Fractional-Order Modeling and Nonlinear Dynamic Analysis of Forward Converter

Abstract

To accurately investigate the nonlinear dynamic characteristics of a forward converter, a fractional-order state-space averaged model of a forward converter in continuous conduction mode (CCM) is established based on the fractional calculus theory. And nonlinear dynamical bifurcation maps which use PI controller parameters and a reference current as bifurcation parameters are obtained. The nonlinear dynamic behavior is analyzed and compared with that of an integral-order forward converter. The results show that under certain operating conditions, the fractional-order forward converter exhibits bifurcations characterized by low-frequency oscillations and period-doubling as certain circuit and control parameters change. Under the same circuit conditions, there is a difference in the stable parameter region between the fractional and integral-order models of the forward converter. The stable zone of the fractional-order forward converter is larger than that of the integral-order one. Therefore, the circuit struggles to enter states of bifurcation and chaos. The stability domain for low-frequency oscillations and period-doubling bifurcations can be accurately predicted by using a small signal model and a predictive correction model of the fractional-order forward converter, respectively. Finally, by performing circuit simulations and hardware-in-the-loop experiments, the rationality and correctness of the theoretical analysis are verified.

1. Introduction

Fractional calculus refers to a field where the order of differentiation and integration can be any arbitrary or fractional number. Employing the theory of fractional calculus can greatly expand our ability to describe systems [1]. Given the fractional-order characteristics of dielectric materials, actual inductors and capacitors are fractional-order components [2,3,4,5]. Therefore, a forward converter that contains fractional-order inductors and capacitors is considered as a fractional-order system. Approximating actual systems as integer-order ones is only viable for systems where there is not a high requirement for accuracy [6]. With the rapid development of power electronic converters, there is a growing demand for high levels of stability, reliability, and accuracy [7,8]. A forward converter is a highly nonlinear time-variant system that presents rich nonlinear dynamics under certain working conditions, directly affecting the converter’s stability and reliability [9,10,11]. These nonlinear dynamic phenomena are mainly attributed to the nonlinearities introduced by switching devices and feedback loops [12]. In power converter design, particular attention must be paid to nonlinear dynamics to ensure system stability and performance. Typically, electronic engineers hope the converter operates in a stable state [13]. Therefore, predicting the system’s bifurcation boundaries to prevent such nonlinear dynamic behavior is a research priority. The dynamical analysis of fractional-order nonlinear systems is a hot research topic [14]. Many existing studies indicate that the models established using fractional calculus closely resemble actual situations [15] and can more accurately describe the physical phenomena of power electronic converters.

In recent years, research has demonstrated that fractional-order models of inductors and capacitors accurately represent their real electrical characteristics. Inductors and capacitors are essential components in power converters, making the modeling and nonlinear dynamic analysis of fractional-order converters practically significant. Studies on the fractional-order modeling of DC-DC converters have been reported in [16,17], mainly focusing on non-isolated converters. For example, the fractional-order modeling of boost converters was examined in [18], and the effect of fractional-order on harmonics was analyzed. A fractional-order model of the Boost converter was established in [19], and the effect of the fractional-order on harmonics was analyzed. Circuit and simulation models for fractional-order Buck–Boost converters were also developed in [20], revealing the influence of fractional-order on harmonic characteristics. These findings suggest that the fractional order can serve as an additional system parameter. However, research on fractional-order models of complex power converters, particularly those with transformers, remains limited. The inclusion of electromagnetic issues complicates their modeling process.

Power electronic converters exhibit rich and complex nonlinear dynamic behaviors, including various bifurcation phenomena and chaotic motions. When bifurcation and chaotic motion occur in a converter, they cause substantial noise and instability, which are detrimental to practical applications. Therefore, it is crucial to avoid bifurcation phenomena and chaotic motion as much as possible. However, designing the system parameters too far from the bifurcation boundary can degrade the system’s dynamic response and other performance metrics. Analyzing the nonlinear phenomena in converters is a significant research focus. The methods for studying bifurcation phenomena in DC-DC converters have been summarized in [21]. A novel discrete mapping method for analyzing nonlinear phenomena in DC-DC converters was proposed in [22]. Permutation entropy was used to characterize the nonlinear phenomena in nonlinear DC-DC converter systems in [23]. The dynamic behavior of a single-stage Boost converter system with memristive load peak current mode control was studied in [24]. Most of these studies focus on integral-order systems, while the nonlinear phenomena and analysis methods for fractional-order systems are considerably more complex. The dynamical behavior of Boost converter systems with memristor loads was analyzed in [25]. A fractional-order model for magnetically coupled Boost converters in CCM mode was established, and the nonlinear dynamic phenomena of this model were explored in [26]. A fractional-order Duffing system was established, and bifurcation diagrams, phase diagrams, Poincaré maps, and time domain waveforms were used to study the nonlinear dynamic behavior of the fractional-order Duffing system in [27]. These studies indicate that fractional-order systems exhibit richer dynamic characteristics, making it challenging to accurately describe the actual operating state of power electronic converters using integral-order models. Thus, studying fractional-order systems is crucial for establishing precise system models.

Much of the current research on fractional-order converters is mostly focused on certain singular nonlinear dynamic behaviors, lacking the systematic analysis of the nonlinear phenomena generated by converters. In fact, different control methods can lead to various nonlinear dynamic behaviors in power electronic converters, all of which can affect the system’s output variables. The analysis of the nonlinear dynamic behavior of fractional-order isolated DC-DC converters is even more complex. The fractional-order modeling of a flyback converter has been proposed in [28], but it did not delve deeply into the nonlinear dynamic characteristics of this model. Similar to the flyback converter, forward converters are also traditional isolated DC-DC converters. The fractional-order modeling and study of its nonlinear dynamic behavior also hold significant research value.

In this paper, a state-space averaged model of the fractional-order forward converter in continuous conduction mode (CCM) is established, and the profound study and analysis of its nonlinear dynamic behavior are conducted. The structure of this paper is as follows: Section 2 establishes the state-space averaged model of the fractional-order forward converter and derives the direct current component, the alternating current component, and the transfer function for this model in CCM. Section 3 conducts simulations of the fractional-order forward converter’s circuit model and compares the results with those of an integral-order forward converter. Section 4 focuses on discussing the low-frequency oscillation phenomena and period-doubling bifurcations of the forward converter, deeply analyzing the nonlinear dynamic behavior of the model, and accurately predicting the stable parameter domain of fractional-order forward converter using small signal model and predictor-corrector methods. In Section 5, the rationality and correctness of theoretical analysis are further validated through hardware-in-the-loop experiments. Finally, conclusions are drawn in Section 6.

