## 2 Cell Testing  
This section describes the test data generated to identify parameters in a battery model by using a fresh cell. The cell was then cycled to produce battery aging test data, which were used to trace changes in parameters and states with the developed cell model and its filter. We generally followed the test conditions used in our prior study [37]. We utilized a LIB cell (Samsung SDI, 18650-35E) with a nominal capacity of 3.5 Ah and a nominal voltage of 3.7 V. The load profiles were simulated by combining a DC electronic load (Kikusui, PLZ1004W) and a DC power supply (Kikusui, PWR800L). Seamless transitions between charging and discharging in the load profiles were achieved by integrating these devices with a charge-discharge system controller (Kikusui, PFX2512). Previously, we estimated current sensor bias to study its effect on SOC estimation, as the bias is a well-known source \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Ref.**} & \multirow{2}{*}{\begin{tabular}{c} **Battery Model** \\ **(Thevenin)** \\ \end{tabular} } & \begin{tabular}{c} **Filter** \\ **(State** \\ **Parameters)** \\ \end{tabular} & \
\begin{tabular}{c} **Estimates** \\ **(State** \\ **Parameters)** \\ \end{tabular} &
\begin{tabular}{c} **Level of** \\ **Aging [\%]** \\ **(Equation to SOH)** \\ \end{tabular} & \multirow{2}{*}{**Notes**} \\ \hline
[31] & 1 RC & \begin{tabular}{c} EKF \\ (multi-scale) \\ \end{tabular} & \
\begin{tabular}{c} SOC \\ All parameters \\ \end{tabular} &
\begin{tabular}{c} 97.5 \\ 90.3 \\ 80 \\ 70.3 \\ \end{tabular} \\ \hline
[32] & 2 RC & \begin{tabular}{c} EKF \\ EKF \\ \end{tabular} & \
\begin{tabular}{c} SOC \\ Capacity, Resistance \\ \end{tabular} &
\begin{tabular}{c} 100 \\ 85 \\ 78 \\ 49 \\ \end{tabular} \\ \hline
[33] & 2 RC & \begin{tabular}{c} AEKF \\ KF \\ \end{tabular} & \
\begin{tabular}{c} SOC \\ Resistance \\ \end{tabular} &
\begin{tabular}{c} 96.5 \\ 94 \\ 92.45 \\ \end{tabular} & Adaptive EKFs \\ \hline
[34] & Fractional \begin{tabular}{c} FOEKF \\ 2 RC \\ \end{tabular} & \begin{tabular}{c} SOC \\ FOEKF \\ \end{tabular} & \
\begin{tabular}{c} SOC \\ Capacity, Resistance \\ \end{tabular} &
\begin{tabular}{c} 93 \\ 88.2 \\ 80.4 \\ \end{tabular} & Fractional-Order EKFs \\ \hline
[35] & Fractional \begin{tabular}{c} FOMIUKF \\ 2 RC \\ \end{tabular} & \begin{tabular}{c} SOC \\ UKF \\ (multi-scale) \\ \end{tabular} & \
\begin{tabular}{c} 98.1 \\ Capacity, Resistance \\ \end{tabular} &
\begin{tabular}{c} Fractional-Order \\ Multi-Innovation \\ 91.5 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Prior studies on the co-estimation of SOC and capacity using a dual filter for aged batteries. of process noise affecting the reliability of the SOC. Since the bias is not subject to change with battery aging, only a fresh cell was used. However, this study requires aged cells with reduced capacity to investigate the impact of capacity variations on SOC estimation. For this purpose, the cell was intentionally aged through continuous cycling, as depicted in Figure 1b. Each cycle involved a constant current charge at 2 C-rate until the voltage reached 4.2 V, followed by a constant voltage charge at this level until the current dropped to 1 A. After resting for an hour, the cycle proceeded with a constant current discharge at 2 C-rate until the voltage fell to 2.5 V. This high current, applied over a wide SOC range, accelerated capacity loss. Before starting this cycle test, the cell was evaluated through an initial reference performance test (RPT), as shown in Figure 1a. The test was repeated every 100 cycles up to cycle 300, and then every 300 cycles up to cycle 2100. The test was designed to mimic the real-world operations of PHEVs, as described below. First, the batteries in the vehicle are plugged in to charge. The electrical energy stored in the batteries is then used to propel the vehicle, gradually depleting SOC. This mode is thus referred to as the charge-depleting (CD) operation.