Research on Fractional Order Modeling and PIλ Control Strategy of V2G Two-Stage Bidirectional Converter

With regard to the insufficiency of independent modeling accuracy in V2G two-stage bi-directional converter in the two-stage system of integer order, it is impossible to accurately describe the actual situation of the poor dynamic performance of the classical PI control strategy as well as the lower power transmission efficiency. Based on the fractional calculus theory, the modeling methods based on the fractional, namely, the state space averaging method and the extended description function method are proposed. The integer inductance and capacitance in the system are extended to fractional inductance and capacitance, and the Fourier decomposition in the extended description function method is extended from integer order to fractional order. The joint mathematical model and small signal model of fractional order for V2G two-stage bidirectional converter are constructed, and the fractional order transfer function of the whole system is obtained. Therefore, a fractional PIλ control strategy for V2G two-stage converter is designed. Finally, a 3 kW prototype is built in the laboratory, which verifies the correctness and feasibility of fractional modeling and fractional PIλ control strategy.

With the goal of "double carbon" and the proposal of "power system with new energy as the main body", China's new energy power generation industry will accelerate its development, and the increment of new energy power generation will continue to rise, which will promote the transformation of power system to clean energy [1], [2].It is estimated that by 2030, the total installed capacity of wind and solar power generation in China will reach more than 1.2 billion kW [3].The resulting intermittence, randomness, load time-space mismatch, load fluctuation and regional interconnection of power grids make China face serious challenges of renewable energy consumption and safe and stable operation of power grid system [4].V2G (Vehicle to Grid) on-vehicle bidirectional converter system is a key technology for vehicle-network interconnection, which uses a large number of energy storage batteries of new energy vehicles as a large-scale distributed storage capacity system to assist in peak shaving and frequency modulation of power grid, consumes a large amount of new energy generation and improves the safety and stability of power grid [5], [6], [7].The main function of V2G system is to realize energy transmission between power grid and new energy vehicles, communicate with the management system of new energy vehicles in real time, and participate in peak shaving and frequency modulation of power grid.The core equipment for realizing bidirectional energy transmission is of a two-stage architecture, which is composed of front-stage bidirectional AC/DC and back-stage bidirectional DC/DC circuits [8], [9], [10], which plays a vital role as a bridge between power grid and new energy vehicles.
At present, many scholars have done a lot of research on V2G two-stage converter.A novel wide-range high-gain bidirectional DC/DC converter for V2G is proposed in Reference [11], which has higher voltage gain and higher power efficiency, and the electrical efficiency reaches 97.2% and 96.8% in boost-buck mode respectively.Reference [12] studies a buck-type bidirectional battery chargers (BBC) to support V2G new energy vehicles to participate in the reactive power compensation service of the power grid, which can realize the simultaneous charging of the battery and the transmission of energy from the battery to the power grid, and has been verified by experiments.Reference [13] adopts a new control strategy based on DAB trinomial AC/DC isolated converter, which improves the reliability and power density of the system and realizes the soft switching characteristics of all semiconductor devices in the system.A scalable bidirectional DC/DC converter is proposed in Reference [14].The converter has higher voltage transfer ratio (VTR), redundancy and modularity in new energy vehicle applications, and its current is expanded to n times in bidirectional operation.It is verified by experiments on a 500 W prototype.Reference [15] introduces a bidirectional CLLLC resonant converter applied to battery charging and discharging.Since all switches in the system can realize soft switching, the system can work at very high frequency and the power density is greatly improved.A 3.5 kW converter prototype is built in the laboratory to verify the effectiveness of this topology.
In the above research, analysis modeling and loop control are all carried out in the integer order system, that is, the capacitor in the V2G two-stage converter is an integer order capacitor and the inductor is of an integer order.However, through the study of capacitance and inductance in the system, it is found that, in practical application, integer order capacitance and inductance do not exist, and the capacitance and inductance in the system are often of fractional order [15].Therefore, it is necessary to establish a fractional mathematical model for V2G two-stage converter, so as to describe the fractional components in the system more accurately [16], [17].Westerlund et al. measured the fractional order of capacitance in impassable dielectric by experiments [18].Petrs uses fractional inductance and fractional capacitance to design a fractional Chua's circuit, which proves the fractional characteristics in the actual system [19].
Compared with integer order calculus and integer order control, fractional order calculus and fractional order control have the following main advantages [20], [21]: 1) Integer order calculus reflects the local characteristics of the function, while fractional order calculus considers the global information of the function in the form of weighting.Fractional order calculus is similar to convolution operator, which can describe both finitedimensional systems and infinite-dimensional systems.Therefore, using fractional calculus to establish the mathematical model of the research object can more accurately describe the essential characteristics and dynamic characteristics of complex systems with history dependent processes or distributed parameters; 2) When using integer order calculus to identify and model a complex system, if you want to ensure high identification accuracy, the identified model commonly has a higher order and complex expression.When introducing fractional order calculus, you can use a fractional order mathematical model that is not an integer but has a more concise form to describe the system [22].For example, for the amplitude frequency characteristic curve whose slope is not 20ndB/dec(nࢠz), the integer order identification is more complex and the system order is higher; if fractional order identification is adopted, the gain slope of a α-order pure integration link is α, and its expression is particularly concise.3) Fractional order control is an extension of integer order control, and integer order control is a special case of fractional order control.The study of fractional order control has more general significance.4) The fractional order controller expands the degree of freedom of control, makes the parameter selection more flexible and the parameter tuning range larger, which can achieve better control performance than the integer order controller, such as stronger robustness and antiinterference.Therefore, this article introduces fractional inductance and fractional capacitance to replace the inductance and capacitance in the traditional PFC and CLLC topology and increase the degree of freedom of the system, so as to construct a fractional V2G two-stage converter system with richer and more flexible operating characteristics and better performance.On this basis, with the help of fractional calculus, the mathematical models of the former PFC and the latter CLLC topologies are established, the working characteristics of PFC and CLLC are explored and analyzed, and a fractional order controller with finer control and better robustness is designed for the established mathematical models of PFC and CLLC to improve the dynamic performance of the system.
To sum up, in the past, the research on mathematical model and control strategy of V2G two-stage converter was based on integer order model, while the actual system was of a fractional order system, and the classical modeling was not accurate enough.In the present article, the bidirectional totem pole circuit in the front stage-PFC (Power Factor Correction, PFC) and the back stage CLLC bidirectional resonant circuit of V2G bidirectional converter are studied, and the fractional order model and the small signal model of fractional order of the front stage and the back stage, as well as the joint mathematical model of the two-stage system, are given, so as to obtain the transfer function of the system.Through Matlab, the fractional order Bode diagram is drawn and the integer order Bode diagram is compared and analyzed.Finally, a 3 kW V2G prototype is built in the laboratory to verify the principle of fractional order system and fractional order control.

