## 4 Results and Discussion  
Figures 5 and 6 present the degradation of the test batteries for the TC1 and the TC2 conditions through D\({}^{2}\)SADT. The capacity loss of the test battery continuously increased under both TC1 and TC2 conditions. It can be normalized by  
\[Normalized\ Capacity\ Loss=\frac{\left|\frac{The\ capacity\ of\ the\ nth\ measurement\ -\ The\ initial\ capacity}{The\ initial\ capacity}\right|}{The\ initial\ capacity} \tag{8}\]  
The failure threshold was defined by a 20% capacity loss, referred from the major carmakers in the market, and the lifetime of each test battery was to be the time to reach the threshold. It is of note that the test battery operated under the hybrid dynamic stress from the charging-discharging test (DST) and temperature cycling (TC) test to accelerate the degradation of them, and both TC1 and TC2 had the same 10 DSTs for each cycle. In the case of TC1, test battery No. 1, No. 2, and No. 3 required 13,882, 13,940, 12,628 hr to reach this threshold, respectively. The test of TC2 conditions was performed after completing the TC1 test, and thus, these two test conditions were operated using the same test and measure equipment to maintain the minimum variance. Compared to TC1, the test batteries in TC2 required a shorter time to reach the threshold associated with 8971, 8139, and 10,393 hr. The average  
Figure 4: Timing diagram of reference power test (RPT). Figure 3: Test profile of the DST. lifetimes of TC1 and TC2 were 13,483 and 9168 h, respectively. However, the significant difference of test conditions between the TC1 and TC2 was the minimum and maximum temperature; it is believed that the latter one will be the major contributor to the degradation of the Li-ion battery. As for the results, the magnitudes of capacity loss of test batteries in TC2 were generally higher than those in TC1 at the same measure time since the maximum temperature of TC2 was higher than that of TC1. Besides, only No. 1 in TC1 and No. 6 in TC2 are displayed for simplicity, though we have performed the prognosis assessment for all samples. Figure 7 presents the relationship between the activation energy and the percentage of capacity loss for the LiFePO4 battery test. It is noted that the activation energy continuously decreased when the capacity loss increased overtime. The lower activation energy represented a higher reaction rate inside the battery, corresponding to speed up Lithium-ion diffusion in electrode materials. From the point of view of long-term reliability, the capacity loss of the test battery operating under certain power and temperature cycling conditions can be accelerated when the charge-discharge cycles increase. Figure 8 shows the lifetime prediction by Monte Carlo (MC) simulation at the 115 th TC in the cases of TC1 and No. 1, and TC2 and No. 6 at the 85 th TC. Because a higher statistic standard associated with a 0.95 confidence level and 0.01 error was setup by the MC simulation, it required higher numbers of runs in order to decrease the uncertainty of the simulation; there are many possible degradation trajectories represented in the figure to reach the threshold. A statistical calculation was performed to determine the mean life of the PDF on the threshold, which was considered the predicted lifetime. Generally speaking, the predicted lifetime was very close to the real lifetime for the test batteries. To quantify the prediction accuracy, the prediction error is simply defined by  
\[Prediction\ Error=\frac{Predicted\ life-Real\ life}{Real\ life}\times 100\% \tag{9}\]  
Figure 5: The capacity loss of the test batteries during D\({}^{2}\)SADT (TC1 condition). When the value is negative, the predicted lifetime was shorter than the real lifetime. However, the predicted lifetime was longer than the real lifetime when the value was positive. For the purpose of prognosis, the optimal result was that the predicted lifetime was as close as possible to the real lifetime, but the predicted lifetime must be shorter.