Research ArticleModeling and Characteristics Analysis for a Buck-BoostConverter in Pseudo-Continuous Conduction Mode Based onFractional Calculus

Ningning Yang12 Chaojun Wu3 Rong Jia12 and Chongxin Liu4

1State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area Xirsquoan University of Technology Xirsquoan 710048 China2Institute of Water Resources and Hydro-Electric Engineering Xirsquoan University of Technology Xirsquoan 710048 China3College of Electronics and Information Xirsquoan Polytechnic University Xirsquoan 710048 China4School of Electrical Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Chaojun Wu chaojunwustuxjtueducn

Received 31 October 2015 Revised 9 January 2016 Accepted 14 January 2016

Academic Editor Luis J Yebra

Copyright copy 2016 Ningning Yang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In recent days fractional calculus (FC) has been accepted as a novel modeling tool that can extend the descriptive power of thetraditional calculus Fractional-order descriptiveness can increase the flexibility and degrees of freedom of the model by meansof fractional parameters Based on the fact that real capacitors and inductors are ldquointrinsicrdquo fractional order fractional calculusis introduced into the modeling process to establish a fractional-order state-space averaging model of the Buck-Boost converterin pseudo-continuous conduction mode (PCCM) Orders of the model are considered as extra parameters and these parametershave significant influences on the performance of the model The inductor current the inductor current ripple the amplitudeof the output voltage and the transfer functions of the fractional-order model are all related to orders The contrast simulationexperiments are conducted to investigate the performance of integer-order and fractional-order Buck-Boost converters in PCCMResults of numerical and circuit simulations demonstrate that the proposed theoretical analysis is effective the fractional-ordermodel of the Buck-Boost converter in PCCM has certain theoretical and practical significance for modeling and performanceanalysis of other electrical or electronic equipment

1 Introduction

Fractional calculus is an oldmathematical topic which can goback to 1695 when LrsquoHospital wrote to Leibniz hementionedthis kind of calculation [1 2] Over the past 300 years severalmathematicians contributed to this mathematical subjectIn recent years fractional calculus has been accepted asa new instrument that can extend the descriptive powerof the traditional calculus It has turned out that manyphenomena in widespread fields of science and engineeringcan be described very successfully by fractional-ordermodels[3ndash9] Comparing with integer-order models the significantadvantage of fractional-order models is that they are char-acterized by hereditary and memory properties And it hasbeen demonstrated that fractional-order models can increase

the flexibility and degrees of freedom by means of fractional-order parameters [10ndash12]

The theory and applications of fractional-order compo-nents had a considerable progress during the last two decadesResearchers have reported the ldquointrinsicrdquo fractional-orderbehavior of real objects such as capacitors and inductorsWesterlund and his colleagues constructed the fractional-order model of the capacitor and measured the orders ofsome real capacitors under different dielectrics by experiment[13 14] Westerlund also demonstrated that real inductorsare fractional order and measured the orders of some realinductors [14] Using different fractal structures fractional-order capacitors with different orders were implementedby Jesus and Machado [3] Based on skin effect Machadoand Galhano proposed that fractional-order inductors with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 6835910 11 pageshttpdxdoiorg10115520166835910

2 Mathematical Problems in Engineering

different orders can be constructed [15] Real fractional-ordercapacitors were created by Haba and his colleagues [16 17]These realizations of fractional-order capacitors and induc-tors indicate the possibility of fractional-order componentsand models being employed in practical applications

Capacitors and inductors are the fundamental part of theBuck-Boost converter Based on the facts that real capacitorsand inductors are all fractional-order components the ideaof establishing the fractional-order model of the Buck-Boostconverter seems to be far more sensible It can be provedthat the order of the model has significant influence onthe performance of the Buck-Boost converter and can beconsidered as an extra parameter Up to date few fractional-order models of converter have been established [18 19]Martinez and his colleagues established the fractional-ordermodel of the Buck-Boost converter but only the capaci-tor was considered as the fractional-order circuit elementWang and Ma constructed the fractional-order model of theBoost converter in continuous conduction mode (CCM) anddiscontinuous conduction mode (DCM) [19] Yang and hiscolleagues proposed the fractional-order model of the Buck-Boost converter in CCM [20] but no circuit simulationsor experiments were presented to verify these models Inthis paper a fractional-order state-space averaging model ofthe Buck-Boost converter in pseudo-continuous conductionmode (PCCM) is established numerical and circuit simula-tion experiments are presented to verify the efficiency of theproposed theoretical analysis

Generally Buck-Boost converters operate in CCM orDCM In recent reports pseudo-continuous conductionmode (PCCM) is proposed which is the third operationmode Comparing with operating in DCM in PCCM thecurrent handling capability of the converter is improvedand the current ripple and voltage ripple are reducedAnd a single-pole behavior in the control-to-output transferfunction is exhibited the load transient response is muchfaster than operating in CCM and DCM [21ndash23] Thereforethe study of the fractional-order model of the Buck-Boostconverter in PCCM is an important theoretical issue and haspractical engineering value

In this paper we established the fractional-order state-space averagingmodel of the Buck-Boost converter in PCCMand analyzed the influence of orders on some dynamicalproperties of the fractional-order model The rest of thispaper is organized as follows Section 2 is the preliminariessome basic concepts of fractional calculus and the basicfractional-order inductor and capacitor models are intro-duced In Section 3 based on fractional calculus we estab-lished a fractional-order state-space averaging model of theBuck-Boost converter in PCCM In Section 4 the quiescentoperation point and transfer functions of the fractional-orderconverter are analyzed Orders of the fractional-order modelare proved to be extra parameters and have influence onthe performance of the converter In Section 5 numericaland circuit simulation experiments are implemented to verifythe correctness of theoretical analysis and the proposedfractional-order model Finally in Section 6 influences oforders on the fractional-order model are summarized

2 Preliminaries

21 Fractional Calculus Fractional calculus allows opera-tions of ordinary differentiations and integrations to non-integer-order It is a generalization of traditional calculus butits applicability ismuchwider [24]The fundamental operator119886119863120572

119905 where 120572 (120572 isin R) is the order and 119886 and 119905 are the bounds

of the operation is defined as

119886119863120572

119905=

119889120572

119889119905120572 120572 gt 0

1 120572 = 0

int

119905

119886

(119889120591)minus120572

120572 lt 0

(1)

The three best known definitions for fractional calculusare Grunwald-Letnikov (GL) definition Riemann-Liouville(RL) definition and Caputo definition [2]

Caputo definition is defined as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)int

119905

119886

119891(119899)

(120591)

(119905 minus 120591)120572minus119899+1

119889120591 (2)

where Γ(sdot) is the Gamma function and 119899 isin 119873 is the firstinteger which is not less than 120572 119899 minus 1 lt 120572 lt 119899

The Laplace transform of Caputo fractional derivativesatisfies

119871 119886119863120572

119905119891 (119905) = 119904

120572

119865 (119904) minus

119899minus1

sum

119896=0

119904120572minus119896minus1

119891(119898)

(0)

119899 minus 1 lt 120572 lt 119899

(3)

For zero initial conditions the Laplace transform offractional derivatives has the following form

119871 119886119863120572

119905119891 (119905) = 119904

120572

119865 (119904) (4)

Theoretical calculations are based on the Caputo deriva-tive and more specifically on the initial condition 119909(0)Hartley and Lorenzo discuss the error incurred in usingthe Caputo-derivative Laplace transform [25] It is now wellestablished that more precisely initial state of a FractionalDifferential Equation can be represented by the weightedintegral of 119911(119908 0) [26ndash30] Nevertheless under the premisethat the error is acceptable and for convenience the Caputoderivative and the initial condition 119909(0) are adopted

22 The Fractional-Order Model of Capacitors and InductorsThe fractional-order model of capacitors stems from Curiersquosempirical law which is proposed in 1889 [13]

119894 (119905) =1198810

ℎ1119905119898 0 lt 119898 lt 1 (5)

where ℎ1is a constant related to the capacitance and the kind

of the dielectric119898 is a constant close to 1 related to the lossesof the capacitor 119881

0is the DC voltage applied at 119905 = 0

Mathematical Problems in Engineering 3

S1

S2

D2

Vin iL

D1

CL R

minus

+

Figure 1 Buck-Boost converter in PCCM

In 1994 based on fractional calculus Westerlund andEkstam developed a new capacitor theory [13]

119894 (119905) = 119862119891

119889120573

119906 (119905)

119889119905120573 0 lt 120573 lt 1 (6)

where 119862119891is the capacitance of the capacitor related to the

kind of dielectric And120573 is the order related to the losses of thecapacitorWesterlund and Ekstamprovided various capacitordielectrics with appropriated order 120573 by experiment

Westerlund also created the fractional-order model ofinductors [14]

119906 (119905) = 119871119891

119889120572

119894 (119905)

119889119905120572 0 lt 120572 lt 1 (7)

where 119871119891is the inductance of the inductor and 120572 is the order

related to the proximity effectIn engineering applications orders of capacitors and

inductors are close to 1 [13] In order to ensure that the powerelectronic devices can work properly we redefine a fractionalorder between 095 and 1

3 The State-Space Averaging Model ofthe Fractional-Order Buck-Boost Converterin PCCM

31 The Fractional-Order Mathematical Model of the Buck-Boost Converter The Buck-Boost converter can operate asBuck (for voltage stepdown) or Boost (for voltage stepup)converter inverting the voltage polarity Connecting aninductor in parallel with a power switch makes the converteroperate in PCCM [31] The circuit diagram is shown inFigure 1 There are two fully controlled switching devices 119878

1

and 1198782 two diodes 119863

1and 119863

2 one DC input voltage 119881in

one load resistor 119877 and two fractional-order energy storageelements 119871 and 119862 in the circuit

In Figure 2 where 119879 is the switching time period 119879on isthe on time 119879off is the off time for the switch and 119889

1+1198892+1198893

= 1 There are three states for the pseudo-continuous mode

State 1 When 0 lt 119905 le 1198891119879 the circuit topology is shown in

Figure 3 Switch 1198781is on 119878

2is off and diodes119863

12are reverse-

biased The inductor current 119894119871rises

iL

L

0

0

0

0

Ton

Ton

Toff

Toff

ΔiL

t

t

t

t

d1T d2T d3T

Pulse 2

Pulse 1

Figure 2 Inductor voltage and current of Buck-Boost converter inPCCM

S1

S2

D2

Vin iL

D1

CL R

minus

+

Figure 3 Schematic diagram of Buck-Boost converter in State 1

The expression of the fractional-order mathematicalmodel is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

= [

[

0 0

0 minus1

119862119877

]

]

[

119894119871(119905)

V119862(119905)] + [

[

1

119871

0

]

]

Vin (119905) (8)

State 2 When 1198891119879 lt 119905 le (119889

1+ 1198892)119879 as shown in Figure 4

switches 11987812

and diode 1198632are off and diode 119863

1is forward-

biased and behaves as a short circuit the inductor current 119894119871

ramps downThe expression of the fractional-order mathematical

model is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01

119871

minus1

119862minus1

119862119877

]]

]

[

119894119871(119905)

V119862(119905)] + [

0

0] Vin (119905) (9)

State 3 When (1198891+1198892)119879 lt 119905 le 119879 switch 119878

1and diode119863

1are

off switch 1198782and diode 119863

2are on and the inductor current

119894119871maintains a constant theoretically The circuit topology is

shown in Figure 5

4 Mathematical Problems in Engineering

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 4 Schematic diagram of Buck-Boost converter in State 2

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 5 Schematic diagram of Buck-Boost converter in State 3

The expression of the fractional-order mathematicalmodel is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

= [

[

0 0

0 minus1

119862119877

]

]

[

119894119871(119905)

V119862(119905)] + [

0

0] Vin (119905) (10)

As shown in (8) (9) and (10) the inductor order 120572 andthe capacitor order 120573 can be considered as extra parametersof the fractional-order mathematical models in PCCM Theinfluence of these extra parameters on the performance of theconverter will be discussed in the next section

32 The Fractional-Order State-Space Averaging Model ofthe Buck-Boost DCDC Converter To remove the switchingharmonics waveforms over one switching period should beaveraged [32] The average value of variable 119909(119905) which is anarbitrary circuit variable of the Buck-Boost converter in oneswitching period is defined as follows

⟨119909 (119905)⟩119879=1

119879int

119905+119879

119905

119909 (120591) 119889120591 (11)

Its fractional-order form is

119889120572

⟨119909 (119905)⟩119879

119889119905120572=

119889120572

((1119879) int119905+119879

119905

119909 (120591) 119889120591)

119889119905120572

=1

119879int

119905+119879

119905

(119889120572

119909 (120591)

119889120591120572)119889120591 = ⟨

119889120572

119909 (119905)

