A Novel Simulation Method for Power Electronics: Discrete State Event Driven Method

Transcription

1 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 1, NO. 3, SEPTEMBER A Novel Simulation Method for Power Electronics: Discrete State Event Driven Method Boyang Li, Zhengming Zhao, Yi Yang, Yicheng Zhu, Zhujun Yu (Invited) Abstract In the analysis of power electronics system, it is necessary to simulate ordinary differential equations (ODEs) with discontinuities and stiffness. However, there are many difficulties in using traditional discretetime algorithms to solve such equations. Kofman and others presented the quantized state systems (QSS) algorithm in the discrete event system specification (DEVS) formalism. The discretization is applied to the state variables instead of time range in QSS. QSS is efficient to solve ODEs, but it is difficulty to be used when simulating actual power electronics systems with controller s and other events. Based on the idea of this numerical algorithm and discrete event, a Discrete State Event Driven (DSED) simulation method is presented in this paper, which is fit for simulation of power electronics system. The method is developed to deal with nonlinearity, stiffness and multitime scale of power electronics systems. The DSED simulation method includes event definition, module seperation and modeling, eventdriven mechanisms, numerical computation based on QSS, and some other operations. Simulation results verified the effectiveness and validity of the proposed method.. Index Terms Discrete state event driven, simulation method, discontinuities and stiffness, power electronics systems I. INTRODUCTION UMERICAL simulation is of great importance in the Nanalysis, design and control of power electronics systems. These systems can be described by ordinary difference equations (ODEs) in mathematics. Due to their intrinsic nature, power electronics systems have characteristics of nonlinearity, strong stiffness, and multitime scale. When using classical numerical methods to simulate these systems, we may encounter many difficulties. These classical methods as Euler, Backward Euler, RungeKutta, Adams, etc., and their variation versions are all based on the discretization of time. In these methods, values of state variables at next timelevel are approximated by a polynomial extrapolation. It means that state variables are updated simultaneously. Every state variable will be calculated in each step. These methods are not fast or efficient when simulating power electronics systems. Firstly, since power electronics systems are usually multitime scaled, there are both fast and slow state variables in them. In order to capture the trajectories of the fast state variables, we need to use This work was supported by a grant from the National Nature Science Foundation of China(No , No ) Boyang Li is with the Department of Electrical Engineering, Tsinghua University, Being, , China.( @163.com). small time steps, which are also required to satisfy stability conditions in the case of approximating fast state variables. Thus it will increase the computation cost to adjust simulation step. Secondly, power electronics systems always contain many discrete events, which leads to discontinuities in mathematics, and the iterations to find points of discontinuity will consume a large amount of computation time. Furthermore, if the method is not well chosen, it will lead to spurious oscillations or divergence in the simulation, which are not expected. Therefore, it s an important topic to find an accurate, fast and highly efficient simulation method for power electronics systems. As early as 1970s, Bernard Zeigler in university of Arizona worked on modeling of discrete event systems and developed the Discrete Event System specification [1]. Compared to discrete time methods, discrete event methods can not only control numerical errors easily, but also reduce the computation amount a lot, especially for systems with discontinuities. Moreover, due to its asynchronism, we can implement distributed parallel simulation to save the total computing time. In 2001, Ernesto Kofman et al. proposed an algorithm based on DEVS, named Quantized State Systems (QSS) algorithm [2]. Different from classical discrete time algorithm, in QSS, state variables are quantized first, then the time which takes from a quantized state to next is calculated to push the simulation. Because QSS is based on DEVS, the errors can be controlled by specifying the quantum, and computation amount can be reduced significantly. Also distributed computation can be easily conducted by applying QSS [3][6]. However, there are still many difficulties when applying QSS algorithm to actual power electronics system with controllers and other events. It s very worthwhile to enrich this numerical algorithm. Based on the idea of discrete event, we expand the QSS algorithm into an integrated simulation