Volume 114 No. 8 217, 297-37 ISSN: 1311-88 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Design of controller for Cuk converter using Evolutionary algorithm via Model Order Reduction S.Suguna 1, * M. Siva Kumar 2 1,2 Dept. Of EEE, Gudlavalleru Engineering college, E-mail:profsivakumar.m@gmail.com Abstract In this paper, the modelling and design of controller for Cuk converter operating in continuous conduction mode (CCM) is proposed. The Cuk converter is a DC- DC converter, operating in step-up as well as stepdown modes based on a switching buck-boost topology. By using the State Space Averaging (SSA) technique, the mathematical model of this converter is carried out and yields to a fourth order system. The feedback compensator design for higher order system is very difficult. In this proposed paper, the fourth order system is reduced to a second order model using Evolutionary algorithm based Particle Swarm Optimization via model order reduction by minimizing the Integral Square Error (ISE)and the controller designed by this proposed method gives the satisfactory results. Keywords: Model Order Reduction, Cuk Converter, State-Space Averaging, Compensator, Integral Square Error (ISE). 1. Introduction Now a days the switched mode dc-dc converters, which converts electrical voltage from one level to another by using the switching action, are mostly used because of 297
their greater efficiency, lighter weight and small size. The Cuk converter is considered as a series combination of both boost and buck converters. Cuk converter consists of excellent properties like negative output voltage, low output current ripple and switching ripple, capacitive energy transfer, smooth input and output currents. Due to the fourth order characteristics the dynamic response is usually affected, which automatically calls for the limitations in band wih of the closed-loop system. Moreover, to decouple the stages of input and output, big energy capacitors are required by stability, which involves complexity in both the control and theoretical implementation. In order to avoid these problems, the order of the transfer function is reduced then after that controller is designed. A Cuk converter comprises of two inductors, two capacitors, a diode and power switch, hence it is a fourth order system which is nonlinear. A linear model is required to design a feedback controller. The linear model is derived by replacing the switch and diode by small signal averaged model. The state space averaging (SSA) technique [2] to model the power stages is used to obtain the analytical description of Cuk converter which is a linear model. By depending on the control to output transfer function, the output voltage is regulated by using PWM controller [6] is designed. Since the feedback compensator design for higher order system is very difficult, to avoid these complexities, the original fourth order system is reduced to a second order model by evolutionary algorithm via Model Order Reduction [6-17].The proposed paper is organized as, The section 2 consists of State Space Averaging (SSA) technique, section consists 3 of Cuk converter Analysis, section 4 consists of controller design of Cuk converter and simulation results and section 5 consists of conclusion and references. 2. SSA Technique The closed loop system's power stage is a non-linear system, which are quite complex to model and also difficult to predict their nature. So it is preferable to approximate it as a linear one. Bode plot is mostly used to design the compensator in feedback loop for the desired response. For this purpose the state space averaging technique is used. The dc-dc converters which are operating in continuous condition mode have mainly two states, one during the switch is on and other when the switch is off. During the switch on; X = A 1 X+B 1 V d <t<dt (2.1) During switch off; X = A 2 X+B 2 V d <t<(1-d)t (2.2) 298