119889119905120572⟩

(12)

where 120572 is the order and 095 lt 120572 lt 1

Let us consider the fractional-order state-space averagingmodel of Buck-Boost converter in PCCM

[[[

[

119889120572

⟨119894119871(119905)⟩119879

119889119905120572

119889120573

⟨V119862(119905)⟩119879

119889119905120573

]]]

]

=[[

[

01198892(119905)

119871

minus1198892(119905)

119862minus1

119862119877

]]

]

[

⟨119894119871(119905)⟩119879

⟨V119862(119905)⟩119879

]

+ [

[

1198891(119905)

119871

0

]

]

⟨Vin (119905)⟩119879

(13)

Average values of circuit variables are defined as thefollowing form

⟨119894119871(119905)⟩119879= 119868119871+ 119871(119905)

⟨V119862(119905)⟩119879= 119881119862+ V119862(119905)

⟨Vin (119905)⟩119879 = 119881in + Vin (119905)

1198891(119905) = 119863

1+ 1(119905)

1198892(119905) = 119863

2+ 2(119905)

(14)

where variables in capital letters represent DC componentsand variables with the symbol ldquordquo above them represent ACcomponents which are much smaller in magnitude than thecorresponding DC component

Equation (13) can be rewritten as

[[[

[

119889120572

119868119871

119889119905120572+119889120572

119871(119905)

119889119905120572

119889120573

119881119862

119889119905120573+119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119868119871+ 119871(119905)

119881119862+ V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

(119881in + Vin (119905))

(15)

DC components in (15) are as follows

[[[

[

119889120572

119868119871

119889119905120572

119889120573

119881119862

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[

119868119871

119881119862

] + [

[

1198631

119871

0

]

]

119881in (16)

The quiescent operation point is

[

119868119871

119881119862

] = minus[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

minus1

[

[

119863

119871

0

]

]

119881in =[[[

[

119881in11986311198632

2119877

minus119881in11986311198632

]]]

]

(17)

The voltage ratio is defined as follows

119872 =119881119862

119881in= minus

1198631

1198632

(18)

Mathematical Problems in Engineering 5

AC components of (15) are

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

Vin (119905)

+

[[[

[

02(119905)

119871

minus2(119905)

1198620

]]]

]

[

119868119871

119881119862

] + [

[

1(119905)

119871

0

]

]

119881in

(19)

where 2(119905)119894119871(119905) 2(119905)V119862(119905) and

1(119905)Vin(119905) are high order

small signal terms which can be omitted Then the smallsignal AC equation of Buck-Boost converter in PCCM canbe expressed as follows

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631

119871

0

]

]

Vin (119905)

+[[

[

01

119871

minus1

1198620

]]

]

[

119868119871

119881119862

] 2(119905)

+ [

[

1

119871

0

]

]

119881in (119905)

(20)

4 Performance Analysis ofthe Fractional-Order Model of theBuck-Boost Converter in PCCM

In this section the quiescent operation point and transferfunctions of the AC small signal model will be analyzedbecause they have practical significance for the designingof various parameters and reflect the performance of theconverter

The inductor current ripple and output voltage rippleof the fractional-order Buck-Boost converter in PCCM areshown in Figure 6

t0

0t

d1T d1Td2T d3T

iLΔiL

C

c(t0)

ΔC

iL(t0)

Figure 6 The inductor current ripple and output voltage ripple

The small ripple approximation is defined as replacingwaveforms with their low-frequency averaged values

119889120572

119894119871(119905)

119889119905120572asymp⟨Vin (119905)⟩119879

119871 (21)

Using Caputo definition we can get the correspondinginitial condition as

119863119896

119894119871(0) = 119894

(119896)

1198710 119896 = 0 1 119898 minus 1 (22)

Solving fractional-order differential equation (21) we canget

119894119871(119905) =

1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

⟨Vin (119905)⟩119879119871

119889120591

+

119898minus1

sum

119896=0

119905119896

119896119894(119896)

1198710(119905)

(23)

In PCCM the initial value of inductor current is given as119894119871(1199050) = 119894119871min

Then

119894119871max = 119894119871 (1198891119879) =

119881in (1198631119879)120572

119871Γ (120572) 120572+ 119894119871min (24)

The inductor current ripple is

Δ119894119871=119881in (1198631119879)

120572

119871Γ (120572) 120572 (25)

The maximum and minimum values of inductor currentare

119894119871max = 119868119871 +

1

2Δ119894119871=119881in11986311198632

2119877+119881in (1198631119879)

120572

2119871Γ (120572) 120572

119894119871min = 119868119871 minus

1

2Δ119894119871=119881in11986311198632

2119877minus119881in (1198631119879)

120572

2119871Γ (120572) 120572

(26)

According to Adomian decomposition scheme one canget

119889120573

119909 (119905)

119889119905120573= 119860119909 (119905) + 119891 (119909) (27)

6 Mathematical Problems in Engineering

In States 2 and 3 the load is powered by the capacitorTheoutput voltage of the converter is negative which satisfies thestate equation and the amplitude is monotone decreasing

119889120573V119862(119905)

119889119905120573= minus

V119862(119905)

119862119877 (28)

where 119860 = minus1119862119877 119891(119909) = 0 and 095 lt 120573 lt 1The solution of (28) is

V119862(119905) = 119864

1205731(119860119905120573

)119881max (29)

where 119881max is the initial value of V119862(119905) which means V119862(1199050) =

119881max for 1199050 = 0When 119905 = (119889

1+ 1198893)119879 the output voltage is

V119862((1198893+ 1198891) 119879) = 119881max1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862) (30)

The variation of the output voltage is

ΔV = 119881max(1 minus 1198641205731(minus((1198893+ 1198891) 119879)120573

119877119862)) (31)

According to (17) and (31) it yields

119881max = 119881119862 +1

2ΔV

=119881in11986311198632

+119881max2

(1 minus 1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862))

=2119881in1198631

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(32)

Taking (32) to (31) yields

ΔV =2119881in1198631 (1 minus 1198641205731 (minus ((1198893 + 1198891) 119879)

120573

119877119862))

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(33)

We can find from (25) and (33) that the inductor currentripple and the output voltage ripple are closely related to theorder of energy storage elements The ripple increases withthe decreasing of the order

Under conditions of zero initial value the Laplace trans-form of converter equations is

119904120573V119862(119904) = minus

1198632119871(119904)

119862minus1198681198712(119904)

119862minusV119862(119904)

119862119877

119904120572

119871(119904) =

1198631Vin (119904) + 119881in1 (119904) + 1198632V119862 (119904) + 1198811198622 (119904)

119871

(34)

Set the duty cycle variation to be zero

1(119904) = 0

2(119904) = 0

(35)

According to (34) one can get the Vin-to-V119862 and the Vin-to-119894119871transfer function as follows

119866V119862Vin (119904) =V119862(119904)

Vin (119904)

100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

= minus11986311198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

119866119894119871Vin (119904) =

119871(119904)

Vin (119904)

1003816100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

=

(11986311198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(36)

Set the AC input voltage variation and the duty cycle 2to be

zero

Vin (119904) = 0

2(119904) = 0

(37)

The 1-to-V119862and the

1-to-119894119871transfer functions are

119866V1198621198891 (119904) =V119862(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

= minus119881in1198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198891

(119904) =119871(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

=

(119881in1198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(38)

Set the AC input voltage variation and the duty cycle 1to be

zero

Vin (119904) = 0

1(119904) = 0

(39)

According to (17) and (34) the 2-to-V119862and the

2-to-119894119871

transfer functions are

119866V1198621198892 (119904) =V119862(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11987111986311198634

2119877) 119904120572

+ 1198811198621198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198892

(119904) =119871(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11986311198771198633

2) (119862119877119904

120573

+ 2)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(40)

Mathematical Problems in Engineering 7

Table 1 Parameters of the Buck-Boost converter

Parameter Symbol ValueInput voltage 119881in 50VCapacitor 119862 100120583FInductance 119871 1mHSwitching frequency 119891 20 kHzDuty cycle 119863

104

Duty cycle 1198633

04The order 120572 0951The order 120573 0951

We can find from (36) (38) and (40) that orders of theinductor and capacitor have influence on transfer functionsThatmeans fractional orders can be seen as extra parametersWhen orders are 1 we can get integer-order transfer functionsfrom the fractional-order model The integer-order model isa special case of the fractional-order model

5 Experimental Researches

In this section some numerical and circuit simulation exper-iments are presented to verify the fractional-ordermodel andthe influence of the order on performances of the converterin PCCM We choose 095 as the order which is close to 1Parameters of the Buck-Boost converter are listed in Table 1

We select the load resistor as 119877 = 20Ω to ensure thatthe converter is operating in PCCM Difference values ofthe quiescent operating point between the integer-order andfractional-order Buck-Boost converter model are shown inTable 2

In Table 2 we can find that the output voltage ripple andthe inductor ripple rise with the decreasing of the orderThatmeans the order of the Buck-Boost converter impacts theoutput characteristics of the converter obviously

51 Numerical Simulation In numerical simulation Oustal-ouprsquos recursive approximation is adopted [33] The switchingfrequency of the converter is selected as 119891 = 20 kHz thecorresponding rotational frequency is 120596 = 2120587119891 = 1256 times

105 and parameters of Oustaloup are 120596

ℎ= 2 times 10

5 120596119887=

5 times 10minus6 and119873 = 6

Figure 7 shows the inductor current the detailed valuesare listed in Table 3 We can see that in the inductor currentripple maximum and minimum values of the inductorcurrent increase significantly in the fractional-order modelThat means the effect of the inductor order on the inductorcurrent should be appropriately considered

The order also influences the output voltage As shownin Figure 8 and Table 4 in the output voltage ripple themaximum and minimum values of output voltage increasesignificantly in the fractional-order model

52 Circuitry Simulation Experiment To further verify theresults of the theoretical analysis the circuit simulationexperiments are presented

03 03 03001 03001 03002 0300223

235

24

245

25

255

26

265

27

Time (s)

IOFO (095)

i L(A

)

Figure 7 The inductor current

03 03 03001 03001 03002 03002minus105

minus100

minus95

Time (s)

IOFO (095)

C(V

)

Figure 8 The output voltage

By using resistorcapacitor or resistorinductor networkswe can construct an approximation circuit of the fractional-order circuit elements We choose the chain structure as thefractance unit to approximately achieve the fractional-ordercircuit elements

The chain fractance unit of fractional-order capacitor isshown in Figure 9

The transfer function model of the capacitor is

119867(119904) = 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (41)

We choose 120573 = 095 119862 = 100 120583F and 119899 = 6We can calculate resistors and inductors in Figure 9 by themethod of undetermined coefficients comparing the input

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

different orders can be constructed [15] Real fractional-ordercapacitors were created by Haba and his colleagues [16 17]These realizations of fractional-order capacitors and induc-tors indicate the possibility of fractional-order componentsand models being employed in practical applications

Capacitors and inductors are the fundamental part of theBuck-Boost converter Based on the facts that real capacitorsand inductors are all fractional-order components the ideaof establishing the fractional-order model of the Buck-Boostconverter seems to be far more sensible It can be provedthat the order of the model has significant influence onthe performance of the Buck-Boost converter and can beconsidered as an extra parameter Up to date few fractional-order models of converter have been established [18 19]Martinez and his colleagues established the fractional-ordermodel of the Buck-Boost converter but only the capaci-tor was considered as the fractional-order circuit elementWang and Ma constructed the fractional-order model of theBoost converter in continuous conduction mode (CCM) anddiscontinuous conduction mode (DCM) [19] Yang and hiscolleagues proposed the fractional-order model of the Buck-Boost converter in CCM [20] but no circuit simulationsor experiments were presented to verify these models Inthis paper a fractional-order state-space averaging model ofthe Buck-Boost converter in pseudo-continuous conductionmode (PCCM) is established numerical and circuit simula-tion experiments are presented to verify the efficiency of theproposed theoretical analysis

Generally Buck-Boost converters operate in CCM orDCM In recent reports pseudo-continuous conductionmode (PCCM) is proposed which is the third operationmode Comparing with operating in DCM in PCCM thecurrent handling capability of the converter is improvedand the current ripple and voltage ripple are reducedAnd a single-pole behavior in the control-to-output transferfunction is exhibited the load transient response is muchfaster than operating in CCM and DCM [21ndash23] Thereforethe study of the fractional-order model of the Buck-Boostconverter in PCCM is an important theoretical issue and haspractical engineering value

