Modeling and Analysis of a Fractional-Order Buck–Boost Converter in CCM Based on the Riemann–Liouville Definition

1. Introduction

In recent years, the proportion of distributed generation in power systems has continued to increase [1]. Among these systems, the performance of DC–DC converters has a direct and significant impact on conversion efficiency and power quality. Owing to its advantages of low ripple, high efficiency, and low switching loss, the interleaved parallel Buck–Boost converter has been widely applied in the field of distributed generation [2,3]. Therefore, to improve the performance of distributed generation systems, it is necessary to further investigate and optimize the modeling and control techniques of interleaved parallel Buck–Boost converters.

As early as the 17th century, the discussion between L’Hôpital and Leibniz laid the foundation for the emergence of fractional calculus. Fractional calculus is a branch of mathematics that studies integration and differentiation of arbitrary order. However, since physical systems and related processes have long been described mainly by classical calculus, the development of fractional calculus was relatively delayed [4]. Since the 1960s, with the breakthrough of fractal theory and the discovery of fractional-order physical models, fractional calculus has gradually been widely applied [5,6]. With the advancement of science and technology and the deepening of human understanding of nature, traditional integer-order calculus has become insufficient to meet the needs of modern scientific research, and fractional calculus has gradually become a research hotspot both domestically and internationally [7,8,9,10,11].

At present, abundant achievements and significant progress have been made in fractional-order modeling of power electronic converters at home and abroad [12,13]. As a typical converter suitable for fractional-order modeling, most current studies on interleaved parallel Buck–Boost converters are still based on integer-order theory. When describing components such as inductors and capacitors, traditional integer-order calculus models ignore their intrinsic fractional-order characteristics, which limits the modeling accuracy and may introduce errors in converter analysis and design. This is inconsistent with the actual fractional-order characteristics of the converter and affects the accuracy of the model as well as the characterization of its dynamic behavior [14].

The continuous development of fractional calculus theory has provided broad prospects for its application in control systems [15,16]. Fractional-order differentiation and integration not only extend traditional integer-order calculus, but also possess memory characteristics that can store historical states and reduce control effort, enabling fractional-order controllers to generate smoother control signals. This provides a new possibility for the optimized control of power electronic converters. Therefore, constructing a fractional-order model of the interleaved parallel Buck–Boost converter by considering fractional-order energy storage elements is crucial for understanding its dynamic characteristics and designing controllers. This can not only provide a solid theoretical basis for the stable operation of the converter, but also improve the accuracy and reliability of its design.

Considering the diversity of fractional calculus definitions, their requirements for initial conditions are also different. Fractional calculus studies based on different definitions often lead to different results. There are three main definitions of fractional calculus, namely the Caputo fractional-order definition, the Riemann–Liouville (R–L) fractional-order definition, and the Grünwald–Letnikov (G–L) fractional-order definition [17,18,19].

This paper focuses on the Caputo fractional-order definition and aims to model and analyze the fractional-order Buck–Boost converter operating in continuous conduction mode (CCM). A fractional-order mathematical model of the converter is established. Then, the steady-state model of the fractional-order converter is analyzed, and the steady-state operating point and ripple characteristics under the Caputo fractional-order definition are calculated. By adopting the improved Oustaloup approximation method, the fractional-order operator is equivalently represented as a finite-dimensional integer-order network. A fractional-order circuit model is then established and compared with mathematical models based on different fractional-order definitions. Furthermore, the influence of the order of fractional-order energy storage elements on the converter is investigated. Finally, an experimental platform is built to verify the correctness of the theoretical analysis.

The main contributions of this paper are summarized as follows. First, an R–L definition-based modeling framework is established for a fractional-order buck–boost converter operating in CCM. Second, a steady-state analysis method is developed to characterize the DC operating point, voltage ratio, and ripple quantities of the converter. Third, the differences between the Caputo-based and R–L-based results are quantitatively evaluated, and the influence of passive-component fractional orders on converter performance is clarified through theoretical analysis, simulation, and experiment.

2. State-Space Model of the CCM Fractional-Order Buck–Boost Converter

The fractional-order buck–boost converter considered in this paper is shown in Figure 1. In this circuit, L₁^α denotes a fractional-order inductor of order α , and C^β denotes a fractional-order capacitor of order β. Here, R is the load resistance, V_in is the input voltage, and both the switch S₁ and S₂ , the diode D₁ and D₂are assumed to be ideal.

