## 3 The New Power System
### Energy Management and Power Sizing  
In the previous investigation, a complex energy management strategy was proposed to allow the engine to work at optimal operating points independently of the propeller speed and torque request. In the new configuration, thanks to the constant speed of the propeller, the energy management can be simplified. The size of the electric machines and the battery were obtained from the power request distribution among the engine and the electric machines according to the three operating modes of the system. * Full electric: the engine is disengaged and no fuel is used;
* Hybrid power assist: the mission is performed with the engine running at 4590 rpm. The EM2 machine is used help the engine when the shaft power overcomes \(51.8\cdot[1.13\sigma(z)-0.13]\) kW, i.e., the maximum power that the engine is able to deliver at that speed at altitude z (see Table 1 and Equation (1));
* Hybrid series: the engine is disengaged from the transmission so that its operating point can be set equal to the OOP of Table 1. The engine power is delivered to the generator (EM1), which in turn supplies electric power to the motor (EM2). If insufficient, additional power is delivered to EM2 from the battery. On the contrary, when the EM1 power is higher than the shaft power request, the excess power is store in the battery. The power distribution among the different machines and the battery according to these operating modes is reported in Figure 9. The mission cannot be performed in thermal mode (using only the engine) with the selected engine speed. However, to obtain a term for a comparison of fuel saving, the engine is assumed to be running at its max power \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **z1** & **z2** & **z3** & **z4** & **z5** & **z6** & **z7** & **\(\tau\)21** & **\(\tau\)43** & **\(\tau\)73** & **\(\tau\)65** \\ \hline Original Honda iMMD & 65 & 19 & 54 & 22 & 39 & 76 & 69 & 3.42 & 2.43 & 0.81 & 0.51 \\ Adapted HHS & 50 & 28 & 36 & 15 & 21 & 32 & 28 & 1.79 & 2.4 & 1.29 & 0.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of teeth (z) and transmission ratios (\(\tau\)). Figure 8: Loss of engine power as a function of flight altitude. rotational speed (7000 rpm), thus obtaining a hypothetical thermal mode where only the engine is used to move the propeller. The plots in Figure 9 were obtained by taking into account the efficiency of the three machines. For the engine, the efficiency was obtained from the map of Figure 5, while the electric machines were modelled with a Willans line model. For details of the modelling approach, please refer to [2]. The results of the mission analysis are reported in Table 3. The table reports the consumption of fuel and electricity of the HHS with the three possible operation modes, the consumption of a hypothetical thermal configuration with the engine allowed to run at 7300 rpm (max power), and the consumptions obtained with the THS over the same mission with the same take-off mass. Compared with the thermal hypothetical mode, the HHS in power assist mode allows a fuel saving of about 20%, while needing a very small battery to account for the electricity consumption. On the other hand, a fuel saving of about 45% can be obtained with the series hybrid mode, but this would require a much larger battery (about 10 kWh). With twice this size of the battery, the mission could be performed in all-electric mode. The improvement in fuel economy guaranteed by the series hybrid mode is due not only to the reduced contribution of the engine in satisfying the shaft power request, but also to the higher and constant efficiency of the engine that works always at its OOP while in power assist mode, the engine works at 4590 rpm (see Table 1). The time trends of engine efficiency in the series hybrid, power assist, and thermal cases are compared in Figure 10.