# Parametric Analysis for Hybrid-Electric Regional Aircraft Conceptual Design and Development
## 2 Mathematical Preliminaries and Methodology  
The study presented in this paper consists of the design of regional hybrid-electric aircraft and a critical analysis of their performance. In order to identify the context of the study, it is clarified that a regional aircraft is an aeroplane designed primarily to cover short-haul air routes and is usually optimised to serve local and regional communities, connecting cities and airports of limited size. In terms of specifications, regional aircraft usually accommodate between 40 and 80 passengers, with a flight range that typically does not exceed 1000 nautical miles. In general, they are designed to operate on shorter runways than long-haul aircraft, thus requiring greater short take-off and landing capability, and are typically equipped with turboprop engines, which are specifically optimised to reduce costs and fuel consumption on short routes. A comprehensive reference that outlines the regional aviation scenario, also posing a focus on its future development, is proposed in [24], whereas insights on current and advanced regional aircraft and regional airport infrastructures are described in [25; 26; 27; 28]. In the proposed research, the top-level aircraft requirements used to steer the design are similar to those of the state-of-the-art benchmark regional aircraft ATR 42, as also proposed in the reference paper [8] and described further along in this section. The design phase was carried out through the use of the in-house-developed software THEA-CODE, whose detailed description is in [29]. In particular, THEA-CODE performs the multidisciplinary design of a hybrid-electric aircraft, starting from a set of top-level aircraft requirements and taking into account the interdisciplinary links between aerodynamic, structural, propulsive, and aeromechanical features [29]. In this context, the aircraft's aerodynamic polar curve is evaluated by calculating the induced drag using the Vortex Lattice Method [30]; the parasitic drag of the lifting surfaces is calculated by integrating the profile drag coefficient on the wing surface, computed using the XFOIL code; while that of the nacelles, fuselage, and pylons is assessed using the model proposed in [31]. The wing structural mass is calculated using surrogate models based on FEM simulations taking into account the static loading conditions [32], while the structural mass of the fuselage, undercarriage, and tailplane is evaluated by using the models described in [33]. The wing non-structural masses are calculated by means of the method in [34],while the semi-empirical relationships proposed in [33] are used to evaluate the masses of the on-board systems and operating items. The hybrid-electric propulsion system is sized through the matching chart, which relates aircraft wing loading \(W/S\) to the specific power \(P/W\) necessary for each flight phase [35]. In this context, the required power is allocated in different proportions between electric and thermal chains according to the hybridisation degree \(H_{P}\), which is a design variable defined as:  
\[H_{P}=\frac{P_{i}^{e}}{P_{i}^{e}+P_{i}^{t}} \tag{1}\]  
where \(P_{i}^{e}\) and \(P_{i}^{t}\) are the electrical and thermal power installed on-board, respectively. For each design iteration, a simulation of the mission is carried out by time integration of aircraft equations of motion in the longitudinal plane, as described in detail in [36]. This specific approach gives the amount of fuel and battery mass required to perform the mission, including diversion and reserves, once the power supplied by thermal engines (or electric motor) is assigned. Accordingly, the fraction of power supplied by thermal engines \(\Phi_{k}^{t}\) is another design variable, defined as:  
\[\Phi_{k}^{t}=\frac{P^{t}}{P_{i}^{t}} \tag{2}\]  
where \(P^{t}\) is the power supplied by thermal engine. Note that \(\Phi_{k}^{t}\) is assigned for each \(k\)-th mission phase (i.e., climb, cruise, descent, etc.)