method discrete state event driven (DSED) method for the simulation of power electronics systems in this article. The method is developed to deal with nonlinearity, stiffness and multitime scale of power electronics systems. The DSED simulation method include event definition, module seperation and modeling, eventdriven mechanisms, numerical computation based on QSS, and some other operations. These procedures make up a complete simulation method for power electronics systems together. The article is organized as follows: Section Ⅰ gives an introduction of background. Section Ⅱ proposes DSED method and explains the architecture of DSED in detail. Properties of DSED are analyzed in Section Ⅲ.Section Ⅳ shows simulation results of a SST device, and the effectiveness and validity of the

2 274 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 1, NO. 3, SEPTEMBER 2017 proposed method are verified. Finally, the conclusion is drawn in Section Ⅴ. II. DISCRETE STATE EVENT DRIVEN METHOD QSS is an numerical algorithm implemented in the framework of DEVS, and is conducted on the base of quantization of state variables. Consider the following ODE system: x() t = f( x(), t u ()) t (1) where x =(x 1, x 2,, x n ) is the state vector, u =(u 1, u 2,, u m ) is the input vector. QSS method approximates this system by: x() t = f( Q(), t u ()) t (2) where Q =(Q 1, Q 2,, Q n ) is the quantization of state vector and is obtained by the following hysteretic quantization function: Qj( t ) qj xj( t) Qj( t ) qj Qj() t = Qj( t ) qj Qj( t ) xj() t ε j (3) Qj ( t ) others where q j is the quantum, ε j is the hysteresis width. The quantum plays a role similar to the time step in classical discrete time methods. The quantums are taken as threshold for change of state variables, Q in the equation(2) is updated only when the change of any state variable reaches its quantum. We must point out that explicit QSS method as QSS1 cannot deal with the stiffness efficiently. However, implicit QSS method as LIQSS, BQSS proposed later can make up the shortfall [7][8]. In spite of advantages as we mentioned before, there are many difficulties for QSS algorithm when dealing with actual power electronics system. Firstly, state equations of systems always change according to the discontinuities, thus we must know how and when they change, which means we cannot apply QSS algorithm directly. Secondly, input events systems are hugely enriched in power electronics systems, so we must identify and handle these events correctly. Thirdly, many problems encountered in power electronics systems, such as cascade connection, cannot yet be solved efficiently by QSS algorithm. So we expand this algorithm to an integrated discrete state event driven (DSED) method, in which eventdriven mechanisms and other procedures are designed and included to solve problems above. A. Definition of Event The Power electronics system is a typical hybrid system, which is a mixture of large time scale and small time scale events, discrete and continuous events. The whole system can be regarded as a piecewise continuous system separated by discontinuous points, which correspond to discrete events. Between adjacent events, the mathematic model of the system can be described by the following state equation: x = AQ Βu (4) where Q is the quantization function of state variables, u is the input including both active and controlled sources, A is the Jacobi matrix and B is the input matrix. Any event, no matter happens in large or small time scale, can be described by the changes in A, B, Q, u when reflected in mathematic description. Common events in power electronics systems can be divided into two categories according to whether they trigger computation initiatively: 1) Active Event: Time of these events is known or can be calculated before they happen. This kind of events, including the change of any state variable reaching the threshold, inputs or loads changing, open or closed loop system sending switch instructions and so on, triggers computation initiatively. 2) Passive Event: Time of these events is unknown before they happen and is only determined by the state of the whole system. The most typical characteristic of these events is that they only happen when system state changes. So they are not considered to be trigger sources and only detected after active events occur. This kind of events includes the turnon or turnoff process of uncontrolled devices, transient process of switches, etc. TABLE I COMMON EVENTS IN POWER ELECTRONICS SYSTEM Variable(s) Category Event changed in state equation Change of any state variable s value reaches its threshold value Q Active event Inputs change U Loads change A, B Open/Closedloop control systems send switching instructions. A, B Passive event Uncontrollable devices turn on/off. A, B Switching devices enter transients. A, B Among all types of active events, closed loop control system s action is