Research Article Modeling and Characteristics Analysis for a Buck-Boost Converter in Pseudo-Continuous Conduction Mode Based on Fractional Calculus

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1 Mahemaical Problems in Engineering Volume 26, Aricle ID 68359, pages hp://dx.doi.org/.55/26/68359 Research Aricle Modeling and Characerisics Analysis for a Buck-Boos Converer in Pseudo-Coninuous Conducion Mode Based on Fracional Calculus Ningning Yang,,2 Chaojun Wu, 3 Rong Jia,,2 and Chongxin iu 4 Sae Key aboraory Base of Eco-Hydraulic Engineering in Arid Area, Xi an Universiy of Technology, Xi an 748, China 2 Insiue of Waer Resources and Hydro-Elecric Engineering, Xi an Universiy of Technology, Xi an 748, China 3 College of Elecronics and Informaion, Xi an Polyechnic Universiy, Xi an 748, China 4 School of Elecrical Engineering, Xi an Jiaoong Universiy, Xi an 749, China Correspondence should be addressed o Chaojun Wu; chaojun.wu@su.xju.edu.cn Received 3 Ocober 25; Revised 9 January 26; Acceped 4 January 26 Academic Edior: uis J. Yebra Copyrigh 26 Ningning Yang e al. This is an open access aricle disribued under he Creaive Commons Aribuion icense, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. In recen days, fracional calculus (FC) has been acceped as a novel modeling ool ha can exend he descripive power of he radiional calculus. Fracional-order descripiveness can increase he flexibiliy and degrees of freedom of he model by means of fracional parameers. Based on he fac ha real capaciors and inducors are inrinsic fracional order, fracional calculus is inroduced ino he modeling process o esablish a fracional-order sae-space averaging model of he Buck-Boos converer in pseudo-coninuous conducion mode (PCCM). Orders of he model are considered as exra parameers, and hese parameers have significan influences on he performance of he model. The inducor curren, he inducor curren ripple, he ampliude of he oupu volage, and he ransfer funcions of he fracional-order model are all relaed o orders. The conras simulaion experimens are conduced o invesigae he performance of ineger-order and fracional-order Buck-Boos converers in PCCM. Resuls of numerical and circui simulaions demonsrae ha he proposed heoreical analysis is effecive; he fracional-order model of he Buck-Boos converer in PCCM has cerain heoreical and pracical significance for modeling and performance analysis of oher elecrical or elecronic equipmen.. Inroducion Fracional calculus is an old mahemaical opic which can go back o 695; when Hospial wroe o eibniz, he menioned his kind of calculaion, 2. Over he pas 3 years, several mahemaicians conribued o his mahemaical subjec. In recen years, fracional calculus has been acceped as a new insrumen ha can exend he descripive power of he radiional calculus. I has urned ou ha many phenomena in widespread fields of science and engineering can be described very successfully by fracional-order models 3 9. Comparing wih ineger-order models, he significan advanage of fracional-order models is ha hey are characerized by herediary and memory properies. And i has been demonsraed ha fracional-order models can increase he flexibiliy and degrees of freedom by means of fracionalorder parameers 2. The heory and applicaions of fracional-order componens had a considerable progress during he las wo decades. Researchers have repored he inrinsic fracional-order behavior of real objecs, such as capaciors and inducors. Weserlund and his colleagues consruced he fracionalorder model of he capacior and measured he orders of some real capaciors under differen dielecrics by experimen 3, 4. Weserlund also demonsraed ha real inducors are fracional order and measured he orders of some real inducors 4. Using differen fracal srucures, fracionalorder capaciors wih differen orders were implemened by Jesus and Machado 3. Based on skin effec, Machado and Galhano proposed ha fracional-order inducors wih