In this paper we established the fractional-order state-space averagingmodel of the Buck-Boost converter in PCCMand analyzed the influence of orders on some dynamicalproperties of the fractional-order model The rest of thispaper is organized as follows Section 2 is the preliminariessome basic concepts of fractional calculus and the basicfractional-order inductor and capacitor models are intro-duced In Section 3 based on fractional calculus we estab-lished a fractional-order state-space averaging model of theBuck-Boost converter in PCCM In Section 4 the quiescentoperation point and transfer functions of the fractional-orderconverter are analyzed Orders of the fractional-order modelare proved to be extra parameters and have influence onthe performance of the converter In Section 5 numericaland circuit simulation experiments are implemented to verifythe correctness of theoretical analysis and the proposedfractional-order model Finally in Section 6 influences oforders on the fractional-order model are summarized

2 Preliminaries

21 Fractional Calculus Fractional calculus allows opera-tions of ordinary differentiations and integrations to non-integer-order It is a generalization of traditional calculus butits applicability ismuchwider [24]The fundamental operator119886119863120572

119905 where 120572 (120572 isin R) is the order and 119886 and 119905 are the bounds

of the operation is defined as

119886119863120572

119905=

119889120572

119889119905120572 120572 gt 0

1 120572 = 0

int

119905

119886

(119889120591)minus120572

120572 lt 0

(1)

The three best known definitions for fractional calculusare Grunwald-Letnikov (GL) definition Riemann-Liouville(RL) definition and Caputo definition [2]

Caputo definition is defined as

119886119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)int

119905

119886

119891(119899)

(120591)

(119905 minus 120591)120572minus119899+1

119889120591 (2)

where Γ(sdot) is the Gamma function and 119899 isin 119873 is the firstinteger which is not less than 120572 119899 minus 1 lt 120572 lt 119899

The Laplace transform of Caputo fractional derivativesatisfies

119871 119886119863120572

119905119891 (119905) = 119904

120572

119865 (119904) minus

119899minus1

sum

119896=0

119904120572minus119896minus1

119891(119898)

(0)

119899 minus 1 lt 120572 lt 119899

(3)

For zero initial conditions the Laplace transform offractional derivatives has the following form

119871 119886119863120572

119905119891 (119905) = 119904

120572

119865 (119904) (4)

Theoretical calculations are based on the Caputo deriva-tive and more specifically on the initial condition 119909(0)Hartley and Lorenzo discuss the error incurred in usingthe Caputo-derivative Laplace transform [25] It is now wellestablished that more precisely initial state of a FractionalDifferential Equation can be represented by the weightedintegral of 119911(119908 0) [26ndash30] Nevertheless under the premisethat the error is acceptable and for convenience the Caputoderivative and the initial condition 119909(0) are adopted

22 The Fractional-Order Model of Capacitors and InductorsThe fractional-order model of capacitors stems from Curiersquosempirical law which is proposed in 1889 [13]

119894 (119905) =1198810

ℎ1119905119898 0 lt 119898 lt 1 (5)

where ℎ1is a constant related to the capacitance and the kind

of the dielectric119898 is a constant close to 1 related to the lossesof the capacitor 119881

0is the DC voltage applied at 119905 = 0

Mathematical Problems in Engineering 3

S1

S2

D2

Vin iL

D1

CL R

minus

+

Figure 1 Buck-Boost converter in PCCM

In 1994 based on fractional calculus Westerlund andEkstam developed a new capacitor theory [13]

119894 (119905) = 119862119891

119889120573

119906 (119905)

119889119905120573 0 lt 120573 lt 1 (6)

where 119862119891is the capacitance of the capacitor related to the

kind of dielectric And120573 is the order related to the losses of thecapacitorWesterlund and Ekstamprovided various capacitordielectrics with appropriated order 120573 by experiment

Westerlund also created the fractional-order model ofinductors [14]

119906 (119905) = 119871119891

119889120572

119894 (119905)

119889119905120572 0 lt 120572 lt 1 (7)

where 119871119891is the inductance of the inductor and 120572 is the order

related to the proximity effectIn engineering applications orders of capacitors and

inductors are close to 1 [13] In order to ensure that the powerelectronic devices can work properly we redefine a fractionalorder between 095 and 1

3 The State-Space Averaging Model ofthe Fractional-Order Buck-Boost Converterin PCCM

31 The Fractional-Order Mathematical Model of the Buck-Boost Converter The Buck-Boost converter can operate asBuck (for voltage stepdown) or Boost (for voltage stepup)converter inverting the voltage polarity Connecting aninductor in parallel with a power switch makes the converteroperate in PCCM [31] The circuit diagram is shown inFigure 1 There are two fully controlled switching devices 119878

1

and 1198782 two diodes 119863

1and 119863

2 one DC input voltage 119881in

one load resistor 119877 and two fractional-order energy storageelements 119871 and 119862 in the circuit

In Figure 2 where 119879 is the switching time period 119879on isthe on time 119879off is the off time for the switch and 119889

1+1198892+1198893

= 1 There are three states for the pseudo-continuous mode

State 1 When 0 lt 119905 le 1198891119879 the circuit topology is shown in

Figure 3 Switch 1198781is on 119878

2is off and diodes119863

12are reverse-

biased The inductor current 119894119871rises

iL

L

0

0

0

0

Ton

Ton

Toff

Toff

ΔiL

t

t

t

t

d1T d2T d3T

Pulse 2

Pulse 1

Figure 2 Inductor voltage and current of Buck-Boost converter inPCCM

S1

S2

D2

Vin iL

D1

CL R

minus

+

Figure 3 Schematic diagram of Buck-Boost converter in State 1

The expression of the fractional-order mathematicalmodel is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

= [

[

0 0

0 minus1

119862119877

]

]

[

119894119871(119905)

V119862(119905)] + [

[

1

119871

0

]

]

Vin (119905) (8)

State 2 When 1198891119879 lt 119905 le (119889

1+ 1198892)119879 as shown in Figure 4

switches 11987812

and diode 1198632are off and diode 119863

1is forward-

biased and behaves as a short circuit the inductor current 119894119871

ramps downThe expression of the fractional-order mathematical

model is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01

119871

minus1

119862minus1

119862119877

]]

]

[

119894119871(119905)

V119862(119905)] + [

0

0] Vin (119905) (9)

State 3 When (1198891+1198892)119879 lt 119905 le 119879 switch 119878

1and diode119863

1are

off switch 1198782and diode 119863

2are on and the inductor current

119894119871maintains a constant theoretically The circuit topology is

shown in Figure 5

4 Mathematical Problems in Engineering

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 4 Schematic diagram of Buck-Boost converter in State 2

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 5 Schematic diagram of Buck-Boost converter in State 3

The expression of the fractional-order mathematicalmodel is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

= [

[

0 0

0 minus1

119862119877

]

]

[

119894119871(119905)

V119862(119905)] + [

0

0] Vin (119905) (10)

As shown in (8) (9) and (10) the inductor order 120572 andthe capacitor order 120573 can be considered as extra parametersof the fractional-order mathematical models in PCCM Theinfluence of these extra parameters on the performance of theconverter will be discussed in the next section

32 The Fractional-Order State-Space Averaging Model ofthe Buck-Boost DCDC Converter To remove the switchingharmonics waveforms over one switching period should beaveraged [32] The average value of variable 119909(119905) which is anarbitrary circuit variable of the Buck-Boost converter in oneswitching period is defined as follows

⟨119909 (119905)⟩119879=1

119879int

119905+119879

119905

119909 (120591) 119889120591 (11)

Its fractional-order form is

119889120572

⟨119909 (119905)⟩119879

119889119905120572=

119889120572

((1119879) int119905+119879

119905

119909 (120591) 119889120591)

119889119905120572

=1

119879int

119905+119879

119905

(119889120572

119909 (120591)

119889120591120572)119889120591 = ⟨

119889120572

119909 (119905)

119889119905120572⟩

(12)

where 120572 is the order and 095 lt 120572 lt 1

Let us consider the fractional-order state-space averagingmodel of Buck-Boost converter in PCCM

[[[

[

119889120572

⟨119894119871(119905)⟩119879

119889119905120572

119889120573

⟨V119862(119905)⟩119879

119889119905120573

]]]

]

=[[

[

01198892(119905)

119871

minus1198892(119905)

119862minus1

119862119877

]]

]

[

⟨119894119871(119905)⟩119879

⟨V119862(119905)⟩119879

]

+ [

[

1198891(119905)

119871

0

]

]

⟨Vin (119905)⟩119879

(13)

Average values of circuit variables are defined as thefollowing form

⟨119894119871(119905)⟩119879= 119868119871+ 119871(119905)

⟨V119862(119905)⟩119879= 119881119862+ V119862(119905)

⟨Vin (119905)⟩119879 = 119881in + Vin (119905)

1198891(119905) = 119863

1+ 1(119905)

1198892(119905) = 119863

2+ 2(119905)

(14)

where variables in capital letters represent DC componentsand variables with the symbol ldquordquo above them represent ACcomponents which are much smaller in magnitude than thecorresponding DC component

Equation (13) can be rewritten as

[[[

[

119889120572

119868119871

119889119905120572+119889120572

119871(119905)

119889119905120572

119889120573

119881119862

119889119905120573+119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119868119871+ 119871(119905)

119881119862+ V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

(119881in + Vin (119905))

(15)

DC components in (15) are as follows

[[[

[

119889120572

119868119871

119889119905120572

119889120573

119881119862

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[

119868119871

119881119862

] + [

[

1198631

119871

0

]

]

119881in (16)

The quiescent operation point is

[

119868119871

119881119862

] = minus[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

minus1

[

[

119863

119871

0

]

]

119881in =[[[

[

119881in11986311198632

2119877

minus119881in11986311198632

]]]

]

(17)

The voltage ratio is defined as follows

119872 =119881119862

119881in= minus

1198631

1198632

(18)

Mathematical Problems in Engineering 5

AC components of (15) are

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

Vin (119905)

+

[[[

[

02(119905)

119871

minus2(119905)

1198620

]]]

]

[

119868119871

119881119862

] + [

[

1(119905)

119871

0

]

]

119881in

(19)

where 2(119905)119894119871(119905) 2(119905)V119862(119905) and

1(119905)Vin(119905) are high order

small signal terms which can be omitted Then the smallsignal AC equation of Buck-Boost converter in PCCM canbe expressed as follows

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631

119871

0

]

]

Vin (119905)

+[[

[

01

119871

minus1

1198620

]]

]

[

119868119871

119881119862

] 2(119905)

+ [

[

1

119871

0

]

]

119881in (119905)

(20)

4 Performance Analysis ofthe Fractional-Order Model of theBuck-Boost Converter in PCCM

In this section the quiescent operation point and transferfunctions of the AC small signal model will be analyzedbecause they have practical significance for the designingof various parameters and reflect the performance of theconverter

The inductor current ripple and output voltage rippleof the fractional-order Buck-Boost converter in PCCM areshown in Figure 6

t0

0t

d1T d1Td2T d3T

iLΔiL

C

c(t0)

ΔC

iL(t0)

Figure 6 The inductor current ripple and output voltage ripple

The small ripple approximation is defined as replacingwaveforms with their low-frequency averaged values

119889120572

119894119871(119905)

119889119905120572asymp⟨Vin (119905)⟩119879

119871 (21)

Using Caputo definition we can get the correspondinginitial condition as

119863119896

119894119871(0) = 119894

(119896)

1198710 119896 = 0 1 119898 minus 1 (22)

Solving fractional-order differential equation (21) we canget

119894119871(119905) =

1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

⟨Vin (119905)⟩119879119871

119889120591

+

119898minus1

sum

119896=0

119905119896

119896119894(119896)

1198710(119905)

(23)

In PCCM the initial value of inductor current is given as119894119871(1199050) = 119894119871min

Then

119894119871max = 119894119871 (1198891119879) =

119881in (1198631119879)120572

119871Γ (120572) 120572+ 119894119871min (24)

The inductor current ripple is

Δ119894119871=119881in (1198631119879)

120572

119871Γ (120572) 120572 (25)

The maximum and minimum values of inductor currentare

119894119871max = 119868119871 +

1

2Δ119894119871=119881in11986311198632

2119877+119881in (1198631119879)

120572

2119871Γ (120572) 120572

119894119871min = 119868119871 minus

1

2Δ119894119871=119881in11986311198632

2119877minus119881in (1198631119879)

120572

2119871Γ (120572) 120572

(26)

According to Adomian decomposition scheme one canget

119889120573

119909 (119905)

119889119905120573= 119860119909 (119905) + 119891 (119909) (27)

6 Mathematical Problems in Engineering

In States 2 and 3 the load is powered by the capacitorTheoutput voltage of the converter is negative which satisfies thestate equation and the amplitude is monotone decreasing

119889120573V119862(119905)

119889119905120573= minus

V119862(119905)

119862119877 (28)

where 119860 = minus1119862119877 119891(119909) = 0 and 095 lt 120573 lt 1The solution of (28) is

V119862(119905) = 119864

1205731(119860119905120573

)119881max (29)

where 119881max is the initial value of V119862(119905) which means V119862(1199050) =

