## 4 Applications of the SECF Model  
From the previous analysis, we can see that with the equation of SoH given by Equation (2) and the k values computed using Equations (3) and (4), one can estimate the SoH of LiB after different charge-discharge cycle. Conversely, one can also determine the cycles where the SoH will reach a certain value due to its closed form. For example, if the LiB cell is operating under 2 C discharge rate at 40 \({}^{\circ}C\), one can have the following equation, with i = 2C, k\({}_{2}\) = 0.00028, k\({}_{3}\) = 0.003557,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Test Condition**} & \multirow{2}{*}{**SoH (Using ECBE)**} & **\% Error in SoH Estimation** & **\% Error in SoH Estimation with k Values** \\  & & **Using Its Own k Values** & **Computed from Equations (3) and (4)** \\ \hline
25 \({}^{\circ}C\)\_1 C & 89.76(after 800 cycles) & 2.7 & 0.89 \\
25 \({}^{\circ}C\)\_3 C & 80.41(after 300 cycles) & 2.51 & 0.77 \\
55 \({}^{\circ}C\)\_1 C & 80.29(after 600 cycles) & 0.67 & 0.59 \\
55 \({}^{\circ}C\)\_3 C & 79.68(after 500 cycles) & 0.68 & 0.41 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Comparison of the % errors in SoH estimation.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sample \#**} & \multirow{2}{*}{**SoH (Using ECBE)**} & **SoH Estimation Using Our** & **\% Error in SoH Estimation with** \\  & & **Generalized k Values** & **Generalized k Values** \\ \hline
1 & 89.06(After 100 cycles) & 88.24 & 0.92 \\ \hline
2 & 91.39(After 100 cycles) & 90.72 & 0.73 \\ \hline
3 & 90.66(After 100 cycles) & 89.94 & 0.79 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Verification of the model for additional three more batteries tested at 55 \({}^{\circ}C\) and 5 C discharging current.  
and \(Q_{max(fresh)}=1.9463\) Ah. Upon solving it, the \(n\) value is 689, which will be the cycles where the SoH reaches 80%, and thus the useful life of the LiB cell at different operating condition can be determined.  
\[80=1-(k_{2}N)-\frac{k_{3}}{Q_{max(fresh)}i} \tag{7}\]  
The above example is an application of the SECF model for the lifetime prediction of a fresh LiB. This model can also be used for the estimation of the RUL as follows.  
Given the \(k_{2}\) and \(k_{3}\) values of a LiB as in this work, and assuming the LiB is used in 1 C discharge rate and has undergone 100 cycles at 30 \({}^{\circ}\)C, the LiB is now to be used in 1.5 C discharge rate at the same temperature of 30 \({}^{\circ}\)C. If this LiB has been through 20 cycles under 1.5 C discharge rate, we can use our SECF model to compute the RUL of the LiB under this 1.5 C discharge rate as follows.  
For the SoH of the LiB after the 100 cycles @1C discharge rate, A = -10/15 and B = -1 according to Equations (5) and (6), and with the values of k\({}_{2}\) and k\({}_{3}\) as computed using Equations (3) and (4), we obtain the SoH after 100 cycles to be 96.25%, using Equation (2). This is the SoH of the LiB at the start of the next application of 1.5 C. The equivalent cycle for the LiB to reach this SoH at 1.5 C and 30 \({}^{\circ}\)C can be computed from the model by having the B = -0.5. A will not change as the temperature remains the same. With this new B, we have new values of k\({}_{2}\) and k\({}_{3}\) and the equivalent cycle is found to be 88 cycles (N\({}_{equivalent}\)).  
We can also compute the total number of cycles at 1.5 C and 30 \({}^{\circ}\)C to reach 80% SoH, denoted as N\({}_{total}\). From our SECF model, the value is found to be 769 cycles. Since the LiB has been through 20 cycles under 1.5 C, the RUL of the LiB will be N\({}_{total}-N_{equivalent}-20=769-88-20=661\) cycles.  
Going forward, if the discharge current varies, one can use the average current, expressed in terms of C rate to compute the B value. If the temperature is varying, one can use the average temperature to compute the A value. Then, using Equations (2)-(4), the SoH or the RUL can be determined. However, this is to be verified as our future work.