Stabilisation strategy based on feedback linearisation for DC microgrid with multi‐converter
2018; Institution of Engineering and Technology; Volume: 2019; Issue: 16 Linguagem: Inglês
10.1049/joe.2018.8727
ISSN2051-3305
AutoresXuanying Shao, Jian Hu, Zhongtian Zhao,
Tópico(s)Smart Grid Energy Management
ResumoThe Journal of EngineeringVolume 2019, Issue 16 p. 1802-1806 Session – Poster CDOpen Access Stabilisation strategy based on feedback linearisation for DC microgrid with multi-converter Xuanying Shao, Xuanying Shao School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, People's Republic of ChinaSearch for more papers by this authorJian Hu, Corresponding Author Jian Hu hujian@sdut.edu.con School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, People's Republic of ChinaSearch for more papers by this authorZhongtian Zhao, Zhongtian Zhao School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, People's Republic of ChinaSearch for more papers by this author Xuanying Shao, Xuanying Shao School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, People's Republic of ChinaSearch for more papers by this authorJian Hu, Corresponding Author Jian Hu hujian@sdut.edu.con School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, People's Republic of ChinaSearch for more papers by this authorZhongtian Zhao, Zhongtian Zhao School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, People's Republic of ChinaSearch for more papers by this author First published: 20 December 2018 https://doi.org/10.1049/joe.2018.8727Citations: 4AboutSectionsPDF 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 Power electronic converters have been widely used in power system and the stability issues of DC microgrid with converters will be discussed in this study. Firstly, an equivalent circuit and the corresponding state space model for DC microgrid with DC–DC converters were established. The model shows that the DC microgrid systems with DC–DC bidirectional converter in boost or buck mode is a non-linear system, and the tightly regulated point-of-load converters may destabilise the system because it introduces negative incremental resistance to DC microgrid. Then the method of exact feedback linearisation, which transforms the non-linear system into the linear system by coordinate transformation, was applied to stabilise DC microgrid system, and the stabilisation strategy of non-linear system was determined by designing the feedback coefficient of the linearised system. Finally, the feasibility and effectiveness of the stabilisation strategy for DC microgrid with DC–DC bidirectional converter have been verified by the results of simulation with MATLAB. 1 Introduction Distributed generators (DGs) have attracted more and more attention due to the shortages of resource and degradation of environment. However, most of the renewable and clean energy, represented by photovoltaic and wind power, have the characteristics of randomness and uncertainty. Microgrid, which merges DGs, loads, and energy storage systems into an overall, is an effective way solving this problem. Microgrid can be divided into DC microgrid and AC microgrid according to the type of bus voltage. Compared with AC microgrid, DC microgrid has higher efficiency and relatively simple control [1]. In addition, there are no reactive power flow issues and no skin effect issues in DC transmission [2]. DC microgrid has become an important research direction due to these advantages over AC microgrid [3-5]. In DC microgrid, stability issues are related with the power electronic converters which are used for achieving the conversion of electrical energy. The sources and loads are connected to the DC microgrid through power converters, as illustrated in Fig. 1. However, the tightly regulated point-of-load (POL) converters behave as constant power loads (CPLs), which may destabilise the system since it introduces negative incremental resistance to DC microgrid [6-8]. Fig. 1Open in figure viewerPowerPoint Structure of DC microgrid The stabilisation of the system has been studied by the scholars from different point of view. The Middlebrook's impedance ration criterion was commonly used for analysing the stabilisation of DC microgrid [9]. According to this criterion, stabilisation methods can be viewed from two perspectives: reshaping the load impedance or the source impedance [10]. From the view of reason of instability, there are two methods for compensating the negative incremental resistance introducing by CPLs: passive and active damping methods. Three different passive damping methods are given in [11]: RC parallel, RL series, and RL parallel. Passive damping methods require adding physical damping elements, which could introduce dissipative losses and increase the cost of the system [12]. Active damping methods are implemented by a feedback loop, which could produce the similar damping as the physical damping elements [13]. Compared with passive damping methods, active damping methods are simple and effective. Active damping method was applied to the buck, boost, and buck–boost converters in [14]. The basic principle is that inductor current of the converter multiplied by a coefficient and then fed to the control voltage. In [15], a non-linear stabilisation block is implemented, which