## 3 Results and Discussion
### Comparison of Estimation Difference for Batch of Cells  
Apart from estimating an individual cell's SoH using their respective extracted k values and average k values, it is also interesting to observe the potential of the semi-empirical fading model to estimate SoH of different cells using the extracted k values from one cell. This can show the practicality of the semi-empirical fading model for use in estimating SoH for a large number of cells in a set by knowing the k values of only one cell's parameters. Thus, k values obtained from an individual cell were used to estimate the SoH of other cells, and the difference percentage was calculated for all the cells. The results for the cell difference estimation are presented in Figure 8 below.  
Figure 8 shows the results of the percentage difference for SoH estimation when k values from one cell are used for various cell's SoH estimation tested from the same batch. The various colors represent the cell number whose k values are used for SoH estimation in Figure 8. For example, the bar represented in the yellow circle represents the difference percentage of cell 1 when its SoH is estimated using the k values extracted from cell 2.  
The results of the percentage difference are generally less than 2.5% for all the cells, as seen from Figure 8 The difference for cell 7 and 8 which were tested for 650 cycles is in an acceptable range using the estimation using the values obtained from 250 cycles, half of the total cycle numbers estimated. This demonstrates that the simplified model can predict or estimate the SoH with good accuracy using the half-life cell data. Thus, the proposed model can save time in testing and provides rapid SoH estimation with acceptable accuracy.  
Figure 7: Mean difference comparison result using averaged K-value of each cell tested after the completion of the test at the respective charge–discharge cycles mentioned in Figure 5.  
To verify the sensitivity and accuracy of the present semi-empirical model when different charge cycles are used, we estimated average error for the samples using maximum and minimum peak values. As an example, we chose cycles, as shown by value 1, 3, and 4 once and value 2, 3, and 4 to observe the sensitivity of the model, as shown in Figure 3. A similar observation was done for other samples as well, and the error difference is shown in the Table below.  
It is clearly observed from Table 5 that the difference in errors estimated using upper and lower peaks is less than or around 1% showing the robustness of the model. Cells were chosen from different sets which show that the model is suitable for charge estimation at the different life cycle of batteries, which is an important factor while estimating charge of Li-ion batteries.  
In fact, since the fluctuation occurs in the first 250 cycles, and battery, in general, will have a lifetime longer than 250 cycles, we can also perform our computation of k1, k2, and k3 using the ECBE model from 250 cycles onward. In so doing, the average error in charge estimation is less than 1%, which is a lot more accurate, as shown in Table 6 below.  
Figure 8: The Histogram of estimation of the mean difference calculated from each cell by their K-value. The “K-values from A1” means the (k1, k2, k3) obtained from the cell A1.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Cell** & **Upper Peak (1, 3, 4)** & **Lower Peak (2, 3, 4)** \\ \hline A1 & 1.6162\% & 1.1857\% \\ A3 & 1.6446\% & 2.6735\% \\ A6 & 1.807\% & 1.5575\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Difference in average error estimation while choosing upper and lower peaks to observe the semi-empirical model sensitivity.