Introduction

In recent years, the findings, of the fractional-order inductive phenomena, in physics, engineering, biology and other fields, have led to a closer relationship between theory and practice [1–4]. Numerous mathematical modeling studies on the passive components in [5, 6], i.e. inductors and capacitors, have shown that, compared to the integer-order modeling, the fractional order modeling methodology can better present the electrical characteristics of the system, making it a widespread research topic [7]. In addition, since inductors and capacitors play important roles in the power converters, it is more accurate to model the power converters with the fractional-order modeling method [8–16]. So far, there are only a few commercially available fractional-order passive components. The fractional-order modeling for passive components are developed from empirical results [17, 18], or equivalent models of the passive components described by the fractional-order definition [19–24].

Fractional-order models for different DC-DC converters in different modes given in [8–13] are using fractional calculus and state space averaging techniques. By utilizing the fractional-order definitions, the mean and ripple values of the steady-state variables can be obtained, but the transient solutions cannot be derived. Because of the inconsistent definitions of the fractional-order derivative, such modeling technique for the fractional-order converters is not always valid. Besides, [12, 13] discussed the modified Oustaloup’s approximation in fractional integral module, which is used to acquire the transient solutions. Without discretizing, this method is a precise engineering simulation for the transient responses analysis, but is not an appropriate solution for the non-linear behaviors analysis due to the frequency domain approximation. [14] offers a fractional means to characterize the non-solid aluminum electrolytic capacitors in DC-DC converters. On the other hand, the Predictor-Corrector Adams-Bashforth-Moulton (PECE-ABM) type numerical method is often used to obtain the solutions of the fractional systems. Due to the “long memory” characteristics of the fractional-order modeling, the derivation of the fractional-order calculus is usually not straightforward. The approach to get the steady-state variables requires processing the whole datapoints between the initial state and the steady state. Therefore, large amounts of computation efforts are involved in the solving process. The time-domain modeling proposed in [15, 16] uses the simplified equivalent small parameter (SESP) method to get the steady state waveforms, instead of using the fractional-order derivative definitions. It is able to solve the steady state variables without circuit simulations or multiple iterations. However, its accuracy still needs improvement and it does not apply to the situation in which the system state variable changes abruptly in one switching cycle. In addition, the transient solutions of the fractional model are not mentioned.

[12], [13] and [25] studied the dynamic behaviors of the fractional-order PWM converters, but their results are obtained via the MATLAB/Simulink simulations using the approximation of the fractional-order components. Furthermore, none of the above-mentioned literatures have investigated the non-linear behaviors via numerical analysis in time-domain. Ref. [26] presented extensive experimental results, which exposes the operating mechanism of the limit cycle behavior in the integer-order boost converters. Nevertheless, its conclusions do not apply to the Grünwald-Letnikov (a.k.a. Riemann-Liouville and Caputo) definition-based systems. These fractional-order systems do not have periodic solutions [27–30], but only have the asymptotically limit cycle behaviors. Considering their impact on the device stress and the system efficiency during the switching periods, such asymptotically limit cycle behaviors require special attention in practical applications.

This paper is focusing on the asymptotically limit cycle behaviors in the fractional-order converters with the most widely used proportional-integral (PI) voltage compensator. Taking the positive Luo converter as an example, a modified time-domain modeling and analyzing methodology for DC-DC converters is proposed. The theoretical foundation of the work presented here is based on Eq (1) from [31], (1) in which, λ is the order, and ω is the angular frequency of above differential operator. Noticeably, λ can be either an integer or a fraction.

Eq (1) indicates that the differential operator only influences the amplitude, but not the angular frequency in the exponential function. Because of this, the equivalent small parameter (ESP) method and the harmonic balance theory can be applied to the fractional differential operation. By extending the conventional ESP method into the fractional domain, a modeling method and the steady state solutions of the fractional-order DC-DC converters can be obtained. Therefore, the final approximate analytical solutions of the PWM converters in the steady state can be represented as the sum of the harmonic contents. Different from basic DC-DC converters, the voltage across the transfer capacitors in the positive Luo converter exhibits huge oscillations when the power switch turns off. The filtering property of the positive Luo converter is not as strong as that in other PWM converters. Since the first three terms, x₀, x₁ and x₂, in [32], Eq (4) cannot ensure the accuracy of the state variables in positive luo converters, a modified algorithm by extending the consideration scope to more equations of the equivalent system is proposed in this paper. Compared with conventional schemes, the proposed method is able to solve the steady state variables of the positive Luo converter more accurately. With the explicit Grünwald-Letnikov (GL) approximation, the proposed method greatly reduces the number of iterations when computing the transient solutions. Therefore, the speed and the accuracy of the computations can be improved at the same time. Since the proposed model is developed in the time domain, and the correction coefficients are used in the case of a homogeneous initial value, it can uncover the nonlinear fractional-order behaviors more comprehensively and realistically.