2 2 Mahemaical Problems in Engineering differen orders can be consruced 5. Real fracional-order capaciors were creaed by Haba and his colleagues 6, 7. These realizaions of fracional-order capaciors and inducors indicae he possibiliy of fracional-order componens and models being employed in pracical applicaions. Capaciors and inducors are he fundamenal par of he Buck-Boos converer. Based on he facs ha real capaciors and inducors are all fracional-order componens, he idea of esablishing he fracional-order model of he Buck-Boos converer seems o be far more sensible. I can be proved ha he order of he model has significan influence on heperformanceofhebuck-boosconvererandcanbe considered as an exra parameer. Up o dae, few fracionalorder models of converer have been esablished 8, 9. Marinez and his colleagues esablished he fracional-order model of he Buck-Boos converer, bu only he capacior was considered as he fracional-order circui elemen. Wang and Ma consruced he fracional-order model of he Boos converer in coninuous conducion mode (CCM) and disconinuous conducion mode (DCM) 9; Yang and his colleagues proposed he fracional-order model of he Buck- BoosconvererinCCM2,bunocircuisimulaions or experimens were presened o verify hese models. In his paper, a fracional-order sae-space averaging model of he Buck-Boos converer in pseudo-coninuous conducion mode (PCCM) is esablished; numerical and circui simulaion experimens are presened o verify he efficiency of he proposed heoreical analysis. Generally, Buck-Boos converers operae in CCM or DCM. In recen repors, pseudo-coninuous conducion mode (PCCM) is proposed which is he hird operaion mode. Comparing wih operaing in DCM, in PCCM, he curren handling capabiliy of he converer is improved and he curren ripple and volage ripple are reduced. And a single-pole behavior in he conrol-o-oupu ransfer funcion is exhibied; he load ransien response is much faser han operaing in CCM and DCM Therefore, he sudy of he fracional-order model of he Buck-Boos converer in PCCM is an imporan heoreical issue and has pracical engineering value. In his paper, we esablished he fracional-order saespace averaging model of he Buck-Boos converer in PCCM and analyzed he influence of orders on some dynamical properies of he fracional-order model. The res of his paper is organized as follows: Secion 2 is he preliminaries; some basic conceps of fracional calculus and he basic fracional-order inducor and capacior models are inroduced. In Secion 3, based on fracional calculus, we esablished a fracional-order sae-space averaging model of he Buck-Boos converer in PCCM. In Secion 4, he quiescen operaion poin and ransfer funcions of he fracional-order converer are analyzed. Orders of he fracional-order model areprovedobeexraparameersandhaveinfluenceon he performance of he converer. In Secion 5, numerical and circui simulaion experimens are implemened o verify he correcness of heoreical analysis and he proposed fracional-order model. Finally, in Secion 6, influences of orders on he fracional-order model are summarized. 2. Preliminaries 2.. Fracional Calculus. Fracional calculus allows operaions of ordinary differeniaions and inegraions o nonineger-order. I is a generalizaion of radiional calculus, bu is applicabiliy is much wider 24. The fundamenal operaor a Dα,whereα(α R) is he order and a and are he bounds of he operaion, is defined as a Dα = d α, dα { α >, α = { (dτ) α, α <. { a The hree bes known definiions for fracional calculus are Grünwald-enikov (G) definiion, Riemann-iouville (R) definiion, and Capuo definiion 2. Capuo definiion is defined as a Dα f () = Γ (n α) a () f (n) (τ) dτ, (2) α n+ ( τ) where Γ( ) is he Gamma funcion and n N is he firs ineger which is no less han α, n <α<n. The aplace ransform of Capuo fracional derivaive saisfies { a D α n f ()} =sα F (s) k= s α k f (m) (), n <α<n. For zero iniial condiions, he aplace ransform of fracional derivaives has he following form: (3) { a D α f ()} =sα F (s). (4) Theoreical calculaions are based on he Capuo derivaive and more specifically on he iniial condiion x(). Harley and orenzo discuss he error incurred in using he Capuo-derivaive aplace ransform 25. I is now well esablished ha more precisely iniial sae of a Fracional Differenial Equaion can be represened by he weighed inegral of z(w, ) Neverheless, under he premise ha he error is accepable, and for convenience, he Capuo derivaive and he iniial condiion x() are adoped The Fracional-Order Model of Capaciors and Inducors. The fracional-order model of capaciors sems from Curie s empirical law which is proposed in 889 3: i () = V, <m<, h m (5) where h is a consan relaed o he capaciance and he kind of he dielecric. m is a consan close o,relaedohelosses of he capacior. V is he DC volage applied a =.