2. Fractional-Order Modeling of Forward Converter

Fractional calculus can be viewed as the generalization of traditional calculus that extends ordinary integrations and differentiations to non-integral orders, thus demonstrating its wider applicability [29].

Fractional calculus can be defined through three main approaches: Riemann–Liouville (RL), Grünwald–Letnikov (GL), and Caputo [30]. The Caputo definition is particularly adept at addressing initial value problems in fractional differential equations. It is expressed as

$$\frac{{\mathrm{d}}^{\mu }x\left(t\right)}{{\mathrm{d}t}^{\mu }}={}_{a}D_t^{\mu }x\left(t\right)=\frac{1}{\Gamma \left(m-\mu \right)}{{\displaystyle \int }}_a^t{\left(t-\mathsf{\zeta }\right)}^{m-\mu -1}{x}^{\left(m\right)}\left(\mathsf{\zeta }\right)\mathrm{d}\mathsf{\zeta }$$

(1)

In this formula, m is an integer such that m − 1 < µ < m, where Γ(•) represents the gamma function, and α denotes the lower bound of the definition.

Using the Laplace transform, the Caputo fractional derivative is defined as

$$\mathcal{L}\left[{}_{0}D_t^{\mu }x\left(t\right)\right]={{\displaystyle \int }}_0^{\infty }{}_{0}D_t^{\mu }x\left(t\right){\times e}^{-st}{\mathrm{d}t=s}^{\mu }X\left(s\right)-{\displaystyle \sum }_{k=0}^{m-1}{s}^k\left[{}_{0}D_t^{\mu -k-1}{x}^k\left(0\right)\right]$$

(2)

In this paper, the Caputo derivative is employed for its acceptable error margin in all calculations [31]. It is essential for accurate results of the initial conditions that are explicitly known before the commencement of calculations.

With zero initial conditions, the Laplace transform of the Caputo fractional derivative is represented as

$$\mathcal{L}\left[{}_{0}D_t^{\mu }x\left(t\right)\right]={{\displaystyle \int }}_0^{\infty }{}_{0}D_t^{\mu }x\left(t\right){\times e}^{-st}{\mathrm{d}t=s}^{\mu }X\left(s\right)$$

(3)

According to Westerlund’s 1994 linear capacitor model [3], the relationship between the current i_C and the voltage v_C in a real capacitor is given by

$$i_{\mathrm{C}}=C\frac{{\mathrm{d}}^{\alpha }{v}_{\mathrm{C}}}{{\mathrm{d}t}^{\alpha }} , 0 < \alpha < 1$$

(4)

where C represents the capacitance, and α denotes the fractional order related to the dielectric type.

Westerlund also asserted that a real inductor inherently possesses fractional properties [4]. Based on existing research, the voltage across a real inductor is

$$v_{\mathrm{L}}=L\frac{{\mathrm{d}}^{\beta }{i}_{\mathrm{L}}}{{\mathrm{d}t}^{\beta }} , 0 < \beta < 1$$

(5)

in which, L is the inductance, and β is the fractional order associated with electromagnetism.

In continuous conduction mode (CCM), the forward converter has two switching states, as shown in Figure 1. The transformer performs voltage transformation, where n denotes the transformer ratio, n = N₂/N₁. A flux reset circuit is formed by inductor L₀ and diode D₀. S is an active switch, D₁ and D₂ are diodes, L is the filter inductor, and C is the filter capacitor, V_in is the input voltage, α is the order of L, and β is the order of C. In Figure 2a, S is closed, V_in provides power to the load resistor, the inductor is in a charging state. In Figure 2b, S is in an open state, the inductor releases energy and transfers it to the load resistor, and the inductor and capacitor maintain the power supply state.

State 1: When S is on (0 ≤ t ≤ dT), the diode is reverse-biased, and the inductor current flows through S, increasing i_L. The state-space equations for State 1 using i_L and v_o as state variables are

$$\left[\begin{array}{c}\frac{{\mathrm{d}}^{\alpha }{i}_{\mathrm{L}}}{{\mathrm{d}t}^{\alpha }}\\ \frac{{\mathrm{d}}^{\beta }{v}_{\mathrm{o}}}{{\mathrm{d}t}^{\beta }}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{L}}\\ v_{\mathrm{o}}\end{array}\right]+\left[\begin{array}{c}\frac{n}{L}\\ 0\end{array}\right]v_{\mathrm{in}}$$

(6)

$${\mathrm{A}}_1=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]{ , \mathrm{b}}_1=\left[\begin{array}{c}\frac{n}{L}\\ 0\end{array}\right]$$

(7)

State 2: When S is off (dT ≤ t < T), D becomes forward-biased, and the inductor current decreases, while flowing through D. The state-space equations for State 2 using i_L and v_o as state variables are

$$\left[\begin{array}{c}\frac{{\mathrm{d}}^{\alpha }{i}_{\mathrm{L}}}{{\mathrm{d}t}^{\alpha }}\\ \frac{{\mathrm{d}}^{\beta }{v}_{\mathrm{o}}}{{\mathrm{d}t}^{\beta }}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{L}}\\ v_{\mathrm{o}}\end{array}\right]+\left[\begin{array}{c}0\\ 0\end{array}\right]v_{\mathrm{in}}$$

(8)

$${\mathrm{A}}_2=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]{ , \mathrm{b}}_2=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

(9)

According to [32], the coefficient matrices and vectors for the complete state equations are

$${A=d\mathrm{A}}_1+\left(1-d\right){\mathrm{A}}_2=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]$$

(10)

$${\mathrm{b}=d\mathrm{b}}_1+\left(1-d\right){\mathrm{b}}_2=\left[\begin{array}{c}\frac{nd}{L}\\ 0\end{array}\right]$$

(11)

In power converters, the current i_L and voltage v_o exhibit high-frequency switching harmonics. To mitigate these harmonics, averaging the waveforms over a switching period is necessary [31]. This can be achieved by defining the sliding average value as

$$⟨y⟩=\frac{1}{T}{{\displaystyle \int }}_t^{t+T}y\left(\tau \right)\mathrm{d}\tau$$

(12)

in which, y represents the circuit variable of the converter.

The principles of fractional calculus lead to the derivation of the following equation.

$$\frac{{\mathrm{d}}^{\gamma }⟨y⟩}{{\mathrm{d}t}^{\gamma }}=\frac{{\mathrm{d}}^{\gamma }\frac{1}{T}{\int }_t^{t+T}y\left(\tau \right)\mathrm{d}\tau }{{\mathrm{d}t}^{\gamma }}=\frac{1}{T}{{\displaystyle \int }}_t^{t+T}\left(\frac{{\mathrm{d}}^{\gamma }y\left(\tau \right)}{{\mathrm{d}t}^{\gamma }}\right)\mathrm{d}\tau =⟨\frac{{\mathrm{d}}^{\gamma }y}{{\mathrm{d}t}^{\gamma }}⟩$$

(13)

where γ is the order, and it satisfies the condition 0 < γ < 1.

