CN104392080A - Lithium-battery variable fractional order and equivalent circuit model and identification method thereof - Google Patents

Abstract

The invention discloses a lithium-battery variable fractional order and equivalent circuit model and an identification method thereof. The lithium-battery variable fractional order and equivalent circuit comprises a run time circuit and a battery I-V characteristic circuit, wherein a capacitor in the battery I-V characteristic circuit is a variable fractional order capacitor. A second order RC circuit model is generalized to a non-integer order, and the model parameters and the number of fractional order of different SOC are identified based on a least square method, so that the fractional order and equivalent circuit varying order according to the SOC is obtained. The instruction of fractional order realizes the continuous change of the order number of the model, so that the model is relatively stable, good in dynamic property and high in precision. The variation of fractional order realizes more freedom and more flexibility and innovation of the model. As the number of RC networks is not increased, the fractional order model effectively solves the contradiction between the accuracy and practicality of the model, is suitable for various working conditions of batteries, and has high practical value. The invention provides a precise battery model easy to realize for precise estimation of SOC.

Description

A fractional-order variable-order equivalent circuit model of a lithium battery and its identification method

technical field

The invention relates to a fractional-order variable-order equivalent circuit model of a lithium battery and an identification method thereof.

Background technique

In order to deal with the energy crisis and environmental pollution, electric vehicles have emerged as the times require and have become the focus of attention all over the world. As a key component of electric vehicles, vehicle-mounted power batteries are crucial to the power, economy and safety of the vehicle, and are a key factor restricting the scale development of electric vehicles. Lithium batteries have the advantages of high energy density, long service life, good cost performance and high single voltage, and gradually become one of the power sources of hybrid electric vehicles or pure electric vehicles. Accurate battery models are of great significance to the rational design and safe operation of vehicle-mounted power lithium batteries, and are the basis of battery SOC (state of charge) and SOH (state of health) estimation methods.

However, it is not easy to establish an accurate and simple battery model, because the chemical reaction inside the lithium battery involves complex conversion of electrical energy, chemical energy, and thermal energy, which is highly nonlinear and uncertain. At present, the commonly used battery models can be divided into the following five categories according to the different modeling mechanisms: ① electrochemical model, ② analytical model, ③ stochastic model, ④ neural network model and ⑤ equivalent circuit model. Among them, the equivalent circuit model has become a new model widely used because of its simple and intuitive form and the advantages of being suitable for electrical design and simulation. In the equivalent circuit model, the second-order RC model has clear physical meaning compared with other equivalent circuit models, the model parameter identification test is easy to perform, the parameter identification method is systematic, and the model accuracy is higher, which can simulate the dynamic characteristics of the battery more accurately and intuitively. . However, the second-order RC model has a large fitting error due to the low order of the model at the initial and final stages of battery charging and discharging, and cannot accurately simulate the dynamic and static characteristics of the battery. Although increasing the series order of RC can improve the accuracy of the battery model and better simulate the charging and discharging characteristics of the power battery, if the order of the power battery model is too high, it will not be conducive to obtaining the parameters in the model, and will also It will greatly increase the calculation amount of the model, and even cause system oscillation, so on the other hand, the order of RC should also be limited. Therefore, the fixed structure equivalent circuit model is difficult to describe the nonlinear voltage characteristics of the steep and flat middle of the lithium battery, and cannot solve the contradiction between the accuracy and practicability of the model.

For this reason, the Chinese invention patent application (application number 201410185885.7) and utility model (patent number ZL201420226360.9) proposed a variable-order RC equivalent circuit model based on the AIC criterion. By slightly increasing the complexity of the model, it can be more accurately Describe the non-linear voltage characteristics of the lithium battery, which is steep at both ends and flat at the middle, and the error is within 0.04V, which effectively solves the contradiction between model complexity and practicability, and has high practical value. However, the model is an integer-order battery model, and the switching of the model can only be an integer-order change. Therefore, the model order fluctuates greatly, which does not conform to the gradual development law in nature, so the accuracy of the model is greatly limited. In fact, the internal electrochemical reaction process of the battery is extremely complex, including the transfer of conductive ions, internal electrochemical reactions, charge and discharge hysteresis effects, and concentration diffusion effects, etc., showing strong nonlinear characteristics, which are more suitable for simulation by fractional order models . Compared with the integer-order model, the fractional-order battery model has more degrees of freedom, greater flexibility and novelty in design. At the same time, their introduction also adds many new phenomena and laws, which have advantages that conventional integer-order battery models cannot achieve.

Contents of the invention

In order to solve the deficiencies in the prior art, the present invention discloses a lithium battery fractional-order variable-order equivalent circuit model and its identification method. According to the electrochemical reaction principle of lithium-ion batteries, the traditional second-order RC equivalent circuit model uses two Two integer order RC networks describe the polarization effect and concentration effect of the battery. The present invention extends the two integer order RC networks of the model to non-integer order (fractional order), and identifies model parameters at different SOCs based on the least squares method and order, thus obtaining a fractional-order equivalent circuit model that varies according to the SOC order. The introduction of the fractional order realizes the continuous change of the model order, making the model more stable, with better dynamic performance and higher precision. Due to the addition of fractional order variable parameters, the model has more degrees of freedom, greater flexibility and novelty. The model is realized on the basis of the traditional second-order RC model, without increasing the number of model RC networks, effectively solving the contradiction between model accuracy and simplicity, and having high practical value.