3 Mahemaical Problems in Engineering 3 S D Pulse T on T off V in S 2 i C + R Pulse 2 T off T on Figure : Buck-Boos converer in PCCM. i Δi In 994, based on fracional calculus, Weserlund and Eksam developed a new capacior heory 3: i () =C f d β u () d β, <β<, (6) where C f is he capaciance of he capacior relaed o he kind of dielecric. And β is he order relaed o he losses of he capacior. Weserlund and Eksam provided various capacior dielecricswihappropriaedorderβ by experimen. Weserlund also creaed he fracional-order model of inducors 4: d α i () u () = f, <α<, (7) dα where f is he inducance of he inducor and α is he order relaed o he proximiy effec. In engineering applicaions, orders of capaciors and inducors are close o 3. In order o ensure ha he power elecronic devices can work properly, we redefine a fracional order beween.95 and. 3. The Sae-Space Averaging Model of he Fracional-Order Buck-Boos Converer in PCCM 3.. The Fracional-Order Mahemaical Model of he Buck- Boos Converer. The Buck-Boos converer can operae as Buck (for volage sepdown) or Boos (for volage sepup) converer, invering he volage polariy. Connecing an inducor in parallel wih a power swich makes he converer operaeinpccm3.thecircuidiagramisshownin Figure. There are wo fully conrolled swiching devices S and S 2, wo diodes D and,onedcinpuvolagev in, one load resisor R, and wo fracional-order energy sorage elemens and C in he circui. In Figure 2, where T is he swiching ime period, T on is heonime,t off isheoffimeforheswich,andd +d 2 +d 3 =. There are hree saes for he pseudo-coninuous mode. Sae. When< d T, he circui opology is shown in Figure 3. Swich S is on, S 2 is off, and diodes D,2 are reversebiased. The inducor curren i rises. d T d 2 T d 3 T Figure 2: Inducor volage and curren of Buck-Boos converer in PCCM. V in S S 2 i D C + Figure 3: Schemaic diagram of Buck-Boos converer in Sae. The expression of he fracional-order mahemaical model is d α i () d α d β V C () = i () V d β CR C () + V in (). (8) Sae 2. Whend T< (d +d 2 )T, asshowninfigure4, swiches S,2 and diode are off and diode D is forwardbiased and behaves as a shor circui; he inducor curren i ramps down. The expression of he fracional-order mahemaical model is d α i () d α d β V C () = d β C i () V C () + V in (). (9) CR Sae 3.When(d +d 2 )T< T,swichS and diode D are off, swich S 2 and diode are on, and he inducor curren i mainains a consan heoreically. The circui opology is shown in Figure 5. R

4 4 Mahemaical Problems in Engineering S S 2 V in D i C R + Figure 4: Schemaic diagram of Buck-Boos converer in Sae 2. e us consider he fracional-order sae-space averaging model of Buck-Boos converer in PCCM: d α i () T d α d β V C () T d β = d 2 () C d 2 () CR d () + V in () T. i () T V C () T (3) Average values of circui variables are defined as he following form: S D i () T =I + i (), V in S 2 i C + R V C () T =V C + V C (), V in () T =V in + V in (), d () =D + d (), (4) d 2 () = + d 2 (), Figure 5: Schemaic diagram of Buck-Boos converer in Sae 3. The expression of he fracional-order mahemaical model is d α i () d α d β V C () = i () V d β CR C () + V in (). () As shown in (8), (9), and (), he inducor order α and he capacior order β can be considered as exra parameers of he fracional-order mahemaical models in PCCM. The influence of hese exra parameers on he performance of he converer will be discussed in he nex secion The Fracional-Order Sae-Space Averaging Model of hebuck-boosdc/dcconverer. To remove he swiching harmonics, waveforms over one swiching period should be averaged32.theaveragevalueofvariablex() which is an arbirary circui variable of he Buck-Boos converer in one