## 2 Lithium-Ion Battery Modeling and Experimental Setup  
### Lithium-Ion Battery Modeling  
The energy stored in a lithium-ion battery is shown as capacity and voltage, and is determined by electrode materials and cell chemical composition. In a lithium-ion battery, the lithium ions are intercalated and de-intercalated in positive and negative electrodes; the amount of lithium ions that are involved determines the capacity, and the potential difference between positive and negative electrodes determines the cell voltage. The separator separates positive and negative electrodes to avoid short circuits, and the electrolyte allows the lithium ions to move between these two electrodes. Figure 1a shows the discharge process of a typical lithium-ion battery. The lithium-ions pass through the separator from the negative electrode to the positive electrode, and the electrons pass through the load circuit. There is a complex system change behind the above simple discharge process. As shown in Figure 1b, the terminal voltage will decrease non-linearly as the discharge current increases, which is caused by the internal resistance of the cell, and the polarization effect near the electrode. OCV is a non-linear function of DOD and will not change with load current. As shown in Figure 1c, the terminal voltage gradually increases and tends to OCV with the weakening polarization effect after the end of discharge. The polarization effect is non-linear and varying with the DOD and the relaxation time. The non-linear OCV and polarization effect are fundamental properties of lithium-ion batteries. Accurately measuring the OCV curve and modeling polarization effects are critical for SOC estimation. The lithium-ion equivalent circuit model is introduced below, and the measurement and construction method of the OCV curve will be proposed in Section 3. The equivalent circuit model is widely used in battery management systems to describe the dynamics of the battery due to its clear physical meaning and balance between computational accuracy and complexity [6; 7; 21]. Over the past decade, researchers have proposed many kinds of equivalent circuit models, such as the Rint model [22], Thevenin model [23], PNGV model [24], and GNL model [25]. Comparing the SOC estimation accuracy and robustness of the above different equivalent circuit models, [PERSON], [PERSON] and [PERSON] [9] and [PERSON], et al. [26] found that the estimation accuracy does not always improve with the increase of the capacitive resistance network. In contrast, the higher-order equivalent circuit models are more prone to overfitting and reduce their robustness. They may only  
Figure 1: (**a**) The discharge process of a typical lithium-ion battery. (**b**) open-circuit voltage (OCV), ohmic resistance voltage, and polarization voltage as functions of discharge current. (**c**) The relaxation process of the lithium-ion battery, where \(\tau\) is the relaxation time. perform well under test conditions but are poor in noisy environments. Then, the Thevenin model is used in this paper to describe the lithium-ion battery, as shown in Figure 2.  
\(V_{ocv}\) is the OCV, \(V_{t}\) is the terminal voltage, \(V_{o}\) is the voltage drop caused by ohmic resistance, \(V_{p}\) is the polarization voltage drop, \(I\) is the current, \(R_{o}\) is the internal ohmic resistance, \(R_{p}\) and \(C_{p}\) are the polarization resistance and the polarization capacitance, respectively. The model equations can be described as follows:  
\[\left\{ \begin{array}{l}\dot{V}_{p}=\frac{I}{C_{p}}-\frac{V_{p}}{C_{p}R_{p}}\\ V_{t}=V_{soc}-V_{p}-V_{o}\end{array} \right.. \tag{1}\]  
After discretization:  
\[\left\{ \begin{array}{l}V_{p,k+1}=\exp\biggl{(}-\frac{\Delta t}{C_{p}R_{p}} \biggr{)}V_{p,k}+\biggl{(}1-\exp\biggl{(}-\frac{\Delta t}{C_{p}R_{p}}\biggr{)} \biggr{)}R_{p}I_{k}\\ V_{t,k}=V_{ocv,k}-V_{p,k}-V_{o}\end{array} \right., \tag{2}\]  
where \(\Delta t\) donates sampling interval, which is set to 1s in this paper. \(V_{t,k}\) is the measured terminal voltage at time \(k\).