## 2 Ultralight Aircraft and the Modeling Approach  
In this investigation, the retrofitting of the Promecec Aerospace Freccia to a hybrid electric ultralight aircraft was performed. The aircraft has a take-off mass of 450 kg and a wing area of 12 m\({}^{2}\) (see Figure 2). The HEPS was designed and simulated according to the realistic flight mission shown in Figure 3 that was obtained from a previous investigation [23].  
The original configuration for the Freccia includes a fixed pitch propeller. This kind of propeller works at high efficiency in a very limited operating range. High pitch propellers  
Figure 3: Mission profile.  
Figure 2: Reference aircraft (Promecc Aerospace Freccia).  
are preferred because they improve efficiency during cruising [24], which is the longest phase of flight and the most significant for fuel consumption. However, the reduced efficiency during ascent leads to a higher power demand that results in the need to increase the installed power and this, in turn, causes a higher takeoff weight [25]. Moreover, the engine efficiency is constrained in a conventional power system by the need to follow the speed and torque demands of the propeller. Therefore, the engine is sized according to the takeoff and ascent power and then works at partial load during the cruise and in the final part of the flight (descent).  
A variable pitch propeller maintains the ideal blade setting for each flight phase by adjusting the pitch position [25] by means of a push-pull rod driven by a servo [26] or a governor. In this way, the propeller rotation speed can be kept constant (constant speed propeller) [27]. Even if constant speed operation does not warrant optimal propeller efficiency, it allows the engine to be operated at constant speed and expands the propeller's operating range, allowing it to absorb the maximum engine power [28].  
In this investigation, a fixed speed propeller with a diameter of 1.78 m with the dimensionless performance map of Figure 4 was considered. By applying the balance of forces acting on the plane along the mission and by using the following polar, \(C_{D}=0.0349+0.0749C_{L}^{2}\) as suggested in [29] for an ultralight aircraft, the propulsive power request was calculated. By dividing the propulsive power by the efficiency of the propeller, it was possible to obtain the power request at the output shaft of the series/parallel power system, which is reported in Figure 4, together with the propulsive power. More details on the evaluation of the propulsive power can be found in [23].  
The shaft power request in Figure 4 must be satisfied each time by the hybrid electric power system according to the selected operating mode (as explained in the next section). The thermal and electric machines are modeled by means of performance curves and efficiency maps available in the scientific literature. In particular, the wide-open throttle torque curve and the efficiency map of the Wankel engine, reported in Figure 5, are obtained from the two-rotor Mazda 13B [30]. The optimal operating line (OOL) shown in the figure collects all the torque and speed combinations that, for a specific power request, guarantee the minimum fuel consumption in steady-state conditions.  
Generic Brushless DC electric machines were considered in this investigation. Their efficiency map is obtained with a simple Willans line approach as suggested in [31]. More details on the simulation approach can be found in [2]. Note that this kind of approach is universally accepted in the preliminary design of hybrid electric power systems when experimental data are not available [31].  
Figure 4: Propeller efficiency map (**left**) and propulsive vs. shaft power (**right**).