## 3 Results and Discussions  
### ECM Development and Parameter Identification  
In the Simulink environment, an ECM incorporating a second-order RC circuit based on the Thevenin model was constructed for the battery simulation (Figure 6). This work focused on ternary lithium batteries, and high-performance pulse discharge tests (HPPC) were executed to gather pulse discharge data spanning from 100% to 0% SOC. These data were used to support the subsequent equivalent circuit modeling. The accuracy of the parameters directly influences the precision of the ECM. We identified the parameters based on the pulse discharge data using the methods described in Section 2.1. This process included determining the battery's internal resistance, polarization resistance, and polarization capacitance. We conducted parameter identification for a 32 Ah ternary lithium battery with an operating voltage range of 3.3 V to 4.2 V. Figure 7 illustrates the variation of different parameters in the ECM as a function of the battery's SOC. Each subplot represents a specific component of the ECM, showing how internal battery characteristics fluctuate during charging and discharging cycles. For the transient response of the battery, the resistance R\({}_{0}\) exhibits a decreasing trend as the SOC decreases, reaching its minimum value at a mid-level SOC. However, as the SOC continues to drop, R\({}_{0}\) begins to increase slightly, which suggests that the internal resistance rises as the battery's charge diminishes further. Regarding the relaxation response, the resistances R\({}_{1}\) and R\({}_{2}\) in the two RC branches show generally opposite trends. The R\({}_{1}\) exhibits an overall downward trend, with a particularly notable decrease to near zero when the SOC exceeds 0.9. Meanwhile, within the SOC range of 0.2 to 0.4, the R\({}_{1}\) undergoes a steep decline followed by a rapid ascent. Conversely, the R\({}_{2}\) generally remains at a low level, except for a notable peak when the R\({}_{2}\) is in a valley. In addition, the R\({}_{2}\) undergoes a severe increase and reaches its maximum value within the SOC range of 0.9 to 1. Meanwhile, the capacitances C\({}_{\text{P}_{1}}\) and C\({}_{\text{P}_{2}}\) display an overall oscillatory upward trend as the SOC increases. At low SOC levels, a notable dip appears in C\({}_{\text{P}_{1}}\) of the first RC branch, whereas C\({}_{\text{P}_{2}}\) in the second RC branch shows a distinct peak in the corresponding SOC range. The magnitude of this peak in C\({}_{\text{P}_{2}}\) approximately matches the decrease observed in C\({}_{\text{P}_{1}}\). This  
Figure 6: Equivalent Circuit Model of a Single-Cell Battery in Simulink. The ECM model is based on Simscape components provided by MATLAB Simulink, the version is 2024b. The ‘Origin’ box represents OCV, the ‘Pink’ box represents R0, and the ‘Green’ box represents the RC circuit. phenomenon suggests that the first RC branch may struggle to fully capture the battery's relaxation response under certain conditions, a gap compensated for the second RC branch. This finding underscores the superior accuracy of higher-order RC models compared to lower-order ones in fitting the battery's response curve, highlighting their robustness and capability in capturing intricate battery behaviors. Figure 8 shows the 14 sets of the SOC and OCV distribution points obtained from the test data. A polynomial fitting method was used to establish the relationship between the SOC and OCV. To balance the fitting accuracy and polynomial complexity, different polynomial orders were attempted, as shown in Table 2. The 5 th-order polynomial yielded an R\({}^{2}\) of 0.9994, the 6 th order reached an R\({}^{2}\) of 0.9997, and the 7 th order achieved an R\({}^{2}\) of 0.9998. Figure 8: Distribution of 14 SOC-OCV relationship points and the corresponding fitting curve.