Research on fractional order modeling and PIλ control strategy of CLLC bi-directional resonant converter

Research on fractional order modeling and PIλ control strategy of CLLC bi-directional resonant converter | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Research on fractional order modeling and PIλ control strategy of CLLC bi-directional resonant converter Di Li, Qingquan Lv, Zhenzhen Zhang, Yunfan Huang, Haiying Dong This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1918353/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract There are problems such as insufficient modeling accuracy of the CLLC bi-directional resonant converter in integer order, failure to accurately describe the actual conditions of the system, poor dynamic performance of classical PI control strategy, and failure of the system to reach the stable state quickly and smoothly. First, based on the fractional order calculus theory, this paper proposes an extended describing function method based on fractional order, establishes the mathematical model of the fractional order and small signal model of the fractional order of the CLLC bi-directional resonant converter, obtains the fractional order transfer function of the system, constructs the PI λ control strategy of the fractional order of the CLLC bi-directional resonant converter, and builds the simulation model of the CLLC bi-directional resonant converter based on Matlab/Simulink software for simulation verification. Finally, a principle prototype was built in the laboratory to verify the correctness and feasibility of the CLLC bi-directional resonant converter of the fractional order and theoretical analysis of the PI λ control strategy of the fractional order. CLLC bi-directional resonant converter Fractional calculus Fractional order modeling Extended description function method Fractional order PIλ control strategy Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 1. Introduction With the rapid development of new energy power generation and its related industries in China, the proportion of new energy continuously increases. Due to the unique randomness, intermittency, regional interconnection of the power grid, and other problems of new energy power generation, China faces serious basic problems of safety and stability such as consumption of renewable energy and power and electric quantity balance of the power grid [ 1 ] . The V2G (Vehicle to Grid) bi-directional converter system is a key technology in which the new energy vehicles participate in the peak regulation and frequency modulation of the power grid as a large-scale distributed energy storage system. It can realize the mutual transmission of electric energy between the power grid and the new energy vehicles to assist with the peak regulation and frequency modulation of the power grid and consume a large amount of new energy power generation. The V2G system has a structure with two stages, which are the front-stage AC/DC and backward-stage DC/DC circuits [ 2 – 4 ] . As a key component of the V2G system, DC/DC converter serves as a bridge in the mutual transmission of electric energy between the power grid and new energy vehicles. By using the CLLC resonant converter, the energy transmission efficiency can be effectively improved and the switching loss can be reduced [ 5 ] . At present, many scholars have done a lot of researches on the CLLC resonance circuit. Because the CLLC resonance circuit has the characteristics of soft switching in a wide load range and does not need to output inductance, it has attracted extensive attention. A bi-directional three-level LLC resonant converter was proposed in Reference [6]. A new pulse width and amplitude modulation control method was used to realize the soft switching of all switching tubes and diodes in the circuit and it was verified by experiment. A kind of Extended Phase-shift control for full-bridge CLLC resonant converter was proposed in Reference [7] to realize the maximum conversion efficiency under the condition of light load and it was proved in the experimental prototype. In Reference [8], a new bi-directional multimode CLLC resonant converter was designed, which could not only meet the needs of new energy vehicles with different voltage grades but also increase the voltage gain of the bi-directional converter. Its feasibility was verified through simulation and experiment. In Reference [9], an integrated magnetic integration structure suitable for the high-frequency resonant inductor-transformer of the CLLC bi-directional resonant converter was designed. The integration of magnetic components in the converter was realized only through a magnetic component, and the feasibility and effectiveness were verified by finite element simulation and experimental prototype. In Reference [10], a unified modeling method suitable for all kinds of the CLLC resonant converter topologies was proposed. This method greatly simplifies the calculated quantity in parameter design and makes bi-directional SR (Synchronous Rectification) less complicated. This method has been verified in symmetric full bridge, symmetric half bridge, asymmetric full bridge, and asymmetric half bridge, respectively. In the above research and analysis, the analysis, modeling, and control were completed in the integral order system. That is, the capacitance in the CLLC bi-directional resonant converter is an integral order capacitance and the inductance is an integral order inductance. However, in the mathematical study of inductance and capacitance, it is found that the inductance and capacitance of the system are of the fractional order in practical application, so a mathematical model of the fractional order should be established to describe the fractional order inductance and fractional order capacitance in the actual system more accurately [ 11 – 13 ] . Among them, Petráš designed a Chua’s circuit of the fractional order by using fractional order inductance and fractional order capacitance and confirmed the characteristics of the fractional order in the actual system [ 14 ] . Avischek designed a parallel resonator of tunable fractional order and fractional order filter, which were verified through simulation and experiment, and gave detailed experimental results. It was found that the experimental data and simulation data were relatively consistent [ 15 ] . In conclusion, the previous research on the mathematical modeling and control strategy of the CLLC bi-directional resonant converter is based on the integral order model, while the actual system is a fractional order system and its description is not accurate enough in the classical modeling. This paper will take the CLLC bi-directional resonant converter as the research object and give the fractional order model and small signal model of the fractional order of the CLLC bi-directional resonant converter to obtain the system transfer function. The Bode diagram of the fractional order system and the Bode diagram of integral order will be drawn through Matlab for analysis and comparison, and the closed-loop simulation of the fractional order system of the CLLC bi-directional resonant converter will be completed in Simulink. Finally, the principle prototype will be built in the laboratory to verify the correctness and feasibility of the fractional order modeling and PI λ control strategy. 2. Topology And Fractional Order Equivalent Topology Of Cllc Bi-directional Resonant Converter The topology of CLLC bi-directional resonant converter is shown in Fig. 1 . When working in the forward direction, the converter consists of Q 1 ~ Q 4 to form the transformer primary side inverter bridge circuit, Q 5 ~ Q 8 to form the transformer secondary side rectifier bridge circuit. Similarly, in the reverse direction, Q 5 ~ Q 8 to form the transformer primary side inverter bridge circuit, and Q 1 ~ Q 4 to form the transformer secondary side rectifier bridge circuit. D 1 ~ D 8 are the body diodes of MOSFET, C 1 and C 2 are the resonant capacitance of primary and secondary sides respectively, L 1 and L 2 are the resonant inductance of primary and secondary sides respectively, L m is the excitation inductance, V i is the circuit input DC voltage, u ab and u cd are the square wave voltage of primary and secondary sides of transformer respectively, R L is the output load, C is the output filter capacitor, i 1 is the resonant current of primary side of transformer, i 2 is the resonant current of secondary side of transformer, i m is the primary excitation inductance current, n is transformer transformation ratio. Since the inductors and capacitors in the actual system are fractional inductors and fractional capacitors, the CLLC bi-directional resonant converter can be more accurately described by establishing a fractional mathematical model. The integer inductors and capacitors in the above system topology are replaced by fractional inductors and fractional capacitors, as shown in Fig. 2 . Where α and β are the actual orders of fractional capacitors and fractional inductors respectively, and the value range is 0 < α , β < 1. By using fractional calculus theory, the fractional mathematical model of the system is established, and more accurate transfer function is obtained and applied to the subsequent control research of the system. 3. Cllc Working Modal Analysis The forward working modal and reverse working modal of CLLC bi-directional resonant converter are basically the same. When working in one direction, it is similar to LLC resonant converter. According to the different resonant network elements, there will be two resonant frequency points f 1 and f 2 . According to the size of the system operating frequency f n , the converter is divided into three working modals: under resonant condition, quasi resonant condition and over resonant condition. This paper mainly analyzes the working modals of CLLC bi-directional resonant converter when it works in the forward direction, and the modulation strategy is PFM(pulse frequency modulation) which can obtain higher electrical transmission efficiency [ 16 ] . According to reference [17], the relationship between the current i L flowing through the fractional inductor and the voltage v L at both ends is as follows: $${v_L}=L\frac{{{\operatorname{d} ^\alpha }{i_L}}}{{\operatorname{d} {t^\alpha }}}$$ 1 Where α is the fractional order of inductance L α , and 0 < α < 1. The relationship between the voltage v o at both ends of the fractional capacitor and the current i c flowing through the capacitor is as follows: $${i_C}=C\frac{{{\operatorname{d} ^\beta }{v_o}}}{{\operatorname{d} {t^\beta }}}$$ 2 Where β Is the fractional order of capacitance C β , and 0 < β < 1. 3.1. f2 < fn < f1 Underresonance When the operating frequency of the CLLC converter is less than the first resonant point, the converter is under resonant condition, and the key waveforms on the primary side of the transformer are shown in Fig. 3 . There are 6 operating modes in the whole switching cycle, and the circuit waveforms in the front and back half cycles are symmetrical. Therefore, only the circuit operating modes in the first half cycle are analyzed in this paper. Mode 1, [ t 0 , t 2 ]: At time t 0 , the switch Q 1 ~ Q 4 is turned on at zero voltage switching(ZVS), the input voltage of the resonance network is equal to the system input voltage v i , the primary side resonance current i 1 and the excitation inductance current im rise rapidly, at time t 1 , the primary side current i 1 becomes positive, the secondary side diodes D 5 and D 8 are turned on, and the excitation inductance L m does not participate in the resonance. The equivalent topology of CLLC bi-directional resonant converter operating mode 1 is shown in Fig. 4 . When all MOSFET in the circuit are ideal devices, the circuit state equation of circuit mode 1 is: $$\left\{ {\begin{array}{*{20}{l}} {{L_1}\frac{{{\operatorname{d} ^\alpha }{i_1}}}{{\operatorname{d} {t^\alpha }}}+{u_1}+{L_m}\frac{{{\operatorname{d} ^\alpha }{i_m}}}{{\operatorname{d} {t^\alpha }}}={V_i}} \\ {{C_1}\frac{{{\operatorname{d} ^\beta }{u_1}}}{{\operatorname{d} {t^\beta }}}={i_1}} \\ {{L_m}\frac{{{\operatorname{d} ^\alpha }{i_m}}}{{\operatorname{d} {t^\alpha }}}=n\left[ {{L_2}\frac{{{\operatorname{d} ^\alpha }n\left( {{i_1} - {i_m}} \right)}}{{\operatorname{d} {t^\alpha }}}+{u_2}+{U_o}} \right]} \\ {{C_2}\frac{{{\operatorname{d} ^\beta }{u_2}}}{{\operatorname{d} {t^\beta }}}={i_2}=n\left( {{i_1} - {i_m}} \right)} \\ {{C_o}\frac{{{\operatorname{d} ^\beta }{V_o}}}{{\operatorname{d} {t^\beta }}}+\frac{{{V_o}}}{{{R_L}}}={i_2}} \end{array}} \right.$$ 3 Mode 2, [ t 2 , t 3 ]: In this mode, the excitation current i m is equal to the primary side current i 1 . At this time, no current flows through the transformer and the secondary side of the transformer is in the cut-off state. In this mode, the resonant inductor L 1 , the resonant capacitor C 1 and the excitation inductor L m participate in the resonance, which is a ternary resonance state. The equivalent topology of mode 2 of CLLC bi-directional resonant converter is shown in Fig. 5 . The equation of state for mode 2 is as follows: $$\left\{ {\begin{array}{*{20}{l}} {{L_1}\frac{{{\operatorname{d} ^\alpha }{i_1}}}{{\operatorname{d} {t^\alpha }}}+{u_1}+{L_m}\frac{{{\operatorname{d} ^\alpha }{i_m}}}{{\operatorname{d} {t^\alpha }}}={V_i}} \\ {{C_1}\frac{{{\operatorname{d} ^\beta }{u_1}}}{{\operatorname{d} {t^\beta }}}={i_1}} \\ {{i_1}={i_m}} \\ {{C_2}\frac{{{\operatorname{d} ^\beta }{u_2}}}{{\operatorname{d} {t^\beta }}}=0} \\ {{C_o}\frac{{{\operatorname{d} ^\beta }{V_o}}}{{\operatorname{d} {t^\beta }}}+\frac{{{V_o}}}{{{R_L}}}=0} \end{array}} \right.$$ 4 Mode 3, [ t 3 , t 4 ]: This mode is about the dead time. At this time, Q 1 ~ Q 4 are all turned off, and the parasitic capacitance of the switch tube begins to discharge. Therefore, the half cycle Q 2 and Q 3 can realize ZVS conduction. The duration of this mode is very short, so the state equation of the system is not considered. The equivalent topology of CLLC resonant converter mode 3 is shown in Fig. 6 . 3.2. fn = f1 Quasi-resanent condition When the system operating frequency f n is equal to the first resonant frequency, the CLLC bi-directional resonant converter operates in quasi resonant mode. The figure under this working condition is shown in Fig. 7 . It can be seen that under quasi resonant working condition, there will be no ternary resonance in the system. When the operating frequency f n is greater than the first resonant frequency point, the system cannot achieve full resonance. At this time, the diode at the secondary side of the transformer cannot achieve ZCS, which increases the switching loss of the converter. Therefore, the converter generally does not work in the over resonant state. This paper will not discuss it more. 4. Fractional Small Signal Modeling And Analysis Of Cllc Bi-directional Resonant Converter 4.1. Extended description function method based on fractional order In the modeling of traditional DC/DC converter, the state space average model of the system is built based on the "small ripple assumption" [ 18 ][ 19 ] , but this assumption needs to meet the requirement that the switching frequency of the system is much higher than the disturbance small signal frequency of the circuit system For CLLC bi-directional resonant converter, the switching frequency of the system is very close to the resonant frequency of the resonant cavity in actual operation, so it is impossible to construct an accurate small signal model by using the state space average method. In this paper, the extended description function method will be used to construct the small signal model of CLLC bi-directional resonant converter. This method was proposed by Dr. E. X. Yang in 1992. It ignores the high-order harmonics of the system state and greatly simplifies the complexity of nonlinear system modeling. So far, it has been widely used to solve the small signal model of resonant converter. The steps of establishing small signal model for CLLC bi-directional resonant converter by using extended description function method include writing state equation, harmonic approximation, extended description function equation and disturbance separation [ 20 ][ 21 ][ 22 ] . In order to introduce the fractional calculus theory into the extended description function method, according to the modeling method based on Fourier analysis proposed by J.Groves [ 23 ] , the following form is obtained by simplifying the mathematical expression: $$jM{\omega _c}{X_M}=\frac{{\partial {F_M}}}{{\partial {X_M}}}{X_M}+\frac{{\partial {F_M}}}{{\partial {U_M}}}{U_M}$$ 5 An approximate linear time invariant small signal model is given in reference [22] and its effectiveness is proved. The model is as follows: $$\frac{{d{{\hat {x}}_0}}}{{dt}}=\frac{{\partial F_{0}^{s}}}{{\partial X_{0}^{s}}}{\hat {x}_0}+\frac{{\partial F_{0}^{s}}}{{\partial U_{0}^{s}}}\hat {u}$$ 6 According to the nature of fractional calculus, the above formula can be rewritten into fractional order form as follows: $$\frac{{{{\text{d}}^\alpha }{{\hat {x}}_0}}}{{\operatorname{d} {t^\alpha }}}=\frac{{{\partial ^\alpha }F_{0}^{s}}}{{\partial {{\left( {X_{0}^{s}} \right)}^\alpha }}}{\hat {x}_0}+\frac{{{\partial ^\alpha }F_{0}^{s}}}{{\partial {{\left( {U_{0}^{s}} \right)}^\alpha }}}\hat {u}$$ 7 The small signal model of fractional extended description function method is constructed by the above formula. 