This paper is organized as follows: Section 2 introduces the mathematical modeling method for the fractional-order positive Luo converters operated in CCM, and their equivalent circuit models based on the ESP method. Section 3 presents the procedure to get the steady state and transient solutions of the fractional-order positive Luo converter. Section 4 provides the detailed numerical simulation results, which demonstrate the effectiveness of the proposed method. Since the fractional orders have great influence on the CCM-operating criterion, the transfer functions and the harmonic amplitudes of the state variables of the converters, these order-related phenomena are discussed in detail in Section 5. The asymptotically limit cycle behavior analysis of the PI controlled fractional-order positive Luo converter is shown in section 6. Section 7 carries out circuit simulations and experiments which further demonstrate the above-mentioned analysis. Section 8 summarizes the conclusion and future work.

Equivalent model of the fractional-order positive Luo converter using the ESP method

According to the linear capacitor model and the inductor model proposed by Westerlund [33, 34], all real capacitors and inductors are fractional essentially. The voltage across a real inductor v_L and the current through a real capacitor i_C can be, respectively, described by (2a) (2b) where L presents the inductor’s inductance, C is the capacitor’s capacitance, and λ, μ are the orders. They have relationships with the “proximity effect” and the kind of the dielectric.

As shown in Fig 1, circuit for the positive output elementary super lift Luo converter consists of the DC supply voltage V_in, the inductor L, the capacitors C_b and C_o, the power switch S, the diodes D₁ and D₂, and the load resistor R_o. During the analysis of the operating process, all the components are assumed to be ideal and running in CCM mode.

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Fig 1. Circuit of the positive Luo converter.

https://doi.org/10.1371/journal.pone.0237169.g001

Let n be any integer, T represent the switching period and D denote the duty ratio in the steady state. Generally speaking, there are two switching states for the positive Luo converter in CCM mode, identified as:

1. State 1: t ∈ (nT, (n + D)T], during which S and D₁ are on, and D₂ is reversed-biased; the equivalent circuit is shown in Fig 2(a).
2. State 2: t ∈ ((n + D)T, (n + 1)T], during which S and D₁ are off, and D₂ is forward-biased; the equivalent circuit is presented in Fig 2(b).

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Fig 2. Operating states of the positive Luo converter: (a) State 1; (b) State 2.

https://doi.org/10.1371/journal.pone.0237169.g002

In State 1, during the on-time period of the switch S, capacitor C_b is charged by the voltage supply, and the current i_L flowing through L increases. In State 2, during the off-time interval of the switch, the inductor L₁ is still conducting current, and the capacitor C_b and C_o are discharged. It is worth mentioning that R_in is the internal resistance of the voltage supply, which is a very small value but cannot be neglected in the model of this paper [35, 36]. To represent the switching state of the PWM converter, a periodic scalar function s(t) is introduced, which is denoted as (3)

Select the inductor current i_L, the capacitor voltages v_b and v_o as the state variables. In State 1, the state equations of the positive Luo converter can be described as (4a) (4b) (4c)

When operating in State 2, the state equations are expressed as (5a) (5b) (5c)

In order to simplify the equations, we use p to replace d/dt. Correspondingly, d^α/dt^α, d^β/dt^β and d^γ/dt^γ are represented by p^α, p^β and p^γ, respectively. In this way, the converter in the steady state can be formulated by the following vector differential equation of the state variable (6)

Here, the vector of the state variables is x = [i_L v_o v_b]^T, and the non-linear vector function is indicated as f(x) = s(t)x. H₁ = [V_in/L 0 0]^T and H₂ = [−V_in/L 0 V_in/(R_in C_b)]^T are the constant vectors. The matrices G₁(p^α, p^β, p^γ) and G₂ are denoted by (7a) (7b)

These equations have shown that the fractional orders can only influence the matrix G₁, but not G₂.

In consideration of the ESP method, the equation solution for Eq (6) can be represented by a series (8) where the zero order approximate vector x₀ is called the main wave and i^th order approximate vectors x_i are called the corrections. Smallness indicator parameter εⁱ is introduced temporarily to supply the order of the magnitude in the terms and to indicate that x_i is much smaller than x₀. More accurately speaking, we have x₀ ≫ εⁱ x_i ≫ εⁱ⁺¹ x_i+1. Similarly, s(t) can be expanded into (9)

Substituting Eqs (8) and (9) into f(x), and merging terms that have the same order εⁱ, the expression of f(x) can be obtained as (10) where (11a) (11b) (11c) (11d) (11e) and so on.