To achieve the above object, the specific scheme of the present invention is as follows:

A fractional-order variable-order equivalent circuit model of a lithium battery, including a running time circuit and a battery IV characteristic circuit, wherein the running time circuit and the battery IV characteristic circuit perform signal transmission through a current-controlled current source and a voltage-controlled voltage source; the running The time circuit includes the self-discharging resistance R _d and the capacitor C _Q of the battery, the resistance R _d and the capacitor C _Q are connected in parallel at the two ends of the current control current source, and one end of the current control current source is grounded; the voltage control voltage of the IV characteristic circuit of the battery The positive terminal of the source is connected to one end of two parallel-connected branches, and the negative terminal is connected to the negative terminal of the battery model, each branch of the two parallel-connected RC network branches includes two series-connected fractions The second-order RC loop and an internal resistance R _o , the other ends of the two parallel branches are connected to the positive terminal of the battery model.

In the two RC network branches connected in parallel in the IV characteristic circuit of the battery, the discharge branch includes a fractional-order RC loop composed of a diode D _d in series, a fractional-order capacitor FOE _1d and a resistor R _1d , and a fractional-order capacitor FOE _2d Fractional-order RC circuit formed with resistance R _2d and resistance R _od ;

The charging branch includes a series-connected reverse diode D _d , a fractional RC loop composed of a fractional capacitor FOE _1c and a resistor R _1c , a fractional RC loop composed of a fractional capacitor FOE _2c and a resistor R _2c , and a resistor R _oc .

The running time circuit and the IV characteristic circuit are connected through a current control current source and a voltage control voltage source. When charging and discharging the battery, the load current i _bat charges and discharges the capacitor C _Q through the current control current source, changing C The power stored by _Q represents the change of battery SOC, and the voltage OCV across C and _Q also changes accordingly. The controlled voltage source OCV of the IV characteristic circuit changes with the change of SOC.

The capacitance C _Q represents the available capacity of the battery, C _Q =3600·C _Ah ·f ₁ ·f ₂ , wherein, C _Ah is the battery capacity in ampere hours, f ₁ and f ₂ are the battery cycle life and Correction factor for temperature.

The current of the current control current source is the terminal current i _bat of the battery. When the battery is charging and discharging, the load current i _bat charges and discharges the capacitor C _Q through the current control current source, and changes the electric quantity stored in the capacitor C _Q , thereby representing Changes in battery SOC.

The voltage across the two ends of the current control current source is the battery open circuit voltage OCV.

The two parallel RC network branches are respectively the RC network discharge branch and the RC network charge branch, and the capacitors in the two RC network branches are all fractional order capacitors.

The orders α and β of the fractional-order elements FOE _1d and FOE _2d in the discharge branch of the RC network vary with the state of the battery SOC, and satisfy 0≤α _d , β _d ≤1. When α _d , β _d =0, the fractional element FOE is equivalent to a resistor; when α _d , β _d =1, the fractional element FOE is equivalent to an integer-order capacitance; when 0<α _d , β _d When <1, the fractional-order element FOE is a fractional-order capacitor;

The orders α and β of the fractional-order elements FOE _1c and FOE _2c in the charging branch of the RC network vary with the state of the battery SOC, and satisfy 0≤α _c , β _c ≤1. When α _c , β _c =0, the FOE of the fractional-order element is equivalent to a resistor; when α _c , β _c =1, the FOE of the fractional-order element is equivalent to an integer-order capacitance; when 0<α _c , β _c When <1, the fractional-order element FOE is a fractional-order capacitor.

An identification method for a fractional-order variable-order equivalent circuit model of a lithium battery, comprising the following steps:

Step 1: Write the expression of the fractional-order mathematical model of the discharge process and static state of the lithium battery;

Step 2: Perform constant current charge and discharge on the lithium battery to obtain the available capacity C _Q and self-discharge resistance R _d of the battery model;

Step 3: Perform a pulse discharge test on the lithium battery to obtain the instantaneous drop of the battery terminal voltage at different SOCs when the battery starts to discharge, the instantaneous jump value of the battery terminal voltage after the discharge, the discharge current, and the zero input response of the battery terminal voltage, etc. data;

Step 4: Based on the data obtained in Step 3, identify the parameters and order of the model based on the least square method;

Step 5: Calculate the open circuit voltage OCV, ohmic internal resistance R _0d , electrochemical polarization internal resistance R _1d , electrochemical polarization fractional order capacitance FOE _1d , and concentration polarization at different SOCs based on the battery model parameters calculated in step 4 Internal resistance R _2d and concentration polarization fractional order capacitance FOE _2d ;

Step 6: Based on the model parameters obtained in step 5, identify the open circuit voltage OCV, ohmic internal resistance R _0d , electrochemical polarization internal resistance R _1d , electrochemical polarization fractional order capacitance FOE _1d , and concentration polarization internal resistance based on the least squares method. The relationship between resistance R _2d and concentration polarization fractional capacitance FOE _2d and SOC;

Step 7: According to the parameters obtained in steps 2 to 6, build a fractional-order variable-order equivalent circuit model of the lithium battery in Matlab.

The terminal voltage of a lithium battery during discharge can be expressed as:

${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; OCV</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the formula, U _bat is the battery terminal voltage; R _0d is the ohmic internal resistance; OCV _d is the discharge open circuit voltage; α, β are the orders of fractional order elements FOE _1d and FOE _2d , satisfying 0<α, β≤1; i _dis is the discharge current; τ _1d , τ _2d are the time constants of the two RC networks respectively.

When α, β=0, the fractional-order element FOE is equivalent to a resistor; when α, β=1, the fractional-order element FOE is equivalent to a capacitor;

U _1d (0+) and U _2d (0+) are the initial values of the terminal voltages of the two fractional RC branches at the end of battery discharge, and their values can be expressed as:

U _1d (0+)＝i _dis R _1d (2)

U _2d (0+)＝i _dis R _2d (3)

After the battery is discharged, the terminal voltage of the battery can be expressed as:

${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; OCV</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

In the formula, the polarization voltage of the battery and It gradually decreases with the increase of time, when t→∞, and Tends to 0, at this time the battery terminal voltage _Ubat is equal to the open circuit voltage OCV of the battery.

