## 2 The Determination of the Battery Model
### Battery Model Selection and Parameter Identification  
The estimation of the adaptive Kalman filter algorithm's estimation of SOC relies on an accurate and intuitive battery model. In this method, the Thevenin equivalent circuit shown in Figure 2 was used as the estimation model. The Thevenin equivalent circuit, also known as the first-order RC  
Figure 1: OCV-SOC curve. equivalent model, consists of a voltage source and an RC parallel network. Among them, the parallel circuit composed of Rp and Cp is used to simulate the dynamic response of the battery, which can accurately simulate the charging and discharging process of the battery; Uoc is the open circuit voltage of the battery and is positively correlated with the SOC of the battery; R\({}_{0}\) is the resistance characterizing battery characteristics; and U\({}_{\rm L}\) is the load voltage of the battery [25; 26; 27]. A discretized battery space model is constructed based on the Thevenin equivalent circuit model shown in Figure 2. Among them, the RC network and SOC constructed by Rp and Cp are used as state variables; the charge and discharge current \(I\) is used as an input variable, and the output voltage U\({}_{\rm L}\) of the battery is used as an output variable. The specific state space model is shown in Equations (3) and (4); \(\Delta\)t is the sampling time interval; Up(\(k\)) is the terminal voltage of capacitance Cp on the \(k\)th sampling point; and SOC(\(k\)) is the current SOC. And \(k\)-\(1\) is the sampled value at the previous moment; \(\tau\) is the time constant of the RC network; \(\eta\) is the coulombic efficiency of charge and discharge; and C\({}_{\rm N}\) is the total ampere capacity of the battery [28; 29; 30]. \[\left[ \begin{array}{c}U_{p}(k)\\ SOC(k)\end{array} \right]=\left[ \begin{array}{c}R_{p}(1-e^{-\frac{\Delta t}{ \tau}})\\ -\frac{\eta\Delta t}{C_{\rm N}}\end{array} \right]I(k)+\left[ \begin{array}{c}e ^{-\frac{\Delta t}{\tau}}&0\\ 0&1\end{array} \right]\left[ \begin{array}{c}U_{p}(k-1)\\ SOC(k-1)\end{array} \right] \tag{3}\]  
\[U_{L}(k)=\left[ \begin{array}{cc}-1&\frac{dU_{loc}(SOC)}{dSOC}\end{array}  \right]\left[ \begin{array}{c}U_{p}(k)\\ SOC(k)\end{array} \right]-R_{0}I(k) \tag{4}\]  
Since OCV and SOC are exponentially corresponding according to Figure 2, the exponential fitting method is adopted to fit the response curve, and then the required model parameters are obtained. The relationship between the discharge current and the output load voltage is obtained by the Thevenin equivalent circuit model:  
\[U_{L}=\left(U_{\infty}-IR_{0}-IR_{p}\right)+IR_{p}e^{-\frac{t}{\tau}} \tag{5}\]  
An expression is obtained by exponential fitting to Figure 2.  
\[U_{L}=c_{1}+c_{2}e^{-c_{3}t} \tag{6}\]  
Among them, \(c_{1}\), \(c_{2}\), and \(c_{3}\) are constants, and the circuit model parameters can be obtained by comparing Equations (5) and (6):  
\[\left\{ \begin{array}{c}R_{p}=\frac{c_{2}}{c_{3}R_{p}}\\ C_{p}=\frac{1}{c_{3}R_{p}}\\ R_{0}=\left(U_{\infty}-c_{1}-c_{2}\right)/I\end{array} \right. \tag{7}\]  
Figure 2: The Thevenin equivalent circuit. It can be known from Equations (6) and (7), that the parameters of the model components cannot be directly measured, and need to be obtained through identification. There are some standard identification methods and specific processes for the equivalent circuit parameter identification method. According to the HPPC (hybrid pulse power characterization) test method in the _FreedomCAR Battery Test Manual for Power-Assist Hybrid Electric Vehicles_, the experimental data of the lithium-ion battery voltage under constant temperature and different SOC conditions are obtained. The curve is fitted to the obtained data by MATLAB 2014b (The MathWorks, Natick, Massachusetts, USA), and the parameter values under a specific SOC condition at a constant temperature can be identified. The specific steps of the test are described in detail in [31; 32; 33; 34].