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

1State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area, Xi’an University of Technology, Xi’an 710048, China 2Institute of Water Resources and Hydro-Electric Engineering, Xi’an University of Technology, Xi’an 710048, China 3College of Electronics and Information, Xi’an Polytechnic University, Xi’an 710048, China 4School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China


Introduction
Fractional calculus is an old mathematical topic which can go back to 1695; when L'Hospital wrote to Leibniz, he mentioned this kind of calculation [1,2].Over the past 300 years, several mathematicians contributed to this mathematical subject.In recent years, fractional calculus has been accepted as a new instrument that can extend the descriptive power of the traditional calculus.It has turned out that many phenomena in widespread fields of science and engineering can be described very successfully by fractional-order models [3][4][5][6][7][8][9].Comparing with integer-order models, the significant advantage of fractional-order models is that they are characterized by hereditary and memory properties.And it has been demonstrated that fractional-order models can increase the flexibility and degrees of freedom by means of fractionalorder parameters [10][11][12].
The theory and applications of fractional-order components had a considerable progress during the last two decades.Researchers have reported the "intrinsic" fractional-order behavior of real objects, such as capacitors and inductors.
Westerlund and his colleagues constructed the fractionalorder model of the capacitor and measured the orders of some real capacitors under different dielectrics by experiment [13,14].Westerlund also demonstrated that real inductors are fractional order and measured the orders of some real inductors [14].Using different fractal structures, fractionalorder capacitors with different orders were implemented by Jesus and Machado [3].Based on skin effect, Machado and Galhano proposed that fractional-order inductors with 2 Mathematical Problems in Engineering different orders can be constructed [15].Real fractional-order capacitors were created by Haba and his colleagues [16,17].These realizations of fractional-order capacitors and inductors indicate the possibility of fractional-order components and models being employed in practical applications.
Capacitors and inductors are the fundamental part of the Buck-Boost converter.Based on the facts that real capacitors and inductors are all fractional-order components, the idea of establishing the fractional-order model of the Buck-Boost converter seems to be far more sensible.It can be proved that the order of the model has significant influence on the performance of the Buck-Boost converter and can be considered as an extra parameter.Up to date, few fractionalorder models of converter have been established [18,19].
Martinez and his colleagues established the fractional-order model of the Buck-Boost converter, but only the capacitor was considered as the fractional-order circuit element.Wang and Ma constructed the fractional-order model of the Boost converter in continuous conduction mode (CCM) and discontinuous conduction mode (DCM) [19]; Yang and his colleagues proposed the fractional-order model of the Buck-Boost converter in CCM [20], but no circuit simulations or experiments were presented to verify these models.In this paper, a fractional-order state-space averaging model of the Buck-Boost converter in pseudo-continuous conduction mode (PCCM) is established; numerical and circuit simulation experiments are presented to verify the efficiency of the proposed theoretical analysis.
Generally, Buck-Boost converters operate in CCM or DCM.In recent reports, pseudo-continuous conduction mode (PCCM) is proposed which is the third operation mode.Comparing with operating in DCM, in PCCM, the current handling capability of the converter is improved and the current ripple and voltage ripple are reduced.And a single-pole behavior in the control-to-output transfer function is exhibited; the load transient response is much faster than operating in CCM and DCM [21][22][23].Therefore, the study of the fractional-order model of the Buck-Boost converter in PCCM is an important theoretical issue and has practical engineering value.
In this paper, we established the fractional-order statespace averaging model of the Buck-Boost converter in PCCM and analyzed the influence of orders on some dynamical properties of the fractional-order model.The rest of this paper is organized as follows: Section 2 is the preliminaries; some basic concepts of fractional calculus and the basic fractional-order inductor and capacitor models are introduced.In Section 3, based on fractional calculus, we established a fractional-order state-space averaging model of the Buck-Boost converter in PCCM.In Section 4, the quiescent operation point and transfer functions of the fractional-order converter are analyzed.Orders of the fractional-order model are proved to be extra parameters and have influence on the performance of the converter.In Section 5, numerical and circuit simulation experiments are implemented to verify the correctness of theoretical analysis and the proposed fractional-order model.Finally, in Section 6, influences of orders on the fractional-order model are summarized.

Preliminaries
2.1.Fractional Calculus.Fractional calculus allows operations of ordinary differentiations and integrations to noninteger-order.It is a generalization of traditional calculus, but its applicability is much wider [24].The fundamental operator     , where  ( ∈ R) is the order and  and  are the bounds of the operation, is defined as The three best known definitions for fractional calculus are Grünwald-Letnikov (GL) definition, Riemann-Liouville (RL) definition, and Caputo definition [2].
Caputo definition is defined as where Γ(⋅) is the Gamma function and  ∈  is the first integer which is not less than ,  − 1 <  < .
The Laplace transform of Caputo fractional derivative satisfies For zero initial conditions, the Laplace transform of fractional derivatives has the following form: Theoretical calculations are based on the Caputo derivative and more specifically on the initial condition (0).Hartley and Lorenzo discuss the error incurred in using the Caputo-derivative Laplace transform [25].It is now well established that more precisely initial state of a Fractional Differential Equation can be represented by the weighted integral of (, 0) [26][27][28][29][30]. Nevertheless, under the premise that the error is acceptable, and for convenience, the Caputo derivative and the initial condition (0) are adopted.