virtually increases the DC-bus capacitance. With this method, the real capacitance can be decreased and the volume of the system can be reduced. A non-linear feedback called loop cancellation is introduced in [16]. Compared with [14], it requires a block to calculate the reciprocal of output voltage without the current-sensing device. However these stabilisation methods are all for a single source with a CPL. A reduced-order model of the system with n parallel sources under droop control is given in [17], which reduces the n differential equations to only one. This model only works on islanding operation for DC microgrid under droop control. Since the stability caused by CPLs is a non-linear problem, the sliding mode control [18], phase-plane analysis [19], and synergetic control [18, 20] are introduced to solve this problem. The stability of DC microgrid with DC–DC bidirectional converter is analysed and exact feedback linearisation based on differential geometry theory is proposed in this paper. The equivalent circuit model and state space model are built in Section 2. The stability conditions based on the Lyapunov indirect method are determined in Section 3. Section 4 illustrates the method of exact feedback linearisation, which transforms the non-linear system into the linear system by coordinate transformation. Simulation results, which verify the effectiveness of the proposed method, are presented in Section 5 and the conclusions are drawn in Section 6. 2 Model of DC microgrid The equivalent model of DC microgrid with DC–DC bidirectional converter is illustrated in Fig. 2. The battery is modelled as an ideal voltage source and the photovoltaics (PVs) are considered as constant power sources (CPS) [17]. The CPS and CPL can be modelled as a lumped CPL, which lumped power P is given by where is the total power of CPLs (POL converters) and the is the total power of CPSs (PVs). is the output voltage of battery and is the bus voltage, which is also the output voltage of DC–DC bidirectional converter. is the current of inductor, C is the sum of the equivalent capacitor of DC bus and regulator capacitor of converter, L is the filter inductor of the converter and R is resistive load. Fig. 2Open in figure viewerPowerPoint Simplified DC microgrid model There are two operation modes for DC–DC bidirectional converter, which are boost and buck modes. When works and does not work, the converter is in the boost operation mode that energy storage unit discharges to the DC microgrid. On the contrary, when does not work and works, the converter is in the buck operation mode and the microgrid turns into a generator to the battery charge. The differential equations in two operation modes can be obtained according to the Kirchhoff law. The state variables are the inductor current of energy storage side and the input variable is the duty cycle of the DC–DC converter The differential equation in boost mode is as follows: (1) The differential equation in buck mode (2) 3 Stability analysis According to (1), the Jacobi matrix of system in boost mode is as follows: (3) The condition of asymptotic stability for a second order system at the equilibrium point according to Lyapunov's indirect method is that the trace of the Jacobian matrix is 0. Thus, the constraint condition of asymptotic stability for DC microgrid system with DC–DC bidirectional converter in boost mode is (4) Similarly, the constraint condition of asymptotic stability for system in buck mode can be obtained, which is the same as the constraint condition in boost mode. Establish the simulation model of DC microgrid system in MATLAB/Simulink, as shown in Fig. 2 and the simulation parameters of the system are shown in Table 1. Table 1. Simulation parameters DC bus voltage, V , V IGBT frequency, kHz DC/DC inductor, mH P, kW 560 250 10 2 300 When t = 0.1 s, the power of CPL is increased by 50 kW, the DC bus voltage increases, thus the DC–DC bidirectional converter operates in buck mode to maintain the power balance of the DC microgrid. The simulation result of DC bus voltage is shown in Fig. 3. Another state is that when t = 0.1 s, DC load is increased by 50 kW and the DC bus voltage decreases, thus the DC–DC bidirectional converter operates in boost mode and the energy storage units discharge to maintain the power balance. The simulation result of DC bus voltage in boost mode is shown in Fig. 4. The DC bus voltage is unstable in both boost and buck modes from the simulation results. Fig. 3Open in figure viewerPowerPoint DC bus voltage in buck mode Fig. 4Open in figure viewerPowerPoint DC bus voltage in boost mode 4 Exact feedback linearisation 4.1 The requirement of feedback linearisation In the differential geometry theory, assume and are smooth vector fields in a non-linear system, is the state space consisting of the state variables. If and only if the following conditions are met: The matrix is linearly independent in . In other words, the rank of matrix consisting of above vectors is invariant and equal to n. is involutory in . There must be a function that makes the relative order of system r equal to the order of the system n. Assume is the output matrix of system and can make the non-linear system be linearised by state feedback. In boost mode, the affine non-linear differential equation according to (1) is given by (5) (5) (6) From above equations, the matrix (7) The first condition of linearisation is satisfied. The system discussed in this paper is a second-order system, so the vector field is involutory, which meet the second condition. Similarly, the same method is used to verify the conditions in buck mode. (8) (9) (10) The two conditions are all satisfied in buck mode according to the above equations. Thus the exact feedback linearisation technique can be applied to the non-linear DC microgrid system with DC–DC bidirectional converter in both boost and buck modes. 