Modeling and dynamics analysis of the fractional-order Buck Boost converter in continuous conduction mode

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1 hin. Phys. B Vol., No. 8 () 8 Modeling and dynamics analysis of the fractional-order Buck Boost converter in continuous conduction mode Yang Ning-Ning( 杨宁宁 ) a)b), iu hong-xin( 刘崇新 ) a)b), and Wu hao-jun( 吴朝俊 ) a)b) a) State Key aboratory of Electrical Insulation and Power Equipment, Xi an Jiaotong University, Xi an 79, hina b) School of Electrical Engineering, Xi an Jiaotong University, Xi an 79, hina (Received ecember ; revised manuscript received 7 February ) In this paper, the fractional-order mathematical model and the fractional-order state-space averaging model of the Buck Boost converter in continuous conduction mode (M) are established based on the fractional calculus and the Adomian decomposition method. Some dynamical properties of the current-mode controlled fractional-order Buck Boost converter are analysed. The simulation is accomplished by using SIMUINK. Numerical simulations are presented to verify the analytical results and we find that bifurcation points will be moved backward as α and β vary. At the same time, the simulation results show that the converter goes through different routes to chaos. Keywords: fractional-order Buck Boost converter, modeling, bifurcation, numerical simulation PAS:.. a,..pq, 8..Jc,..Hj OI:.88/67-6//8/8. Introduction The idea of fractional calculus has been known since the development of the regular calculus. It is accepted today as a new tool that extends the descriptive power of the conventional calculus, supporting mathematical models that, in many cases, describe more accurately the dynamic response of actual systems in various applications. Nowadays, the concept is employed in physics, engineering, biology, economics, and other scientific fields. ] The real objects are generally fractional. 6] The importance of fractional-order models is that they yield a more accurate description and give a deeper insight into the physical processes underlying a long range memory behaviour. The main reason for using the integer-order models is due to the lack of a method of solving the fractional differential equations. In previous studies, the researchers in most cases used the integral-order model of the converter. However, the capacitor and the inductor are all fractional in nature. Jonscher 7] demonstrated that the ideal capacitor cannot exist in nature, because an impedance of the form /(jω) would violate causality. In fact, the dielectric materials exhibit a fractional behavior, yielding electrical impedances of the form /(jωf ) α, with α R +. Westerlund ] proposed that better capacitor impedance could be Z(jω) /(jω) α, and they experimentally determined the order of the fractional-order capacitor in different dielectrics. Moreover, Westerlund ] demonstrated that the inductor is also fractional in nature. Petras 8] designed fractional-order hua s system. Haba et al. 9 ] created the fractional-order capacitor. Many real dynamical circuits were better characterized by using non-integer order models based on fractional calculus.,] The modeling of the fractional-order converter has become the subject which has important significance in theory and practice. In this paper, we will analyse the modeling and some dynamical properties of the fractional-order Buck Boost converter in continuous conduction mode (M). Firstly, we will give the fractional-order mathematical model and the fractional-order state-space averaging model of the Buck Boost converter. Then we will reveal the difference between the integer model and the fractional model. Finally, we will analyse the nonlinear dynamic behaviour of the current-mode controlled fractional-order model, and an illustrative simulation result will be given to demonstrate the cor- Project supported by the National Natural Science Foundation of hina (Grant No. 777) and the Specialized Research Fund for the octoral Program of Higher Education, hina (Grant No. ). orresponding author. ningning.yang@stu.xjtu.edu.cn hinese Physical Society and IOP Publishing td