According to Eq (1), the order of the fractional differential operator does not affect the angular frequency. Similar to the method of [15], the terms in expansion (8) can be represented as (12)

Here, m is an integer, the normalized time τ = ωt (ω = 2π/T). The terms a_0i are the DC components, a_mi represent the m^th harmonic magnitudes. The spectral content set of {E₀} of the vector x₀, which is a group of numbers denoting relevant frequencies of harmonics, depends on the physical phenomena of object. Considering the low-pass filtering properties of the power converters, the first term x₀ only has the DC components of the state variables, implying that x₀ = a₀₀ and {E₀} = {0}. Likewise, the spectral content {E_i} for each x_i in the DC-DC converters is unknown in advance. {E_i} can be obtained via the iterated operation, which starts with x₁ and moves to higher order corrections.

Like x₁ that uses normalized time τ, the switching function s(τ) can be depicted in Fourier series as (13) where , b_l = (α_l − jβ_l)/2, and stands for the conjugate complex of b_l, in which (14a) and (14b)

The coefficient b_l decreases with the increasing of l. According to the definition described in [15], s_i(τ) can be chosen as (15a) (15b)

If Eqs (8) and (9) are introduced to (10), the spectral content of each term f_i gets wider in comparison with that of x_i. Then, f_i can be denoted as (16) where (17a) (17b)

In the above equations, the spectral content of the term f_0k has the same harmonics with the spectral content of the term x₀, while the additional harmonics outside the set {E₀} belong to R₁. Generally speaking, the harmonics of R₁ have magnitudes that are smaller than the magnitudes of the harmonics in f_0k. As in [15], the set {E₁} is determined by the newly generated harmonics in f₀. In another word, the spectral content of the term f_1k includes the same harmonics with the spectral content of the correction x₁. In the same manner, the spectral contents of the correction x_i+1 is determined by the spectral contents in R_i+1, and the term R_i+1 is considered smaller compared to f_ik [37]. Thus, Eq (16) can be rewritten as (18)

Introducing Eqs (18) into (10), we can obtain that (19)

Then substituting Eqs (8) and (19) into (6) and equating the terms with the same εⁱ on the both sides, we can obtain the following differential equations: (20a) (20b) (20c) (20d) (20e) and so on.

Because the influence of the exponential functions has been eliminated, these equations are all linear. Using the harmonic balance method, the solutions of Eq (20) in the steady state can be found. The amplitudes of the main wave x₀ can be solved by using the Eq (20a). Likewise, the harmonic amplitudes of the corrections x₁, x₂, x₃, x₄,…, etc can be obtained by using Eq (20b) and the following equations.

When the sufficient components are found, the steady-state solution of x is acquired by a simple summing of these components, which is approximated by (21)

Therefore, the parameter ε was used essentially to point out the order of the equations. The right side of Eqs (Eq (20a))–(20e) can be adjusted according to the order of the exponent function in the left side.

Solutions for the state variables of the fractional-order positive Luo converter

Comparison and simulation of the conventional schemes and the proposed method

Comparison of the steady-state solutions

In order to get the analytical solutions of the fractional-order DC-DC converters in the steady state, there are many methods proposed by previous researchers. Among them, the predictor-corrector method [38] is the most widely used used numerical scheme based on the fractional-order definition and the time-domain analysis. It can solve the fractional-order equations in each operating state. By running the routines programmed in MATLAB, the numerical analysis of the fractional-order DC-DC converters can be accomplished cycle by cycle. Through this process, the final results would include the complete contents of the state variables from the initial state to the stable state. On the other hand, the whole information can also be obtained by performing MATLAB/Simulink simulations with the modified Oustaloup’s approximation method [39]. The fractional integral module in the simulation is replaced by the approximated equations. As mentioned in the introduction, the modified Oustaloup’s approximation method is a precise engineering simulation running in the frequency domain. Therefore, a modified Oustaloup’s approximation method proposed in this section will be used to verify the validity of the proposed method, and be used as the comparison criterion for other methods.

In this subsection, the circuit parameters of the fractional-order positive Luo converter are listed as follows: the input voltage V_in = 10 V, the switch frequency f = 20 kHz, the duty ratio D = 0.5, L = 1 mH, C_b = 47 μF, C_o = 10 μF, and R_o = 50 Ω. This part compares the results of the modified Oustaloup’s approximation method, the PECE-ABM method and the proposed scheme. The accuracy of the state variables is the main comparing object. In the modified Oustaloup’s method, there are three key parameters: the filter order 2N + 1, the lower limit ω_b and the upper limit ω_h of the fitting frequency. Normally, ω_b ω_h = 1. We choose ω_h = 5 × 10⁵ rad/s, ω_b = 2 × 10⁻⁶ rad/s and N = 8 for the fractional-order positive Luo converter in this paper. The results gotten by these three methods are listed in Table 1.