The specific process of the fifth step is: due to the existence of the ohmic internal resistance of the battery, when the battery is discharged, the battery terminal voltage will drop instantaneously, and its value is recorded as ΔU ₁ ; when the battery stops discharging, the battery terminal voltage will instantly jump, and its The value is recorded as ΔU ₂ , therefore, the ohmic internal resistance R ₀ of the battery can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

The electrochemical polarization internal resistance R _1d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Concentration polarization internal resistance R _2d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

The electrochemical polarization fractional order capacitance FOE _1d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; FOE</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

Concentration polarization fractional capacitance FOE _2d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; FOE</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

In the step six: there is a nonlinear relationship between the open circuit voltage OCV and the SOC, and the specific relationship is:

$\mathrm{<span\; class="notranslate">\; OCV</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; SOC</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; SOC</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; SOC</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; SOC</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

In the formula, a ₀ -a ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between battery ohmic internal resistance R _od and SOC is:

R _o (SOC) = b ₀ · e - ^SOC + b ₁ + b ₂ · SOC - b ₃ · SOC ² + b ₄ · SOC ³ (11)

In the formula, b ₀ -b ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between electrochemical polarization internal resistance R _1d and SOC is:

R _1d (SOC)=c ₀ ·e ^−SOC +c ₁ +c ₂ ·SOC−c ₃ ·SOC ² +c ₄ ·SOC ³ (12)

In the formula, c ₀ -c ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between electrochemical polarization fractional capacitance FOE _1d and SOC is:

FOE _1d (SOC)＝d ₀ SOC ⁵ +d ₁ SOC ⁴ +d ₂ SOC ³ +d ₃ SOC ² +d ₄ SOC+d ₅ (13)

In the formula, d ₀ -d ₅ are constants, which are identified from the experimental data based on the least square method.

The relationship between concentration polarization internal resistance R _2d and SOC is:

R _2d (SOC)=e ₀ ·e- ^SOC +e ₁ +e ₂ ·SOC-e ₃ ·SOC ² +e ₄ ·SOC ³ (14)

In the formula, e ₀ -e ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between concentration polarization fractional capacitance FOE _2d and SOC is:

FOE _2d (SOC) = f ₀ SOC ⁵ + f ₁ SOC ⁴ + f ₂ SOC ³ + f ₃ SOC ² + f ₄ SOC + f ₅ (15)

In the formula, f ₀ -f ₅ are constants, which are identified from the experimental data based on the least square method.

The relationship between the electrochemical polarization fractional order capacitance FOE _1d order and SOC is:

α(SOC)=g ₀ ·SOC ⁴ +g ₁ ·SOC ³ +g ₂ ·SOC ² +g ₃ ·SOC+g ₄ (16)

In the formula, g ₀ -g ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between concentration polarization fractional order capacitor FOE _2d order and SOC is:

β(SOC)=h ₀ ·SOC ⁴ +h ₁ ·SOC ³ +h ₂ ·SOC ² +h ₃ ·SOC+h ₄ (17)

In the formula, h ₀ -h ₄ are constants, which are identified from the experimental data based on the least square method.

Beneficial effects of the present invention:

1. Extend the traditional second-order RC equivalent circuit model to the fractional order, and identify the model parameters and orders at different SOCs based on the least squares method, and obtain a fractional-order equivalent circuit model that varies according to the SOC order;

2. Due to its special material and chemical properties, lithium batteries exhibit fractional-order kinetic behavior. The accuracy of describing battery characteristics with integer orders is greatly limited, while using fractional-order calculus to describe those with fractional-order It can better describe the essential characteristics of the object and its behavior when it is an object with specific characteristics;

3. Due to the addition of the unknown parameter of fractional order, the model has gained more degrees of freedom, greater flexibility and novelty;

4. Since the fractional calculus has a certain memory function, and is more in line with the simple philosophical view of the universal continuity in nature, the fractional-order variable-order equivalent circuit model thus obtains higher precision, better dynamic performance and stability;

5. Compared with the traditional second-order RC model, since the number of RC networks is not increased, the present invention effectively solves the contradiction between model accuracy and practicability, has high practical value, and is suitable for constant current charging of batteries Discharge, pulse charge and discharge, and UDDS cycle conditions provide an accurate and easy-to-implement battery model for accurate SOC estimation.

Description of drawings

Fig. 1 is a schematic structural diagram of a fractional-order variable-order equivalent circuit model of a lithium battery of the present invention, wherein the c mark represents charging, and the d mark represents discharging;

Fig. 2 is the response process diagram of battery cell voltage under the pulse charge of the present invention;

Fig. 3 is the response process diagram of battery cell voltage under the pulse discharge of the present invention;

Fig. 4 is the comparison chart of recovery response of battery terminal voltage after variable order fractional order, integer order and fixed fractional order model simulation pulse discharge of the present invention, wherein (a) is the whole picture, (b) is the partial enlarged picture;

Fig. 5 is the relationship diagram of open circuit voltage OCV and SOC under the pulse charging of the present invention;

Fig. 6 is the relation figure of ohmic internal resistance R ₀ and SOC under pulse charging of the present invention;

Fig. 7 is the relationship figure of electrochemical polarization internal resistance R _1c and SOC under pulse charging of the present invention;

Fig. 8 is the relationship diagram of electrochemical polarization fractional order capacitance FOE _1c and SOC under the pulse charging of the present invention;

Fig. 9 is the relationship diagram between concentration polarization internal resistance R _2d and SOC under pulse charging of the present invention;

Fig. 10 is a graph showing the relationship between concentration polarization fractional order capacitance FOE _2c and SOC under pulse charging according to the present invention;

