## 3 Simulation Results
### Effect of the Inertial Penalty Factor  
The main original contribution of this paper is the introduction of the inertial penalty factor \(P_{I}\) into the definition of the dynamic power-weighted efficiency to discriminate the most suitable torque distribution. This contribution allows to penalize all the admissible solutions where a considerable amount of input power would be exploited to accelerate one or more powertrain components instead of accelerating the vehicle and/or overcome its motion resistances. To proof the efficacy of this effect, the resulting maps of ICE, MG1 and MG2 angular accelerations are compared against the case where the actuators torque distribution is chosen based on the lower power-weighted efficiency \(\eta_{PW}\) instead of \(\eta_{PW}^{\text{b}}\). The simulation results, obtained for a null road slope, are shown in Figure 11.  
Figure 8: ICE **(left)**, MG1 **(center)**, and MG2 **(right)** operating points (green circles) for the Input Split **(top)** and Compound Split **(bottom)** modes and road slope \(\alpha=0\).  
Figure 9: \(P_{butt}\) maps for the Input Split **(left)** and Compound Split **(right)** modes and road slope \(\alpha=0\).  
The color bar scale has been adapted for each map, in order to center the null angular acceleration within a green zone between the maximum (red zone) and the minimum (blue zone) angular accelerations for each powertrain actuator. Both red and blue zones are considered as undesirable since part of the input power would be wasted to increase the kinetic energy of one of the three actuators inertia. The presence of the penalty factor \(P_{I}\) automatically discards the admissible but undesirable solutions by reducing their corresponding \(\eta_{PW}^{p}\), as it is evident by the larger green zone for the top maps compared to the bottom ones, especially for ICE and MG1. Instead, MG2 angular acceleration is not affected by the introduction of the penalty factor since it is kinematically correlated to the vehicle acceleration \(a_{v}\) for the Input Split mode (clutch 2 engaged).  
A drawback correlated to the penalty factor is that the resulting map of the power-weighted efficiency \(\eta_{PW}\) assumes a different shape, as shown in Figure 12: when the penalty factor is activated there are some regions where the power-weighted efficiency is smaller.  
Figure 11: ICE **(left)**, MG1 **(center)** and MG2 **(right)** angular accelerations to achieve the lower power-weighted efficiency with **(top)** and without **(bottom)** the activation of the penalty factor \(P_{I}\) for the Input Split mode and road slope \(\alpha=0\).  
Figure 10: Resulting powertrain mode map for the two-mode hybrid system with road slope \(\alpha=0\).  
The penalty factor discharges some admissible solutions with higher power-weighted efficiency values since it does not fully exploit the kinetic energy stored within the power-ertrain components. However, this drawback does not affect in a significant way the power-weighted efficiency, as proved by Figure 12. A similar consideration can be drawn with the Compound Split mode as reported by Figures 13 and 14.  
Figure 12: Power-weighted efficiency with (**left**) and without (**right**) the activation of the penalty factor \(P_{I}\) for the Input Split mode and road slope \(\alpha=0\).  
Figure 13: ICE (**left**), MG1 (**center**) and MG2 (**right**) angular accelerations to achieve the lower power-weighted efficiency with (**top**) and without (**bottom**) the activation of the penalty factor \(P_{I}\) for the Compound Split mode and road slope \(\alpha=0\).