1 Modeling, Analysis and Control of an Isolated Boost Converter for System Level Studies Bijan Zahedi, Student Member, IEEE, and Lars E. Norum, Senior Member, IEEE Abstract-- This paper performs a modeling of an isolated full bridge boost converter that suits the system level studies of power electronic-based systems. For system level studies, in which large signal models must be developed, linearized average value models are invalid. Large signal analysis can be carried out by nonlinear State Space models. In this paper generalized state space averaging method is adopted to include ripple modeling for higher accuracy. The obtained model is compared to a number of other averaging methods and also verified by the detailed switching model. Nonlinear control is carried out by feedback linearization approach. Large signal stability and zero steady state error of the closed loop system are validated by simulation results of the detailed and generalized state space models. Index Terms-- boost converter, feedback linearization control, generalized averaging method, large signal analysis, state space representation. I. INTRODUCTION n essential step to study, design and verification of Amodern power systems is modeling and simulation of power electronic modules within the system. With the development of simulation tools the power electronic devices have been readily modeled by their detailed switching behavior in a number of standard software packages. However, the detailed switching models need very small time steps during the simulation that leads to excessive simulation run time especially for large systems. A solution to the above challenge is to use average value modeling (AVM) which approximates the system by averaging over the switching period. This makes the simulation orders of magnitude faster than that of the detailed model. Small signal studies can be carried out by linearized AVM, which models the module by a few Transfer functions [4]. Since classical control theory is applicable to Linearized AVM, this modeling approach facilitates the controller design [5]. However, for system level studies, in which large signal models must be developed, linearized AVMs are invalid [1]. Large signal analysis can be carried out by nonlinear AVM or nonlinear State Space AVM (SSAVM). State space averaging has been successfully implemented for converter modeling, but its limitation is This work is supported by the Norwegian Research Council and Det Norske Veritas. B. Zahedi is with Power Engineering Department, Norwegian University of Technology, Trondheim, Norway (e-mail: bijan.zahedi@ntnu.no). L. E. Norum is with Power Engineering Dept., Norwegian University of Technology, Trondheim, Norway (e-mail: norum@ntnu.no). neglecting the ripples normally produced in power electronic circuits. In order to consider ripples, the generalized averaging method is used [3]. This paper provides a generalized state space averaging model for an isolated boost converter. The model is compared with some other modeling methods, and then validated by the detailed switching model. To control the converter a nonlinear control approach must be adopted, if the analysis is done on system level. This paper proposes feedback linearization method that is applied to the GSSAM and also the detailed model. Finally the modeling and control performance are validated by simulation results of detailed switching model. II. CONVERTER MODELING Fig.1 shows the schematic diagram of an isolated boost converter under study with the specifications in Table I. The circuit differential equations are shown by (1) 1 1, 1 1, 1, where, Fig. 1- the topological circuit of the boost converter under study 1 To analyze such a circuit a number of software package are available. However, due to using the detailed switching models they are too slow for analysis of rather large power electronic-based systems. Therefore, some methods are used to provide appropriate models which we discuss here. A. Average value modeling To study the dynamic behavior of converters one important approach is deriving the small signal average value model. It is performed by taking the average from the differential equations of (1) over switching period intervals [8]
2 1, 2 1, where in equation (2) is the average of variable x over the switching period. Variable d is the duty cycle of the switching signal, and d =1-d. Based on (2) the small signal average model will be obtained as (3), according to which a small signal ac equivalent circuit can be developed (Fig. 2)., 1, 3 where the capital letter notations are the design operating point values. Fig. 2- small signal ac equivalent circuit for the isolated boost converter The converter can also be shown in the block diagram form which is suitable for linear control design. In this case different transfer functions should be deduced from the small signal equivalent circuit. A comprehensive block diagram for the converter small signal model is shown in Fig. 3. These small signal models are valid provided that the system variables don t exceed the linear range around the operating point. However, in design and analysis of a large system usually it is required to study large variations. Therefore, large signal models should be derived for such studies which are nonlinear. The state space average model can also be expressed in a nonlinear form to be appropriate for system level analysis. In order to model the ripples for higher accuracy here we analyze the generalized state space average modeling. Then in the following part we will compare the different averaging methodologies with generalized state space averaging method. B. Generalized average modeling Generalized averaging is based on the idea that a variable can be approximated by Fourier series (4). In order to provide the generalized average model, we need to calculate the coefficients of Fourier series (4) for the circuit state variables based on (5). To get better precision one can take into consideration higher order harmonics i.e. higher n., 4 where is the Fourier coefficient for k th order harmonic, and is obtained from 1, 5 where In this paper first harmonic approximation is performed, which means setting n = 1. The Fourier coefficients are complex values which can be defined by (6) and (7). [1],,,,,, 6 7 The circuit state variables can be written in terms of the above Fourier coefficients by Fig. 3- Block diagram for the small signal model of the isolated boost converter based on transfer functions The small signal average value model of the converter can be also represented through state space equations. The details of deriving such a state space model are given in [8]. 