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Table 1. The DC parts of I_L and V_o acquired by using different schemes.

https://doi.org/10.1371/journal.pone.0237169.t001

From Table 1, we can see that the DC components of these three schemes keep good consistency. Evidently, both i_L and v_o are dependent of the orders of the capacitor and the inductor. No matter which method is selected, the DC components of the state variables decrease with the decreasing of α, β and γ. In order to further compare these three methods, the steady state ripples are shown in Figs 3 and 4, where the black dash lines, the blue dotted lines and the red solid lines stand for the results of the PECE-ABM method, the modified Oustaloup’s approximation method and the proposed scheme, respectively.

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Fig 3. Comparison of the steady state ripples in i_L under different fractional orders: (a) (α, β, γ) = (1, 1, 0.95); (b) (α, β, γ) = (1, 0.8, 0.95); (c) (α, β, γ) = (0.95, 1, 0.95); (d) (α, β, γ) = (0.95, 0.8, 0.95); (e) (α, β, γ) = (0.9, 1, 0.95); (f) (α, β, γ) = (0.9, 0.8, 0.95).

https://doi.org/10.1371/journal.pone.0237169.g003

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Fig 4. Comparison of the steady state ripples in v_o under different fractional orders: (a) (α, β, γ) = (1, 1, 0.95); (b) (α, β, γ) = (1, 0.9, 0.95); (c) (α, β, γ) = (1, 0.8, 0.95); (d) (α, β, γ) = (0.9, 1, 0.95); (e) (α, β, γ) = (0.9, 0.9, 0.95); (f) (α, β, γ) = (0.9, 0.8, 0.95).

https://doi.org/10.1371/journal.pone.0237169.g004

As shown in Figs 3 and 4, the waveforms from these three schemes are consistent with each other, and the steady state ripples are influenced by the fractional orders. Specifically, the steady state ripples of i_L and v_o increase with the decreasing of α and β respectively. In general, the steady state ripples gotten by the proposed scheme are more closely resemble to the simulation results obtained by the modified Oustaloup’s method, especially with smaller values of α and β. For the PECE-ABM method, the rangeabilities of both the DC components and the AC components are all undersized when α and β changes. This proves that the proposed method can track the dramatic changes more closely compared to the PECE-ABM method. Reducing the step size can improve the accuracy of the PECE-ABM, but the number of iterative computations will be greatly increased at the same time. For the modified Oustaloup’s method, although it is more accurate, complex fractional-order component approximation and circuit simulation should be used to obtain the steady state solutions.

In order to show that the proposed method can obtain more accurate state variables, of the positive Luo converters, compared to the simplified equivalent small parameter (SESP) method in [15, 16], the tolerance error index [40] is used to indicate the end of the iteration process. This index is the ratio of the matrix-norm of a_mi (m^th harmonic magnitude of X_i) to that of a₀₀. Thus, the tolerance of a_mi is denoted as (36)

The 1% tolerance error index is utilized to control the iteration process of x_i. The correcting process ends once the tolerance error index is smaller than 1%. For most DC-DC converters, x in the steady state approximating by x₀, x₁ and x₂ is accurate enough. However, for the positive Luo converter, with the decreasing of β, the tolerance error index of a₃₂ can reach 1% or even larger, as represented in Fig 5, implying that the harmonic magnitudes in the high order correction x_i(i ≥ 3) are correspondingly high compared to the DC component. Thus, the iteration process of x_i should be continued. Fig 5 also shows the tolerance error index of a₁₃ and a₀₄. This index of a₁₃ is inversely proportional to α and β, and can be close to 1%. Because the computational efforts for calculating the amplitude of the first three order harmonic amplitude are relatively small, this paper only computes a₁₃ and a₀₄. As depicted in Fig 5, the tolerance error index of a₀₄ with different fractional orders keeps below 0.6%. Therefore, the approximate solution of x calculated by the proposed method is precise enough for the positive Luo converter shown in Fig 1. And the tolerance error index at the end of the iteration process decreases from larger than 1.3% to 0.6%.

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Fig 5. The tolerance error index of m^th harmonic magnitude in x_i (a₃₂, a₁₃ and a₀₄).

https://doi.org/10.1371/journal.pone.0237169.g005

In order to comprehensively compare the steady state solutions obtained by the SESP method, the Oustaloup’s approximation method and the proposed scheme, we consider the case in which (α, β, γ) = (0.9, 0.8, 0.95). Fig 6 shows the comparison of these three schemes, which obviously shows that the steady state solutions obtained by the proposed method can be more closely resemble to the results gotten by the Oustaloup’s approximation method, compared to the SESP method, especially for v_b. When the power switch turns off, the gap between the results gotten by the SESP method and the proposed scheme can be greater than 3%. Similar to the tolerance error index of a_mi, the tolerance error indexes of each state variable are calculated. It can be obtain that Ta13 = [Ti_L13 Tv_o13 Tv_b13]^T = [1.81% 0.88% 0.76%]^T and Ta04 = [Ti_L04 Tv_o04 Tv_b04]^T = [0.02% 0.01% 0.67%]^T. For v_b, the lack of significant decrease in tolerance error index indicates that the number of iterations for SESP method is not enough. Thus, the corrections of DC value and main wave proposed in this paper are necessary.

