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In this engineering note we derive a simple range equation for hybrid-electric aircraft with a constant power split. The equation is shown to be identical to the traditional Breguet range equation for the limit case of zero supplied power ratio. Analogously, for fully-electric configurations, the equation matches the expressions found in literature...
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... propulsive power included on the left-hand side of Eq. 5 can be related to the thrust required in the current flight condition, since P p = T · V. To this end, Fig. 2 indicates the main forces acting on the aircraft in steady symmetric flight. The thrust vector is assumed to be aligned with the velocity vector, and the aircraft is assumed to fly at a constant lift coefficient and velocity, such that the lift-to-drag ratio is maintained. Given that the weight of the aircraft decreases over time, the ...
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... Welstead et al. [18] and Capristan and Welstead [19] described the Layered and Extensible Aircraft Performance System (LEAPS), being developed at NASA to expand and improve upon the capabilities of the legacy Flight Optimization System (FLOPS) [9]. De Vries et al. [20] developed a range equation specifically for generic hybrid powertrains with constant power split and showed that its limiting cases were range relationships for fuel-powered and allelectric propulsion. De Vries et al. [21] showed a sizing method for hybrid-electric fixed-wing aircraft that considered aeropropulsive interactions from distributed electric propulsion and used a generalized matrix to represent power flows in several hybrid-electric powertrains. ...
Conventional aircraft sizing methods face challenges in analyzing all-electric or hybrid-electric novel aircraft configurations, such as those for urban air mobility applications. The vast design space containing both continuous and discrete design variables and competing design objectives necessitates searching for not necessarily a unique optimal design but rather an array of Pareto-optimal designs. This paper uses the Parametric Energy-Based Aircraft Configuration Evaluator, an aircraft sizing framework for novel aircraft and propulsion system architectures, to pursue multi-objective optimization of a lift-plus-cruise urban air mobility aircraft with all-electric, hybrid-electric, and turbo-electric propulsion system architectures using Nondominated Sorting Genetic Algorithm II. The optimization cases considered include multiple objective functions, such as maximum takeoff mass, mission time, energy and propulsion mass fraction, and energy used per unit distance per unit payload. Optimization was performed for trip distances of 80, 120, and 150 km and battery specific energy levels of 350 and 400 Wh/kg. Except when mission time was an objective function, Pareto-optimal designs occurred near the lower bound of the velocity range. Raising that bound had an impact on the architectural composition of the final generation. All-electric architectures appeared exclusively for lower trip distances, hybrid-electric designs appeared for longer trip distances, and turbo-electric designs only appeared for combinations of longer trip distances, a higher minimum cruise speed, and a lower battery specific energy level. The sizing results were sensitive to assumptions regarding the overload capacity of lift propulsor motors in postfailure conditions.
... Because the level of modeling for these electrical characteristics governs the analysis accuracy and design feasibility, more sophisticated modeling of an EPS is essential in the conceptual design of eVTOL aircraft. However, previous studies [12][13][14][15][16][17][18][19][20][21][22][23][24] have shown a limited ability to faithfully reflect the electrical characteristics of EPSs in the analysis and design of eVTOL aircraft. ...
... De Vries et al. [13][14][15] established a sizing method to incorporate various EPSs into a single matrix form by using the constant efficiency coefficient for each electrical device. Finger et al. [16][17][18] modernized the classical conceptual design method by adding a hybridization factor to the point and mission performance modules. ...
... 1) Approach 1 (low-fidelity, Refs. [13][14][15][16][17][18]29]): This is the most straightforward approach used to analyze and design an EPS, involving little consideration of the electrical characteristics. It assumes that the EPSs' efficiencies are constant throughout a given mission and cannot reflect the specifications of each component when sizing the EPS. ...
This paper proposes novel enhancements to the electric propulsion system analysis method for electric vertical takeoff and landing (eVTOL) aircraft conceptual design by considering the electrical characteristics of each electrical device in a more accurate manner. To this end, three modules for motor, inverter, and battery analysis are constructed using equivalent circuits and semi-empirical models. First, the motor analysis module is developed using equivalent circuit analysis with the operation control strategy for a permanent-magnet synchronous motor. Second, the inverter analysis module is built using average loss models for the switching and conduction losses. Third, the battery analysis module is improved using the near-linear discharge model to consider the voltage drop during operation. Moreover, additional modules, such as those for calculating the battery stack in series and parallel, as well as regression models for the motor and inverter performance parameters and consideration of the drive system type (direct or indirect, including a reduction gear), are implemented. A comparative study is presented from an analysis and design perspective while changing the drive system type (direct or indirect) and gear ratio, which highlights the importance of modeling electrical characteristics during eVTOL aircraft conceptual design.