In state-space modeling, the input voltage v_in and the duty ratio d are also considered. $⟨i_{\mathrm{L}}⟩$, $⟨v_{\mathrm{o}}⟩$, $⟨v_{\mathrm{in}}⟩$, and $⟨d⟩$, are the mean values of i_L, v_o, v_in, and d, respectively, and are depicted as

$$⟨i_{\mathrm{L}}⟩=I_{\mathrm{L}}+{\hat{i}}_{\mathrm{L}} , ⟨v_{\mathrm{o}}⟩{=V}_{\mathrm{o}}+{\hat{v}}_{\mathrm{o}} , ⟨v_{\mathrm{in}}⟩{=V}_{\mathrm{in}}+{\hat{v}}_{\mathrm{in}} , ⟨d⟩=D+\hat{d}$$

(14)

in the equations; the variables with uppercase letters indicate the DC components in $⟨i_{\mathrm{L}}⟩$, $⟨v_{\mathrm{o}}⟩$, $⟨v_{\mathrm{in}}⟩$, and $⟨d⟩$, while the AC components are represented by ${\hat{i}}_{\mathrm{L}}$, ${\hat{v}}_{\mathrm{o}}$, ${\hat{v}}_{\mathrm{in}}$, and $\hat{d}$ represent the corresponding AC components. Assuming small signals, the AC components are significantly smaller in magnitude compared to the DC parts.

The state-space averaged model of the fractional-order forward converter can be derived as

$$\left[\begin{array}{c}\frac{{\mathrm{d}}^{\alpha }⟨i_{\mathrm{L}}⟩}{{\mathrm{d}t}^{\alpha }}\\ \frac{{\mathrm{d}}^{\beta }⟨v_{\mathrm{o}}⟩}{{\mathrm{d}t}^{\beta }}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]\left[\begin{array}{c}⟨i_{\mathrm{L}}⟩\\ ⟨v_{\mathrm{o}}⟩\end{array}\right]+\left[\begin{array}{c}\frac{nd}{L}\\ 0\end{array}\right]⟨v_{\mathrm{in}}⟩$$

$$=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]\left[\begin{array}{c}{I}_{\mathrm{L}}+{\hat{i}}_{\mathrm{L}}\\ V_{\mathrm{o}}+{\hat{v}}_{\mathrm{o}}\end{array}\right]+\left[\begin{array}{c}D+\hat{d}\\ 0\end{array}\right]\left(V_{\mathrm{in}},+,{\hat{v}}_{\mathrm{in}}\right)$$

(15)

The quiescent operation point is formulated into

$$\left[\begin{array}{c}{I}_{\mathrm{L}}\\ V_{\mathrm{o}}\end{array}\right]=\left[\begin{array}{c}\frac{{\mathit{nDV}}_{\mathrm{in}}}{R}\\ {\mathit{nDV}}_{\mathrm{in}}\end{array}\right]$$

(16)

According to (6), the increment of i_L during (0, dT) can be calculated as Δi_L.

$${\mathsf{\Delta }i}_{\mathrm{L}}=\frac{\left({\mathit{nV}}_{\mathrm{in}}-V_{\mathrm{o}}\right){\left(DT\right)}^{\alpha }}{\alpha L\Gamma \left(\alpha \right)}$$

(17)

where Γ(•) denotes the Gamma function.

According to (17), the current ripple Δi_L is dependent on variables such as the input voltage V_in, the duty cycle D, the switching period T, and the inductor order α. Incorporating (16) and (17), the peak current i_Lmax and minimum current i_Lmin can be obtained as

$$i_{\mathrm{Lmax}}{=I}_{\mathrm{L}}+\frac{1}{2}\Delta i_{\mathrm{L}}=\frac{{\mathit{nDV}}_{\mathrm{in}}}{R}+\frac{\left({\mathit{nV}}_{\mathrm{in}}-V_{\mathrm{o}}\right){\left(DT\right)}^{\alpha }}{2\alpha L\Gamma \left(\alpha \right)}$$

(18)

$$i_{\mathrm{Lmin}}{=I}_{\mathrm{L}}-\frac{1}{2}\Delta i_{\mathrm{L}}=\frac{{\mathit{nDV}}_{\mathrm{in}}}{R}-\frac{\left({\mathit{nV}}_{\mathrm{in}}-V_{\mathrm{o}}\right){\left(DT\right)}^{\alpha }}{2\alpha L\Gamma \left(\alpha \right)}$$

(19)

Likewise, based on (16), the reduction in capacitor voltage within the timeframe of (0, dT) can be computed. Δv_o can be described as

$$\Delta v_{\mathrm{o}}=\left[1-E_{\beta }\left(-\frac{D^{\beta }{T}^{\beta }}{RC}\right)\right]V_{\mathrm{OT}}$$

(20)

where V_OT is the initial voltage value when the switch S is turned on, and E_β(•) is a Mittag-Leffler function. V_OT can be approximated as

$$V_{\mathrm{OT}}=\frac{1}{2}{\mathsf{\Delta }v}_{\mathrm{o}}{+V}_{\mathrm{o}}$$

(21)

The expression for voltage ripple can be derived from (14), (20), and (21) as

$${\mathsf{\Delta }v}_{\mathrm{o}}=\frac{{2\mathit{nDV}}_{\mathrm{in}}}{R}\frac{1-E_{\beta }\left(-\frac{D^{\beta }{T}^{\beta }}{RC}\right)}{{1+E}_{\beta }\left(-\frac{D^{\beta }{T}^{\beta }}{RC}\right)}$$

(22)

In order to ensure the stable operation of the forward converter in CCM mode, the value of the inductor current i_L needs to be greater than or equal to 0 at the moment of switch cycle initiation or termination, namely

$$I_{\mathrm{L}}-\frac{1}{2}{\mathsf{\Delta }i}_{\mathrm{L}} \ge 0$$

(23)

When substituting (16) and (17) into (23), the result is

$$\frac{{\mathit{nDV}}_{\mathrm{in}}}{R}\ge \frac{\left({\mathit{nV}}_{\mathrm{in}}-V_{\mathrm{o}}\right){\left(DT\right)}^{\alpha }}{2\alpha L\Gamma \left(\alpha \right)}$$

(24)

As can be derived from (18), the conditions for the stable operation of the forward converter in CCM mode are related to the load resistance R, the duty cycle D, the period T, excitation inductance L, and the order α. The larger α is, the easier it is for the forward converter to operate in CCM mode. When α equals one, it corresponds to the same conditions for continuous operation in integral-order modeling as stated.

