## 3 Cell Modeling  
This section formulates a cell model and identifies its parameters that can predict the terminal voltage despite battery aging. We compared a typical Thevenin model to its variant, an enhanced self-correcting (ESC) model [38], and selected the model with higher fidelity to the test data acquired from aged cells. The Thevenin model is one of the most widely utilized equivalent circuit models for battery state estimation, representing the time-dependent polarization voltages using one or more parallel resistor-capacitor (RC) elements. The ESC model extends the Thevenin model with an additional element that can describe the SOC-varying and instantaneous hysteresis voltages. These models were developed through an evolutionary process, progressively integrating states and their relevant parameters. Basically, the output voltage \(v_{T}\) was predicted based on the input current \(i\) flowing through the ohmic resistance \(R_{0}\). The states for the SOC \(z\) and the current \(i_{R_{j}}\) flowing through the polarization resistance \(R_{j}\) paired with capacitance \(C_{j}\) were incorporated, building the Thevenin model. To complete the ESC model in Figure 4, a state for the hysteresis voltage \(h\) was added, using the instantaneous hysteresis voltage \(s\) as an input. Figure 3: Variations in terminal voltage and SOC measurements over time under the (**a**) City, (**b**) Highway, and (**c**) High-speed profiles, representing stop-and-go city, highway, and aggressive highway driving, respectively. Figure 2: Variations in capacity measurements over cycles. As detailed in our previous study [37], the model parameters were identified sequentially. Rather than identifying all parameters at once, the time constant \(\tau_{j}\) was directly solved using suboptimal linear optimization. Based on the initial guess of the decay rate of hysteresis \(\gamma\), the states were computed. Subsequently, the maximum hysteresis \(M\), the maximum instantaneous hysteresis \(M_{0}\), \(R_{0}\), and \(R_{j}\) were iteratively solved by non-negative least-squares. Finally, \(\gamma\) was updated iteratively. This sophisticated method, referred to as a sequential estimator in this study, allows for effectively estimating many parameters with only one measurement, \(v_{T}\). In consequence, a total of 12 cases were created by applying the three load profiles to the four cell models. Both models utilize either one or two RC elements, referred to as Thevenin 1 RC, Thevenin 2 RC, ESC 1 RC, and ESC 2 RC. These models are named according to the load profiles applied during cell modeling. For instance, a Thevenin or ESC model developed using the High-speed profile is referred to as the High-speed model in this context. The load profiles used include City, Highway, and High-speed. Figure 5 plots changes in the error between the measured and predicted terminal voltages, comparing modeling accuracy across the type of load profiles, the type of cell models, and the number of cycles. First, regarding the effect of load type, the error exhibits no appreciable difference among them. The High-speed models operating solely in CD mode produce RMSEs similar to those achieved by the City or Highway models, which operate in both CD and CS modes. Regarding the effect of cycle number, the error generally increases with decreasing capacity, irrespective of load and model types. This observation can be accounted for by limitations in representing the physicochemical changes in aged cells. Regarding the effect of model type, the ESC models evidently outperform the Thevenin models in predicting the terminal voltage. The ESC model maintains higher accuracy throughout all of the cycles due to the incorporation of the hysteresis voltage state and its associated parameter. Here, hysteresis refers to the phenomenon where SOC is prone to a lagging effect based on previous charging and discharging. The original intention of integrating the hysteresis element is to improve SOC accuracy by modeling this phenomenon; however, unintended advantages occurred.