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Fig 6. The steady state solutions obtained by using the SESP method (black dash lines), the proposed method (red solid lines) and the Oustaloup’s approximation method (blue solid lines): (a) i_L; (b) v_o; (c) v_b.

https://doi.org/10.1371/journal.pone.0237169.g006

Transient solutions comparison

To verify the transient solutions obtained with the proposed method, a comparison is made between them and other simulation results obtained by using the SESP method and the modified Oustaloup’s method. Circuit parameters in this subsection are listed as follows: V_in = 10 V, f = 20 kHz, D = 0.4, L = 1 mH, C_b = 47 μF, C_o = 10 μF, R_o = 50 Ω, (α, β, γ) = (0.9, 0.8, 0.95).

The comparison of the state variables versus n (n = (1000f)⁻¹ t), between the SESP method, the Oustaloup’s approximation method and the proposed scheme, is depicted in Fig 7. It can be seen that, the transient solutions of the proposed method are in good accordance with those from the Oustaloup’s approximation method. Based on the analysis of the previous subsection, using the proposed method, the numerical solution of each cycle is more accurate than SESP method. Because the explicit Grünwald-Letnikov (GL) approximation is used in the transient solution calculation, this error will accumulate with the increase of iterations, and be more apparent in the steady state. Therefore, the steady state solutions obtained by the proposed method are more accurate compared to the SESP method, especially for v_b. When the power switch turns on, the gap between the SESP method and the proposed method is more obvious than that shown in Fig 6. The error between the maximum values of v_b obtained by the two methods can exceed 1%. Moreover, the v_b waveform obtained by the SESP method during the switching on conduction appears obvious distortion, which does not conform to the actual situation. Expect for the increased accuracy, the number of iterations is greatly reduced using proposed method, since the step size is not required to be smaller than the switching cycle.

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Fig 7. Transient waveforms comparison of solutions obtained by the SESP method (black dash lines), the proposed method (red solid lines) and the Oustaloup’s approximation method (blue solid lines): (a) i_L; (b) v_o; (c) v_b.

https://doi.org/10.1371/journal.pone.0237169.g007

Analysis of fractional order related phenomena

CCM operating boundary

According to the operating criterion of the positive Luo converter, when the circuit of Fig 1 operates in continuous-conduction mode (CCM), the current i_L must be continuous, meaning that the inductor current is always greater or equal to zero. The CCM operating criterion ccan be obtained from two parameters, namely and Δi_L, which denote the average value of the inductor current and the peak-peak ripple, respectively. can be approximated by the combination of I₀₀, I₀₂ and I₀₄. Δi_L can be computed using the inductor volt-second balance principle and the definition of fractional-order derivative given by Caputo. As R_in is a very small value, v_b is very close to the input voltage V_in during the ON-state of the switch. Due to this and Eq (4a), the CCM operating formula of the positive Luo converter with the fractional order can be concluded as (37) where Γ(•) represents the gamma function. The fractional-order positive Luo converter is operating in CCM mode when Ψ ≥ 0. In other words, the converter goes to discontinuous conduction mode (DCM) when Ψ < 0.

According to the calculation method derived in the previous Section, the 3D plot of the CCM operating criterion can be found in Fig 8 utilizing the circuit parameters in subsection A of Section 4. The space beneath the surface in Fig 8 indicate the DCM mode, while the above part represents the CCM mode. As shown in this figure, the CCM operating criterion is mainly determined by the order α, while β partly affects it, and the impact of γ on the boundary can be neglected.

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Fig 8. Operating criterion of fractional-order positive Luo converter.

https://doi.org/10.1371/journal.pone.0237169.g008

Transfer functions of the converter

The transfer functions of the converter can be obtained by the small-signal perturbation analysis based on Eq (6). Perturbed values can be represented as (38a) (38b) (38c) where , and represent three perturbations. And X, D and V_in are the DC parts corresponding to the perturbed values. Substituting Eqs (38a)–(38c) into (6) gives expression (39)

Neglecting the infinitely small 2^th perturbation can yield the following equation (40)

In order to get the transfer functions from to , should be assumed. Hence, Eq (40) can be changed into (41)

Then, is depicted as (42)

Similarly, by letting , the transfer functions from to can be obtained. Thus, Eq (40) can be rewritten as (43) where X ≈ a₀₀. In this case, is denoted by (44)

In this way, the final transfer functions are equal to the results gotten with the state-space averaging model. However, the derivation is much simpler. From the Fig 9(a) and 9(c), it can be seen that α significantly affects and G_iL d(s) with the same fractional order. With the decrease of α, the open loop amplitude margin increases. In Fig 9(b) and 9(d), both α and β can affect and G_v0 d(s). While, the order γ has very little influence on the bode diagrams.