special to some degree. In power electronics system, the power transformation is usually completed via PWM control. In each control cycle, the sampling is completed at first, followed by the calculation of duty ratio to determine the turnon and turnoff time of devices according to specific control algorithm. Therefore, in eventdriven simulation, the closed loop control system should also contain the corresponding three events: sampling, turnon action and turnoff action. Sampling is timely triggered by the sampling module. Then, according to duty ratio and carrier waveform, the timetable of switching action in the present cycle can be obtained, and turnon events and turnoff events are performed successively until next control cycle. The order of events can be controlled and ensured by a state machine in program. B. Module Seperation and Modeling Cascaded systems are common in practical use. In a cascaded system, different stages undertake different tasks and cooperate with each other to complete the goal of power transformation. However, cascaded connection also brings the following

3 LI et al. : A NOVEL SIMULATION METHOD FOR POWER ELECTRONICS: DISCRETE STATE EVENT DRIVEN METHOD 275 challenges for simulation: 1) Combination amount of switch states is huge and modeling is very complex. The total amount of combinations will be the product of combination amount at each stage. When the multilevel modules such as MMC is contained, the total combination amount will be enormous, which will occupy too much storage space and make modeling very complex. 2) Repeated computation influences simulation efficiency. Any event happening in any stage leads to the change of the whole state equations, and recalculation of derivatives of all state variables will bring too much repeated computation. Therefore, in DSED method proposed in this article, the system is not modeled as a whole like conventional methods. In contrast, each stage is encapsulated as an independent module. Computation is performed independently in each separate module, and data exchange between modules is implemented via interfaces. The following example of a typical backtoback rectifierinverter cascaded system illustrates the module seperation, the modeling method and data exchange process in our simulation method. Fig.1 (a) shows the topology of a backtoback rectifierinverter system, which is composed of the input source, rectifier, capacitor, inverter and load. The capacitor divides the system into rectifier stage and inverter stage, as shown in Fig.1 (b) and Fig.1 (c), respectively. For inverter stage, the influence of the rectifier stage is equivalent to a voltage source, whose value is equal to the terminal voltage of the capacitor. While for the rectifier stage, the influence of the inverter stage is equivalent to a current source, whose value is equal to the input current of the inverter. Thus, the data interfaces are made up of output voltage of the previous stage and input current of the later stage. Every time when the change of any state variable reaches its corresponding threshold value, Q 1 (k) and Q 2 (k) (subscript 1 represents the previous stage and subscript 2 represents the later stage) are calculated first. Then, interface data I s1 (k) and U s2 (k) can be updated because I 2 (k) and U CH (k) are linear combination of Q 1 (k) and Q 2 (k) : I = I = f U Uc f ( k) ( k) ( k) ( k) s1 2 2 ( Q2 ) ( k) ( k) ( k) ( k) s2 = H = 1 Q1 ( ) where f 1 (k) and f 2 (k) are output equations for each stage, determined by the topology and switching states. Substituting I s1 (k) and U s2 (k) to the state equations, it can be derived that: ( k) ( k) ( k) ( k) ( k) U s1 x 1 =A1 Q1 B1 ( k) I s1 ( k) ( k) ( k) ( k) ( k) x 2 =A2 Q2 B2 Us2 It should be pointed out that the interface variables are linear function of Q 1 (k) and Q 2 (k), and Q is updated when the change of any state variable reaches its threshold, thus the interface variables need to be updated only at this moment. In other cases, they remain the same. So unnecessary repeated computation is avoided. (5) (6) U s Rectifier Inverter Load C H (a) Topology of a backtoback rectifierinverter system U s Rectifier C H (b) Equivalent topology of the first rectifier circuit I 2 I s1 U s2 Inverter Load (c) Equivalent topology of the second inverter circuit Fig. 1. Module seperation of the rectifierinverter system. This seperation and modeling method can be easily extended from twostage systems into multistage systems, which shows its good scalability. Under this modeling method, the combination amount of the whole system becomes the sum of each stage s amount rather than their product, which will greatly reduce time and space cost of modeling. In addition, when an event occurs, only the state equations of corresponding stages needs to updated, thus repeated computation can be avoided. What s more, in the simulation of largescale network, this modular modeling