4.2. Small signal modeling Through the above modal analysis of CLLC bidirectional resonant converter, the fractional order equivalent model of the system can be obtained, as shown in Fig. 8 . It can be seen that the transformer secondary side resonance current i 2 has the following relationship with the transformer primary side resonance current i 1 and excitation current i m : $$n\left( {{i_1} - {i_m}} \right)={i_2}$$ 8 According to the fractional order equivalent circuit of the CLLC bi-directional resonant converter, the transformer primary side resonant current i 1 , resonant capacitor voltage u 1 , excitation inductance current i m , transformer secondary side resonant current i 2 , resonant capacitor voltage u 2 and output filter capacitor voltage V o are selected as state variables, and the nonlinear state equation of the system is obtained as follows: $$\left\{ {\begin{array}{*{20}{l}} {{L_1}\frac{{{\operatorname{d} ^\alpha }{i_1}}}{{\operatorname{d} {t^\alpha }}}+{u_1}+{L_m}\frac{{{\operatorname{d} ^\alpha }{i_m}}}{{\operatorname{d} {t^\alpha }}}={V_{AB}}} \\ {{C_1}\frac{{{\operatorname{d} ^\beta }{u_1}}}{{\operatorname{d} {t^\beta }}}={i_1}} \\ {{L_m}\frac{{{\operatorname{d} ^\alpha }{i_m}}}{{\operatorname{d} {t^\alpha }}}=n\left[ {{L_2}\frac{{{\operatorname{d} ^\alpha }n\left( {{i_1} - {i_m}} \right)}}{{\operatorname{d} {t^\alpha }}}+{u_2}+{V_o}\operatorname{sgn} \left( {{i_1} - {i_m}} \right)} \right]} \\ {{C_2}\frac{{{\operatorname{d} ^\beta }{u_2}}}{{\operatorname{d} {t^\beta }}}={i_2}=n\left( {{i_1} - {i_m}} \right)} \\ {{C_o}\frac{{{\operatorname{d} ^\beta }{V_o}}}{{\operatorname{d} {t^\beta }}}+\frac{{{V_o}}}{{{R_L}}}=n|{i_1} - {i_m}|} \end{array}} \right.$$ 9 Where U AB is the square wave voltage generated by the transformer primary input voltage V i through the inverter bridge, which is the input voltage of the resonant network, in which the nonlinear part is U AB , sgn ( i 1 - i m ), | i 1 - i m |, and 0 < α , β < 1. It can be seen that the above formula is complex, which is not conducive to subsequent analysis and calculation. By calculating the separated variables, only one differential term is retained in each equation, and the coefficient is 1. Finally, the above formula is simplified to formula (10): $$\left\{ {\begin{array}{*{20}{l}} {\frac{{{\operatorname{d} ^\alpha }{i_1}}}{{\operatorname{d} {t^\alpha }}}= - \frac{{{L_m}+{L_1}}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_1} - \frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_2} - \frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_o}\operatorname{sgn} \left( {{i_2}} \right)+\frac{{{L_m}+{L_1}}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_{AB}}} \\ {\frac{{{\operatorname{d} ^\alpha }{i_2}}}{{\operatorname{d} {t^\alpha }}}= - \frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_1} - \frac{{{n^2}\left( {{L_m}+{L_1}} \right)}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_2} - \frac{{{n^2}\left( {{L_m}+{L_1}} \right)}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_o}\operatorname{sgn} \left( {{i_2}} \right)+\frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_{AB}}} \\ {{C_1}\frac{{{\operatorname{d} ^\beta }{u_1}}}{{\operatorname{d} {t^\beta }}}={i_1}} \\ {{C_2}\frac{{{\operatorname{d} ^\beta }{u_2}}}{{\operatorname{d} {t^\beta }}}={i_2}} \\ {{C_o}\frac{{{\operatorname{d} ^\beta }{V_o}}}{{\operatorname{d} {t^\beta }}}+\frac{{{V_o}}}{{{R_L}}}=n|{i_2}|} \end{array}} \right.$$ 10 According to the extended description function method in reference [20], the above state variables can be Fourier decomposed through harmonic approximation, and the sine and cosine components in the fundamental wave can be retained, as shown in the following formula: $$\left\{ {\begin{array}{*{20}{l}} {{i_1}(t)={i_{1\operatorname{s} }}\sin \left( {{\omega _n}t} \right)+{i_{1c}}\cos \left( {{\omega _n}t} \right)} \\ {{i_2}(t)={i_{2s}}\sin \left( {{\omega _n}t} \right)+{i_{2c}}\cos \left( {{\omega _n}t} \right)} \\ {{u_1}(t)={u_{1s}}\sin \left( {{\omega _n}t} \right)+{u_{1c}}\cos \left( {{\omega _n}t} \right)} \\ {{u_2}(t)={u_{2s}}\sin \left( {{\omega _n}t} \right)+{u_{2c}}\cos \left( {{\omega _n}t} \right)} \end{array}} \right.$$ 11 According to the definition of fractional derivative, the fractional derivative satisfies Leibniz's law [ 24 ] . Therefore, both sides of Eq. ( 11 ) are differentiated at the same time to obtain α -order and β -order differential forms, where i 1 and i 2 correspond to the resonant inductors L 1 and L 2 in the system, u 1 and u 2 correspond to the resonant capacitors C 1 and C 2 in the system, and 0 < α , β < 1. obtain Eq. ( 12 ). $$\left\{ {\begin{array}{*{20}{l}} {\frac{{{\operatorname{d} ^\alpha }{i_1}}}{{\operatorname{d} {t^\alpha }}}=\frac{{{\operatorname{d} ^\alpha }{i_{1s}}}}{{\operatorname{d} {t^\alpha }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\alpha }{i_{1s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\alpha }{i_{1c}}}}{{\operatorname{d} {t^\alpha }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\alpha }{i_{1c}}\sin \left( {{\omega _n}t} \right)} \\ {\frac{{{\operatorname{d} ^\alpha }{i_1}}}{{\operatorname{d} {t^\alpha }}}=\frac{{{\operatorname{d} ^\alpha }{i_{2s}}}}{{\operatorname{d} {t^\alpha }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\alpha }{i_{2s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\alpha }{i_{2c}}}}{{\operatorname{d} {t^\alpha }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\alpha }{i_{2c}}\sin \left( {{\omega _n}t} \right)} \\ {\frac{{{\operatorname{d} ^\beta }{u_1}}}{{\operatorname{d} {t^\alpha }}}=\frac{{{\operatorname{d} ^\beta }{u_{1s}}}}{{\operatorname{d} {t^\alpha }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\beta }{u_{1s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\beta }{u_{1c}}}}{{\operatorname{d} {t^\beta }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\beta }{u_{1c}}\sin \left( {{\omega _n}t} \right)} \\ {\frac{{{\operatorname{d} ^\beta }{u_2}}}{{\operatorname{d} {t^\beta }}}=\frac{{{\operatorname{d} ^\beta }{u_{2s}}}}{{\operatorname{d} {t^\beta }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\beta }{u_{2s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\beta }{u_{2c}}}}{{\operatorname{d} {t^\beta }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\beta }{u_{2c}}\sin \left( {{\omega _n}t} \right)} \end{array}} \right.$$ 12 The nonlinear part of the system state equation is approximated as the superposition of sine component and cosine component, as follows: $$\left\{ {\begin{array}{*{20}{l}} {{V_{AB}}=\frac{{4 \cdot {V_i}}}{\pi }\sin (\pi \cdot d)=\frac{{4 \cdot {V_i}}}{\pi }\sin \left( {{\omega _n}t} \right)} \\ \begin{gathered} \operatorname{sgn} \left( {{i_1} - {i_m}} \right)=\frac{1}{n}\operatorname{sgn} \left( {{i_2}} \right) \hfill \\ =\frac{4}{\pi }\frac{{{i_{2s}}}}{{{i_p}}}\sin \left( {{\omega _n}t} \right)+\frac{4}{\pi }\frac{{{i_{2c}}}}{{{i_p}}}\cos \left( {{\omega _n}t} \right) \hfill \\ \end{gathered} \\ {n\left| {{i_1} - {i_m}} \right|=\left| {{i_2}} \right|=\frac{2}{\pi } \cdot {i_p}} \\ {{i_p}=\sqrt {{{\left( {{i_{2s}}} \right)}^2}+{{\left( {{i_{2c}}} \right)}^2}} } \end{array}} \right.$$ 13 Bring equations ( 11 ), ( 12 ) and (13) into Eq. ( 10 ) to obtain Eq. ( 14 ): $$\left\{ {\begin{array}{*{20}{c}} \begin{gathered} \frac{{{\operatorname{d} ^\alpha }{i_{1s}}}}{{\operatorname{d} {t^\alpha }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\alpha }{i_{1s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\alpha }{i_{1c}}}}{{\operatorname{d} {t^\alpha }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\alpha }{i_{1c}}\sin \left( {{\omega _n}t} \right)= - {L_{e1}}\left[ {{u_{1s}}\sin \left( {{\omega _n}t} \right)+{u_{1c}}\cos \left( {{\omega _n}t} \right)} \right] \hfill \\ - {L_{{\text{e}}2}}\left[ {{u_{2s}}\sin \left( {{\omega _n}t} \right)+{u_{2c}}\cos \left( {{\omega _n}t} \right)} \right] - {L_{{\text{e}}2}}{V_o}\left[ {\frac{4}{\pi }\frac{{{i_{2s}}}}{{{i_p}}}\sin \left( {{\omega _n}t} \right)+\frac{4}{\pi }\frac{{{i_{2c}}}}{{{i_p}}}\cos \left( {{\omega _n}t} \right)} \right]+{L_{{\text{e}}1}}\frac{{4 \cdot {V_i}}}{\pi }\sin \left( {{\omega _n}t} \right) \hfill \\ \end{gathered} \\ \begin{gathered} \frac{{{\operatorname{d} ^\alpha }{i_{2s}}}}{{\operatorname{d} {t^\alpha }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\alpha }{i_{2s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\alpha }{i_{2c}}}}{{\operatorname{d} {t^\alpha }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\alpha }{i_{2c}}\sin \left( {{\omega _n}t} \right)= - {L_{{\text{e}}2}}\left[ {{u_{1s}}\sin \left( {{\omega _n}t} \right)+{u_{1c}}\cos \left( {{\omega _n}t} \right)} \right] \hfill \\ - {L_{{\text{e}}3}}\left[ {{u_{2s}}\sin \left( {{\omega _n}t} \right)+{u_{2c}}\cos \left( {{\omega _n}t} \right)} \right] - {L_{{\text{e}}3}}{V_o}\left[ {\frac{4}{\pi }\frac{{{i_{2s}}}}{{{i_p}}}\sin \left( {{\omega _n}t} \right)+\frac{4}{\pi }\frac{{{i_{2c}}}}{{{i_p}}}\cos \left( {{\omega _n}t} \right)} \right]+{L_{{\text{e}}2}}\frac{{4 \cdot {V_i}}}{\pi }\sin \left( {{\omega _n}t} \right) \hfill \\ \end{gathered} \\ {{C_1}\left[ {\frac{{{\operatorname{d} ^\beta }{u_{1s}}}}{{\operatorname{d} {t^\beta }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\beta }{u_{1s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\beta }{u_{1c}}}}{{\operatorname{d} {t^\beta }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\beta }{u_{1c}}\sin \left( {{\omega _n}t} \right)} \right]={i_{1s}}\sin \left( {{\omega _n}t} \right)+{i_{1c}}\cos \left( {{\omega _n}t} \right)} \\ {{C_2}\left[ {\frac{{{\operatorname{d} ^\beta }{u_{2s}}}}{{\operatorname{d} {t^\beta }}}\sin \left( {{\omega _n}t} \right)+\omega _{n}^{\beta }{u_{2s}}\cos \left( {{\omega _n}t} \right)+\frac{{{\operatorname{d} ^\beta }{u_{2c}}}}{{\operatorname{d} {t^\beta }}}\cos \left( {{\omega _n}t} \right) - \omega _{n}^{\beta }{u_{2c}}\sin \left( {{\omega _n}t} \right)} \right]={i_{2s}}\sin \left( {{\omega _n}t} \right)+{i_{2c}}\cos \left( {{\omega _n}t} \right)} \\ {\frac{{{\operatorname{d} ^\beta }{V_o}}}{{\operatorname{d} {t^\beta }}}+\frac{{{V_o}}}{{C{R_L}}}=\frac{2}{{C \cdot \pi }}{i_p}} \end{array}} \right.$$ 14 In the above formula, L e 1 , L e 2 and L e 3 are as follows: \({L_{{\text{e}}1}}=\frac{{{L_m}+{L_1}}}{{2{L_1}{L_m}+L_{1}^{2}}}\) ,, \({L_{{\text{e}}2}}=\frac{{n{L_m}}}{{2{L_1}{L_m}+L_{1}^{2}}}\) , \({L_{{\text{e}}3}}=\frac{{{n^2}\left( {{L_m}+{L_1}} \right)}}{{2{L_1}{L_m}+L_{1}^{2}}}\) (15) According to the principle of harmonic balance, the coefficients of sine component and cosine component in the above formula are equal (The amplitude is equal), and the large signal model of the system is analyzed and sorted out: $$\left\{ {\begin{array}{*{20}{l}} {\frac{{{\operatorname{d} ^\alpha }{i_{1s}}}}{{\operatorname{d} {t^\alpha }}}= - {L_{{\text{eq}}1}}{u_{1s}} - {L_{{\text{eq}}2}}{u_{2s}} - {L_{{\text{eq}}2}}{V_o}\frac{4}{\pi }\frac{{{i_{2s}}}}{{{i_p}}}+{L_{{\text{eq}}1}}\frac{{4 \cdot {V_i}}}{\pi }+\omega _{n}^{\alpha }{i_{1c}}} \\ {\frac{{{\operatorname{d} ^\alpha }{i_{1c}}}}{{\operatorname{d} {t^\alpha }}}= - {L_{{\text{eq}}1}}{u_{1c}} - {L_{{\text{eq}}2}}{u_{2c}} - {L_{{\text{eq}}2}}{V_o}\frac{4}{\pi }\frac{{{i_{2c}}}}{{{i_p}}} - \omega _{n}^{\alpha }{i_{1s}}} \\ {\frac{{{\operatorname{d} ^\alpha }{i_{2s}}}}{{\operatorname{d} {t^\alpha }}}= - {L_{{\text{eq}}2}}{u_{1s}} - {L_{{\text{eq}}3}}{u_{2s}} - {L_{{\text{eq}}3}}{V_o}\frac{4}{\pi }\frac{{{i_{2s}}}}{{{i_p}}}+{L_{{\text{eq}}2}}\frac{{4 \cdot {V_i}}}{\pi }+\omega _{n}^{\alpha }{i_{2c}}} \\ {\frac{{{\operatorname{d} ^\alpha }{i_{2c}}}}{{\operatorname{d} {t^\alpha }}}= - {L_{{\text{eq}}2}}{u_{1c}} - {L_{{\text{eq}}3}}{u_{2c}} - {L_{{\text{eq}}3}}{V_o}\frac{4}{\pi }\frac{{{i_{2c}}}}{{{i_p}}} - \omega _{n}^{\alpha }{i_{2s}}} \\ {\frac{{{\operatorname{d} ^\beta }{u_{1s}}}}{{\operatorname{d} {t^\beta }}}=\frac{{{i_{1s}}}}{{{C_1}}}+\omega _{n}^{\beta }{u_{1c}},\frac{{{\operatorname{d} ^\beta }{u_{1c}}}}{{\operatorname{d} {t^\beta }}}=\frac{{{i_{1c}}}}{{{C_1}}} - \omega _{n}^{\beta }{u_{1s}}} \\ {\frac{{{\operatorname{d} ^\beta }{u_{2s}}}}{{\operatorname{d} {t^\beta }}}=\frac{{{i_{2s}}}}{{{C_2}}}+\omega _{n}^{\beta }{u_{2c}},\frac{{{\operatorname{d} ^\beta }{u_{2c}}}}{{\operatorname{d} {t^\beta }}}=\frac{{{i_{2c}}}}{{{C_2}}} - \omega _{n}^{\beta }{u_{2s}}} \\ {\frac{{{\operatorname{d} ^\beta }{V_o}}}{{\operatorname{d} {t^\beta }}}=\frac{2}{{C\pi }}{i_p} - \frac{{{V_o}}}{{C{R_L}}}} \end{array}} \right.$$ 16 The extended description function method is used to approximate treatment the nonlinear part, and the large signal model of the system is obtained According to the definition of Caputo fractional derivative, any fractional derivative of a constant is equal to zero [ 25 ] . When the system operates at the steady-state operating point, the state variables in the model can be regarded as stable, that is, the left side of Eq. ( 16 ) is 0, with A st · X st + B st = 0. Among: $$A=\left[ {\begin{array}{*{20}{c}} 0&{\omega _{n}^{\alpha }}&{ - {L_{e2}}\frac{8}{{{\pi ^2}}}{R_L}}&0&{ - {L_{e1}}}&0&{ - {L_{e2}}}&0 \\ { - \omega _{n}^{\alpha }}&0&0&{ - {L_{e2}}\frac{8}{{{\pi ^2}}}R}&0&{ - {L_{e1}}}&0&{ - {L_{e2}}} \\ 0&0&{ - {L_{e3}}\frac{8}{{{\pi ^2}}}R}&{\omega _{n}^{\alpha }}&{ - {L_{e2}}}&0&{ - {L_{e3}}}&0 \\ 0&0&{ - \omega _{n}^{\alpha }}&{ - {L_{e3}}\frac{8}{{{\pi ^2}}}R}&0&{ - {L_{e2}}}&0&{ - {L_{e3}}} \\ {\frac{1}{{{C_1}}}}&0&0&0&0&{\omega _{n}^{\beta }}&0&0 \\ 0&{\frac{1}{{{C_1}}}}&0&0&{ - \omega _{n}^{\beta }}&0&0&0 \\ 0&0&{\frac{1}{{{C_2}}}}&0&0&0&0&{\omega _{n}^{\beta }} \\ 0&0&0&{\frac{1}{{{C_2}}}}&0&0&{ - \omega _{n}^{\beta }}&0 \end{array}} \right]$$ 17 $$B{\text{=}}{\left[ {\begin{array}{*{20}{c}} {{L_{e1}}\frac{{4 \cdot {V_i}}}{\pi }}&o&{{L_{e2}}\frac{{4 \cdot {V_i}}}{\pi }}&0&0&0&0&0 \end{array}} \right]^T}$$ 18 $${X_{st}}={\left[ {\begin{array}{*{20}{c}} {{I_{1s}}}&{{I_{1c}}}&{{I_{2s}}}&{{I_{2c}}}&{{U_{1s}}}&{{U_{1c}}}&{{U_{2s}}}&{{U_{2c}}} \end{array}} \right]^T}$$ 19 It can be seen that when the system operates at a certain stable operating point, the steady-state solution X st of the system can be obtained by bringing in the input voltage V i , load resistance R L and switching frequency f s : $${X_{st}}= - {\left( {{A_{st}}} \right)^{ - 1}}{B_{st}}$$ 20 Finally, small signal disturbance is added to the steady-state operating point of the system. Through disturbance separation, it is divided into DC component and AC component, that is, each state variable can be expressed as the superposition of steady-state DC and AC small signal disturbance, as shown below: $$x={X_{st}}{\text{+}}\hat {x}$$ 21 $$\hat {x}={\left[ {\begin{array}{*{20}{c}} {{{\hat {i}}_{1s}}}&{{{\hat {i}}_{1c}}}&{{{\hat {i}}_{2s}}}&{{{\hat {i}}_{2c}}}&{{{\hat {u}}_{1s}}}&{{{\hat {u}}_{1c}}}&{{{\hat {u}}_{2s}}}&{{{\hat {u}}_{2c}}}&{{{\hat {v}}_o}} \end{array}} \right]^T}$$ 22 Bring Eq. ( 22 ) into Eq. ( 16 ) and ignore the second-order disturbance in the equation, then the small signal model of CLLC bidirectional resonant converter is as follows: $$\left\{ {\begin{array}{*{20}{l}} {\frac{{{\operatorname{d} ^{\alpha ,\beta }}\hat {x}}}{{\operatorname{d} {t^{\alpha ,\beta }}}}=A\hat {x}+B\hat {u}} \\ {\hat {y}=C\hat {x}} \end{array}} \right.