## 3 Modeling and Formulation of Vehicle Subsystems  
To optimize the EMS, first, it is needed to establish the vehicle's model before incorporating it in an optimization algorithm. This section goes through the modeling process of the individual components as subsystems of the vehicle model. The MATLAB/Simulink(r) version 2016b environment was used to perform the modeling, simulation, and optimization procedures in the present study. Regarding the modeling approach, the backward calculation method was employed since it combines the advantages of simplicity and low computational cost when it comes to integrating the model into the optimization procedure [37; 48; 49; 50]. Figure 2: Hybrid electric vehicles (HEV) equipped with EVT topology. The required speed and acceleration time series besides the vehicle's constant parameters (Table 1) are used in the vehicle longitudinal dynamic subsystem to calculate the tractive forces. To this end, this subsystem considers resistance forces corresponding to drag, rolling, gradient, and inertia as follows:  
\[F_{T}=\frac{1}{2}\rho v^{2}C_{D}A+C_{r}mg\cos a+mg\sin a+\text{mC}_{\text{J}} \frac{dv}{dt} \tag{1}\]  
The outputs of the vehicle dynamic subsystem are the wheels' required torque and rotational speed, which can be readily calculated knowing the wheel's radius:  
\[T_{w}=F_{T}R_{w} \tag{2}\]  
\[\omega_{w}=\frac{v}{R_{w}} \tag{3}\]  
An input-output approach using torque-speed pairs based on the efficiency and the fuel rate maps are stored in look-up tables for the ICE subsystem. The non-scaled efficiency map of a generic ICE [37] was considered for the present study. In the fuel tank subsystem, the consumed fuel (liter) over the driving cycle can be modeled based on Equation (4), where \(\overset{\bullet}{m}_{f}\) (g/s) stands for the fuel consumption rate and \(\rho_{f}\) (kg/m\({}^{3}\)) represents the fuel density. \[Fuel=\int\limits_{0}^{t}\frac{\overset{\bullet}{m}_{f}}{\rho_{f}}dt \tag{4}\]  
To model the battery pack, the elements of a first-order Thevenin equivalent circuit as a function of SoC were identified by using the experimental data [51] and were stored in the Simulink(r) look-up tables. The terminal voltage (\(V_{\text{{hatt}}}\)) and SoC can be mathematically expressed through Equations (5)-(8), where \(V_{\alpha c}\) is the open circuit voltage; \(R_{\text{{int}}}\) stands for the internal resistance; \(C_{p}\) and \(R_{p}\) represent the polarization capacitance and polarization resistance, respectively; and \(N_{\text{{Batt}}}\) stands for the number of the batteries. A LiFePO\({}_{4}\) (LFP) battery type with the specifications given in Table 2 was considered in this study. \[I_{\text{{hatt}}}=\frac{I_{\text{{load}}}}{N_{\text{{Batt}}}} \tag{5}\]  
\[\frac{dV_{\text{{cp}}}}{dt}=\frac{-V_{\text{{cp}}}}{C_{p}R_{p}}+\frac{I_{ \text{{Batt}}}}{C_{p}} \tag{6}\]  
\[V_{\text{{Batt}}}=N_{\text{{Batt}}}(V_{\text{{oc}}}-I_{\text{{Batt}}}R_{\text{ int}}-V_{\text{{cp}}}) \tag{7}\]  
\[SoC=SoC_{0}+\frac{1}{3600}\int\frac{I_{\text{{Batt}}}}{C_{p}}dt \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Parameter (Unit)** & **Quantity** \\ \hline Drag coefficient & \(C_{D}\) & 0.24 \\ Rolling resistance coefficient & \(C_{r}\) & 0.009 \\ Rotational inertia coefficient & \(C_{\text{J}}\) & 1.075 \\ Frontal area & \(A\) (m\({}^{2}\)) & 1.74 \\ Wheel radius & \(R_{w}\) (m) & 0.287 \\ Air Density & \(\rho\) (kg/m\({}^{3}\)) & 1.2 \\ Auxiliary load & \(P\) (W) & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Constant parameters for the vehicle dynamic calculations. Regarding the power converters, the power-efficiency pairs were stored in their corresponding look-up tables and the power flow directions were considered in the calculations. To this end, for the traction mode (while \(p>0\)), the efficiency operator \(\beta=-1\), and for the braking mode (while \(p<0\)), the efficiency operator \(\beta=1\) are applied in Equation (9). \[P_{out}=P_{in}\eta^{\beta} \tag{9}\]  
The EVT subsystem consists of two concentric rotors.