The battery model describes the mathematical relationship between the influencing factors of the battery and its working characteristics. The factors considered are voltage, current, power, SOC, temperature, internal resistance, internal pressure, cycle times and self-discharge. A large number of research literature on battery models are synthesized , the battery model is divided into four types: electrochemical model, thermal model, coupled model and performance model. The electrochemical model uses a mathematical method to describe the reaction process inside the battery based on the electrochemical theory. These models mainly describe the voltage characteristics of the battery, the current distribution of the battery electrodes, the separator, and the overpotential changes. The battery thermal model is used to study the heat generation and heat transfer process of the battery. Since the electrochemical reaction of the battery and the heat generation of the battery are mutually influenced, the establishment of a coupled model of the electrochemical process and the heat generation process has become a new method to study the working process of the battery. The battery performance model describes the external characteristics of the battery during operation. The battery performance model combined with mathematical methods can estimate the SOC of the battery, and the battery thermal model can be used to guide the design of the battery thermal management system.

**1. Battery performance model**Electric vehicle battery performance models can be further divided into simplified electrochemical models, equivalent circuit models, neural network models and specific factor models. The following are some typical battery performance models.

(1) Simplified electrochemical model

Electrochemical models are too complex to be applied in electric vehicles. Electric vehicles use simplified electrochemical models that estimate battery SOC and voltage changes.

①Peukert equation

The Peukert equation shown in the formula below is a classic battery performance model proposed in 1898. The Peukert equation states that the usable capacity of a battery decreases as the discharge current increases.

I^{n}T_{i}=K

In the formula, I is the discharge current; n is the Peukert constant, which is related to the battery structure, and is generally about 1.35 for lead-acid batteries; Ti is the discharge time of the current I; K_{i} a constant, indicating the theoretical capacity of the battery.

The Peukert equation and its application are illustrated with the discharge curve of a typical lead-acid battery shown in Figure 1. The discharge currents in the figure are 3C, 2C, 1C, C/1.5, C/2, C/5, C/10 and C/20 respectively. It can be seen that the larger the discharge current, the lower the battery termination voltage, and the shorter the time it takes for the battery to reach the termination voltage. This indicates that the larger the discharge current, the corresponding reduction in the available capacity of the battery.

Peukert constant n reflects the quality of battery performance in a sense. The closer n is to 1, the better the performance of the battery at high currents. The larger the value of n, the greater the capacity loss of the battery when the battery is discharged at a large current. It can be seen from Figure 2 that the two batteries with n=1.1 and n=1.3, even if the theoretical capacity is the same, when the discharge current is 25A, the capacity of the battery with n=1.3 is only half of that of the battery with n=1.1.

Since there is only one theoretical capacity of the same battery, and the discharge duration T_{1} and T_{2} under different discharge currents I_{1} and I_{2} are measured respectively, the Peukert constant n can be calculated from the following formula.

T_{1}I^{n}_{1}=T_{2}I^{n}_{2}=K

LgT_{1}+nlgI_{1}=lgT_{2}+nlgI_{2}

n=(lgT_{2}-lgT_{1})/(lgI_{2}-lgI_{1})

Simple battery parameter selection and calibration can be performed using the Peukert equation. For example, an electric car is equipped with 10 12V batteries, which are supplied by the battery manufacturer with a Peukert constant n of 1.1, which work in series. Assuming that the drive motor needs to draw 155A of current from the battery pack in a certain driving condition, if the vehicle speed under this driving condition is 60km/h and it is required to be able to travel 14km, and the battery can only discharge 80%, what theoretical capacity should be selected what about the battery?

The time required to travel 14km at 60km/h is t=14/60=0.233h. At this time, the battery is only discharged 80%, and the time corresponding to 100% discharge is T=0.233/0.8=0.292h; the corresponding discharge time at 155A is 0.292h The theoretical capacity of the battery is KI^{n}T=155^{1.1} × 0.291667=75A·h, the selected theoretical capacity K of the battery should be greater than 75A·h

②Shepherd model

The Shepherd model shown in the formula below is often used in hybrid electric vehicle analysis. This model was proposed in 1965 to describe the electrochemical behavior of the battery according to the battery voltage and current, and it is often used together with the Peukert equation to calculate the battery voltage and SOC at different power requirements.

E_{t}=E_{o}-R_{i}I-(K_{i}/1-f)

In the formula, E_{t} is the terminal voltage of the battery; E_{o} is the open circuit voltage when the battery is fully charged; Ri is the ohmic internal resistance; K_{i} is the polarization internal resistance; I is the instantaneous current; .

(2) Equivalent circuit model

Due to the needs of electric vehicle simulation technology, researchers have designed a large number of equivalent circuit battery performance models. The equivalent circuit model uses a circuit network to describe the working characteristics of the battery based on the working principle of the battery, and is suitable for a variety of batteries. According to the characteristics of circuit components, it can be divided into linear equivalent circuit model and nonlinear equivalent circuit model. In the simulation software ADVISOR developed by the National Renewable Energy Laboratory (NREL), several typical equivalent circuit models are integrated.

①Basic circuit model

The basic circuit model is the basis for other complex equivalent circuit models. The Thevenin model, shown in Figure 3, is the most representative circuit model. Capacitor C is connected in parallel with resistor R_{2} (to describe overpotential) and then in series with voltage source U_{oc} (to describe open circuit voltage) and resistor R_{1} (to describe battery internal resistance). Since the parameters of the Thevenin battery model cannot be changed with the change of battery operating conditions and SOC, the accuracy is poor.