119881max for 1199050 = 0When 119905 = (119889

1+ 1198893)119879 the output voltage is

V119862((1198893+ 1198891) 119879) = 119881max1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862) (30)

The variation of the output voltage is

ΔV = 119881max(1 minus 1198641205731(minus((1198893+ 1198891) 119879)120573

119877119862)) (31)

According to (17) and (31) it yields

119881max = 119881119862 +1

2ΔV

=119881in11986311198632

+119881max2

(1 minus 1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862))

=2119881in1198631

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(32)

Taking (32) to (31) yields

ΔV =2119881in1198631 (1 minus 1198641205731 (minus ((1198893 + 1198891) 119879)

120573

119877119862))

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(33)

We can find from (25) and (33) that the inductor currentripple and the output voltage ripple are closely related to theorder of energy storage elements The ripple increases withthe decreasing of the order

Under conditions of zero initial value the Laplace trans-form of converter equations is

119904120573V119862(119904) = minus

1198632119871(119904)

119862minus1198681198712(119904)

119862minusV119862(119904)

119862119877

119904120572

119871(119904) =

1198631Vin (119904) + 119881in1 (119904) + 1198632V119862 (119904) + 1198811198622 (119904)

119871

(34)

Set the duty cycle variation to be zero

1(119904) = 0

2(119904) = 0

(35)

According to (34) one can get the Vin-to-V119862 and the Vin-to-119894119871transfer function as follows

119866V119862Vin (119904) =V119862(119904)

Vin (119904)

100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

= minus11986311198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

119866119894119871Vin (119904) =

119871(119904)

Vin (119904)

1003816100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

=

(11986311198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(36)

Set the AC input voltage variation and the duty cycle 2to be

zero

Vin (119904) = 0

2(119904) = 0

(37)

The 1-to-V119862and the

1-to-119894119871transfer functions are

119866V1198621198891 (119904) =V119862(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

= minus119881in1198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198891

(119904) =119871(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

=

(119881in1198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(38)

Set the AC input voltage variation and the duty cycle 1to be

zero

Vin (119904) = 0

1(119904) = 0

(39)

According to (17) and (34) the 2-to-V119862and the

2-to-119894119871

transfer functions are

119866V1198621198892 (119904) =V119862(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11987111986311198634

2119877) 119904120572

+ 1198811198621198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198892

(119904) =119871(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11986311198771198633

2) (119862119877119904

120573

+ 2)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(40)

Mathematical Problems in Engineering 7

Table 1 Parameters of the Buck-Boost converter

Parameter Symbol ValueInput voltage 119881in 50VCapacitor 119862 100120583FInductance 119871 1mHSwitching frequency 119891 20 kHzDuty cycle 119863

104

Duty cycle 1198633

04The order 120572 0951The order 120573 0951

We can find from (36) (38) and (40) that orders of theinductor and capacitor have influence on transfer functionsThatmeans fractional orders can be seen as extra parametersWhen orders are 1 we can get integer-order transfer functionsfrom the fractional-order model The integer-order model isa special case of the fractional-order model

5 Experimental Researches

In this section some numerical and circuit simulation exper-iments are presented to verify the fractional-ordermodel andthe influence of the order on performances of the converterin PCCM We choose 095 as the order which is close to 1Parameters of the Buck-Boost converter are listed in Table 1

We select the load resistor as 119877 = 20Ω to ensure thatthe converter is operating in PCCM Difference values ofthe quiescent operating point between the integer-order andfractional-order Buck-Boost converter model are shown inTable 2

In Table 2 we can find that the output voltage ripple andthe inductor ripple rise with the decreasing of the orderThatmeans the order of the Buck-Boost converter impacts theoutput characteristics of the converter obviously

51 Numerical Simulation In numerical simulation Oustal-ouprsquos recursive approximation is adopted [33] The switchingfrequency of the converter is selected as 119891 = 20 kHz thecorresponding rotational frequency is 120596 = 2120587119891 = 1256 times

105 and parameters of Oustaloup are 120596

ℎ= 2 times 10

5 120596119887=

5 times 10minus6 and119873 = 6

Figure 7 shows the inductor current the detailed valuesare listed in Table 3 We can see that in the inductor currentripple maximum and minimum values of the inductorcurrent increase significantly in the fractional-order modelThat means the effect of the inductor order on the inductorcurrent should be appropriately considered

The order also influences the output voltage As shownin Figure 8 and Table 4 in the output voltage ripple themaximum and minimum values of output voltage increasesignificantly in the fractional-order model

52 Circuitry Simulation Experiment To further verify theresults of the theoretical analysis the circuit simulationexperiments are presented

03 03 03001 03001 03002 0300223

235

24

245

25

255

26

265

27

Time (s)

IOFO (095)

i L(A

)

Figure 7 The inductor current

03 03 03001 03001 03002 03002minus105

minus100

minus95

Time (s)

IOFO (095)

C(V

)

Figure 8 The output voltage

By using resistorcapacitor or resistorinductor networkswe can construct an approximation circuit of the fractional-order circuit elements We choose the chain structure as thefractance unit to approximately achieve the fractional-ordercircuit elements

The chain fractance unit of fractional-order capacitor isshown in Figure 9

The transfer function model of the capacitor is

119867(119904) = 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (41)

We choose 120573 = 095 119862 = 100 120583F and 119899 = 6We can calculate resistors and inductors in Figure 9 by themethod of undetermined coefficients comparing the input

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

S1

S2

D2

Vin iL

D1

CL R

minus

+

Figure 1 Buck-Boost converter in PCCM

In 1994 based on fractional calculus Westerlund andEkstam developed a new capacitor theory [13]

119894 (119905) = 119862119891

119889120573

119906 (119905)

119889119905120573 0 lt 120573 lt 1 (6)

where 119862119891is the capacitance of the capacitor related to the

kind of dielectric And120573 is the order related to the losses of thecapacitorWesterlund and Ekstamprovided various capacitordielectrics with appropriated order 120573 by experiment

Westerlund also created the fractional-order model ofinductors [14]

119906 (119905) = 119871119891

119889120572

119894 (119905)

119889119905120572 0 lt 120572 lt 1 (7)

where 119871119891is the inductance of the inductor and 120572 is the order

related to the proximity effectIn engineering applications orders of capacitors and

inductors are close to 1 [13] In order to ensure that the powerelectronic devices can work properly we redefine a fractionalorder between 095 and 1

3 The State-Space Averaging Model ofthe Fractional-Order Buck-Boost Converterin PCCM

31 The Fractional-Order Mathematical Model of the Buck-Boost Converter The Buck-Boost converter can operate asBuck (for voltage stepdown) or Boost (for voltage stepup)converter inverting the voltage polarity Connecting aninductor in parallel with a power switch makes the converteroperate in PCCM [31] The circuit diagram is shown inFigure 1 There are two fully controlled switching devices 119878

1

and 1198782 two diodes 119863

1and 119863

2 one DC input voltage 119881in

one load resistor 119877 and two fractional-order energy storageelements 119871 and 119862 in the circuit

In Figure 2 where 119879 is the switching time period 119879on isthe on time 119879off is the off time for the switch and 119889

1+1198892+1198893

= 1 There are three states for the pseudo-continuous mode

State 1 When 0 lt 119905 le 1198891119879 the circuit topology is shown in

Figure 3 Switch 1198781is on 119878

2is off and diodes119863

12are reverse-

biased The inductor current 119894119871rises

iL

L

0

0

0

0

Ton

Ton

Toff

Toff

ΔiL

t

t

t

t

d1T d2T d3T

Pulse 2

Pulse 1

Figure 2 Inductor voltage and current of Buck-Boost converter inPCCM

S1

S2

D2

Vin iL

D1

CL R

minus

+

Figure 3 Schematic diagram of Buck-Boost converter in State 1

The expression of the fractional-order mathematicalmodel is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

= [

[

0 0

0 minus1

119862119877

]

]

[

119894119871(119905)

V119862(119905)] + [

[

1

119871

0

]

]

Vin (119905) (8)

State 2 When 1198891119879 lt 119905 le (119889

1+ 1198892)119879 as shown in Figure 4

switches 11987812

and diode 1198632are off and diode 119863

1is forward-

biased and behaves as a short circuit the inductor current 119894119871

ramps downThe expression of the fractional-order mathematical

model is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01

119871

minus1

119862minus1

119862119877

]]

]

[

119894119871(119905)

V119862(119905)] + [

0

0] Vin (119905) (9)

State 3 When (1198891+1198892)119879 lt 119905 le 119879 switch 119878

1and diode119863

1are

off switch 1198782and diode 119863

2are on and the inductor current

119894119871maintains a constant theoretically The circuit topology is

shown in Figure 5

4 Mathematical Problems in Engineering

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 4 Schematic diagram of Buck-Boost converter in State 2

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 5 Schematic diagram of Buck-Boost converter in State 3

The expression of the fractional-order mathematicalmodel is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

= [

[

0 0

0 minus1

119862119877

]

]

[

119894119871(119905)

V119862(119905)] + [

0

0] Vin (119905) (10)

As shown in (8) (9) and (10) the inductor order 120572 andthe capacitor order 120573 can be considered as extra parametersof the fractional-order mathematical models in PCCM Theinfluence of these extra parameters on the performance of theconverter will be discussed in the next section

32 The Fractional-Order State-Space Averaging Model ofthe Buck-Boost DCDC Converter To remove the switchingharmonics waveforms over one switching period should beaveraged [32] The average value of variable 119909(119905) which is anarbitrary circuit variable of the Buck-Boost converter in oneswitching period is defined as follows

⟨119909 (119905)⟩119879=1

119879int

119905+119879

119905

119909 (120591) 119889120591 (11)

Its fractional-order form is

119889120572

⟨119909 (119905)⟩119879

119889119905120572=

119889120572

((1119879) int119905+119879

119905

119909 (120591) 119889120591)

119889119905120572

=1

119879int

119905+119879

119905

(119889120572

119909 (120591)

119889120591120572)119889120591 = ⟨

119889120572

119909 (119905)

119889119905120572⟩

(12)

where 120572 is the order and 095 lt 120572 lt 1

Let us consider the fractional-order state-space averagingmodel of Buck-Boost converter in PCCM

[[[

[

119889120572

⟨119894119871(119905)⟩119879

119889119905120572

119889120573

⟨V119862(119905)⟩119879

119889119905120573

]]]

]

=[[

[

01198892(119905)

119871

minus1198892(119905)

119862minus1

119862119877

]]

]

[

⟨119894119871(119905)⟩119879

⟨V119862(119905)⟩119879

]

+ [

[

1198891(119905)

119871

0

]

]

⟨Vin (119905)⟩119879

(13)

Average values of circuit variables are defined as thefollowing form

⟨119894119871(119905)⟩119879= 119868119871+ 119871(119905)

⟨V119862(119905)⟩119879= 119881119862+ V119862(119905)

⟨Vin (119905)⟩119879 = 119881in + Vin (119905)

1198891(119905) = 119863

1+ 1(119905)

1198892(119905) = 119863

2+ 2(119905)

(14)

where variables in capital letters represent DC componentsand variables with the symbol ldquordquo above them represent ACcomponents which are much smaller in magnitude than thecorresponding DC component

Equation (13) can be rewritten as

[[[

[

119889120572

119868119871

119889119905120572+119889120572

119871(119905)

119889119905120572

119889120573

119881119862

119889119905120573+119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119868119871+ 119871(119905)

119881119862+ V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

(119881in + Vin (119905))

(15)

DC components in (15) are as follows

[[[

[

119889120572

119868119871

119889119905120572

119889120573

119881119862

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[

119868119871

119881119862

] + [

[

1198631

119871

0

]

]

119881in (16)

The quiescent operation point is

[

119868119871

119881119862

] = minus[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

minus1

[

[

119863

119871

0

]

]

119881in =[[[

[

119881in11986311198632

2119877

minus119881in11986311198632

]]]

]

(17)

The voltage ratio is defined as follows

119872 =119881119862

119881in= minus

1198631

1198632

(18)

Mathematical Problems in Engineering 5

AC components of (15) are

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

Vin (119905)

+

[[[

[

02(119905)

119871

minus2(119905)

1198620

]]]

]

[

119868119871

119881119862

] + [

[

1(119905)

119871

0

]

]

119881in

(19)

where 2(119905)119894119871(119905) 2(119905)V119862(119905) and

1(119905)Vin(119905) are high order

small signal terms which can be omitted Then the smallsignal AC equation of Buck-Boost converter in PCCM canbe expressed as follows

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631

119871

0

]

]

Vin (119905)

+[[

[

01

119871

minus1

1198620

]]

]

[

119868119871

119881119862

] 2(119905)

+ [

[

1

119871

0

]

]

119881in (119905)

(20)