$$ 23 Where A and B are the coefficient matrix composed of the coefficients of each state variable. Through calculation and derivation, the transfer function of output voltage disturbance and system switching frequency disturbance can be obtained, as follows: $$G(s)=\frac{{{{\hat {v}}_o}(s)}}{{{{\hat {f}}_s}(s)}}=C{(S - A)^{ - 1}}B$$ 24 Matrix A, B, C and S are shown in formula (25–28): $$A={\left( {{a_{ij}}} \right)_{9 \times 9}}=\left\{ {\begin{array}{*{20}{l}} {{a_{12}}={{\left( {2\pi {f_n}} \right)}^\alpha };{a_{13}}= - 4{{\left( {{I_{{2_c}}}} \right)}^2}{V_o}{L_{e2}}/\pi {{\left( {{I_p}} \right)}^3};{a_{14}}=4{I_{{2_s}}}{I_{{2_c}}}{V_o}{L_{e2}}/\pi {{\left( {{I_p}} \right)}^3};{a_{15}}= - {L_{e1}}} \\ {{a_{17}}= - {L_{e2}};{a_{19}}= - 4{I_{2s}}{L_{e2}}/\pi {I_p};{a_{21}}= - {{\left( {2\pi {f_n}} \right)}^\alpha };{a_{23}}=4{I_{{2_s}}}{I_{2c}}{V_o}{L_{e2}}/\pi {{\left( {{I_p}} \right)}^3};} \\ {{a_{24}}= - 4{{\left( {{I_{{2_s}}}} \right)}^2}{V_o}{L_{e2}}/\pi {{\left( {{I_p}} \right)}^3};{a_{26}}= - {L_{e1}};{a_{28}}= - {L_{e2}};{a_{29}}= - 4{I_{2c}}{L_{e2}}/\pi {I_p};} \\ {{a_{33}}= - 4{{\left( {{I_{{2_c}}}} \right)}^2}{U_o}{L_{e3}}/\pi {{\left( {{I_p}} \right)}^3};{a_{34}}={{\left( {2\pi {f_n}} \right)}^\alpha }+4{I_{{2_s}}}{I_{{2_c}}}{V_o}{L_{e3}}/\pi {{\left( {{I_p}} \right)}^3};{a_{35}}= - {L_{e2}};} \\ {{a_{37}}= - {L_{e3}};{a_{39}}= - 4{I_{2s}}{L_{e3}}/\pi {I_p};{a_{43}}= - {{\left( {2\pi {f_n}} \right)}^\alpha }+4{I_{{2_s}}}{I_{{2_c}}}{V_o}{L_{e3}}/\pi {{\left( {{I_p}} \right)}^3};} \\ {{a_{44}}= - 4{{\left( {{I_{{2_s}}}} \right)}^2}{V_o}{L_{e3}}/\pi {{\left( {{I_p}} \right)}^3};{a_{46}}= - {L_{e2}};{a_{48}}= - {L_{e3}};{a_{49}}= - 4{I_{2c}}{L_{e3}}/\pi {I_p};} \\ {{a_{51}}=1/{C_1};{a_{56}}={{\left( {2\pi {f_n}} \right)}^\beta };{a_{62}}=1/{C_1};{a_{65}}= - {{\left( {2\pi {f_n}} \right)}^\beta };{a_{73}}=1/{C_2};{a_{78}}={{\left( {2\pi {f_n}} \right)}^\beta };{a_{84}}=1/{C_2};} \\ {{a_{87}}= - {{\left( {2\pi {f_n}} \right)}^\beta };{a_{93}}=2{I_{{2_s}}}/{I_p}\pi {C_o};{a_{94}}=2{I_{{2_c}}}/{I_p}\pi {C_o};{a_{99}}= - 1/{R_L}{C_o};} \end{array}} \right.$$ 25 $$B={\left[ {\begin{array}{*{20}{c}} {{{\left( {2\pi {f_n}} \right)}^\alpha }{I_{1c}}}&{ - {{\left( {2\pi {f_n}} \right)}^\alpha }{I_{1s}}}&{{{\left( {2\pi {f_n}} \right)}^\alpha }{I_{2c}}}&{ - {{\left( {2\pi {f_n}} \right)}^\alpha }{I_{2s}}}&{{{\left( {2\pi {f_n}} \right)}^\beta }{U_{1c}}}&{ - {{\left( {2\pi {f_n}} \right)}^\beta }{U_{1s}}}&{{{\left( {2\pi {f_n}} \right)}^\beta }{U_{2c}}}&{ - {{\left( {2\pi {f_n}} \right)}^\beta }{U_{2s}}}&0 \end{array}} \right]^T}$$ 26 $$C=\left[ {\begin{array}{*{20}{c}} 0&0&0&0&0&0&0&0&1 \end{array}} \right]$$ 27 $$S=\left( {{s_{ij}}} \right)=\left\{ {\begin{array}{*{20}{c}} {{s_{11}}={s^\alpha },{s_{22}}={s^\alpha },{s_{33}}={s^\alpha },} \\ {{s_{44}}={s^\alpha },{s_{55}}={s^\beta },{s_{66}}={s^\beta },} \\ {{s_{77}}={s^\beta },{s_{88}}={s^\beta },{s_{99}}={s^\beta }} \end{array}} \right.$$ 28 The non-zero elements in matrix A are shown in Eq. ( 25 ), and matrix B is the sine and cosine component amplitudes of each circuit state u 1 , u 2 , i 1 , i 2 . Each non-zero element in matrix S is specifically expressed as formula (27). The matrix is the fractional order power corresponding to each state variable, and 0 < α , β < 1。 I p is the sine and cosine composite amplitude of i 2 , and U o is the output voltage. It can be seen that in Eq. ( 23 – 27 ), the system transfer function is not only related to various parameters in the circuit, but also strongly related to fractional order α and β . when α = β = 1, the result of the above equation is consistent with the integer order system transfer function described in reference [16]. The parameters selected in this paper are V i = 400V, L 1 = 20µH, L 2 = 20µH, C 1 = 140nF, C 2 = 140nF, L m =205µH, f 1 = 94kHz, C = 630µF, R L = 26.6Ω. The system frequency is equal to the first resonant frequency point, i.e. f n = f 1 . Bring various parameters into the above equation to obtain Eq. ( 29 ): $$G\left( s \right)=\frac{\begin{gathered} {a_1}{s^{3\alpha +4\beta }}+{a_2}{s^{2\alpha +4\beta }}+{a_3}{s^{3\alpha +3\beta }}+{a_4}{s^{2\alpha +3\beta }}+{a_5}{s^{3\alpha +2\beta }}+{a_6}{s^{\alpha +4\beta }}+{a_7}{s^{2\alpha +2\beta }}+{a_8}{s^{\alpha +3\beta }}+{a_9}{s^{3\alpha +\beta }}+{a_{10}}{s^{4\beta }} \hfill \\ +{a_{11}}{s^{\alpha +2\beta }}+{a_{12}}{s^{2\alpha +\beta }}+{a_{13}}{s^{3\alpha }}+{a_{14}}{s^{3\beta }}+{a_{15}}{s^{\alpha +\beta }}+{a_{16}}{s^{2\alpha }}+{a_{17}}{s^{2\beta }}+{a_{18}}{s^\alpha }+{a_{19}}{s^\beta }+{a_{20}} \hfill \\ \end{gathered} }{\begin{gathered} {b_1}{s^{4\alpha +5\beta }}+{b_2}{s^{4\alpha +4\beta }}+{b_3}{s^{3\alpha +5\beta }}+{b_4}{s^{2\alpha +5\beta }}+{b_5}{s^{4\alpha +3\beta }}+{b_6}{s^{3\alpha +4\beta }}+{b_7}{s^{2\alpha +4\beta }}+{b_8}{s^{4\alpha +2\beta }}+{b_9}{s^{3\alpha +3\beta }}+{b_{10}}{s^{\alpha +5\beta }} \hfill \\ +{b_{11}}{s^{2\alpha +3\beta }}+{b_{12}}{s^{3\alpha +2\beta }}+{b_{13}}{s^{\alpha +4\beta }}+{b_{14}}{s^{4\alpha +\beta }}+{b_{15}}{s^{5\beta }}+{b_{16}}{s^{2\alpha +2\beta }}+{b_{17}}{s^{\alpha +3\beta }}+{b_{18}}{s^{3\alpha +\beta }}+{b_{19}}{s^{4\beta }}+{b_{20}}{s^{\alpha +2\beta }} \hfill \\ +{b_{21}}{s^{2\alpha +\beta }}+{b_{22}}{s^{3\alpha }}+{b_{23}}{s^{3\beta }}+{b_{24}}{s^{\alpha +\beta }}+{b_{25}}{s^{2\alpha }}+{b_{26}}{s^{2\beta }}+{b_{27}}{s^\alpha }+{b_{28}}{s^\beta }+{b_{29}} \hfill \\ \end{gathered} }$$ 29 Since the transfer function obtained is extremely complex and related to fractional order α and β , the coefficient is defined as follows: $$\left\{ {\begin{array}{*{20}{c}} {{a_i}={f_i}\left( {\alpha ,\beta } \right){\text{ }}\left( {i=1,2....20} \right)} \\ {{b_j}={g_j}\left( {\alpha ,\beta } \right){\text{ }}\left( {j=1,2....30} \right)} \end{array}} \right.$$ 30 When α = β = 0.8, the system transfer function with both fractional capacitance and fractional inductance of 0.8 can be obtained, as shown in Eq. ( 31 ): $$G\left( s \right)=\frac{\begin{gathered} 4.36 \times {10^{90}}{s^{5.6}} - 1.762 \times {10^{115}}{s^{4.8}} - 1.453 \times {10^{123}}{s^4}+ \hfill \\ 6.197 \times {10^{127}}{s^{3.2}} - 5.384 \times {10^{133}}{s^{2.4}}+6.4 \times {10^{137}}{s^{1.6}} - 1.18 \times {10^{142}}{s^{0.8}} \hfill \\ \end{gathered} }{\begin{gathered} 3.499 \times {10^{100}}{s^{7.2}}+2.888 \times {10^{108}}{s^{6.4}} - 9.35 \times {10^{112}}{s^{5.6}}+1.181 \times {10^{120}}{s^{4.8}} - 4.197 \times {10^{124}}{s^4} \hfill \\ +5.396 \times {10^{130}}{s^{3.2}} - 6.648 \times {10^{134}}{s^{2.4}}+5.437 \times {10^{140}}{s^{1.6}}+9.467 \times {10^{143}}+1.005 \times {10^{146}} \hfill \\ \end{gathered} }$$ 31 According to the FOTF toolbox designed based on MATLAB in reference [17], the fractional order Bode diagram can be directly drawn, and its implementation process has completed the approximate fitting of the fractional transfer function to the integer order. As shown in Fig. 9 , for the open-loop transfer function Bode diagram when α = β = 1, α = β = 0.9 and α = β = 0.8, it can be clearly seen that the asymptote slope of the fractional order system is no longer an integral multiple of 20dB/dec, and presents a smooth transition curve with the gradual decrease of α and β . However, when the fractional order is equal to 0.8, because there are many passive components in the system, the fractional order is not only reflected in the order of the transfer function, but also significantly affects the coefficients in the transfer function. When the fractional order is 0.8, the Bode diagram of the system changes greatly. Therefore, using fractional order theory modeling, we can get a more accurate mathematical model of the system and get better design results. 5. Fractional Order Pi Control Strategy For Cllc Bi-directional Resonant Converter According to the Fractional order PI λ D µ controller proposed by Professor podlubny in reference [26], the Fractional Order PI λ D µ closed-loop control strategy of CLLC bi-directional resonant converter is constructed. Compared with the classical PID control strategy, it has two more adjustable parameters λ and µ , with a wider adjustable range and better control effect. Its mathematical form is: $${G_C}\left( s \right)={K_P}+\frac{{{K_i}}}{{{s^\lambda }}}+{K_d}{s^\mu }$$ 32 5.1. Double closed loop control strategy based on fractional order PI λ When the CLLC bi-directional resonant converter works, the PFM modulation strategy is used, and the double closed-loop control of current inner loop and voltage outer loop is adopted. The specific control block diagram is shown in Fig. 10 . The PI λ controller is obtained by deleting the differential part in the PID controller. The calculation has the following definitions, α = β = λ = 0.9, H i = H v = 0.1 and H s = 1. In order to meet the requirements of system stability, the fractional mathematical expressions of voltage loop and current loop are obtained by analysis. The open-loop transfer function is shown in Eq. ( 32 ): $$\left\{ {\begin{array}{*{20}{c}} {{G_i}\left( s \right)={H_i}{H_s} \cdot PI_{i}^{\lambda }\left( s \right){G_{if}}\left( s \right)} \\ {{G_v}\left( s \right)={H_v} \cdot PI_{i}^{\lambda }\left( s \right){G_{vf}}\left( s \right){G_i}\left( s \right)} \end{array}} \right.$$ 32 Since the order of G vd ( s ) is fractional, it is impossible to describe the mathematical model of the system with a definite expression. In reference [27], an improved oustaloup filter is proposed to realize the approximate fitting of fractional operator s in the integer order within the frequency band ( ω b , ω h ), and ω b ω h = 1. G vd ( s ) is approximately fitted by MATLAB to obtain its integer order model. See Appendix A. After obtaining the approximate fitted integer order transfer function, the following voltage loop compensation controller is designed through analysis and calculation: $$PI_{v}^{\lambda }\left( s \right)=1.49+\frac{{0.501}}{{{s^{0.9}}}}$$ 33 It can be seen that the dynamic response capability of the system has been significantly improved after the introduction of current loop closed-loop control. As shown in Fig. 11 . 6. Simulation Analysis Of Cllc Bi-directional Resonant Converter Based On Fractional Order Control According to the above analysis, the circuit simulation model of CLLC bi-directional resonant converter is built in Simulink, as shown in Fig. 12 . Its control module adopts fractional PI λ controller in Fomcon toolbox to complete fractional double closed-loop control strategy of bidirectional CLLC bidirectional resonant converter by constructing double closed-loop control of voltage outer loop and current inner loop. The specific simulation parameters are shown in Table 1 . Table 1 Converter parameters. Parameters Value DC input voltage 500V DC output voltage 500V DC output current < 20A Transformer ratio K 1/1 Switching frequency f s 90-100kHz Load resistance R 26.6Ω Resonant inductors L 1 and L 2 20µH Resonant capacitor C 1 and C 2 140µF Excitation inductance L m 205µH Output filter capacitor C 630µF When the given voltage V ref = 500V, the performance of the converter using fractional PI λ control strategy is compared with the classical PI control strategy, as shown in Fig. 13 . Under the same given voltage, the PI parameters are the same, and the fractional order λ = 0.9 starts the system. Under fractional order control strategy, the system reaches steady state at t = 0.6ms, with peak δ = 512.9V and overshoot б % = 2.58%; under classical PI control strategy, the system reaches steady state at t = 0.10ms, with peak δ = 572.9V and overshoot б % = 14.58%. It can be seen that the fractional order PI λ control strategy can make the system get better dynamic response ability, improve the system response speed, greatly reduce the overshoot and reduce the system loss. As shown in Fig. 14 , the system is started at a given voltage of 500V. When t = 0.1s, the output load of the system suddenly changes, and the load r suddenly decreases from 80Ω to 40Ω. It can be seen from the waveform diagram that under the fractional order control strategy, the voltage fluctuation is 3.5V, and after stabilization, the system voltage reaches 496.5V, with a static error of 3.5V; Under the classical PID control strategy, the voltage fluctuation is 13.1V. After stabilization, the system voltage reaches 486.9V, and there is a static error of 13.1V. It can be seen that the fractional order control strategy has better robustness. As shown in Fig. 15 , the voltage waveforms and driving waveforms at both ends of the switch tube at the primary side of the transformer are shown respectively. It can be seen that under the fractional order control strategy, when the voltage at both ends of the transformer's primary side switch drops to zero, the switch receives the driving signal and turns on, realizing ZVS. It is proved that the fractional order control strategy can achieve good soft switching characteristics. Figure 15 Soft switching waveform(a)Driving voltage and voltage at both ends of switch tube;(b) Partial enlarged drawing. 7. Experimental Verification In order to verify the effectiveness and correctness of the above theoretical analysis, a 3kW CLLC bidirectional resonant converter prototype is built in the laboratory for experimental verification, as shown in Fig. 16 . The prototype adopts the mixed modulation strategy of frequency conversion and phase shift, which realizes the step-up by changing the working frequency of the system and the step-down by changing the phase shift angle [ 28 ][ 29 ] . The control chip adopts DSP28335 of TI company. A fractional order PI λ control strategy applied to DSP is designed to compare with the classical PI control strategy to verify that the fractional order control strategy has better dynamic response ability, stronger robustness and soft switching characteristics. Since the forward and directional operating waveforms of the converter are consistent, only the forward operating mode of the system is analyzed. The specific parameters of the prototype are as follows: Table 2 Converter parameters. Parameters Value DC input voltage 400-450V DC output voltage 400V DC output current < 6.5A Transformer ratio K 1/1 Switching frequency f s 80-100kHz Resonant inductors L 1 and L 2 20µH Resonant capacitor C 1 and C 2 140µF Excitation inductance L m 205µH Output filter capacitor C 630µF As shown in Fig. 16 , the system starts under the working condition of input voltage 400V and given voltage 400V. It can be seen that the rise time of the classical PI controller is 145ms, the peak voltage δ = 446V, and the overshoot б % = 11.15%. After reaching the steady state, the system output voltage fluctuates greatly. The rise time of fractional order PI λ controller is 125ms, the peak voltage δ = 404v, and the overshoot б % = 1%. After reaching the steady state, the system output voltage fluctuates slightly. It can be seen that the fractional order PI λ control strategy has faster rise time, lower overshoot and smaller voltage fluctuation than the classical PI control strategy. As shown in Fig. 17 , the system starts under the given voltage of 400V and load resistance of 200Ω. Under the classical PI control, at time t 1 , the output load resistance of the system suddenly decreases from 200Ω to 100Ω. It can be seen that the output current increases from 2A to 4A. At this time, the output voltage fluctuation increases. At time t 2 , the load resistance suddenly increases from 100Ω to 200Ω, and the voltage fluctuation decreases. Under fractional PI λ control, at time t 1 , the output load resistance of the system suddenly decreases from 200Ω to 100Ω. It can be seen that the output current increases from 2A to 4A. At this time, the output voltage fluctuation has no obvious change, and the output current fluctuation is less than that of the classical PI control. As shown in Fig. 18 , when the system is started at a given voltage of 400V, the input voltage increases from 400V to 450V at time t 1 , and the output voltage remains unchanged. At this time, the input voltage is higher than the output voltage in the step-down mode. Using the frequency conversion phase shift hybrid control, it can be seen that the classical PI control voltage waveform fluctuation increases. The voltage fluctuation of fractional order PI λ control increases after t 1 , but it is smaller than that of classical PI control, which proves that fractional order control has stronger anti-interference ability. To sum up, the validity and feasibility of fractional order PI λ control strategy are verified by building the principle prototype of CLLC bi-directional resonant converter in the laboratory. 