## 3 Comprehensive Model Development
### Butler-Volmer Equation  
Capturing of the battery's dynamic responses can be done through the equivalent circuit model descriptions. For the battery's electrochemical reaction phenomena, Butler-Volmer kinetic equation descriptions can be employed, as shown in (5). Charge transfer kinetics of lithium intercalation obeys the Butler-Volmer equation [55; 56; 57]. It describes the current from basic electrochemical reactions varying with activation overvoltage \(\eta_{act}\), such that  
\[J=J_{0}\cdot\left\{e^{[\frac{\alpha\eta_{act}}{R_{B}T}]}-e^{[\frac{(\alpha-1) \eta_{act}}{R_{B}T}]}\right\}, \tag{5}\]  
where \(J\) is the electrode current density in A/m\({}^{2}\), \(J_{0}\) is the exchange current density in A/m\({}^{2}\), \(\eta_{act}\) is the surface overpotential in volts, \(k_{B}\) is Boltzmann constant, \(T\) is the absolute temperature, \(q~{}=~{}1.602\times 10^{-19}\)(C), \(\alpha\) is the anodic charge transfer coefficient, and \((\alpha-1)\) is the cathodic charge transfer coefficient, where \(\alpha\) is assumed to be 0.5 in a battery.  
Figure 4: The dual RC equivalent-circuit model for a LiFePO\({}_{4}\) battery. (a) OCV-SoC characteristics; (b) Voltage/current characteristics.  
The surface overpotential is as follows: \(\eta_{act}=(E-E_{rev})\), where \(E\) is the difference of the mean electrostatic potentials of ions and electronsand \(E_{rev}\) is the Nernst equilibrium potential, which is strongly related to the local activity of lithium.  
Closed circuit voltage (CCV) is defined as \(V=V_{ideal}-\eta_{act}\), where \(V_{ideal}\) is he standard potential defined by the open circuit voltage plateau (\(V_{ideal}\) = 3.42 V for Li metal) [5]. The size of \(\eta_{act}\) is dependent on reaction kinetics, meaning \(J_{0}\) influences \(\eta_{act}\) and it is in turn affected by the reaction temperature. \(\eta_{act}\) represents the voltage loss necessary in overcoming the activation energy barrier of electrochemical reactions. Thus, to obtain a greater current from the battery, a larger voltage loss is inevitable. Changes in electro-chemical reactions can be seen as changes in kinetic parameters, such as \(\alpha\) and \(J_{0}\). If \(J_{0}\) is low for any specific net current, the kinetics will become sluggish and the activation overvoltage will be even greater. If \(J_{0}\) is large, a large current can be supplied. In [58], it was stated that since \(J_{0}\) is difficult to be measured from the outside, Equation (5) can be rewritten as:  
\[\frac{1}{J_{0}\cdot S}\cdot I(t)=\left[e^{\frac{i\pi\eta_{act}}{E_{B}t}}-e^{ \frac{(\alpha-1)\eta_{act}}{E_{B}t}}\right]; \tag{6}\]  
\[I(t)=J_{(t)}\cdot S, \tag{7}\]  
where \(S\) is the effective area.  
In [54], the overvoltage \(\eta_{act}(t)\) caused by current variation is represented as:  
\[\eta_{act}(t)=\frac{2 RT}{F}\ln(\frac{1}{2J_{0}\cdot S}\cdot I(t)+\sqrt{\left( \frac{1}{2J_{0}\cdot S}\cdot I(t)\right)^{2}+1}) \tag{8}\]  
Although there are still a variety of electrochemical variables that cannot be obtained externally, it is possible to acquire a specific change of potential excited by diverse currents. Thus, it can be seen that \(J_{0}\) and \(S\) are closely related to SoC, reflecting the complexity during the polarization establishment. Increasing the current while fixing other available variables will result in a sharp deviation from the equilibrium state.