②Nonlinear circuit model

Most of the circuit elements in nonlinear circuit models are not constants, but functions of voltage, temperature, or SOC. The PNGV model shown in Figure 4 is the standard battery model in the 2001 “PNGV Battery Test Manual”, and is also used as the standard battery model in the 2003 “FreedomCAR Battery Test Manual”. In the model, U_{oc} is an ideal voltage source, representing the open-circuit voltage of the battery; R_{o} is the ohmic resistance of the battery; Rp is the polarization resistance of the battery; C_{p} is the parallel capacitor next to R_{p}; I_{p} is the current on the polarization resistance; Capacitance C_{b }describes the change in open circuit voltage as the load current accumulates over time.

(3) Neural network model

The battery is a highly nonlinear system, and so far there is no analytical mathematical model that can describe the battery characteristics in all operating ranges. Neural network has nonlinear basic characteristics, parallel structure and learning ability, and can give corresponding output response to external excitation, which is suitable for battery modeling.

The ADVISOR software has been using the neural network model since 1999. The model was designed by Professor R. Mahajan of the University of Colorado. It is a two-layer neural network. The input is demand power and SOC, and the output is current and voltage. The model parameters are based on lead-acid battery test data at 25°C, and the accuracy can reach 5%.

The selection and number of input variables of the neural network affect the accuracy and computational complexity of the model. The error of the neural network method is greatly affected by the training data and the training method. All battery test data can be used to train the model and optimize the performance of the model, and the neural network model trained with this data can only be used within the range of the original training data. Networks are more suitable for mass-produced mature products.

(4) Specific factor model

The researchers designed a battery model with influencing factors as the research object, such as temperature and cycle life.

①Temperature capacity model When the battery works outside its optimal operating temperature range, the capacity will decay. The following formula is the most commonly used model to describe the effect of temperature on battery capacity.

C_{T}=C_{25}[1-α(25-T)]

In the formula, C_{T }is the capacity of the battery at temperature T; C_{25 }is the capacity of the battery at 25°C; α is the temperature coefficient, A h/°C; T is the working temperature of the battery. Different types or types of batteries have different temperature coefficients, which need to be obtained through experiments.

②Cycle life model The researchers established a battery cycle life model shown in formula , which describes the relationship between battery life and depth of discharge DOD, and the battery cycle life is characterized by the number of cycles.

Life=Life_{0}·e^{（M·DOD）}

In the formula, Life is the cycle life of the battery under a certain DOD; Life_{0} is the cycle life when the DOD extrapolated from the experimental data is zero; the coefficient M is different when the battery is different.

**2. Battery thermal model**The battery thermal model describes the laws of battery heat generation, heat transfer, and heat dissipation, and can calculate the temperature change of the battery in real time; the battery temperature field information calculated based on the battery thermal model can not only provide guidance for the design and optimization of the battery pack thermal management system, but also provide The optimization of battery cooling performance provides a quantitative basis.

The working battery pack itself is a heat source in an electric vehicle, and its heat dissipation environment is provided by the battery pack thermal management system. The internal heat generation rate of the battery pack is affected by the working current, internal resistance and SOC. The working current of the electric vehicle battery pack has no definite variation law, so the heat generation and heat dissipation process of the electric vehicle battery pack is a typical unsteady heat conduction process with a time-varying internal heat source. The thermal models of various power batteries can be described by the energy conservation equation for unsteady heat transfer shown in the following formula. The application object of the battery thermal model is any micro-element inside the battery. The left side of the thermal model represents the increase in the thermodynamic energy of the battery cell in unit time (unsteady term), and the first item on the right side represents the increase in the energy of the battery cell per unit time due to heat transfer through the interface ( Diffusion term), the second term on the right ^{·}q is the battery heat generation rate (source term). In the following formula, ρ_{k} is the density of the battery cell; C_{p·k} is the specific heat capacity of the battery cell and Ak is the thermal conductivity of the battery cell. As shown in the formula below,^{ ·}q is composed of the combination of heat generation caused by different heat generation factors. The thermal model in the form of rectangular coordinates shown in the above formula is often used for the calculation of the internal temperature field of the prismatic battery.

In order to reduce the complexity of the numerical calculation of the battery temperature field, the following assumptions are usually made for the battery: the various materials that make up the battery are uniform in medium and have the same density, the specific heat capacity of the same material is the same value, and the thermal conductivity of the same material is the same everywhere in the same direction. ; The specific heat capacity and thermal conductivity of various materials that make up the battery are not affected by changes in temperature and SOC; when the battery is charged and discharged, the current density in the core area of the battery is uniform and the heat generation rate is consistent. Based on the above assumptions, the simplified three-dimensional thermal model of the rectangular coordinate system shown in the above formula is obtained.

It can be seen from the above analysis that the essence of calculating the internal temperature field of the battery is to solve the thermal conductivity differential equation shown in the above formula. To solve the differential equation of thermal conductivity, three key problems need to be solved: accurate acquisition of thermophysical parameters ρ, C_{p}, λ; the accurate expression of the heat generation rate ^{·}q; the accurate determination of the solution conditions (initial conditions and boundary conditions). Thermophysical parameters, heat generation rate and fixed solution conditions constitute the three elements of the battery thermal model.