## 2 Materials and Methods
### Nissan Leaf Battery Modeling  
In this article, a 2011 Nissan Leaf battery pack based on lithium-ion batteries (LIBs), commonly used in EVs and HEVs, is utilized as proposed in our previous studies. The LIBs have gained significant research attention in recent years and have become the preferred choice for EVs, both currently and in future research, due to their exceptionally high energy and power density features [38]. Therefore, the superiority of LIBs in this regard makes them a suitable candidate in the field of electric vehicles, driving the widespread adoption of LIBs in the automotive industry. The 2011 Nissan Leaf battery pack comprises 48 modules, and in each module, there are four cells, each containing four cells arranged in two series pairs in parallel, totalling 192 cells in the pack. The battery pack has a 24 kWh rated energy capacity and a nominal voltage of 360 V. Moreover, the individual cell specifications include a nominal voltage of 3.75 V, maximum voltage of 4.2 V, discharge voltage of 2.5 V, and current capacity of 33.1 Ah. The pack achieves a maximum voltage of 403.2 V, a current capacity of 66.2 Ah, and a power output exceeding 90 kW [39]. To develop a model of the Nissan Leaf battery, it is assumed to be a spontaneous electrochemical redox reaction that does electrical work, converting chemical energy into electromotive force (EMF). This perspective enables us to model the Nissan Leaf battery as an energy conversion system. By applying fundamental principles of electrochemistry and thermodynamics, we derive a mathematical model that captures the dynamic behavior of the battery, including its voltage, current, and state of charge (SOC) as expressed in [39] as follows:  
\[ \begin{cases}V_{r}^{0}=-\frac{\Delta C^{0}}{nF}\\ V_{b}=V_{r}^{0}-\frac{RT}{nF}log_{e}Q_{R}-\Omega_{d}\\ V_{b}=V_{r}^{0}-\frac{RT}{nF}log_{e}Q_{R}-R_{b}l_{b};\Omega_{d}=R_{b}l_{b} \end{cases}  \tag{4}\]  
where \(R\) is the ideal gas constant [J/K/mol], \({V_{r}}^{0}\) is the cell voltage [V] (no-load open circuit), is the ohmic drop \(\Omega_{d}\), \(\Delta G^{0}\) is the change in the Gibbs free energy [J/mol], \(n\) is the number of moles of electron transferred according balanced equation of the redox reaction [39], \(F\) is the [PERSON]'s constant (96,485 J/V-mol), \(Q_{R}\) is the reaction quotient, \(R_{b}\) is the cell resistance [\(\Omega\)], \(l_{b}\) is the cell current [A], and \(V_{b}\) is the cell voltage [V], \(T\) is the temperature [K]. The battery cell voltage can be expressed as:  
\[V_{b}(l_{b},y)={V_{r}}^{0}-{A}log_{e}(B\cdot y)-K\cdot y-Fe^{G(y-y_{3})}-R_{b} l_{b} \tag{5}\]  
Figure 6: VW Crafter Closed-Loop Overview. This represents the overall overview of the subsystems of the hybrid vehicle model described in this section. where \(y\) is a function of the battery depth of discharge (DOD), the battery SOC, and the capacity. In addition, \(y_{3}\) is the value when the exponential drop starts, \(A\), \(B\), \(K\), \(F\), and \(G\) are the constants [39]. Equation (5) can be expressed as a function of the DOD as follows:  
\[ \begin{cases}V_{b}(I_{b},DOD)={V_{r}}^{0}-{Alog_{e}}(B\cdot DOD)-K\cdot DOD-{Fe^ {G(DOD-DOD_{3})}}-R_{b}I_{b}\\ V_{b}(I_{b},DOD)={V_{b}}(nl)(DOD)-R_{b}I_{b}\\ V_{b}(nl)(DOD)={V_{r}}^{0}-{Alog_{e}}(B\cdot DOD)-K\cdot DOD-{Fe^{G(DOD-DOD_{3} )}}\end{cases}  \tag{6}\]  
Therefore, these equations is presented by the general equivalent circuit consisting of the voltage source in series with the resistance as studied in [39]. To obtain the model of the whole pack, we connect the 48 modules in series and determined the pack parameters if \(I_{b}\) and \(V_{b}\) are the cell current and voltage, based on the following formulations, respectively:  
\[ \begin{cases}V_{bp}=2\cdot N_{m}V_{b}\\ I_{bp}=2\cdot I_{b}\\ R_{bp}=N_{m}R_{b}\\ C_{b}=N_{c}V_{b}I_{b}\\ C_{r}=\frac{C_{r}\cdot C_{l}}{C_{Ah}}\end{cases}