Efficient Methodology for Power Management Optimization of Hybrid-Electric Aircraft ^†

Abstract

This paper presents an effective simplified model to optimize the mission power management supply for hybrid-electric aircraft in the conceptual design phase. The main aim is to show that, by using simplified representations of the aircraft dynamics, it is possible to achieve reliable results and identify trends useful for early-stage design, avoiding the use of more expensive and advanced methods. This model has been integrated into a multidisciplinary design framework, where the mission analysis, based on a simplified point mass dynamic model, focuses on splitting the power supply between electric and thermal power throughout the flight. An optimization algorithm identifies the time profiles of the supplied power, thermal and electric, to minimize fuel consumption. The power supplied by the thermal engine, modeled as a time piecewise function, is a design variable; a parametric study on the number of intervals composing this function is performed. The framework is used to propose a generalized approach for hybrid-electric power management optimization during the conceptual design iterations. This study showed that, for regional hybrid-electric aircraft, dividing the airborne mission into climb, cruise and descent is sufficient to define the optimum power split supply profiles. This allows for the avoiding of finer mission discretization, or the adoption of more complex simulative models, providing a very efficient model.

1. Introduction

Aviation is facing a relevant transformative phase [1,2], led by the urgent necessity to deal with its impact on the climate [3,4,5,6], which is going to increase in the near future according to the current forecast [7,8]. This phase is driven by the investigation of novel technological solutions aimed at improving the performance of currently established tube-and-wing aircraft [9]. Examples are hybrid-electric propulsion [10,11,12,13,14], hydrogen propulsion [15,16,17], high bypass ratio turbofans [18,19], innovative engine–airframe integrations such as distributed electric propulsion and boundary layer ingestion [19,20,21] and non-conventional airframes such as box-wing [22,23], truss-braced [24,25,26] and blended-wing body [27]. In the field of novel propulsion systems, electric power is reaching technological maturity and can be used for aeronautical applications, but the well-known limitations imposed by the low gravimetric energy density of batteries (see [28,29,30,31]) restrict such applications to small aircraft outside the transport class. To extend the outreach of electric propulsion, electric power systems are to be coupled with thermal ones, defining the so-called hybrid-electric propulsion. Its practical integration into aircraft is currently widely investigated up to the regional class [32,33,34,35,36,37,38,39,40,41,42,43], since applications on larger aircraft may not result in performance benefits, especially in terms of fuel consumption [44]. When dealing with the design of hybrid-electric aircraft powertrains, two aspects are relevant: the installed power and the related power supply management throughout the mission [11]. The latter aspect has been investigated in the literature; specifically, several authors have used techniques belonging to the field of optimal control to find the optimum power supply profile which minimizes fuel consumption [45,46,47,48,49,50]. To do so, the general approach is to model the aircraft as a point mass to derive simplified aircraft dynamics and power equilibrium equations; the equations for aircraft dynamics are used to calculate the power required for flight, while the power equilibrium equations determine the power supplied by the electric and thermal systems at each time step of the mission. Optimal control techniques permit researchers to find the optimum time power profile, i.e., the amount of power supplied by the electric and thermal chains in each time step of the mission. Generally, the time step is of the order of seconds, whereas the flight mission lasts thousands of seconds; this aspect highlights that the optimal control techniques could present high computational costs and are generally not suitable in the early stages of the design process. A simpler way to reduce computational costs is to significantly enlarge the time discretization of the simulated mission and adopt gradient-based algorithms; specifically, the approach used in [51] is based on considering the power supply time profile as a piecewise function composed of three stages, corresponding to each airborne phase of the mission, namely climb, cruise and descent. The research proposed in [51] highlights the key role of power management in optimizing hybrid-electric aircraft performance. A similar approach has been used in [52] on a 50 pax regional turboprop with a parallel hybrid-electric architecture, obtaining analogous results. Similar conclusions emerge from the study presented in [53], which proposes a discussion on the performance analysis of regional hybrid-electric aircraft, aiming at identifying the optimal power split profile between thermal and electric sources to minimize the fuel consumption.