4 Performance Analysis ofthe Fractional-Order Model of theBuck-Boost Converter in PCCM

In this section the quiescent operation point and transferfunctions of the AC small signal model will be analyzedbecause they have practical significance for the designingof various parameters and reflect the performance of theconverter

The inductor current ripple and output voltage rippleof the fractional-order Buck-Boost converter in PCCM areshown in Figure 6

t0

0t

d1T d1Td2T d3T

iLΔiL

C

c(t0)

ΔC

iL(t0)

Figure 6 The inductor current ripple and output voltage ripple

The small ripple approximation is defined as replacingwaveforms with their low-frequency averaged values

119889120572

119894119871(119905)

119889119905120572asymp⟨Vin (119905)⟩119879

119871 (21)

Using Caputo definition we can get the correspondinginitial condition as

119863119896

119894119871(0) = 119894

(119896)

1198710 119896 = 0 1 119898 minus 1 (22)

Solving fractional-order differential equation (21) we canget

119894119871(119905) =

1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

⟨Vin (119905)⟩119879119871

119889120591

+

119898minus1

sum

119896=0

119905119896

119896119894(119896)

1198710(119905)

(23)

In PCCM the initial value of inductor current is given as119894119871(1199050) = 119894119871min

Then

119894119871max = 119894119871 (1198891119879) =

119881in (1198631119879)120572

119871Γ (120572) 120572+ 119894119871min (24)

The inductor current ripple is

Δ119894119871=119881in (1198631119879)

120572

119871Γ (120572) 120572 (25)

The maximum and minimum values of inductor currentare

119894119871max = 119868119871 +

1

2Δ119894119871=119881in11986311198632

2119877+119881in (1198631119879)

120572

2119871Γ (120572) 120572

119894119871min = 119868119871 minus

1

2Δ119894119871=119881in11986311198632

2119877minus119881in (1198631119879)

120572

2119871Γ (120572) 120572

(26)

According to Adomian decomposition scheme one canget

119889120573

119909 (119905)

119889119905120573= 119860119909 (119905) + 119891 (119909) (27)

6 Mathematical Problems in Engineering

In States 2 and 3 the load is powered by the capacitorTheoutput voltage of the converter is negative which satisfies thestate equation and the amplitude is monotone decreasing

119889120573V119862(119905)

119889119905120573= minus

V119862(119905)

119862119877 (28)

where 119860 = minus1119862119877 119891(119909) = 0 and 095 lt 120573 lt 1The solution of (28) is

V119862(119905) = 119864

1205731(119860119905120573

)119881max (29)

where 119881max is the initial value of V119862(119905) which means V119862(1199050) =

119881max for 1199050 = 0When 119905 = (119889

1+ 1198893)119879 the output voltage is

V119862((1198893+ 1198891) 119879) = 119881max1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862) (30)

The variation of the output voltage is

ΔV = 119881max(1 minus 1198641205731(minus((1198893+ 1198891) 119879)120573

119877119862)) (31)

According to (17) and (31) it yields

119881max = 119881119862 +1

2ΔV

=119881in11986311198632

+119881max2

(1 minus 1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862))

=2119881in1198631

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(32)

Taking (32) to (31) yields

ΔV =2119881in1198631 (1 minus 1198641205731 (minus ((1198893 + 1198891) 119879)

120573

119877119862))

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(33)

We can find from (25) and (33) that the inductor currentripple and the output voltage ripple are closely related to theorder of energy storage elements The ripple increases withthe decreasing of the order

Under conditions of zero initial value the Laplace trans-form of converter equations is

119904120573V119862(119904) = minus

1198632119871(119904)

119862minus1198681198712(119904)

119862minusV119862(119904)

119862119877

119904120572

119871(119904) =

1198631Vin (119904) + 119881in1 (119904) + 1198632V119862 (119904) + 1198811198622 (119904)

119871

(34)

Set the duty cycle variation to be zero

1(119904) = 0

2(119904) = 0

(35)

According to (34) one can get the Vin-to-V119862 and the Vin-to-119894119871transfer function as follows

119866V119862Vin (119904) =V119862(119904)

Vin (119904)

100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

= minus11986311198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

119866119894119871Vin (119904) =

119871(119904)

Vin (119904)

1003816100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

=

(11986311198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(36)

Set the AC input voltage variation and the duty cycle 2to be

zero

Vin (119904) = 0

2(119904) = 0

(37)

The 1-to-V119862and the

1-to-119894119871transfer functions are

119866V1198621198891 (119904) =V119862(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

= minus119881in1198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198891

(119904) =119871(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

=

(119881in1198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(38)

Set the AC input voltage variation and the duty cycle 1to be

zero

Vin (119904) = 0

1(119904) = 0

(39)

According to (17) and (34) the 2-to-V119862and the

2-to-119894119871

transfer functions are

119866V1198621198892 (119904) =V119862(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11987111986311198634

2119877) 119904120572

+ 1198811198621198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198892

(119904) =119871(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11986311198771198633

2) (119862119877119904

120573

+ 2)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(40)

Mathematical Problems in Engineering 7

Table 1 Parameters of the Buck-Boost converter

Parameter Symbol ValueInput voltage 119881in 50VCapacitor 119862 100120583FInductance 119871 1mHSwitching frequency 119891 20 kHzDuty cycle 119863

104

Duty cycle 1198633

04The order 120572 0951The order 120573 0951

We can find from (36) (38) and (40) that orders of theinductor and capacitor have influence on transfer functionsThatmeans fractional orders can be seen as extra parametersWhen orders are 1 we can get integer-order transfer functionsfrom the fractional-order model The integer-order model isa special case of the fractional-order model

5 Experimental Researches

In this section some numerical and circuit simulation exper-iments are presented to verify the fractional-ordermodel andthe influence of the order on performances of the converterin PCCM We choose 095 as the order which is close to 1Parameters of the Buck-Boost converter are listed in Table 1

We select the load resistor as 119877 = 20Ω to ensure thatthe converter is operating in PCCM Difference values ofthe quiescent operating point between the integer-order andfractional-order Buck-Boost converter model are shown inTable 2

In Table 2 we can find that the output voltage ripple andthe inductor ripple rise with the decreasing of the orderThatmeans the order of the Buck-Boost converter impacts theoutput characteristics of the converter obviously

51 Numerical Simulation In numerical simulation Oustal-ouprsquos recursive approximation is adopted [33] The switchingfrequency of the converter is selected as 119891 = 20 kHz thecorresponding rotational frequency is 120596 = 2120587119891 = 1256 times

105 and parameters of Oustaloup are 120596

ℎ= 2 times 10

5 120596119887=

5 times 10minus6 and119873 = 6

Figure 7 shows the inductor current the detailed valuesare listed in Table 3 We can see that in the inductor currentripple maximum and minimum values of the inductorcurrent increase significantly in the fractional-order modelThat means the effect of the inductor order on the inductorcurrent should be appropriately considered

The order also influences the output voltage As shownin Figure 8 and Table 4 in the output voltage ripple themaximum and minimum values of output voltage increasesignificantly in the fractional-order model

52 Circuitry Simulation Experiment To further verify theresults of the theoretical analysis the circuit simulationexperiments are presented

03 03 03001 03001 03002 0300223

235

24

245

25

255

26

265

27

Time (s)

IOFO (095)

i L(A

)

Figure 7 The inductor current

03 03 03001 03001 03002 03002minus105

minus100

minus95

Time (s)

IOFO (095)

C(V

)

Figure 8 The output voltage

By using resistorcapacitor or resistorinductor networkswe can construct an approximation circuit of the fractional-order circuit elements We choose the chain structure as thefractance unit to approximately achieve the fractional-ordercircuit elements

The chain fractance unit of fractional-order capacitor isshown in Figure 9

The transfer function model of the capacitor is

119867(119904) = 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (41)

We choose 120573 = 095 119862 = 100 120583F and 119899 = 6We can calculate resistors and inductors in Figure 9 by themethod of undetermined coefficients comparing the input

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 4 Schematic diagram of Buck-Boost converter in State 2

S1

S2

D2

Vin

D1

CL R

minus

+

iL

Figure 5 Schematic diagram of Buck-Boost converter in State 3

The expression of the fractional-order mathematicalmodel is

[[[

[

119889120572

119894119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

= [

[

0 0

0 minus1

119862119877

]

]

[

119894119871(119905)

V119862(119905)] + [

0

0] Vin (119905) (10)

As shown in (8) (9) and (10) the inductor order 120572 andthe capacitor order 120573 can be considered as extra parametersof the fractional-order mathematical models in PCCM Theinfluence of these extra parameters on the performance of theconverter will be discussed in the next section

32 The Fractional-Order State-Space Averaging Model ofthe Buck-Boost DCDC Converter To remove the switchingharmonics waveforms over one switching period should beaveraged [32] The average value of variable 119909(119905) which is anarbitrary circuit variable of the Buck-Boost converter in oneswitching period is defined as follows

⟨119909 (119905)⟩119879=1

119879int

119905+119879

119905

119909 (120591) 119889120591 (11)

Its fractional-order form is

119889120572

⟨119909 (119905)⟩119879

119889119905120572=

119889120572

((1119879) int119905+119879

119905

119909 (120591) 119889120591)

119889119905120572

=1

119879int

119905+119879

119905

(119889120572

119909 (120591)

119889120591120572)119889120591 = ⟨

119889120572

119909 (119905)

119889119905120572⟩

(12)

where 120572 is the order and 095 lt 120572 lt 1

Let us consider the fractional-order state-space averagingmodel of Buck-Boost converter in PCCM

[[[

[

119889120572

⟨119894119871(119905)⟩119879

119889119905120572

119889120573

⟨V119862(119905)⟩119879

119889119905120573

]]]

]

=[[

[

01198892(119905)

119871

minus1198892(119905)

119862minus1

119862119877

]]

]

[

⟨119894119871(119905)⟩119879

⟨V119862(119905)⟩119879

]

+ [

[

1198891(119905)

119871

0

]

]

⟨Vin (119905)⟩119879

(13)

Average values of circuit variables are defined as thefollowing form

⟨119894119871(119905)⟩119879= 119868119871+ 119871(119905)

⟨V119862(119905)⟩119879= 119881119862+ V119862(119905)

⟨Vin (119905)⟩119879 = 119881in + Vin (119905)

1198891(119905) = 119863

1+ 1(119905)

1198892(119905) = 119863

2+ 2(119905)

(14)

where variables in capital letters represent DC componentsand variables with the symbol ldquordquo above them represent ACcomponents which are much smaller in magnitude than thecorresponding DC component

Equation (13) can be rewritten as

[[[

[

119889120572

119868119871

119889119905120572+119889120572

119871(119905)

119889119905120572

119889120573

119881119862

119889119905120573+119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119868119871+ 119871(119905)

119881119862+ V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

(119881in + Vin (119905))

(15)

DC components in (15) are as follows

[[[

[

119889120572

119868119871

119889119905120572

119889120573

119881119862

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[

119868119871

119881119862

] + [

[

1198631

119871

0

]

]

119881in (16)

The quiescent operation point is

[

119868119871

119881119862

] = minus[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

minus1

[

[

119863

119871

0

]

]

119881in =[[[

[

119881in11986311198632

2119877

minus119881in11986311198632

]]]

]

(17)

The voltage ratio is defined as follows

119872 =119881119862

119881in= minus

1198631

1198632

(18)

Mathematical Problems in Engineering 5

AC components of (15) are

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

Vin (119905)

+

[[[

[

02(119905)

119871

minus2(119905)

1198620

]]]

]

[

119868119871

119881119862

] + [

[

1(119905)

119871

0

]

]

119881in

(19)

where 2(119905)119894119871(119905) 2(119905)V119862(119905) and

1(119905)Vin(119905) are high order

small signal terms which can be omitted Then the smallsignal AC equation of Buck-Boost converter in PCCM canbe expressed as follows

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631

119871

0

]

]

Vin (119905)

+[[

[

01

119871

minus1

1198620

]]

]

[

119868119871

119881119862

] 2(119905)

+ [

[

1

119871

0

]

]

119881in (119905)

(20)

4 Performance Analysis ofthe Fractional-Order Model of theBuck-Boost Converter in PCCM

In this section the quiescent operation point and transferfunctions of the AC small signal model will be analyzedbecause they have practical significance for the designingof various parameters and reflect the performance of theconverter

The inductor current ripple and output voltage rippleof the fractional-order Buck-Boost converter in PCCM areshown in Figure 6

t0

0t

d1T d1Td2T d3T

iLΔiL

C

c(t0)

ΔC

iL(t0)

Figure 6 The inductor current ripple and output voltage ripple

The small ripple approximation is defined as replacingwaveforms with their low-frequency averaged values

119889120572

119894119871(119905)

119889119905120572asymp⟨Vin (119905)⟩119879

119871 (21)