II. TOPOLOGY AND FRACTIONAL ORDER EQUIVALENT TOPOLOGY OF CLLC BI-DIRECTIONAL RESONANT CONVERTER
The topology of V2G two-stage converter used in this article is the front stage bidirectional totem pole bridgeless PFC circuit and the rear stage CLLC bidirectional resonant circuit,  which has the advantages of less devices, high power transmission efficiency, soft switching characteristics in a wide load range and so on.
The specific topology is shown in Fig. 1.The front PFC circuit is composed of switches S 1 ∼S 4 , V i is the AC input, L is the AC inductance, C is the DC bus capacitance, V bus is the DC bus voltage, and the rear CLLC resonant current route switches Q 1 ∼Q 4 to form the primary side rectifier and inverter circuit, and switches Q 5 ∼Q 8 to form the secondary side rectifier and inverter circuit, L 1 and C 1 are the primary side resonant inductance and capacitance, L 2 and C 2 are the secondary side resonant inductance and capacitance, L m is the excitation inductance, i 1 is the primary side current, i 2 is the secondary side current, i m is the excitation inductance current, u 1 , u 2 are the primary and secondary side resonant capacitor voltage, V o is the output voltage.When the system works in the forward direction, S 1 ∼S 2 and Q 5 ∼Q 8 participate in the circuit operation as rectifier diodes.Similarly, when working in reverse, S 3 ∼S 4 and Q 1 ∼Q 4 participate in the circuit operation as rectifier diodes.
Replace the integer order inductance and capacitance in the system with fractional order inductance and capacitance to obtain the fractional order equivalent topology of the system, as shown in Fig. 2, where α and β are the fractional order of inductance and capacitance, and 0<α, β< 1.
By constructing the fractional order model, the fractional order mathematical model of the system is established by using the fractional order calculus theory, and the fractional order transfer function of the system is obtained, which provides a theoretical basis for subsequent research.