Fig. 11 is the relationship diagram between the order of electrochemically polarized fractional order capacitor FOE _1c and SOC under the pulse charging of the present invention;

Fig. 12 is a graph showing the relationship between the concentration polarization fractional order capacitor FOE _2c order and SOC under pulse charging according to the present invention;

Fig. 13 is the relationship diagram of open circuit voltage OCV and SOC under the pulse discharge of the present invention;

Fig. 14 is the relation figure of ohmic internal resistance R ₀ and SOC under the pulse discharge of the present invention;

Fig. 15 is the relationship figure of electrochemical polarization internal resistance R _1d and SOC under the pulse discharge of the present invention;

Fig. 16 is the relationship diagram between electrochemical polarization fractional order capacitance FOE _1d and SOC under the pulse discharge of the present invention;

Fig. 17 is the relationship figure between concentration polarization internal resistance R _2d and SOC under the pulse discharge of the present invention;

Fig. 18 is a graph showing the relationship between concentration polarization fractional order capacitance FOE _2d and SOC under pulse discharge according to the present invention;

Fig. 19 is a graph showing the relationship between the FOE _1d order and the SOC of the electrochemically polarized fractional-order capacitor under the pulse discharge of the present invention;

Fig. 20 is a graph showing the relationship between the FOE _2d order and the SOC of the concentration polarization fractional-order capacitor under the pulse discharge of the present invention;

Fig. 21 is a voltage output diagram of the equivalent circuit model of the pulse discharge down-variable fractional-order single battery of the present invention;

Fig. 22 is a voltage output diagram of the equivalent circuit model of the pulse charging down-variable fractional-order single battery of the present invention;

Fig. 23 is the voltage output diagram of the equivalent circuit model of the variable-order fractional-order single battery under the constant current discharge of the present invention;

Fig. 24 is a voltage output diagram of the equivalent circuit model of the constant current charging down-variable fractional order single battery of the present invention.

Detailed ways:

The present invention is described in detail below in conjunction with accompanying drawing:

Building a battery model refers to applying mathematical theory to describe the response characteristics and internal characteristics of the actual battery as comprehensively as possible. The so-called response characteristic refers to the corresponding relationship between the terminal voltage of the battery and the load current; the internal characteristic refers to the relationship between the internal variable ohmic internal resistance, polarization internal resistance and polarization voltage of the battery, SOC and temperature.

As shown in Figure 1, the fractional-order variable-order equivalent circuit model of the lithium battery disclosed by the present invention includes a running time circuit and an IV characteristic circuit, wherein the IV characteristic circuit includes two branches, and each branch includes two groups of one fraction A fractional-order RC loop composed of an order capacitor FOE and a resistor in parallel. The running time circuit includes battery self-discharging resistor R _d , capacitor C _Q and a current control current source circuit. The resistor R _d and capacitor C _Q are connected in parallel at both ends of the controlled source of the current control current source, and one end of the independent power supply is grounded.

The IV characteristic circuit includes ohmic internal resistance R ₀ , electrochemical polarization internal resistance R ₁ , electrochemical polarization fractional order capacitance FOE ₁ , concentration polarization internal resistance R ₂ , concentration polarization fractional order capacitance FOE ₂ and current control Current source, voltage control voltage source circuit, wherein:

The anode of the controlled source of the voltage control voltage source circuit is connected to two circuits, one path is connected to the diode D _d , and then connected to the resistor R _1d , resistor R _2d , and resistor R _od , and then connected to the positive electrode of the battery model, and one path is reversely connected to the diode D _c , and then connected to the resistor R _1c , resistor R _2c , and resistor R _oc are connected to the positive pole of the battery model. The fractional capacitor FOE _1d is connected in parallel to both ends of the resistor R _1d ; the fractional capacitor FOE _1c is connected in parallel to both ends of the resistor R _1c ; the fractional capacitor FOE _2d is connected in parallel to both ends of the resistor R _2d ; the fractional capacitor FOE _2c is connected in parallel to the resistor The two ends of R _2c ; the voltage between the positive and negative poles of the controlled source of the voltage control voltage source circuit is the battery open circuit voltage OCV.

The running time circuit and the IV characteristic circuit are connected through a current-controlled current source and a voltage-controlled voltage source. When charging and discharging the battery, the load current i _bat charges and discharges the capacitor C _Q through the current-controlled current source, changing C _Q The stored power represents the change of battery SOC, and the voltage OCV across C _Q also changes accordingly, and the controlled voltage source OCV of the IV characteristic circuit changes with the change of SOC.

Capacitance C _Q represents the available capacity of the battery, C _Q =3600 · C _Ah · f ₁ · f ₂ , where C _Ah is the battery capacity in ampere hours, f ₁ and f ₂ are the cycle life and temperature of the battery, respectively correction factor.

The current of the controlled source of the current control current source is the terminal current i _bat of the battery. When the battery is charging and discharging, the load current i _bat charges and discharges the capacitor C _Q through the current control current source, changing the power stored in the capacitor C _Q , To characterize the change of battery SOC.

The voltage across the controlled source of the current controlled current source is the battery open circuit voltage OCV.