The present paper resumes and expands on what was discussed in ref. [53], aiming to investigate the potential impact of the finer discretization of mission profiles on the design and optimization of hybrid-electric powertrains for regional aircraft. This is a key step to understanding if coarse approximations may fail to capture critical variations in power demand and supply during different mission phases, leading to suboptimal design choices. This study seeks to address this gap by discussing the effect of mission discretization refinement. Moreover, this work represents a reliable preliminary step toward integrating more advanced control theory formulations, particularly those aimed at real-time power management and energy efficiency optimization. By proposing a generalized and simplified model for optimizing the power supply time profile, this study aims to provide a robust yet computationally efficient tool that can be applied from the earliest stages of aircraft conceptual design. Such an approach enables the reliable and rapid evaluation of both aircraft design and performance, enhancing the decision-making process for hybrid-electric concepts.

In summary, this study introduces a simplified yet effective approach tailored to the conceptual design phase of regional hybrid-electric aircraft. The proposed methodology, developed in MATLAB 2022b, relies on a point mass representation of the aircraft, which, while avoiding the complexities of advanced simulation techniques, delivers reliable and readily usable results. This is particularly critical during the initial design stages, where fast evaluations and access to large datasets are essential to guide exploration and conceptual choices. The airborne mission is represented through a limited number of phases, i.e., climb, cruise and descent, and the aircraft flight dynamics are simulated via the Euler integration method. This enables the effective analysis of power management with sufficient detail to identify the most promising configurations, without imposing excessive implementation complexity or computational demands. Hence, this work underscores that generalized tools, such as the one proposed, do not compromise the quality of the information required to drive the conceptual design process. On the contrary, they provide a robust and efficient starting point, enabling rapid design space exploration and preliminary performance evaluations. These features facilitate the identification of optimal solutions, while maintaining a balance between computational efficiency and the depth of insight needed for informed decision-making in the early stages of hybrid-electric aircraft design.

This paper is structured as follows: Section 2 describes the methodology, Section 3 details the aircraft requirements, Section 4 reports the main results, which are discussed in Section 5 and Section 6 provides the conclusions.

2. Methodology

2.1. Conceptual Design Tool

The conceptual design and performance analysis of the hybrid-electric aircraft were carried out using the THEA-CODE software [54]; the design framework, sketched in Figure 1, combines the main conceptual aircraft design disciplines such as aerodynamics, propulsion, flight mechanics and weight estimation. All the main blocks are inserted in an iterative cycle that terminates when the value of the maximum take-off weight (MTOW) converges under a prescribed tolerance ε.

The selected design variables are the wing loading W/S, the degree of hybridization $H_P$ and the thermal fraction of supplied power ${\Phi }_t$. $H_P$, defined in accordance with [55], is the ratio between the installed power of the electric motors and the total installed power, calculated as the sum of the installed power of the electric motors and thermal engines. ${\Phi }_t$ is a function of time and directly relates to the power supply management strategy. In this study, it has been considered as a piecewise time function; accordingly, the thermal fraction of supplied power of the k-th stage Φ_t,k is defined by Equation (1):

$${\Phi }_{t,k}=P_{t,k}/P_t^i$$

(1)

where $P_{t,k}$ is the power supplied by thermal engine at the k-th stage and $P_t^i$ is the total installed thermal power. The aerodynamics, engine sizing, mission analysis and weight estimation blocks of the workflow of Figure 1 provide the aircraft’s polar drag, the required installed power, the aircraft performance and the mass breakdown, respectively. The aerodynamics block receives as an input the geometry of the wing through dimensionless geometric parameters that univocally identify the lifting system, and computes the aircraft polar drag through a combination of the Vortex-Lattice method and viscous drag coefficient estimation. The method is widely described in ref. [54]. The engine sizing block calculates the total installed power through the matching chart, a specific power vs. wing loading chart that includes the power constraints related to the current aircraft regulations and the TLARs. In particular, once the wing loading is selected, the total required installed power is calculated and can be split between thermal and electric chain according to the selected value of $H_P$. The mission analysis block assesses the fuel consumption and the battery mass for a specific mission profile, which is thoroughly detailed in the following paragraph. The weight estimation block computes the weight breakdown of the aircraft, i.e., the structural mass, the onboard systems mass, the operating items mass, the propulsion system mass and the payload mass. The input and output of each block are reported in

Table 1. Input/output of each block of THEA-CODE.