... Similarly, the Stanford University Aerospace Vehicle Environment (SUAVE) [9] was developed for unconventional aircraft configurations and augments regressions with low-fidelity physics-based models. Other noteworthy investigations and research that can improve the conceptual design of eVTOL aircraft is summarized as follows: a) sizing of hybrid-electric propulsion architectures [10][11][12][13] b) geometry-centric modeling and design [14,15] c) trajectory optimization [16][17][18][19] d) preliminary sizing and weight estimation [20][21][22] e) fast vehicle conceptual design [23][24][25][26] and f) large-scale multidisciplinary design optimization [27][28][29][30][31]. ...
... In this process, battery-electric propulsion is generally considered as an unrealistic pathway for sustainable aviation. While hybrid architectures with a low or zero degree of hybridization have received ample attention [3][4][5], large battery-electric aircraft have been considered infeasible when relying on current and near-future expected battery performance [6][7][8]. Small battery-electric aircraft have been developed and are in operation (see e.g. the Pipistrel Velis Electro). ...
... This is maximum cruise range is determined without considering energy required for take-off, climb and descent, for non-propulsive systems, or to cover reserves. For battery-electric aircraft, the formula for this maximum cruise range ( max ), derived from the well-known Breguet range equation, is [3,12]: max = elec p bat 1 max EM MTOM (1) In this equation, elec and p are the efficiency of the electric powertrain and the propeller, respectively. is the gravitational acceleration, bat is the mass-specific energy of the battery, ( / ) max is the maximum lift-to-drag ratio, EM is the energy mass, and MTOM is the maximum take-off mass. ...
... The authors of this paper have also endorsed such views in previous work [3,4]. However, in the following sections we challenge these statements. ...
Thus far, battery-electric propulsion has not been considered a promising pathway to climate-neutral aviation. Given current and expected battery technology, in most literature battery-electric aircraft are only considered feasible for short ranges (< 400 km) and small payloads (< 19 pax). As a result, battery-electric aircraft development focuses on new aviation segments such as regional and urban air mobility. However, little effort has been made to develop battery-electric aircraft that can replace existing larger aircraft. This paper re-examines the assumptions that lead to the conclusion of limited applicability of battery-electric aircraft.
Starting from the range equation, this paper assesses the drivers of two key parameters: the ratio between energy mass and maximum take-off mass, and the maximum lift-to-drag ratio.
This assessment, based on Class-I mass and aerodynamic-efficiency estimates, shows that there is a design space where these two parameters can reach significantly higher values than often assumed in the open literature. Based on this finding, several parametric aircraft designs are evaluated, relying on Class-II mass and aerodynamics methods. These parametric studies validate the conclusion from the Class-I assessment. This implies that battery-electric passenger aircraft can play a larger role in climate-neutral aviation than was previously envisioned.
... In order to compare different electric hybrid solutions, some simplified sizing methods are required during the preliminary feasibility study phase [10][11][12][13][14]. Any range equation based on the "energy approach" implies that the airplane's mass depends on the energy split ratio. Let us take the example of the formulas provided by De Vries et al. in [15]. As the authors claim the proposed equation (Equation (17) in [15]) should be used to calculate the necessary energy to achieve a required range depending on the power split. ...
... Let us take the example of the formulas provided by De Vries et al. in [15]. As the authors claim the proposed equation (Equation (17) in [15]) should be used to calculate the necessary energy to achieve a required range depending on the power split. In [16], the battery and the fuel weights are constrained to the power split and for this reason: if the power split increases, then the fuel mass decreases, and the battery mass increases. ...
... Switching from weights to weight fractions is an everyday necessity, but this is quite complicated in energy-based formulas where the weight fractions depend on the power split. For this reason, the authors of [15] use the Operative Empty weight and the Payload weight expressed in Newtons. The same goes for Roachs et al. in [16]. ...
