**2. Establishment of Lithium-Ion Battery Model**

The ECMs (integer-order models and FOMs) are the most widely used type of battery models in various battery-related research because of their clear physical representation, ease of mathematical analysis, and simple parameter identification [34]. Among these models, considering the trade-off between prediction accuracy and structural complexity, the DPM stands out from all integer-order models [35]. However, the integer-order models cannot accurately reflect the electrochemical reactions inside the battery. Therefore, in [36], the authors have used fractional-order impedance elements to improve the integer-order model further. This is because, from the perspective of EIS, a circuit composed of fractional-order impedance elements can better fit the impedance characteristics of a lithium-ion battery, and thus has better applications in battery principle analysis, battery modeling, and state estimation. To investigate the effects of different operating conditions on the battery SOC estimation, the SOC estimation experiments under different operating conditions were conducted in this study, based on the DPM and FOM.

Figure 1a shows the structure of the DPM, and Figure 1b shows the structure of the FOM, where *Ud* represents the battery terminal voltage, *UOCV* stands for the OCV, the current is denoted by *I*, *R*<sup>0</sup> indicates the Ohmic internal resistance, the polarization internal resistances are represented by *R*<sup>1</sup> and *R*2, *C*<sup>1</sup> and *C*<sup>2</sup> stand for the polarization capacitances, the constant phase elements are denoted by *CPE*<sup>1</sup> and *CPE*2, and *W* indicates the Wahlberg element.

(a)Structure of the dual-polarization model (DPM)

**Figure 1.** Structural diagram of equivalent circuit models (ECMs).
