## 2 Powertrain Modeling  
### Vehicle Dynamic Model  
The CEV is modeled as a 3-DOF bicycle model, as illustrated in Figure 2. Right and left wheels on the same axle are lumped into one element, both for the front and rear. The resulting rear wheel is fixed, whereas the front one can steer at an angle \(\delta\). This angle is input for the vehicle dynamic model. The center of gravity \(C\), mass \(m\), polar moment of inertia \(l_{zz}\) with respect to the \(z\) axis, and wheelbase \(L\) of the vehicle are constant parameters. The wheelbase is the distance between axles, thus \(L=l_{f}+l_{r}\).  
The dynamic model of the vehicle is governed by the following equations:These expressions show the coupled dynamic behavior between the longitudinal (\(x\)), lateral (\(y\)), and yaw (\(\psi\)) DOFs. The motion of the vehicle is mainly affected by the longitudinal force on the front (motoring) axle (\(F_{xf}\)). Moreover, external lateral forces on the front (\(F_{yf}\)) and rear (\(F_{yr}\)) axles are further contributions to vehicle dynamics. Roll, pitch, and vertical dynamics are assumed to be uncoupled from this representation. Aerodynamic contributions can be neglected even at the maximum speed of the class-B-sized vehicle (14 m/s). cby the powertrain propulsion minus the rolling resistance. On the rear axle, the longitudinal force is only given by the rolling resistance term:  
\[F_{xf} =\frac{T_{m}\tau_{gb}}{\tau_{t}}-\frac{F_{c}}{2}\text{sign}\left( \dot{x}\right) \tag{4}\] \[F_{xr} =-\frac{F_{c}}{2}\text{sign}\left(\dot{x}\right) \tag{5}\]  
where \(T_{m}\) is the electric motor torque, \(\tau_{gb}\) the gear box gain, \(r_{t}\) the tire radius, \(\dot{x}\) the longitudinal speed of the vehicle, and \(F_{c}\) the Coulomb friction. Aerodynamic effects are neglected in the longitudinal model due to the low velocity values expected in urban environments (\(\dot{x}<60\,\text{km/h}\)) [50]. The propulsion force is another input of the vehicle model. The cornering stiffness of the front and rear tires (\(C_{xf}\) and \(C_{xr}\)) contribute to the lateral forces:  
\[F_{yf} =2C_{xf}\Big{(}\delta-\theta_{vf}\Big{)} \tag{6}\] \[F_{yr} =2C_{xr}(-\theta_{yr}) \tag{7}\]  
In addition, tire velocity angles are given by  
\[\theta_{vf} =\arctan\left(\frac{\dot{y}+l_{f}\dot{\psi}}{\dot{x}}\right) \tag{8}\] \[\theta_{vr} =\arctan\left(\frac{\dot{y}-l_{r}\dot{\psi}}{\dot{x}}\right) \tag{9}\]  
To favor faster computation, tire velocity angles were linearized assuming small angle deviations:  
\[\theta_{vf} \approx\frac{\dot{y}+l_{f}\dot{\psi}}{\dot{x}} \tag{10}\] \[\theta_{vr} \approx\frac{\dot{y}-l_{r}\dot{\psi}}{\dot{x}} \tag{11}\]  
Figure 2: The 3-DOF bicycle CEV model.  
The nonlinear model in Equations (1)-(3) describe the lateral dynamics [51]. The global expressions are obtained by substituting Equations (10) into (6), and (11) into (7). Finally, resulting Equations (6) and (7) are substituted into (1) and (2) to yield the following expressions:  
\[m\ddot{y} =-m\dot{x}\dot{y}+2\left[C_{af}\left(\delta-\frac{\dot{y}+l_{f} \dot{\psi}}{\dot{x}}\right)+C_{ar}\frac{l_{r}\dot{\psi}-\dot{y}}{\dot{x}}\right] \tag{12}\] \[l_{zz}\ddot{\psi} =2\left[l_{f}C_{af}\left(\delta-\frac{\dot{y}+l_{f}\dot{\psi}}{ \dot{x}}\right)-l_{r}C_{ar}\frac{l_{r}\dot{\psi}-\dot{y}}{\dot{x}}\right] \tag{13}\]  
The parameters for the vehicle model used in this study were extracted from the vehicle templates in \(\text{CarSIM}^{\text{\tiny{TM}}}\) of a B-class hatchback vehicle in electric powertrain configuration, front-wheel drive. The values are listed in Table 2. The presented model is used to reproduce the dynamic behavior of the CEV in the rule-based and FIS GLOSA/RBS framework. This representation has been extensively used due its accurate vehicle dynamic behavior [52].