swiching period is defined as follows: x () T = +T T x (τ) dτ. () Is fracional-order form is d α x () T d α = d α ((/T) +T x (τ) dτ) d α = T +T where α is he order and.95 < α <. ( dα x (τ) dτ α )dτ= dα x () d α, (2) where variables in capial leers represen DC componens and variables wih he symbol above hem represen AC componens which are much smaller in magniude han he corresponding DC componen. Equaion (3) can be rewrien as d α I d α d β V C d β + dα i () d α + dβ V C () d β = + d 2 () C + d 2 () CR D + d () + (V in + V in ()). DC componens in (5) are as follows: I + i () V C + V C () (5) d α I D d α d β V = 2 C d β C I D + V in. (6) V C CR The quiescen operaion poin is I D V = 2 D in D V C C V in = D2 2R CR V ind. (7) The volage raio is defined as follows: M= V C V in = D. (8)

5 Mahemaical Problems in Engineering 5 AC componens of (5) are d α i () d α d β V C () d β = + d 2 () C + d 2 () CR D + d () + V in () + d 2 () C d 2 () i () V C () I d () + V C V in, (9) where d 2 () i (), d 2 () V C (), and d () V in () are high order smallsignalermswhichcanbeomied.thenhesmall signal AC equaion of Buck-Boos converer in PCCM can be expressed as follows: d α i () D d α d β V C () = 2 D i () 2 d β C V C () CR D + V in () + I d 2 () C V C + V in d (). 4. Performance Analysis of he Fracional-Order Model of he Buck-Boos Converer in PCCM (2) Inhissecion,hequiescenoperaionpoinandransfer funcions of he AC small signal model will be analyzed, because hey have pracical significance for he designing of various parameers and reflec he performance of he converer. The inducor curren ripple and oupu volage ripple of he fracional-order Buck-Boos converer in PCCM are shown in Figure 6. i C i ( ) d T d 2 T d 3 T c ( ) d T Δi Δ C Figure 6: The inducor curren ripple and oupu volage ripple. The small ripple approximaion is defined as replacing waveforms wih heir low-frequency averaged values: d α i () d α V in () T. (2) Using Capuo definiion, we can ge he corresponding iniial condiion as ge D k i () =i (k), k=,,...,m. (22) Solving fracional-order differenial equaion (2), we can i () = Γ (α) + m k k! i(k) k= ( τ) α V in () T dτ (). (23) In PCCM, he iniial value of inducor curren is given as i ( )=i min. Then i max =i (d T) = V in (D T) α Γ (α) α +i min. (24) The inducor curren ripple is Δi = V in (D T) α Γ (α) α. (25) The maximum and minimum values of inducor curren are i max =I + 2 Δi = V ind D2 2R + V α in (D T) 2Γ (α) α, i min =I 2 Δi = V ind D2 2R V α in (D T) 2Γ (α) α. (26) According o Adomian decomposiion scheme, one can ge d β x () d β =Ax() +f(x). (27)

6 6 Mahemaical Problems in Engineering InSaes2and3,heloadispoweredbyhecapacior.The oupu volage of he converer is negaive which saisfies he sae equaion and he ampliude is monoone decreasing: d β V C () d β = V C () CR, (28) where A= /CR, f(x) =,and.95 < β <. The soluion of (28) is V C () =E β, (A β ) V max, (29) where V max is he iniial value of V C (),whichmeansv C ( )= V max for =. When =(d +d 3 )T,heoupuvolageis V C ((d 3 +d )T)=V max E β, ( ((d 3 +d )T) β ). (3) RC The variaion of he oupu volage is ΔV =V max ( E β, ( ((d 3 +d )T) β )). (3) RC Accordingo(7)and(3),iyields V max =V C + 2 ΔV According o (34), one can ge he V in -o-v C and he V in - o-i ransfer funcion as follows: G VC V in (s) = V C (s) V in (s) d (s)=, d 2 (s)= D = / (C/D2 2)sα+β + (/RD2 2)sα +, G i V in (s) = i (s) V in (s) d (s)=, d 2 (s)= = (D /R 2 )(CRsβ +) (C/ 2 )sα+β + (/R 2 )sα +. (36) Se he AC inpu volage variaion and he duy cycle d 2 o be zero: V in (s) =, d 2 (s) =. The d -o-v C and he d -o-i ransfer funcions are G VC d (s) = V C (s) d (s) Vin (s)=, d 2 (s)= (37) = V ind + V max 2 ( E β, ( ((d 