## 2 Battery State Space Model  
The EKF algorithm is used to estimate the dynamic SoC of a battery. A nonlinear observer is often combined with a battery model. The selection of a battery model has an important impact on obtaining an accurate SoC value. The common SoC estimation models include the electrochemical model (EM), the electrochemical impedance model (EIM), and the equivalent circuit model (ECM). It must be considered that it is difficult for the electrochemical model to determine all parameters, and the corresponding calculation is complex and time consuming [21], while the electrochemical impedance model is difficult,complex, and limited in terms of the actual matching process [22]. The equivalent circuit model can more accurately describe the dynamic behavior of a battery, and it is easy to implement and readily improves the accuracy in SoC estimation. For the Rint model [23], Thevenin model [15], and the second-order resistance capacitance parallel model [24] in the equivalent circuit model, fully considering the complexity and accuracy of the model, the commonly used Thevenin equivalent circuit model with a first-order RC circuit was selected to establish the battery state space model, as Figure 1 illustrates. Based on the Thevenin equivalent battery model, and according to Kirchhoff's law, the continuous state space equation of the battery is obtained, that is, the voltage and current are as follows (2) [25]:  
\[\left\{ \begin{array}{l}U_{L}(t)=U_{\text{OCV}}(t)-U_{b}(t)-I(t)\cdot R_{a}(t) \\ I(t)=\frac{U_{b}(t)}{K_{b}}+C_{b}\cdot\frac{d(U_{b}(t))}{dt}\end{array} \right. \tag{2}\]  
In the formula, \(U_{L}(t)\) represents the output voltage at time \(t\); \(U_{\text{OCV}}(t)\) is the battery open circuit voltage at time \(t\), which is a nonlinear function varying with SoC; \(U_{b}(t)\) represents the voltage of the RC circuit at time \(t\); \(R_{a},R_{b}\), and \(C_{b}\) represent the internal resistance of the battery, the resistance of the battery circuit, and the capacitance of the battery circuit, respectively; \(I(t)\) represents the total charging and discharging current at time \(t\). From Equation (2), the discrete state space Equation (3) [25] can be derived  
\[\left\{ \begin{array}{l}U_{b}(k)=U_{b}(k-1)\cdot e^{-\frac{At}{Rb+b}}+I(k-1) \cdot R_{b}(1-e^{-\frac{At}{Rb+b}})\\ U_{L}(k)=U_{\text{OCV}}(k)-I(k)\cdot R_{a}-U_{b}(k)\end{array} \right. \tag{3}\]  
where \(\Delta t\) is the sampling interval time, which is usually \(1\). \(k\) represents the number of sampling times. Then, the SoC value of lithium battery can be estimated using a modified Ampere-Hour integral method [26]. \[SoC(t)=SoC(0)-\frac{\int_{0}^{t}T_{k}\eta Idt}{Q} \tag{4}\]  
where \(\eta\) is the charging and discharging efficiency of the battery, \(T_{k}\) is the temperature correction factor, and \(Q\) indicates the rated capacity of the battery. They are all definite values in the numerical experiments. The discretization of Equation (4) [26] can be obtained as follows:  
\[SoC(k+1)=SoC(k)-\frac{T_{k}\eta\Delta t}{Q}I(k) \tag{5}\]  
To sum up, the equation expression of the model's state space can be deduced, which can be seen in Equation (6):  
Figure 1: [PERSON] battery model. \[\left\{ \begin{array}{l}\left[\begin{array}{c}\ SoC(k)\\ l_{b}(k)\end{array}\right]=\left[\begin{array}{cc}1&0\\ 0&e^{-\frac{Mt}{R_{b}^{2}c_{b}}}\end{array}\right]\left[\begin{array}{c}SoC(k-1 )\\ l_{b}(k-1)\end{array}\right]+\left[\begin{array}{c}-\frac{\mathbb{T}_{k}\mu \Delta t}{Q}\\ R_{b}(1-e^{-\frac{Mt}{R_{b}^{2}c_{b}}})\end{array}\right]I(k-1)+\mathbf{w}(k-1)\\ U_{L}(k)=U_{OCV}(k)-U_{b}(k)-R_{a}I(k)+v(k)\end{array} \right. \tag{6}\]  
In the equation, \(\mathbf{w}(k)\),\(v(k)\) represents the process noise and observation noise of the system at time \(k\), and the corresponding variances are \(\mathbf{q}_{k}\) and \(r_{k}\), respectively. In this paper, 18,650 individual lithium-ion power batteries are used.