔV capb, ΔV capc Δθ, Δω, ΔM f i f]T, the input signal vector U = [ΔT m, ΔQ set, Δω n, Δv m, Δv r, Δi Lsa, Δi Lsb, Δi Lsc]T, the linearised model of the closed-loop system can is given by (39). The HSS model of the controller can also be represented by the matrix format in the frequency domain: (34) where (A − N) is given by (35) (35) with A 1 = diag( − R s /L s I − N), A 2 = diag( − 1/L s I), A 4 = diag( − R g /L g I − N), A T5 = diag(1/L g I), A T6 = diag(1/CI), A T7 = diag( − 1/CI), (36) (37) (38) Assuming that the three-phase of the VSM is balanced, the phase difference of θ 0, θ b0 and θ c0 is 120°, and the value of 〈, 〉 is constant. It is worth to mention that all the input and output signals and state variables are harmonic matrices, which are made up of Fourier coefficients, and the size depends on the harmonics considered during the modelling procedure. The output vector can be chosen by users themselves, because the output signals do not affect the stability analysis of the system, the stability analysis is based on the eigenvalues of the closed-loop system matrix (A − N). 4 Simulation results The HSS model and the stability theory proposed in the previous section are validated though simulation results in this section. By studying the eigenvalue loci of the system matrix (A −N), the stable state of VSM can be evaluated. In this section, the truncation order is chosen to be 40, P set = 40 kW and Q set = 30 kVar; the other parameters are shown in Table 1. Fig. 3 shows the simulation results with different integration coefficients. Case 1 K = 10,000, all of the eigenvalues are in the LHP as shown in Fig. 3 e, and the magnified picture is shown in Fig. 3 g. In this case, the system is stable, the output power P and Q track the set value and the current is non-distorted as shown in the time simulation results in Figs. 3 a and c. Case 2 K = 7000, some of the eigenvalues are in the right half-plane, as shown in Figs. 3 f and h, so the system is unstable, which is in agreement with the time-domain simulation, as shown in Figs. 3 b and d. It is worth to mention that, in this paper, not only integration coefficient K but also inertia J and damping coefficients D p have an essential impact on the stability of VSM. (39) Fig. 3Open in figure viewerPowerPoint Simulation results of VSM under different controlling parameters (a), (c), (e), (g) Stable system with K = 10,000, (b), (d), (f), (h) Unstable system with K = 7000, (a), (b) Current i g; (c), (d) Active power P (red) and reactive power Q (blue); (e), (f) Eigenvalue loci; (g), (h) Eigenvalue loci (zoomed-in) 5 Conclusion In this paper, an HSS modelling method of VSM is presented, which is based on the LTP theory and considers the important harmonics of the system. The stability analysis is based on the eigenvalues of the closed-loop system matrix. 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