4.2 Exact feedback linearisation The DC microgrid system with DC–DC bidirectional converter is a second-order system. Then the express of linear system by a coordinate transformation is as follows: (11) where z is the state variable of linear system and v is the control variable of linear system. The corresponding coordinate transformation is as follows: Boost (12) Buck (13) In the new coordinates, the control variable of original non-linear system can be expressed as (14) For the state (14), the state feedback control law can be assumed to be as follows: (15) where and are the feedback coefficients of corresponding feedback variable. The control block diagram of state feedback linearisation is shown in Fig. 5. Fig. 5Open in figure viewerPowerPoint Control block diagram of exact feedback linearisation 4.3 Design of feedback coefficients The value of feedback coefficient will affect the stability and dynamic performance of the whole system. There are many methods to design the state feedback coefficient of linear system. The feedback coefficients are designed by the quadratic optimal control theory in [21]. The coefficients need to be adjusted and its calculation is difficult. A method of placing poles to the expected place to determine the coefficients is proposed in this paper. If the linear time-invariant system (16) is completely controllable, the poles of closed-loop system can be placed to any places by state feedback. The expression of closed-loop system is as follows: (17) The pole of closed-loop system is determined by the eigenvalues of matrix . Thus as long as the system matrix , input matrix and the expected place of closed-loop poles are determined, a suitable feedback gain matrix can be obtained to place poles to the expected place. It is a complex problem to select the expected pole of closed-loop system. The poles of second-order underdamping system is (18) where is damping ratio of system and is oscillation frequency of undamped system (19) (20) where is the overshoot of system and is the settling time. Through the above formulas, the place of expected pole can be determined according to the required dynamic quality indicators. The state equations of the linear system through state feedback linearisation are as follows: (21) (22) (23) Suppose (24) (25) It is selected through adjustment the expected poles It can be seen that the DC microgrid system is completely controllable. The DC microgrid system can be linearised through state feedback linearisation technique. Set the state feedback matrix (26) (27) command It can be seen that and . Thus the control variable of linear system is shown as (28) Boost mode: (29) Buck mode: (30) From (14), the control variable of original non-linear system is as follows: Boost mode: (31) Buck mode: (32) The express of control variable of non-linear DC microgrid system is a complex non-linear function. 5 Simulation results The simulation parameters are shown in Table 1 and the feedback coefficients . When t = 1 s, the power of CPL is increased by 50 kW, the DC bus voltage increases, thus the DC–DC bidirectional converter operates in buck mode to maintain the power balance of DC microgrid. The exact feedback linearisation strategy is applied in the converter. The simulation result of DC bus voltage is shown in Fig. 6. Fig. 6Open in figure viewerPowerPoint Bus voltage in buck mode Another state is that when t = 1 s, DC load is increased by 50 kW and the DC bus voltage decreases, thus the DC–DC bidirectional converter operates in boost mode and the energy storage units discharge to maintain the power balance. The exact feedback linearisation strategy is applied. The simulation result of DC bus voltage in boost mode is shown in Fig. 7. Fig. 7Open in figure viewerPowerPoint Bus voltage in boost mode The feedback linearisation technique can achieve the stability of DC microgrid system with bidirectional converter in both boost and buck modes from the simulation results. 6 Conclusion The stability strategy of DC microgrid was discussed and the exact feedback linearisation technique was implemented on the DC–DC bidirectional converter. The conclusions obtained from the above analysis and simulation are as follows: the analysis based on Lyapunov method provided stability conditions of DC microgrid with CPL. However the stability conditions are only valid near steady-state operating point. The exact feedback linearisation technique was applied on the DC–DC bidirectional converter. An advantage of this method is its relatively simple control. The control variable of non-linear system can be regulated by changing the control variable of linear system. 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