After isolating the AC component in (13) and ignoring the higher-order signals, the state-space equations for the fractional-order AC small signal model are obtained

$$\left[\begin{array}{c}\frac{{\mathrm{d}}^{\alpha }{\hat{i}}_{\mathrm{L}}}{{\mathrm{d}t}^{\alpha }}\\ \frac{{\mathrm{d}}^{\beta }{\hat{v}}_{\mathrm{o}}}{{\mathrm{d}t}^{\beta }}\end{array}\right]=\left[\begin{array}{cc}0& -\frac{1}{L}\\ \frac{1}{C}& -\frac{1}{RC}\end{array}\right]\left[\begin{array}{c}{\hat{i}}_{\mathrm{L}}\\ {\hat{v}}_{\mathrm{o}}\end{array}\right]+\left[\begin{array}{c}\frac{D}{L}\\ 0\end{array}\right]{\hat{v}}_{\mathrm{in}}+\left[\begin{array}{c}\frac{\hat{d}}{L}\\ 0\end{array}\right]V_{\mathrm{in}}$$

(25)

Through the application of the Laplace transform based on fractional calculus, the following is derived.

$$s^{\alpha }{\hat{i}}_{\mathrm{L}}\left(s\right)=-\frac{1}{L}{\hat{v}}_{\mathrm{o}}\left(s\right)+\frac{\mathit{nD}}{L}{\hat{v}}_{\mathrm{in}}\left(s\right)+\frac{n{V}_{\mathrm{in}}}{L}\hat{d}\left(s\right)$$

(26)

$$s^{\beta }{v}_{\mathrm{o}}\left(s\right)=\frac{1}{C}{\hat{i}}_{\mathrm{L}}\left(s\right)-\frac{1}{RC}{\hat{v}}_{\mathrm{o}}\left(s\right)$$

(27)

According to (26) and (27), the transfer function from ${\hat{v}}_{\mathrm{in}}$ to ${\hat{v}}_{\mathrm{o}}$ can be expressed as

$$G_{v_{\mathrm{o}}{v}_{\mathrm{in}}}\left(s\right)={\left.\frac{{\hat{v}}_{\mathrm{o}}\left(s\right)}{{\hat{v}}_{\mathrm{in}}\left(s\right)}\right|}_{\hat{d}\left(s\right)=0}=\frac{\frac{\mathit{nD}}{\mathit{LC}}}{s^{\alpha +\beta }+\frac{1}{RC}{s}^{\alpha }+\frac{1}{\mathit{LC}}}$$

(28)

The transfer function from $\hat{d}$ to ${\hat{v}}_{\mathrm{o}}$ can be expressed as

$$G_{v_{\mathrm{o}}d}\left(s\right)={\left.\frac{{\hat{v}}_{\mathrm{o}}\left(s\right)}{\hat{d}\left(s\right)}\right|}_{{\hat{v}}_{\mathrm{in}}\left(s\right)=0}=\frac{\frac{{\mathit{nV}}_{\mathrm{in}}}{\mathit{LC}}}{s^{\alpha +\beta }+\frac{1}{RC}{s}^{\alpha }+\frac{1}{\mathit{LC}}}$$

(29)

The transfer function from ${\hat{v}}_{\mathrm{in}}$ to ${\hat{i}}_{\mathrm{L}}$ can be denoted as

$$G_{i_{\mathrm{L}}{v}_{\mathrm{in}}}\left(s\right)={\left.\frac{{\hat{i}}_{\mathrm{L}}\left(s\right)}{{\hat{v}}_{\mathrm{in}}\left(s\right)}\right|}_{\hat{d}\left(s\right)=0}=\frac{\frac{\mathit{nD}}{L}{s}^{\beta }+\frac{\mathit{nD}}{\mathit{LCR}}}{s^{\alpha +\beta }+\frac{1}{RC}{s}^{\alpha }+\frac{1}{\mathit{LC}}}$$

(30)

The transfer function from $\hat{d}$ to ${\hat{i}}_{\mathrm{L}}$ can be indicated as

$$G_{i_{\mathrm{L}}d}\left(s\right)={\left.\frac{{\hat{i}}_{\mathrm{L}}\left(s\right)}{\hat{d}\left(s\right)}\right|}_{{\hat{v}}_{\mathrm{in}}\left(s\right)=0}=\frac{\frac{{\mathit{nV}}_{\mathrm{in}}}{L}{s}^{\beta }+\frac{{\mathit{nV}}_{\mathrm{in}}}{\mathit{LCR}}}{s^{\alpha +\beta }+\frac{1}{RC}{s}^{\alpha }+\frac{1}{\mathit{LC}}}$$

(31)

The relationship between the transfer functions in (28)–(31) and the fractional orders of the inductor and capacitor can be observed. By assigning a value of one to all the orders, the model transitions to integral-order transfer functions.

3. Simulation Results

To validate the fractional-order models mentioned above, this section simulates the fractional-order model of the forward converter and compares the simulation results with those of the integral-order model. The circuit parameters of the forward converter are as follows: V_in = 5 V, n = 1, D = 0.7, L = 0.125 mH, C = 200 μF, R = 5 Ω, f = 10 kHz, α = 0.95, and β = 0.95. In order to further validate the research results, this paper establishes an approximate circuit for fractional-order inductors and fractional order capacitors using the Oustaloup algorithm and employing resistor/capacitor or resistor/inductor networks. Fractional-order elements are not commercially available, but researchers have suggested constructing equivalent circuits using the fractional properties of elements [33,34].

The impedance chain of the fractional-order inductance is shown in Figure 3. Parallel impedance can be obtained from the parallel relationship between resistance and inductance, and the model structure of fractional-order inductance impedance chain can be obtained using (32).

$${ \mathit{Ls}}^{\alpha }={\displaystyle \sum }_{i=1}^n\frac{R_{\mathit{Li}}s}{s+\frac{R_{\mathit{Li}}}{L_{\mathit{Li}}}}{+R}_{\mathit{Lin}}$$

(32)

The values of R_Li and L_Li for the inductor in the order α = 0.95 are shown in

Table 1. The values of R_Li, L_Li, R_Ci, and C_Ci for fractional-order components with α = 0.95 and β = 0.95.