Aerodynamics	Input	Aircraft weight Geometric parameters of the lifting system, i.e., aspect ratio, taper ratio, sweep angle, W/S TLARs, i.e., cruise altitude and related Mach number
	Output	Aircraft polar drag
Engine Sizing	Input	Aircraft polar drag and weight Current regulations, i.e., FAR W/S and $H_P$ TLARs, i.e., cruise altitude and related Mach number
	Output	Electric and thermal installed power
Mission Analysis	Input	Aircraft polar drag and weight Mission profile Power management parameters, i.e., ${\Phi }_{t,k}$ Powertrain model
	Output	Fuel consumption and battery weight, mission parameters
Weight Estimation	Input	Aircraft weight and geometry TLARs
	Output	Aircraft weight breakdown

. For further details, the reader can refer to ref. [54].

For a deeper insight into the overall design procedure, the reader can refer to [11,54], where the multidisciplinary conceptual design procedure is widely detailed; hereafter, only a brief insight into the mission analysis block is provided, as it is the core of the following discussion. The mission analysis module computes the aircraft performance by means of the flight simulation, which is set according to the following simplified assumptions: (i) the aircraft is a point mass, (ii) the mission trajectory lies in the vertical plane and (iii) the thrust force is always aligned with the aircraft velocity. Accordingly, Equations (2)–(6) describe the motion of the aircraft in the vertical plane during the airborne phases of the mission; W is the aircraft weight, $\dot{W}$ is its time derivative, L is aircraft lift, P is the required power, D is aircraft drag, V is flight speed, $\gamma$ is flight trajectory angle, $P_b$ is the power supplied by battery pack, $k_c$ is the brake-specific fuel consumption, ${\eta }_t$ and ${\eta }_e$ are the thermal and electric chain efficiency and $k_c$, ${\eta }_t$ and ${\eta }_e$ are considered to be constant throughout the mission. It is important to underline that the hybrid-electric powertrain architecture is parallel; see refs. [11,32] for a description of the different architectures. In the parallel powertrain, the power to the fan or propeller can be delivered by the thermal chain, the electric chain, or a combination of both in different shares. This is managed by a control power unit, responsible for regulating the power split and managing the power supply throughout the flight.

$$\dot{W}=-k_c{P}_t$$

(2)

$$L=W$$

(3)

$$P=DV+\gamma WV$$

(4)

$$P=P_t{\eta }_t+P_b{\eta }_e$$

(5)

$$P_t={ \Phi }_t{P}_t^i$$

(6)

The Euler forward method is used for the numerical integration of the differential equations. Such a method is proposed in Equation (7) for a generic y function of time t, where $\dot{y}$ is the time derivative of the considered function and $\Delta t$ is a finite timestep.

$$y(t+\Delta t)=y\left(t\right)+\dot{ y}\left(t\right)\Delta t$$

(7)

The mission is split into different phases, as depicted in Figure 2. Regarding the power management, it has been assumed that the electric chain is off during diversion, whereas the thermal chain is off during taxi operations. The first choice avoids the need to introduce unnecessarily large battery weight increases and ineffective gains in terms of emission reductions, as diversion rarely happens. The second choice is related to the reduction in the ground pollution. The whole available power is supplied for take-off. During the airborne phases of the mission, i.e., climb, cruise and descent, the power is split between thermal and electric chains according to Equations (5) and (6). Each airborne phase can in turn be divided into a different number of stages, enabling a more flexible selection of the thermal and electrical power supply profiles; Figure 2 shows a generic example of ${\Phi }_t$ divided into 7 and 16 stages. The power required during the flight must be continuously balanced by the combined output of the thermal and electrical systems, accounting for losses. Given the required power P and a fixed value for ${\Phi }_t$, it is possible to uniquely determine the time profile of both the thermal power supply, according to Equation (6), and the electrical power supply, according to Equation (5), in each k-th stage of the mission.

The mission simulation allows for the weight of the fuel and the battery required to accomplish the mission and diversion to be assessed; in addition, a fuel reserve of 5% and a residual battery state of charge of 20% are considered.

