## 2 Plug-In Hybrid Architecture Modelling
\begin{split} S_{fuel}(k)=c_{1 ice}.T_{ice}(k).W_{ice}{}^{2}+c_{2 ce}.W_{ice}{}^{2}+c_{3 ice}.T_{ice}{}^{2}(k).W_{ice}(k)+c_{4 icec}.T_{ice}(k).W_{ice} \\ +c_{3 ice}.W_{ice}(k).+c_{6 ice}.T_{ice}(k)^{2}+c_{7 ice}.T_{ice}(k)+c _{8 ice}\end{split}  \tag{1}\]  
\[ \begin{split} P_{BATT}(k)=T_{em}(k).W_{em}+c_{lem}.T_{em}(k).W_{ em}{}^{2}+c_{2 em}.W_{em}{}^{2}+c_{3 em}.T_{em}{}^{2}(k).W_{em}(k)+c_{4 em}.T_{em}(k).W_{ em}\\ +c_{5 em}.W_{em}(k).+c_{6 em}.T_{em}(k)^{2}+c_{7 em}.T_{em}(k)+c_{8 em} \end{split}  \tag{2}\]  
Figure 5: The global EMS-Parallel-PHEV model presenting four main sub-blocks. where \(T_{ice}\), \(T_{em}\) are the engine and electric motor torques, \(W_{ice}\), \(W_{em}\) are the engine and electric motor speeds and \(c_{lice}\) and \(c_{tem}\) for \(1\leq I\leq 8\) are the polynomial interpolation coefficients of the internal combustion engine and the electric motor. The battery is a complex chemical device difficult to be mathematically described. In our case, it is modeled with a simple model of an equivalent electric circuit, which presents an open circuit voltage (OCV) connected in series with internal resistance, depending on the state of charge of the battery (SoC). As its evolution cannot be measured easily and accurately by sensors, SoC is defined as the ratio between the remaining capacity and its nominal capacity as presented in Equation (3) where \(i_{batt}\) is the battery current, \(C_{n}\) is the  
Figure 6: Electric motor cartography presenting power loss within EM torque and speed limits. Figure 7: ICE cartography presenting fuel flow within engine torque and speed limits. nominal capacity (Ah), and \(\eta_{F}\) is the Faraday efficiency. The main battery characteristics are represented in Table 1.  
\[SoC(t)=SoC(t_{0})-\frac{100\int_{t_{0}}^{T}i_{batt}(t)\eta_{F}(sign\;i_{batt})dt}{ 3600C_{n}} \tag{3}\]  
\[\eta_{F}(sign\;i_{batt})=\left\{ \begin{array}{cc}0.95,&i_{batt}<0\\ 1,&i_{batt}\geq 0\end{array} \right. \tag{4}\]  
Furthermore, the vehicle is considered as an urban passenger car with a chassis, wheels, and the external environment (resistance forces) then, Equation (5) defines its longitudinal dynamics [22]. Here, the gearbox model considers a five-speed gearbox that can be changed instantly. The main vehicle parameters are depicted in Table 2. The provided vehicle model and engine performance maps (Figure 7) allow for the calculation of speed and torque over time. Then the fuel consumption rate can be deduced using fuel flow projections \(S_{fuel}\)\((T_{ice}(k).W_{ice})\). \[\dot{w}_{wh}(t)=\frac{1}{M_{\rm{p}}R_{t}}(T_{wh}(t)-T_{res}(t)-T_{br}(t)) \tag{5}\]  
\[T_{res}=(m_{\rm{p}}g(\sin\alpha+(a+bv^{2})\cos\alpha)+\frac{1}{2}\rho_{a}C_{d} A_{f}v^{2})R_{t} \tag{6}\]  
where \(\dot{w}_{wh}\) is the angular acceleration of the wheels, \(T_{wh}\) is the powertrain torque at the wheels (Nm) \((T_{wh}=R(k)(T_{ice}+\rho T_{em}))\), \(T_{br}\) is the brake torque (Nm), \(M_{\rm{p}}\) is the total vehicle mass (kg) (covering the unladen vehicle mass (1155 kg), the mass of battery (280 kg), the engine (140 kg)) and the electric motor combined with the converters(59 kg)), \(R_{t}\) is wheel radius (m), \(R(k)\) the gearbox speed ratio and \(\rho\) reduction ratio between the electric and thermal motors, \(T_{res}\) is the resistant torque presented in Equation (6) [1], \(g\) is the gravity acceleration (m/s\({}^{2}\)), \(v\) is the vehicle velocity (m/s), \(A_{f}\) is the vehicle front area (m\({}^{2}\)), \(\rho\) is the air density (kg/m\({}^{3}\)), \(Cd\) is the aerodynamic drag coefficient, \(\alpha\) is the road slope angle (rd) and \(a\) and \(b\) are rolling coefficients [1]. Moreover, the role of the supervision calculator EMS is to provide the system with the appropriate control of all the traction chain components: the electric torque, the thermal