The Fractional-Order Model of Capacitors and Inductors.
The fractional-order model of capacitors stems from Curie's empirical law which is proposed in 1889 [13]: where ℎ 1 is a constant related to the capacitance and the kind of the dielectric. is a constant close to 1, related to the losses of the capacitor. 0 is the DC voltage applied at  = 0.
In 1994, based on fractional calculus, Westerlund and Ekstam developed a new capacitor theory [13]: where   is the capacitance of the capacitor related to the kind of dielectric.And  is the order related to the losses of the capacitor.Westerlund and Ekstam provided various capacitor dielectrics with appropriated order  by experiment.
Westerlund also created the fractional-order model of inductors [14]: where   is the inductance of the inductor and  is the order related to the proximity effect.
In engineering applications, orders of capacitors and inductors are close to 1 [13].In order to ensure that the power electronic devices can work properly, we redefine a fractional order between 0.95 and 1.

The State-Space Averaging Model of the Fractional-Order Buck-Boost Converter in PCCM
3.1.The Fractional-Order Mathematical Model of the Buck-Boost Converter.The Buck-Boost converter can operate as Buck (for voltage stepdown) or Boost (for voltage stepup) converter, inverting the voltage polarity.Connecting an inductor in parallel with a power switch makes the converter operate in PCCM [31].The circuit diagram is shown in Figure 1.There are two fully controlled switching devices  1 and  2 , two diodes  1 and  2 , one DC input voltage  in , one load resistor , and two fractional-order energy storage elements  and  in the circuit.
In Figure 2, where  is the switching time period,  on is the on time,  off is the off time for the switch, and  1 +  2 +  3 = 1.There are three states for the pseudo-continuous mode.
State 1.When 0 <  ≤  1 , the circuit topology is shown in Figure 3. Switch  1 is on,  2 is off, and diodes  1,2 are reversebiased.The inductor current   rises.The expression of the fractional-order mathematical model is State 2. When  1  <  ≤ ( 1 +  2 ), as shown in Figure 4, switches  1,2 and diode  2 are off and diode  1 is forwardbiased and behaves as a short circuit; the inductor current   ramps down.The expression of the fractional-order mathematical model is State 3. When ( 1 +  2 ) <  ≤ , switch  1 and diode  1 are off, switch  2 and diode  2 are on, and the inductor current   maintains a constant theoretically.The circuit topology is shown in Figure 5.
The expression of the fractional-order mathematical model is As shown in ( 8), (9), and (10), the inductor order  and the capacitor order  can be considered as extra parameters of the fractional-order mathematical models in PCCM.The influence of these extra parameters on the performance of the converter will be discussed in the next section.

The Fractional-Order State-Space Averaging Model of the Buck-Boost DC/DC Converter.
To remove the switching harmonics, waveforms over one switching period should be averaged [32].The average value of variable () which is an arbitrary circuit variable of the Buck-Boost converter in one switching period is defined as follows: Its fractional-order form is where  is the order and 0.95 <  < 1.
Let us consider the fractional-order state-space averaging model of Buck-Boost converter in PCCM: Average values of circuit variables are defined as the following form: where variables in capital letters represent DC components and variables with the symbol "̂" above them represent AC components which are much smaller in magnitude than the corresponding DC component.
Equation ( 13) can be rewritten as DC components in (15) are as follows: The quiescent operation point is The voltage ratio is defined as follows: AC components of (15) are where d2 () î (), d2 ()V  (), and d1 ()V in () are high order small signal terms which can be omitted.Then the small signal AC equation of Buck-Boost converter in PCCM can be expressed as follows:

Performance Analysis of the Fractional-Order Model of the Buck-Boost Converter in PCCM
In this section, the quiescent operation point and transfer functions of the AC small signal model will be analyzed, because they have practical significance for the designing of various parameters and reflect the performance of the converter.
The inductor current ripple and output voltage ripple of the fractional-order Buck-Boost converter in PCCM are shown in Figure 6.
Figure 6: The inductor current ripple and output voltage ripple.
The small ripple approximation is defined as replacing waveforms with their low-frequency averaged values: Using Caputo definition, we can get the corresponding initial condition as Solving fractional-order differential equation ( 21), we can get In PCCM, the initial value of inductor current is given as   ( 0 ) =   min . Then The inductor current ripple is The maximum and minimum values of inductor current are According to Adomian decomposition scheme, one can get In States 2 and 3, the load is powered by the capacitor.The output voltage of the converter is negative which satisfies the state equation and the amplitude is monotone decreasing: where  = −1/, () = 0, and 0.95 <  < 1.
The solution of ( 28) is where  max is the initial value of V  (), which means V  ( 0 ) =  max for  0 = 0.
When  = ( 1 +  3 ), the output voltage is The variation of the output voltage is According to ( 17) and ( 31), it yields Taking (32) to (31) yields We can find from ( 25) and ( 33) that the inductor current ripple and the output voltage ripple are closely related to the order of energy storage elements.The ripple increases with the decreasing of the order.
Under conditions of zero initial value, the Laplace transform of converter equations is Set the duty cycle variation to be zero: According to (34), one can get the V in -to-V  and the V into-  transfer function as follows: Set the AC input voltage variation and the duty cycle d2 to be zero: The d1 -to-V  and the d1 -to-  transfer functions are Set the AC input voltage variation and the duty cycle d1 to be zero: According to ( 17) and ( 34), the d2 -to-V  and the d2 -to-  transfer functions are We can find from (36), (38), and (40) that orders of the inductor and capacitor have influence on transfer functions.That means fractional orders can be seen as extra parameters.When orders are 1, we can get integer-order transfer functions from the fractional-order model.The integer-order model is a special case of the fractional-order model.