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Fig 9. Bode diagrams of , , G_iL d(s) and G_vo d(s) with different α, β and γ: (a) ; (b) ; (c) G_iL d(s); (d) G_vo d(s).

https://doi.org/10.1371/journal.pone.0237169.g009

Harmonics of the state variables

Based on the results gotten by the proposed scheme, we further analyze the harmonics of the inductor current i_L and the output voltage v_o with different orders. In a word, with the decreasing of α, β and γ, the RMS values of the harmonics increase as displayed in Fig 10. This directly results in the increasing of the inductor current ripples and the output voltage ripples, which are generally undesired in the DC-DC converters design. Separately, α has a significant impact on the first and third harmonics of i_L, while β exerts considerable influence on the first and third harmonics of v_o. However, their effects on the second harmonic are reversed. The order α decreases with the increasing of the second harmonic amplitude in v_o. Similarly, β decreases when the second harmonic amplitude of i_L increases. Normally, the order γ has very little influence on i_L and v_o, relatively speaking. It mainly affects i_L by influencing the value of v_b.

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Fig 10. RMS values of harmonics in i_L and v_o with different fractional orders: (a) i_L; (b) v_o.

https://doi.org/10.1371/journal.pone.0237169.g010

Dynamical behavior analysis of the PI controlled fractional-order positive Luo converter

In this section, whether the non-existence isolated period oscillations can be observed in fractional-order model is used to establish the validity of proposed model in analyzing dynamical behavior.

In order to analyze dynamical behaviors in the fractional-order model, the block diagram of the positive Luo converter with PI voltage compensator are described in Fig 11. The control voltage v_vf can be expressed as (45)

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Fig 11. Block diagram showing the positive Luo converter with PI voltage compensator.

https://doi.org/10.1371/journal.pone.0237169.g011

The integral coefficient k_I of the controller is treated as the bifurcation parameter, which is critical in the practical design. The component values and the control parameters of the test bench, as shown in Fig 11, are chosen as: V_in = 10 V, f = 20 kHz, L = 4 mH, C_b = 47 μF, C_o = 10 μF, R_o = 50 Ω, V_ref = 18 V, k_v = 1, k_p = 0.01.

With the parameter k_I varying from 10 to 110, the bifurcation graphs of the integer-order and the fractional-order circuits are depicted in Fig 12, which are only constructed with the steady state data at the beginning of each switching cycle. Take the integer-order system as an example. As presented in Fig 12(a), the converter remains stable in period-1 orbit when k_I varies from 10 to 34. Once k_I reaches to 34, the Hopf bifurcation appears, forming the bifurcation points. Fig 12(a)–12(d) have shown that the bifurcation points are shifting backward with the decreasing fractional order α of converters, given a fixed β value. In other words, with the same parameters, the bifurcation occurs in the integer-order circuit, while the fractional-order converter remains asymptotically stable. Similarly, the value of k_I at the time of bifurcation increases when β decreases, which effect is dominating. The analysis discussed above shows that the fractional-order converter is easier to keep stable.

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Fig 12. Bifurcation diagrams (v_o vs. k_I) of the integer-order and the fractional-order converters: (a) (α, β, γ) = (1, 1, 1); (b) (α, β, γ) = (0.95, 1, 1); (c) (α, β, γ) = (1, 0.95, 1); (d) (α, β, γ) = (0.95, 0.95, 1).