3 +d )T) β )) RC (32) V = in / (C/D2 2)sα+β + (/RD2 2)sα +, G i d (s) = i (s) d (s) Vin (s)=, d 2 (s)= (38) 2V = in D. ( + E β, ( ((d 3 +d )T) β /RC)) Taking (32) o (3) yields 2V in D ( E β, ( ((d 3 +d )T) β /RC)) ΔV =. (33) ( + E β, ( ((d 3 +d )T) β /RC)) We can find from (25) and (33) ha he inducor curren rippleandheoupuvolageripplearecloselyrelaedohe order of energy sorage elemens. The ripple increases wih he decreasing of he order. Under condiions of zero iniial value, he aplace ransform of converer equaions is s β V C (s) = i (s) C I d 2 (s) C V C (s) CR, s α i (s) = D V in (s) +V in d (s) + V C (s) +V C d2 (s). Se he duy cycle variaion o be zero: d (s) =, d 2 (s) =. (34) (35) = (V in /R 2 )(CRsβ +) (C/ 2 )sα+β + (/R 2 )sα +. Se he AC inpu volage variaion and he duy cycle d o be zero: V in (s) =, d (s) =. (39) According o (7) and (34), he d 2 -o-v C and he d 2 -o-i ransfer funcions are G VC d 2 (s) = V C (s) d 2 (s) Vin (s)=, d (s)= = (V ind /D 4 2 R) sα +V C / (C/ 2 )sα+β + (/R 2 )sα +, G i d 2 (s) = i (s) d 2 (s) Vin (s)=, d (s)= (V in D /RD 3 2 = )(CRsβ +2) (C/D2 2)sα+β + (/RD2 2)sα +. (4)

7 Mahemaical Problems in Engineering 7 Table : Parameers of he Buck-Boos converer. Parameer Symbol Value Inpu volage V in 5 V Capacior C μf Inducance mh Swiching frequency f 2kHz Duy cycle D.4 Duy cycle D 3.4 The order α.95/ The order β.95/ We can find from (36), (38), and (4) ha orders of he inducor and capacior have influence on ransfer funcions. Tha means fracional orders can be seen as exra parameers. When orders are, we can ge ineger-order ransfer funcions from he fracional-order model. The ineger-order model is a special case of he fracional-order model. i (A) Time (s) IO FO (.95) Figure 7: The inducor curren. 5. Experimenal Researches In his secion, some numerical and circui simulaion experimens are presened o verify he fracional-order model and he influence of he order on performances of he converer in PCCM. We choose.95 as he order which is close o. Parameers of he Buck-Boos converer are lised in Table. We selec he load resisor as R = 2Ω o ensure ha he converer is operaing in PCCM. Difference values of he quiescen operaing poin beween he ineger-order and fracional-order Buck-Boos converer model are shown in Table 2. InTable2,wecanfindhaheoupuvolagerippleand heinducorripplerisewihhedecreasingofheorder.tha means he order of he Buck-Boos converer impacs he oupu characerisics of he converer obviously. 5.. Numerical Simulaion. In numerical simulaion, Ousaloup s recursive approximaion is adoped 33. The swiching frequency of he converer is seleced as f = 2kHz, he corresponding roaional frequency is ω = 2πf =.256 5, and parameers of Ousaloup are ω h = 2 5, ω b = 5 6,andN=6. Figure 7 shows he inducor curren; he deailed values arelisedintable3.wecanseeha,inheinducorcurren ripple, maximum and minimum values of he inducor curren increase significanly in he fracional-order model. Tha means he effec of he inducor order on he inducor curren should be appropriaely considered. The order also influences he oupu volage. As shown in Figure 8 and Table 4, in he oupu volage ripple, he maximum and minimum values of oupu volage increase significanly in he fracional-order model Circuiry Simulaion Experimen. To furher verify he resuls of he heoreical analysis, he circui simulaion experimens are presened. C (V) Time (s) IO FO (.95) Figure 8: The oupu volage. By using resisor/capacior or resisor/inducor neworks, we can consruc an approximaion circui of he fracionalorder circui elemens. We choose he chain srucure as he fracance uni o approximaely achieve he fracional-order circui elemens. The chain fracance uni of fracional-order capacior is shown in Figure 9. The ransfer funcion model of he capacior is n /C H (s) =R in + k (s + /R k C k ). (4) k= We choose β =.95, C= μf, and n = 6. We can calculae resisors and inducors in Figure 9 by he mehod of undeermined coefficiens, comparing he inpu