An identification method using the above-mentioned fractional-order variable-order equivalent circuit model of a lithium battery, taking battery discharge as an example, the charging identification method is the same as that of discharging, and will not be repeated here. Include the following steps:

Step 1: Write the expression of the fractional-order mathematical model of the discharge process and static state of the lithium battery;

Step 2: Perform constant current charge and discharge on the lithium battery to obtain the available capacity C _Q and self-discharge resistance R _d of the battery model;

Step 3: Perform a pulse discharge test on the lithium battery to obtain the instantaneous drop of the battery terminal voltage at different SOCs when the battery starts to discharge, the instantaneous jump value of the battery terminal voltage after the discharge, the discharge current, and the zero input response of the battery terminal voltage, etc. data;

Step 4: Based on the data obtained in Step 3, identify the parameters and order of the model based on the least square method;

Step 5: Calculate the open circuit voltage OCV, ohmic internal resistance R _0d , electrochemical polarization internal resistance R _1d , electrochemical polarization fractional order capacitance FOE _1d , and concentration polarization at different SOCs based on the battery model parameters calculated in step 4 Internal resistance R _2d and concentration polarization fractional order capacitance FOE _2d ;

Step 6: Based on the model parameters obtained in step 5, identify the open circuit voltage OCV, ohmic internal resistance R _0d , electrochemical polarization internal resistance R _1d , electrochemical polarization fractional order capacitance FOE _1d , and concentration polarization internal resistance based on the least squares method. The relationship between resistance R _2d and concentration polarization fractional capacitance FOE _2d and SOC;

Step 7: According to the parameters obtained in steps 2 to 6, build a fractional-order variable-order equivalent circuit model of the lithium battery in Matlab.

As shown in Figure 2, it is the response process figure of the battery cell voltage under the pulse charging of the present invention; As shown in Figure 3, it is the response process figure of the battery cell voltage under the pulse discharge of the present invention; The terminal voltage of the battery in the pulse discharge process Can be expressed as:

${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; OCV</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the formula, U _bat is the battery terminal voltage; R _0d is the ohmic internal resistance; OCV _d is the discharge open circuit voltage; α, β are the orders of fractional order elements FOE _1d and FOE _2d , satisfying 0<α, β≤1. When α, β=0, the fractional-order element FOE is equivalent to a resistor, and when α, β=1, the fractional-order element FOE is equivalent to a capacitor. U _1d (0+) and U _2d (0+) are the initial values of the terminal voltages of the two fractional RC branches at the end of battery discharge, and their values can be expressed as:

U _1d (0+)＝i _dis R _1d (2)

U _2d (0+)＝i _dis R _2d (3)

After the battery is discharged, the terminal voltage of the battery can be expressed as:

${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; OCV</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

In the formula, the polarization voltage of the battery and It gradually decreases with the increase of time, when t→∞, and Tends to 0, at this time the battery terminal voltage _Ubat is equal to the open circuit voltage OCV of the battery.

The specific method of step 5 is: due to the existence of the ohmic internal resistance of the battery, when the battery is discharged, the battery terminal voltage will drop instantaneously, and its value is recorded as ΔU ₁ ; when the battery stops discharging, the battery terminal voltage will instantly jump, its The value is noted as ΔU ₂ . Therefore, the ohmic internal resistance R ₀ of the battery can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

The electrochemical polarization internal resistance R _1d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Concentration polarization internal resistance R _2d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

The electrochemical polarization fractional order capacitance FOE _1d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; FOE</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

Concentration polarization fractional capacitance FOE _2d can be obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; FOE</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

The specific method of the step 6 is: there is a nonlinear relationship between the open circuit voltage OCV and the SOC, and the specific relationship is:

$\mathrm{<span\; class="notranslate">\; OCV</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; SOC</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; SOC</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; SOC</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; SOC</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

In the formula, a ₀ -a ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between battery ohmic internal resistance R _od and SOC is:

R _o (SOC) = b ₀ · e - ^SOC + b ₁ + b ₂ · SOC - b ₃ · SOC ² + b ₄ · SOC ³ (11)

In the formula, b ₀ -b ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between electrochemical polarization internal resistance R _1d and SOC is:

R _1d (SOC)=c ₀ ·e ^−SOC +c ₁ +c ₂ ·SOC−c ₃ ·SOC ² +c ₄ ·SOC ³ (12)

In the formula, c ₀ -c ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between electrochemical polarization fractional capacitance FOE _1d and SOC is:

FOE _1d (SOC)＝d ₀ SOC ⁵ +d ₁ SOC ⁴ +d ₂ SOC ³ +d ₃ SOC ² +d ₄ SOC+d ₅ (13)

In the formula, d ₀ -d ₅ are constants, which are identified from the experimental data based on the least square method.

The relationship between concentration polarization internal resistance R _2d and SOC is:

R _2d (SOC)=e ₀ ·e- ^SOC +e ₁ +e ₂ ·SOC-e ₃ ·SOC ² +e ₄ ·SOC ³ (14)

In the formula, e ₀ -e ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between concentration polarization fractional capacitance FOE _2d and SOC is:

FOE _2d (SOC) = f ₀ SOC ⁵ + f ₁ SOC ⁴ + f ₂ SOC ³ + f ₃ SOC ² + f ₄ SOC + f ₅ (15)

In the formula, f ₀ -f ₅ are constants, which are identified from the experimental data based on the least square method.

The relationship between the electrochemical polarization fractional order capacitance FOE _1d order and SOC is:

α(SOC)=g ₀ ·SOC ⁴ +g ₁ ·SOC ³ +g ₂ ·SOC ² +g ₃ ·SOC+g ₄ (16)

In the formula, g ₀ -g ₄ are constants, which are identified from the experimental data based on the least square method.

The relationship between concentration polarization fractional order capacitor FOE _2d order and SOC is:

β(SOC)=h ₀ ·SOC ⁴ +h ₁ ·SOC ³ +h ₂ ·SOC ² +h ₃ ·SOC+h ₄ (17)

In the formula, h ₀ -h ₄ are constants, which are identified from the experimental data based on the least square method.

1. Experiment setup

Experiments and simulations were carried out on Hite's 10-parallel 16-series cylindrical 26650 lithium iron phosphate power battery, with a nominal capacity of 23Ah and a nominal voltage of 51.2V. The battery test platform consists of advanced AVL battery simulation/test cabinet, AVL InMotion hardware-in-the-loop test platform, AVL control cabinet, temperature control box and AVL Lynx control software. The experiment records the battery voltage, current and SOC and other operating conditions, and the sampling frequency is set to 1Hz.