8. Conclusion This paper based on fractional calculus theory, the fractional mathematical model and fractional PI λ control strategy of CLLC bi-directional resonant converter are established. The following conclusions are obtained through theoretical and simulation analysis: 1. In the mathematical modeling of CLLC resonant converter, the classical integer order modeling method can not accurately describe the system, there are errors and poor flexibility. By introducing the integral operator s α , the integral order inductance and capacitance in the system are extended to fractional order inductance and capacitance, which improves the accuracy of the mathematical model and more accurately describes the actual system. Through mathematical analysis, it is proved that fractional order α and β not only affect the order of the transfer function, but also have a great impact on the coefficients in the transfer function. 2. By designing fractional order PI λ control strategy applied to CLLC bi-directional resonant converter, the voltage and current double closed-loop control of the system is realized. Compared with the classical PI control strategy, the fractional order control strategy has the advantages of better dynamic response, fast rise time, small overshoot, strong robustness and so on. Compared with the classical PI control strategy, the fractional order control strategy has the advantages of better dynamic response, fast rise time, small overshoot, strong robustness and so on. And it can get good soft switching characteristics of the primary switch, and realize zero voltage conduction of the switch. 3. Finally, a prototype is built in the laboratory, and a fractional order PI λ controller is designed to realize the closed-loop control of CLLC bi-directional resonant converter. Compared with the classical PI controller, the correctness and effectiveness of the fractional order modeling and fractional order PI λ control strategy of CLLC bi-directional resonant converter proposed in this paper are verified. The datasets generated during and/or analysed during the current study are not publicly available due [REASON(S) WHY DATA ARE NOT PUBLIC] but are available from the corresponding author on reasonable request. Declarations Acknowledgements Fund Project:State Grid Gansu Electric Power Company Projects (W22KJ2722005). References [1] CHEN Guoping, LIANG Zhifeng, DONG Yu. Analysis and Reflection on the Marketization Construction of Electric Power With Chinese Characteristics Based on Energy Transformation [J]. Proceedings of the CSEE,2020,40(02):369-379(in chinese). [2] C. Tan, Q. Chen, L. Zhang and K. Zhou, "Frequency-Adaptive Repetitive Control for Three-Phase Four-Leg V2G Inverters," in IEEE Transactions on Transportation Electrification, vol. 7, no. 4, pp. 2095-2103, Dec. 2021. [3] C. Tan, Q. Chen, K. Zhou and L. 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Groves, "Small-signal analysis using harmonic balance methods," PESC '91 Record 22nd Annual IEEE Power Electronics Specialists Conference, 1991, pp. 74-79. [24] Podlubny I. Fractional Differential Equations. San Diego: Academic Press,1999. [25]K. Singh, R. Saxena and S. Kumar, "Caputo-Based Fractional Derivative in Fractional Fourier Transform Domain," in IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 3, no. 3, pp. 330-337, Sept. 2013. [26]Podlubny I. Fractional-order systems and PI/sup/spl lambda//D/sup/spl mu//-controllers[J]. IEEE Transactions on automatic control, 1999, 44(1): 208-214. [27]Xue D, Zhao C, Chen Y Q. A modified approximation method of fractional order system[C]//2006 International conference on mechatronics and automation. IEEE, 2006: 1043-1048. [28]WU Hongfei，DING Shun，SUN Kai，et al．Bidirectional soft-switching series-resonant converter with simple PWM control and load-independent voltage-gain characteristics for energy storage system in DC microgrids[J]. IEEE Journal of Emerging and Selected Topics in Power Electronics,2017,5(3):995-1007. [29]YU Zhiyuan, WU Hongfei, HUA Wenmin, et al. A dual-transformer-based LLC resonant converter with phase-shift control for hold-up time compensation application[C]//2018 IEEE Energy Conversion Congress and Exposition (ECCE). Portland: IEEE,2018:5961-5966. Supplementary Files AppendixA.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1918353","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":126822247,"identity":"a3f4cbdb-405c-4b3e-aee5-8cc10756b819","order_by":0,"name":"Di 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22:09:42","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1121759,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1918353/v1/d4a43508-5c87-45f4-af74-390456e94a5e.pdf"},{"id":24850869,"identity":"51a879d7-2444-43d4-b736-ad454cb5eb14","added_by":"auto","created_at":"2022-08-05 21:59:37","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":24570,"visible":true,"origin":"","legend":"","description":"","filename":"AppendixA.docx","url":"https://assets-eu.researchsquare.com/files/rs-1918353/v1/ff087adff9d672027789d8f8.docx"}],"financialInterests":"","formattedTitle":"Research on fractional order modeling and PIλ control strategy of CLLC bi-directional resonant converter","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eWith the rapid development of new energy power generation and its related industries in China, the proportion of new energy continuously increases. Due to the unique randomness, intermittency, regional interconnection of the power grid, and other problems of new energy power generation, China faces serious basic problems of safety and stability such as consumption of renewable energy and power and electric quantity balance of the power grid\u003csup\u003e[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]\u003c/sup\u003e. The V2G (Vehicle to Grid) bi-directional converter system is a key technology in which the new energy vehicles participate in the peak regulation and frequency modulation of the power grid as a large-scale distributed energy storage system. It can realize the mutual transmission of electric energy between the power grid and the new energy vehicles to assist with the peak regulation and frequency modulation of the power grid and consume a large amount of new energy power generation. The V2G system has a structure with two stages, which are the front-stage AC/DC and backward-stage DC/DC circuits\u003csup\u003e[\u003cspan additionalcitationids=\"CR3\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]\u003c/sup\u003e. As a key component of the V2G system, DC/DC converter serves as a bridge in the mutual transmission of electric energy between the power grid and new energy vehicles. By using the CLLC resonant converter, the energy transmission efficiency can be effectively improved and the switching loss can be reduced\u003csup\u003e[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eAt present, many scholars have done a lot of researches on the CLLC resonance circuit. Because the CLLC resonance circuit has the characteristics of soft switching in a wide load range and does not need to output inductance, it has attracted extensive attention. A bi-directional three-level LLC resonant converter was proposed in Reference [6]. A new pulse width and amplitude modulation control method was used to realize the soft switching of all switching tubes and diodes in the circuit and it was verified by experiment. A kind of Extended Phase-shift control for full-bridge CLLC resonant converter was proposed in Reference [7] to realize the maximum conversion efficiency under the condition of light load and it was proved in the experimental prototype. In Reference [8], a new bi-directional multimode CLLC resonant converter was designed, which could not only meet the needs of new energy vehicles with different voltage grades but also increase the voltage gain of the bi-directional converter. Its feasibility was verified through simulation and experiment. In Reference [9], an integrated magnetic integration structure suitable for the high-frequency resonant inductor-transformer of the CLLC bi-directional resonant converter was designed. The integration of magnetic components in the converter was realized only through a magnetic component, and the feasibility and effectiveness were verified by finite element simulation and experimental prototype. In Reference [10], a unified modeling method suitable for all kinds of the CLLC resonant converter topologies was proposed. This method greatly simplifies the calculated quantity in parameter design and makes bi-directional SR (Synchronous Rectification) less complicated. This method has been verified in symmetric full bridge, symmetric half bridge, asymmetric full bridge, and asymmetric half bridge, respectively.\u003c/p\u003e \u003cp\u003eIn the above research and analysis, the analysis, modeling, and control were completed in the integral order system. That is, the capacitance in the CLLC bi-directional resonant converter is an integral order capacitance and the inductance is an integral order inductance. However, in the mathematical study of inductance and capacitance, it is found that the inductance and capacitance of the system are of the fractional order in practical application, so a mathematical model of the fractional order should be established to describe the fractional order inductance and fractional order capacitance in the actual system more accurately\u003csup\u003e[\u003cspan additionalcitationids=\"CR12\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]\u003c/sup\u003e. Among them, Petr\u0026aacute;š designed a Chua\u0026rsquo;s circuit of the fractional order by using fractional order inductance and fractional order capacitance and confirmed the characteristics of the fractional order in the actual system\u003csup\u003e[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]\u003c/sup\u003e. Avischek designed a parallel resonator of tunable fractional order and fractional order filter, which were verified through simulation and experiment, and gave detailed experimental results. It was found that the experimental data and simulation data were relatively consistent\u003csup\u003e[\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eIn conclusion, the previous research on the mathematical modeling and control strategy of the CLLC bi-directional resonant converter is based on the integral order model, while the actual system is a fractional order system and its description is not accurate enough in the classical modeling. This paper will take the CLLC bi-directional resonant converter as the research object and give the fractional order model and small signal model of the fractional order of the CLLC bi-directional resonant converter to obtain the system transfer function. The Bode diagram of the fractional order system and the Bode diagram of integral order will be drawn through Matlab for analysis and comparison, and the closed-loop simulation of the fractional order system of the CLLC bi-directional resonant converter will be completed in Simulink. Finally, the principle prototype will be built in the laboratory to verify the correctness and feasibility of the fractional order modeling and PI\u003csup\u003eλ\u003c/sup\u003e control strategy.\u003c/p\u003e"},{"header":"2. Topology And Fractional Order Equivalent Topology Of Cllc Bi-directional Resonant Converter","content":"\u003cp\u003e \u003c/p\u003e \u003cp\u003eThe topology of CLLC bi-directional resonant converter is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. When working in the forward direction, the converter consists of Q\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;Q\u003csub\u003e4\u003c/sub\u003e to form the transformer primary side inverter bridge circuit, Q\u003csub\u003e5\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;Q\u003csub\u003e8\u003c/sub\u003e to form the transformer secondary side rectifier bridge circuit. Similarly, in the reverse direction, Q\u003csub\u003e5\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;Q\u003csub\u003e8\u003c/sub\u003e to form the transformer primary side inverter bridge circuit, and Q\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;Q\u003csub\u003e4\u003c/sub\u003e to form the transformer secondary side rectifier bridge circuit. D\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;D\u003csub\u003e8\u003c/sub\u003e are the body diodes of MOSFET, \u003cem\u003eC\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eC\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e are the resonant capacitance of primary and secondary sides respectively, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e are the resonant inductance of primary and secondary sides respectively, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e is the excitation inductance, \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e is the circuit input DC voltage, \u003cem\u003eu\u003c/em\u003e\u003csub\u003e\u003cem\u003eab\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eu\u003c/em\u003e\u003csub\u003e\u003cem\u003ecd\u003c/em\u003e\u003c/sub\u003e are the square wave voltage of primary and secondary sides of transformer respectively, \u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e is the output load, \u003cem\u003eC\u003c/em\u003e is the output filter capacitor, \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e is the resonant current of primary side of transformer, \u003cem\u003ei\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e is the resonant current of secondary side of transformer, \u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e is the primary excitation inductance current, \u003cem\u003en\u003c/em\u003e is transformer transformation ratio.\u003c/p\u003e \u003cp\u003eSince the inductors and capacitors in the actual system are fractional inductors and fractional capacitors, the CLLC bi-directional resonant converter can be more accurately described by establishing a fractional mathematical model. The integer inductors and capacitors in the above system topology are replaced by fractional inductors and fractional capacitors, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Where \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e are the actual orders of fractional capacitors and fractional inductors respectively, and the value range is 0\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eα\u003c/em\u003e, \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;1.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBy using fractional calculus theory, the fractional mathematical model of the system is established, and more accurate transfer function is obtained and applied to the subsequent control research of the system.\u003c/p\u003e"},{"header":"3. Cllc Working Modal Analysis","content":"\u003cp\u003eThe forward working modal and reverse working modal of CLLC bi-directional resonant converter are basically the same. When working in one direction, it is similar to LLC resonant converter. According to the different resonant network elements, there will be two resonant frequency points \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e. According to the size of the system operating frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e, the converter is divided into three working modals: under resonant condition, quasi resonant condition and over resonant condition. This paper mainly analyzes the working modals of CLLC bi-directional resonant converter when it works in the forward direction, and the modulation strategy is PFM(pulse frequency modulation) which can obtain higher electrical transmission efficiency\u003csup\u003e[\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eAccording to reference [17], the relationship between the current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e flowing through the fractional inductor and the voltage \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e at both ends is as follows:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${v_L}=L\\frac{{{\\operatorname{d} ^\\alpha }{i_L}}}{{\\operatorname{d} {t^\\alpha }}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cem\u003eα\u003c/em\u003e is the fractional order of inductance \u003cem\u003eL\u003c/em\u003e\u003csup\u003e\u003cem\u003eα\u003c/em\u003e\u003c/sup\u003e, and 0\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eα\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;1.