# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 3 Results and Discussion
### Battery-Failure Warning Results of the Proposed Linear Aging Model  
Measurements and estimations of (quality and quantity) the remaining useful capacity of parallel samples are shown in Figure 6. Approximately 450 Ah as 80% of the rated design capacity (560 Ah) was set up as a red line of battery failure for the linear-aging model. Hysteresis of measurement of the remaining capacity near 400 Ah appeared in the fifth and sixth tests of E-52#. The fifth and sixth tests of E-02# near 210 Ah also showed hysteresis. The value of the remaining capacity of E-20# continuously reduced during aging experiments. During fifth to seventh tests, measurement of remaining useful capacity of E-11# around 450 Ah showed hysteresis. Prediction of quantity of the remaining capacity was according to model in Section 2.6. Prediction of the fifth test was very close to measured value which was above red line. The model carried out the sixth prediction and determined battery failure; however, the capacity verification confirmed that the measured remaining capacity was still above red line. The model adjusted its prediction of the seventh test with a result of 458 Ah. However, at this round, the measured remaining capacity (418 Ah) was below the red line. The model of Section 2.6 continuously showed a false alarm at the sixth test and missing the alarm at the seventh alarm. The missing alarm was a serious error of class-1E nuclear backup batteries. The hysteresis of the remaining useful capacity--especially at the end of battery service life--was an obvious difficult problem for the typical aging model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cell** & \(\mathbf{R^{2}}\) & **MBE** & **RMSE** \\ \hline E-52\# 01 st & 0.9976 & \(-\)0.0004 & 0.0059 \\ E-52\# 02 nd & 0.9977 & 0.0010 & 0.0058 \\ E-52\# 03 rd & 0.9971 & \(-\)0.0006 & 0.0073 \\ E-52\# 04 th & 0.9992 & \(-\)0.0049 & 0.0058 \\ E-52\# 05 th & 0.9974 & \(-\)0.0128 & 0.0141 \\ E-52\# 06 th & 0.9975 & \(-\)0.0129 & 0.0142 \\ E-52\# 07 th & 0.9956 & \(-\)0.0131 & 0.0154 \\ E-11\# 01 st & 0.9977 & \(-\)0.0019 & 0.0065 \\ E-11\# 02 nd & 0.9981 & \(-\)0.0028 & 0.0062 \\ E-11\# 03 rd & 0.9983 & \(-\)0.0031 & 0.0067 \\ E-11\# 04 th & 0.9992 & \(-\)0.0062 & 0.0068 \\ E-11\# 05 th & 0.9992 & \(-\)0.0077 & 0.0082 \\ E-11\# 06 th & 0.9994 & \(-\)0.0048 & 0.0057 \\ E-11\# 07 th & 0.9985 & \(-\)0.0109 & 0.0119 \\ E-20\# 01 st & 0.9974 & \(-\)0.0037 & 0.0084 \\ E-20\# 02 nd & 0.9956 & 0.0032 & 0.0117 \\ E-20\# 03 rd & 0.9975 & 0.0059 & 0.0134 \\ E-20\# 04 th & 0.9984 & \(-\)0.0096 & 0.0105 \\ E-20\# 05 th & 0.9953 & \(-\)0.0144 & 0.0160 \\ E-20\# 06 th & 0.9895 & \(-\)0.0206 & 0.0234 \\ E-20\# 07 th & 0.9854 & \(-\)0.0219 & 0.0251 \\ E-02\# 01 st & 0.9974 & \(-\)0.0026 & 0.0077 \\ E-02\# 02 nd & 0.9950 & 0.0068 & 0.0159 \\ E-02\# 03 rd & 0.9980 & 0.0051 & 0.0120 \\ E-02\# 04 th & 0.9986 & \(-\)0.0100 & 0.0109 \\ E-02\# 05 th & 0.9848 & \(-\)0.0233 & 0.0271 \\ E-02\# 06 th & 0.9857 & \(-\)0.0209 & 0.0247 \\ \hline \hline \end{tabular} \(\text{R}^{2}\): correlation coefficient; MBE: mean biased error; RMSE: root mean square error.  
\end{table} 
Table 1: Fitting errors of proposed Thevenin model. The proposed linear-aging model accurately predicted the battery failures of E-52#, E-20# and E-02#. There were no false alarms at the fourth test, and no missing alarms at the fifth test. The proposed linear-aging model continued to present failure warnings for E-11# from the fifth to the seventh tests. E-11# presented saturation of the morphology correction factor from the fifth test. The proposed linear-aging model successfully predicted this saturation and the failure warning of E-11# at the fifth test was an effective alarm of the quality of the remaining capacity. This alarm should not be judged as a false alarm, although measurement of the remaining capacity was slightly more than the red line. The red line of measurement of the remaining capacity was average and was individually different.