## 4 Analysis of Energy Storage System
### SOC Estimation  
Usually, the battery state of charge and the remaining useful life are considered as two important parameters to quantify and monitor the present battery state. In this study, long-term low temperature is the main factor affecting the battery capacity and the remaining useful life. References [21; 22] have reported joint/dual extended Kalman filter and unscented Kalman filter with an enhanced self-correcting model, which can simultaneously estimate the SOC and capacity. The SOC estimation in this study is realized based on study on low-temperature characteristics of battery. The relationship between battery voltage and battery capacity at low temperatures is shown in Figure 5.  
The mathematical model of the relationship between battery voltage and battery capacity at different temperatures can be expressed as follows. \[\left\{ \begin{array}{l}U_{-50}\,{}^{\circ}\mathrm{C}=5.5\times 10^{-8}\times SOC ^{6}_{-50}\,{}^{\circ}\mathrm{C}-2.51\times 10^{-5}\times SOC^{5}_{-50}\,{}^{ \circ}\mathrm{C}+4.72\times 10^{-3}\times SOC^{4}_{-50}\,{}^{\circ}\mathrm{C}-0.471 \times SOC^{3}_{-50}\,{}^{\circ}\mathrm{C}\\ +26.24\times SOC^{2}_{-50}\,{}^{\circ}\mathrm{C}-711.8\times SOC_{-50}\,{}^{ \circ}\mathrm{C}+9362.2\\ U_{-40}\,{}^{\circ}\mathrm{C}=2.9\times 10^{-8}\times SOC^{6}_{-40}\,{}^{ \circ}\mathrm{C}-1.36\times 10^{-5}\times SOC^{5}_{-40}\,{}^{\circ}\mathrm{C}+2.58 \times 10^{-3}\times SOC^{4}_{-40}\,{}^{\circ}\mathrm{C}-0.261\times SOC^{3}_{-40 }\,{}^{\circ}\mathrm{C}\\ +14.64\times SOC^{2}_{-40}\,{}^{\circ}\mathrm{C}-434.1\times SOC_{-40}\,{}^{ \circ}\mathrm{C}+5306.9\\ U_{-30}\,{}^{\circ}\mathrm{C}=1.3\times 10^{-8}\times SOC^{6}_{-30}\,{}^{ \circ}\mathrm{C}-6.04\times 10^{-6}\times SOC^{5}_{-30}\,{}^{\circ}\mathrm{C}+1.15 \times 10^{-3}\times SOC^{4}_{-30}\,{}^{\circ}\mathrm{C}-0.116\times SOC^{3}_{-30 }\,{}^{\circ}\mathrm{C}\\ +6.487\times SOC^{2}_{-30}\,{}^{\circ}\mathrm{C}-192.4\times SOC_{-30}\,{}^{ \circ}\mathrm{C}+2353.8\\ U_{-20}\,{}^{\circ}\mathrm{C}=1.5\times 10^{-8}\times SOC^{6}_{-20}\,{}^{ \circ}\mathrm{C}-6.82\times 10^{-6}\times SOC^{5}_{-20}\,{}^{\circ}\mathrm{C}+1.31 \times 10^{-3}\times SOC^{4}_{-20}\,{}^{\circ}\mathrm{C}-0.132\times SOC^{3}_{-20 }\,{}^{\circ}\mathrm{C}\\ +7.508\times SOC^{2}_{-20}\,{}^{\circ}\mathrm{C}-244.2\times SOC_{-20}\,{}^{ \circ}\mathrm{C}+2760.1\\ U_{-10}\,{}^{\circ}\mathrm{C}=8.0\times 10^{-9}\times SOC^{6}_{-10}\,{}^{ \circ}\mathrm{C}-3.72\times 10^{-6}\times SOC^{5}_{-10}\,{}^{\circ}\mathrm{C}+7.13 \times 10^{-4}\times SOC^{4}_{-10}\,{}^{\circ}\mathrm{C}-0.072\times SOC^{3}_{-10 }\,{}^{\circ}\mathrm{C}\\ +4.098\times SOC^{2}_{-10}\,{}^{\circ}\mathrm{C}-122.4\times SOC_{-10}\,{}^{ \circ}\mathrm{C}+1508.8\\ U_{0}\,{}^{\circ}\mathrm{C}=4.66\times 10^{-9}\times SOC^{6}_{0}\,{}^{ \circ}\mathrm{C}-2.13\times 10^{-6}\times SOC^{5}_{0}\,{}^{\circ}\mathrm{C}+4.01 \times 10^{-4}\times SOC^{4}_{0}\,{}^{\circ}\mathrm{C}-0.0401\times SOC^{3}_{0 }\,{}^{\circ}\mathrm{C}+\\ 2.229\times SOC^{2}_{0}\,{}^{\circ}\mathrm{C}-25.5895\times SOC_{0}\,{}^{ \circ}\mathrm{C}+797.2\end{array} \right. \tag{11}\]  
where \(U_{-50}\,{}^{\circ}\mathrm{C}\), \(U_{-40}\,{}^{\circ}\mathrm{C}\), \(U_{-30}\,{}^{\circ}\mathrm{C}\), \(U_{-20}\,{}^{\circ}\mathrm{C}\), \(U_{-10}\,{}^{\circ}\mathrm{C}\) and \(U_{0}\,{}^{\circ}\mathrm{C}\) are the battery voltages at \(-50\)\({}^{\circ}\)C, \(-40\)\({}^{\circ}\)C, \(-30\)\({}^{\circ}\mathrm{C}\), \(-20\)\({}^{\circ}\mathrm{C}\), \(-10\)\({}^{\circ}\mathrm{C}\), \(0\)\({}^{\circ}\mathrm{C}\), respectively; \(SOC_{-50}\,{}^{\circ}\mathrm{C}\), \(SOC_{-40}\,{}^{\circ}\mathrm{C}\), \(SOC_{-30}\,{}^{\circ}\mathrm{C}\), \(SOC_{-20}\,{}^{\circ}\mathrm{C}\), \(SOC_{-10}\,{}^{\circ}\mathrm{C}\) and \(SOC_{0}\,{}^{\circ}\mathrm{C}\) are the values of state of charge at \(-50\)\({}^{\circ}\)C, \(-40\)\({}^{\circ}\)C, \(-30\)\({}^{\circ}\mathrm{C}\), \(-20\)\({}^{\circ}\mathrm{C}\), \(-10\)\({}^{\circ}\mathrm{C}\), \(0\)\({}^{\circ}\mathrm{C}\), respectively.