method also helps parallel processing. It naturally divides the computation tasks by modules, and computation is performed independently by parallel with only two interface variables to be communicated between tasks, which can greatly improve the efficiency of parallel computing. C. PostProcessing of Derivatives DSED method has large space for manual intervention, thus numerical algorithm can be more tightly combined with power electronics system. Based on the electrical stress in power electronics system, we can determine the maximum amplitude of the derivates. In some rigid systems, it may occur that the derivative modulus of a certain state variable is too large so that the simulation step is very small. Multiple occurrence of such situation will result in high frequency unexpected oscillation in simulation result. Apart from backward algorithm which enhances the numerical stability [9], another simple yet effective way to solve this problem is to limit the derivate modulus [10]. State variables in power electronics systems are choosen from the inductor s voltage and the capacitor s current, and the physical meanings of their derivatives are corresponding to the electrical stress. Therefore, the potential maximum electrical stress in the system can be used to limit

4 276 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 1, NO. 3, SEPTEMBER 2017 derivates modulus. The goal is to effectively suppress high frequency oscillation while calculation accuracy can also be ensured. Thus, a upper boundary x max is set to limit derivative modulus, and x max is determined by the electrical stress of the converter. In specific, x max is related to the voltage or current peaks, equivalent capacitance and inductance values: U I x (7) max max max = β max{, } Lmin Cmin where U max and I max are the maximum peak values of capacitor s voltage and inductor s current, and β is the limiting factor. To ensure simulation accuracy, β must be bigger than 1. Increase of β can reduce the impacts brought by limiting derivative amplitudes, but it also weakens the ability to suppress oscillation. So trial and error is necessary to obtain a proper value of β. D. EventDriven Mechanisms The main difference between DSED method and discrete time simulation method is that DSED method is driven by discrete events rather than time. In DSED, computation is only performed when a discrete event occurs, and the process of simulation is entirely pushed by the handling of discrete events. That is to say, the core of DSED is to update A B Q u according to different event types listed above and find the time when the next event occurs. Assuming that the system has n stages, the system is divided into several submodules based on the modeling method proposed above. Then the state variables are determined for each submodule and the Jacobi matrix and input matrix of submodules under different situations of switching states are restored. Denote x as the jth state variable of the ith submodule and discretize it with the Q function shown in equation (3). The value range of x is equally divided into m intervals. Denote q as the length of the discrete interval, and then the quantization function Q is a piecewise constant function. The function value on each subdomain is the upper bound of each interval: Q ( t ) q x ( t) Q ( t ) = q Q ()= t Q ( t ) q Q ( t ) x () t = ε Q ( t ) Otherwise Denote x i as the state variable vector, Q i as the vector of Q function and x i as the derivative vector of the ith module, and the external input is known at any time. So, if the initial values of x i, Q i and the initial switching states are given, we can get the initial state equations of every module, whose initial Jacobi matrix A i and output matrix B i are determined by: A = B i gai Si i = gbi Si ( ) ( ) (8) (9) where S i denotes the switching state of the ith stage, including both controlled and uncontrolled switches, and ga i as well as gb i are determined by the specific topology of the ith module. Then x i can be calculated by equation (5) and (6) and modified by the limiting method proposed above. According to the value of x i, the time t for the state variable x to reach its threshold can be calculated: t Q q x x = Q ε x x x >0 x <0 x = 0 (10) The next work is to initialize two vectors: t q = { t q1, t q2 t qn }, which is used to record the shortest time needed for the change of a state variable to reach its threshold value in each stage, and t c = { t c1, t c2 t cn },which is used to record the time intervals between the two actions from control systems in each stage. t ci is the first time when controllers in the ith stage act (usually it is the first sampling time), and t qi can be obtained by the results of equation (10): t = min{ t, t... t...