Using Caputo definition we can get the correspondinginitial condition as

119863119896

119894119871(0) = 119894

(119896)

1198710 119896 = 0 1 119898 minus 1 (22)

Solving fractional-order differential equation (21) we canget

119894119871(119905) =

1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

⟨Vin (119905)⟩119879119871

119889120591

+

119898minus1

sum

119896=0

119905119896

119896119894(119896)

1198710(119905)

(23)

In PCCM the initial value of inductor current is given as119894119871(1199050) = 119894119871min

Then

119894119871max = 119894119871 (1198891119879) =

119881in (1198631119879)120572

119871Γ (120572) 120572+ 119894119871min (24)

The inductor current ripple is

Δ119894119871=119881in (1198631119879)

120572

119871Γ (120572) 120572 (25)

The maximum and minimum values of inductor currentare

119894119871max = 119868119871 +

1

2Δ119894119871=119881in11986311198632

2119877+119881in (1198631119879)

120572

2119871Γ (120572) 120572

119894119871min = 119868119871 minus

1

2Δ119894119871=119881in11986311198632

2119877minus119881in (1198631119879)

120572

2119871Γ (120572) 120572

(26)

According to Adomian decomposition scheme one canget

119889120573

119909 (119905)

119889119905120573= 119860119909 (119905) + 119891 (119909) (27)

6 Mathematical Problems in Engineering

In States 2 and 3 the load is powered by the capacitorTheoutput voltage of the converter is negative which satisfies thestate equation and the amplitude is monotone decreasing

119889120573V119862(119905)

119889119905120573= minus

V119862(119905)

119862119877 (28)

where 119860 = minus1119862119877 119891(119909) = 0 and 095 lt 120573 lt 1The solution of (28) is

V119862(119905) = 119864

1205731(119860119905120573

)119881max (29)

where 119881max is the initial value of V119862(119905) which means V119862(1199050) =

119881max for 1199050 = 0When 119905 = (119889

1+ 1198893)119879 the output voltage is

V119862((1198893+ 1198891) 119879) = 119881max1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862) (30)

The variation of the output voltage is

ΔV = 119881max(1 minus 1198641205731(minus((1198893+ 1198891) 119879)120573

119877119862)) (31)

According to (17) and (31) it yields

119881max = 119881119862 +1

2ΔV

=119881in11986311198632

+119881max2

(1 minus 1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862))

=2119881in1198631

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(32)

Taking (32) to (31) yields

ΔV =2119881in1198631 (1 minus 1198641205731 (minus ((1198893 + 1198891) 119879)

120573

119877119862))

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(33)

We can find from (25) and (33) that the inductor currentripple and the output voltage ripple are closely related to theorder of energy storage elements The ripple increases withthe decreasing of the order

Under conditions of zero initial value the Laplace trans-form of converter equations is

119904120573V119862(119904) = minus

1198632119871(119904)

119862minus1198681198712(119904)

119862minusV119862(119904)

119862119877

119904120572

119871(119904) =

1198631Vin (119904) + 119881in1 (119904) + 1198632V119862 (119904) + 1198811198622 (119904)

119871

(34)

Set the duty cycle variation to be zero

1(119904) = 0

2(119904) = 0

(35)

According to (34) one can get the Vin-to-V119862 and the Vin-to-119894119871transfer function as follows

119866V119862Vin (119904) =V119862(119904)

Vin (119904)

100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

= minus11986311198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

119866119894119871Vin (119904) =

119871(119904)

Vin (119904)

1003816100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

=

(11986311198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(36)

Set the AC input voltage variation and the duty cycle 2to be

zero

Vin (119904) = 0

2(119904) = 0

(37)

The 1-to-V119862and the

1-to-119894119871transfer functions are

119866V1198621198891 (119904) =V119862(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

= minus119881in1198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198891

(119904) =119871(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

=

(119881in1198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(38)

Set the AC input voltage variation and the duty cycle 1to be

zero

Vin (119904) = 0

1(119904) = 0

(39)

According to (17) and (34) the 2-to-V119862and the

2-to-119894119871

transfer functions are

119866V1198621198892 (119904) =V119862(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11987111986311198634

2119877) 119904120572

+ 1198811198621198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198892

(119904) =119871(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11986311198771198633

2) (119862119877119904

120573

+ 2)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(40)

Mathematical Problems in Engineering 7

Table 1 Parameters of the Buck-Boost converter

Parameter Symbol ValueInput voltage 119881in 50VCapacitor 119862 100120583FInductance 119871 1mHSwitching frequency 119891 20 kHzDuty cycle 119863

104

Duty cycle 1198633

04The order 120572 0951The order 120573 0951

We can find from (36) (38) and (40) that orders of theinductor and capacitor have influence on transfer functionsThatmeans fractional orders can be seen as extra parametersWhen orders are 1 we can get integer-order transfer functionsfrom the fractional-order model The integer-order model isa special case of the fractional-order model

5 Experimental Researches

In this section some numerical and circuit simulation exper-iments are presented to verify the fractional-ordermodel andthe influence of the order on performances of the converterin PCCM We choose 095 as the order which is close to 1Parameters of the Buck-Boost converter are listed in Table 1

We select the load resistor as 119877 = 20Ω to ensure thatthe converter is operating in PCCM Difference values ofthe quiescent operating point between the integer-order andfractional-order Buck-Boost converter model are shown inTable 2

In Table 2 we can find that the output voltage ripple andthe inductor ripple rise with the decreasing of the orderThatmeans the order of the Buck-Boost converter impacts theoutput characteristics of the converter obviously

51 Numerical Simulation In numerical simulation Oustal-ouprsquos recursive approximation is adopted [33] The switchingfrequency of the converter is selected as 119891 = 20 kHz thecorresponding rotational frequency is 120596 = 2120587119891 = 1256 times

105 and parameters of Oustaloup are 120596

ℎ= 2 times 10

5 120596119887=

5 times 10minus6 and119873 = 6

Figure 7 shows the inductor current the detailed valuesare listed in Table 3 We can see that in the inductor currentripple maximum and minimum values of the inductorcurrent increase significantly in the fractional-order modelThat means the effect of the inductor order on the inductorcurrent should be appropriately considered

The order also influences the output voltage As shownin Figure 8 and Table 4 in the output voltage ripple themaximum and minimum values of output voltage increasesignificantly in the fractional-order model

52 Circuitry Simulation Experiment To further verify theresults of the theoretical analysis the circuit simulationexperiments are presented

03 03 03001 03001 03002 0300223

235

24

245

25

255

26

265

27

Time (s)

IOFO (095)

i L(A

)

Figure 7 The inductor current

03 03 03001 03001 03002 03002minus105

minus100

minus95

Time (s)

IOFO (095)

C(V

)

Figure 8 The output voltage

By using resistorcapacitor or resistorinductor networkswe can construct an approximation circuit of the fractional-order circuit elements We choose the chain structure as thefractance unit to approximately achieve the fractional-ordercircuit elements

The chain fractance unit of fractional-order capacitor isshown in Figure 9

The transfer function model of the capacitor is

119867(119904) = 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (41)

We choose 120573 = 095 119862 = 100 120583F and 119899 = 6We can calculate resistors and inductors in Figure 9 by themethod of undetermined coefficients comparing the input

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

AC components of (15) are

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=

[[[

[

01198632+ 2(119905)

119871

minus1198632+ 2(119905)

119862minus1

119862119877

]]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631+ 1(119905)

119871

0

]

]

Vin (119905)

+

[[[

[

02(119905)

119871

minus2(119905)

1198620

]]]

]

[

119868119871

119881119862

] + [

[

1(119905)

119871

0

]

]

119881in

(19)

where 2(119905)119894119871(119905) 2(119905)V119862(119905) and

1(119905)Vin(119905) are high order

small signal terms which can be omitted Then the smallsignal AC equation of Buck-Boost converter in PCCM canbe expressed as follows

[[[

[

119889120572

119871(119905)

119889119905120572

119889120573V119862(119905)

119889119905120573

]]]

]

=[[

[

01198632

119871

minus1198632

119862minus1

119862119877

]]

]

[119871(119905)

V119862(119905)

]

+ [

[

1198631

119871

0

]

]

Vin (119905)

+[[

[

01

119871

minus1

1198620

]]

]

[

119868119871

119881119862

] 2(119905)

+ [

[

1

119871

0

]

]

119881in (119905)

(20)

4 Performance Analysis ofthe Fractional-Order Model of theBuck-Boost Converter in PCCM

In this section the quiescent operation point and transferfunctions of the AC small signal model will be analyzedbecause they have practical significance for the designingof various parameters and reflect the performance of theconverter

The inductor current ripple and output voltage rippleof the fractional-order Buck-Boost converter in PCCM areshown in Figure 6

t0

0t

d1T d1Td2T d3T

iLΔiL

C

c(t0)

ΔC

iL(t0)

Figure 6 The inductor current ripple and output voltage ripple

The small ripple approximation is defined as replacingwaveforms with their low-frequency averaged values

119889120572

119894119871(119905)

119889119905120572asymp⟨Vin (119905)⟩119879

119871 (21)

Using Caputo definition we can get the correspondinginitial condition as

119863119896

119894119871(0) = 119894

(119896)

1198710 119896 = 0 1 119898 minus 1 (22)

Solving fractional-order differential equation (21) we canget

119894119871(119905) =

1

Γ (120572)int

119905

0

(119905 minus 120591)120572minus1

⟨Vin (119905)⟩119879119871

119889120591

+

119898minus1

sum

119896=0

119905119896

119896119894(119896)

1198710(119905)

(23)

In PCCM the initial value of inductor current is given as119894119871(1199050) = 119894119871min

Then

119894119871max = 119894119871 (1198891119879) =

119881in (1198631119879)120572

119871Γ (120572) 120572+ 119894119871min (24)

The inductor current ripple is

Δ119894119871=119881in (1198631119879)

120572

119871Γ (120572) 120572 (25)

The maximum and minimum values of inductor currentare

119894119871max = 119868119871 +

1

2Δ119894119871=119881in11986311198632

2119877+119881in (1198631119879)

120572

2119871Γ (120572) 120572

119894119871min = 119868119871 minus

1

2Δ119894119871=119881in11986311198632

2119877minus119881in (1198631119879)

120572

2119871Γ (120572) 120572

(26)

According to Adomian decomposition scheme one canget

119889120573

119909 (119905)

119889119905120573= 119860119909 (119905) + 119891 (119909) (27)

6 Mathematical Problems in Engineering

In States 2 and 3 the load is powered by the capacitorTheoutput voltage of the converter is negative which satisfies thestate equation and the amplitude is monotone decreasing

119889120573V119862(119905)

119889119905120573= minus

V119862(119905)

119862119877 (28)

where 119860 = minus1119862119877 119891(119909) = 0 and 095 lt 120573 lt 1The solution of (28) is

V119862(119905) = 119864

1205731(119860119905120573

)119881max (29)

where 119881max is the initial value of V119862(119905) which means V119862(1199050) =

119881max for 1199050 = 0When 119905 = (119889

1+ 1198893)119879 the output voltage is

V119862((1198893+ 1198891) 119879) = 119881max1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862) (30)

The variation of the output voltage is

ΔV = 119881max(1 minus 1198641205731(minus((1198893+ 1198891) 119879)120573

119877119862)) (31)

According to (17) and (31) it yields

119881max = 119881119862 +1

2ΔV

=119881in11986311198632

+119881max2

(1 minus 1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862))

=2119881in1198631

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(32)

Taking (32) to (31) yields

ΔV =2119881in1198631 (1 minus 1198641205731 (minus ((1198893 + 1198891) 119879)

120573

119877119862))

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(33)

We can find from (25) and (33) that the inductor currentripple and the output voltage ripple are closely related to theorder of energy storage elements The ripple increases withthe decreasing of the order

Under conditions of zero initial value the Laplace trans-form of converter equations is

119904120573V119862(119904) = minus

1198632119871(119904)

119862minus1198681198712(119904)

119862minusV119862(119904)

119862119877

119904120572

119871(119904) =

1198631Vin (119904) + 119881in1 (119904) + 1198632V119862 (119904) + 1198811198622 (119904)

119871

(34)

Set the duty cycle variation to be zero

1(119904) = 0

2(119904) = 0

(35)

According to (34) one can get the Vin-to-V119862 and the Vin-to-119894119871transfer function as follows

119866V119862Vin (119904) =V119862(119904)

Vin (119904)

100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

= minus11986311198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

119866119894119871Vin (119904) =

119871(119904)

Vin (119904)

1003816100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

=

(11986311198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(36)

Set the AC input voltage variation and the duty cycle 2to be

zero

Vin (119904) = 0

2(119904) = 0

(37)

The 1-to-V119862and the

1-to-119894119871transfer functions are

119866V1198621198891 (119904) =V119862(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