2 cos 2 sin, 2 cos 2 sin, 8 9 Taking the Fourier transform of (1), replacing (6) and (7) into it and substituting the Fourier coefficients of u(t)
3 2 1;, 1 a new set of state space equations are obtained whose state vector is X = [x 1 x 2 x 6 ] T. 11 V in OPEN LOOP RESPONSE TO INPUT VOLTAGE STEP CAHNGES 15 1 5 State space matrices A and B for the resulted state space model can be seen in (12). Providing a time-invariant large signal state space model, software packages like Matlab/Simulink can be used to implement and control such a nonlinear model. In order to investigate the explained methods, the models are simulated in Matlab/Simulink. The simulation results are illustrated in Figs. 4-6. Figs. 4 and 5 show the open loop responses of the models to step changes of input voltage and load respectively when the duty cycle is at the design value. All models at this condition comply well with the detailed switching model. Fig. 6 shows the open loop time response of the above mentioned methods to duty cycle step changes. It can be seen that at the beginning, since the circuit is operating on the design duty cycle, all the models comply with the detailed switching model. The only difference at the first part is about the ripple modeling. The generalized methods are modeling the first order harmonic and are following the ripples of the actual wave forms; while the averaging methods are following only the DC value of the wave forms. As soon as a step change takes place on the duty cycle, the small signal averaging models deviate from the detailed switching model. A certain error can be seen at the steady state for the small signal (linear) models. However the generalized state space model follows the detailed model with a good precision and shows the most accurate behavior. TABLE I DESIGN PARAMETERS OF THE CONVERTER UNDER STUDY Fig. 4- Open loop response to input voltage step changes by different models Fig. 5 - Open loop load step response by different modeling methods Parameters Values Parameters Values 62 R 19.22 (Ω) V in 1 C 267 µf D.516 L 516 µh f sw 2 (khz) n 3 R L 5 1 5 time (sec) 3 2 OPEN LOOP RESPONSE TO LOAD STEP CHANGES 1 5 1 5 (12)
4 (v) OPEN LOOP RESPONSE TO DUTY CYCLE STEP CHANGES.8 detailed model AVM.6 SSAVM Generalized SS.4.4.5.6.7.8.9.1.11.12 4 2.4.5.6.7.8.9.1.11.12 8 6 4.4.5.6.7.8.9.1.11.12 Fig. 6- Comparison of Open loop duty cycle step responses of different models III. NONLINEAR CONTROL METHODOLOGY The straight forward way to control the converters is to use the small signal transfer functions and design the controller based on linear control theory. However, for large signal models as is required for the system level studies a nonlinear control method should be adopted. There are numerous nonlinear methods for converters control that are utilized in the literature such as adaptive control, robust control and state feedback control. This paper aims at utilizing feedback linearization approach for the generalized state space average model. Feedback linearization method takes state variable feedback from nonlinear systems in such a way that a linear closed loop system will be obtained. The condition of using this method is that the state feedback can be expressed in the canonical form of (13). 13 In converters these state feedbacks are normally inductors currents and capacitors voltages. If we pay close attention to the generalized state space equations for x 5 and x 6 (12) though there are sinusoidal terms in the equation, they are multiplied by x 1 to x 4 which have small amplitude compared to x 5 and x 6. In addition x 1 to x 4 are oscillatory terms with an average of zero over a switching period. So those nonlinear terms are eliminated in the feedback loop. Therefore, this method is applicable to our model. This will be also proved by the closed loop simulation results later on. In other words since the target of control is the average value of currents and voltages, it is simpler to neglect the ripples in the control system design. This means the controller design can be done for a nonlinear state space average model which is thoroughly carried out in [9]. According to given explanations the state equations can be approximated with Taylor expansion of matrix A(d) and form the canonical form of 14 With functions f(x) and g(x) as follows ; 15 16 To build the feedback loop, the first step is to define function h(x) in equation (13) which determines the feedback term. It can be shown that the choice of output voltage as the variable of h(x) leads to a positive feedback and cannot support a stable operation [9]. So the inductor current is chosen as the state feedback variable. This will lead to a stable zero dynamics which is acceptable. Furthermore, in order to eliminate steady state error on output voltage an additional integrative state variable (x 7 ) is defined into the loop. Therefore, where, 17 This additional state provides an integral control on the output voltage to provide zero steady state error. Now nonlinear feedback can be applied by Where.,,, 1, And Lie derivatives along f and g are calculated by 18 19 2 21,. 22,. 23 Where 1 Fig. 7 shows the block diagram of the control system.