2 hin. Phys. B Vol., No. 8 () 8 rectness of the proposed theoretical analysis and the efficiency of the fractional modeling.. Fractional calculus and the mathematical modeling of the fractional-order converter Buck Boost In this section, first, we introduce some basic concepts of the fractional calculus. The idea of fractional calculus is not new and is, in fact, as old as its integerorder counterpart. Fractional calculus has been used for modeling different physical phenomena ] and in control theory.,] We can notice that the systems in nature have fractional behaviours, but many of them have very low fractionalities. ] The Riemann iouville definition which is well known for the fractional differential operator is given as d α f(t) Γ(n α) d n t dt n f(τ) dτ, () (t τ) α n+ where n < α < n, n is an integer number, and Γ( ) is the gamma function. Another alternative definition of the Riemann iouville function was reported by aputo as follows: d α f(t) Γ(n α) t f (n) (τ) dτ. () (t τ) α n+ Then we create the fractional-order mathematical model of the Buck Boost converter by using the concept of fractional calculus. Westerlund ] in 99 proposed a new linear capacitor model. It is based on urie s empirical law developed in 889 which states that the current through a capacitor is i(t) V h t m, where h and m are constant, V is the voltage applied at t, and < m <. For a general input voltage V (t), the current is d α u(t) i(t) f, < α <, () dtα where f is the capacitance of the capacitor. It is related to the kind of dielectric. Another constant α (order) is related to the loss of the capacitor. Westerlund provided in his work the table of various capacitor dielectric materials with the values of appropriated constant α which had been obtained experimentally. Westerlund in his work also described the behaviour of a real inductor. ] For a general current in the inductor, the voltage is d β i(t) u(t) f, < β <, () dtβ where f is the inductance of the inductor and β (order) is the constant related to the proximity effect. The Buck Boost converter, sometimes called a step-up/down power stage, is an inverting power stage topology. The current-mode Buck Boost converter is shown in Fig.. 6] v in S i R Fig.. Buck Boost converter. In the M, there are two switching modes. ) When switch S is on, the inductor current passes through the switch, and the diode is reverse-biased with the inductor current i rising. ) When the switch S is turned off, the inductor maintains current flowing in the same direction so that the diode is forward-biased. The state equations of the converter are as follows. When S turns on, d α i (t) d β v o (t) + A, B When S turns off, d α i (t) d β v o (t) A v o i (t) v o (t) v in (t), () ; i (t) v o (t) + v in (t), (6), B. 8-

3 hin. Phys. B Vol., No. 8 () 8 In comparison with the integer-order model, the fractional-order model of the Buck Boost converter in the M described by Eqs. () and (6) is relevant to α and β.. State-space averaging model of fractional-order Buck Boost converter According to the operational features of the Buck Boost converter, the circuit variables in the Buck Boost converter, such as inductance current i and the output voltage v o, have all the high frequency switching harmonics. These harmonics can be removed by averaging a circuit variable over one switching period, i.e., x(t) T T t+t t x(τ)dτ, (7) where x is an arbitrary circuit variable of the Buck Boost converter. In light of the properties of fractional calculus, we obtain d α x(t) T d α ( T t+t T t d α x(t) t+t t ) x(τ)dτ ( d α x(τ) ) dτ dτ α, (8) where α is the order and < α <. Above all, we begin to create the fractional statespace averaging model of the Buck Boost converter in M. Then theoretical analysis on this model will be carried out. The average values of the circuit variables i (t), v o (t), v in (t), and d(t) can be described in the following forms: i (t) I + î (t), v o (t) V o + ˆv o (t), v in (t) V in + ˆv in (t), d(t) + ˆd(t), (9) where I, V o,, and V in are the components of i (t), v o (t), d(t), and v in (t) respectively, î (t), ˆv o (t), ˆd(t), and ˆv in (t) are the A components respectively. In circuit the A component of each variable is much smaller than the corresponding amplitude of the component (under small-signal assumption), that is, î (t) is far less than I, ˆv o (t) is far less than V o, ˆd(t) is far less than, and ˆv in (t) is far less than V in. According to Ref. 6], we can obtain A d(t)a + ( d(t))a, d(t) A d(t) B d(t)b + ( d(t))b, B, () d(t). () Using the results of Ref. 6], is the steady state value of d(t), we can obtain the quiescent operation point as follows: is I V o A BV in V in ( ) R V in V in. () The fractional-order state-space averaging model d α i (t) T d β v o (t) T d(t) d(t) + d(t) v in (t) ˆd(t) I + î (t) V o + ˆv o (t) + ˆd(t) + i (t) v o (t) ˆd(t) (V in + ˆv in (t)). () Using the definition of fractional-order derivative given by aputo ] and considering the input voltage V in to be a constant, we can obtain the inductor cur- 8-

