## 5 Case Study  
As a case study, the model developed in the present work is applied to a 12-volts lithium-ion battery. In fact, for this type of battery, the terminal voltage varies during charging and discharging, and its sensitivity coefficient reflects how it changes in response to current flow. The charging terminal voltage increases from 13.2 V to 14.6 V as energy is stored, while the discharging terminal voltage decreases from 12.9 V to 12 V as energy is released. Furthermore, the energy transfer coefficient characterizes the efficiency of energy conversion from electrical to chemical forms during charging or from chemical to electrical forms during discharging processes. All these parameter variations influence the battery SOC. Therefore, we have chose to vary these parameters in the finite range in the case of charging and discharging, and the results are summarized in Tables 3 and 4. Figure 2 shows the evolution of the state of charge, \(SOC(t)\), of the lithium-ion battery as a function of time for different values of the rate of charging \(\alpha_{1}\) and for \(V_{n}=12.8\) V, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Different** & \begin{tabular}{c} **Charging** \\ **Terminal** \\ **Voltage-Sensitive** \\ **Coefficient, \(A\)** \\ **[V\%] \(\times\)10\({}^{-3}\)** \\ \end{tabular} & \begin{tabular}{c} **Charging Energy** \\ **Transfer** \\ **Coefficient, \(\gamma_{1}\)** \\ **[min\({}^{-1}\)]** \\ **[min\({}^{-1}\)]** \\ \end{tabular} & \begin{tabular}{c} **Charging Rate \(\alpha_{1}\)** \\ **(min)** \\ **[min\({}^{-1}\)]** \\ \end{tabular} & \
\begin{tabular}{c} **Charging Time** \\ **(min)** \\ **[min\({}^{-1}\)]** \\ \end{tabular} &
\begin{tabular}{c} **Corresponding** \\ **State of Charge,** \\ **SOC [\%]** \\ \end{tabular} \\ \hline
12.9 V & 3.395 & 861.453 & 1.15 & 10 min & 20\% \\
12.8 V & 4.706 & 756.58 & 1.4 & 10 min & 17\% \\
12 V & 12.5 & 579.84 & 2.85 & 10 min & 9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation parameters for the discharging condition. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Specification** \\ \hline Charging cut-off voltage \((U_{ch})\) & 14.6 V \\ Nominal voltage \((V_{n})\) & 12.8 V \\ Discharging cut-off voltage \((U_{dis})\) & 12 V \\ Charging efficiency \((\beta_{ch})\) & 80\% \\ Discharging efficiency \((\beta_{dis})\) & 95\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Battery basic parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Specification** \\ \hline Charging cut-off voltage \((U_{ch})\) & 14.6 V \\ Nominal voltage \((V_{n})\) & 12.8 V \\ Discharging cut-off voltage \((U_{dis})\) & 12 V \\ Charging efficiency \((\beta_{ch})\) & 80\% \\ Discharging efficiency \((\beta_{dis})\) & 95\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters for the charging condition. \(U_{ch}=14.6\) V, \(\beta_{ch}=80\%\), and \(\mu_{0}=20\%\). Each curve corresponds to \(\alpha_{1}=0.365\) (blue curve), \(\alpha_{1}=0.57\) (orange curve), and \(\alpha_{1}=1.2\) (green curve). In Figure 2, we observe that at higher terminal charging voltages \(U_{b}^{ch}\), voltage-sensitive coefficients \(A\) and energy transfer coefficients \(\gamma_{1}\) lead to faster increases in time, making them advantageous for applications requiring fast charging. However, it should be noted that large values of these parameters can lead to drawbacks, such as increased battery degradation, overheating, and wasted energy. In addition, practical consideration of charging time, which varies from 8 minutes at a charging rate of 1.2 for \(SOC=100\%\) to 16 min at rates of 0.365 and 0.57 for \(SOC=70\%\) and \(SOC=99\%\), respectively, highlights the trade-off between fast charging and time efficiency. These results are summarized in Table 3.  
Figure 3 shows the discharging profile of the lithium-ion battery as a function of time for different values of discharging rate \(\alpha_{2}\) and for \(V_{n}=12.8\) V, \(U_{dis}=12\) V, \(\beta_{dis}=95\%\), and \(\mu_{100}=100\%\).