α/β	0.95		0.95
i	RLi (Ω)	LLi (mH)	RCi (Ω)	CCi (mF)
in	0.00000002	/	0.000000000399	/
1	61.875	0.06588	0.000000000504	0.952
2	0.7125	0.00981	0.00000000614	1.021
3	0.05875	0.01038	0.0000000704	1.176
4	0.00513	0.01175	0.000000804	1.296
5	0.00045	0.01338	0.00000902	1.492
6	0.00004	0.01513	0.000104	1.694
7	0.0000035	0.01725	0.00116	1.924
8	0.0000003	0.01951	0.0142	2.041
9	0.000000025	0.02063	1.24	0.304

.

Similarly, for capacitance possessing fractional-order characteristics, the structure of their impedance chain is depicted in Figure 4.

$$\frac{1}{C}{s}^{-\beta }={\displaystyle \sum }_{i=1}^n\frac{\frac{1}{C_{\mathit{Ci}}}}{s+\frac{1}{R_{\mathit{Ci}}{C}_{\mathit{Ci}}}}{+R}_{\mathit{Cin}}$$

(33)

The component parameters in the fractional-order capacitor impedance chain are also provided in Table 1.

Figure 5 presents the Bode plots for constructed fractional-order inductance and capacitance. As observed, the Oustaloup method of fitting the impedance chain is ideal, except at both ends of the fitting band. This is an inherent characteristic of Oustaloup approximation. Using too low an order for fitting might not offer the necessary accuracy, while too high an order can increase the scale of impedance network. Ninth-order fitting provides a good balance between accuracy and complexity, effectively capturing the characteristics of fractional-order elements. This paper uses ninth-order fitting for s^0.95 and s^−0.95, setting the switch frequency of the forward converter to 10 kHz. By appropriately configuring the fitting interval, it demonstrates favorable frequency characteristics at the switching frequency. From the diagram, within the defined frequency range, the magnitude lines fit well, and the phase diagram fluctuates within a certain band around the theoretical curve.

By replacing inductance L and capacitance C in Figure 1 with the approximate circuits for fractional-order inductance and capacitance shown in Figure 3 and Figure 4, we can observe from the circuit simulation results in Figure 6 that the ripples of i_L and v_o increase significantly as the orders of inductance and capacitance decrease. As it can be observed in Figure 6, the inductor current in the fractional-order model is higher than that in the integral-order model. This phenomenon is triggered by the fractional-order characteristic of the capacitors. When the capacitor’s order is 0.95, from Figure 4 it can be deduced that the angle between its voltage and current is approximately −85°. Consequently, the power factor cosφ is non-zero, with φ being the angle between the voltage and current, which expands the total output power of the circuit, and subsequently leads to an escalation in the input current, inducing an increase in the inductor current. The circuit simulation results are shown in

Table 2. Simulation results of the forward converter.

Parameter	Symbol	(α, β) = (1, 1)	(α, β) = (0.95, 0.95)
Maximum of inductor current	iL_max	1.121 A	4.204 A
Minimum of inductor current	iL_min	0.274 A	2.641 A
Maximum of output voltage	vo_max	3.517 V	3.511 V
Minimum of output voltage	vo_min	3.464 V	3.443 V

.

The bode diagrams of $G_{i_{\mathrm{L}}{v}_{\mathrm{in}}}\left(s\right)$ and $G_{v_{\mathrm{o}}{v}_{\mathrm{in}}}\left(s\right)$ with (α, β) = (1, 1) and (α, β) = (0.95, 0.95), respectively, are shown in Figure 7. The transfer functions $G_{i_{\mathrm{L}}{v}_{\mathrm{in}}}\left(s\right)$ and $G_{v_{\mathrm{o}}{v}_{\mathrm{in}}}\left(s\right)$ derived from the state-space averaged model are provided in (27) and (34), respectively. The majority of the differences seen in the circuit models can be attributed to adjustments made to the fractional-order inductors and capacitors. When compared with the integral-order model of the forward converter, due to the fractional-order properties of the impedance chain, the amplitude-frequency characteristic curve of the fractional-order model shifts to the right, and the phase-frequency characteristic curve exhibits a leading phase angle, which suggests a superior dynamic performance and stability for the fractional-order model of the forward converter.

The dynamic characteristics of the fractional-order model and integral-order model of the forward converter can be observed in Figure 8. It can be found that when the orders of the energy storage components in the circuit transition to fractional orders, the circuit’s response is more immediate, the transient state fluctuations lessen, and it enters the steady-state phase more rapidly. This is due to the introduction of fractional-order chain, which alters the internal parameters of the circuit. As illustrated by the Bode plot in Figure 7, the change in the order of the energy storage elements modifies the system’s transfer function. Fractional-order circuits exhibit a better dynamic performance, which is reflected in the system’s regulation time, response speed, and overshoot. All these factors imply that relative to the integral-order model, the fractional-order model of the forward converter delivers a better dynamic performance and stability. The dynamic performance of the circuit is shown in

Table 3. The setting times, overshoot, peak time, and peak voltage of the output voltage.

(α, β)	Setting Time (ms)	Overshoot (%)	Peak Time (ms)	Peak Voltage (V)
(1, 1)	5.834	80.91	0.4841	6.241
(0.95, 0.95)	1.541	36.31	0.4811	4.702

. It can be observed that there is a significant difference in the overshoot between the two. This is because the peak voltage of the integral-order forward converter far exceeds the steady-state output voltage of the circuit. However, due to the reduction in the order of the energy storage components, the peak voltage of the fractional-order forward converter has significantly decreased, becoming relatively close to the steady-state voltage of the circuit. As a result, the system’s overshoot is significantly improved. The data in Table 3 further illustrate that compared to the integral-order forward converter, the fractional-order forward converter exhibits certain advantages.

4. Nonlinear Dynamic Analysis

As a nonlinear time-varying system, the forward converter encompasses a rich set of nonlinear dynamical behaviors. Low-frequency oscillations usually refer to oscillations within the frequency range of several tens of hertz to several thousand hertz. Such oscillations can lead to fluctuations in the output voltage, affecting the stability and reliability of the power supply system. Period-doubling bifurcation refers to the phenomenon where the oscillation frequency of the system’s motion doubles or halves as the system parameters change, which similarly affects the stability and reliability of power electronic converters. To explore the nonlinear dynamical behavior of the forward converter, MATLAB/Simulink (version: 2018a) circuit simulation is used to compare the nonlinear dynamical phenomena of fractional-order and integral-order models of the forward converter and to analyze their stable parameter domains. The state-space average model and predictive-correction model can accurately predict the stable parameter domain of the fractional-order forward converter.