A new Range Equation for a hybrid-electric propeller-driven aircraft was formulated by an original derivation based on the comparison of Virtual Electrical Aircraft (VEA) and Virtual Thermal Aircraft (VTA) range equations. The new formulation makes it possible to study the range of a hybrid aircraft with pre-established values of electric motor usage rate. The fuel and battery mass are defined "a priori", and do not depend on the power split, so even the aircraft’s total mass is constant. The comparison with the typical range formulas available for hybrid aircraft was made on the basis of a reference composite VLA category aircraft manufactured by the CFM Air company. The analysis carried out shows that there is an optimum hybridization level as a function of the pre-set specific energy of the batteries system.
... Cinar et al. [12] built a matrix-based analysis framework capable of evaluating any propulsion architecture. De Vries et al. [13], derived the traditional Breguet range equation that allows exploring cruise performance of various hybrid propulsion configurations based on branches and nodes network. This equation represents a precious tool to appreciate the major phenomena involved in hybrid propulsion design and performance. ...
Hybrid electric propulsion is one of the alternative solutions for reducing fuel burn and lower [Formula: see text] emissions while keeping a reasonable battery mass for regional turboprop aircraft operating on short routes. Most studies reporting fuel burn reductions evaluate the aircraft on the design mission, although regional transport aircraft rarely operate under these conditions. Therefore, considering its off-design performance is essential for providing a more complete understanding of aircraft capabilities under various operating conditions. Under these flight conditions, the multi-energy management aspect of hybrid propulsion and the fixed size of the batteries could have a significant impact on the system robustness in off-design operation. In this study, the off-design performance of an existing regional turboprop aircraft retrofitted with a parallel hybrid electric powertrain is assessed. Fuel burn benefits are evaluated on the payload–range diagram for an initial hybrid design and compared to the baseline aircraft. Then, using a novel sizing approach, considering a typical mission operation, this study shows an average improvement of [Formula: see text] percentage point on fuel burn benefits relative to the initial hybrid aircraft, creating a more robust design.
... Conversely, in a parallel-hybrid powertrain, the propeller is connected to a gearbox which is fed either by the gas turbine, the electric motor powered by a battery or a combination of both. The gearbox is an addition in this paper, which was not considered in [11]. The parallel hybrid-electric powerplant is depicted in Figure 2. ...
... Conversely, in a parallel-hybrid powertrain, the propeller is connected to a gearbox which is fed either by the gas turbine, the electric motor powered by a battery or a combination of both. The gearbox is an addition in this paper, which was not considered in [11]. The parallel hybrid-electric powerplant is depicted in Figure 2. The symbols in the figure are defined as follows: f stands for the fuel, gt is the gas turbine, bat is the battery, em is the electric motor, prop is the aircraft propeller, P denotes power and η is the efficiency. ...
... Likewise, the electrical power P2 is a result of the energy stored in the battery and the efficiency of converting this into work. Without considering the efficiencies of the components (as was the case in [11]), the power is not split adequately, and the definition remains incomplete. A new efficiencybased definition for the degree of hybridization is shown in Equation (2). ...
This paper proposes a new range equation for hybrid-electric aircraft. The paper revisits the theory of the range equation for a hybrid-electric aircraft with constant power split published earlier in the literature and proposes a new efficiency-based definition of the degree of hybridization (φ), one which includes the efficiencies of the electric or fuel-powered drivetrain. The paper shows that the efficiencies of the respective drivetrains play a significant role in the range estimation of the hybrid-electric aircraft. The paper makes use of a case study to show the relationship between battery energy density, powertrain efficiency and modification in the definition of the degree of hybridization φ with aircraft range. We show that for every aircraft design, there is a battery energy density threshold, for which the aircraft range becomes independent of the degree of hybridization. Below this threshold, the range decreases with an increase in the degree of hybridization. Conversely, beyond this threshold, the aircraft range increases with the degree of hybridization. Our study finds that the new definition of φ has shifted this threshold significantly upwards compared to earlier publications in the literature. This makes the design of an aircraft with a high degree of hybridization less optimistic.