III. JOINT FRACTIONAL ORDER MATHEMATICAL MODELING OF V2G TWO-STAGE BIDIRECTIONAL CONVERTER
The forward and reverse working modes of V2G converter are basically the same, and it can be regarded as PFC+LLC circuit when it works in the forward direction.This article mainly analyzes the mathematical model of the system when it works in the forward direction.The mathematical model of the front stage PFC circuit is established by the fractional order state space average method and the mathematical model of the rear stage CLLC circuit is established by the fractional order extended description function method.On this basis, in order to solve the lack of accuracy caused by the independent modeling of the two-level system in the traditional modeling, it cannot reflect the coupling relationship of the two-level system.A joint optimization modeling method of two-stage system based on fractional order is proposed.
According to Reference [23], the relationship between the current i L flowing through the fractional inductor and the voltage v L at both ends is: 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: Where β is the fractional order of capacitor C β , and 0 < β <1.

A. JOINT MATHEMATICAL MODEL OF PFC
Because the topology of totem pole bridgeless PFC is relatively simple, the expressions of fractional steady-state large signal model and small signal model of the system obtained by using the state space average method will be directly given below, the state variables of the system are separated into DC component and AC component, as shown in (3): Formula ( 4) after separating the DC component of the system, where R eq is the equivalent resistance of the later CLLC circuit.
⎧ ⎨ ⎩ Separate the AC component to obtain (5): According to Caputo's fractional order definition [24], any fractional derivative of the constant is equal to zero, that is, the left side of the equal sign in (4) is equal to zero.The steadystate working point formula (6) of the system can be obtained.
The transfer function of the system is obtained by fractional Laplace transform of AC component, as shown in formula (7).
Among them, the later CLLC circuit generally works at the first resonant frequency point, and the circuit is a purely resistive load, so the CLLC equivalent load R eq is obtained as (8).Bring ( 8) and ( 6) into (7) to obtain (9).