4 hin. Phys. B Vol., No. 8 () 8 rent ripple i in (, T ) as i V in T ] α αγ(α), () where Γ( ) is the gamma function. ] The inductor current ripple i is related not only to the inductance, the input voltage V in, the duty ratio, and the switching period T, but also to the order of inductor. When α is increased, i is reduced. Especially when α, we can obtain the integral-order model mentioned in Ref. 7]. From Eqs. () and (), we can obtain the peak inductor current in the following form: i max I + i V in ( ) R + V in(t ) α αγ(α). () The continuous-conduction mode of operation occurs when the current through the inductor in the circuit of Fig. is continuous. That means the inductor current is always greater than zero, i.e., I > i. (6) Substituting Eqs. () and () into inequality (6), we have the following inequality: RT α > α ( ) αγ(α). (7) In order to make the Buck Boost converter operate in the M, we should consider not only the values of inductor, load resistance R, duty ratio, and switching period T, but also the value of α. When α increases, it is easier for the Buck Boost converter to work in the M. Especially when α, we can obtain the integral-order model mentioned in Ref. 8]. Then we consider the variation of output voltage v o. Using the Adomian decomposition method, we can solve the following fractional differential equation: α x(t) Ax(t) + f(t), (8) where < α <, < t < T. According to Eq. (), we can have the following equation: β v o (t) v o(t), (9) where A / and f(t). Using the results of Ref. 9] we can have v o (t) E β, (At β )V os, () where E β ( ) is the Mittag effler function ] and V os is the initial value of the output voltage. When t, the switch is on, so v o () E β, ()V os V os ; () when t T, the output voltage is v o (T ) E β, ] V os. () Then, we can obtain the output voltage ripple v o as v o v o () v o (T ) V os E β, { E β, The approximate expression of V os is ] V os ]} V os. () V os V o + v o. () Substituting Eq. () into Eq. () yields v o { E β, ]} ( V o + v ) o. () Substituting Eq. () into Eq. (), we have the following equation: ]} { E β, v o V o ]} { + E β, ]} { V E β, in ]}. (6) ( ) { + E β, The output voltage ripple is related not only to capacitor, load resistance R, input constant voltage V in, duty ratio, and switching period T, but also to β. When β increases, v o is reduced. Especially when β, we can have the integral-order model mentioned in Ref. 7]. Separate the A component in Eq. (), then we will obtain the fractional-order A small signal state equations as d α î (t) d βˆv o (t) d(t) d(t) î(t) ˆv o (t) 8-

5 + d(t) hin. Phys. B Vol., No. 8 () 8 ˆv in (t) + I ˆd(t) + V o + + V in ˆd(t) î(t) ˆd(t) ˆv o (t) ˆv in (t) ˆd(t). (7) To neglect the higher order terms ˆd(t)ˆvo (t), ˆd(t)î (t), ˆv in (t) ˆd(t), the following state equations can be obtained d α î (t) d βˆv o (t) î(t) ˆv o (t) + ˆv in (t) + I ˆd(t) V o + V in ˆd(t). (8) By using aplace transform based on the F, we obtain s βˆv o (s) ( )î (s) + ˆd(s)I ˆv o(s), (9) s α î (s) V ˆd(s) in + ˆv in (s) + ( )ˆv o (s) V o ˆd(s). () The transfer function of ˆv o (t) to ˆv in (t) is G vov in (s) ˆv o(s) ˆv in (s) ˆd(s) ( ) s α+β +. () R sα + ( ) The transfer function of ˆv o (t) to ˆd(t) is G vod(s) ˆv o(s) ˆd(s) ˆvin(s) ( V in ) ( ) sα s α+β +. () R sα + ( ) The transfer function of î (t) to ˆv in (t) is G i v in (s) î(s) ˆv in (s) ˆd(s) ( ) sα s α+β +. () R sα + ( ). Simulation study and dynamical property analysis of the current-mode controlled fractional-order converter in M Buck Boost Firstly, the mathematical model of the fractionalorder Buck Boost converter in M is built by using SIMUINK based on the fractional differential equation of the fractional-order Buck Boost converter which has been derived in Section. In the simulation we use the following parameters: f. khz, v in V, d.6, R Ω,. H, and 7 µf. Therefore, the modified Oustaloup s method,] is used to solve the system of fractional differential equations. Because the switching frequency of the converter is f. khz and the corresponding rotational frequency is ω πf.7 rad/s, we choose the parameters of Oustaloup to be ω b rad/s, ω h rad/s, N 8. When α and β, according to Eqs. () and (6), we can obtain I.7 A, i. A, i max.87 A, V o V, v o 7.6 V, and v omin.8 V. According to Fig., the practical simulation results are i max.8 A and v omin.6 V. When α.8 and β.9, according to Eqs. () and (6), we obtain I.7 A, i. A, i max. A, V o V, v o.9 V, and v min.79 V. According to Fig., the practical simulation results are i max. A and v omin.7 V. Then some dynamical properties of the model will be analysed. We wish to obtain a bifurcation diagram for the current-mode controlled Buck Boost converter. The operation of the system can be described as shown in Fig.. 8-

6 hin. Phys. B α.8, β.9 α, β... i/a i/a Vol., No. 8 () α.8, β.9 α, β - vo/v - vo/v Fig.. Time evolutions of i (t) and vo (t). current mode controlled Buck-Boost converter clock (. khz) S i Q I R + ref vin i i S Iref R sweep generator sample & hole X i(n) bifureation diagrams Y vo + Fig.. Block diagram of the system for displaying bifurcation diagrams. We use reference current Iref as the bifurcation parameter. When α and β, the bifurcation diagram of the fractional-order Buck Boost converter in M is shown in Fig. (a), and when α.8 and β.9, the bifurcation diagram is shown in Fig. (b). The fractional-order Buck Boost converter goes through period-, period-, period-, and eventually to chaos as reference current Iref varies, which is similar to the case in Fig.. At the same time, bifurcation points will be moved backward when α and β vary and it shows going through diﬀerent routes to chaos. α.8, β.9 α, β (a) (b) i/a i/a Iref/A - Iref/A Fig.. Bifurcation diagrams for the current-mode controlled Buck Boost converter. (a) α, β (b) α.8, β.9 8-6

