## 2 Case Study and HEV Model
According to the powertrain architecture in Figure 2, the power provided by the ICE and motor must be equal to the mechanical power required by the vehicle to follow the driving cycle; moreover, due to the mechanical link, the motor and ICE share the same speed and, therefore, the following relation between the torque provided by both machines should be fulfilled:  
\[T_{g}=T_{m}+T_{ICE} \tag{2}\]  
where \(T_{g}\), \(T_{m}\) and \(T_{ICE}\) are the torque in the powertrain transmission, motor and ICE, respectively. Including the inertia of powertrain elements in an equivalent vehicle mass (\(m\)), the wheel torque required to drive the vehicle is given by:  
\[T_{w}=(m\dot{v}-mg\epsilon cos\theta-mgsin\theta-\frac{1}{2}\rho Ac_{d}v^{2})R _{w} \tag{3}\]  
where \(\mu\) is the rolling coefficient, \(\theta\) is the angle due to the road slope (that can be neglected in the considered route), \(\rho\) is the air density, \(Ac_{d}\) the product of frontal area and aerodynamic coefficient of the vehicle, \(R_{w}\) the wheel radius and, finally, \(v\) and \(\dot{v}\) the vehicle speed and acceleration, respectively. Note that while motor and ICE speeds are proportional to the wheel speed through the gear ratio of the selected gear, their joint torque in Equation (2) (\(T_{g}\)) is proportional to the wheel torque through the inverse of the gear ratio. In this sense, if the demanded vehicle speed is known, the demanded acceleration can be computed and the requested torque for the powertrain (\(T_{g}\)) can be estimated from Equation (3) and the gear ratio. Then, defining the control action as \(u=T_{m}\), Equation (2) can be reformulated as:  
\[T_{ICE}=T_{g}-T_{m} \tag{4}\]  
Figure 2: System layout, nomenclature and sign criteria for the parallel HEV architecture considered in this paper. and, therefore, provided a vehicle-speed demand and a decision on \(u\), the ICE torque is defined. Regarding the ICE, the model employed is based on the quasi-static (QS) approach introduced in [32], using experimental data to map the fuel consumption depending on the engine speed (\(\omega_{g}\)) and torque (\(T_{ICE}\)), so:  
\[m_{f}^{QS}=g(\omega_{g},T_{ICE}) \tag{5}\]  
The quasi-static approach assumes that the engine can move instantaneously from one operating condition to another without an increase in the fuel consumption and, particularly, there is not any fuel associated to start the ICE. In this sense, the simplicity of the quasi-static approach comes at the price of having frequent switching between ICE on and off that can be unrealistic. To avoid this issue, an additional cost is used for the engine start as:  
\[m_{f}=\left\{ \begin{array}{c}(1+\beta)m_{f}^{QS}\;if\;ICE_{on}=0\\ m_{f}^{QS}\;if\;ICE_{on}=1\end{array} \right. \tag{6}\]  
where \(\beta\) is a positive value to be calibrated and the integer variable \(ICE_{on}\) represents if the engine is on (1) or off (0) according to the following dynamics:  
\[ICE_{on}(t)=\left\{ \begin{array}{c}1\;if\;m_{f}(t-dt)>0\\ 0\;else\end{array} \right. \tag{7}\]  
Regarding the energy stored in the battery (\(E_{b}\)), its dynamic equation is:  
\[\dot{E}_{b}=-P_{b} \tag{8}\]  
with \(P_{b}\) being the battery power (considered positive when the battery is drained and negative when the battery is being charged according to the sign criteria defined in Figure 2). The battery power \(P_{b}\) is obtained from the motor power demand and the motor efficiency by means of a model. The quasi-static approach is employed for the electric motor, while in this case the map used considers positive and negative torques to model battery recharging.