## 2 Powertrain Model  
The ECMS implementation is based on a powertrain model, which is used to perform a local optimization in order to determine the powertrain's control signals. Since the optimization needs to be applied in real time, a powertrain model with low computational effort and yet a sufficient level of detail is required. This is obtained by the so-called backward approach [42]. Starting with the requested acceleration \(a_{\text{veh}}\) and speed \(v_{\text{veh}}\) of the vehicle, the operating states of each powertrain component are determined backwards (see Figure 3). A vehicle dynamics model determines the required torque \(T_{\text{FD}}\) and angular velocity \(\omega_{\text{FD}}\) at the final drive. In order to satisfy these requirements, the DHT model determines the corresponding torques and angular velocities of the ICE and electric drive considering the transmission's control variables \(s_{\text{m}}\) and \(u_{\text{ED}}\). It is always assumed that the ICE and the electric drive are capable of generating the required torque and satisfying the requested vehicle acceleration, respectively. The submodels of the DHT, ICE, electric drive, and battery consider stationary states only, as no controllers are required to operate these components. Only the operation mode of the powertrain must be controlled via the DHT. Therefore, \(s_{\text{m}}\) defines the mode (CVT, PAR, or EM mode) and \(u_{\text{ED}}\) the torque or angular velocity of the electric drive, definable due to the resulting degree of freedom in the PAR and CVT modes, respectively. Both control variables are determined by the ECMS and are considered here as inputs of the powertrain model. Since no dynamic behavior is considered, the operation mode changes immediately without any transition (e.g., continuously changing the state of a clutch from engaged to disengaged). The output of the powertrain simulation is the mass flow rate of the fuel \(\dot{m}_{\text{f}}\) and the state of charge of the battery \(SoC\). Figure 4 shows the submodels of the powertrain and their most significant parameters. The models of the ICE and electric drive are map-based, whereby only the state behavior of the components is considered. To determine the specific fuel consumption \(b_{\text{e}}\) and power loss \(P_{\text{ED,loss}}\), the corresponding torques and speeds are used. Figure 3: Powertrain model according to the backward approach and the basic concept shown in Figure 1.  
The vehicle dynamics are considered in a longitudinal direction only. The torque \(T_{\text{FD}}\) and angular velocity \(\omega_{\text{FD}}\) at the final drive are determined according to the following equation:  
\[T_{\text{FD}}=r_{\text{w}}\cdot\left(F_{\text{drag}}(v_{\text{veh}})+F_{\text{ roll}}+F_{\text{downhill}}+m_{\text{veh}}.4_{\text{veh}}\right), \tag{1}\]  
with the radius of the wheels \(r_{\text{w}}\), the vehicle mass \(m_{\text{veh}}\), the vehicle speed \(v_{\text{veh}}\), and the vehicle acceleration \(a_{\text{veh}}\) (extracted from the driving cycle). The forces represent the driving resistances comprising the drag force \(F_{\text{drag}}\), the rolling friction force \(F_{\text{roll}}\), and the downhill force \(F_{\text{downhill}}\). In order to consider the inertias of the wheels and the flywheel mass of the ICE, equivalent masses are added to the overall vehicle mass. In the case of the flywheel mass, the average transmission ratio is used for parameter conversion. By means of the torque \(T_{\text{FD}}\) and the rotational speed \(\omega_{\text{FD}}\) at the final drive, the model of the DHT is evaluated (i.e., the torques and rotational speeds of the ICE and the electric drive are determined by considering the operation mode):  
\[s_{\text{m}}=\left\{ \begin{array}{cl}j&\text{for CVT-Mode in $j$-th gear,}\\ j+2&\text{for PAR-Mode in $j$-th gear,}\\ j+4&\text{for EM-Mode in $j$-th gear,}\end{array} \right. \tag{2}\]  
where \(j\) denotes the gear in which the corresponding mode is driven.