\u003c/p\u003e \u003cp\u003eThe relationship between the voltage \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003eo\u003c/em\u003e\u003c/sub\u003e at both ends of the fractional capacitor and the current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e flowing through the capacitor is as follows:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${i_C}=C\\frac{{{\\operatorname{d} ^\\beta }{v_o}}}{{\\operatorname{d} {t^\\beta }}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cem\u003eβ\u003c/em\u003e Is the fractional order of capacitance \u003cem\u003eC\u003c/em\u003e\u003csup\u003e\u003cem\u003eβ\u003c/em\u003e\u003c/sup\u003e, and 0\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eβ\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;1.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1. f2\u0026thinsp;\u0026lt;\u0026thinsp;fn\u0026thinsp;\u0026lt;\u0026thinsp;f1 Underresonance\u003c/h2\u003e \u003cp\u003eWhen the operating frequency of the CLLC converter is less than the first resonant point, the converter is under resonant condition, and the key waveforms on the primary side of the transformer are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. There are 6 operating modes in the whole switching cycle, and the circuit waveforms in the front and back half cycles are symmetrical. Therefore, only the circuit operating modes in the first half cycle are analyzed in this paper.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eMode 1, [\u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e,\u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e]:\u003c/p\u003e \u003cp\u003eAt time \u003cem\u003et\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e, the switch Q\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;Q\u003csub\u003e4\u003c/sub\u003e is turned on at zero voltage switching(ZVS), the input voltage of the resonance network is equal to the system input voltage \u003cem\u003ev\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e, the primary side resonance current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and the excitation inductance current im rise rapidly, at time \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, the primary side current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e becomes positive, the secondary side diodes D\u003csub\u003e5\u003c/sub\u003e and D\u003csub\u003e8\u003c/sub\u003e are turned on, and the excitation inductance \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e does not participate in the resonance. The equivalent topology of CLLC bi-directional resonant converter operating mode 1 is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWhen all MOSFET in the circuit are ideal devices, the circuit state equation of circuit mode 1 is:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {{L_1}\\frac{{{\\operatorname{d} ^\\alpha }{i_1}}}{{\\operatorname{d} {t^\\alpha }}}+{u_1}+{L_m}\\frac{{{\\operatorname{d} ^\\alpha }{i_m}}}{{\\operatorname{d} {t^\\alpha }}}={V_i}} \\\\ {{C_1}\\frac{{{\\operatorname{d} ^\\beta }{u_1}}}{{\\operatorname{d} {t^\\beta }}}={i_1}} \\\\ {{L_m}\\frac{{{\\operatorname{d} ^\\alpha }{i_m}}}{{\\operatorname{d} {t^\\alpha }}}=n\\left[ {{L_2}\\frac{{{\\operatorname{d} ^\\alpha }n\\left( {{i_1} - {i_m}} \\right)}}{{\\operatorname{d} {t^\\alpha }}}+{u_2}+{U_o}} \\right]} \\\\ {{C_2}\\frac{{{\\operatorname{d} ^\\beta }{u_2}}}{{\\operatorname{d} {t^\\beta }}}={i_2}=n\\left( {{i_1} - {i_m}} \\right)} \\\\ {{C_o}\\frac{{{\\operatorname{d} ^\\beta }{V_o}}}{{\\operatorname{d} {t^\\beta }}}+\\frac{{{V_o}}}{{{R_L}}}={i_2}} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMode 2, [\u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e,\u003cem\u003et\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e]:\u003c/p\u003e \u003cp\u003eIn this mode, the excitation current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e is equal to the primary side current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e. At this time, no current flows through the transformer and the secondary side of the transformer is in the cut-off state. In this mode, the resonant inductor \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, the resonant capacitor \u003cem\u003eC\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and the excitation inductor \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e participate in the resonance, which is a ternary resonance state. The equivalent topology of mode 2 of CLLC bi-directional resonant converter is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe equation of state for mode 2 is as follows:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {{L_1}\\frac{{{\\operatorname{d} ^\\alpha }{i_1}}}{{\\operatorname{d} {t^\\alpha }}}+{u_1}+{L_m}\\frac{{{\\operatorname{d} ^\\alpha }{i_m}}}{{\\operatorname{d} {t^\\alpha }}}={V_i}} \\\\ {{C_1}\\frac{{{\\operatorname{d} ^\\beta }{u_1}}}{{\\operatorname{d} {t^\\beta }}}={i_1}} \\\\ {{i_1}={i_m}} \\\\ {{C_2}\\frac{{{\\operatorname{d} ^\\beta }{u_2}}}{{\\operatorname{d} {t^\\beta }}}=0} \\\\ {{C_o}\\frac{{{\\operatorname{d} ^\\beta }{V_o}}}{{\\operatorname{d} {t^\\beta }}}+\\frac{{{V_o}}}{{{R_L}}}=0} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMode 3, [\u003cem\u003et\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e,\u003cem\u003et\u003c/em\u003e\u003csub\u003e4\u003c/sub\u003e]:\u003c/p\u003e \u003cp\u003eThis mode is about the dead time. At this time, Q\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;Q\u003csub\u003e4\u003c/sub\u003e are all turned off, and the parasitic capacitance of the switch tube begins to discharge. Therefore, the half cycle Q\u003csub\u003e2\u003c/sub\u003e and Q\u003csub\u003e3\u003c/sub\u003e can realize ZVS conduction. The duration of this mode is very short, so the state equation of the system is not considered. The equivalent topology of CLLC resonant converter mode 3 is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2. fn\u0026thinsp;=\u0026thinsp;f1 Quasi-resanent condition\u003c/h2\u003e \u003cp\u003eWhen the system operating frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e is equal to the first resonant frequency, the CLLC bi-directional resonant converter operates in quasi resonant mode. The figure under this working condition is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. It can be seen that under quasi resonant working condition, there will be no ternary resonance in the system.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eWhen the operating frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e is greater than the first resonant frequency point, the system cannot achieve full resonance. At this time, the diode at the secondary side of the transformer cannot achieve ZCS, which increases the switching loss of the converter. Therefore, the converter generally does not work in the over resonant state. This paper will not discuss it more.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Fractional Small Signal Modeling And Analysis Of Cllc Bi-directional Resonant Converter","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e4.1. Extended description function method based on fractional order\u003c/h2\u003e \u003cp\u003eIn the modeling of traditional DC/DC converter, the state space average model of the system is built based on the \"small ripple assumption\"\u003csup\u003e[\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e][\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]\u003c/sup\u003e, but this assumption needs to meet the requirement that the switching frequency of the system is much higher than the disturbance small signal frequency of the circuit system For CLLC bi-directional resonant converter, the switching frequency of the system is very close to the resonant frequency of the resonant cavity in actual operation, so it is impossible to construct an accurate small signal model by using the state space average method.\u003c/p\u003e \u003cp\u003eIn this paper, the extended description function method will be used to construct the small signal model of CLLC bi-directional resonant converter. This method was proposed by Dr. E. X. Yang in 1992. It ignores the high-order harmonics of the system state and greatly simplifies the complexity of nonlinear system modeling. So far, it has been widely used to solve the small signal model of resonant converter. The steps of establishing small signal model for CLLC bi-directional resonant converter by using extended description function method include writing state equation, harmonic approximation, extended description function equation and disturbance separation\u003csup\u003e[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e][\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e][\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eIn order to introduce the fractional calculus theory into the extended description function method, according to the modeling method based on Fourier analysis proposed by J.Groves \u003csup\u003e[\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]\u003c/sup\u003e, the following form is obtained by simplifying the mathematical expression:\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$jM{\\omega _c}{X_M}=\\frac{{\\partial {F_M}}}{{\\partial {X_M}}}{X_M}+\\frac{{\\partial {F_M}}}{{\\partial {U_M}}}{U_M}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAn approximate linear time invariant small signal model is given in reference [22] and its effectiveness is proved. The model is as follows:\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\frac{{d{{\\hat {x}}_0}}}{{dt}}=\\frac{{\\partial F_{0}^{s}}}{{\\partial X_{0}^{s}}}{\\hat {x}_0}+\\frac{{\\partial F_{0}^{s}}}{{\\partial U_{0}^{s}}}\\hat {u}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the nature of fractional calculus, the above formula can be rewritten into fractional order form as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\frac{{{{\\text{d}}^\\alpha }{{\\hat {x}}_0}}}{{\\operatorname{d} {t^\\alpha }}}=\\frac{{{\\partial ^\\alpha }F_{0}^{s}}}{{\\partial {{\\left( {X_{0}^{s}} \\right)}^\\alpha }}}{\\hat {x}_0}+\\frac{{{\\partial ^\\alpha }F_{0}^{s}}}{{\\partial {{\\left( {U_{0}^{s}} \\right)}^\\alpha }}}\\hat {u}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe small signal model of fractional extended description function method is constructed by the above formula.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e4.2. Small signal modeling\u003c/h2\u003e \u003cp\u003eThrough the above modal analysis of CLLC bidirectional resonant converter, the fractional order equivalent model of the system can be obtained, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIt can be seen that the transformer secondary side resonance current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e has the following relationship with the transformer primary side resonance current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and excitation current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$n\\left( {{i_1} - {i_m}} \\right)={i_2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the fractional order equivalent circuit of the CLLC bi-directional resonant converter, the transformer primary side resonant current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, resonant capacitor voltage \u003cem\u003eu\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, excitation inductance current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e, transformer secondary side resonant current \u003cem\u003ei\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, resonant capacitor voltage \u003cem\u003eu\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e and output filter capacitor voltage \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003eo\u003c/em\u003e\u003c/sub\u003e are selected as state variables, and the nonlinear state equation of the system is obtained as follows:\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {{L_1}\\frac{{{\\operatorname{d} ^\\alpha }{i_1}}}{{\\operatorname{d} {t^\\alpha }}}+{u_1}+{L_m}\\frac{{{\\operatorname{d} ^\\alpha }{i_m}}}{{\\operatorname{d} {t^\\alpha }}}={V_{AB}}} \\\\ {{C_1}\\frac{{{\\operatorname{d} ^\\beta }{u_1}}}{{\\operatorname{d} {t^\\beta }}}={i_1}} \\\\ {{L_m}\\frac{{{\\operatorname{d} ^\\alpha }{i_m}}}{{\\operatorname{d} {t^\\alpha }}}=n\\left[ {{L_2}\\frac{{{\\operatorname{d} ^\\alpha }n\\left( {{i_1} - {i_m}} \\right)}}{{\\operatorname{d} {t^\\alpha }}}+{u_2}+{V_o}\\operatorname{sgn} \\left( {{i_1} - {i_m}} \\right)} \\right]} \\\\ {{C_2}\\frac{{{\\operatorname{d} ^\\beta }{u_2}}}{{\\operatorname{d} {t^\\beta }}}={i_2}=n\\left( {{i_1} - {i_m}} \\right)} \\\\ {{C_o}\\frac{{{\\operatorname{d} ^\\beta }{V_o}}}{{\\operatorname{d} {t^\\beta }}}+\\frac{{{V_o}}}{{{R_L}}}=n|{i_1} - {i_m}|} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cem\u003eU\u003c/em\u003e\u003csub\u003e\u003cem\u003eAB\u003c/em\u003e\u003c/sub\u003e is the square wave voltage generated by the transformer primary input voltage \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e through the inverter bridge, which is the input voltage of the resonant network, in which the nonlinear part is \u003cem\u003eU\u003c/em\u003e\u003csub\u003e\u003cem\u003eAB\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003esgn\u003c/em\u003e(\u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e-\u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e), |\u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e-\u003cem\u003ei\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e|, and 0\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eα\u003c/em\u003e, \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;1.\u003c/p\u003e \u003cp\u003eIt can be seen that the above formula is complex, which is not conducive to subsequent analysis and calculation. By calculating the separated variables, only one differential term is retained in each equation, and the coefficient is 1. Finally, the above formula is simplified to formula (10):\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {\\frac{{{\\operatorname{d} ^\\alpha }{i_1}}}{{\\operatorname{d} {t^\\alpha }}}= - \\frac{{{L_m}+{L_1}}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_1} - \\frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_2} - \\frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_o}\\operatorname{sgn} \\left( {{i_2}} \\right)+\\frac{{{L_m}+{L_1}}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_{AB}}} \\\\ {\\frac{{{\\operatorname{d} ^\\alpha }{i_2}}}{{\\operatorname{d} {t^\\alpha }}}= - \\frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_1} - \\frac{{{n^2}\\left( {{L_m}+{L_1}} \\right)}}{{L_{1}^{2}+2{L_1}{L_m}}}{u_2} - \\frac{{{n^2}\\left( {{L_m}+{L_1}} \\right)}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_o}\\operatorname{sgn} \\left( {{i_2}} \\right)+\\frac{{n{L_m}}}{{L_{1}^{2}+2{L_1}{L_m}}}{V_{AB}}} \\\\ {{C_1}\\frac{{{\\operatorname{d} ^\\beta }{u_1}}}{{\\operatorname{d} {t^\\beta }}}={i_1}} \\\\ {{C_2}\\frac{{{\\operatorname{d} ^\\beta }{u_2}}}{{\\operatorname{d} {t^\\beta }}}={i_2}} \\\\ {{C_o}\\frac{{{\\operatorname{d} ^\\beta }{V_o}}}{{\\operatorname{d} {t^\\beta }}}+\\frac{{{V_o}}}{{{R_L}}}=n|{i_2}|} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the extended description function method in reference [20], the above state variables can be Fourier decomposed through harmonic approximation, and the sine and cosine components in the fundamental wave can be retained, as shown in the following formula:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {{i_1}(t)={i_{1\\operatorname{s} }}\\sin \\left( {{\\omega _n}t} \\right)+{i_{1c}}\\cos \\left( {{\\omega _n}t} \\right)} \\\\ {{i_2}(t)={i_{2s}}\\sin \\left( {{\\omega _n}t} \\right)+{i_{2c}}\\cos \\left( {{\\omega _n}t} \\right)} \\\\ {{u_1}(t)={u_{1s}}\\sin \\left( {{\\omega _n}t} \\right)+{u_{1c}}\\cos \\left( {{\\omega _n}t} \\right)} \\\\ {{u_2}(t)={u_{2s}}\\sin \\left( {{\\omega _n}t} \\right)+{u_{2c}}\\cos \\left( {{\\omega _n}t} \\right)} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the definition of fractional derivative, the fractional derivative satisfies Leibniz's law\u003csup\u003e[\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]\u003c/sup\u003e. Therefore, both sides of Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e) are differentiated at the same time to obtain \u003cem\u003eα\u003c/em\u003e-order and \u003cem\u003eβ\u003c/em\u003e-order differential forms, where \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003ei\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e correspond to the resonant inductors \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e in the system, \u003cem\u003eu\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eu\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e correspond to the resonant capacitors \u003cem\u003eC\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eC\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e in the system, and 0\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eα\u003c/em\u003e, \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;1. obtain Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e12\u003c/span\u003e).