Based on the fractional-order component models reported in [17], the constitutive relations for the inductor voltage v_L and capacitor current i_c are written as, where $\alpha \in \left(0,\left.1\right)\right.$,$\beta \in \left(0,\left.1\right)\right.$.

$$\left\{\begin{array}{c}{v}_L(t)=L{\displaystyle \frac{d^{\alpha }{i}_L(t)}{d{t}^{\alpha }}}\\ i_C(t)=C{\displaystyle \frac{d^{\beta }{v}_C(t)}{d{t}^{\beta }}}\end{array}\right.$$

(1)

When the converter operates in continuous conduction mode (CCM), it exhibits four switching states, which are described as follows. where n is an integer, T denotes the switching period, and D represents the duty ratio, i.e., the ratio of the turn-on interval of S_T to the switching period. In steady state, the converter operates by alternating periodically between States 1 and 4.

State 1: S₁ and S₂ ON , D₁ and D₂ OFF, for (t₀<t<t₀+d₁T) , The state equations of the converter are given by:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{L1}}{d{t}^{\alpha }}}={\displaystyle \frac{V_{\mathrm{in}}}{L_1}}\\ {\displaystyle \frac{d^{\alpha }{i}_{L2}}{d{t}^{\alpha }}}={\displaystyle \frac{V_{in}}{L_2}}\\ {\displaystyle \frac{d^{\beta }{v}_C}{d{t}^{\beta }}}=-{\displaystyle \frac{v_C}{RC}}\end{array}\right.$$

(2)

State 2: S₁ OFF , S₂ ON , D₁ ON and D₂ OFF, for（t₀+d₁T<t<t₀+d₁T+d₂T）, The state-space representation of the converter is expressed as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{L1}}{d{t}^{\alpha }}}=-{\displaystyle \frac{v_C}{L_1}}\\ {\displaystyle \frac{d^{\alpha }{i}_{L2}}{d{t}^{\alpha }}}={\displaystyle \frac{V_{in}}{L_2}}\\ {\displaystyle \frac{d^{\beta }{v}_C}{d{t}^{\beta }}}={\displaystyle \frac{i_{L1}}{C}}-{\displaystyle \frac{v_C}{RC}}\end{array}\right.$$

(3)

State 3: S₁ and S₂ ON , D₁ and D₂ OFF, for（t₀+d₁T+d₂T <t<t₀+d₁T+d₂T+ d₃T）, The state-space representation of the converter is expressed the same as State 1.

State 4: S₁ ON, S₂ OFF , D₁ OFF and D₂ ON, for(t₀+d₁T+d₂T+d₃T <t<t₀+T), The state-space representation of the converter is expressed the same as State 1.

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{i}_{L1}}{d{t}^{\alpha }}}={\displaystyle \frac{V_{in}}{L_1}}\\ {\displaystyle \frac{d^{\alpha }{i}_{L2}}{d{t}^{\alpha }}}=-{\displaystyle \frac{v_C}{L_2}}\\ {\displaystyle \frac{d^{\beta }{v}_C}{d{t}^{\beta }}}={\displaystyle \frac{i_{L2}}{C}}-{\displaystyle \frac{v_C}{RC}}\end{array}\right.$$

(4)

Since State 1 and 3 are identical, d₁=d₃. Moreover, S₁ and S₂ are operated interleavingly with a 180° phase shift and equal conduction durations. Thus, their duty ratios are identical and both are denoted by d. Accordingly:

$$d_1+d_2+d_3=d$$

(5)

$$d_1+d_2+d_3+d_4=1$$

(6)

From (5) and (6), it follows that the turn-off interval of switch S₂ within one switching period T is d₄=1- d. Since d₂ represents the turn-off interval of switch S₁ over the same period, one obtains d₂= d₄=1- d.

Among the four operating modes within one switching period, Modes 1 and 3 are identical. Hence, it is assumed that d₁= d₃.

