## 2 Model Description  
### Vehicle Model  
In this section, the platoon model is introduced. A number of N heavy-duty electric vehicles (HDEVs) moving on a straight and sloped road is considered. Figure 1 shows the scheme of the mechanical model of the vehicle platoon: the dark blue vehicle is called the lead vehicle, whereas the others are referred to as the followers. Each vehicle is characterised by inertial and geometrical properties: a mass \(m_{i}\) and the total vehicle length \(l_{i}\). Only the longitudinal motion is accounted for, whereas the lateral and pitch ones are neglected. The position, velocity, and acceleration (\(x_{i},v_{i},\dot{v}_{i}\)) of each vehicle are computed by using a fixed reference system \(X\). The quantity \(d_{i-1,i}\) refers to the inter-vehicle distance (i.e., the distance between the rear and front bumpers of two consecutive vehicles). The electric motor provides the amount of torque necessary to overcome the rolling, aerodynamic, and climbing resistances, as well as the inertial torque for vehicle and powertrain acceleration. The amount of torque is computed by the control system, and it is transmitted to the driveshafts by a single-speed transmission, modelled as a constant transmission ratio and assuming rigid shafts. Even though the pure rolling motion is assumed for each wheel, the actual tyre-road friction condition has been considered by implementing a simple traction control system that saturates the tyre forces to their maximum transmissible limit, depending on road grip \(\mu\) and the normal load \(F_{z}\). In this way, the tyres never exceed their maximum transmissible force, \(F_{x}<\mu F_{z}\). The equation of motion of the \(i_{th}\) vehicle is:  
Figure 1: Vehicle platooning on straight and sloped roads. \[m_{tot,i}\dot{v}_{F,i}=\frac{\eta\tau T_{m,i}}{r_{w,i}}-\left[m_{ig}\text{sin}( \alpha)+m_{i}gf_{0,i}\text{cos}(\alpha)+\left(m_{ig}f_{2,i}\text{cos}(\alpha)+ \frac{1}{2}\rho c_{x,i}(d_{i-1,i})A_{f,i}\right)v_{F,i}^{2}\right] \tag{1}\]  
where \(m_{tot,i}\) is the equivalent translating mass of the vehicle, accounting for the translating and rotating components, \(f_{0,i}\) and \(f_{2,i}\) are the rolling resistance coefficients that are generally characterised experimentally through quadratic regression applied to the coast-down test, \(r_{w,i}\) is the wheel rolling radius, \(\tau\) and \(\eta\) are the reduction ratio and the efficiency of the transmission, \(\rho\) is the air density, \(A_{f,i}\) is the vehicle frontal area, \(c_{x,i}\) is the aerodynamic drag coefficient, and \(\alpha\) represents the longitudinal road slope. Once the torque requested for the electric motor is known, the power request can be derived as the aforementioned torque multiplied by the angular velocity of the motor. Electric motor losses are modelled as a function of their torque and speed through a 2D map. The power absorbed by the auxiliaries is assumed to be constant. The battery power request is the sum of the electric motor power and the abovementioned contributions. The behaviour of the battery is represented by the following equations, which evaluate the amount of current flowing through the battery and the instantaneous change in the state of charge (\(\dot{\sigma}\)):  
\[I_{batt}=\frac{V_{batt}-\sqrt{V_{batt}^{2}-4R_{batt}P_{batt}}}{2R_{batt}} \tag{2}\]  
\[\dot{\sigma}=\frac{I_{batt}}{Q_{batt}\cdot\Delta t} \tag{3}\]  
where \(I_{batt}\) is the battery current, \(V_{batt}\) and \(R_{batt}\) are its open-circuit voltage and internal resistance, and \(Q_{batt}\) and \(\Delta t\) are the battery maximum capacity and the simulation timestep, respectively.