https://doi.org/10.1371/journal.pone.0237169.g012

To observe the causes of the asymptotically limit cycle behavior, we keep k_I = 105. To reveal the differences of the non-linear behaviors between the fractional and the integer order converters, the phase portraits, the harmonic spectrums and the time-domain curves are utilized. The simulation results can be found in Figs 13–15. Because the steady state solutions in the proposed model are acquired without using the definitions of fractional calculus, the vital difference between the essence of regular oscillations in the integer and the fractional order systems can be evinced. Unlike their integer order corresponding systems under the same condition, the limit cycle behaviors do not exist in the fractional-order converters. Take the fractional-order converter with (α, β, γ) = (0.95, 0.95, 1) as an example. Fig 15(b) illustrates the time-domain waveforms of the output voltage v_o, the inductor current i_L, the control voltage v_vf and the triangular wave v_ramp. It can be seen that in the steady state, the control signal v_vf (red line) reaches the lower limit of v_ramp (blue line) several times, but the inductor current i_L remains continuous all the time. The phase graph (Fig 15(a)) seems to be a trajectory of the periodic oscillation (solar to the limit cycle), while actually this oscillation is not perfectly periodic but is generating non-periodic items obviously. If the approximation of the fractional-order components in [13] is used, the phase graph will be a closed curve like Fig 13(a), which does not satisfy the well-known fact that the fractional-order systems cannot have any periodic solutions. Fig 15(c) shows the most significant harmonic component occurs at a frequency of 1159 Hz, indicating the frequency of the asymptotically limit cycle f_acyc under this condition. It is lower than the switching frequency. From above analysis and the plots in Figs 13–15, it can be revealed that f_acyc increases with the decreasing of the fractional order. Besides, the influence of β is more obvious. In addition, since the transient solutions are obtained using the explicit Grünwald-Letnikov (GL) approximation, the limit cycle behaviors, which normally appear in the time-domain analysis, do not exist in the converters when using the proposed method. The only existence of the asymptotically period oscillation can further demonstrate the validity of the proposed modeling of the non-linear behaviors.

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Fig 13. Simulation results of asymptotically limit cycle behavior with (α, β, γ) = (0.95, 1, 1): (a) phase portrait; (b) time-domain waveform; (c) harmonic spectrum.

https://doi.org/10.1371/journal.pone.0237169.g013

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Fig 14. Simulation results of asymptotically limit cycle behavior with (α, β, γ) = (1, 0.95, 1): (a) phase portrait; (b) time-domain waveform; (c) harmonic spectrum.

https://doi.org/10.1371/journal.pone.0237169.g014

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Fig 15. Simulation results of asymptotically limit cycle behavior with (α, β, γ) = (0.95, 0.95, 1): (a) phase portrait; (b) time-domain waveform; (c) harmonic spectrum.

https://doi.org/10.1371/journal.pone.0237169.g015

Circuit simulation and experimental results

As there are no commercially available fractional-order components, many schemes are designed to build the equivalent circuit of the fractional-order components. In order to further analyze the simulation results listed in previous section, according to the method in [41], the approximation circuits of the inductor and the capacitor with fractional-order properties are built by using the resistor/inductor or resistor/capacitor networks (as shown in Fig 16(a) and 16(b) respectively). The parameters in the approximation circuit of the fractional-order components are deduced in S2 Appendix. And the circuit parameters are the same with the simulation in subsection A of Section 4.

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Fig 16. The approximate circuit of components with fractional-order property: (a) fractional-order inductor; (b) fractional-order capacitor.

https://doi.org/10.1371/journal.pone.0237169.g016

Circuit simulation results

Let L, C_b and C_o in Fig 1 be replaced by the fractional-order inductor and capacitor units shown in Fig 16(a) and 16(b) correspondingly, the circuit simulation results can be found in Figs 17 and 18. All these simulations are accomplished by PLECS Standalone.

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Fig 17. Bode diagram obtained by the theoretical analysis and the circuit simulation with (α, β, γ) = (0.9, 0.9, 0.95) and (α, β, γ) = (0.9, 0.8, 0.95): (a) ; (b) G_vod(s).

https://doi.org/10.1371/journal.pone.0237169.g017

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Fig 18. State variables in the steady state under different fractional orders: (a) i_L in CCM; (b) v_o in CCM; (c) i_L in CCM and DCM; (d) v_o in CCM and DCM.

https://doi.org/10.1371/journal.pone.0237169.g018

For example, the bode diagrams of and G_vo d(s) with (α, β, γ) = (0.9, 0.9, 0.95) and (α, β, γ) = (0.9, 0.8, 0.95) respectively are shown in Fig 17. These curves are obtained by the theoretical analysis in Section 3 and the PLECS circuit simulation separately. The theoretical equations of and G_vo d(s) gotten by the proposed fractional-order model have been expressed by (42) and (44). As depicted in Fig 17, the circuit simulation results are basically like the theoretical analysis. The discrepancies are mainly caused by the approximation of circuit for the fractional-order inductor and capacitor.

As shown in the steady state waveforms in Fig 18, with decreasing of the fractional-order α and β, the ripple of i_L and v_o increase significantly, which is consistent with the results of the theoretical analysis and the numerical simulation in the previous sections.

Especially, as shown in Fig 18(c), when α = 0.8, the converter enters into the DCM mode, in which v_o changes greatly. All of these phenomena agree very well with the analysis of the CCM operating criterion,as shown in Fig 8.