B. JOINT MATHEMATICAL MODEL OF CLLC
According to Reference [25], the equivalent topology of CLLC resonant converter in integer order is extended to fractional order equivalent topology, as shown in Fig. 3.The fractional order state equation of the system is obtained through analysis, as shown in formula (10).
Among them, the nonlinear part is V ab , sgn(i 1 -i m ), |i 1 -i m | and 0 < α, β < 1, and the formula V ab is the square wave voltage generated by the primary input voltage V ab of the transformer through the inverter bridge, which is the input voltage of the resonant network.
According to the extended description function method in Reference [26], the above state variables are decomposed by fractional Fourier transform, as shown in (11) and (12).
The nonlinear part of the system state equation is approximated as the superposition of sine component and cosine component: Among them, the bus voltage V bus of the system can be rewritten as follows through the analysis of PFC circuit in the previous section: Bring ( 11), ( 12), ( 13) and ( 14) into (10) to make the sine and cosine components on both sides of the equation equal.In order to simplify the equation, the following definitions are given: ) Through sorting and simplification, get (16): Finally, the above formula is decomposed into DC component and AC component, and x is any state variable in the system.
x = X st + x (17) x = î1s î1c î2s î2c û1s û1c û2s û2c vo A st •X st + B st = 0 is obtained from the fact that any fractional derivative is zero.The parameter matrix in the formula is shown in (19), (20) and (21).
It can be seen from the above formula that the steady-state solution X st of the system is (22).
In the V2G converter designed in this article, the input voltage V i of the front PFC circuit is 220 v/50 hz, the duty cycle D = 0.56 (the DC bus voltage V bus is 500 V), and the AC inductance L = 475 μH, DC bus capacitance C = 3460 μF.Taking the above parameters into the separated AC components, due to the tedious calculation, this article directly gives the joint transfer function model of V2G two-stage converter system.
Among them, the transfer function of the front PFC circuit is shown in (23): 4.6316 • 10 5 s 0.9 + 2.5066 • 10 7 s 1.8 + 13.4s 0.9 + 1.178 The transfer function of the later CLLC system is shown in (24).    in Figs. 4 and 5.It can be seen from the baud diagram of the system transfer function that the dynamic performance of the system is poor in the open-loop case, and the cut-off frequencies of the front PFC and the rear CLLC are greater than 100 kHz.It is necessary to design a fractional order control strategy of V2G two-stage converter to improve the dynamic response ability of the system.
In reference [16], Jesus et al. manufactured fractional capacitors with 0.59 and 0.42 orders; in reference [27], Macha et al. also pointed out that inductive elements of any order can be manufactured based on skin effect.However, in practical applications, the fractional inductance and fractional capacitance are of order approximately one.For the CLLC topology in this article has the characteristics of multiple passive devices, the influence of fractional order on the transfer function is also reflected in the coefficients.When the fractional order is 0.8 or 0.7, the transfer function of the system is considerably affected and the designed controller performs poorly.Therefore, in this article we only discuss the system transfer function for α = β = 0.9.

IV. FRACTIONAL ORDER PI λ CONTROL STRATEGY OF V2G TWO-STAGE BIDIRECTIONAL CONVERTER
According to the Fractional Order PI λ D μ controller proposed by Professor podlubny in Reference [23], the Fractional Order PI λ D μ closed-loop control strategy of V2G two-stage bidirectional converter is constructed.Compared with the classical PID control strategy, it has two more adjustable parameters λ, μ, the adjustment range of the system is larger and the effect is better.Its mathematical form is:

A. FRACTIONAL ORDER PI λ CONTROL STRATEGY OF FRONT STAGE PFC
The front PFC circuit adopts double closed-loop control, as shown in Fig. 6, and the PI controller is obtained by removing the differential link.Its current inner loop open-loop transfer function is shown in (28).
The calculation has the following definitions, α = β = λ = 0.9, Because the G id_PFC transfer function is of fractional order, it is impossible to design the optimal control parameters of the system.In Reference [27], an improved Oustaloup filter is proposed to realize the approximate fitting of fractional operator s α in integer order in frequency band (ω b ,ω h ), and ω b ω h = 1.After the transfer function is fitted to integer order by MATLAB, the current loop compensation controller of the system is designed, as shown in (27).
As shown in Fig. 7, the dynamic response ability of the system has been significantly improved after the introduction of the closed-loop control strategy.

B. FRACTIONAL ORDER PI λ CONTROL STRATEGY OF POST STAGE CLLC
The closed-loop control block diagram of the later CLLC is shown in Fig. 8, and the expression of the open-loop transfer function can be obtained, as shown in (28).
By approximating the fractional transfer function to an integer order, the current loop compensation controller of the system is designed as follows:

V. EXPERIMENTAL VERIFICATION
In order to verify the correctness and effectiveness of fractional modeling theory and fractional control, a prototype of  V2G two-stage bidirectional converter with power of 3 kW and bidirectional transmission function is built in the laboratory.The prototype is mainly composed of a front-stage bidirectional PFC circuit and a rear-stage CLLC resonant circuit.The controller uses TMS320F28335 DSP as the main control chip to realize the front-stage power factor correction and the rear-stage PFM modulation strategy, as shown in Fig. 10.Prototype parameters are shown in Tables 1 and 2 The interface of the prototype is divided into DC input/output interface and AC input/output interface, as shown in Fig. 11.When working in the forward direction, the AC interface is in the input mode and connected to the urban power 220 V/50 Hz.The DC interface is in output mode and connected to DC load (or lithium battery).Complete the conversion of AC-DC, and simulate the power grid to charge new energy vehicles in this mode.Similarly, in reverse operation, the AC interface is in output mode and the DC interface is in input mode.At this time, new energy vehicles transmit energy to the power grid to assist the power grid in peak load and frequency regulation.
As shown in Fig. 12, when working in the forward direction, the system starts the circuit at a given bus voltage of 500 V and a DC output voltage of 450 V.Under the classical PI control strategy, the bus voltage V bus reaches the steadystate voltage of 500 V for 210 ms, the peak voltage is 530 V, and the overshoot б% = 6%, the time for the output voltage V o to reach the steady-state voltage of 450 V is 402 ms, the peak voltage is 500 V, and the overshoot б% = 11.1%.Under the fractional PI λ control strategy, the peak voltage of bus voltage V bus is 530 V, which is overshoot б% = 5%, the time for the output voltage Vo to reach the steady-state voltage of 450 V is 360 ms, the peak voltage is 492 V, and the overshoot б% = 9.3%.It can be seen that the fractional order control As shown in Fig. 13, the system starts under the given output voltage of 450 V and output load of 200 .Under the classical PI control, after the system reaches the steady state, the output current is 2.109 A, the system power reaches 971.55 W, and the electric energy transmission efficiency is 97.1%.At time t 1 , the load suddenly decreases to 100 , the output current of the system increases to 4.31 A, the output power reaches 1939.5 W, and the electric energy transmission efficiency is 95.7%.At time t 2 , the system load suddenly increased to 200 , and the system output current changed back to 2.159 A. The current fluctuated greatly during operation.Under fractional order PI λ control, after the system reaches steady state, the output current is 2.2 A, the system power reaches 990 W, and the electric energy transmission efficiency is 97.7%.At time t 1 , the load suddenly decreases to 100 , the output current of the system increases to 4.36 A, the output power reaches 1962 W, and the power transmission efficiency is 96.8%.At time t 2 , the system load suddenly increases to 200 , and the system output current changes back to 2.2 A. The current fluctuation is small during operation.It can be seen that the output current fluctuation under fractional order control is smaller, the system loss is smaller, and higher output power and higher power transmission efficiency can be achieved.
As shown in Fig. 14, when working in reverse, the system is DC input and AC output.Under the given voltage of 220 V, the power factor at the output end is greater than 99% in all modes.It can be seen that the fractional order control strategy can well achieve power factor correction.
As shown in Fig. 15, the system starts when the output load is 200 and the given voltage is 400 V.After reaching steady state with classical PI control strategy, the system voltage reaches 403.1 V, the system current reaches 2.05 A, and the steady state ripple of the voltage is v 1 = 16 V, steady ripple of current is i 1 = 0.23 A, at t 1 , the load suddenly decreases to 100 , and the current rises to 3.92 A. After reaching the steady state, the voltage ripple remains unchanged, and the current ripple i 2 = 0.22 A. Under the fractional control strategy, the system voltage reaches 401.2 V and the system current reaches 2.02 A when the steady state is first reached.At this time, the steady-state ripple of the voltage is v 1 = 8 V, steady ripple of current is i 1 = 0.13 A, at t 2 , the load suddenly decreases to 100 , and the current rises to 3.92 A. After reaching the steady state, the voltage ripple remains unchanged, and the current ripple i 2 = 0.11 A. It can be seen that the static error of the fractional control strategy is definitely smaller than that of the classical PI control strategy, which has better steady-state performance.
As shown in Fig. 16, when the system is in V bus = 350 V steady state operation, the bus voltage is increased to 400 V.Under the classical PI control strategy, the time for the system to reach the steady state is T2 = 35 ms, with a peak voltage of 426 V.Under the fractional control strategy, the time to reach the steady state is T1 = 23 ms, and the peak voltage is 416 V.It can be seen that the fractional control strategy has a stronger dynamic response.As shown in Fig. 17, when V2G is in reverse operation, Vo is the system input voltage, V i is the system output voltage, and V bus is the intermediate DC bus voltage.The system is started by setting the input voltage Vo to 400 V, the given bus voltage V bus to 400 V, and the output voltage V i to 220 V/50 Hz.It can be seen that under the classical PID control strategy, the system bus voltage response time T1 = 145 ms, the system peak voltage is 452 V, and the overshoot is б% = 13%.Moreover, the i L harmonic distortion rate of the system output current is large.Using fractional order control strategy, the system bus voltage response time  T1 = 125 ms, the system peak voltage is 405 V, and the overshoot is б% = 1.25%.Moreover, the i L harmonic distortion rate of the current is significantly improved.
As shown in Fig. 18, The waveform Figure is the local waveform of the frequency conversion control with an initial starting frequency of 40 K.After launching the system, the operating frequency of the system quickly rises to 90 K, enabling closed-loop control of the system bus voltage.It can be seen that after the system reaches a stable frequency of 90 K, the system bus voltage also reaches the system's given voltage.