= minus119881in1198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198891

(119904) =119871(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

=

(119881in1198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(38)

Set the AC input voltage variation and the duty cycle 1to be

zero

Vin (119904) = 0

1(119904) = 0

(39)

According to (17) and (34) the 2-to-V119862and the

2-to-119894119871

transfer functions are

119866V1198621198892 (119904) =V119862(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11987111986311198634

2119877) 119904120572

+ 1198811198621198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198892

(119904) =119871(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11986311198771198633

2) (119862119877119904

120573

+ 2)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(40)

Mathematical Problems in Engineering 7

Table 1 Parameters of the Buck-Boost converter

Parameter Symbol ValueInput voltage 119881in 50VCapacitor 119862 100120583FInductance 119871 1mHSwitching frequency 119891 20 kHzDuty cycle 119863

104

Duty cycle 1198633

04The order 120572 0951The order 120573 0951

We can find from (36) (38) and (40) that orders of theinductor and capacitor have influence on transfer functionsThatmeans fractional orders can be seen as extra parametersWhen orders are 1 we can get integer-order transfer functionsfrom the fractional-order model The integer-order model isa special case of the fractional-order model

5 Experimental Researches

In this section some numerical and circuit simulation exper-iments are presented to verify the fractional-ordermodel andthe influence of the order on performances of the converterin PCCM We choose 095 as the order which is close to 1Parameters of the Buck-Boost converter are listed in Table 1

We select the load resistor as 119877 = 20Ω to ensure thatthe converter is operating in PCCM Difference values ofthe quiescent operating point between the integer-order andfractional-order Buck-Boost converter model are shown inTable 2

In Table 2 we can find that the output voltage ripple andthe inductor ripple rise with the decreasing of the orderThatmeans the order of the Buck-Boost converter impacts theoutput characteristics of the converter obviously

51 Numerical Simulation In numerical simulation Oustal-ouprsquos recursive approximation is adopted [33] The switchingfrequency of the converter is selected as 119891 = 20 kHz thecorresponding rotational frequency is 120596 = 2120587119891 = 1256 times

105 and parameters of Oustaloup are 120596

ℎ= 2 times 10

5 120596119887=

5 times 10minus6 and119873 = 6

Figure 7 shows the inductor current the detailed valuesare listed in Table 3 We can see that in the inductor currentripple maximum and minimum values of the inductorcurrent increase significantly in the fractional-order modelThat means the effect of the inductor order on the inductorcurrent should be appropriately considered

The order also influences the output voltage As shownin Figure 8 and Table 4 in the output voltage ripple themaximum and minimum values of output voltage increasesignificantly in the fractional-order model

52 Circuitry Simulation Experiment To further verify theresults of the theoretical analysis the circuit simulationexperiments are presented

03 03 03001 03001 03002 0300223

235

24

245

25

255

26

265

27

Time (s)

IOFO (095)

i L(A

)

Figure 7 The inductor current

03 03 03001 03001 03002 03002minus105

minus100

minus95

Time (s)

IOFO (095)

C(V

)

Figure 8 The output voltage

By using resistorcapacitor or resistorinductor networkswe can construct an approximation circuit of the fractional-order circuit elements We choose the chain structure as thefractance unit to approximately achieve the fractional-ordercircuit elements

The chain fractance unit of fractional-order capacitor isshown in Figure 9

The transfer function model of the capacitor is

119867(119904) = 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (41)

We choose 120573 = 095 119862 = 100 120583F and 119899 = 6We can calculate resistors and inductors in Figure 9 by themethod of undetermined coefficients comparing the input

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

In States 2 and 3 the load is powered by the capacitorTheoutput voltage of the converter is negative which satisfies thestate equation and the amplitude is monotone decreasing

119889120573V119862(119905)

119889119905120573= minus

V119862(119905)

119862119877 (28)

where 119860 = minus1119862119877 119891(119909) = 0 and 095 lt 120573 lt 1The solution of (28) is

V119862(119905) = 119864

1205731(119860119905120573

)119881max (29)

where 119881max is the initial value of V119862(119905) which means V119862(1199050) =

119881max for 1199050 = 0When 119905 = (119889

1+ 1198893)119879 the output voltage is

V119862((1198893+ 1198891) 119879) = 119881max1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862) (30)

The variation of the output voltage is

ΔV = 119881max(1 minus 1198641205731(minus((1198893+ 1198891) 119879)120573

119877119862)) (31)

According to (17) and (31) it yields

119881max = 119881119862 +1

2ΔV

=119881in11986311198632

+119881max2

(1 minus 1198641205731(minus

((1198893+ 1198891) 119879)120573

119877119862))

=2119881in1198631

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(32)

Taking (32) to (31) yields

ΔV =2119881in1198631 (1 minus 1198641205731 (minus ((1198893 + 1198891) 119879)

120573

119877119862))

1198632(1 + 119864

1205731(minus ((119889

3+ 1198891) 119879)120573

119877119862))

(33)

We can find from (25) and (33) that the inductor currentripple and the output voltage ripple are closely related to theorder of energy storage elements The ripple increases withthe decreasing of the order

Under conditions of zero initial value the Laplace trans-form of converter equations is

119904120573V119862(119904) = minus

1198632119871(119904)

119862minus1198681198712(119904)

119862minusV119862(119904)

119862119877

119904120572

119871(119904) =

1198631Vin (119904) + 119881in1 (119904) + 1198632V119862 (119904) + 1198811198622 (119904)

119871

(34)

Set the duty cycle variation to be zero

1(119904) = 0

2(119904) = 0

(35)

According to (34) one can get the Vin-to-V119862 and the Vin-to-119894119871transfer function as follows

119866V119862Vin (119904) =V119862(119904)

Vin (119904)

100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

= minus11986311198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

119866119894119871Vin (119904) =

119871(119904)

Vin (119904)

1003816100381610038161003816100381610038161003816100381610038161198891(119904)=01198892(119904)=0

=

(11986311198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(36)

Set the AC input voltage variation and the duty cycle 2to be

zero

Vin (119904) = 0

2(119904) = 0

(37)

The 1-to-V119862and the

1-to-119894119871transfer functions are

119866V1198621198891 (119904) =V119862(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

= minus119881in1198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198891

(119904) =119871(119904)

1(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198892(119904)=0

=

(119881in1198771198632

2) (119862119877119904

120573

+ 1)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(38)

Set the AC input voltage variation and the duty cycle 1to be

zero

Vin (119904) = 0

1(119904) = 0

(39)

According to (17) and (34) the 2-to-V119862and the

2-to-119894119871

transfer functions are

119866V1198621198892 (119904) =V119862(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11987111986311198634

2119877) 119904120572

+ 1198811198621198632

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

1198661198941198711198892

(119904) =119871(119904)

2(119904)

100381610038161003816100381610038161003816100381610038161003816Vin(119904)=01198891(119904)=0

= minus

(119881in11986311198771198633

2) (119862119877119904

120573

+ 2)

(1198711198621198632

2) 119904120572+120573 + (119871119877119863

2

2) 119904120572 + 1

(40)

Mathematical Problems in Engineering 7

Table 1 Parameters of the Buck-Boost converter

Parameter Symbol ValueInput voltage 119881in 50VCapacitor 119862 100120583FInductance 119871 1mHSwitching frequency 119891 20 kHzDuty cycle 119863

104

Duty cycle 1198633

04The order 120572 0951The order 120573 0951

We can find from (36) (38) and (40) that orders of theinductor and capacitor have influence on transfer functionsThatmeans fractional orders can be seen as extra parametersWhen orders are 1 we can get integer-order transfer functionsfrom the fractional-order model The integer-order model isa special case of the fractional-order model

5 Experimental Researches

In this section some numerical and circuit simulation exper-iments are presented to verify the fractional-ordermodel andthe influence of the order on performances of the converterin PCCM We choose 095 as the order which is close to 1Parameters of the Buck-Boost converter are listed in Table 1

We select the load resistor as 119877 = 20Ω to ensure thatthe converter is operating in PCCM Difference values ofthe quiescent operating point between the integer-order andfractional-order Buck-Boost converter model are shown inTable 2

In Table 2 we can find that the output voltage ripple andthe inductor ripple rise with the decreasing of the orderThatmeans the order of the Buck-Boost converter impacts theoutput characteristics of the converter obviously

51 Numerical Simulation In numerical simulation Oustal-ouprsquos recursive approximation is adopted [33] The switchingfrequency of the converter is selected as 119891 = 20 kHz thecorresponding rotational frequency is 120596 = 2120587119891 = 1256 times

105 and parameters of Oustaloup are 120596

ℎ= 2 times 10

5 120596119887=

5 times 10minus6 and119873 = 6

Figure 7 shows the inductor current the detailed valuesare listed in Table 3 We can see that in the inductor currentripple maximum and minimum values of the inductorcurrent increase significantly in the fractional-order modelThat means the effect of the inductor order on the inductorcurrent should be appropriately considered

The order also influences the output voltage As shownin Figure 8 and Table 4 in the output voltage ripple themaximum and minimum values of output voltage increasesignificantly in the fractional-order model

52 Circuitry Simulation Experiment To further verify theresults of the theoretical analysis the circuit simulationexperiments are presented

03 03 03001 03001 03002 0300223

235

24

245

25

255

26

265

27

Time (s)

IOFO (095)

i L(A

)

Figure 7 The inductor current

03 03 03001 03001 03002 03002minus105

minus100

minus95

Time (s)

IOFO (095)

C(V

)

Figure 8 The output voltage

By using resistorcapacitor or resistorinductor networkswe can construct an approximation circuit of the fractional-order circuit elements We choose the chain structure as thefractance unit to approximately achieve the fractional-ordercircuit elements

The chain fractance unit of fractional-order capacitor isshown in Figure 9

The transfer function model of the capacitor is

119867(119904) = 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (41)

We choose 120573 = 095 119862 = 100 120583F and 119899 = 6We can calculate resistors and inductors in Figure 9 by themethod of undetermined coefficients comparing the input

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Parameters of the Buck-Boost converter

Parameter Symbol ValueInput voltage 119881in 50VCapacitor 119862 100120583FInductance 119871 1mHSwitching frequency 119891 20 kHzDuty cycle 119863

104

Duty cycle 1198633

04The order 120572 0951The order 120573 0951

We can find from (36) (38) and (40) that orders of theinductor and capacitor have influence on transfer functionsThatmeans fractional orders can be seen as extra parametersWhen orders are 1 we can get integer-order transfer functionsfrom the fractional-order model The integer-order model isa special case of the fractional-order model

5 Experimental Researches

In this section some numerical and circuit simulation exper-iments are presented to verify the fractional-ordermodel andthe influence of the order on performances of the converterin PCCM We choose 095 as the order which is close to 1Parameters of the Buck-Boost converter are listed in Table 1

We select the load resistor as 119877 = 20Ω to ensure thatthe converter is operating in PCCM Difference values ofthe quiescent operating point between the integer-order andfractional-order Buck-Boost converter model are shown inTable 2

In Table 2 we can find that the output voltage ripple andthe inductor ripple rise with the decreasing of the orderThatmeans the order of the Buck-Boost converter impacts theoutput characteristics of the converter obviously

51 Numerical Simulation In numerical simulation Oustal-ouprsquos recursive approximation is adopted [33] The switchingfrequency of the converter is selected as 119891 = 20 kHz thecorresponding rotational frequency is 120596 = 2120587119891 = 1256 times

105 and parameters of Oustaloup are 120596

ℎ= 2 times 10

5 120596119887=

5 times 10minus6 and119873 = 6

Figure 7 shows the inductor current the detailed valuesare listed in Table 3 We can see that in the inductor currentripple maximum and minimum values of the inductorcurrent increase significantly in the fractional-order modelThat means the effect of the inductor order on the inductorcurrent should be appropriately considered

The order also influences the output voltage As shownin Figure 8 and Table 4 in the output voltage ripple themaximum and minimum values of output voltage increasesignificantly in the fractional-order model

52 Circuitry Simulation Experiment To further verify theresults of the theoretical analysis the circuit simulationexperiments are presented

03 03 03001 03001 03002 0300223

235

24

245

25

255

26

265

27

Time (s)

IOFO (095)

i L(A

)

Figure 7 The inductor current

03 03 03001 03001 03002 03002minus105

minus100

minus95

Time (s)

IOFO (095)

C(V

)

Figure 8 The output voltage

By using resistorcapacitor or resistorinductor networkswe can construct an approximation circuit of the fractional-order circuit elements We choose the chain structure as thefractance unit to approximately achieve the fractional-ordercircuit elements

The chain fractance unit of fractional-order capacitor isshown in Figure 9

The transfer function model of the capacitor is

119867(119904) = 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (41)