To verify the mechanism of the asymptotically limit cycle behavior, a circuit simulation of the positive Luo converter with the PI voltage compensator (schematically presented in Fig 11) has been performed. The simulation setup has the same parameters as listed in Section 6. Fig 19 shows the time-domain waveforms of the inductor current and the output voltage when k_I = 105. The green and the red curves present the simulation results of the converters with (α, β, γ) = (1, 1, 1) and (α, β, γ) = (0.95, 0.95, 1) respectively. As shown in Fig 19, the waveforms have a good agreement with the numerical simulation results described in Section 6.

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Fig 19. Circuit simulation results of (a) v_o; (b) i_L.

https://doi.org/10.1371/journal.pone.0237169.g019

Experimental results

In order to further verify the effectiveness of the proposed method and to realize the practical performance of fractional-order components, a positive Luo converter prototype is built. Fig 20 shows the photograph of the positive Luo converter. In this prototype, power switch and diode in Fig 1 choose STB18N20 and MUR160 respectively. To reduce the interference of inductors in the equivalent implementation circuit of fractional-order inductor, some shielded techniques, such as shielded power inductors are utilized.

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Fig 20. Photograph of the prototype positive Luo converter.

https://doi.org/10.1371/journal.pone.0237169.g020

Fig 21 shows the steady-state wave forms of i_L and v_o, which is obtained by experimental test with different fractional orders. Experimental measurements of these state variables are listed in Table 2. From Fig 21(a)–21(d) and Table 2, it can be seen that the RMS and peak-peak value of i_L and v_o are basically consistent with the simulation analysis. Considering the influence of parasitic parameters and electromagnetic interference, the error between them can be ignored.

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Fig 21. Experimental results under different fractional orders: (a) (α, β, γ) = (1, 1, 1); (b) (α, β, γ) = (0.9, 1, 1); (c) (α, β, γ) = (1, 0.9, 1); (d) (α, β, γ) = (0.9, 0.9, 1).

https://doi.org/10.1371/journal.pone.0237169.g021

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Table 2. Experimental results under different fractional orders.

https://doi.org/10.1371/journal.pone.0237169.t002

Conclusion

A modified fractional-order modeling and asymptotically limit cycle behavior analysis method for the positive Luo converters is proposed in this paper. With the proposed method, the approximate steady state solutions can be obtained without utilizing the definitions of fractional calculus. In addition, it eliminates the needs for the circuit simulations or multiple iterations. Based on the particularity of the positive Luo converter, the improved algorithm can ensure the accuracy without adding more harmonics in the approximate expression, compared to [32]. The final transient solution can be acquired by replacing the DC part by the primary transient components. Therefore, the proposed modeling methodology provides a more convenient and faster approach to the optimal design of the fractional-order positive Luo converters. Furthermore, the proposed method, with the time-domain model using the explicit Grünwald-Letnikov (GL) approximation, can analyze the nonlinear behaviors via the numerical analysis more accurately. The analysis of the dynamic behaviors in this paper confirms that nonexistence of the periodic solutions in continuous-time fractional-order is a remarkable difference between the integer-order converters and the fractional-order converters.

As the DC parts and the harmonics of the state variables in the converters are directly related to the fractional orders, changing orders influences the characteristics in the steady state. The most obvious phenomenon is that the ripples of the state variables are markedly depended on the fractional orders. More specifically, as presented in the simulation results, the steady state ripples of i_L and v_o increase with the decreasing of α and β, respectively. Moreover, the CCM-operating criterion is also order-dependent, and is mainly determined by the order α of the inductor. At the meantime, the line-to-output and duty cycle-to-output transfer functions of the DC-DC converters can be obtained from the analysis discussed in this paper, which both show a close relationship to the fractional orders. The generation condition of the asymptotically limit cycle behaviors, in the PI controlled fractional-order positive Luo converters, are also affected by the fractional orders. A smaller α or β, can help the converter to keep stability, and to increase the frequency of the asymptotically limit cycle f_acyc. In this paper, all the phenomena discussed above have been analyzed by the proposed method, of which the effectiveness has been verified by simulations and experiments.

In recent years, study results of the mathematical modeling of the passive components have shown that both the inductors and the capacitors are essentially of the fractional order. The modeling of the fractional-order converters has received widespread acceptance in engineering applications. Due to the great influence of the fractional orders to the properties of the power converters and their nonlinear dynamical behaviors, there is a big room for development and improvement in this research area. Based on the fact that the equivalent small parameter method and the principle of the harmonic balance are suitable for all DC-DC converters, the modified method proposed in this paper can be expanded to and cope with other fractional-order DC-DC converters. Moreover, with more precise equivalent circuit models of the electrolytic capacitors and the coil inductors, the benefits of using fractional-order modeling methodology can be further explored in the future.

Supporting information