\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {\\frac{{{\\operatorname{d} ^\\alpha }{i_1}}}{{\\operatorname{d} {t^\\alpha }}}=\\frac{{{\\operatorname{d} ^\\alpha }{i_{1s}}}}{{\\operatorname{d} {t^\\alpha }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\alpha }{i_{1s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\alpha }{i_{1c}}}}{{\\operatorname{d} {t^\\alpha }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\alpha }{i_{1c}}\\sin \\left( {{\\omega _n}t} \\right)} \\\\ {\\frac{{{\\operatorname{d} ^\\alpha }{i_1}}}{{\\operatorname{d} {t^\\alpha }}}=\\frac{{{\\operatorname{d} ^\\alpha }{i_{2s}}}}{{\\operatorname{d} {t^\\alpha }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\alpha }{i_{2s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\alpha }{i_{2c}}}}{{\\operatorname{d} {t^\\alpha }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\alpha }{i_{2c}}\\sin \\left( {{\\omega _n}t} \\right)} \\\\ {\\frac{{{\\operatorname{d} ^\\beta }{u_1}}}{{\\operatorname{d} {t^\\alpha }}}=\\frac{{{\\operatorname{d} ^\\beta }{u_{1s}}}}{{\\operatorname{d} {t^\\alpha }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\beta }{u_{1s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\beta }{u_{1c}}}}{{\\operatorname{d} {t^\\beta }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\beta }{u_{1c}}\\sin \\left( {{\\omega _n}t} \\right)} \\\\ {\\frac{{{\\operatorname{d} ^\\beta }{u_2}}}{{\\operatorname{d} {t^\\beta }}}=\\frac{{{\\operatorname{d} ^\\beta }{u_{2s}}}}{{\\operatorname{d} {t^\\beta }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\beta }{u_{2s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\beta }{u_{2c}}}}{{\\operatorname{d} {t^\\beta }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\beta }{u_{2c}}\\sin \\left( {{\\omega _n}t} \\right)} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe nonlinear part of the system state equation is approximated as the superposition of sine component and cosine component, as follows:\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {{V_{AB}}=\\frac{{4 \\cdot {V_i}}}{\\pi }\\sin (\\pi \\cdot d)=\\frac{{4 \\cdot {V_i}}}{\\pi }\\sin \\left( {{\\omega _n}t} \\right)} \\\\ \\begin{gathered} \\operatorname{sgn} \\left( {{i_1} - {i_m}} \\right)=\\frac{1}{n}\\operatorname{sgn} \\left( {{i_2}} \\right) \\hfill \\\\ =\\frac{4}{\\pi }\\frac{{{i_{2s}}}}{{{i_p}}}\\sin \\left( {{\\omega _n}t} \\right)+\\frac{4}{\\pi }\\frac{{{i_{2c}}}}{{{i_p}}}\\cos \\left( {{\\omega _n}t} \\right) \\hfill \\\\ \\end{gathered} \\\\ {n\\left| {{i_1} - {i_m}} \\right|=\\left| {{i_2}} \\right|=\\frac{2}{\\pi } \\cdot {i_p}} \\\\ {{i_p}=\\sqrt {{{\\left( {{i_{2s}}} \\right)}^2}+{{\\left( {{i_{2c}}} \\right)}^2}} } \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eBring equations (\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e), (\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e12\u003c/span\u003e) and (13) into Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e) to obtain Eq.\u0026nbsp;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e):\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{c}} \\begin{gathered} \\frac{{{\\operatorname{d} ^\\alpha }{i_{1s}}}}{{\\operatorname{d} {t^\\alpha }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\alpha }{i_{1s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\alpha }{i_{1c}}}}{{\\operatorname{d} {t^\\alpha }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\alpha }{i_{1c}}\\sin \\left( {{\\omega _n}t} \\right)= - {L_{e1}}\\left[ {{u_{1s}}\\sin \\left( {{\\omega _n}t} \\right)+{u_{1c}}\\cos \\left( {{\\omega _n}t} \\right)} \\right] \\hfill \\\\ - {L_{{\\text{e}}2}}\\left[ {{u_{2s}}\\sin \\left( {{\\omega _n}t} \\right)+{u_{2c}}\\cos \\left( {{\\omega _n}t} \\right)} \\right] - {L_{{\\text{e}}2}}{V_o}\\left[ {\\frac{4}{\\pi }\\frac{{{i_{2s}}}}{{{i_p}}}\\sin \\left( {{\\omega _n}t} \\right)+\\frac{4}{\\pi }\\frac{{{i_{2c}}}}{{{i_p}}}\\cos \\left( {{\\omega _n}t} \\right)} \\right]+{L_{{\\text{e}}1}}\\frac{{4 \\cdot {V_i}}}{\\pi }\\sin \\left( {{\\omega _n}t} \\right) \\hfill \\\\ \\end{gathered} \\\\ \\begin{gathered} \\frac{{{\\operatorname{d} ^\\alpha }{i_{2s}}}}{{\\operatorname{d} {t^\\alpha }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\alpha }{i_{2s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\alpha }{i_{2c}}}}{{\\operatorname{d} {t^\\alpha }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\alpha }{i_{2c}}\\sin \\left( {{\\omega _n}t} \\right)= - {L_{{\\text{e}}2}}\\left[ {{u_{1s}}\\sin \\left( {{\\omega _n}t} \\right)+{u_{1c}}\\cos \\left( {{\\omega _n}t} \\right)} \\right] \\hfill \\\\ - {L_{{\\text{e}}3}}\\left[ {{u_{2s}}\\sin \\left( {{\\omega _n}t} \\right)+{u_{2c}}\\cos \\left( {{\\omega _n}t} \\right)} \\right] - {L_{{\\text{e}}3}}{V_o}\\left[ {\\frac{4}{\\pi }\\frac{{{i_{2s}}}}{{{i_p}}}\\sin \\left( {{\\omega _n}t} \\right)+\\frac{4}{\\pi }\\frac{{{i_{2c}}}}{{{i_p}}}\\cos \\left( {{\\omega _n}t} \\right)} \\right]+{L_{{\\text{e}}2}}\\frac{{4 \\cdot {V_i}}}{\\pi }\\sin \\left( {{\\omega _n}t} \\right) \\hfill \\\\ \\end{gathered} \\\\ {{C_1}\\left[ {\\frac{{{\\operatorname{d} ^\\beta }{u_{1s}}}}{{\\operatorname{d} {t^\\beta }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\beta }{u_{1s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\beta }{u_{1c}}}}{{\\operatorname{d} {t^\\beta }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\beta }{u_{1c}}\\sin \\left( {{\\omega _n}t} \\right)} \\right]={i_{1s}}\\sin \\left( {{\\omega _n}t} \\right)+{i_{1c}}\\cos \\left( {{\\omega _n}t} \\right)} \\\\ {{C_2}\\left[ {\\frac{{{\\operatorname{d} ^\\beta }{u_{2s}}}}{{\\operatorname{d} {t^\\beta }}}\\sin \\left( {{\\omega _n}t} \\right)+\\omega _{n}^{\\beta }{u_{2s}}\\cos \\left( {{\\omega _n}t} \\right)+\\frac{{{\\operatorname{d} ^\\beta }{u_{2c}}}}{{\\operatorname{d} {t^\\beta }}}\\cos \\left( {{\\omega _n}t} \\right) - \\omega _{n}^{\\beta }{u_{2c}}\\sin \\left( {{\\omega _n}t} \\right)} \\right]={i_{2s}}\\sin \\left( {{\\omega _n}t} \\right)+{i_{2c}}\\cos \\left( {{\\omega _n}t} \\right)} \\\\ {\\frac{{{\\operatorname{d} ^\\beta }{V_o}}}{{\\operatorname{d} {t^\\beta }}}+\\frac{{{V_o}}}{{C{R_L}}}=\\frac{2}{{C \\cdot \\pi }}{i_p}} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the above formula, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e1\u003c/sub\u003e, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e2\u003c/sub\u003e and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e3\u003c/sub\u003e are as follows:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({L_{{\\text{e}}1}}=\\frac{{{L_m}+{L_1}}}{{2{L_1}{L_m}+L_{1}^{2}}}\\)\u003c/span\u003e \u003c/span\u003e,, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L_{{\\text{e}}2}}=\\frac{{n{L_m}}}{{2{L_1}{L_m}+L_{1}^{2}}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L_{{\\text{e}}3}}=\\frac{{{n^2}\\left( {{L_m}+{L_1}} \\right)}}{{2{L_1}{L_m}+L_{1}^{2}}}\\)\u003c/span\u003e\u003c/span\u003e (15)\u003c/p\u003e \u003cp\u003eAccording to the principle of harmonic balance, the coefficients of sine component and cosine component in the above formula are equal (The amplitude is equal), and the large signal model of the system is analyzed and sorted out:\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {\\frac{{{\\operatorname{d} ^\\alpha }{i_{1s}}}}{{\\operatorname{d} {t^\\alpha }}}= - {L_{{\\text{eq}}1}}{u_{1s}} - {L_{{\\text{eq}}2}}{u_{2s}} - {L_{{\\text{eq}}2}}{V_o}\\frac{4}{\\pi }\\frac{{{i_{2s}}}}{{{i_p}}}+{L_{{\\text{eq}}1}}\\frac{{4 \\cdot {V_i}}}{\\pi }+\\omega _{n}^{\\alpha }{i_{1c}}} \\\\ {\\frac{{{\\operatorname{d} ^\\alpha }{i_{1c}}}}{{\\operatorname{d} {t^\\alpha }}}= - {L_{{\\text{eq}}1}}{u_{1c}} - {L_{{\\text{eq}}2}}{u_{2c}} - {L_{{\\text{eq}}2}}{V_o}\\frac{4}{\\pi }\\frac{{{i_{2c}}}}{{{i_p}}} - \\omega _{n}^{\\alpha }{i_{1s}}} \\\\ {\\frac{{{\\operatorname{d} ^\\alpha }{i_{2s}}}}{{\\operatorname{d} {t^\\alpha }}}= - {L_{{\\text{eq}}2}}{u_{1s}} - {L_{{\\text{eq}}3}}{u_{2s}} - {L_{{\\text{eq}}3}}{V_o}\\frac{4}{\\pi }\\frac{{{i_{2s}}}}{{{i_p}}}+{L_{{\\text{eq}}2}}\\frac{{4 \\cdot {V_i}}}{\\pi }+\\omega _{n}^{\\alpha }{i_{2c}}} \\\\ {\\frac{{{\\operatorname{d} ^\\alpha }{i_{2c}}}}{{\\operatorname{d} {t^\\alpha }}}= - {L_{{\\text{eq}}2}}{u_{1c}} - {L_{{\\text{eq}}3}}{u_{2c}} - {L_{{\\text{eq}}3}}{V_o}\\frac{4}{\\pi }\\frac{{{i_{2c}}}}{{{i_p}}} - \\omega _{n}^{\\alpha }{i_{2s}}} \\\\ {\\frac{{{\\operatorname{d} ^\\beta }{u_{1s}}}}{{\\operatorname{d} {t^\\beta }}}=\\frac{{{i_{1s}}}}{{{C_1}}}+\\omega _{n}^{\\beta }{u_{1c}},\\frac{{{\\operatorname{d} ^\\beta }{u_{1c}}}}{{\\operatorname{d} {t^\\beta }}}=\\frac{{{i_{1c}}}}{{{C_1}}} - \\omega _{n}^{\\beta }{u_{1s}}} \\\\ {\\frac{{{\\operatorname{d} ^\\beta }{u_{2s}}}}{{\\operatorname{d} {t^\\beta }}}=\\frac{{{i_{2s}}}}{{{C_2}}}+\\omega _{n}^{\\beta }{u_{2c}},\\frac{{{\\operatorname{d} ^\\beta }{u_{2c}}}}{{\\operatorname{d} {t^\\beta }}}=\\frac{{{i_{2c}}}}{{{C_2}}} - \\omega _{n}^{\\beta }{u_{2s}}} \\\\ {\\frac{{{\\operatorname{d} ^\\beta }{V_o}}}{{\\operatorname{d} {t^\\beta }}}=\\frac{2}{{C\\pi }}{i_p} - \\frac{{{V_o}}}{{C{R_L}}}} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe extended description function method is used to approximate treatment the nonlinear part, and the large signal model of the system is obtained\u003c/p\u003e \u003cp\u003eAccording to the definition of Caputo fractional derivative, any fractional derivative of a constant is equal to zero \u003csup\u003e[\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]\u003c/sup\u003e. When the system operates at the steady-state operating point, the state variables in the model can be regarded as stable, that is, the left side of Eq.\u0026nbsp;(\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e16\u003c/span\u003e) is 0, with \u003cem\u003eA\u003c/em\u003e\u003csub\u003e\u003cem\u003est\u003c/em\u003e\u003c/sub\u003e \u0026middot; \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003est\u003c/em\u003e\u003c/sub\u003e + \u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003est\u003c/em\u003e\u003c/sub\u003e = 0. Among:\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$$A=\\left[ {\\begin{array}{*{20}{c}} 0\u0026amp;{\\omega _{n}^{\\alpha }}\u0026amp;{ - {L_{e2}}\\frac{8}{{{\\pi ^2}}}{R_L}}\u0026amp;0\u0026amp;{ - {L_{e1}}}\u0026amp;0\u0026amp;{ - {L_{e2}}}\u0026amp;0 \\\\ { - \\omega _{n}^{\\alpha }}\u0026amp;0\u0026amp;0\u0026amp;{ - {L_{e2}}\\frac{8}{{{\\pi ^2}}}R}\u0026amp;0\u0026amp;{ - {L_{e1}}}\u0026amp;0\u0026amp;{ - {L_{e2}}} \\\\ 0\u0026amp;0\u0026amp;{ - {L_{e3}}\\frac{8}{{{\\pi ^2}}}R}\u0026amp;{\\omega _{n}^{\\alpha }}\u0026amp;{ - {L_{e2}}}\u0026amp;0\u0026amp;{ - {L_{e3}}}\u0026amp;0 \\\\ 0\u0026amp;0\u0026amp;{ - \\omega _{n}^{\\alpha }}\u0026amp;{ - {L_{e3}}\\frac{8}{{{\\pi ^2}}}R}\u0026amp;0\u0026amp;{ - {L_{e2}}}\u0026amp;0\u0026amp;{ - {L_{e3}}} \\\\ {\\frac{1}{{{C_1}}}}\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;{\\omega _{n}^{\\beta }}\u0026amp;0\u0026amp;0 \\\\ 0\u0026amp;{\\frac{1}{{{C_1}}}}\u0026amp;0\u0026amp;0\u0026amp;{ - \\omega _{n}^{\\beta }}\u0026amp;0\u0026amp;0\u0026amp;0 \\\\ 0\u0026amp;0\u0026amp;{\\frac{1}{{{C_2}}}}\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;{\\omega _{n}^{\\beta }} \\\\ 0\u0026amp;0\u0026amp;0\u0026amp;{\\frac{1}{{{C_2}}}}\u0026amp;0\u0026amp;0\u0026amp;{ - \\omega _{n}^{\\beta }}\u0026amp;0 \\end{array}} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$$B{\\text{=}}{\\left[ {\\begin{array}{*{20}{c}} {{L_{e1}}\\frac{{4 \\cdot {V_i}}}{\\pi }}\u0026amp;o\u0026amp;{{L_{e2}}\\frac{{4 \\cdot {V_i}}}{\\pi }}\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0 \\end{array}} \\right]^T}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$${X_{st}}={\\left[ {\\begin{array}{*{20}{c}} {{I_{1s}}}\u0026amp;{{I_{1c}}}\u0026amp;{{I_{2s}}}\u0026amp;{{I_{2c}}}\u0026amp;{{U_{1s}}}\u0026amp;{{U_{1c}}}\u0026amp;{{U_{2s}}}\u0026amp;{{U_{2c}}} \\end{array}} \\right]^T}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIt can be seen that when the system operates at a certain stable operating point, the steady-state solution \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003est\u003c/em\u003e\u003c/sub\u003e of the system can be obtained by bringing in the input voltage \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e, load resistance \u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e and switching frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e:\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$${X_{st}}= - {\\left( {{A_{st}}} \\right)^{ - 1}}{B_{st}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFinally, small signal disturbance is added to the steady-state operating point of the system. Through disturbance separation, it is divided into DC component and AC component, that is, each state variable can be expressed as the superposition of steady-state DC and AC small signal disturbance, as shown below:\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\n$$x={X_{st}}{\\text{+}}\\hat {x}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e\n$$\\hat {x}={\\left[ {\\begin{array}{*{20}{c}} {{{\\hat {i}}_{1s}}}\u0026amp;{{{\\hat {i}}_{1c}}}\u0026amp;{{{\\hat {i}}_{2s}}}\u0026amp;{{{\\hat {i}}_{2c}}}\u0026amp;{{{\\hat {u}}_{1s}}}\u0026amp;{{{\\hat {u}}_{1c}}}\u0026amp;{{{\\hat {u}}_{2s}}}\u0026amp;{{{\\hat {u}}_{2c}}}\u0026amp;{{{\\hat {v}}_o}} \\end{array}} \\right]^T}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eBring Eq.\u0026nbsp;(\u003cspan refid=\"Equ21\" class=\"InternalRef\"\u003e22\u003c/span\u003e) into Eq.