8 8 Mahemaical Problems in Engineering Table 2: The quiescen operaing poin of he Buck-Boos DC/DC converer. Parameer Symbol Ineger order Fracional order (.95) Oupu volage V C V V Oupu volage ripple ΔV C 2V 3.38V Minimum values of he oupu volage V C min V.69 V Maximum values of he oupu volage V C max 99 V 98.3 V Inducor curren I 25 A 25 A Inducor curren ripple Δi A.75A Minimum values of he inducor curren i min 24.5 A 24.2 A Maximum values of he inducor curren i max 25.5 A A Table 3: Values of inducor curren. Order of he model Inducor curren ripple Minimum values of he inducor curren Maximum values of he inducor curren Δi =A i min = 24.5 A i max =25.5A.95 Δi =.89A i min = 24.4 A i max =25.93A Table 4: Values of he oupu volage. Order of he model Oupu volage ripple Minimum values of he oupu volage Maximum values of he oupu volage ΔV C =.96V V C min =.9 V V C max = V.95 ΔV C =3.66V V C min =.7 V V C max = 98.4 V Cs β R in C C 2 C n R R 2 R n Figure 9: The approximae circui of he fracional-order capacior (C= μf, β =.95). impedance of he approximae circui model and Ousaloup s approximaion: n /C C s β =R in + k (s + /R k C k ). (42) k= Solving (42), parameers of he chain fracance uni of fracional-order capacior are as follows: R =.36 Ω, R 2 = 2.9 Ω, R 3 = Ω, R 4 = 8.32 kω, R 5 =.46 MΩ, R 6 =. 3 MΩ, R in = 2.2 mω, C = μf, C 2 = μf, C 3 = μf, C 4 = μf, C 5 = 62.4 μf, and C 6 = μf. Bode diagrams of he fracional-order capacior by numerical simulaions and circui simulaions are compared in Figure ; hey are approximae fiing. The chain fracance uni of fracional-order inducor is shown in Figure. The ransfer funcion model of he inducor is H (s) = n k= ((/ k)/(s+r k / k )+/R in ). (43) We choose α =.95, = mh, and n = 6.We can calculae resisors and inducors in Figure by he mehod of undeermined coefficiens, comparing he inpu Magniude (db) Phase (deg.) Bode diagram Frequency (Hz) Frequency (Hz) /Cs β (C = μf, β =.95) Ousaloup approximaion Fracance circui approximaion Figure : The Bode diagram of he.95-order capacior. impedance of he approximae circui model and Ousaloup s approximaion: s α = n / k (s + R k / k ) +. (44) R in k= Parameers of he chain fracance uni of he fracionalorder capacior are as follows: R = Ω, R 2 = 3.44 Ω,

9 Mahemaical Problems in Engineering 9 V R in R R 2 R n + s α 2 n A Figure : The approximae circui of he fracional-order inducor ( =mh, α =.95). Magniude (db) Bode diagram Frequency (Hz) C (V) Figure 3: The.95-order Buck-Boos converer by Psim Fracional-order model Ineger-order model Phase (deg.) Frequency (Hz) s α ( = mh, α =.95) Ousaloup approximaion Fracance circui approximaion Figure 2: The Bode diagram of he.95-order inducor. R 3 = 43.4 mω, R 4 = μω, R 5 = 6.87 μω, R 6 = μω, R in = 2.2 mω, = 2.5 mh, 2 = 3. mh, 3 = 3.9 mh, 4 = 4.9 mh, 5 = 6. mh, and 6 =.6 mh. Bode diagrams of he fracional-order capacior by numerical simulaions and circui simulaions are compared in Figure 2; hey are approximae fiing. In Figure 3, we implemened he circui simulaion by Psim. The load resisor is seleced as R=2Ω;dueononideal circui elemens, he oupu volage and inducor curren are slighly differen from he numerical simulaion. As shown in Figures 4 and 5, wih he decrease of he order of energy sorage elemens, he response of he converer becomes faser and he overshoo becomes smaller. All circui simulaion experimens are consisen wih resuls of he heoreical analysis and numerical simulaions. The influence of he fracional order on he performance Time (s) Figure 4: The oupu volage. of he converer is obvious; ha means he order should be considered as exra parameers. 