## 2 Equivalent Circuit Modeling of the Lithium-Ion Batteries  
### Modeling of the Equivalent Circuit Model and Parameter Identification of the Lithium-Ion Battery  
In general, the Thevenin ECM parameters of LiBs are identified through EIS analysis and time-domain extraction using the HPPC test. The impedance of the Thevenin ECM is obtained by applying a small-amplitude sine waveform of the AC voltage and current to the electrodes. In the AC impedance method, the impedance parameters are identified by generating Nyquist plots based on the EIS data measured with a potentiostat. However, this method necessitates additional equipment, specifically a potentiostat, alongside a charge-discharge cycler. In contrast, the time-domain-based Thevenin ECM method utilizes voltage and current time-series data. This approach is advantageous because it eliminatesthe need for costly impedance measurements using a potentiostat. Furthermore, battery cells, packs, and modules can be characterized and modeled directly based on the data collected solely with batter charge-discharge cycler equipment. The Thevenin ECM is illustrated in Figure 1.  
In this study, the utilized LiB was the Panasonic NCR18650 PF LiB with the specifications listed in Table 1.  
The experimental battery charging/discharging cycler system is shown in Figure 2. The WBCS3000S battery cycler and its operating software is Smart Interface software Ver. 1.8, which were manufactured by WonATech Corp. (Seoul, Republic of Korea). The charging temperatures of the battery were measured using thermocouples attached to the battery, which was fixed in a low-temperature chamber. The desired charging and discharging conditions were regulated by Smart Interface software installed on a desktop computer. The measured voltage, current, and temperature of the battery were transferred to the desktop via the battery cycler. The single HPPC test involved a cycle 1C discharge/charge current held for 20 s, followed by a 3C discharge current held for 60 s. Ten cycles were spaced at 1.0, 0.9, 0.8,..., 0.2, and 0.1 of the SOC, as depicted in Figure 3.  
The corresponding curve of the HPPC experimental voltage variation for the current pulse input variation is shown in Figure 4. Ten open-circuit voltage values corresponding to the SOC values from 1.0 to 0.1 could be obtained directly, as illustrated in Figure 4a. The analysis in Figure 4b provided the following four features: (1) As the discharge started at \(t_{1}\), the terminal voltage of the battery dropped suddenly from \(V_{1}\) to \(V_{2}\) because of the voltage change caused by the resistance of the battery. (2) From \(t_{2}\) to \(t_{3}\), the terminal voltage of the battery decreased slowly from \(V_{2}\) to \(V_{3}\) due to the battery's polarization effect. (3) From \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Nominal Capacity & 2900 mAh \\ Nominal Voltage & 3.6 V \\ Cut-off Voltage & 2.5 V \\ Standard Charge & CC-CV, 1375 mA, 4.2 V, 4.0 h \\ Dimensions & 18.5 mm (diameter), 65.3 mm (height) \\ Charge : & 0 to +45 °C \\ Temperature & Discharge : \(-\) 20 to +60 °C \\ Storage : & \(-\) 20 to +50 °C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the Panasonic NCR18650 PF LiB. Figure 1: Thevenin equivalent circuit model, where \(R_{0}\) denotes the internal resistance, \(R_{p}\) and \(C_{p}\) represent the polarization resistance and capacitance, respectively, which characterize the rapid electrode reaction to the battery, and \(C_{b}\) indicates the battery capacitance. The equivalent internal resistance is defined by \(R_{eq}=R_{0}+R_{p}\). to \(t_{4}\), the terminal voltage of the battery rose suddenly from \(V_{3}\) to \(V_{4}\) because of the voltage changes caused by the internal resistance of the battery. (4) The terminal voltage of the battery increased slowly from \(V_{4}\) to \(V_{5}\) for \(t_{4}\) to \(t_{5}\), which represents the discharge process of the polarization capacitance through the polarization resistance.