\u0026nbsp;(\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e16\u003c/span\u003e) and ignore the second-order disturbance in the equation, then the small signal model of CLLC bidirectional resonant converter is as follows:\u003cdiv id=\"Equ22\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{l}} {\\frac{{{\\operatorname{d} ^{\\alpha ,\\beta }}\\hat {x}}}{{\\operatorname{d} {t^{\\alpha ,\\beta }}}}=A\\hat {x}+B\\hat {u}} \\\\ {\\hat {y}=C\\hat {x}} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere A and B are the coefficient matrix composed of the coefficients of each state variable. Through calculation and derivation, the transfer function of output voltage disturbance and system switching frequency disturbance can be obtained, as follows:\u003cdiv id=\"Equ23\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ23\" name=\"EquationSource\"\u003e\n$$G(s)=\\frac{{{{\\hat {v}}_o}(s)}}{{{{\\hat {f}}_s}(s)}}=C{(S - A)^{ - 1}}B$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMatrix A, B, C and S are shown in formula (25\u0026ndash;28):\u003cdiv id=\"Equ24\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ24\" name=\"EquationSource\"\u003e\n$$A={\\left( {{a_{ij}}} \\right)_{9 \\times 9}}=\\left\\{ {\\begin{array}{*{20}{l}} {{a_{12}}={{\\left( {2\\pi {f_n}} \\right)}^\\alpha };{a_{13}}= - 4{{\\left( {{I_{{2_c}}}} \\right)}^2}{V_o}{L_{e2}}/\\pi {{\\left( {{I_p}} \\right)}^3};{a_{14}}=4{I_{{2_s}}}{I_{{2_c}}}{V_o}{L_{e2}}/\\pi {{\\left( {{I_p}} \\right)}^3};{a_{15}}= - {L_{e1}}} \\\\ {{a_{17}}= - {L_{e2}};{a_{19}}= - 4{I_{2s}}{L_{e2}}/\\pi {I_p};{a_{21}}= - {{\\left( {2\\pi {f_n}} \\right)}^\\alpha };{a_{23}}=4{I_{{2_s}}}{I_{2c}}{V_o}{L_{e2}}/\\pi {{\\left( {{I_p}} \\right)}^3};} \\\\ {{a_{24}}= - 4{{\\left( {{I_{{2_s}}}} \\right)}^2}{V_o}{L_{e2}}/\\pi {{\\left( {{I_p}} \\right)}^3};{a_{26}}= - {L_{e1}};{a_{28}}= - {L_{e2}};{a_{29}}= - 4{I_{2c}}{L_{e2}}/\\pi {I_p};} \\\\ {{a_{33}}= - 4{{\\left( {{I_{{2_c}}}} \\right)}^2}{U_o}{L_{e3}}/\\pi {{\\left( {{I_p}} \\right)}^3};{a_{34}}={{\\left( {2\\pi {f_n}} \\right)}^\\alpha }+4{I_{{2_s}}}{I_{{2_c}}}{V_o}{L_{e3}}/\\pi {{\\left( {{I_p}} \\right)}^3};{a_{35}}= - {L_{e2}};} \\\\ {{a_{37}}= - {L_{e3}};{a_{39}}= - 4{I_{2s}}{L_{e3}}/\\pi {I_p};{a_{43}}= - {{\\left( {2\\pi {f_n}} \\right)}^\\alpha }+4{I_{{2_s}}}{I_{{2_c}}}{V_o}{L_{e3}}/\\pi {{\\left( {{I_p}} \\right)}^3};} \\\\ {{a_{44}}= - 4{{\\left( {{I_{{2_s}}}} \\right)}^2}{V_o}{L_{e3}}/\\pi {{\\left( {{I_p}} \\right)}^3};{a_{46}}= - {L_{e2}};{a_{48}}= - {L_{e3}};{a_{49}}= - 4{I_{2c}}{L_{e3}}/\\pi {I_p};} \\\\ {{a_{51}}=1/{C_1};{a_{56}}={{\\left( {2\\pi {f_n}} \\right)}^\\beta };{a_{62}}=1/{C_1};{a_{65}}= - {{\\left( {2\\pi {f_n}} \\right)}^\\beta };{a_{73}}=1/{C_2};{a_{78}}={{\\left( {2\\pi {f_n}} \\right)}^\\beta };{a_{84}}=1/{C_2};} \\\\ {{a_{87}}= - {{\\left( {2\\pi {f_n}} \\right)}^\\beta };{a_{93}}=2{I_{{2_s}}}/{I_p}\\pi {C_o};{a_{94}}=2{I_{{2_c}}}/{I_p}\\pi {C_o};{a_{99}}= - 1/{R_L}{C_o};} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ25\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ25\" name=\"EquationSource\"\u003e\n$$B={\\left[ {\\begin{array}{*{20}{c}} {{{\\left( {2\\pi {f_n}} \\right)}^\\alpha }{I_{1c}}}\u0026amp;{ - {{\\left( {2\\pi {f_n}} \\right)}^\\alpha }{I_{1s}}}\u0026amp;{{{\\left( {2\\pi {f_n}} \\right)}^\\alpha }{I_{2c}}}\u0026amp;{ - {{\\left( {2\\pi {f_n}} \\right)}^\\alpha }{I_{2s}}}\u0026amp;{{{\\left( {2\\pi {f_n}} \\right)}^\\beta }{U_{1c}}}\u0026amp;{ - {{\\left( {2\\pi {f_n}} \\right)}^\\beta }{U_{1s}}}\u0026amp;{{{\\left( {2\\pi {f_n}} \\right)}^\\beta }{U_{2c}}}\u0026amp;{ - {{\\left( {2\\pi {f_n}} \\right)}^\\beta }{U_{2s}}}\u0026amp;0 \\end{array}} \\right]^T}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ26\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ26\" name=\"EquationSource\"\u003e\n$$C=\\left[ {\\begin{array}{*{20}{c}} 0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;0\u0026amp;1 \\end{array}} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e27\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ27\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ27\" name=\"EquationSource\"\u003e\n$$S=\\left( {{s_{ij}}} \\right)=\\left\\{ {\\begin{array}{*{20}{c}} {{s_{11}}={s^\\alpha },{s_{22}}={s^\\alpha },{s_{33}}={s^\\alpha },} \\\\ {{s_{44}}={s^\\alpha },{s_{55}}={s^\\beta },{s_{66}}={s^\\beta },} \\\\ {{s_{77}}={s^\\beta },{s_{88}}={s^\\beta },{s_{99}}={s^\\beta }} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e28\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe non-zero elements in matrix A are shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ24\" class=\"InternalRef\"\u003e25\u003c/span\u003e), and matrix B is the sine and cosine component amplitudes of each circuit state \u003cem\u003eu\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003eu\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u003cem\u003ei\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ei\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e. Each non-zero element in matrix S is specifically expressed as formula (27). The matrix is the fractional order power corresponding to each state variable, and 0\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eα\u003c/em\u003e, \u003cem\u003eβ\u003c/em\u003e\u0026thinsp;\u0026lt;\u0026thinsp;1。 \u003cem\u003eI\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e is the sine and cosine composite amplitude of \u003cem\u003ei\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, and \u003cem\u003eU\u003c/em\u003e\u003csub\u003e\u003cem\u003eo\u003c/em\u003e\u003c/sub\u003e is the output voltage.\u003c/p\u003e \u003cp\u003eIt can be seen that in Eq.\u0026nbsp;(\u003cspan refid=\"Equ22\" class=\"InternalRef\"\u003e23\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Equ26\" class=\"InternalRef\"\u003e27\u003c/span\u003e), the system transfer function is not only related to various parameters in the circuit, but also strongly related to fractional order \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e. when \u003cem\u003eα\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, the result of the above equation is consistent with the integer order system transfer function described in reference [16].\u003c/p\u003e \u003cp\u003eThe parameters selected in this paper are \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = 400V, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;20\u0026micro;H, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;20\u0026micro;H, \u003cem\u003eC\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;140nF, \u003cem\u003eC\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;140nF, \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e =205\u0026micro;H, \u003cem\u003ef\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;94kHz, \u003cem\u003eC\u003c/em\u003e\u0026thinsp;=\u0026thinsp;630\u0026micro;F, \u003cem\u003eR\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e = 26.6Ω. The system frequency is equal to the first resonant frequency point, i.e. \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003ef\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e. Bring various parameters into the above equation to obtain Eq.\u0026nbsp;(\u003cspan refid=\"Equ28\" class=\"InternalRef\"\u003e29\u003c/span\u003e):\u003cdiv id=\"Equ28\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ28\" name=\"EquationSource\"\u003e\n$$G\\left( s \\right)=\\frac{\\begin{gathered} {a_1}{s^{3\\alpha +4\\beta }}+{a_2}{s^{2\\alpha +4\\beta }}+{a_3}{s^{3\\alpha +3\\beta }}+{a_4}{s^{2\\alpha +3\\beta }}+{a_5}{s^{3\\alpha +2\\beta }}+{a_6}{s^{\\alpha +4\\beta }}+{a_7}{s^{2\\alpha +2\\beta }}+{a_8}{s^{\\alpha +3\\beta }}+{a_9}{s^{3\\alpha +\\beta }}+{a_{10}}{s^{4\\beta }} \\hfill \\\\ +{a_{11}}{s^{\\alpha +2\\beta }}+{a_{12}}{s^{2\\alpha +\\beta }}+{a_{13}}{s^{3\\alpha }}+{a_{14}}{s^{3\\beta }}+{a_{15}}{s^{\\alpha +\\beta }}+{a_{16}}{s^{2\\alpha }}+{a_{17}}{s^{2\\beta }}+{a_{18}}{s^\\alpha }+{a_{19}}{s^\\beta }+{a_{20}} \\hfill \\\\ \\end{gathered} }{\\begin{gathered} {b_1}{s^{4\\alpha +5\\beta }}+{b_2}{s^{4\\alpha +4\\beta }}+{b_3}{s^{3\\alpha +5\\beta }}+{b_4}{s^{2\\alpha +5\\beta }}+{b_5}{s^{4\\alpha +3\\beta }}+{b_6}{s^{3\\alpha +4\\beta }}+{b_7}{s^{2\\alpha +4\\beta }}+{b_8}{s^{4\\alpha +2\\beta }}+{b_9}{s^{3\\alpha +3\\beta }}+{b_{10}}{s^{\\alpha +5\\beta }} \\hfill \\\\ +{b_{11}}{s^{2\\alpha +3\\beta }}+{b_{12}}{s^{3\\alpha +2\\beta }}+{b_{13}}{s^{\\alpha +4\\beta }}+{b_{14}}{s^{4\\alpha +\\beta }}+{b_{15}}{s^{5\\beta }}+{b_{16}}{s^{2\\alpha +2\\beta }}+{b_{17}}{s^{\\alpha +3\\beta }}+{b_{18}}{s^{3\\alpha +\\beta }}+{b_{19}}{s^{4\\beta }}+{b_{20}}{s^{\\alpha +2\\beta }} \\hfill \\\\ +{b_{21}}{s^{2\\alpha +\\beta }}+{b_{22}}{s^{3\\alpha }}+{b_{23}}{s^{3\\beta }}+{b_{24}}{s^{\\alpha +\\beta }}+{b_{25}}{s^{2\\alpha }}+{b_{26}}{s^{2\\beta }}+{b_{27}}{s^\\alpha }+{b_{28}}{s^\\beta }+{b_{29}} \\hfill \\\\ \\end{gathered} }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e29\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSince the transfer function obtained is extremely complex and related to fractional order \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e, the coefficient is defined as follows:\u003cdiv id=\"Equ29\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ29\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{c}} {{a_i}={f_i}\\left( {\\alpha ,\\beta } \\right){\\text{ }}\\left( {i=1,2....20} \\right)} \\\\ {{b_j}={g_j}\\left( {\\alpha ,\\beta } \\right){\\text{ }}\\left( {j=1,2....30} \\right)} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e30\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhen \u003cem\u003eα\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.8, the system transfer function with both fractional capacitance and fractional inductance of 0.8 can be obtained, as shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ30\" class=\"InternalRef\"\u003e31\u003c/span\u003e):\u003cdiv id=\"Equ30\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ30\" name=\"EquationSource\"\u003e\n$$G\\left( s \\right)=\\frac{\\begin{gathered} 4.36 \\times {10^{90}}{s^{5.6}} - 1.762 \\times {10^{115}}{s^{4.8}} - 1.453 \\times {10^{123}}{s^4}+ \\hfill \\\\ 6.197 \\times {10^{127}}{s^{3.2}} - 5.384 \\times {10^{133}}{s^{2.4}}+6.4 \\times {10^{137}}{s^{1.6}} - 1.18 \\times {10^{142}}{s^{0.8}} \\hfill \\\\ \\end{gathered} }{\\begin{gathered} 3.499 \\times {10^{100}}{s^{7.2}}+2.888 \\times {10^{108}}{s^{6.4}} - 9.35 \\times {10^{112}}{s^{5.6}}+1.181 \\times {10^{120}}{s^{4.8}} - 4.197 \\times {10^{124}}{s^4} \\hfill \\\\ +5.396 \\times {10^{130}}{s^{3.2}} - 6.648 \\times {10^{134}}{s^{2.4}}+5.437 \\times {10^{140}}{s^{1.6}}+9.467 \\times {10^{143}}+1.005 \\times {10^{146}} \\hfill \\\\ \\end{gathered} }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e31\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the FOTF toolbox designed based on MATLAB in reference [17], the fractional order Bode diagram can be directly drawn, and its implementation process has completed the approximate fitting of the fractional transfer function to the integer order. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e, for the open-loop transfer function Bode diagram when \u003cem\u003eα\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1, \u003cem\u003eα\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.9 and \u003cem\u003eα\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.8, it can be clearly seen that the asymptote slope of the fractional order system is no longer an integral multiple of 20dB/dec, and presents a smooth transition curve with the gradual decrease of \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e. However, when the fractional order is equal to 0.8, because there are many passive components in the system, the fractional order is not only reflected in the order of the transfer function, but also significantly affects the coefficients in the transfer function. When the fractional order is 0.8, the Bode diagram of the system changes greatly. Therefore, using fractional order theory modeling, we can get a more accurate mathematical model of the system and get better design results.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5. Fractional Order Pi Control Strategy For Cllc Bi-directional Resonant Converter","content":"\u003cp\u003eAccording to the Fractional order PI\u003csup\u003eλ\u003c/sup\u003eD\u003csup\u003e\u0026micro;\u003c/sup\u003e controller proposed by Professor podlubny in reference [26], the Fractional Order PI\u003csup\u003eλ\u003c/sup\u003eD\u003csup\u003e\u0026micro;\u003c/sup\u003e closed-loop control strategy of CLLC bi-directional resonant converter is constructed. Compared with the classical PID control strategy, it has two more adjustable parameters \u003cem\u003eλ\u003c/em\u003e and \u003cem\u003e\u0026micro;\u003c/em\u003e, with a wider adjustable range and better control effect. Its mathematical form is:\u003cdiv id=\"Equ31\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ31\" name=\"EquationSource\"\u003e\n$${G_C}\\left( s \\right)={K_P}+\\frac{{{K_i}}}{{{s^\\lambda }}}+{K_d}{s^\\mu }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e32\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e5.1. Double closed loop control strategy based on fractional order PI\u003csup\u003eλ\u003c/sup\u003e\u003c/h2\u003e \u003cp\u003eWhen the CLLC bi-directional resonant converter works, the PFM modulation strategy is used, and the double closed-loop control of current inner loop and voltage outer loop is adopted. The specific control block diagram is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e. The PI\u003csup\u003eλ\u003c/sup\u003e controller is obtained by deleting the differential part in the PID controller.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe calculation has the following definitions, \u003cem\u003eα\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eβ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eλ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.9, \u003cem\u003eH\u003c/em\u003e\u003csub\u003e\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e = \u003cem\u003eH\u003c/em\u003e\u003csub\u003e\u003cem\u003ev\u003c/em\u003e\u003c/sub\u003e = 0.1 and \u003cem\u003eH\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e = 1.\u003c/p\u003e \u003cp\u003eIn order to meet the requirements of system stability, the fractional mathematical expressions of voltage loop and current loop are obtained by analysis. The open-loop transfer function is shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ32\" class=\"InternalRef\"\u003e32\u003c/span\u003e):\u003cdiv id=\"Equ32\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ32\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{c}} {{G_i}\\left( s \\right)={H_i}{H_s} \\cdot PI_{i}^{\\lambda }\\left( s \\right){G_{if}}\\left( s \\right)} \\\\ {{G_v}\\left( s \\right)={H_v} \\cdot PI_{i}^{\\lambda }\\left( s \\right){G_{vf}}\\left( s \\right){G_i}\\left( s \\right)} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e32\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSince the order of \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003evd\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003es\u003c/em\u003e) is fractional, it is impossible to describe the mathematical model of the system with a definite expression. In reference [27], an improved oustaloup filter is proposed to realize the approximate fitting of fractional operator s in the integer order within the frequency band (\u003cem\u003eω\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e,\u003cem\u003eω\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e), and \u003cem\u003eω\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003cem\u003eω\u003c/em\u003e\u003csub\u003e\u003cem\u003eh\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1. \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003evd\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003es\u003c/em\u003e) is approximately fitted by MATLAB to obtain its integer order model. See Appendix A.\u003c/p\u003e \u003cp\u003eAfter obtaining the approximate fitted integer order transfer function, the following voltage loop compensation controller is designed through analysis and calculation:\u003cdiv id=\"Equ33\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ33\" name=\"EquationSource\"\u003e\n$$PI_{v}^{\\lambda }\\left( s \\right)=1.49+\\frac{{0.501}}{{{s^{0.9}}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e33\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIt can be seen that the dynamic response capability of the system has been significantly improved after the introduction of current loop closed-loop control. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e"},{"header":"6. Simulation Analysis Of Cllc Bi-directional Resonant Converter Based On Fractional Order Control","content":"\u003cp\u003e \u003c/p\u003e \u003cp\u003eAccording to the above analysis, the circuit simulation model of CLLC bi-directional resonant converter is built in Simulink, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e. Its control module adopts fractional PI\u003csup\u003eλ\u003c/sup\u003e controller in Fomcon toolbox to complete fractional double closed-loop control strategy of bidirectional CLLC bidirectional resonant converter by constructing double closed-loop control of voltage outer loop and current inner loop. The specific simulation parameters are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eConverter parameters.