6. Conclusions Fracional-order models can increase he flexibiliy and degrees of freedom by means of fracional parameers. They are discussed in many fields and generae some new conceps. In his paper, based on fracional calculus, we esablished he fracional-order sae-space averaging model of he Buck- Boos converer in PCCM. And some influences of he order on he quiescen operaing poin and he low-frequency characerisics of he converer are invesigaed by heoreical analyses and numerical and circui simulaions. We have proved ha fracional componens increase he wo ripples (he oupu volage ripple ΔV C and he inducor curren ripple Δi ), bu wih he decrease of he order of energy sorage elemens, he response of he converer becomes faser and he overshoo becomes smaller. In heory, all he acual energy sorage elemens should be fracional order. Tha may be an imporan reason why he acual device is always imperfec. Therefore, how o deermine he order of energy sorage elemens and he relaionship beween he fracional

10 Mahemaical Problems in Engineering i (A) Ineger-order model Fracional-order model Time (s) Figure 5: The inducor curren. order and performances of he acual energy sorage elemens are our fuure research direcions. Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens This work was suppored by he Naional Naural Science Foundaion of China (Gran nos , 5777, and 52796), Scienific Research Program Funded by Shaanxi Provincial Educaion Deparmen (Program no. 5JK537), and Docoral Scienific Research Foundaion of Xi an Polyechnic Universiy (Gran no. BS525). References K. B. Oldham and J. Spanier, The Fracional Calculus,Academic Press, New York, NY, USA, I. Podlubny, Fracional Differenial Equaions, Academic Press, San Diego, Calif, USA, I.S.JesusandJ.A.T.Machado, Developmenoffracional order capaciors based on elecrolye processes, Nonlinear Dynamics,vol.56,no.-2,pp.45 55,29. 4 F.-Q. Wang and X.-K. Ma, Transfer funcion modeling and analysis of he open-loop Buck converer using he fracional calculus, Chinese Physics B, vol.22,no.3,aricleid356, 23. 5A.Jalloul,J.-C.Trigeassou,K.Jelassi,andP.Melchior, Fracional order modeling of roor skin effec in inducion machines, Nonlinear Dynamics, vol. 73, no. -2, pp. 8 83, I. Peráš, Chaos in he fracional-order Vola s sysem: modeling and simulaion, Nonlinear Dynamics, vol. 57, no. -2, pp. 57 7, S. I. R. Arias, D. R. Muñoz, J. S. Moreno, S. Cardoso, R. Ferreira, and P. J. P. Freias, Fracional modeling of he AC largesignal frequency response in magneoresisive curren sensors, Sensors,vol.3,no.2,pp ,23. 8 A. Charef, Modeling and analog realizaion of he fundamenal linear fracional order differenial equaion, Nonlinear Dynamics,vol.46,no.-2,pp.95 2,26. 9F.Q.WangandX.K.Ma, Modelingandanalysisofhe fracional order Buck converer in DCM operaion by using fracional calculus and he circui-averaging echnique, Journal of Power Elecronics,vol.3,no.6,pp.8 5,23. A. G. Radwan and M. E. Fouda, Opimizaion of fracionalorder RC filers, Circuis, Sysems, and Signal Processing, vol. 32,no.5,pp ,23. I. Podlubny, Fracional-order sysems and PI-lambda-D-muconrollers, IEEE Transacions on Auomaic Conrol, vol. 44, no., pp , I. Peráš, Chaos in fracional-order populaion model, Inernaional Bifurcaion and Chaos,vol.22,no.4,Aricle I572, S. Weserlund and. Eksam, Capacior heory, IEEE Transacions on Dielecrics and Elecrical Insulaion, vol.,no.5,pp , S. Weserlund, Dead Maer has Memory!, Causal Consuling, Kalmar, Sweden, 22. 