## 3 Simulation Results
\right.\] (8)
* Evaluation of the transmission output resistance torque \(T_{load}\): \[T_{load}=\frac{R_{W}}{\tau_{F}}\bigg{(}Mgf_{0}\cos\alpha+Mg\sin\alpha+\frac{ 1}{2}\rho C_{x}S_{F}V^{2}\bigg{)}\] (9) where \(f_{0}\) is the constant tire rolling resistance coefficient, \(g\) is the gravitational acceleration, \(\rho\) is the air density, \(C_{x}\) is the aerodynamic drag coefficient, \(S_{F}\) the vehicle frontal area; It is important to remark that the vehicle braking is only entrusted to the electric generators, thus not considering the presence of a conventional hydraulic braking system (see the results presented by [PERSON] et al. [41] for further details about hydraulic braking system modelling). * For each set of boundary conditions (\(\alpha\), \(V\), \(a_{v}\)), there exist infinite solutions for the distribution torque vector \(T\), which satisfy Equation (4) (number of actuators larger than the number of dofs). By ranging the two MGs torques, \(T_{MG1}\) and \(T_{MG2}\), between their minimum and maximum values at \(\omega_{MG1}\) and \(\omega_{MG2}\), respectively, the system of equations in Equation (4) can be numerically discretized and solved to elaborate the resulting ICE torque \(T_{ICE}\) and angular acceleration \(\omega_{ICE}\). The angular accelerations of the two MGs, \(\omega_{MG1}\) and \(\omega_{MG2}\), are then calculated from \(\omega_{out}\) and \(\omega_{ICE}\) through the PGS kinematic relations in Equation (8). Finally, the procedure computes a discrete number of solutions, identified by a combination of \(\omega_{ICE}\), \(T_{MG1}\), and \(T_{MG2}\), that satisfy the boundary conditions of road slope, vehicle speed, and acceleration. An admissible solution is accepted, to calculate the output power and efficiency of each actuator, only if the following constraints are satisfied: \[\left\{ \begin{array}{l}\omega_{ICE,min}\leq\omega_{ICE}\leq\omega_{ICE,max} \\ \omega_{MG1,min}\leq\omega_{MG1}\leq\omega_{MG1,max}\\ \omega_{MG2,min}\leq\omega_{MG2}\leq\omega_{MG2,max}\\ \end{array} \right.\quad\left\{ \begin{array}{l}0\leq P_{ICE}\leq P_{ICE,max} (\omega_{ICE})\\ |P_{MG1}|\leq P_{MG1,max}(\omega_{MG1})\\ |P_{MG2}|\leq P_{MG2,max}(\omega_{MG2})\\ P_{but,min}\leq P_{but}\leq P_{butt,max}\end{array} \right.\] (10)  
The classic definition of the power-weighted efficiency in Equation (5) is only valid for steady-state working conditions of the powertrain components (negligible influence of inertial parameters) and cannot be directly adopted as discriminant factor to select the optimal combination of \(\omega_{ICE}\), \(T_{MG1}\), and \(T_{MG2}\), for a given road slope, vehicle speed, and acceleration. Consequently, a modified version of power-weighted efficiency, \(\eta_{PW}^{p}\), is defined to penalize the admissible solutions with a high inertial power \(P_{I}\) of the powertrain components:  
\[\eta_{PW}=\left\{\begin{array}{l}\frac{P_{0}I_{0}I_{0}I_{0,max}+\frac{P_{0}I _{0}I_{0,max}}{P_{0,max}I_{0,max}I_{0,max}I_{0,max}I_{0,max}I_{0,max}I_{0,max}I _{0,  
The inertial power enters as a penalty factor in the definition of the power-weighted efficiency but it physically represents the variation of kinetic energy accumulated or released to equilibrate the power distribution of the transmission in Figure 3: \(P_{I}=P_{c2}\eta_{C}\eta_{M}+P_{c3}+\mu P_{Butt}\eta_{butt}\eta_{M}-P_{load}\). The admissible solution with a combination of \(\omega_{ICE}\), \(T_{MG1}\) and \(T_{MG2}\) that guarantees the lowest value of \(\eta_{PW}^{p}\) is then selected as the optimal choice, in terms of actuators power distribution, for that boundary conditions of road slope, vehicle speed, and acceleration. The optimal solution is then represented through 2D maps reporting the value of speed, torque and power for each powertrain component as function of vehicle speed and acceleration, calculated for a fixed road slope, as shown in Figure 6 for the Input Split mode and \(\alpha=0\).