## 2 Materials and Methods
### ICE Dynamic Modeling  
In this article, the engine is modeled based on the generic model in Simscape and the other mathematical formulations have been adapted from [40]. Therefore, the ICE model is parameterized on the basis of a third-order polynomial as expressed in [41] as follows:  
\[ \begin{cases}\omega_{N}(\omega)\equiv\frac{\omega}{\omega_{pp}}\\ P(\omega(\omega_{N}))=P_{P}p_{N}(\omega_{N})\\ p_{N}(\omega_{N})=s_{1}\omega_{N}+s_{2}\omega_{N}^{2}+s_{3}\omega_{N}^{3}\\ \end{cases}  \tag{8}\]  
where \(\omega_{N}(\omega)\), \(\omega\), and \(\omega_{PP}\), are the normalized ICE speed, current ICE speed [rpm], and ICE speed [rpm] at the peak power. Moreover, \(P(\omega(\omega_{N}))\), \(P_{P}\), and \(p_{N}\) are the current ICE power [kW], peak ICE power [kW], and normalized ICE power. Finally, \(s_{1}\), \(s_{2}\), and \(s_{3}\) are the constant coefficients of the polynomial [41]. For wide open throttle condition, the ICE torque is expressed as follows:  
\[ \begin{cases}T(\omega(\omega_{N}))=p_{1}+p_{2}\omega_{N}+p_{3}\omega_{N}^{2}\\ p_{1}\equiv\frac{s_{1}P_{P}}{\omega_{PP}}\\ p_{2}\equiv\frac{s_{2}P_{P}}{\omega_{PP}}\\ p_{3}\equiv\frac{s_{3}P_{P}}{\omega_{PP}}\\ \omega_{NPP}=\omega_{N}(\omega_{NPP})=1\\ \end{cases}  \tag{9}\]  
where \(\omega_{NPP}\) is the normalized engine speed aligned with peak power. If we substitute the normalized engine speed (\(\omega_{NPT}\)) at peak power (\(P_{P}\)) [kW] into \(p_{N}\) and set \(p_{N}((\omega_{PP}))\) to zero, we have:  
\[ \begin{cases}s_{1}+s_{2}+s_{3}=1\\ s_{1}+2s_{2}+3s_{3}=0;p_{N}=0\\ \omega_{NPT}=\omega_{N}(\omega_{PT})=\frac{\omega_{PT}}{\omega_{PP}}<1\\ \end{cases}  \tag{10}\]  
where speed of the peak power is greater than the speed of the peak torque. Based on the polynomial formula, we can find the normalized, dimensionless torque (\(\tau_{N}\)) and set the slope to zero and solve for \(s_{i}\)[41]:  
Figure 9: Battery Discharge Characteristics. \[ \begin{cases}\frac{d\tau_{\text{N}}(\omega_{NPT})}{d\omega}=s_{2}+2s_{3}\omega_{ NPT}=0\\ s_{1}=\frac{3-4\omega_{NPT}}{2(1-\omega_{NPT})}\\ s_{2}=-\frac{\omega_{NPT}}{(\omega_{NPT}-1)}\\ s_{3}=\frac{1}{2(\omega_{NPT}-1)}\end{cases}  \tag{11}\]  
Therefore, the engine torque \(T(\omega(\omega_{N}))\), the peak torque (\(T_{p}\)), normalized engine speed at peak torque (\(\omega_{NPT}\)), engine speed at peak power (\(\omega_{PP}\)), and engine speed at peak torque (\(\omega_{PT}\)) are related as follows [41]:  
\[ \begin{cases}T(\omega(\omega_{N}))=\frac{2T_{p}}{3-\omega_{NPT}}(s_{1}+s_{2} \omega_{N}+s_{3}\omega_{N}^{2})\\ 2T_{p}\omega_{PP}^{2}-3P_{p}\omega_{PP}+P_{P}\omega_{PT}\\ \omega_{PP}=\frac{3P_{p}+\sqrt{P_{p}(9P_{p}-8T_{p}\omega_{PT})}}{4T_{p}}> \omega_{PT}\end{cases}  \tag{12}\]  
where (\(T_{P}\omega_{PT}\)) is equivalent to power at peak torque (\(P_{PT}\)), which is always less than peak power (\(P_{p}\)) [41]. For the partially opened throttle condition, the following equation can be described:  
\[ \begin{cases}\Pi=max(\Pi_{i},\Pi_{c})\\ \frac{d\Pi_{c}}{dx}=\frac{0.5(1-\tanh(4\frac{\omega_{N}-\omega_{N}}{\omega_{T}} ))-\Pi_{c}}{\tau}\end{cases}  \tag{13}\]  
where \(\Pi\) is the throttle [%], \(\Pi_{i}\) is the input throttle [%], \(\Pi_{c}\) is the controller throttle, \(\omega\) is the engine speed [rpm], \(\omega_{r}\) is the partial or idle speed reference [rpm], \(\omega_{t}\) is the controller speed threshold [rpm], and \(\tau\) is the controller time constant [seconds] [41]. To compute the other parameters in this research, such as the engine efficiency, fuel flow rate, etc., we adapted the formulations described in [40], and as adapted for our previous paper [36]. Therefore, the ICE torque \(T(\omega(\omega))\) [Nm] is linked to the engine power (\(P(\omega(\omega_{N}))\)) [kW], and the rotational speed \(\omega\) [rpm]. \[