... A simplification with respect to the time-stepping approach by de Vries et al. [66] is used, instead using the hybrid range equation derived by de Vries et al. [71] is used to calculate the required energy during the cruise phase of the mission. The energy required during the other mission phases is obtained using fuel and energy fractions, combustion and electrical efficiencies and a constant power split value. ...
... In this case, the range equation can better be expressed as a trade-off between payload mass and aircraft energy. de Vries et al. [71] derived this equation for hybrid-electric aircraft with constant power split, as shown in Equation 28. ...
... This equation (Breguet Range Equation) was originally established for kerosene aircraft, as presented in Equation (36). This work uses the hybrid-electric and fully electric aircraft derived by de Vries et al. [71] presented in Equations (28) and (37). Since these equations only need a set of parameters to "create" new aircraft variants, this method can be coupled with a strategic airline planning model. ...
... Therefore, we introduce a simple energy-based sizing methodology for hybrid-electric VTOL aircraft. The method is based on a generic representation of the powertrain architecture developed in [17]. ...
... For verification, the approach is applied to an academic case study to determine the preliminary characteristics of the energy storage system of a hybrid-electric VTOL aircraft considering a typical example mission. In this context we also present an equation for calculating the hover endurance of VTOL aircraft with HEP, analogous to the range equation for hybrid-electric, fixed-wing aircraft in [17]. ...
... As mentioned in section I, we use the generic power system architecture developed in [17], which provides the basis for our energy-based sizing methodology. Therefore, the architecture can cover conventional (fuel-only) powertrains, serial/parallel hybrid-electric configurations, as well as fully electric propulsion [17]. ...
Electric vertical take-off and landing (VTOL) aircraft are considered a promising solution for fast and sustainable Urban Air Mobility (UAM), as they combine the advantages of vertical flight with the highly efficient flight of fixed-wing aircraft. However, the specific energies of today’s batteries still significantly limit the range and endurance of such vehicles. A more viable intermediate step towards sustainable UAM is represented by hybrid-electric aircraft. However, in the preliminary design phase, sizing of dual-power systems is a major challenge, particularly for complex and widely-varying VTOL configurations. To address this challenge, we present a simple, configuration-independent sizing methodology to estimate the relative weight and size of the propulsion system of hybrid-electric VTOL aircraft for a set of given mission parameters. The approach is demonstrated using a comprehensive case study. Since we introduce weight-specific parameters such as the energy-change-to-weight ratio, the presented methodology does not require prior knowledge of the initial aircraft weight or any other in-depth knowledge of the aircraft configuration. In this context, equations to calculate the hover endurance and the vertical climb performance are also derived. The proposed design methodology can provide valuable insight during the early design process of new VTOL concepts by showing which hybrid-electric VTOL configurations are feasible in terms of the required energy storage characteristics. Since the methodology is configuration-independent, it can provide a basis for new preliminary development tools for hybrid-electric VTOL aircraft.
... Other significant research that can benefit the design of eVTOL vehicles is summarized as follows: sizing of novel aircraft concepts [12,13], geometry-centric design [9,14], unconventional propulsion architectures [15,16], dynamic loads analysis [17], physics-based structural sizing [18], and safety, reliability and robust design [19,20]. ...
The conceptual design of eVTOL aircraft is a high-dimensional optimization problem that involves large numbers of continuous design parameters. Therefore, eVTOL design method would benefit from numerical optimization algorithms capable of systematically searching these high-dimensional parameters spaces, using comprehensive and multidisciplinary models of the aircraft. By leveraging recent progress in sensitivity analysis methods, a computational framework called the Comprehensive Aircraft high-Dimensional DEsign Environment (CADDEE) has been developed for large-scale multidisciplinary design optimization (MDO) of electric air taxis. CADDEE uses a geometry-centric approach that propagates geometry changes in a differentiable manner to meshes for physics-based models of arbitrary fidelity level. The paper demonstrates the capabilities of this new aircraft design tool, by presenting large-scale MDO results for NASA's Lift+Cruise eVTOL concept. MDO with over 100 design variables, 17 constraints, and low-fidelity predictive models for key disciplines is demonstrated with an optimization time of less than one hour with a desktop computer. The results show a reduction in gross weight of 11.4% and suggest that CADDEE can be valuable in the conceptual design and optimization of eVTOL aircraft.