The battery mass $m_b$ is modeled according to the simplified methodology proposed in ref. [56]. Specifically, the battery mass is calculated according to Equation (8).

$$m_b={\displaystyle \frac{{\int }_0^t{P}_{b}dt}{\left(SO{C}_i-SO{C}_f\right)BED}}$$

(8)

where $SO{C}_i$, $SO{C}_f$ and BED are the initial state of charge, the final state of charge and the gravimetric energy density of the battery pack, respectively. Ref. [56] compares this simplified model with a more detailed one which assesses the layout of the battery pack, i.e., the number of cells which are arranged in parallel and in series, and the current supplied by the battery pack. The battery mass estimation of this more refined model is aligned with the simplified one, as the difference between the two models is below 2%; therefore, the general recommendation of ref. [56] is to use the simplified approach to assess the mass of the battery pack. Accordingly, this approach has been adopted in this paper.

A final assumption of the power supply is that for two consecutive stages, the variation is considered to be a step, without any smoothing connection function between them.

2.2. Optimization Framework

The mathematical formulation of the optimization problem is as follows:

$$\left\{\begin{array}{c}min\left(FoM\right(x\left)\right)\\ 0< H_P<{H_P}_{\mathit{max}}\\ 0< {\Phi }_{t,k}< {\phi }_{t,k}\\ MTOW\left(\mathit{x}\right)\le { \mathit{MTOW}}_{\mathit{max}}\\ {\Phi }_{e,k}\le {\phi }_{e,k}\end{array}\right.$$

(9)

THEA-CODE is used for both the assessment of the objective function and the constraints. For each run, the optimizer gives the set of the current design variables to THEA-CODE, which iterates an aircraft design until the MTOW converges. The objective function, which depends on the vector of the design variables x constituted by $H_P$ and ${\Phi }_{t,k}$, can be selected in a pool of suitable figure of merits (FoMs), as detailed in [57]. A constraint on the maximum MTOW is imposed; ${\phi }_{t,k}$ and ${\phi }_{e,k}$ represent the maximum thermal and electric power fractions which can be supplied in the k-th stage of the mission; ${\phi }_{t,k}$ is computed according to Equations (10) and (11):

$${\phi }_{t,k}= P_{t,k}^a/P_t^i$$

(10)

$$P_{t,k}^a={\mathit{\mu }}_t{P}_t^i{\left(\rho /{\rho }_0\right)}^{0.75}$$

(11)

where $P_{t,k}^a$ is the thermal available power at the k-th stage, i.e., the maximum power which can be supplied by the engine at a specific altitude; ${\mathit{\mu }}_t$ is a constraint parameter set to 0.9 to avoid engine overheating; $\rho$ is the air density at a specific altitude; ${\rho }_0$ is the air density at sea level, as defined by the ICAO and ${\phi }_{e,k}$ is equal to 1 and is constant in each phase, since electric motor performance does not depend on altitude. Climb, cruise and descent are divided into multiple stages having the same spatial length (see the generic example of Figure 2); accordingly, since the mission profile is fixed a priori, the altitude of each k-th stage and its corresponding air density are known. ${\Phi }_{e,k}$ represents the fraction of electric supplied power, and it is computed by means of Equations (12) and (13); ${\eta }_b$, ${\eta }_{inv}$ and ${\eta }_{em}$ are the battery, inverter and electric motor efficiency, respectively and $P_{e,k}$ and $P_e^i$ are the power supplied by the electric motor at the k-th stage of the mission and the installed electric motor power, respectively. $P_{e,k}$ is computed by Equation (13), which is related to the value of ${\Phi }_{t,k}$ and $P_{b,k}$; specifically, by fixing the value of ${\Phi }_{t,k}$, hence the power supplied by the thermal engines, $P_{b,k}$, can be calculated through Equations (2)–(6). Hence, it is sufficient to use ${\Phi }_{t,k}$ as a design variable and to compute accordingly the related ${\Phi }_{e,k}$.

$${\Phi }_{e,k}=P_{e,k}/ P_e^i$$

(12)

$$P_{e,k}=P_{b,k}{\eta }_b{\eta }_{inv}{\eta }_{em}$$

(13)

The optimization procedure has been developed in MATLAB, and it is based on local gradient-based algorithms, i.e., the sequential quadratic programming (via fmincon built-in function), coupled with a multi-start approach; in this work, n = 10 different starting points have been considered for each optimization case. The initial guess points are randomly chosen within the boundaries of the design variables. The local gradient-based method was chosen for its computational efficiency and suitability in the early conceptual design phase. It provides reliable results with relatively low computational costs, which is ideal for exploring a wide design space quickly. To overcome the limitations derived from the search for a local solution, which are typical of this algorithm, it has been coupled with a multi-start solution. A single-objective optimization suffices to provide clear insights, while a multi-objective approach would complicate the analysis without significant added value in this context. For further general insights on optimization techniques, the reader can refer to refs. [58,59].