} (11) qi i1 i2 (k) (k) During the simulation process, t c and t q are updated at each calculation step. When initialization is finished, the time of the next calculation point is just the time when the first upcoming active event occurs. The time interval t can be derived by: t = min{min{ t }, min{ t },Δ t } (12) q c sl (k) where t sl is the shortest time interval between the upcoming other active events (like input or load changes) and present calculation step. Then the active event occurs, simulation time is pushed from t to t (1) and the values of state variables are updated: (1) t = t t x = x x ( i=1,, n) (1) i i i t (13) Now the simulation process has made a step forward, and the present event needs to be identified and processed according to its type: 1) If the present event is triggered by a controller, and denote s as the number of the module where the action occurs, then how to deal with this event depends on its specific action. If the action is sampling, control algorithm will be performed to calculate the duty ratio. Based on it the turnon and turnoff action timetable of switches in this cycle can be obtained (the end of the timetable is just the

5 LI et al. : A NOVEL SIMULATION METHOD FOR POWER ELECTRONICS: DISCRETE STATE EVENT DRIVEN METHOD 277 beginning of the next control cycle and the header value of the timetable is recorded as t cs_head ), and the switching states remain the same. Otherwise, which means the action is to turn on or turn off switches, S s should be updated. The header pointer of the timetable should also be moved one bit backward and t cs_head is updated as the new header. When the event has been processed, the new value of the t c (1) can be correspondingly updated: Events Definition Start Module Separation& Modeling Initial Calculation Initialization ( ) (1) Δ tc j j s Δt c ( j) = (14) tcs _ head j = s k=1 Since there are no state variables reaching their threshold values, Q (1) s and the interfaces variables will not change (1) after this event. Substituting the new switching state S s to equation (9), the new A (1) s and B (1) s can be calculated. Then the new derivative vector x s (1) of the sth module is recalculated according to the equation(6), while derivative vectors of other modules remain the same. Finally, the new t (1) q can be updated by equation (10) and equation (11), which means the new t (1) q and t (1) c are all obtained till now. Identify the Type of Current Event Handle the Event and Update State Equations Numerical Calculation Calculate the Time When the Next Event Occurs k=k1 2) If the present event is the change of a state variable or some state variables reaching their threshold values, the Q function vectors of state variables should be updated firstly: Update Simulation Time and State Variables Simulation Ends No Q = Q _ fuction ( x ) ( i = 1, 2,3...n) (15) (1) (1) i i Then, whether there is a passive event happening is checked. If not, the Jacobi matrix and input matrix of each stage remain the same. Otherwise, they need to be updated like situation(1). According to the new Q (1) (1) i, A i and B (1) i, x i (1) can be recalculated and substituted to equation (10) and equation (11) to update t (1) q, while t (1) c remains the same: t = t (16) (1) c c 3) If the present event is input or load changing, then only the state equations of corresponding modules need to be updated. All we need to do is to recalculate the derivative vectors and update t q (1) and t sl (1), while t c (1) still remains the same. When the present event has been processed, the happening time and type of the next event (k=2) can be obtained according to the new t (1) q, t (1) c and t (1) sl.then the simulation process is pushed forward by repeating steps above until the simulation time reaches terminal. The flow chart of the DSED method is illustrated in Fig.2. End Fig. 2. Flowchart of the DSED method. Yes III. PROPERTIES OF DSED DSED method is based on the idea of discrete event, whose core is eventdriven mechanisms. It has some advantages over classical timedriven method: 1) DSED method is a variablestep method in nature. The time from one quantized state to next quantized state is determined by the derivative of the state variable and the relevant quantum. Usually quantum is a given value, when the derivative is large, then the time step calculated is a small value. On the contrary, small derivative determines a large time step. Thus, no extral computation is cost in finding a suitable step. And from this point of view, we can reduce the computation amount. What s more, in power electronics systems, a state variable usually experiences steady states and transient states alternately. By using DSED method, we will get large time steps in steady states and small time steps in transient states. Thus we can capture the rapid changes in transient states and reduce the computation amount in steady state. 2) DSED is driven by event with no iteration. All kinds of events, no matter active events or passive events, lead to discontinuities in simulation, while traditional timedriven