Experimental Researches
In this section, some numerical and circuit simulation experiments are presented to verify the fractional-order model and the influence of the order on performances of the converter in PCCM.We choose 0.95 as the order which is close to 1. Parameters of the Buck-Boost converter are listed in Table 1.
We select the load resistor as  = 20 Ω to ensure that the converter is operating in PCCM.Difference values of the quiescent operating point between the integer-order and fractional-order Buck-Boost converter model are shown in Table 2.
In Table 2, we can find that the output voltage ripple and the inductor ripple rise with the decreasing of the order.That means the order of the Buck-Boost converter impacts the output characteristics of the converter obviously.
Figure 7 shows the inductor current; the detailed values are listed in Table 3.We can see that, in the inductor current ripple, maximum and minimum values of the inductor current increase significantly in the fractional-order model.That means the effect of the inductor order on the inductor current should be appropriately considered.
The order also influences the output voltage.As shown in Figure 8 and Table 4, in the output voltage ripple, the maximum and minimum values of output voltage increase significantly in the fractional-order model.

Circuitry Simulation Experiment.
To further verify the results of the theoretical analysis, the circuit simulation experiments are presented.By using resistor/capacitor or resistor/inductor networks, we can construct an approximation circuit of the fractionalorder circuit elements.We choose the chain structure as the fractance unit to approximately achieve the fractional-order circuit elements.
The chain fractance unit of fractional-order capacitor is shown in Figure 9.
The transfer function model of the capacitor is We choose  = 0.95,  = 100 F, and  = 6.We can calculate resistors and inductors in Figure 9 by the method of undetermined coefficients, comparing the input impedance of the approximate circuit model and Oustaloup's approximation: Solving (42), parameters of the chain fractance unit of fractional-order capacitor are as follows:  1 = 0.036 Ω,  2 = 2.9 Ω,  Bode diagrams of the fractional-order capacitor by numerical simulations and circuit simulations are compared in Figure 10; they are approximate fitting.
The chain fractance unit of fractional-order inductor is shown in Figure 11.
The transfer function model of the inductor is We choose  = 0.95,  = 1mH, and  = 6.We can calculate resistors and inductors in Figure 11

Oustaloup approximation
Fractance circuit approximation impedance of the approximate circuit model and Oustaloup's approximation: Parameters of the chain fractance unit of the fractionalorder capacitor are as follows:  1 = 276.32Ω,  2 = 3.44 Ω, Bode diagrams of the fractional-order capacitor by numerical simulations and circuit simulations are compared in Figure 12; they are approximate fitting.
In Figure 13, we implemented the circuit simulation by Psim.
The load resistor is selected as  = 20 Ω; due to nonideal circuit elements, the output voltage and inductor current are slightly different from the numerical simulation.As shown in Figures 14 and 15, with the decrease of the order of energy storage elements, the response of the converter becomes faster and the overshoot becomes smaller.
All circuit simulation experiments are consistent with results of the theoretical analysis and numerical simulations.The influence of the fractional order on the performance  of the converter is obvious; that means the order should be considered as extra parameters.

Conclusions
Fractional-order models can increase the flexibility and degrees of freedom by means of fractional parameters.They are discussed in many fields and generate some new concepts.In this paper, based on fractional calculus, we established the fractional-order state-space averaging model of the Buck-Boost converter in PCCM.And some influences of the order on the quiescent operating point and the low-frequency characteristics of the converter are investigated by theoretical analyses and numerical and circuit simulations.We have proved that fractional components increase the two ripples (the output voltage ripple ΔV  and the inductor current ripple Δ  ), but with the decrease of the order of energy storage elements, the response of the converter becomes faster and the overshoot becomes smaller.In theory, all the actual energy storage elements should be fractional order.That may be an important reason why the actual device is always imperfect.Therefore, how to determine the order of energy storage elements and the relationship between the fractional Mathematical Problems in Engineering order and performances of the actual energy storage elements are our future research directions.

Figure 2 :
Figure 2: Inductor voltage and current of Buck-Boost converter in PCCM.

Figure 3 :
Figure 3: Schematic diagram of Buck-Boost converter in State 1.
by the method of undetermined coefficients, comparing the input

Table 1 :
Parameters of the Buck-Boost converter.