Considering the difference in charge and discharge parameters, the hybrid pulse test in the HPPC hybrid pulse test (Hybrid Pulse Power Characterization Test, HPPC) was changed to a unidirectional pulse test, that is, the power battery pulse charge test and pulse discharge test. The so-called pulse discharge is to discharge a fully charged battery at a room temperature of 25 degrees to an SOC of 95% with a current of 0.2C, stop discharging and let it stand for 45 minutes, then discharge at the same current to an SOC of 90%, and so on until End when SOC is 0%. The pulse charging process is similar to the pulse discharging process, and will not be repeated here.

2. Model order and parameter identification

As shown in FIG. 4 , it is a comparison chart of recovery response of battery terminal voltage after pulse discharge simulation by the variable order fractional order, integer order and fixed fractional order models of the present invention. It can be seen from the figure that the simulation accuracy of the fractional-order variable-order equivalent circuit model is the highest.

(1) Open circuit voltage OCV model

According to the battery open circuit voltage measured at different SOC (3.45%, 5%, 10%...90%, 95%, 98.89%), the charging open circuit voltage and discharging open circuit voltage OCV of the battery can be respectively fitted, as shown in Figure 5 and Figure 13 shows. And according to the formula (10), the parameters a ₀ -a ₄ can be identified by using the Matlab cftool toolbox, as shown in Table 1.

Table 1 The open circuit voltage OCV parameters obtained by fitting the Matlab cftool toolbox

parameter a ₀ a ₁ a ₂ a ₃ a ₄ Charging identification value -1.015 1.915 -0.02047 -0.006009 0.03582 Discharge identification value 2.588 -1.499 -0.007343 -0.01424 0.04207

(2) Ohm internal resistance R ₀ model

According to the ohmic internal resistance of the battery measured at different SOC (3.45%, 5%, 10%...90%, 95%, 98.89%), the charging ohmic internal resistance and discharging ohmic internal resistance R ₀ of the battery can be respectively obtained by fitting, As shown in Figure 6 and Figure 14. And according to the formula (11), the parameters b ₀ -b ₄ can be identified by using the Matlab cftool toolbox, as shown in Table 2.

Table 2 The ohmic internal resistance R ₀ parameters obtained by fitting the Matlab cftool toolbox

parameter b _{0 b1} b ₂ b ₃ b ₄ Charging identification value 0.5612 -0.5588 0.5546 -0.2568 0.05642 Discharge identification value -0.638 0.6397 -0.634 0.2967 -0.0663

(3) Electrochemical polarization internal resistance R ₁ model

According to the battery electrochemical polarization internal resistance measured at different SOC (3.45%, 5%, 10%, 15%..., 95%), the charging electrochemical polarization internal resistance and discharge voltage of the battery can be fitted respectively. The chemical polarization internal resistance R ₁ is shown in Fig. 7 and Fig. 15 . And according to the formula (12), the parameters c ₀ -c ₄ can be identified by using the Matlab cftool toolbox, as shown in Table 3.

Table ₃ The electrochemical polarization internal resistance R1 parameters obtained by fitting with the Matlab cftool toolbox

parameter c ₀ c ₁ c ₂ c ₃ c ₄ Charging identification value 3.891 -3.887 3.87 -1.835 0.4279 Discharge identification value 10.26 -10.24 10.1 -4.607 0.9788

(4) Electrochemically polarized fractional order capacitor FOE ₁ model

According to the electrochemical polarization fractional order capacitance measured at different SOC (3.45%, 5%, 10%...90%, 95%, 98.89%), the charging electrochemical polarization fractional order capacitance and Discharge electrochemical polarization fractional order capacitance FOE ₁ , as shown in Fig. 8 and Fig. 16 . And according to the formula (13), the parameters d ₀ -d ₅ can be identified by using the Matlab cftool toolbox, as shown in Table 4.

Table 4 FOE ₁ parameters of electrochemically polarized fractional order capacitance obtained by fitting with Matlab cftool toolbox

parameter d ₀ d ₁ d ₂ d ₃ d ₄ d ₅ Charging identification value -3.775e ⁶ 8.913e ⁶ -7.623e ⁶ 2.976e ⁶ -5.542e ⁵ 9.351e ⁴ Discharge identification value 1.387e ⁸ -5.26e ⁸ 7.008e ⁸ -4.023e ⁸ 9.138e ⁷ -3.683e ⁶

(5) Concentration polarization internal resistance R ₂ model

According to the concentration polarization internal resistance of the battery measured at different SOC (3.45%, 5%, 10%, 15%..., 95%), the charging concentration polarization internal resistance and the discharge concentration polarization internal resistance of the battery can be fitted respectively. Differential polarization internal resistance R ₂ , as shown in Figure 9 and Figure 17. And according to the formula (14), the parameters e ₀ -e ₄ can be identified by using the Matlab cftool toolbox, as shown in Table 5.

Table ₅ Concentration polarization internal resistance R2 parameters obtained by fitting with Matlab cftool toolbox

parameter e ₀ e ₁ e ₂ e ₃ e ₄ Charging identification value 3.267 -3.265 3.271 -1.594 0.3971 Discharge identification value -4.582 4.584 -4.535 2.102 -0.4632

(6) Concentration polarization fractional capacitance FOE ₂ model

According to the concentration polarization fractional order capacitance of the battery measured at different SOC (3.45%, 5%, 10%, 15%..., 95%), the charging concentration polarization fractional order capacitance and The discharge concentration polarization fractional order capacitor FOE ₂ is shown in Fig. 10 and Fig. 18 . And according to the formula (12), the parameters f ₀ -f ₅ can be identified by using the Matlab cftool toolbox, as shown in Table 6.

