## 3 Results
### Effect of Temperature  
2.1 SOH Estimation for Batteries at 25 \({}^{\circ}\)C using 55 \({}^{\circ}\)C Battery's Parameter Values (55 \({}^{\circ}\)C to 25 \({}^{\circ}\)C Case)  
Similar to our previous approach, we use the k values from the batteries with similar C-rate but apply them to SoH estimation of cells discharged at different temperatures. First, SoH estimation is performed for batteries tested at 25 \({}^{\circ}\)C and the k values obtained from battery tested at 55 \({}^{\circ}\)C are used for the SoH estimation, and the results are shown in Figure 6.  
It can be observed from Figure 7 that the highest SoH estimation percentage error of 8.4% is obtained for the case when k values of battery tested at 3 C rate and 55 \({}^{\circ}\)C is used to estimate the SoH for battery tested at 1 C rate and 25 \({}^{\circ}\)C. This is due to the largest cell temperature difference between the cells, which is approximately 28.14 \({}^{\circ}\)C in this case as can be seen in Table 9. The smallest estimation error for the estimation of battery tested at 3 C and 25 \({}^{\circ}\)C is corresponding to the small difference in cell temperature for cell discharge at 1 C and 55 \({}^{\circ}\)C and cell discharge at 3 C 25 \({}^{\circ}\)C. This finding supports our postulation that cell temperature difference is an important factor in the estimation accuracy. \begin{table}
\begin{tabular}{c c c} \hline  & **1 C Rate** & **3 C Rate** \\ \hline
25 \({}^{\circ}\)C Ambient & 34.86 \({}^{\circ}\)C & 42.32 \({}^{\circ}\)C \\ \hline
55 \({}^{\circ}\)C Ambient & 58 \({}^{\circ}\)C & 63 \({}^{\circ}\)C \\ \hline \end{tabular}
\end{table} 
Table 9: Cell temperature when discharged at different C rate under two ambient conditions. 3.2.2. SOH Estimation for Batteries at 55 \({}^{\circ}\)C using 25 \({}^{\circ}\)C Battery's Parameter Values (25 \({}^{\circ}\)C to 55 \({}^{\circ}\)C Case)  
Now we study the opposite situation where estimation is made using parameters values from low temperature to high temperature, and the results are shown in Figure 8.  
From Figure 8, we can see that the smallest percentage error is the case where the parameters values from 3 C and 25 \({}^{\circ}\)C is applied to 1 C at 55 \({}^{\circ}\)C. This is expected with our postulation where the cell temperature difference between the two cells is only 15.68 \({}^{\circ}\)C in this case. Likewise, the largest  
Figure 8: Estimation % error using k values from 25 \({}^{\circ}\)C.  
Figure 7: Estimate % error using k’s values from 55 \({}^{\circ}\)C.  
percentage error is observed for the case where the parameters values from 1 C at 25 \({}^{\circ}\)C is applied to 3 C at 55 \({}^{\circ}\)C and the temperature difference is as high as 28.14 \({}^{\circ}\)C.  
From the above investigation, the interaction of discharge current and cell temperature needs to be identified. The statistical design of experiment (DoE) method, a commonly used method in industry for studying the effect of multiple factors, is thus employed. The detail of the DoE method can be found in reference [23]. We summarized the k values at the four test conditions as shown in Table 3. The interaction of the discharge current and temperature as reflected in the k values can be plotted as shown in Figures 9 and 10, as part of the DoE analysis. Using these values and applying the \(2^{2}\) factorial design analysis, we obtain the equations for k\({}_{2}\) and k\({}_{3}\) as given in Equations (3) and (4). k\({}_{1}\) is zero as our experiments do not involve high temperatures. From the DoE analysis, we obtain the following equations for k\({}_{2}\) and k\({}_{3}\). \[\mathrm{k}_{2}=0.000287-0.000115\;\mathrm{A}-0.000080\;\mathrm{B}-0.000032\; \mathrm{A}\times\mathrm{B} \tag{3}\]  
\[\mathrm{k}_{3}=0.003557+0.002207\;\mathrm{A}+0.002843\;\mathrm{B}+0.001493\; \mathrm{A}\times\mathrm{B} \tag{4}\]  
Figure 10: Interaction of current density and temperature in the k\({}_{3}\) value. Figure 9: Interaction of current density and temperature in the k\({}_{2}\) value.