4.1. Low-Frequency Oscillation Phenomena

4.1.1. Low-Frequency Oscillation Phenomena in Fractional-Order Forward Converter

To analyze the low-frequency oscillation phenomena of the fractional-order forward converter, this study establishes a fractional-order model of the voltage-mode-controlled forward converter. With PI controller parameters as bifurcation parameters, a comparison of low-frequency oscillation phenomena when the forward converter (α, β) = (1, 1) and (α, β) = (0.95, 0.95) is conducted. The phase diagrams of the model in stable and low-frequency oscillation states are plotted with inductor current and output voltage as coordinate axes. The system’s block diagram is illustrated in Figure 9.

Initially, K_i in the PI controller is used as the bifurcation parameter, and the bifurcation diagram of the forward converter for (α, β) = (1, 1) is shown in Figure 10a. When (α, β) = (0.95, 0.95), as illustrated in Figure 10b, similar to the integral-order forward converter, as K_i changes, the system exhibits lower-frequency oscillation. The system’s motion transitions from a fixed point to periodic oscillation. As the orders decrease, the bifurcation point shifts backward, indicating that the stable parameter domain enlarges.

To further describe the low-frequency oscillation phenomena of the fractional-order forward converter, Figure 11 presents the time-domain waveform of the inductor current at the bifurcation point. Figure 11a shows the time-domain waveform of the inductor current i_L for a forward converter with (α, β) = (1, 1) when K_i = 320. Figure 11b shows the time-domain waveform of the inductor current i_L for a forward converter with (α, β) = (0.95, 0.95) when K_i = 885. It can be observed that when the converter operates at the bifurcation point, due to the occurrence of low-frequency oscillations, the inductor current i_L drops to zero during certain switching cycles, putting the system in a critical state of continuous inductor current mode. That is, the system alternates between continuous conduction mode (CCM) and Discontinuous Conduction Mode (DCM). When the converter experiences low-frequency oscillations, the ripple of the inductor current significantly increases, worsening the operating performance of the system.

Figure 12 shows the phase diagram formed by the inductor current and the output voltage. Using K_i as the bifurcation parameter for a forward converter with (α, β) = (1, 1), when K_i < 320, the converter operates in a period-1 stable manner, as illustrated in Figure 12a, where the phase diagram is a closed, smooth loop. When K_i ≥ 320, the system undergoes low-frequency oscillations, forming a limit cycle in the phase diagram, as shown in Figure 12b. Similarly, for a forward converter with (α, β) = (0.95, 0.95), when K_i < 885, the system operates stably, forming a closed, smooth loop in the phase diagram, as depicted in Figure 12c. When K_i ≥ 885, the converter exhibits low-frequency oscillation behavior, and the corresponding phase diagram shown in Figure 12d is a limit cycle. The appearance of limit cycles in the phase diagram of DC-DC converters is indicative of low-frequency oscillations in the circuit. It is observed that when the circuit undergoes low-frequency oscillations, both the inductor current and the output voltage ripple significantly increase. These observations are consistent with the conclusions drawn from the bifurcation diagrams and time domain graphs.

4.1.2. Predicting the Low-Frequency Oscillation Characteristics of the Fractional-Order Forward Converter

The small-signal averaging approach is a modelling strategy extensively adopted in power electronics systems. Despite the fact that the small-signal averaging model ignores the influence of switching characteristics in the converter, it considers the low-frequency oscillation dynamics of the system. Hence, the small-signal averaging model can be effectively used to analyze the low-frequency oscillation phenomena in the system. The small-signal block diagram of the system is shown in Figure 13.

G_v(s) represents

$$G_{\mathrm{v}}\left(s\right)=\frac{{\hat{v}}_{\mathrm{o}}}{\hat{d}}=\frac{\frac{{\mathit{nV}}_{\mathrm{in}}}{\mathit{LC}}}{s^{\alpha +\beta }+\frac{1}{RC}{s}^{\alpha }+\frac{1}{\mathit{LC}}}$$

(34)

The transfer function of the PI voltage control loop is

$$G_{\mathrm{c}}\left(s\right)=\frac{{\hat{v}}_{\mathrm{con}}}{{\hat{v}}_{\mathrm{e}}}{=K}_p{+K}_i\frac{1}{s}$$

(35)

The transfer function of the PWM stage is

$$G_{\mathrm{d}}\left(s\right)=\frac{\hat{d}}{{\hat{v}}_{\mathrm{con}}}=\frac{1}{V_{\mathrm{M}}}$$

(36)

in which, V_M = V_U − V_V.

With the use of (31), (32), and (33), we can determine the open-loop and closed-loop transfer functions of the system, expressed as (34) and (35), respectively.

$$G_{\mathrm{k}}\left(s\right){=G}_{\mathrm{v}}\left(s\right)G_{\mathrm{c}}\left(s\right)G_{\mathrm{d}}\left(s\right)$$

$$=\frac{b_0{s+b}_1}{a_0{s}^{\alpha +\beta +1}{+a}_1{s}^{\alpha +1}{+a}_{2}s}$$

(37)

in which, b₀ = nV_inK_p/LC, b₁ = nV_inK_i/LC, a₀ = V_M, a₁ = V_M/RC, and a₂ = V_M/LC.

Table 4. The gain margin, phase margin, and crossover frequency of the (α, β) = (1, 1) and (α, β) = (0.95, 0.95) forward converter as K_i varies.

(α, β) = (1, 1)					(α, β) = (0.95, 0.95)
Ki	A(fg) (dB)	φ(fc) (°)	fc(Hz)	State	Ki	A(fg) (dB)	φ(fc) (°)	fc (Hz)	State
280	0.98	7.21	6251	stable	845	0.33	4.61	10,009	stable
290	0.68	4.93	6290	stable	855	0.24	3.63	10,080	stable
300	0.38	2.81	6331	stable	865	0.13	2.25	10,130	stable
310	0.11	1.24	6370	stable	875	0.05	0.74	10,195	stable
320	−0.23	−1.25	6408	unstable	885	−0.07	−1.12	10,260	unstable

shows the gain margin, phase margin, and crossover frequency of the fractional-order forward converter as K_i gradually increases. It is observed that as K_i increases, both the gain margin and phase margin of the system progressively decrease until the system loses stability. For the forward converter with (α, β) = (1, 1), the system transitions from stable to unstable as K_i increases from 310 to 320. Similarly, for the forward converter with (α, β) = (0.95, 0.95), the system is on the verge of stable operation as K_i changes from 875 to 885. Notably, these critical points coincide with the bifurcation points obtained from the circuit model. This demonstrates that the small-signal model established based on the state-space averaging method can accurately predict the low-frequency oscillation boundaries of the fractional-order forward converter.