Table 6 Concentration polarization fractional order capacitor FOE ₂ parameters obtained by fitting with Matlab cftool toolbox

parameter f ₀ f ₁ f ₂ f ₃ f ₄ f ₅ Charging identification value -1.842e ⁵ 4.507e ⁵ -4.116e ⁵ 1.704e ⁵ -2.857e ⁴ 4.572e ⁴ Discharge identification value -4.529e ⁵ 1.188e ⁶ -1.153e ⁶ 5.072e ⁵ -9.649e ⁴ 8.68e ⁴

(7) Electrochemically polarized fractional-order capacitor FOE ₁ -order model

According to the battery electrochemical polarization fractional capacitance order measured at different SOC (3.45%, 5%, 10%, 15%..., 95%), the charging electrochemical polarization fractional order of the battery can be fitted separately Capacitance FOE ₁ order and discharge electrochemical polarization fractional capacitance FOE ₁ order, as shown in Figure 11 and Figure 19. And according to the formula (16), the parameters g ₀ -g ₄ can be identified by using the Matlab cftool toolbox, as shown in Table 7.

Table 7 FOE _first -order parameters of electrochemically polarized fractional-order capacitors fitted by Matlabcftool toolbox

parameter g ₀ g ₁ g ₂ g ₃ g ₄ Charging identification value 21.24 -46.26 33.74 -8.854 0.9861 Discharge identification value -15.45 27.57 -14.44 1.862 0.6595

(8) Concentration polarization fractional-order capacitor FOE ₂ -order model

According to the battery concentration polarization fractional capacitance order measured at different SOC (3.45%, 5%, 10%, 15%..., 95%), the charging concentration polarization fraction of the battery can be fitted separately The _2nd order of capacitor FOE and the _2nd order of discharge concentration polarization fractional capacitor FOE are shown in Figure 12 and Figure 20. And according to the formula (17), the parameters h ₀ -h ₄ can be identified by using the Matlabcftool toolbox, as shown in Table 8.

Table 8 Concentration polarization fractional order capacitance FOE _2nd order parameters obtained by fitting with Matlab cftool toolbox

parameter h ₀ h ₁ h ₂ h ₃ h ₄ Charging identification value 2.752 -6.434 5.085 -1.319 0.4986 Discharge identification value 4.813 -10.91 8.4 -2.468 0.6779

3. Simulation and experimental verification

In order to verify the accuracy of the battery model, constant current charge and discharge and pulse charge and discharge experiments were carried out on the battery. As shown in Figure 21-22, it is the comparison between the experimental results of the battery terminal voltage obtained under pulse charging and discharging and the simulation results of the model. It can be seen from the figure that the fractional-order variable-order equivalent circuit model proposed by the present invention can better reflect the pulse charging and discharging process of the battery, indicating that the model is accurate. Among them, the error generated in the static stage is larger than the error generated in the constant current charge and discharge stage.

As shown in Figure 23-24, it is the comparison between the experimental results of the battery terminal voltage obtained under constant current charging and discharging and the simulation results of the model. It can be seen from the figure that the model error is relatively large in the initial and final stages of charging and discharging; while in the middle stage of charging and discharging, the output value of the model almost coincides with the experimental value. This is completely consistent with the nonlinear voltage characteristics of the lithium battery, which is steep at both ends and flat in the middle.

In summary, the simulation results obtained by the variable-order fractional-order equivalent circuit model disclosed in the present invention basically conform to the experimental data, the maximum error of the model is within 0.05V, and it is suitable for pulse charging and discharging and constant current charging and discharging of electric vehicles, etc. working conditions. The effectiveness of the variable-order fractional-order model is proved.

Although the specific implementation of the present invention has been described above in conjunction with the accompanying drawings, it does not limit the protection scope of the present invention. Those skilled in the art should understand that on the basis of the technical solution of the present invention, those skilled in the art do not need to pay creative work Various modifications or variations that can be made are still within the protection scope of the present invention.

Claims (10)