Applying the state-space averaging method to (2)–(4), the state-space model of the fractional-order interleaved parallel buck–boost converter is obtained as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{〈i_{L1}〉}_T}{d{t}^{{\alpha }_1}}}=-{\displaystyle \frac{d^{\prime }{〈v_c〉}_T}{L_1}}+{\displaystyle \frac{d{〈V_{in}〉}_T}{L_1}}\hfill \\ {\displaystyle \frac{d^{\alpha }{〈i_{L2}〉}_T}{d{t}^{{\alpha }_2}}}=-{\displaystyle \frac{d^{\prime }{〈v_c〉}_T}{L_2}}+{\displaystyle \frac{d{〈V_{in}〉}_T}{L_2}}\hfill \\ {\displaystyle \frac{d^{\beta }{〈v_c〉}_T}{d{t}^{\beta }}}={\displaystyle \frac{d^{\prime }{〈i_{L1}〉}_T}{C}}+{\displaystyle \frac{d^{\prime }{〈i_{L2}〉}_T}{C}}-{\displaystyle \frac{{〈v_c〉}_T}{RC}}\hfill \end{array}\right.$$

(7)

where d^’=1- d, denote the average values of, and, respectively. Each variable is decomposed into a dc component and a small-signal ac component.

Let the dc components of i_L1、i_L2、v_c and v_in be I_L1、I_L2、V_c and V_in respectively, and let their corresponding small-signal ac components be ${\widehat{i}}_{L1}$、${\widehat{i}}_{L2}$、${\widehat{v}}_c$and${\widehat{v}}_{in}$,. Then, (5) can be rewritten as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }\left(I_{L1}+{\widehat{i}}_{L1}\right)}{d{t}^{\alpha }}}=-{\displaystyle \frac{\left(D^{\prime }-\widehat{d}\right)\left(V_c+{\widehat{v}}_c\right)}{L_1}}+{\displaystyle \frac{\left(D+\widehat{d}\right)\left(V_{in}+{\widehat{v}}_{in}\right)}{L_1}}\hfill \\ {\displaystyle \frac{d^{\alpha }\left(I_{L2}+{\widehat{i}}_{L2}\right)}{d{t}^{\alpha }}}=-{\displaystyle \frac{\left(D^{\prime }-\widehat{d}\right)\left(V_c+{\widehat{v}}_c\right)}{L_2}}+{\displaystyle \frac{\left(D+\widehat{d}\right)\left(V_{in}+{\widehat{v}}_{in}\right)}{L_2}}\hfill \\ {\displaystyle \frac{d^{\beta }\left(V_c+{\widehat{v}}_c\right)}{d{t}^{\beta }}}={\displaystyle \frac{\left(D^{\prime }-\widehat{d}\right)\left(I_{L1}+{\widehat{i}}_{L1}\right)}{C}}+{\displaystyle \frac{\left(D^{\prime }-\widehat{d}\right)\left(I_{L2}+{\widehat{i}}_{L2}\right)}{C}}-{\displaystyle \frac{\left(V_c+{\widehat{v}}_c\right)}{RC}}\hfill \end{array}\right.$$

(8)

After separating the dc and small-signal components in (9) and omitting the higher order small-signal terms, the following expression is obtained:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{I}_L}{d{t}^{\alpha }}}+{\displaystyle \frac{d^{\alpha }{\widehat{i}}_L}{d{t}^{\alpha }}}={\displaystyle \frac{\left[D{V}_{in}-D^{\prime }{V}_c\right]+\left[D{\widehat{v}}_{in}+\widehat{d}\left(V_{in}+V_c\right)-D^{\prime }{\widehat{v}}_c\right]}{L_1}}\hfill \\ {\displaystyle \frac{d^{\alpha }{I}_L}{d{t}^{\alpha }}}+{\displaystyle \frac{d^{\alpha }{\widehat{i}}_L}{d{t}^{\alpha }}}={\displaystyle \frac{\left[D{V}_{in}-D^{\prime }{V}_c\right]+\left[D{\widehat{v}}_{in}+\widehat{d}\left(V_{in}+V_c\right)-D^{\prime }{\widehat{v}}_c\right]}{L_2}}\hfill \\ {\displaystyle \frac{d^{\beta }{V}_C}{d{t}^{\beta }}}+{\displaystyle \frac{d^{\beta }{\widehat{v}}_C}{d{t}^{\beta }}}=\left[{\displaystyle \frac{R{D}^{\prime }\left(I_{L1}+I_{L2}\right)-V_c}{RC}}\right]+\left[{\displaystyle \frac{R{D}^{\prime }\left({\widehat{i}}_{L1}+{\widehat{i}}_{L2}\right)-R\widehat{d}\left(I_{L1}+I_{L2}\right)-{\widehat{v}}_c}{RC}}\right]\hfill \end{array}\right.$$

