## 2 Battery Modeling Methods
### Equivalent Circuit Model  
ECM consists of a voltage source related to SOC, an internal resistor, and Resistance-Capacitance (RC) networks, which is able to describe the electrical relationship between the inputs (current, SOC, and temperature) and the terminal voltage [37]. Compared to empirical models, ECMs are much easier for the understanding of the electrical characteristic of the battery. Moreover, due to the plentiful circuit components and their combinations, ECM gives researchers sufficient freedom to design a  
\begin{table}
\begin{tabular}{c c} \hline
**Reference** & **Model Expression** \\ \hline
[29] & \(y_{k}=E_{0}-K\cdot\frac{Q}{Q-i_{k}\cdot T}+A\cdot e^{-B\cdot i_{k}\cdot T}\) \\  & \(Q\) is the battery capacity, \(A\) is the exponential zone amplitude, \(B\) is the time constant inverse of \\  & the exponential zone, \(K\) is the polarization voltage. \\ \hline
[30] & Discharge: \(y_{k}=E_{0}-R\cdot i_{k}-K\cdot\frac{Q}{Q-i_{k}\cdot T}\cdot(i_{k}\cdot T+i^{*} )+A\cdot e^{-B\cdot i_{k}\cdot T}\) \\  & Charge: \(y_{k}=E_{0}-R\cdot i_{k}-K\cdot\frac{Q}{i_{k}\cdot T-Q\cdot i_{k}\cdot T}\cdot i ^{*}-K\cdot\frac{Q}{Q-i\cdot i}\cdot i_{k}\cdot T+A\cdot e^{-B\cdot i_{k} \cdot T}\) \\  & \(i^{*}\) is the filtered current through the polarization resistance. \\ \hline
[31; 32] & \(y_{k}=E_{0}-R\cdot i_{k}-R_{pol}\cdot i_{k}^{*}-K\cdot Q\cdot\left(\frac{1}{ z_{1}+z_{0}}-1\right)+A\cdot e^{-B\cdot(1-z_{k})}\) \\  & \(\tau\frac{d\cdot i^{*}}{dt}+i^{*}=i\) \\  & \(R_{pol}= \begin{cases}\frac{K}{z_{k}}\\ \frac{K}{x-z_{k}}\end{cases} \) discharge \\  & \(R_{pol}\) is the polarization resistance. \\ \hline
[27] & \(y_{k}=E_{0}-R\cdot i_{k}-K_{2}\cdot\ln(\tau_{1}+z_{k})+K_{3}\cdot\ln(\tau_{2} +1-z_{k})\) \\  & \(\tau_{1}\) and \(\tau_{2}\) are the two additional constants. \\ \hline
[35] & \(y_{k}=E_{0}-R\cdot i_{k}-K_{2}\cdot\ln(z_{k})+K_{3}\cdot\ln(1-z_{k})+s_{k}\cdot M\) \\  & \(s_{k}= \begin{cases}1&i_{k}>\epsilon\\ -1&i_{k}<-\epsilon\\ s_{k-1}&|i_{k}|\leq\epsilon\end{cases} \) \\  & \(\epsilon\) is a small positive number, \(M\) is the correction term. \\ \hline
[33;suitable structure for their applications. Generally, in ECM, the resistor indicates the self-discharge and the high-valued capacitor or the voltage source stands for the battery OCV. The RC pairs with different time constants stand for the diffusion process in electrolyte and porous electrodes and the charge transfer and double-layer effect in the electrode. The _Rint_ model, _Thevenin_ model, Partnership for a New Generation of Vehicles (PNGV) model, and General Non-Linear (GNL) model are the four common ECMs in the literature, and are illustrated in Table 3.  
As seen from Table 3, the structures of the different ECMs have some connections. By adding an additional RC network to the _Rint_ model, the _Thevenin_ model is achieved and can better capture the dynamic terminal voltage of battery. Considering the effects of OCV variation, the PNGV model is obtained after adding a capacitor \(C_{cap}\) to the _Thevenin_ model [39; 40]. \(C_{cap}\) describes the OCV variation by the accumulation of the discharging current [43]. Adding more RC networks helps with describing the terminal voltage in more detail; the PNGV model usually has better accuracy than the _Rint_ and _Thevenin_ models. Compared with the _Thevenin_ model, one more RC network is included in the GNL model by taking into account the concentration polarization effect. In terms of SOC estimation area, ECMs with one RC or two RC pairs are especially popular for modeling the battery. Twelve battery models including the empirical models and ECMs with different RCs are compared in terms of modeling accuracy, which proves that one RC ECM is more suitable for the LiNMC battery and one RC ECM with one-state hysteresis is the best choice for the LiFePO\({}_{4}\) battery [44]. Ten lumped parameter models for Li-ion battery are compared in terms of modeling accuracy and the effects of the