## 3 Discussion
### Charge and Discharge at High Power  
In this case study, the initial voltage of the cell is also considered to be 2.7 V. A charge is preceded by a discharge as well. The discharge is assumed to occur at 200 W for 10 s, whereas the charge is kept at 400 W for 5 s. The latter is around 40% of the maximum allowed power given by the manufacturer, thus representing a high-power value. Table 3 shows the data for this case, the external voltage at the beginning of each process, \(\mathrm{u_{co(0)}}\) has also been included. The electrical and thermal parameters of the cell are the ones presented in Table 1. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline
**Process** & **P (W)** & **Duration (s)** & \(\mathrm{T_{0}}\) (\(\mathrm{\SIUnitSymbolDegree}\)) & \(\mathrm{T_{final}}\) (\(\mathrm{\SIUnitSymbolDegree}\)) & \(\mathrm{U_{0}}\) (V)** & \(\mathrm{g_{1(0)}}\) & \(\mathrm{k_{\mathrm{\theta}1}}\)(\(\mathrm{\SIUnitSymbolDegree}\)) & \(\mathrm{k_{\mathrm{\theta}1}}\)(\(\mathrm{\SIUnitSymbolDegree}\)) & \(\mathrm{k_{\mathrm{\theta}2}}\)(\(\mathrm{\SIUnitSymbolDegree}\)) \\ \hline Discharge & 20 & 100 & 20 & 20.05 & 2.7 & 453.623 & 54.849 & \(-\)27.374 \\ Charge & \(-\)40 & 50 & 20.05 & 20.15 & 1.0516 & \(-\)36.528 & 52.419 & 54.748 \\ \hline  & & & & & & & \\ \hline Discharge & 20 & 100 & 20 & 20.047 & 2.7 & - & - & - \\ Charge & \(-\)40 & 50 & 20.047 & 20.140 & - & - & - & - \\ \hline  & & & & & & & \\ \hline  & & & & & & & \\ \hline Discharge & 20 & 100 & 0 & \(4.168\times 10^{-4}\) & - & - & - & - \\ Charge & \(-\)40 & 50 & \(3\times 10^{-3}\) & \(1\times 10^{-2}\) & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Charge and discharge parameters at a low power. Figure 6 is the counterpart of Figure 5 for this case study. As can be observed, the results rendered by the approximate formulas diverge from the exact solution, more markedly during the charging process. Therefore, the exact formulation proposed in this study is the most suitable for high-power processes. It must be noticed that if more charge and discharge cycles were added after those 15 s, the accumulation of errors in each cycle would give rise to even greater differences between the approximate equations and the proposed exact solution. Figure 7 shows the evolution of the variable g1 over time according to Equations (1) and (2). The values of g1 are positive during the discharge, whereas they are negative in the charge. Figure 8 shows the evolution of the cell internal and external voltages over time according to Equations (1), (2), (6) and (35). The discontinuity that can be observed in the external voltage is due to the voltage drop at the internal resistance, as the reference of the current is reversed. Figure 9 shows the evolution of the cell temperature as a function of the variable g1 obtained by means of Equations (28)-(34). Figure 10 shows the evolution of the cell temperature as a function of the external voltage obtained from Equations (16), (28)-(30) and (36). As depicted in the graph, both variables increase during the charge, whereas the external voltage decreases and the temperature increases in the discharge. Figure 5: Top chart: Temperature of the cell obtained by using Equations (28)–(34) and the approximate equations given in [56] for discharge and charge at low or moderate power. Bottom chart: Temperature difference between both solutions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & & \\ \hline \hline
**Process** & **P (W)** & **Duration (s)** & **T\({}_{0}\) (\({}^{\circ}\)C)** & **T\({}_{\text{final}}\) (\({}^{\circ}\)C)** & **U\({}_{0}\) (V)** & **u\({}_{\text{co(0)}}\) (V)** & **g\({}_{\text{1(0)}}\)** & **k\({}_{01}\)(\({}^{\circ}\)C)** & **k\({}_{02}\)(\({}^{\circ}\)C)** \\ \hline Discharge & 200 & 10 & 20 & 20.71 & 2.7 & 2.639 & 43.539 & 0.00623 & \(-\)0.2737 \\ Charge & \(-\)400 & 5 & 20.71 & 21.74 & 0.848 & 1.131 & \(-\)3.998 & 0.84887 & 0.5475 \\ \hline \hline \end{tabular}
\end{table}