(9)

3. Steady-State Analysis of the Interleaved Parallel Buck–Boost Converter Based on the Riemann–Liouville Definition

According to the Riemann–Liouville (R–L) definition:

$${}_{a}D{}_{{}_t}{}^{\alpha }f(t)=\left\{\begin{array}{l}{\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle \frac{d^n}{d{t}^n}}{\displaystyle {\int }_a^t\frac{f(\tau )}{{(t-\tau )}^{\alpha -n+1}}}d\tau ,\alpha >0\\ {\displaystyle \frac{1}{\Gamma (-\alpha )}}{\displaystyle {\int }_a^t\frac{f(\tau )}{{(t-\tau )}^{\alpha +1}}}d\tau ,\alpha <0\end{array}\right.$$

(11)

where $n\in N^{+}$,$n-1<\alpha <n$.

Separating the dc components of the converter state variables from (10) yields

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{I}_{L1}}{d{t}^{\alpha }}}={\displaystyle \frac{D{V}_{in}-D^{\prime }{V}_c}{L}}\hfill \\ {\displaystyle \frac{d^{\alpha }{I}_{L2}}{d{t}^{\alpha }}}={\displaystyle \frac{D{V}_{in}-D^{\prime }{V}_c}{L}}\hfill \\ {\displaystyle \frac{d^{\beta }{V}_C}{d{t}^{\beta }}}={\displaystyle \frac{R{D}^{\prime }\left(I_{L1}+I_{L2}\right)-V_c}{RC}}\hfill \end{array}\right.$$

(12)

The left-hand side of (12) is rewritten as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d^{\alpha }{I}_{L1}}{d{t}^{\alpha }}}={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\displaystyle {\int }_{nT}^{(n+1)T}\frac{I_{L1}}{{(t-\tau )}^{\alpha }}}d\tau ={\displaystyle \frac{I_{L1}{t}_s{}^{-\alpha }}{\Gamma (1-\alpha )}}\hfill \\ {\displaystyle \frac{d^{\alpha }{I}_{L2}}{d{t}^{\alpha }}}={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle \frac{d}{dt}}{\displaystyle {\int }_{nT}^{(n+1)T}\frac{I_{L2}}{{(t-\tau )}^{\alpha }}}d\tau ={\displaystyle \frac{I_{L2}{t}_s{}^{-\alpha }}{\Gamma (1-\alpha )}}\hfill \\ {\displaystyle \frac{d^{\beta }{V}_C}{d{t}^{\beta }}}={\displaystyle \frac{1}{\Gamma (1-\beta )}}{\displaystyle \frac{d}{dt}}{\displaystyle {\int }_{nT}^{(n+1)T}\frac{V_C}{{(t-\tau )}^{\beta }}}d\tau ={\displaystyle \frac{V_C{t}_s{}^{-\beta }}{\Gamma (1-\beta )}}\hfill \end{array}\right.$$

(13)

where t_s denotes the settling time of the converter. From (12) and (13), the expression for the steady-state operating point of the converter under the Riemann–Liouville (R–L) definition can be obtained as:

$$\left\{\begin{array}{c}{I}_{L1}{}^{R-L}={\displaystyle \frac{D{V}_{in}\Gamma (1-\alpha )\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}{2{D^{\prime }}^{2}R\Gamma (1-\beta )\Gamma (1-\alpha )+L{t}_s{}^{-\alpha }\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}}\\ I_{L2}{}^{R-L}={\displaystyle \frac{D{V}_{in}\Gamma (1-\alpha )\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}{2{D^{\prime }}^{2}R\Gamma (1-\beta )\Gamma (1-\alpha )+L{t}_s{}^{-\alpha }\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}}\\ V_C{}^{R-L}={\displaystyle \frac{2D{D}^{\prime }R{V}_{in}\Gamma (1-\beta )\Gamma (1-\alpha )}{2{D^{\prime }}^{2}R\Gamma (1-\beta )\Gamma (1-\alpha )+L{t}_s{}^{-\alpha }\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}}\end{array}\right.$$

(14)

where the subscript “R–L” denotes the quantities obtained under the Riemann–Liouville (R–L) definition.

