## 2 Experimental Setup and Procedures
### Capacity Fade Analysis of a Battery under Different Temperatures and Discharge Rates for 400 Cycles  
Under environmental temperatures of 20 and 30 \({}^{\circ}\)C, the results of 400 cycles of different discharge rates are listed in Table 1. The experimental results indicate that at the 400 th cycle for rates 0.5, 1, and 2 C, the \(SoH_{N}\) is still higher than 93%. For 3 C rate at the 400 th cycle, the \(SoH_{N}\) is lower than 80%. At 20 and 30 \({}^{\circ}\)C, the battery performance is very stable and with a very long cycle life. The work in [50] has shown that the discharge capacity measured at 25 \({}^{\circ}\)C shows 15.5% loss after 600 cycles; the experimental results are similar to the references [50; 51; 52]. Since over 2000 charge and discharge cycles are expected to be undergone by the LiFePO4 battery, considering the experimental cost, the number of battery tests was set to 400 cycles.  
At 40 \({}^{\circ}\)C and 50 \({}^{\circ}\)C, the results of 400 cycles of different discharge rates are listed in Table 2. Under high temperature, owing to rapid electrochemical reactions, the CE increases. After 400 cycles at 50 \({}^{\circ}\)C and with a 0.5 C rate, \(SoH_{N}\) still has a value of 104.83%. When the discharge rate is raised to 3 C, it can be seen that \(SoH_{N}\) drops rapidly at 400 cycles to only 49.33%. The experimental results indicate that under working conditions of 50 \({}^{\circ}\)C and a 3 C rate, the battery life will quickly decay.  
Under environmental temperatures of \(-\)10, 0, and 10 \({}^{\circ}\)C, the results of 400 cycles of different discharge rates are listed in Table 3. Under low temperature CE is not high. With \(-\)10 \({}^{\circ}\)C and 3 C rate as example of the working condition, at the first cycle the \(SoH_{N}\) is only about 91.08%. As the number of cycles increase, the compound influence from low temperature and high discharge rates gradually become obvious. At 400 cycles, the \(SoH_{N}\) is only 26.4%, far lower than the battery life under room temperature. This indicates that under low temperature the CE of LiFePO4 is not high and the capacity fades rapidly.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & & \multicolumn{3}{c}{\(SoH_{N}\) (\%)} & & & \\
**C Rate** & \multicolumn{2}{c}{**0.5 C**} & \multicolumn{2}{c}{**1 C**} & \multicolumn{2}{c}{**2 C**} & \multicolumn{2}{c}{**3 C**} \\
**T (\({}^{\circ}\)C)** & **20** & **30** & **20** & **30** & **20** & **30** & **20** & **30** \\
**Cycle** & & & & & & & & \\ \hline
1 & 108.11 & 108.29 & 109.00 & 107.69 & 106.94 & 104.28 & 108.33 & 101.58 \\
100 & 105.97 & 106.15 & 106.42 & 106.14 & 104.06 & 100.56 & 103.17 & 98.08 \\
200 & 104.61 & 103.78 & 102.83 & 105.00 & 100.17 & 97.67 & 93.75 & 90.33 \\
300 & 103.85 & 100.51 & 102.11 & 100.47 & 96.56 & 95.22 & 86.92 & 85.67 \\
400 & 102.86 & 98.97 & 99.92 & 98.36 & 93.22 & 93.56 & 78.42 & 81.83 \\ \hline \end{tabular}
\end{table}
Table 1: Capacity change with 400 cycles under different rates of discharge (environmental temperatures 20 and 30 \({}^{\circ}\)C).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & & \multicolumn{3}{c}{\(SoH_{N}\) (\%)} & & & \\
**C Rate** & \multicolumn{2}{c}{**0.5 C**} & \multicolumn{2}{c}{**1 C**} & \multicolumn{2}{c}{**2 C**} & \multicolumn{2}{c}{**3 C**} \\
**T (\({}^{\circ}\)C)** & **40** & **50** & **40** & **50** & **40** & **50** & **40** & **50** \\
**Cycle** & & & & & & & & \\ \hline
1 & 107.03 & 114.50 & 107.17 & 108.94 & 108.94 & 108.22 & 109.67 & 105.08 \\
100 & 105.25 & 114.31 & 106.25 & 108.83 & 104.61 & 106.33 & 106.25 & 102.58 \\
200 & 102.12 & 113.60 & 104.56 & 107.44 & 103.17 & 99.06 & 103.50 & 93.17 \\
300 & 100.51 & 110.54 & 99.33 & 87.33 & 97.83