## 3 Results
### Effect of C-Rate at Constant Temperature Conditions  
#### 3.1.1 SOH Estimation for Batteries under Different Discharge C-Rate at 55 \({}^{\circ}\)C  
Experimental SoH is calculated for each cell tested for different number of cycles at two C-rates and 55 \({}^{\circ}\)C, and the results are shown in Table 4. SoH decreases with the cycle number and the rate of decrease of SoH is higher for battery discharged at 3 C rate as expected. The sample with 1 C-rate reaches the cut-off point of around 80% SoH after 600 cycles, while the sample with 3 C-rate reaches the cut-off point before 500 cycles.  
The calculated SoH values shown in Table 4 are used to obtain the k's values of the SECF model, and the values are shown in Table 5. The very small negative values of k\({}_{1}\) is likely due to fitting approximation error and hence they are set to zero. In fact, k\({}_{1}\) value should be zero as our ambient temperature is not high as expected from the work by [21].  
To verify the accuracy of the SECF model, the k-values obtained from one battery is applied to obtain the SoH of same LiB. The percentage error is then evaluated by comparing the estimated SoH with the experimental SoH. Figure 2 shows such comparison and the result is satisfactory.  
To study the effect of C-rate on the estimation accuracy, the k-values obtained from one battery is applied to obtain the SoH of other battery tested under same temperature but at other C-rate. Figure 5 summarizes the percentage errors at other discharge conditions. It is to be noted that the maximum percentage error on estimating SoH using semi-empirical model is around 3.14% using the k-values of battery tested at 1 C and applied to battery tested at 3 C, and the percentage error is around 2.11% for the reverse case. Both errors are within the acceptable limit of 10%.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cycle Number** & **1 C** & **3 C** \\ \hline
0 & 100 & 100 \\
100 & 96.86 & 95.24 \\
200 & 93.55 & 91.29 \\
300 & 90.21 & 87.68 \\
400 & 87.46 & 82.98 \\
500 & 83.88 & 79.67 \\
600 & 80.28 & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Experimental SoH at 55 \({}^{\circ}\)C based on the Qm computed from the ECBE model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{k}_{1}\) & \(\mathbf{k}_{2}\) & \(\mathbf{k}_{3}\) \\ \hline
1 C-rate of 55 \({}^{\circ}\)C (A) & \(-1.61478\times 10^{-7}\) & \(3.5497\times 10^{-4}\) & 0 \\ \hline
3 C-rate of 55 \({}^{\circ}\)C (B) & \(-3.7322\times 10^{-7}\) & \(4.5086\times 10^{-4}\) & \(1.43376\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: The k-values of 1 and 3 C-rate, respectively, at 55 \({}^{\circ}\)C.  
From Figure 5, we can see that the estimation error of using the SECF model is sufficiently accurate for predicting the SoH of its own cell at larger cycle number, and this has been elaborated in the work by [PERSON] et al. [17]. We can also see that it is fairly accurate when it is applied to the cells at different C rates. In comparing the percentage errors of A\({}^{\prime\prime}\) and B\({}^{\prime}\), although the different in the C rate is the same, the largest percentage error is observed for the case of B\({}^{\prime}\). This is expected because of the largest temperature change of the cells from the reference cell to the cell of interest in the case of B\({}^{\prime}\). We will discuss this in more detail later. Hence, we see that both the different in the discharge rate and the cell temperature can affect the estimation accuracy.  
In terms of the selection of the cycles to estimate the k's values, we select a different set of the three cycles for the SoH estimation of battery tested at 55 \({}^{\circ}\)C and 1 C rate. Table 6 shows the comparison, and one can see that the effect of the selection on the percentage error in SoH estimation is not significant.