methodology. The closed loop time responses for large step changes of the input voltage and load are studied by simulation. The simulation results show that the control method supports the large signal stability of the converter operation and steady state regulation of the output voltage. 5 Fig. 7- Block diagram of the system with feedback linearization control Setting the feedback parameters k p and k ip on 1 and 15 respectively, the closed loop simulation results are obtained as shown in Fig. 8 and 9. The closed loop response to input voltage step changes are illustrated in Fig. 8 for both the detailed model and GSSAM. Fig. 9 shows the closed loop response to load step changes for both the detailed model and GSSAM. The control variable as shown in the block diagram of Fig. 7 is the duty cycle. The controller varies the duty cycle so that the output voltage is kept fixed as can be seen in Figs. 8 and 9. The simulation results prove the validity of the model and the control method as well. V in CLOSED LOOP RESPONSE TO INPUT VOLTAGE STEP CHANGES 15 1.8.6.4 4 3 2 1 8 6 Fig. 8- Evaluation of the model and the proposed control method by detailed model simulation of the closed loop system response to large signal input voltage step changes IV. CONCLUSION Detailed model GSSAM A large signal modeling based on generalized averaging method is provided for an isolated boost converter. The obtained model is compared to small signal average value models and detailed switching model. The advantages of using such a model over detailed switching model and other averaging methods are discussed and its validity is proved by open loop simulation results of the detailed model. Feedback linearization method is chosen as a nonlinear control R L I out 3 CLOSED LOOP RESPONSE TO LOAD STEP CHANGES 2 Detailed model GSSAM 1.6.4 4 3 2 1 6 4 2 8 6 Fig. 9- Evaluation of the model and the proposed control method by detailed model simulation of the closed loop system response to large signal load step changes V. REFERENCES [1] Emadi, A.; Modeling and analysis of multiconverter DC power electronic systems using the generalized state-space averaging method, IEEE Transactions on Industrial Electronics, vol.51, no.3, pp. 661-668, June 24. [2] Soltine, J.; Li, W.; Applied Nonlinear Control, Prentice Hall [3] Sanders, S.R. ; Noworolski, J.M. ; Liu, X.Z. ; Verghese, G.C. Generalized averaging method for power conversion circuits, IEEE Trans. on Power Electronics, 1991 [4] Il-Yop Chung, et al.; Integration of a bi-directional dc-dc converter model into a large-scale system simulation of a shipboard MVDC power system, IEEE Electric Ship Technologies Symposium, pp. 318-325, April 29 [5] Cho, H.Y.; Santi, E.; Modeling and stability analysis in multiconverter systems including positive feedforward control, IEEE IECON Conference, pp. 839-844, Nov. 28 [6] Chiniforoosh, S., et al.; Definitions and Applications of Dynamic Average Models for Analysis of Power Systems, IEEE Trans. on Power Delivery, Vol. 25, no. 4, pp.2655-2669, Oct. 21. [7] Sh. Dingxin, XIE Yunxiang, WANG Xiaogang; Optimal Control of Buck Converter by State Feedback Linearization, Proceedings of the 7 th World Congress on Intelligent Control and Automation, June 25-27, 28, Chongqing, China [8] Robert W. Erickson; Fundamentals of Power Electronics, Kluwer Academic Publishers, 1999. [9] Ramón Leyva, Pedro Garcés, Javier Calvente, and Luis Martínez Salamero; Feedback Linearization Control Applied to a Boost Power Converter [1] Mahdavi, J.; Emaadi, A.; Bellar, M.D.; Ehsani, M.;, "Analysis of power electronic converters using the generalized state-space averaging approach," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,, vol.44, no.8, pp.767-77, Aug 1997 [11] Davoudi, A.; Jatskevich, J.; De Rybel, T.;, "Numerical state-space average-value modeling of PWM DC-DC converters operating in DCM and CCM," IEEE Transactions on Power Electronics, vol.21, no.4, pp. 13-112, July 26