V = C 1 X during dt (2.3) V = C 2 X during (1-d)T (2.4) The averaged model for the Cuk converter can be produced over above mentioned switching period, the state space equations corresponding to the two states are time weighted and time averaged, resulting in below mentioned equations X = [A 1 d + A 2 (1 d)]x+[b 1 d + B 2 (1 d)]v d (2.5) V = [C 1 d + C 2 (1 d)]x (2.6) 3. Cuk Convert Analysis 3.1 Cuk converter modelling by state space technique: The Cuk converter comprises two capacitors C1 and C2 with equivalent series resistances rc1, rc2 respectively, two inductors L1 and L2 with equivalent series resistances rl1 and rl2 respectively, switch S, diode D and resistance R is represented as load. The converter mainly exchanges the energy between capacitors and inductors to achieve the conversion from one level of voltage to another. From the voltage source, input voltage Vd is applied to the converter circuit through L1.During the ON position of switch S, the current flowing through L1, il1 increases at the same instant the voltage across the capacitor VC1, turns off the diode by reverse biasing it. The capacitor C1, discharges its energy into the circuit C1, C2, L2 and R. During the OFF position of switch S, in order to produce the uninterrupted current the voltage across the inductor L1 will reverse its polarity. The diode D is forward biased, the capacitor C1 is charged by Vd, and the energy stored in input conductor. The load current is supplied by the energy stored in the inductor L2 and also the capacitor C2. Under the assumption that the voltage VC1 is constant, the Sum of the currents il1 and il2 must be equal to zero in the steady state. The relation between Vc and Vd for the ideal converter is given V c = d V d 1 d Where d is the duty cycle. From these equations the output voltage V can be controlled by controlling the duty cycle. The duty cycle of the converter can be varied by using a controller and the circuit can also be made to reject disturbances. 299
3.2 State space equation of Cuk converter: The state space equations for Cuk converter during switch on and off are During switch is ON di L1 di L2 = r L1i L1 + V d L 1 L 1 (3.1) = V C1 L 2 = (r L2+r C1 +r C2 ) i L2 L2 (3.2) ( r C2 r C2 +R -1)V C2 C 2 dv C1 = i L2 C 1 (3.3) dv C2 V C2 (r C2 +R)C 2 (r C2 +R)C 2 = Ri L2 (3.4) When the switch is off di L1 = (r L1+r C1 )i L1 L 1 - V C1 L 1 (3.5) di L2 = (r L2+R r C2 )i C2 + ( L 2 dv C1 dv C2 = i L1 C 1(3.7) = Ri L2 (r C2 +R)C 2 - (3.6) + V d L 1 r C2 R+r C2 1)V C2 V C2 (r C2 +R)C 2 L 2 (3.8) According to the above mentioned equations, we can write the averaged matrices for the steady-state and linear small-signal state-space equations A 1 = r L1 L 1 1 (r L2 +r C1 +r C2 R) L2 1 L 2 ( r C2 R+r C2 1) C 1 (3.9) R (r C2 +R)C 2 C 2 1 (r C2 +R)C 2 3
A 2 = r L1 +r C1 L 1 1 (r L2 +r C2 R) L2 1 L 2 ( r C2 R+r C2 1) C 2 C 1 R (r C2 +R)C 2 (3.1) 1 (r C2 +R)C 2 1 B 1 = B 2 = B = L 1 C 1 = C 2 = C = 1 E 1 = E 2 = E = () 3.3 Transfer function: With the state space matrices defined above, the control to output transfer function can be calculated as G vd = C(SI A) 1 B d + E d (3.11) Where B d = (A 1 A 2 )X+(B 1 B 2 )V d (3.12) Output to input transfer function G vg = C(SI A) 1 B (3.13) X = -CA 1 V d (3.14) 3.4 Particle Swarm Optimization: PSO[5] is an evolutionary algorithm that can be used for solving the nonlinear equations. It is a kind of swarm intelligence that is based on social-psychological principles and provides insights into social behaviour, as well as contributing to engineering applications. In PSO, the velocity and position are randomly chosen for a set of particles. During the start, the initial position is taken as the best position and the velocity is updated. The main purpose of this optimization method is a) A global optimum for the nonlinear system may be found, b) It can produce a many number of solutions, c) There are no mathematical limitations on the formulation of the problem, d) Comparatively very simple in execution and e) Numerically strong. 31