4.2. Period-Doubling Bifurcation

4.2.1. The Period-Doubling Bifurcation in Fractional-Order Forward Converter

In order to analyze the period-doubling bifurcations of the fractional-order forward converter, this study has established a bifurcation diagram for a current-controlled fractional-order forward converter. With the reference current as a bifurcation parameter, the comparison of period-doubling bifurcation when the forward converter (α, β) = (1, 1) and (α, β) = (0.95, 0.95) is conducted. The phase diagrams of the model in a stable, period-doubling bifurcation state and a chaotic state are plotted with the inductor current and output voltage as coordinate axes. The system’s block diagram is illustrated in Figure 14.

The reference current is used as a parameter to describe the bifurcation. As the I_ref varies, the bifurcation diagram for a forward converter with (α, β) = (1, 1) under continuous inductor current mode is shown in Figure 15a. Figure 15b shows the bifurcation diagram for a forward converter with (α, β) = (0.95, 0.95) under continuous inductor current mode. It is observed that as I_ref changes, the forward converter first passes through period-1, and then period-2, followed by period-4, and it eventually reaches the chaotic mode. When the system is in chaos, the inductor current i_L no longer has periodicity and exhibits large fluctuations. Comparing Figure 15a and b, the bifurcation points move backward as the order of the fractional-order converter decreases. The fractional-order forward converter possesses a larger stable parameter domain.

To better observe the period-doubling bifurcation phenomenon of the fractional-order forward converter, Figure 16 illustrates the time-domain waveforms of the inductor current at the bifurcation points for the fractional-order forward converter. Figure 16a shows the inductor current waveform for a forward converter with (α, β) = (1, 1) at I_ref = 5.7 A. At this point, the converter is at the bifurcation point, and the circuit operates in a period-2 state. Figure 16b depicts the time-domain graph of the inductor current at the bifurcation point for a forward converter with (α, β) = (0.95, 0.95). When I_ref = 8.2 A, the circuit undergoes period-doubling bifurcation, transitioning the system’s motion from period T to period 2T. Period-doubling bifurcation is one of the pathways leading to chaotic behavior. As I_ref gradually increases, the fractional-order forward converter transitions to a chaotic state, cascading through the period-doubling pathways.

To further observe the impact of the reference current on the system, phase diagrams are utilized to describe the four states of the fractional-order forward converter as I_ref varies. As shown in Figure 17, with the increase in I_ref, the forward converter with (α, β) = (1, 1) displays four different states. It is observed that when I_ref = 5 A, the phase diagram exhibits a single closed, smooth loop, indicating that the system is in a stable operating condition, namely the period-1 state. When I_ref increases to 6 A, the phase diagram still shows a closed loop, but with indentations, suggesting the occurrence of a period-doubling bifurcation. At I_ref = 7 A, the heart-shaped closed loop in the phase diagram undergoes another period-doubling bifurcation, with the inductor current and output voltage transforming into signals four times the period of the ramp signal, indicating that the system operates in the period-4 state. When I_ref reaches 8 A, the trajectories in the phase diagram become chaotic and disordered, signifying that the system has entered a chaotic state. Figure 8 presents the phase diagrams of the forward converter with (α, β) = (0.95, 0.95), illustrating the changes with varying I_ref. Similar to the integral-order circuit, as I_ref gradually increases, the system transitions from a stable period-1 operation to a period-2 state. Subsequently, the system undergoes another period-doubling bifurcation, moving into the period-4 state. Eventually, the trajectories in the phase diagram become chaotic, indicating the system’s entry into a chaotic state. Compared to the integral-order model, the fractional-order model with the same parameters exhibits different dynamical behaviors. For instance, at I_ref = 7 A, the forward converter with (α, β) = (1, 1) is in the period-4 state, as shown in Figure 17c, whereas the forward converter with (α, β) = (0.95, 0.95) remains in the period-1 state, as depicted in Figure 18a. Similarly, it can be deduced from the phase diagrams that fractional-order forward converters are more resistant to entering a chaotic state.

4.2.2. Prediction of the Period-Doubling Bifurcation Boundary for Fractional-Order Forward Converter

The fractional-order predictor-corrector model is a method used for handling nonlinear systems. It accurately characterizes the dynamic properties of a system and can predict the period-doubling bifurcation boundaries of a fractional-order forward converter. This algorithm is a time-domain approximation method in the fractional-order domain for solving fractional calculus. It can discretize differential equations of any order to compute numerical solutions and conduct stability analysis of these solutions. Equations (6) and (8) describe two operating states of the fractional-order forward converter in continuous conduction mode. This model is applied to discretize the fractional-order forward converter operating in CCM. The discretized form of the fractional-order forward converter is shown in (38) and (39). Equation (40) represents the nonlinear switching function S(t), where S = 1 means the switch is on, and S = 0 means the switch is off. Equation (41) describes the switching conditions of the converter.

$$i_{m+1}{=i}_0+\frac{h^{q_1}}{\Gamma \left(q_1+2\right)}\left(\frac{{\mathit{nV}}_{\mathrm{in}}}{L}S-\frac{u_{m+1}^p}{L}\right)+\frac{h^{q_1}}{\Gamma \left(q_1+2\right)}{\displaystyle \sum }_{i=0}^m{a}_{i,m+1}^{q_1}\left(\frac{{\mathit{nV}}_{\mathrm{in}}}{L}S-\frac{u_i}{L}\right)$$

(38)

$$u_{m+1}{=u}_0+\frac{h^{q_2}}{\Gamma \left(q_2+2\right)}\left(\frac{i_{m+1}^p}{C}-\frac{u_{m+1}^p}{RC}\right)+\frac{h^{q_2}}{\Gamma \left(q_2+2\right)}{\displaystyle \sum }_{i=0}^m{a}_{i,m+1}^{q_2}\left(\frac{i_i}{C}-\frac{u_i}{RC}\right)$$

(39)

$$S\left(t\right)=\left\{\begin{array}{c}1, \mathit{mT}<t<\left(m+d\right)T \\ 0, \left(m+d\right)T<t<\left(m+1\right)T\end{array}\right.$$

(40)

$${\sigma =i}_{\mathrm{L}}\left(t\right)-I_{\mathrm{ref}}=0$$

(41)

In the formula, q₁ and q₂ represent the orders of the inductor and capacitor, respectively. i₀ and u₀ are the initial values of the inductor current and capacitor voltage, respectively. The correction coefficients for the inductor current and capacitor voltage are denoted as a_i,m+1^q1 and a_i,m+1^q2, respectively. m represents the number of iterations. The initial estimate approximations for the inductor current and capacitor voltage are represented as i_m+1^p and u_m+1^p, and their expressions are provided in (42)–(45).

