## 4 Results
### Capacity Fade Forecasting at Different Discharge Conditions  
This section analyzes the evolution of the capacity fade of a fifth cell that was discharged at a discharge C-rate of 2, i.e., at a discharge current of 4 A instead of 2 A, as was performed in the previous section. Figure 9 shows three ARIMA predictions at different cycles for each analyzed cell. The training range varies between 30% (left), 50% (middle), and 80% (right) of the cell lifetime. Each plot shows the behavior of the measured capacity fade \(\Delta C\) (black line, which corresponds to the objective time series) and the forecasting results using the proposed health indicator \(\Delta V_{RMS}\) to estimate \(\Delta C\) (blue line) and the forecast directly using \(\Delta C\) (red line). The solid lines of \(\Delta V_{RMS}\) and \(\Delta C\) refer to known values, while the dashed lines refer to predicted values. \begin{table}
\begin{tabular}{c c c c c} \hline
**Cell** & **B0005** & **B0006** & **B0007** & **B0018** \\ \hline
**Forecast Start Point (\%** **of Total Useful Life**) & **RMSE** & **RMSE** & **RMSE** & **RMSE** \\ \hline
25\% & 0.221 & 0.172 & 0.101 & 0.134 \\
50\% & 0.125 & 0.145 & 0.075 & 0.077 \\
80\% & 0.135 & 0.616 & 0.068 & 0.307 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparative results between the experimental measurements of the cell terminal voltage vs. SOC and the Nernst equation adjusted with the proposed degradation model with a discharge current of 2 A (C-rate of 1). Figure 8: Prediction of future discharge cycles at different stages of cell aging. (**a**) Prediction of future cycle using 25% useful life data. (**b**) Prediction of future cycle using 50% useful life data. (**c**) Prediction of future cycle using 80% useful life data. Table 5 shows the root-mean square error (RMSE) of the blue (\(\Delta V_{RMS}\) prediction) and red lines (\(\Delta C\) prediction) of Figure 9, which is calculated against the black line (objective data series) at different cycles. The trend in the results presented in Table 5 is that the prediction accuracy improves with the number of cycles in the training set. The mean values of the RMSE for the LF forecasting are 0.076 and 0.074 for \(\Delta V_{RMS}\) and \(\Delta C\), respectively. In the case of ARIMA the mean values are 0.080 and 0.043 for \(\Delta V_{RMS}\) and \(\Delta C\), respectively. Since the mean RMSE values of \(\Delta V_{RMS}\) and \(\Delta C\) are similar, it can be concluded that \(\Delta V_{RMS}\), can be used as an estimator of the capacity fade \(\Delta C\), even with a C-rate of 2. Figure 10 shows the experimental values of the terminal voltage of cell B0034 vs. the SOC at the reference cycle (blue line, using the Nernst fitting from experimental data) and at different cycles (black dashed lines) and the predictions of future discharge cycles (solid magenta line) using the Nernst model described by (11) and \(\Delta V_{RMS,n}\) and \(\Delta C_{n}\). The SOC is calculated by applying (13) once the estimated value of \(\Delta C_{n}\) is known, while the cell terminal voltage in Figure 8 is the voltage obtained from (11) minus \(\Delta V_{RMS,n}\). The magenta line (predicted future discharge cycles) should be very close to the black dashed line corresponding to the objective data. \begin{table}
\begin{tabular}{c c c c} \hline
**Forecast Start Point (\% of** & & & \\
**Total Useful Life)** & **Method** & \(\Delta V_{RMS}\) **[Ah]** & \(\Delta C\) **[Ah]** \\ \hline \multirow{2}{*}{30\%} & LF & 0.149 & 0.052 \\  & ARIMA & 0.161 & 0.034 \\  & LF & 0.101 & 0.058 \\  & ARIMA & 0.094 & 0.046 \\  & LF & 0.072 & 0.095 \\  & ARIMA & 0.070 & 0.043 \\  & LF & 0.061 & 0.093 \\  & ARIMA & 0.057 & 0.044 \\  & LF & 0.047 & 0.083 \\  & ARIMA & 0.050 & 0.032 \\  & LF & 0.023 & 0.065 \\  & ARIMA & 0.051 & 0.058 \\ \hline \end{tabular}
\end{table}