## 2 Numerical Simulations
The main differences between gasoline- and hydrogen-fueled engine simulations are that the gasoline engine is operated with a rich A/F mixture and limited SA near full load conditions (Figure 1) to avoid knocking phenomena, and this lowers the engine BTE compared to hydrogen configurations; and the combustion duration of a gasoline-fueled engine is set to around 80 CAD [16; 22; 23] while the hydrogen combustion duration is set to around 50 CAD (roughly 60% lower than gasoline as stated above), and this further increases the hydrogen engine BTE compared to gasoline operation. As already stated before, the hybridization obtains the best results for a conventional engine, and the scope of this work is to compare the hydrogen- and gasoline-fueled engines on their best operating conditions. The ICE is supposed to be implemented in a series-parallel hybrid vehicle equipped with a power-split device and two electric motor-generators in order to obtain the advantages of both series and parallel configurations as well as a high voltage battery pack (140 V) able to both recover the kinetic energy of the decelerating vehicle and drive the electric motor during transient operations. The electric motors maximum power can be set to around 80% of the ICE maximum power (36 kW), and the battery pack energy capacity can be set to around 900 Wh. The above-described hybrid vehicle specifications correspond to that of a light duty passenger car [29]. Figure 1: \(\lambda\) (**a**) and SA (**b**) maps of the gasoline-fueled engine vs. engine speed [rpm] and MAP [bar]. Figure 2: SA maps of the TC (**a**) and VCSC (**b**) hydrogen-fueled engine vs. engine speed and MAP. In order to test the three hybridized engines (gasoline, VCSC and TC hydrogen) in a realistic situation, the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) was considered. The cycle is composed of 1800 speed and acceleration values divided by a one second time interval (a fine discretization). For each speed value of the WLTC, the power required by the vehicle (PRV) was evaluated as the product between the speed and the force required to push the vehicle at that speed, sum of the acceleration force \(F_{a}\) and the drag force \(F_{d}\). Since both the speed and acceleration are known, for each operating point of the WLTC, it is possible to evaluate \(F_{a}\) with the following equation:  
\[F_{a}\ =\ (m\ +m_{r})a \tag{5}\]  
where \(m\) is the vehicle curb weight, a is its acceleration and \(m_{r}\) is the total equivalent mass of the rotating elements. \(F_{d}\) is the sum of aerodynamic and rolling resistance forces and can be evaluated with the following equation:  
\[F_{d}\ =\ \frac{1}{2}A_{d}\rho v^{2}\ +\ mgf_{r} \tag{6}\]  
where \(\rho\) is the air density, \(v\) is the vehicle speed, \(A_{d}\) is the drag area (product of the vehicle frontal area and the drag coefficient), \(g\) is the gravitational acceleration and \(f_{r}\) is the wheels' rolling friction coefficient. Assuming \(m_{r}\) = 45 kg, \(m\) = 1000 kg, \(A_{d}\) = 0.75 m\({}^{2}\) (mean values of a standard passenger car [29]) and \(f_{r}\) = 0.01, the PRV was evaluated for each point of the WLTC. The power supplied by the engine (PSE) was evaluated with the following equation:  
\[\text{PSE}\ =\ \frac{\text{PRV}}{\eta_{t}} \tag{7}\]  
where \(\eta_{t}\) is the transmission efficiency for which a value of 0.9 can be set [30; 31; 32; 33]. Assuming that in a split-power HEV the engine is always operated in its best efficiency points, for each WLTC operating point a specific procedure (described in the following section) was used to link a specific engine efficiency to each _PSE_ and then to evaluate the power required by the engine (PRE); integrating the PRE values along the 1800 s of the WLTC duration, the total required energy was found.