According to (15), the steady-state operating points I_L1、I_L2 and V_C depend not only on the orders α and β of the energy-storage elements, but also on the converter settling time t_s. By assuming that steady state is reached within one switching period (t_s=T), the variations of the output voltage Vo and the inductor current I_L1with the orders of the energy-storage elements are illustrated in Figure 2 and Figure 3, respectively.

It can be observed from Figure 2(a) that the output voltage V_o shows a concave dependence on the orders of the energy-storage elements, V_o decreases initially and then increases as the energy storage element orders increase. As illustrated in Figure 2(b): 1) when both α and β approach 1, the fractional-order results are nearly the same as those obtained from the conventional integer-order analysis, whereas a noticeable deviation appears when α and β are approximately 0.9; 2) for a fixed inductor order α , V_o first decreases and then increases with increasing capacitor order β; 3) for a fixed capacitor order β, V_o likewise first decreases and then increases with increasing inductor order α.

Figure 3(a) shows that the inductor current I_L1 depends on both the inductor order α and the capacitor order β . As the orders of the energy-storage elements vary, I_L1 ranges from 0.6 A to 3 A.

As further illustrated in Figure 3(b): 1) with the other converter parameters fixed, I_L1 increases initially and then decreases as β increases; 2) with the remaining parameters unchanged, I_L1 decreases first and then increases as α increases.

The voltage conversion ratio is given by

$$M^{R-L}={\displaystyle \frac{V_C{}^{R-L}}{V_{in}}}={\displaystyle \frac{2D{D}^{\prime }R\Gamma (1-\beta )\Gamma (1-\alpha )}{2{D^{\prime }}^{2}R\Gamma (1-\beta )\Gamma (1-\alpha )+L{t}_s{}^{-\alpha }\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}}$$

(15)

According to (15), the voltage conversion ratio derived under the fractional-order Riemann–Liouville (R–L) definition is determined not only by the orders α and β of the energy-storage elements, but also by the time t_s required for the fractional-order interleaved parallel buck–boost converter to reach steady state. This result differs from the voltage conversion ratios obtained from the integer-order model and the fractional-order Caputo formulation. Assuming t_s =Tt , the dependence of the voltage conversion ratio M on the orders α and β is shown in Figure 4.

Figure 4(a) clearly shows that the voltage conversion ratio M has a concave dependence on the orders of the energy-storage elements, with a minimum located in the intermediate region. As illustrated in Figure 4(b), for a fixed capacitor order β, M decreases slowly at first and then increases sharply as the inductor order α . The value of M ranges from 0.7 to 1.3, and the minimum occurs at approximately α=0.9 . Figure 4(c) indicates that, when the inductor order α is fixed and α≠1 , the variation of M with the capacitor order β is similar to that shown in Figure 4(b). By contrast, when α=1 , M remains unchanged with varying β . As further shown in Figure 4(d), for fixed energy-storage-element orders, M increases rapidly first and then more gradually with increasing load resistance R. For a fixed load resistance, M decreases initially and then increases as the element orders increase. When α=β=1, the voltage conversion ratio reduces to that of the integer-order converter and becomes independent of R. The current ripple of the converter is given by:

$$\left\{\begin{array}{c}\Delta i^{R-L}(t)={\displaystyle \frac{V_{in}{\left[\left(2D-1\right)T\right]}^{\alpha }}{L\alpha \Gamma (\alpha )}}\\ \Delta i_{L1}^{R-L}=\Delta i_{L2}^{R-L}={\displaystyle \frac{V_{in}{\left[DT\right]}^{\alpha }}{L\alpha \Gamma (\alpha )}}\end{array}\right.$$

(16)

