#### Abstract

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

#### 1. 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 [3–9]. 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 fractional-order parameters [10–12].

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 fractional-order 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, fractional-order capacitors with different orders were implemented by Jesus and Machado [3]. Based on skin effect, Machado and Galhano proposed that fractional-order inductors with 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 fractional-order 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 [21–23]. 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 state-space 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.

#### 2. Preliminaries

##### 2.1. Fractional Calculus

Fractional calculus allows operations of ordinary differentiations and integrations to non-integer-order. It is a generalization of traditional calculus, but its applicability is much wider [24]. The fundamental operator , where () 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 aswhere is the Gamma function and is the first integer which is not less than , .

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 . 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 [26–30]. Nevertheless, under the premise that the error is acceptable, and for convenience, the Caputo derivative and the initial condition are adopted.

##### 2.2. 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 is a constant related to the capacitance and the kind of the dielectric. is a constant close to , related to the losses of the capacitor. is the DC voltage applied at .

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.

#### 3. 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 and , two diodes and , one DC input voltage , one load resistor , and two fractional-order energy storage elements and in the circuit.

In Figure 2, where is the switching time period, is the on time, is the off time for the switch, and . There are three states for the pseudo-continuous mode.

*State 1*. When , the circuit topology is shown in Figure 3. Switch is on, is off, and diodes are reverse-biased. The inductor current rises.

The expression of the fractional-order mathematical model is

*State 2*. When , as shown in Figure 4, switches and diode are off and diode is forward-biased and behaves as a short circuit; the inductor current ramps down.

The expression of the fractional-order mathematical model is

*State 3*. When , switch and diode are off, switch and diode 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.

##### 3.2. 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 iswhere is the order and .

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) arewhere , , and 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:

#### 4. 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.

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 .

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 , , and .

The solution of (28) iswhere is the initial value of , which means for .

When , the output voltage isThe variation of the output voltage is

According to (17) and (31), it yieldsTaking (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 -to- and the -to- transfer function as follows:Set the AC input voltage variation and the duty cycle to be zero:

The -to- and the -to- transfer functions areSet the AC input voltage variation and the duty cycle to be zero:

According to (17) and (34), the -to- and the -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.

#### 5. 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 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.

##### 5.1. Numerical Simulation

In numerical simulation, Oustaloup’s recursive approximation is adopted [33]. The switching frequency of the converter is selected as kHz, the corresponding rotational frequency is , and parameters of Oustaloup are , , and .

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.

##### 5.2. 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 fractional-order 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 *μ*F, and . 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: Ω, Ω, Ω, kΩ, MΩ, MΩ, mΩ, *μ*F, *μ*F, *μ*F, *μ*F, *μ*F, and *μ*F.

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 , mH, and . We can calculate resistors and inductors in Figure 11 by the method of undetermined coefficients, comparing the input impedance of the approximate circuit model and Oustaloup’s approximation:

Parameters of the chain fractance unit of the fractional-order capacitor are as follows: Ω, Ω, mΩ, *μ*Ω, *μ*Ω, *μ*Ω, mΩ, mH, mH, mH, mH, mH, and mH.

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 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.

#### 6. 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 and the inductor current ripple ), 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 order and performances of the actual energy storage elements are our future research directions.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 51507134, 51177117, and 51279161), Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 15JK1537), and Doctoral Scientific Research Foundation of Xi’an Polytechnic University (Grant no. BS15025).