We choose 120573 = 095 119862 = 100 120583F and 119899 = 6We can calculate resistors and inductors in Figure 9 by themethod of undetermined coefficients comparing the input

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 The quiescent operating point of the Buck-Boost DCDC converter

Parameter Symbol Integer order Fractional order (095)Output voltage 119881

119862minus100V minus100V

Output voltage ripple ΔV119862

2V 338VMinimum values of the output voltage V

119862min minus101 V minus10169VMaximum values of the output voltage V

119862max minus99V minus9831 VInductor current 119868

11987125A 25A

Inductor current ripple Δ119894119871

1 A 175AMinimum values of the inductor current 119894

119871min 245 A 2412 AMaximum values of the inductor current 119894

119871max 255 A 2588A

Table 3 Values of inductor current

Order of the model Inductor current ripple Minimum values of the inductor current Maximum values of the inductor current1 Δ119894

119871= 1A 119894

119871min = 245 A 119894119871max = 255 A

095 Δ119894119871= 189A 119894

119871min = 2404A 119894119871max = 2593A

Table 4 Values of the output voltage

Order of the model Output voltage ripple Minimum values of the output voltage Maximum values of the output voltage1 ΔV

119862= 196V V

119862min = minus1009V V119862max = minus9896V

095 ΔV119862= 366V V

119862min = minus1017 V V119862max = minus9804V

1

Cs120573cong

RinR1 R2 Rn

C1 C2 Cn

middot middot middot

Figure 9 The approximate circuit of the fractional-order capacitor(119862 = 100 120583F 120573 = 095)

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119862119904minus120573

= 119877in +119899

sum

119896=1

1119862119896

(119904 + 1119877119896119862119896) (42)

Solving (42) parameters of the chain fractance unit offractional-order capacitor are as follows 119877

1= 0036Ω 119877

2=

29Ω 1198773= 23064Ω 119877

4= 1832 kΩ 119877

5= 146MΩ

1198776= 111 times 10

3MΩ 119877in = 22mΩ 1198621 = 24627 120583F 1198622 =30698 120583F 119862

3= 38642 120583F 119862

4= 48448 120583F 119862

5= 61241 120583F

and 1198626= 15858 120583F

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 10 they are approximate fitting

The chain fractance unit of fractional-order inductor isshown in Figure 11

The transfer function model of the inductor is

119867(119904) =1

sum119899

119896=1((1119871

119896) (119904 + 119877

119896119871119896) + 1119877in)

(43)

We choose 120572 = 095 119871 = 1mH and 119899 = 6 Wecan calculate resistors and inductors in Figure 11 by themethod of undetermined coefficients comparing the input

minus200

20406080

100120

Mag

nitu

de (d

B)

minus120

minus90

minus60

Phas

e (de

g)

Bode diagram

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Frequency (Hz)10minus2 10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

1Cs120573 (C = 100120583F 120573 = 095)

Figure 10 The Bode diagram of the 095-order capacitor

impedance of the approximate circuit model and Oustalouprsquosapproximation

1

119871119904minus120572

=

119899

sum

119896=1

1119871119896

(119904 + 119877119896119871119896)+1

119877in (44)

Parameters of the chain fractance unit of the fractional-order capacitor are as follows 119877

1= 27632Ω 119877

2= 344Ω

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Ls120572 cong

Rin R1 R2 Rn

L1 L2 Ln

middot middot middot

middot middot middot

Figure 11 The approximate circuit of the fractional-order inductor(119871 = 1mH 120572 = 095)

minus80minus60minus40minus20

0204060

Mag

nitu

de (d

B)

60

90

120

Bode diagram

Phas

e (de

g)

Frequency (Hz)10minus1 100 101 102 103 104

Oustaloup approximationFractance circuit approximation

Frequency (Hz)10minus1 100 101 102 103 104

Ls120572 (L = 1mH 120572 = 095)

Figure 12 The Bode diagram of the 095-order inductor

1198773= 434mΩ 119877

4= 54584 120583Ω 119877

5= 687 120583Ω 119877

6= 898 times

10minus3

120583Ω 119877in = 22mΩ 1198711= 25mH 119871

2= 31mH 119871

3=

39mH 1198714= 49mH 119871

5= 61mH and 119871

6= 16mH

Bode diagrams of the fractional-order capacitor bynumerical simulations and circuit simulations are comparedin Figure 12 they are approximate fitting

In Figure 13 we implemented the circuit simulation byPsim

The load resistor is selected as 119877 = 20Ω due to nonidealcircuit elements the output voltage and inductor current areslightly different from the numerical simulation As shown inFigures 14 and 15 with the decrease of the order of energystorage elements the response of the converter becomesfaster and the overshoot becomes smaller

All circuit simulation experiments are consistent withresults of the theoretical analysis and numerical simulationsThe influence of the fractional order on the performance

+minus

A

V

Figure 13 The 095-order Buck-Boost converter by Psim

0

minus50

minus100

minus150

0 002 004 006 008 01

Integer-order model

minus96

minus98

minus100

minus102

minus104007 007005 00701 007015 00702 007025

Time (s)

C(V

)

Fractional-order model

Figure 14 The output voltage

of the converter is obvious that means the order should beconsidered as extra parameters

6 Conclusions

Fractional-order models can increase the flexibility anddegrees of freedom by means of fractional parameters Theyare discussed inmany fields and generate some new conceptsIn this paper based on fractional calculus we establishedthe fractional-order state-space averagingmodel of the Buck-Boost converter in PCCM And some influences of the orderon the quiescent operating point and the low-frequencycharacteristics of the converter are investigated by theoreticalanalyses and numerical and circuit simulations We haveproved that fractional components increase the two ripples(the output voltage rippleΔV

119862and the inductor current ripple

Δ119894119871) but with the decrease of the order of energy storage

elements the response of the converter becomes faster andthe overshoot becomes smaller In theory all the actualenergy storage elements should be fractional order Thatmay be an important reason why the actual device is alwaysimperfect Therefore how to determine the order of energystorage elements and the relationship between the fractional

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

40

30

20

10

0

0 002 004 006 008 01

Fractional-order model

Integer-order model

00723

24

25

26

27

007005 00701 007015 00702 007025

Time (s)

i L(A

)

Figure 15 The inductor current

order and performances of the actual energy storage elementsare our future research directions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 51507134 51177117 and51279161) Scientific Research Program Funded by ShaanxiProvincial Education Department (Program no 15JK1537)and Doctoral Scientific Research Foundation of Xirsquoan Poly-technic University (Grant no BS15025)

References

[1] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[2] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[3] I S Jesus and J A T Machado ldquoDevelopment of fractionalorder capacitors based on electrolyte processesrdquo NonlinearDynamics vol 56 no 1-2 pp 45ndash55 2009

[4] F-Q Wang and X-K Ma ldquoTransfer function modeling andanalysis of the open-loop Buck converter using the fractionalcalculusrdquo Chinese Physics B vol 22 no 3 Article ID 0305062013

[5] A Jalloul J-C Trigeassou K Jelassi and P Melchior ldquoFrac-tional order modeling of rotor skin effect in inductionmachinesrdquo Nonlinear Dynamics vol 73 no 1-2 pp 801ndash8132013

[6] I Petras ldquoChaos in the fractional-order Voltarsquos system mod-eling and simulationrdquo Nonlinear Dynamics vol 57 no 1-2 pp157ndash170 2009

[7] S I R Arias D R Munoz J S Moreno S Cardoso R Ferreiraand P J P Freitas ldquoFractional modeling of the AC large-signal frequency response in magnetoresistive current sensorsrdquoSensors vol 13 no 12 pp 17516ndash17533 2013

[8] A Charef ldquoModeling and analog realization of the fundamentallinear fractional order differential equationrdquoNonlinear Dynam-ics vol 46 no 1-2 pp 195ndash210 2006

[9] F Q Wang and X K Ma ldquoModeling and analysis of thefractional order Buck converter in DCM operation by usingfractional calculus and the circuit-averaging techniquerdquo Journalof Power Electronics vol 13 no 6 pp 1008ndash1015 2013

[10] A G Radwan and M E Fouda ldquoOptimization of fractional-order RLC filtersrdquo Circuits Systems and Signal Processing vol32 no 5 pp 2097ndash2118 2013

[11] I Podlubny ldquoFractional-order systems and PI-lambda-D-mu-controllersrdquo IEEE Transactions on Automatic Control vol 44no 1 pp 208ndash214 1999

[12] I Petras ldquoChaos in fractional-order population modelrdquo Inter-national Journal of Bifurcation and Chaos vol 22 no 4 ArticleID 1250072 2012

[13] S Westerlund and L Ekstam ldquoCapacitor theoryrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 1 no 5 pp826ndash839 1994

[14] S Westerlund Dead Matter has Memory Causal ConsultingKalmar Sweden 2002

[15] J A T Machado and A M S F Galhano ldquoFractional orderinductive phenomena based on the skin effectrdquo NonlinearDynamics vol 68 no 1-2 pp 107ndash115 2012

[16] T C Haba G Ablart T Camps and F Olivie ldquoInfluence ofthe electrical parameters on the input impedance of a fractalstructure realised on siliconrdquo Chaos Solitons amp Fractals vol 24no 2 pp 479ndash490 2005

[17] T C Haba G L Loum andG Ablart ldquoAn analytical expressionfor the input impedance of a fractal tree obtained by a micro-electronical process and experimentalmeasurements of its non-integral dimensionrdquoChaos Solitonsamp Fractals vol 33 no 2 pp364ndash373 2007

[18] R Martinez Y Bolea A Grau and H Martinez ldquoFractionalDCDC converter in solar-powered electrical generation sys-temsrdquo in Proceedings of the IEEE Conference on Emerging Tech-nologies amp Factory Automation (ETFA rsquo09) pp 1ndash6 MallorcaSpain September 2009

[19] F-QWang and X-K Ma ldquoFractional order modeling and sim-ulation analysis of Boost converter in continuous conductionmode operationrdquo Acta Physica Sinica vol 60 no 7 Article ID070506 2011

[20] N-N Yang C-X Liu and C-J Wu ldquoModeling and dynamicsanalysis of the fractional-order Buck-Boost converter in contin-uous conductionmoderdquo Chinese Physics B vol 21 no 8 ArticleID 080503 2012

[21] DMa andW-H Ki ldquoFast-transient PCCM switching converterwith freewheel switching controlrdquo IEEETransactions onCircuitsand Systems II Express Briefs vol 54 no 9 pp 825ndash829 2007

[22] F Zhang and J P Xu ldquoA novel PCCM boost PFC converterwith fast dynamic responserdquo IEEE Transactions on IndustrialElectronics vol 58 no 9 pp 4207ndash4216 2011

[23] Y Yang L Sun and X B Wu ldquoA single-inductor dual-output buck converter with self-adapted PCCM methodrdquo inProceedings of the IEEE International Conference of ElectronDevices and Solid-State Circuits (Edssc rsquo09) pp 87ndash90 XirsquoanChina December 2009

[24] I Petras Fractional Order Nonlinear SystemsmdashModeling Anal-ysis and Simulation Springer New York NY USA 2011

[25] T T Hartley and C F Lorenzo ldquoThe error incurred in usingthe caputo-derivative laplace-transformrdquo in Proceedings of the

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

ASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conference vol4 pp 271ndash278 San Diego Calif USA September 2009

[26] J Sabatier M Merveillaut R Malti and A Oustaloup ldquoHowto impose physically coherent initial conditions to a fractionalsystemrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 5 pp 1318ndash1326 2010

[27] J C Trigeassou NMaamri J Sabatier andAOustaloup ldquoStatevariables and transients of fractional order differential systemsrdquoComputers ampMathematics with Applications vol 64 no 10 pp3117ndash3140 2012

[28] T T Hartley R J Veillette J L Adams and C F LorenzoldquoEnergy storage and loss in fractional-order circuit elementsrdquoIET Circuits Devices amp Systems vol 9 no 3 pp 227ndash235 2015

[29] T T Hartley R J Veillette C F Lorenzo and J L AdamsldquoOn the energy stored in fractional-order electrical elementsrdquoin Proceedings of the ASME International Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference 2013 vol 4 ASME 2014

[30] T T Hartley C F Lorenzo J-C Trigeassou and NMaamri ldquoEquivalence of history-function based and infinite-dimensional-state initializations for fractional-order operatorsrdquoJournal of Computational and Nonlinear Dynamics vol 8 no4 Article ID 041014 2013

[31] D S Ma W-H Ki and C-Y Tsui ldquoA pseudo-CCMDCMSIMO switching converter with freewheel switchingrdquo IEEEJournal of Solid-State Circuits vol 38 no 6 pp 1007ndash1014 2003

[32] R W Erickson Fundamentals of Power Electronics Chapmanand Hall New York NY USA 1997

[33] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of