The minimum value of the total current is given by

$$i^{R-L}{}_{\min }={\displaystyle \frac{2D{V}_{in}\Gamma (1-\alpha )\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}{2{D^{\prime }}^{2}R\Gamma (1-\beta )\Gamma (1-\alpha )+L{t}_s{}^{-\alpha }\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}}-{\displaystyle \frac{V_{in}{\left[\left(2D-1\right)T\right]}^{\alpha }}{2L\alpha \Gamma (\alpha )}}$$

(17)

Let i^R-L_min=0, the critical condition of the fractional-order interleaved parallel buck–boost converter can be derived as:

$${\displaystyle \frac{2D\Gamma (1-\alpha )\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}{2{D^{\prime }}^{2}R\Gamma (1-\beta )\Gamma (1-\alpha )+L{t}_s{}^{-\alpha }\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}}-{\displaystyle \frac{{\left[\left(2D-1\right)T\right]}^{\alpha }}{2L\alpha \Gamma (\alpha )}}=0$$

(18)

It follows from (15) that the boundary condition of the converter is determined not only by the circuit parameters, but also by the energy-storage-element orders α and β , as well as the settling time t_s of the converter. The relationship between the operating modes of the fractional-order interleaved parallel buck–boost converter and the inductance together with the orders α and β is illustrated in Figure 5.

Figure 5 indicates that the operating mode of the converter depends on both energy-stor age-element orders α and β. As these orders decrease, the inductance must be increased in order to maintain CCM operation. The ripple of the output voltage v_o can be expressed as :

$$\begin{array}{l}\Delta v_o^{R-L}={\displaystyle \frac{2V_c^{R-L}\left\{\mathrm{RC}+{(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[{(DT)}^{\beta }\right]\right\}}{\left\{\mathrm{RC}-{(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[{(DT)}^{\beta }\right]\right\}}}\\ -{\displaystyle \frac{\frac{2DR{V}_{in}{t}_s{}^{-\alpha }\Gamma (1-\alpha )\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right){(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[{(DT)}^{\beta }\right]}{2{D^{\prime }}^{2}R{t}_s{}^{-\alpha }\Gamma (1-\beta )\Gamma (1-\alpha )+L{\left(t_s{}^{-\alpha }\right)}^2\left(\Gamma (1-\beta )+RC{t}_s{}^{-\beta }\right)}}{\left\{\mathrm{RC}-{(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[{(DT)}^{\beta }\right]\right\}}}\\ -{\displaystyle \frac{R\left\{\frac{V_{in}{\left[DT\right]}^{\alpha }}{2L\alpha \Gamma (\alpha )}{(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[{(DT)}^{\beta }\right]\right\}}{\left\{\mathrm{RC}-{(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[{(DT)}^{\beta }\right]\right\}}}\end{array}$$

(19)

where E_β,_β(▪) is the function of Mittag–Leffle, When α=β=1, Equations (14)–(19) have the same form as those of the integer-order converter.

Table 1 lists the expressions for the steady-state operating point, total current ripple Δi , output-voltage ripple Δvo, and voltage conversion ratio M obtained from the integer-order formulation, the fractional-order Riemann–Liouville (R–L) definition, and the fractional-order Caputo definition. It can be seen from Table 1 that the steady-state operating point (Vo,I) and the voltage conversion ratio M obtained under the Caputo definition are the same as those derived from the integer-order model, and are independent of the orders of the energy-storage elements. In contrast, the steady-state operating point obtained under the R–L definition depends not only on the orders of the energy-storage elements, but also on the time required for the converter to reach steady state. Moreover, the output-voltage ripple derived from both the R–L and Caputo definitions is affected by α and β, whereas the current ripple obtained under the Caputo definition depends only on α.

where, ${\gamma }_1=\Gamma (1-\beta )\Gamma (1-\alpha )$，${\gamma }_2=1-{(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[A_1{(DT)}^{\beta }\right]$，${\gamma }_3={(DT)}^{\beta -1}{E}_{\beta ,\beta }\left[A_2{(DT)}^{\beta }\right]$.