$$a_{i,m+1}^{q_1}=\left\{\begin{array}{c}{m}^{q_1+1}-\left(m-q_1\right){\left(m+1\right)}^{q_1},i=0\\ {\left(m-i+2\right)}^{q_1+1}+{\left(m-i\right)}^{q_1+1}-2{\left(m-i+1\right)}^{q_1+1,1\le i\le m}\\ i,i=m+1\end{array}\right.$$

(42)

$$a_{i,m+1}^{q_2}=\left\{\begin{array}{c}{m}^{q_2+1}-\left(m-q_2\right){\left(m+1\right)}^{q_2},i=0\\ {\left(m-i+2\right)}^{q_2+1}+{\left(m-i\right)}^{q_2+1}-2{\left(m-i+1\right)}^{q_2+1,1\le i\le m}\\ i,i=m+1\end{array}\right.$$

(43)

$$i_{m+1}^p{=i}_0+\frac{1}{\Gamma \left(q_1\right)}{\displaystyle \sum }_{i=0}^m{b}_{i,m+1}^{q_1}\left(\frac{{\mathit{nV}}_{\mathrm{in}}}{L}S-\frac{u_i}{L}\right)$$

(44)

$$u_{m+1}^p{=u}_0+\frac{1}{\Gamma \left(q_2\right)}{\displaystyle \sum }_{i=0}^m{b}_{i,m+1}^{q_2}\left(\frac{i_{\mathrm{L}}}{C}-\frac{u_i}{RC}\right)$$

(45)

The fractional-order predictor-corrector model is used to describe the states of the forward converter under different I_ref values. After m iterations, the waveform of the inductor current can be obtained. Figure 19 shows the inductor current waveforms of the forward converter with (α, β) = (1, 1). As depicted in Figure 19a, when I_ref = 5.4 A, the circuit operates in a stable state. When I_ref increases to 5.7 A, the inductor current undergoes period-doubling bifurcation, as shown in Figure 19b, and the circuit enters the period-2 state. Figure 20 presents the inductor current waveforms for the forward converter with (α, β) = (0.95, 0.95). At I_ref = 7.9 A, the circuit is in the period-1 state, as illustrated in Figure 20a. As shown in Figure 20b, when I_ref reaches 8.2 A, the inductor current exhibits period-doubling bifurcation. Clearly, the bifurcation points obtained using the fractional-order predictor-corrector model are consistent with the results of circuit simulations. This indicates that the model can accurately predict the period-doubling bifurcation boundaries of a fractional-order forward converter.

5. Experimental Results

To further verify the validity of theoretical analysis, a hardware-in-the-loop experiment was conducted. The experimental system consists of an NI PXIe-1082 chassis (ModelingTech, Shanghai, China), a digital signal processor (TMS320F28335, Texas Instruments, Dallas, The USA) and an oscilloscope. The experimental circuit parameters of the forward converter are shown in

Table 5. Circuit parameters of the forward converter.

Variable	Signification	Value
Vin	Input voltage	10 V/5 V
f	Switching frequency	10 kHz
L	Inductance	0.125 mH
C	Capacitance	200 μF
R	Load resistance	3.5 Ω/1 Ω
n	Transformer ratio	1
α	Inductor order	0.95/1
β	Capacitor order	0.95/1

. The experimental results are shown in Figure 21, Figure 22 and Figure 23; it is evident that as the order of the fractional-order forward converter decreases, the ripple of the inductor current and the output voltage ripple greatly increase. This aligns with the circuit simulation results of the forward converter. The order of the fractional-order component has a significant impact on the forward converter. As the order decreases, the ripple suppression capability of the energy storage component weakens.

The parameter K_i of the PI controller is used as the bifurcation parameter. By adjusting K_i, the low-frequency oscillation of the forward converter is observed. The experimental results are shown in Figure 21 and Figure 22. Figure 21 shows the experimental results of the forward converter with (α, β) = (1, 1). When K_i = 300, the circuit is in a stable state. As K_i increases to 340, the circuit begins to oscillate at a low frequency. Figure 22 shows the experimental results of the forward converter with (α, β) = (0.95, 0.95). As K_i increases from 870 to 910, the circuit transitions from a stable state to low-frequency oscillations. It can be observed that when the forward converter experiences low-frequency oscillations, the ripple of the circuit’s inductor current and output voltage significantly increase. Comparing the experimental results with the MATLAB/Simulink simulation results, the same conclusion can be drawn. Therefore, the reasonableness and correctness of the theoretical analysis are validated. When the order of the fractional-order forward converter decreases, the bifurcation point will move backward, and the parameter stability region of the system is greater, which means the circuit cannot easily enter the low-frequency oscillation state.

Utilizing the reference current as a parameter to describe bifurcations. By adjusting I_ref, the period-doubling bifurcation phenomenon of the forward converter can be obtained. The experimental results are shown in Figure 23. Figure 23a,b shows the experimental results of the forward converter with (α, β) = (1, 1). When I_ref = 5.6 A, the circuit is in a stable state. As I_ref increases to 6.1 A, the circuit undergoes period-doubling bifurcation, transitioning from a period-1 state to a period-2 state. Figure 23c,d shows the experimental results of the forward converter with (α, β) = (0.95, 0.95). As I_ref increases from 7.4 to 7.9 A, the circuit transitions from a stable state to period-doubling bifurcation. It can be observed that when the forward converter experiences period-doubling bifurcation, the inductor current undergoes fission, with the period changing from T to 2T. The experimental results closely correspond to the simulation outcomes, thereby affirming the accuracy and validity of theoretical analysis. As the order of the fractional forward converter decreases, the period-doubling bifurcation point moves backward. This implies that, compared to the integral-order forward converter, the fractional-order forward converter cannot easily enter into period-doubling bifurcation.

6. Conclusions

In this article, a fractional-order forward converter model is established based on the fractional calculus theory and state-space averaging method. The orders of the inductor and capacitor affect the performance of the system. As the orders decrease, the system responds faster and has smaller overshoot in transient processes, indicating a more better dynamic performance and stability. The reason for this is revealed in the right-shifted amplitude-frequency characteristic curve and the leading amplitude-frequency characteristic curve. As an integral-order forward converter, the fractional-order forward converter also exhibits nonlinear dynamical phenomena. However, the fractional-order forward converter cannot easily enter the states of bifurcation and chaos, and therefore has a wider range of control parameter values to stabilize the system. The bifurcation diagrams, time-domain simulation results, and phase diagrams all prove the correctness of analysis. The bifurcation boundary is predicted by using the fractional-order system stability determination method. The validity of theoretical analysis is further verified by a hardware-in-the-loop experimental study.