6 278 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 1, NO. 3, SEPTEMBER 2017 methods just cannot integrate cross the discontinuity points. Thus, iteration must be operated to find where the discontinuity points are, and usually the most of the computation time is cost on finding discontinuity points when simulating power electronics systems. However in DSED, there is no iteration because DSED is driven by active events and all state variables are approximated by piecewise constant Q function. Time intervals between two adjacent steps can be easily calculated by simple linear operations, which means high efficiency when simulating systems with a lot of points discontinuity. 3) DSED method is naturally asynchronous. Each state variable is updated only at the time its quantized state changes by a quantum. That is to say, every state variable update its values independently and separately. While in discrete time methods, all state variables update their values whenever the simulation arrive a new time level. Considering the typical power electronics systems, there are both fast and slow state variables. If we use a discrete time method to simulate this kind of systems, in order to capture trajectories of fast state variables, we must choose a small time step. Then the slow state variables are also approximated with this small time step. But obviously we can approximate the slow state variables with a larger time step because they change more smoothly. However, if we use the DSED method to simulate these systems, because of the asynchronism, we can approximate the fast state variables with a smaller time step, and approximate the slow state variables with a larger time step. In this way, we can reduce a lot of computation amount. When the size of system increase too large, the contrast of these two kinds of methods will be more significant. 4) It is very convenient to implement DSED method on parallel computers due to its modeling method. Since DSED method divides computation tasks naturally, and allow us to minimize the communication bandwidth between different subsystems. So DSED method is a good candidate for realtime simulation and is fit for the simulation of huge power electronics systems. 5) DSED has excellent mathematical properties because its numerical algorithm is based on QSS. For example, global errors in DSED method can be controlled by quantum, which means solutions will not differ from the exact solutions by an amount proportional to the quantum. In addition, DSED also has good numerical stability. In a word, DSED method can deal with power electronics systems efficiently. Properties stated as before can definitely reduce computation amount and computing time when simulating these systems. IV. SIMULATION RESULTS In this chapter, the proposed DSED simulation method is applied to a 50kW SST and DSED simulation results are compared with that of experiment Simulink simulation to verify the validity and effectiveness of DSED method. A. Topology, Parameters and Control Strategy of the 50kW SST As is illustrated in Fig. 3, the 50 kw SST consists of three modules: an ACDC module, a DCDC module and a DCAC module, whose circuit parameters are shown in Table II. TABLE II CIRCUIT PARAMETERS OF 50KW SST ACDC Module Input threephase voltage (RMS) e a, e b, e c 380V Threephase input inductance L s 2mH Output capacitor C H1, C H2 13.6mF Switching frequency f s(acdc) 4kHz DCDC Module DAB low voltage side capacitor C L 9.4mF Transformation ratio n 2 Switching frequency f s(dcdc) 20kHz Highvoltage side DC link voltage U DABH 700V Lowvoltage side DC link voltage U DABL 330V DCAC Module Output singlephase voltage (RMS) u O 220V Output inductance L O 1.5mH Output resistance R O 0.1Ω Output capacitance C O 94μF Switching frequency f s(dcac) 10kHz Load Module Load Resistance R Load 7.5Ω QA1 DA1 QB1 DB1 QC1 DC1 P ea eb ec QA2 Ls Ls Ls QA3 QA4 DA2 a DZA1 QB2 DA3 QB3 DZA2 DA4 QB4 b DB2 DZB1 QC2 DB3 QC3 DZB2 DB4 QC4 c DC2 DC3 ACDC Threelevel Threephase Rectifier DZC1 Z DZC2 DC4 N CH1 CH2 UDABH QH1 QH3 DH1 DH3 QL1 n:1 QH2 DH2 T QH4 DH4 QL2 DCDC Dual Active Bridge DL1 QL3 DL2 QL4 DL3 CL DL4 QO1 UDABL QO2 DO1 QO3 DO2 QO4 DO3 LO DO4 RO CO DCAC Singlephase Inverter ico uo RLoad Fig. 3. Topology of 50 kw SST.

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10 282 CES TRANSACTIONS ON ELECTRICAL MACHINES AND SYSTEMS, VOL. 1, NO. 3, SEPTEMBER 2017 Yi Yang received the B.S. degree in physics from Tsinghua University, Being, in 2010 and the Ph.D degree in mathematics from Tsinghua University, Being, in He is currently a postdoctor in the Department of Electrical Engineering at Tsinghua University, Being, China. From 2016 to 2017, he carry out his postdoctoral work at the State Key Laboratory of Control and Simulation of Power Systems and Generation Equipments. His research interest includes modeling of power device, simulation of power electronics systems and high performance algorithms.