In practice, when studying the response of dynamic systems to periodic signals in engineering applications, the lower limit of integration must be set to negative infinity. In the Caputo definition, the lower limit of integration is a constant. Although the use of the Caputo definition simplifies the calculation in stability analysis, it reduces the accuracy of the results. By contrast, in the Riemann–Liouville (R–L) definition of fractional calculus, the lower limit of integration in the time domain is negative infinity, and the obtained solution is a variable related to both the fractional order and time, reflecting the long-memory property of fractional-order systems. Therefore, analyzing the interleaved parallel Buck–Boost converter based on the R–L fractional calculus definition can yield more accurate results than the Caputo definition and is more consistent with the actual operating conditions of the converter.

4. Simulation Results and Analysis

4.1. Fractional-Order Circuit Model

In order to verify fractional-order model obtained above, several simulations are carried out in this Section. The circuit parameters of the interleaved parallel Buck–Boost converter are shown in Table 2.

This paper uses the improved Oustaloup approximation method to construct fractional-order energy storage elements, and its mathematical model is given by:

$$G(s)=s^{\gamma }\approx {\left({\displaystyle \frac{d{\omega }_{\mathrm{h}}}{b}}\right)}^{\gamma }\left({\displaystyle \frac{d{s}^2+b{\omega }_{\mathrm{h}}s}{d(1-\gamma )s^2+b{\omega }_{\mathrm{h}}s+d\gamma }}\right)K{\displaystyle \prod _{i=1}^N\frac{s+{{\omega }^{\prime }}_i}{s+{\omega }_i}}$$

(20)

where ${{\omega }^{\prime }}_i={\omega }_b{\omega }_u^{(2i-1-\gamma )/N}$，${\omega }_i={\omega }_b{\omega }_u^{(2i-1+\gamma )/N}$，$K={\omega }_h^{\gamma }$，${\omega }_u=\sqrt{{\omega }_h/{\omega }_b}$，and 0<γ<1. Generally, d=9,b=10 are selected.

The lower fitting frequency${\omega }_b$ , the upper fitting frequency ${\omega }_h$, and the order of the energy storage element 2N+1 are three important parameters in the improved Oustaloup approximation method. In simulation, ${\omega }_b\times {\omega }_h=1$, they are selected as follows: $f_s=10Hz$,${\omega }_b=5\times {10}^{-6}rad/s$, ${\omega }_h=2\times {10}^{5}rad/s$,$N=8$. The topology of the fractional-order energy storage elements in the fractional-order interleaved parallel Buck–Boost converter is shown in Figure 6.

The values of the integer-order resistors $R_i$, inductors $L_i$, and capacitors$C_i$ obtained by the improved Oustaloup approximation method in Figure 6 are listed in Table 3, In the table $\mu F/{(S)}^{1-\beta }$ denotes the unit of the fractional-order capacitor.

The mathematical model and state-space model of the fractional-order interleaved par allel Buck–Boost converter built in Simulink are shown in Figure 7 and Figure 8 respectively. In Figure 7 and Figure 8, the “Fractional” module is constructed using the fractional-order Caputo integra.

The simulated waveforms of the mathematical model and circuit model of the fractional-order converter are shown in Figure 9. In Figure 9, the blue dashed line and the orange dashed line represent the simulated waveforms of the fractional-order mathematical model and circuit model of the converter, respectively.

The DC values and ripple values of the inductor current and output voltage are listed in Table 4, such as$I_L$,$V_C$,$\Delta i_L$,$\Delta v$can be measured from Figure 9.

According to Figure 9 and Table 4, the inductor current and output voltage waveforms obtained from the two fractional-order converter models are similar. The value of $i_{L1}$ obtained from the circuit model simulation is 0.17 A higher than that obtained from the mathematical model, while the output voltage $V_o$obtained from the circuit model is 0.19 V lower than that obtained from the mathematical model.The numerical errors in the inductor current and output voltage obtained from the two fractional-order models are caused by the fact that the energy storage elements are constructed using integer-order components based on the improved Oustaloup approximation method. In addition, the switching devices and diodes in the circuit model are non-ideal, which also introduces errors. Meanwhile, the differences in the inductor current ripple and output voltage ripple are 0.09 A and 0.16 V, respectively. These differences are relatively small and within the allowable error range, which verifies the correctness of applying fractional calculus theory to the study of the converter.