VI. CONCLUSION
Based on fractional order theory, this article establishes the fractional order mathematical model and fractional order PI λ control strategy of V2G two-stage bidirectional converter.Through theoretical analysis and experimental verification, the following conclusions are obtained: 1) In the mathematical modeling of V2G two-stage bidirectional converter, the integer order modeling method can not accurately describe the system, and there are errors.In this article, the integer order inductance and capacitance in the system are extended to fractional order inductance and capacitance, and the former PFC circuit and the latter CLLC circuit are jointly modeled to obtain a more accurate system model, and the fractional order a, its influence on the system mathematical model and the correlation between the former and the latter are explained.2) A 3 kW principle prototype is built in the laboratory.
A fractional order PI λ controller is designed to realize the closed-loop control of CLLC bi-directional resonant converter.By comparing with the classical PI controller, it is verified that the fractional order PI λ control strategy has better dynamic response ability, smaller system loss, higher power transmission efficiency and good quality factor.

( 24 )
The Bode diagram of the open-loop transfer function of the system obtained from the analysis is shown in Fig.4, and is compared with the integer order open-loop transfer function

FIGURE 4 .
FIGURE 4. Bode diagram of fractional-order open-loop transfer function.

FIGURE 5 .
FIGURE 5. Bode diagram of integer order open-loop transfer function.

FIGURE 6 .
FIGURE 6. Closed loop control block diagram of PFC.

FIGURE 7 .
FIGURE 7. Closed loop control bode diagram of PFC.

FIGURE 8 .
FIGURE 8. Closed loop control block diagram of CLLC.

FIGURE 15 .
FIGURE 15.Steady state ripple of voltage and current.

FIGURE 17 .
FIGURE 17. System waveform diagram of reverse operation.

DI
LI received the B.S. degree in electrical engineering in 2018 from Weinan Normal University, Weinan, China, where she is currently working toward the M.S. degree.Her research interests include the fractional order model and fractional order control strategy.KUN DING received the Master of Electrical Engineering degree from Tsinghua University, Beijing, China, in 2009.He is currently the Director of the System Analysis Room of the School of Electrical Science and Technology.He participated in the completion of "wind farm, photovoltaic power plant cluster control system research and development" and other national 863 projects, completed two science and technology projects in Gansu Province and six science and technology projects of State Grid Corporation, and responsible for four science and technology projects of state grid gansu electric power company.YI XIA was the Deputy General Manager and Senior Engineer with State Grid Linxia Power Supply Company, and presided over a number of national and provincial science and technology projects.HAIYING DONG was born in Lintong, China, in 1966.He received the B.S. degree from Beihang University, Beijing, China, in 1989, the M.S. degree from Lanzhou Jiaotong University, Lanzhou, China, in 1998, and the Ph.D. degree from Xi'an Jiaotong University, Xi'an, China, in 2003.He is currently a Professor with Lanzhou Jiaotong University.His research interests include the optimization and intelligent control of power systems, and optimal control of renewable energy generation.