7 hin. Phys. B Vol., No. 8 () 8 ompared with the integral-order model, the fractional-order model shows the different dynamic behaviours at the same I ref. At I ref A, the phase portrait of period- is presented in Fig. (a) when α and β. Figure (b) shows the phase portrait of period- when α.8 and β.9. At I ref A, the phase portrait of the converter is shown in Fig. (c), and it can be observed that there is a strange attractor in the phase portrait for α and β, that is, the integral-order Buck Boost converter is chaotic. However, as shown in Fig. (d), whenα.8 and β.9, the phase portrait is period (a) α, β.6.8. i A α, β (c).. i A - -7 α.8, β i A (b) α.8, β.9 (d) i A Fig.. Phase portraits from Buck Boost operating under current-mode control: (a) I ref A, α, β, (b) I ref A, α.8, β.9, (c) I ref A, α, β, and (b) I ref A, α.8, β.9.. onclusions According to the theorem of fractional calculus, we created the mathematical model and the statespace averaging model of the fractional-order Buck Boost converter. Then we gave the parameter condition of the Buck Boost converter in M. Finally, some dynamical properties of the model were analysed. Through the theoretical analysis, we can obtain the following conclusions: in M, the component of the output voltage and the inductor current are independent of α and β. The inductor current ripple, the peak inductor current, the output voltage ripple, the initial value of the output voltage and the transfer functions (G vo v in (s), G vo d(s), and G i v in (s)) are all related to fractional-order α or β. When α increases, the inductor current ripple and the peak inductor current decrease. When other parameters are fixed, the output voltage ripple and the initial value of the output voltage decrease with β increasing. According to simulation results, we find that when α and β vary, bifurcation points will be moved back and the converter goes through the different routes to chaos. References ] Nigmatullin R R, Arbuzov A A, Salehli F, Giz A, Bayrak I and atalgil G 7 Physica B 88 8 ] Westerlund S 99 IEEE Transactions on ielectrics and Electrical Insulation 86 ] Wu J, Zhang Y B and Yang N N hin. Phys. B 6 ] Westerlund S ead Matter Has Memory! (Kalmar, Sweden: ausal onsulting) hap. 7 ] Podlubny I 999 Fractional ifferential Equations (San iego: Academic Press) hap. 6] Nakagava M and Sorimachi K 99 IEIE Trans. Fundam. E7-A 8 7] Jonscher A K 98 ielectric Relaxation in Solids (ondon: helsea ielectric Press) hap. 8] Petras I 8 haos, Solitons and Fractals 8 9] isse Haba T, Ablart G, amps T and Olivie F haos, Solitons and Fractals 79 ] isse Haba T, oum G and Ablart G 7 haos, Solitons and Fractals 6 ] isse Haba T, oum G, Zoueu J T and Ablart G 8 J. Appl. Sci. 8 9 ] Jesus I S and Tenreiro M J A 9 Nonlinear yn. 6 ] Wang F Q and Ma X K Acta Phys. Sin (in hinese) ] Tenreiro M J A 997 Journal Systems Analysis Modelling Simulation 7 7 ] Torvik P J and Bagley R 98 Trans. ASME 9 6] Zhang W P The Modeling and ontrol of the Switching Power Supply (Beijing: hina Electric Power Press) hap. (in hinese) 7] Zhang Z S and ai X S 6 The Principle and esign of Switching Power Supply (Beijing: Publishing House of Electronics Industry) hap. (in hinese) 8] Yang X, Pei Y Q and Wang Z A Switching Power Supply Technology (Beijing: hina Machine Press) hap. (in hinese) 9] uan J S, An J Y and Xu M Y 7 Appl. Math. J. hin. Univ. Ser. B 7 ] Xue Y, Zhao N and hen Y Q 6 Proceedings of IEEE International onference on Mechatronics and Automation p. ] Xue Y and hen Y Q 7 MATAB Solution to Mathematical Problems in ontrol (Beijing: Tsinghua University Press) hap. 9 (in hinese) 8-7

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Commun. Theor. Phys. (Beijing, China) 5 (2010) pp. 1105 1110 c Chinese Physical Society and IOP Publishing Ltd Vol. 5, No. 6, June 15, 2010 Construction of a New Fractional Chaotic System and Generalized