4.2. Simulation Results Under Different Fractional-Order Definitions

The order of the energy storage elements is set to 0.8, and the initial value of the fractional-order Caputo definition is set to 0. The simulated waveforms of the converter under integer-order calculus, fractional-order Riemann–Liouville (R–L) definition, and Caputo definition are shown in Figure 10. In the figure, the fractional-order R–L definition is represented by the red solid line, the fractional-order Caputo definition is represented by the blue dotted line, and the fractional-order state-space averaging model is represented by the green dashed line.

According to Figure 10, the simulated waveforms of the output voltage $v_O$ and inductor current $i_{L1}$obtained under the fractional-order Caputo definition and the Riemann–Liouville (R–L) definition are basically the same. The output results of the converter under the two definitions show only slight differences. Based on the analysis in this paper, when the Caputo definition is adopted for the converter, the formulas for the output voltage ripple and inductor current ripple are much simpler than those derived using the R–L definition.

4.3. Influence of the Order of Fractional-Order Energy Storage Elements on the Converter

Let the orders of the fractional-order inductors $L_1$ and $L_2$ be ${\alpha }_1$and ${\alpha }_2$, respectively, and let the order of the fractional-order capacitor $C$ be $\beta$. For the fractional-order interleaved parallel Buck–Boost converter, when ${\alpha }_1={\alpha }_2=\beta =0.8$, ${\alpha }_1={\alpha }_2=\beta =0.85$, and ${\alpha }_1={\alpha }_2=\beta =0.9$, the waveforms of the output voltage and inductor current obtained through simulation of the fractional-order mathematical model are shown in Figure 11.

It can be seen from Figure 11 that as the order of the energy storage elements increases, the overshoot of the converter output voltage and inductor current increases, and the settling time also becomes longer. The ripple magnitudes of the converter output voltage and inductor current are negatively correlated with the order of the energy storage elements. Therefore, if the order of the energy storage elements continues to decrease, the valley value of the total converter current will become less than zero, and the operating mode of the converter will change from CCM to DCM.

Therefore, when designing the controller of a fractional-order converter, selecting an appropriate order of the energy storage elements can not only reduce the overshoot of the dynamic response but also shorten the settling time.

5. Experiment

To verify the accuracy of the theoretical analysis, a prototype was built for experimental validation, as shown in Figure 12. In the figure, the fractional-order energy storage elements are constructed using integer-order resistors, inductors, and capacitors based on the improved Oustaloup approximation method and the fractal chain method.

The waveforms of the output voltage and inductor current are shown in Figure 13. Since the fractional-order energy storage elements are obtained by approximate fitting using integer-order elements, certain errors exist in the components. As a result, the filtering performance of the capacitor deteriorates, which further leads to some irregular ripple in the converter output voltage.

For ease of analysis, the measured results of the experimental waveforms in Figure 13 are compared with the simulated results of the converter under the fractional-order Caputo and R–L definitions in the preceding section, as shown in Table 5. Here, ,${\epsilon }^{R-L}$ and ${\epsilon }^{caputo}$ represent the deviations of the experimental results from the simulation results obtained under the R–L and Caputo definitions, respectively.

As shown in Table 5, the output voltage $v_o$ and inductor current $i_{L1}$obtained from simulations based on different fractional-order definitions are almost identical. The experimental inductor current $i_{L1}$ is larger than the simulated value, whereas the experimental output voltage $v_o$ is lower than the simulated value. The discrepancies between the experimental and simulation results are mainly attributed to the non-ideal characteristics of the converter components, electromagnetic interference, and the approximate fitting of the fractional-order elements.

6. Conclusions

Simulation and experimental results show that the inductor current and output voltage waveforms obtained from the fractional-order mathematical model and the fractional-order circuit model are basically consistent, with relatively small ripple errors, which verifies the accuracy of the fractional-order modeling method. The converter output results under the Caputo definition and the R–L definition show only slight differences; however, the ripple formulas derived under the Caputo definition are simpler and more convenient for engineering analysis. The order of the energy storage elements has a significant influence on the dynamic response and ripple characteristics of the converter. A proper selection of the order of the energy storage elements can reduce the overshoot of the dynamic response and shorten the settling time, thereby providing a theoretical basis for the controller design of the fractional-order interleaved parallel Buck–Boost converter.