## 2 Battery Modeling  
For control-oriented battery online monitoring, a precise battery state-space model must be available. As mentioned, the ECM is the most widely used choice in combination with the KFs. The ECM is based on the [PERSON]'s theorem, which approximates the battery's electrical behavior through a voltage source and some resistances and capacitors. The accuracy of ECM was enhanced by adding extra resistance-capacitance terms (RC network) into the circuits. However, the complication of the model structure could lead to inefficient parameters' identification and low real-time computation. Obviously, a reasonable model must consider simulation accuracy, parameterization efficiency, and computation burden. Additionally, hysteresis is also a very significant variable for lithium-ion batteries [25; 26; 27; 28]. However, studies for this variable are lacking as it is rarely being considered. In this paper, the hysteresis model was used. As shown in Figure 1, the proposed model consists of an Electro-Motive Force (EMF) voltage resource, a hysteresis voltage resource, a resistor \(R_{0}\), and a RC network connected in series. The EMF voltage resource and the hysteresis voltage resource are together to form a controlled voltage source \(-U_{\text{oc}}\) (OCV). \(R_{0}\) is the ohmic resistance. It represents the instantaneous voltage variation caused by the electrolyte and the active mass. \(R_{1}\) and \(C_{1}\) represent the polarization resistance and polarization capacitance, respectively. These are also used to depict the transient response of the cell caused by double-layer capacity effects. The hysteresis effect between the charge and discharge boundary curves is illustrated in Figure 2a. The curves are obtained by charging/discharging the cell at C/25 rate (a current corresponding to the manufacturer's rated capacity for a 25 h discharge to low cut-off voltage) at a temperature of 30 \({}^{\circ}\)C; 30 \({}^{\circ}\)C is the standard temperature according to the latest U. S. Advanced Battery Consortium (USABC) battery test manual [29]. The charge voltage is the upper curve and the discharge voltage is the lower curve. At such low rate, we believed that the voltage drop was caused by ohmic resistance, and polarization is small enough to ignore (less than 1 mV in this case). Thus, the voltage difference at each SOC is caused by the hysteresis phenomenon. The space function equations corresponding to Figure 1 are derived as follows:  
First, according to the Kirchhoff voltage law, the governing equation of the cell model can be derived as follows:  
\[\left\{ \begin{array}{c}U_{L}=U_{oc}(SOC,T)+IR_{0}+U_{1}+h\\ \overline{U_{1}}=-\frac{1}{R_{1}C_{1}}U_{1}+\frac{1}{C_{1}}I\end{array} \right. \tag{1}\]  
where \(U_{L}\) is the terminal voltage of the cell. EMF is just a theoretical concept, where the EMF is approximated by \(U_{oc}(SOC,T)\) after an hour rest, which is the function of SOC and temperature. \(U_{1}\) is the voltage over the RC network and \(h\) is the voltage caused by hysteresis. \(I\) is the main circuit current. The charging process is positive, and the discharging process is negative. \(R_{1}\) is the polarization resistance and \(C_{1}\) is the polarization capacitance. \(h\) is the hysteresis voltage, it is the function of the SOC and temperature. The SOC of the cell can be expressed as follows:  
\[\text{SOC(k)}=\text{SOC(0)}+\frac{\int_{0}^{k}\eta Idt}{\text{C}_{\text{N}}} \tag{2}\]  
where SOC(0) is the initial SOC value of the cell; SOC(\(k\)) is the current SOC value at time \(k\); C\({}_{\text{N}}\) is the nominal capacity and the function of the discharge rate and temperature; \(\eta\) is the coulomb efficiency, the function of current and temperature [30].