The parameters c1 and c2 determine the relative pull of the pbest and gbest and r1 and r2 helps to stochastically unreliable these pulls. v t+1 ij = wv t t ij + c 1 r 1j (3.15) ᵡj (t+1).g = ᵡj (t) (t+1).g + v j.g (3.16) t p best,i t x ij + c 2 r t 2j [G best x t ij ] With j =1,2,3,..,n and g =1,2,3,.,m Where n = number of particles in the group, m = number of components for the vectors and, t = pointer of iterations (generations), v (t+1) j.g = velocity of g th component of particle j at iteration t, w = inertia weight factor, w = w max w max w min k 1 N 1 (3.17) Where K=current iteration number and N= maximum number iterations. c 1, c 2 = acceleration constant and r 1, r 2 = random numbers between and 1 ᵡj (t).g =current position of g th component of particle j at iteration t, pbest j = Best previous position of j th particle., gbest = Best particle among all the particle in the population. In Table 1, the typical parameters for PSO optimization routines, used in the present study are given. TABLE1 Typical parameters used by PSO Name Value(type) Number of generations 1 Population size 5 Maximum Particle 2 velocity Epoch 1 Termination method Maximum Generation 3.5 Performance index: The arrangement of the lower order system is established by the performance index principle. In the present study, PSO is applied to minimize the Integral square error between the transient part of step response of original system. ISE is frequently employed for the performance evaluation because of ease of achievement. 32
ISE = y t y r (t) 2 (3.18) Mathematically, the integral square error can be represented as ISE = y t y r t 2 M i= (3.19) where, y (t) represents the step response of higher order yr (t) represents the step response of reduced order model at the t th instant in the time interval t M, where M is to be chosen. 4. Control of Cuk Converter 4.1 PWM feedback control: The PWM control for the converter [3-4] is shown in fig 2(a). The voltage at the output V is compared with Vref. The error voltage Ve between V and Vref, is fed to the compensator, Gc(s) to generate the control signal Vc, and then to be compared with saw tooth voltage VM by using PWM comparator. As shown in Fig. 2(b), the Switch S is turned on when the control signal Vc is larger than saw tooth voltage Vsaw, and it is turned off when VC is smaller compared to Vsaw. If V is varied, the feedback control will adjust Vc and then duty cycle until V is again equal to Vref Fig. 2.a PWM control Fig. 2.b waveform of PWM comparator Fig 3 shows the closed loop block diagram of the converter. Gvd represents the power stage transfer functions. The PWM comparator transfer function is given by FM = 1 V M Where VM is the amplitude of saw tooth voltage. The open loop transfer function is determined as T(s) = GC(s) Gvd(s) FM 33
Fig.3 Block diagram of converter T(s) is defined as the product of the small signal gain in the forward and feedback paths of the feedback loop. 4.2. Numerical Example: The transfer function is derived from (11) is as follows: Gvd = 814S3 +2.456 1 7 S 2 1.232 1 12 S+2.154 1 16 (4.1) S 4 +149.4S 3 +4.922 1 8 S 2 +6.25 1 1 S+2.2 114 This is a fourth order transfer function which consists of and three zero in the RHP and two pair of complex pole Poles Zeros are -63.5356 +637.7841i 2156.5294-63.5356-637.7841i - 4995.229 +35864.4541i -11.1685 +22175.8678i 4995.229-35864.4541i -11.1685-22175.8678i 4.3 Model Order Reduction: Using Particle Swarm Optimization [5], the reduced order model for the converter is obtained as follows G 1vd = 2512S+4.378 17 S 2 +126.2S+4.15 1 5 (4.2) Zeros and poles of reduced order system are Zeros Poles 17425.388-63.1+637.5879i -63.1-637.5879i Fig. 4 Step response of original and reduced order model 34