1. A lithium battery fractional-order variable-order equivalent circuit model is characterized in that it comprises a running time circuit and a battery IV characteristic circuit, and the described running time circuit and the battery IV characteristic circuit are carried out by a current control current source and a voltage control voltage source Signal transmission; the running time circuit includes battery self-discharging resistor R _d and capacitor C _Q , resistor R _d and capacitor C _Q are connected in parallel at both ends of the current control current source, and one end of the current control current source is grounded; the battery IV The positive end of the voltage control voltage source of the characteristic circuit is connected to one end of two parallel RC network branches, the negative end is connected to the negative end of the battery model, and each branch of the two parallel RC network branches Each includes two series-connected fractional-order RC loops and an internal resistance R _o , and the other ends of the two parallel-connected RC network branches are connected to the positive terminal of the battery model. 2. a kind of lithium battery fractional-order variable-order equivalent circuit model as claimed in claim 1, is characterized in that, two RC network branches connected in parallel are respectively RC network discharge branch and RC network charging branch, two The capacitors in each RC network branch are fractional order capacitors. 3. a kind of lithium battery fractional-order variable-order equivalent circuit model as claimed in claim 2, is characterized in that, the order α of described RC network discharge branch fractional-order element FOE _1d and FOE _2d , β changes with battery SOC It varies with different states and satisfies 0≤α _d , β _d ≤1. When α _d , β _d =0, the fractional-order element FOE is equivalent to a resistance. When α _d , β _d =1, the fractional-order element FOE is equivalent to an integer-order capacitance; when 0<α _d , β _d <1, the fractional-order element FOE is a fractional-order capacitance; The order α and β of the fractional-order elements FOE _1c and FOE _2c in the charging branch of the RC network vary with the state of the battery SOC, and satisfy 0≤α _c , β _c ≤1, when α _c , β _c =0 , the fractional-order element FOE is equivalent to a resistor, when α _c , β _c =1, the fractional-order element FOE is equivalent to an integer-order capacitance; when 0<α _c , β _c <1, the fractional-order element FOE is A fractional capacitance. 4. A kind of lithium battery fractional-order variable-order equivalent circuit model as claimed in claim 1, is characterized in that, in the two RC network branches that are connected in parallel in the IV characteristic circuit of the battery, the discharge branch includes successively series-connected Diode D _d , fractional order RC circuit composed of fractional order capacitance FOE _1d and resistance R _1d , fractional order RC circuit composed of fractional order capacitance FOE _2d and resistance R _2d , and resistance R _od ; The charging branch includes a series-connected reverse diode D _c , a fractional RC loop composed of a fractional capacitor FOE _1c and a resistor R _1c , a fractional RC loop composed of a fractional capacitor FOE _2c and a resistor R _2c , and a resistor R _oc . 5. a kind of lithium battery fractional-order variable-order equivalent circuit model as claimed in claim 1, is characterized in that, described run-time circuit and battery IV characteristic circuit establish connection by a current control current source and voltage control voltage source, When charging and discharging the battery, the load current i _bat charges and discharges the capacitor C _Q through the current control current source, changes the power stored in C _Q , and represents the change of battery SOC, and the voltage OCV at both ends of C _Q also changes accordingly, IV The controlled voltage source OCV of the characteristic circuit changes with the change of SOC. 6 . The fractional-order variable-order equivalent circuit model of a lithium battery as claimed in claim 1 , wherein the voltage at both ends of the current-controlled current source is the battery open-circuit voltage OCV. 6 . 7. The identification method of a fractional-order variable-order equivalent circuit model of a lithium battery as claimed in any one of claims 1 to 6, characterized in that it comprises the following steps: Step 1: Write the expression of the fractional-order mathematical model of the discharge process and static state of the lithium battery; Step 2: Perform constant current charge and discharge on the lithium battery to obtain the available capacity C _Q and self-discharge resistance R _d of the battery model; Step 3: Perform a pulse discharge test on the lithium battery to obtain the instantaneous drop of the battery terminal voltage at different SOCs when the battery starts to discharge, the instantaneous jump value of the battery terminal voltage after the discharge, the discharge current, and the zero input response of the battery terminal voltage, etc. data; Step 4: Based on the data obtained in Step 3, identify the parameters and order of the model based on the least square method; Step 5: Calculate the open circuit voltage OCV, ohmic internal resistance R _0d , electrochemical polarization internal resistance R _1d , electrochemical polarization fractional order capacitance FOE _1d , and concentration polarization at different SOCs based on the battery model parameters calculated in step 4 Internal resistance R _2d and concentration polarization fractional order capacitance FOE _2d ; Step 6: Based on the model parameters obtained in step 5, identify the open circuit voltage OCV, ohmic internal resistance R _0d , electrochemical polarization internal resistance R _1d , electrochemical polarization fractional order capacitance FOE _1d , and concentration polarization internal resistance based on the least squares method. The relationship between resistance R _2d and concentration polarization fractional capacitance FOE _2d and SOC; Step 7: According to the parameters obtained in steps 2 to 6, build a fractional-order variable-order equivalent circuit model of the lithium battery. 8. The identification method of a fractional-order variable-order equivalent circuit model of a lithium battery as claimed in claim 7, wherein the terminal voltage of the lithium battery during discharge can be expressed as: ${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; OCV</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ In the formula, U _bat is the battery terminal voltage; R _0d is the ohmic internal resistance; OCV _d is the discharge open circuit voltage; α, β are the orders of fractional order elements FOE _1d and FOE _2d , satisfying 0<α, β≤1; i _dis is the discharge current; τ _1d and τ _2d are the time constants of the two RC networks; U _1d (0+) and U _2d (0+) are the initial terminal voltages of the two fractional RC branches at the end of the battery discharge . 9. The identification method of a fractional-order variable-order equivalent circuit model of a lithium battery as claimed in claim 8, wherein the values of U _1d (0+) and U _2d (0+) can be expressed as: U _1d (0+)＝i _dis R _1d (2) U _2d (0+)＝i _dis R _2d (3) After the battery is discharged, the terminal voltage of the battery can be expressed as: ${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; OCV</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; t</span>}}^{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ In the formula, the polarization voltage of the battery and It gradually decreases with the increase of time, when t→∞, and Tends to 0, at this time the battery terminal voltage _Ubat is equal to the open circuit voltage OCV of the battery. 10. A method for identifying a fractional-order variable-order equivalent circuit model of a lithium battery as claimed in claim 9, wherein the specific process of step five is: due to the existence of the ohmic internal resistance of the battery, when the battery is discharged , the battery terminal voltage will drop instantaneously, and its value is recorded as ΔU ₁ ; when the battery stops discharging, the battery terminal voltage will instantly jump, and its value is recorded as ΔU ₂ , therefore, the ohmic internal resistance R ₀ of the battery can be obtained by the following formula: ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&Delta;\; U</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&Delta;\; U</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; bat</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ The electrochemical polarization internal resistance R _1d can be obtained by the following formula: ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$ Concentration polarization internal resistance R _2d can be obtained by the following formula: ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; i</span>}}_{\mathrm{<span\; class="notranslate">\; dis</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$ The electrochemical polarization fractional order capacitance FOE _1d can be obtained by the following formula: ${\mathrm{<span\; class="notranslate">\; FOE</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$ Concentration polarization fractional capacitance FOE _2d can be obtained by the following formula: ${\mathrm{<span\; class="notranslate">\; FOE</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$