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameters\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDC input voltage\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e500V\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDC output voltage\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e500V\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDC output current\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;20A\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTransformer ratio K\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1/1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSwitching frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e90-100kHz\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLoad resistance \u003cem\u003eR\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e26.6Ω\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eResonant inductors \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e20\u0026micro;H\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eResonant capacitor \u003cem\u003eC\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eC\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e140\u0026micro;F\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExcitation inductance \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e205\u0026micro;H\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOutput filter capacitor \u003cem\u003eC\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e630\u0026micro;F\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWhen the given voltage \u003cem\u003eV\u003c/em\u003e\u003csub\u003e\u003cem\u003eref\u003c/em\u003e\u003c/sub\u003e = 500V, the performance of the converter using fractional PI\u003csup\u003eλ\u003c/sup\u003e control strategy is compared with the classical PI control strategy, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e13\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eUnder the same given voltage, the PI parameters are the same, and the fractional order \u003cem\u003eλ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.9 starts the system. Under fractional order control strategy, the system reaches steady state at \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.6ms, with peak \u003cem\u003eδ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;512.9V and overshoot \u003cem\u003eб\u003c/em\u003e% = 2.58%; under classical PI control strategy, the system reaches steady state at \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.10ms, with peak \u003cem\u003eδ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;572.9V and overshoot \u003cem\u003eб\u003c/em\u003e% = 14.58%. It can be seen that the fractional order PI\u003csup\u003eλ\u003c/sup\u003e control strategy can make the system get better dynamic response ability, improve the system response speed, greatly reduce the overshoot and reduce the system loss.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e14\u003c/span\u003e, the system is started at a given voltage of 500V. When \u003cem\u003et\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.1s, the output load of the system suddenly changes, and the load r suddenly decreases from 80Ω to 40Ω. It can be seen from the waveform diagram that under the fractional order control strategy, the voltage fluctuation is 3.5V, and after stabilization, the system voltage reaches 496.5V, with a static error of 3.5V; Under the classical PID control strategy, the voltage fluctuation is 13.1V. After stabilization, the system voltage reaches 486.9V, and there is a static error of 13.1V. It can be seen that the fractional order control strategy has better robustness.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e, the voltage waveforms and driving waveforms at both ends of the switch tube at the primary side of the transformer are shown respectively. It can be seen that under the fractional order control strategy, when the voltage at both ends of the transformer's primary side switch drops to zero, the switch receives the driving signal and turns on, realizing ZVS. It is proved that the fractional order control strategy can achieve good soft switching characteristics.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e15\u003c/span\u003e Soft switching waveform(a)Driving voltage and voltage at both ends of switch tube;(b) Partial enlarged drawing.\u003c/p\u003e"},{"header":"7. Experimental Verification","content":"\u003cp\u003eIn order to verify the effectiveness and correctness of the above theoretical analysis, a 3kW CLLC bidirectional resonant converter prototype is built in the laboratory for experimental verification, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe prototype adopts the mixed modulation strategy of frequency conversion and phase shift, which realizes the step-up by changing the working frequency of the system and the step-down by changing the phase shift angle \u003csup\u003e[\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e][\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e]\u003c/sup\u003e. The control chip adopts DSP28335 of TI company. A fractional order PI\u003csup\u003eλ\u003c/sup\u003e control strategy applied to DSP is designed to compare with the classical PI control strategy to verify that the fractional order control strategy has better dynamic response ability, stronger robustness and soft switching characteristics. Since the forward and directional operating waveforms of the converter are consistent, only the forward operating mode of the system is analyzed.\u003c/p\u003e \u003cp\u003eThe specific parameters of the prototype are as follows:\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eConverter parameters.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameters\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eValue\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDC input voltage\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e400-450V\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDC output voltage\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e400V\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDC output current\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;6.5A\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTransformer ratio K\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1/1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSwitching frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e80-100kHz\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eResonant inductors \u003cem\u003eL\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eL\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e20\u0026micro;H\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eResonant capacitor \u003cem\u003eC\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eC\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e140\u0026micro;F\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eExcitation inductance \u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003em\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e205\u0026micro;H\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOutput filter capacitor \u003cem\u003eC\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e630\u0026micro;F\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e16\u003c/span\u003e, the system starts under the working condition of input voltage 400V and given voltage 400V. It can be seen that the rise time of the classical PI controller is 145ms, the peak voltage \u003cem\u003eδ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;446V, and the overshoot \u003cem\u003eб\u003c/em\u003e% = 11.15%. After reaching the steady state, the system output voltage fluctuates greatly. The rise time of fractional order PI\u003csup\u003eλ\u003c/sup\u003e controller is 125ms, the peak voltage \u003cem\u003eδ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;404v, and the overshoot \u003cem\u003eб\u003c/em\u003e% = 1%. After reaching the steady state, the system output voltage fluctuates slightly. It can be seen that the fractional order PI\u003csup\u003eλ\u003c/sup\u003e control strategy has faster rise time, lower overshoot and smaller voltage fluctuation than the classical PI control strategy.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig17\" class=\"InternalRef\"\u003e17\u003c/span\u003e, the system starts under the given voltage of 400V and load resistance of 200Ω. Under the classical PI control, at time \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, the output load resistance of the system suddenly decreases from 200Ω to 100Ω. It can be seen that the output current increases from 2A to 4A. At this time, the output voltage fluctuation increases. At time \u003cem\u003et\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, the load resistance suddenly increases from 100Ω to 200Ω, and the voltage fluctuation decreases. Under fractional PI\u003csup\u003eλ\u003c/sup\u003e control, at time \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, the output load resistance of the system suddenly decreases from 200Ω to 100Ω. It can be seen that the output current increases from 2A to 4A. At this time, the output voltage fluctuation has no obvious change, and the output current fluctuation is less than that of the classical PI control.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig18\" class=\"InternalRef\"\u003e18\u003c/span\u003e, when the system is started at a given voltage of 400V, the input voltage increases from 400V to 450V at time \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, and the output voltage remains unchanged. At this time, the input voltage is higher than the output voltage in the step-down mode. Using the frequency conversion phase shift hybrid control, it can be seen that the classical PI control voltage waveform fluctuation increases. The voltage fluctuation of fractional order PI\u003csup\u003eλ\u003c/sup\u003e control increases after \u003cem\u003et\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, but it is smaller than that of classical PI control, which proves that fractional order control has stronger anti-interference ability.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo sum up, the validity and feasibility of fractional order PI\u003csup\u003eλ\u003c/sup\u003e control strategy are verified by building the principle prototype of CLLC bi-directional resonant converter in the laboratory.\u003c/p\u003e"},{"header":"8. Conclusion","content":"\u003cp\u003eThis paper based on fractional calculus theory, the fractional mathematical model and fractional PI\u003csup\u003eλ\u003c/sup\u003e control strategy of CLLC bi-directional resonant converter are established. The following conclusions are obtained through theoretical and simulation analysis:\u003c/p\u003e \u003cp\u003e1. In the mathematical modeling of CLLC resonant converter, the classical integer order modeling method can not accurately describe the system, there are errors and poor flexibility. By introducing the integral operator \u003cem\u003es\u003c/em\u003e\u003csup\u003e\u003cem\u003eα\u003c/em\u003e\u003c/sup\u003e, the integral order inductance and capacitance in the system are extended to fractional order inductance and capacitance, which improves the accuracy of the mathematical model and more accurately describes the actual system. Through mathematical analysis, it is proved that fractional order \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eβ\u003c/em\u003e not only affect the order of the transfer function, but also have a great impact on the coefficients in the transfer function.\u003c/p\u003e \u003cp\u003e2. By designing fractional order PI\u003csup\u003eλ\u003c/sup\u003e control strategy applied to CLLC bi-directional resonant converter, the voltage and current double closed-loop control of the system is realized. Compared with the classical PI control strategy, the fractional order control strategy has the advantages of better dynamic response, fast rise time, small overshoot, strong robustness and so on. Compared with the classical PI control strategy, the fractional order control strategy has the advantages of better dynamic response, fast rise time, small overshoot, strong robustness and so on. And it can get good soft switching characteristics of the primary switch, and realize zero voltage conduction of the switch.\u003c/p\u003e\u003cp\u003e3. Finally, a prototype is built in the laboratory, and a fractional order PI\u003csup\u003eλ\u003c/sup\u003e controller is designed to realize the closed-loop control of CLLC bi-directional resonant converter. Compared with the classical PI controller, the correctness and effectiveness of the fractional order modeling and fractional order PI\u003csup\u003eλ\u003c/sup\u003e control strategy of CLLC bi-directional resonant converter proposed in this paper are verified.\u003c/p\u003e \u003cp\u003eThe datasets generated during and/or analysed during the current study are not publicly available due [REASON(S) WHY DATA ARE NOT PUBLIC] but are available from the corresponding author on reasonable request.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eFund Project:State Grid Gansu Electric Power Company Projects (W22KJ2722005).\u003c/p\u003e"},{"header":"References","content":"\u003cp\u003e[1]\u0026nbsp;CHEN Guoping, LIANG Zhifeng, DONG Yu. Analysis and Reflection on the Marketization Construction of Electric Power With Chinese Characteristics Based on Energy Transformation [J]. 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Portland: IEEE,2018:5961-5966. \u0026nbsp;\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"CLLC bi-directional resonant converter, Fractional calculus, Fractional order modeling, Extended description function method, Fractional order PIλ control strategy","lastPublishedDoi":"10.21203/rs.3.rs-1918353/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1918353/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"There are problems such as insufficient modeling accuracy of the CLLC bi-directional resonant converter in integer order, failure to accurately describe the actual conditions of the system, poor dynamic performance of classical PI control strategy, and failure of the system to reach the stable state quickly and smoothly. First, based on the fractional order calculus theory, this paper proposes an extended describing function method based on fractional order, establishes the mathematical model of the fractional order and small signal model of the fractional order of the CLLC bi-directional resonant converter, obtains the fractional order transfer function of the system, constructs the PI λ control strategy of the fractional order of the CLLC bi-directional resonant converter, and builds the simulation model of the CLLC bi-directional resonant converter based on Matlab/Simulink software for simulation verification. Finally, a principle prototype was built in the laboratory to verify the correctness and feasibility of the CLLC bi-directional resonant converter of the fractional order and theoretical analysis of the PI λ control strategy of the fractional order.","manuscriptTitle":"Research on fractional order modeling and PIλ control strategy of CLLC bi-directional resonant converter","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2022-08-05 21:54:35","doi":"10.21203/rs.3.rs-1918353/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"89c61467-61fb-4c2b-8366-fd8833288a45","owner":[],"postedDate":"August 5th, 2022","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2022-08-05T21:54:35+00:00","versionOfRecord":[],"versionCreatedAt":"2022-08-05 21:54:35","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-1918353","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-1918353","identity":"rs-1918353","version":["v1"]},"buildId":"rHA-KDH7Qsr4HCuvH75dn","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}