## 2 Methodology
### FSEV Battery Pack Modelling  
The FSEV was designed as per the vehicle characteristics specified in the Formula Bharat rule book [44]. The features of the designed vehicle (Table 1) and the Formula Bharat track parameters at the Kari Motor Speedway, Coimbatore, India were fed into OptimumLap software to obtain the required drive cycle. Drive cycle characteristics and FSEV features were put into the vehicle powertrain model created in Simulink, MATLAB. The powertrain model resulted in the input current profile (Figure 1) for further computations. The charging current was limited to 20 A as an additional safety margin considering the 40 A limit specified by the cell manufacturer design guide (Table 2). The FSEV battery pack was designed as per the constraints and specifications provided by the competition rulebook and the A123 cell manufacturer (based on real experimentation datasets) (Tables 2 and 3). Figure 1: Current profile for a single cell obtained from powertrain model. The FSEV battery pack was designed to have 90 cells connected in series, all drawing the same amount of current. In the current study, SOC was estimated for one cell assuming ideal operating conditions over one lap of the Formula Bharat track. For cell parameter estimation, the series internal resistance (\(R_{0}\)) was first approximated as per the direct current resistance (DCR) \(-\%\) SOC characteristics prescribed in the A123 manufacturer design guide (based on actual experiments) [46]. The remaining cell parameters, namely, parallel internal resistance (\(R_{1}\)) and capacitance (\(C_{1}\)), were determined by mutual substitution in the following equations (also derived from actual experiments) [47,48]  
\[R_{0} =-0.000513(SOC)+0.002733 \tag{1}\] \[R_{1} =0.001426(SOC)^{-0.771947}\] (2) \[C_{1} =3297.55 log(SOC)+13,481.96 \tag{3}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter/Specification** & **Value** & **Parameter/Specification** & **Value** \\ \hline Vehicle Mass & 300 kg & Tire Rolling Drag & 0.03 \\ Drag coefficient & 0.7 & Power Scaling Factor & 95\% \\ Downforce coefficient & 0.5 & Aero Scaling Factor & 90\% \\ Frontal area & 0.9 m\({}^{2}\) & Grip Scaling Factor & 90\% \\ Divertain efficiency & 90\% & Mass Lateral Friction & 260 kg \\ Tire rolling radius & 0.203 m & Mass Longitudinal Friction & 260 kg \\ Air density & 1.23 kg/m\({}^{3}\) & Aero Efficiency & 0.7142 \\ Final drive ratio & 5 & Motor Thermal Efficiency & 80\% \\ Longitudinal friction & 1.4 & Motor Torque Data & 90 N\(\cdot\)m \\ Lateral friction & 1.5 & Motor RPM Data & 6500 rpm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: FSEV vehicle design parameters [45]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Lower Limit** & **Upper Limit** \\ \hline Charge current & 0 A & 40 A \\ Discharge current & 0 A & 300 A \\ Cell voltage & 2 V & 3.65 V \\ Cell temperature & 0 \({}^{\circ}\)C & 55 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 2: BMS constraints. \begin{table}
\begin{tabular}{c c} \hline \hline
**Specifications** & **Value** \\ \hline Drive cycle distance & 22 km \\ Pack voltage & 300 V \\ Range & 22 km \\ Cell voltage nominal & 3.3 V \\ Cell capacity & 20 Ah \\ Pack capacity & 20 Ah \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Battery pack specifications. These \(R_{0}\), \(R_{1}\), \(C_{1}\) values along with the input current profile were utilised to compute voltage drops across \(R_{0}\), \(R_{1}\) and \(C_{1}\). Subsequently, these voltage drops and the input current profile were employed to obtain the total PL due to the cell components \(R_{0}\), \(R_{1}\) and \(C_{1}\). Thereafter, battery pack temperature was estimated using total PL (\(P_{loss}\)) as per the following relation  
\[C_{T}*\frac{dT}{dt}=(\text{ambient Temp}-\text{current Temp})*R_{T}+P_{loss} \tag{4}\]  
where \(C_{T}\) is specific heat capacity (J m\({}^{-3}\) K\({}^{-1}\)), and \(R_{T}\) is convection resistance (W m\({}^{-2}\) K\({}^{-1}\)) of the cell.