## 2 Data Description and Pre-Processing  
The experimental data used in this work is derived from the MIT-Stanford Toyota Research Center battery dataset, originally measured by [PERSON] et al. [18]. The dataset focuses on the APR18650M1A battery cells, manufactured by A123 Systems, Hangzhou, China. Table 1 provides an overview of the key parameters and usage recommendations for the APR18650M1A battery. Notably, the testing was conducted at a constant temperature of 30 \({}^{\circ}\)C, employing a one-step or two-step fast charging strategy. The charging protocol follows a constant-current-constant-voltage (CCCV) approach with an upper cutoff voltage of 3.6 V and a lower cutoff voltage of 2.0 V, as recommended by the manufacturer. For our study, we select 60 battery samples from the dataset and use the data of their first 400 charge-discharge cycles to validate the proposed clustering approach. All samples undergo 4C-discharge rates to ensure a uniform discharge process. The fundamental goal is to repurpose these retired or partially used cells for second-life applications, making it critical to quantify their current condition accurately and group them accordingly. From the available measurements, we focus on three key parameters [19], capacity, internal resistance (IR), and remaining useful life (RUL), as these factors play a dominant role in determining battery health, performance, and viability for second-life usage. Battery capacity is a primary indicator of the SOH for lithium-ion batteries. The discharge capacity \(C_{n}\) of the nth cycle can be expressed by the following:  
\[C_{n}=C_{n}^{H}-C_{n}^{L} \tag{1}\]  
where \(C_{n}^{H}\) denotes the total capacity after the battery is fully discharged and \(C_{n}^{L}\) is the capacity at the start of discharge for cycle \(n\). As the number of cycles increases, \(C_{n}\) typically declines, adversely affecting the battery's usable lifetime. Figure 1 illustrates how the discharge capacity decreases over the cycles. Internal resistance (IR) is another critical parameter that impacts the performance of battery and thermal characteristics. A higher IR leads to increased energy losses and faster degradation. When the battery starts discharging, there is an instantaneous voltage drop [20]. Let \(t\) be the moment the battery begins discharging, and \(U\) be the interpolated voltage at that instant. If t\({}_{1}\) and t\({}_{2}\) are two closely spaced sampling points with voltages U\({}_{1}\) and U\({}_{2}\), the ohmic internal resistance \(R_{n}\) during the nth discharge cycle can be approximated by the following:  
\[R_{n}=\frac{U_{n}}{I_{n}}\;\textit{where}\;U_{n}=\left|\frac{U_{2}-U_{1}}{t_{ 2}-t_{1}}\right|_{t} \tag{2}\]  
where \(U_{n}\) is the effective voltage change caused by internal resistance and \(I_{n}\) is the discharge current of the nth cycle. Figure 2 illustrates the rapid voltage-current fluctuations occurring during instantaneous discharge of the power lithium-ion batteries due to internal resistance. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Nominal Capacity & 1.1 Ah \\ Nominal Voltage & 3.3 V \\ Recommended standard charge method & 1.5 A to 3.6 V (CCCV) \\ Recommended charge and cut-off voltage at 25\({}^{\circ}\) & 3.6 to 2.0 V \\ Operating temperature range & –30\(\sim\)60 \({}^{\circ}\)C \\ \hline \end{tabular}
\end{table} 
Table 1: Specifications and cycling conditions of the battery cell. Figure 1: Relationship between capacity and voltage. RUL represents the projected service life or the number of additional cycles a battery can deliver before reaching a specified end-of-life threshold. In this dataset, we can calculate RUL directly from the measured capacity. When a battery's capacity decays to 80% of its rated capacity \(C_{R}\), it is considered retired [21]. That is, the condition for retirement is as follows:  
\[C\leq 0.8C_{R} \tag{3}\]  
where \(C\) is the actual capacity of the battery. Figure 3 shows the variation trend of battery capacity with the number of cycles.