5J.A.T.MachadoandA.M.S.F.Galhano, Fracionalorder inducive phenomena based on he skin effec, Nonlinear Dynamics,vol.68,no.-2,pp.7 5,22. 6 T. C. Haba, G. Ablar, T. Camps, and F. Olivie, Influence of he elecrical parameers on he inpu impedance of a fracal srucure realised on silicon, Chaos, Solions & Fracals,vol.24, no. 2, pp , T.C.Haba,G..oum,andG.Ablar, Ananalyicalexpression for he inpu impedance of a fracal ree obained by a microelecronical process and experimenal measuremens of is noninegral dimension, Chaos, Solions & Fracals,vol.33,no.2,pp , 27. 8R.Marinez,Y.Bolea,A.Grau,andH.Marinez, Fracional DC/DC converer in solar-powered elecrical generaion sysems, in Proceedings of he IEEE Conference on Emerging Technologies & Facory Auomaion (ETFA 9), pp. 6,Mallorca, Spain, Sepember F.-Q. Wang and X.-K. Ma, Fracional order modeling and simulaion analysis of Boos converer in coninuous conducion mode operaion, Aca Physica Sinica, vol. 6, no. 7, Aricle ID 756, 2. 2 N.-N. Yang, C.-X. iu, and C.-J. Wu, Modeling and dynamics analysis of he fracional-order Buck-Boos converer in coninuous conducion mode, Chinese Physics B, vol. 2,no. 8, Aricle ID 853, D. Ma and W.-H. Ki, Fas-ransien PCCM swiching converer wih freewheel swiching conrol, IEEE Transacions on Circuis and Sysems II: Express Briefs,vol.54,no.9,pp , F. Zhang and J. P. Xu, A novel PCCM boos PFC converer wih fas dynamic response, IEEE Transacions on Indusrial Elecronics, vol. 58, no. 9, pp , Y. Yang,. Sun, and X. B. Wu, A single-inducor dualoupu buck converer wih self-adaped PCCM mehod, in ProceedingsofheIEEEInernaionalConferenceofElecron Devices and Solid-Sae Circuis (Edssc 9), pp.87 9,Xi an, China, December I. Peras, Fracional Order Nonlinear Sysems Modeling, Analysis and Simulaion, Springer, New York, NY, USA, T. T. Harley and C. F. orenzo, The error incurred in using he capuo-derivaive laplace-ransform, in Proceedings of he

11 Mahemaical Problems in Engineering ASME Inernaional Design Engineering Technical Conferences and Compuers and Informaion in Engineering Conference,vol. 4, pp , San Diego, Calif, USA, Sepember J. Sabaier, M. Merveillau, R. Mali, and A. Ousaloup, How o impose physically coheren iniial condiions o a fracional sysem? Communicaions in Nonlinear Science and Numerical Simulaion,vol.5,no.5,pp ,2. 27 J. C. Trigeassou, N. Maamri, J. Sabaier, and A. Ousaloup, Sae variables and ransiens of fracional order differenial sysems, Compuers & Mahemaics wih Applicaions,vol.64,no.,pp , T. T. Harley, R. J. Veillee, J.. Adams, and C. F. orenzo, Energy sorage and loss in fracional-order circui elemens, IET Circuis Devices & Sysems,vol.9,no.3,pp , T. T. Harley, R. J. Veillee, C. F. orenzo, and J.. Adams, On he energy sored in fracional-order elecrical elemens, in Proceedings of he ASME Inernaional Design Engineering Technical Conferences and Compuers and Informaion in Engineering Conference, 23,vol.4,ASME,24. 3 T. T. Harley, C. F. orenzo, J.-C. Trigeassou, and N. Maamri, Equivalence of hisory-funcion based and infiniedimensional-sae iniializaions for fracional-order operaors, Compuaional and Nonlinear Dynamics, vol.8,no. 4, Aricle ID 44, D. S. Ma, W.-H. Ki, and C.-Y. Tsui, A pseudo-ccm/dcm SIMO swiching converer wih freewheel swiching, IEEE Solid-Sae Circuis, vol. 38, no. 6, pp. 7 4, R. W. Erickson, Fundamenals of Power Elecronics, Chapman and Hall, New York, NY, USA, A. Ousaloup, F. evron, B. Mahieu, and F. M. Nano, Frequency-band complex nonineger differeniaor: characerizaion and synhesis, IEEE Transacions on Circuis and Sysems I: Fundamenal Theory and Applicaions, vol.47,no., pp.25 39,2.

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