Integral Square Error (ISE) between the original and reduced order model is.3463. 5. Conclusion This paper proposes the design of the controller for Cuk converter. By applying the SSA technique the linear model for the Cuk converter in terms of the ratio of duty cycle to output voltage (Gvd) is determined and it yields to a higher order system. Since the feedback compensator design for the fourth order system is quite complex, the fourth order function of the Cuk converter is reduced to a second order model by using PSO technique via model order reduction and it is observed that the reduced and original system s step response is almost similar. By comparing with the original system, the controller designed for the reduced order system gives the satisfactory results. References [1] Brijesh Kumar Kushwaha and Mr.Anirudha Narain Controller Design for Cuk Converter Using Model Order Reduction International Conference on Power, Control and Embedded Systems.978-1-4673-149-9IEEE(212). [2] R. D. Middlebook and S. Cuk, "A General Unified Approach to Modeling Switching Converter Power Stages" International Journal of Electronics, vol. 42, (1977) pp. 521-55. [3] R. Ridley, "Analyzing the sepic converter," Power Systems Design Europe Magazine, (26)pp. 14-18. [4]A. J. Forsyth and S.V. Mollov, "Modelling and control of DC- DC converters," IEEE Power Engineering Journal, (1998)pp. 229-236. [5].Kennedy J, Eberhart R.C..Particle swarm optimization.proc IEEE int l. Conf. on Neural networks,4, (1995) 1942-1948. [6] Maurice F.Hutton and Bernard Friedland, Routhapproximation for reducing order of linear, time-invariant systems IEEE Trans. Auto. Contr. Vol. AC-2, No.3, (1975) pp. 329-337. [7] G.Vasu,M.SivaKumar and M.RamalingaRaju, A novel method for optimal model simplification of large scale linear discrete time systems, International journals.automation and control, vol.1,no.2(216) pp. 12 141. [8] G.VasuanaM.SivaKumar and M.RamalingaRaju, Optimal Least Squares Model Approximation For Large Scale Discrete-Time Systems,Transactions of the Institueof Measurement and control,sage publications, (216)DOI:1177/14233121664923. 35
[9] Vasu.G,SivaKumar.M and RamalingaRaju.M(Feb217)A novel Model Order Reduction Technique for Linear Continous-Time Systems Using PSO-DV Alorithm, Journal of Control, Automation and Electrical Systems,SpringerVol.28,issue:1,pp.68-77 [1] Sastry G V K R and SivaKumar M(28) High -order MIMO interval system reduction using direct routh approximation method,international Journal of Engineering Reasearch&Indus.Appls.,vol.1,no.4,pp 45-54,28. [11] Sastry G V K R and SivaKumar M(21) Direct Routh Approximation method for linear SISO uncertain systems Reduction,International Journal of Applied Engineering Reasearch,vol.5,no.1.pp.99-98. [12] SivaKumar M VijayaAnand N and Srinivas Rao R(211) Model Reduction of Linear Interval Systems Using Kharitonov s Polynomial,IEEE International Conference on Energy Automation and Signal(ICEAS),211. [13] SivaKumar M VijayaAnand N and Srinivas Rao R(216), Impluse Energy Approximation of Higher Order Interval systems using Kharitonov s polynomials,transactions of the Institute of Measurement and Control,vol 39,no 6,pp.1225 1235. [14] SivaKumar M and Gulushad BEGUM (216) A New Biased Model Order Reduction for Higher Order Interval Systems.Advances in Electrical and Electronics Engineering,volume 14,no 2,pp.145-152. [15] G. V. Nagesh Kumar, J. Amarnath and B.P.Singh "Behavior of Metallic Particles in a Single Phase Gas Insulated Electrode Systems with Dielectric Coated Electrodes", IEEE International Conference on Condition Monitoring and Diagnosis CMD-28 Beijing, P. R. China., April 21-24, 28, Page(s):377-38. [16] Kumar, G.V. Nagesh,Amarnath, J.,Singh, B.P., Srivastava, K.D., " Particle initiated discharges in gas insulated substations by random movement of particles in electromagnetic fields, International Journal of Applied Electromagnetics and Mechanics, vol. 29, no. 2, pp. 117-129, 29 [17] Siva kumar M and and Gulushad BEGUM (216)model order reduction of time Interval Systems using Stability Equation Method and a Soft computing technique Advances in Electrical And Electronics Engineering, volume 14,no 2,pp.153-161. 36
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