3. Top-Level Aircraft Requirements and Main Aircraft Data

This section reports the design requirements and the assumptions considered for a regional hybrid-electric aircraft with an entry into service in 2035. These assumptions pose limitations on the performance of the electric components of the powertrain, whose forecast has been collected in [11], and also described in [12,13,60], and is here assumed as a reference for the definition of their main features. Specifically, the efficiency and power density of electric motors are 0.96 and 16 kW/kg, respectively; the efficiency and power density of the inverters are 0.98 and 19 kW/kg, respectively and the efficiency and gravimetric energy density of the battery are 0.95 and 500 Wh/kg, respectively. Regarding the internal combustion engine, the brake-specific fuel consumption is assumed to be equal to 0.2675 kg/kWh. In this study, design requirements similar to those of the ATR 42 have been adopted, namely, having a number of seats equal to 40, a take-off and landing distance equal to 1100 m, a cruise Mach of 0.4, an altitude of 6100 m and a design range of 600 nm. Also, other design ranges, i.e., 400 and 800 nm, have been investigated. The main parameters of the design mission are reported in

Table 2. Assumptions of the design mission.

Mission Phase	Assumption
Climb	IAS = 170 kt, rate of climb = 900 ft/min
Cruise	M = 0.4 @ flight level = 200
Descent	IAS = 220 kt, rate of descent = −1100 ft/min

.

In a similar manner, the geometry of the designed hybrid-electric aircraft resembles the ATR 42; specifically, the geometrical parameters of the lifting systems, such as the aspect ratio, taper ratio and fuselage length, are not changed. For comparison purposes, as discussed later in Section 4, a set of full thermal regional aircraft have also been designed following the same TLARs. The data of the reference thermal-powered tube-and-wing aircraft are presented in

Table 3. Data of the reference thermal aircraft.

Range [nm]	400	600	800
Wingspan [m]	20.5	20.9	21.9
Wing surface [m2]	46.6	48.2	50.3
Tail surface [m2]	13	13.5	14
MTOW [kgf]	15,153	15,731	16,365
OEW [kgf]	10,361	10,550	10,766
Total fuel [kg]	1042	1432	1837
Block fuel [kg]	733	1103	1487
Installed power [MW]	4	4.15	4.3

.

4. Results

In this section, the main results of the application of the methodology outlined in Section 2 are detailed; specifically, in Section 4.1, a parametric analysis of the design variables is carried out, and the general correlations between the design variables and the block fuel, selected as a figure of merit in this work, are discussed. Section 4.2 provides the results of the optimization problem presented in Section 2.2, considering the minimum airborne mission discretization, i.e., climb, cruise and descent. The focus of the study on the effect of increasing the number of stages in which the power supply profile function is divided, i.e., the number of intervals of a piecewise function of time which represents the power supply time profile of the thermal engine, as detailed in Section 2.1, is instead proposed in Section 4.3: this enables the analysis of the optimal power profile throughout the mission. To provide a quantitative performance assessment, a comparison with a reference full-thermal configuration designed using the same design methodology is carried out.

4.1. Parametric Analysis of the Design Variables

The parametric analysis relating to the aircraft performance and its design variables is a preliminary step which is useful to provide the general context to properly understand the subsequent optimization outcomes. In this preliminary parametric study, a number of stages equal to two have been considered to divide the power supply profile; this means that ${\Phi }_t$ can be set in two airborne phases of the design mission, namely climb and cruise, to simplify the formulation of the parametric problem, permitting an easier interpretation of the results; the descent is set to be accomplished by using only thermal power. The chosen values for the parametric analysis are defined as follows: $H_P$ values are taken from a vector equally spaced between 0.2 and 0.6 and ${\Phi }_{t,1}$ (climb) and ${\Phi }_{t,2}$ (cruise) values are taken from vectors equally spaced between 0.1 and 0.4 and 0.1 and 0.5, respectively. The main outcomes of the parametric study are reported in 2D plots where the x-axis is the MTOW of the designed configuration and the y-axis is the selected FoM, the block fuel mass $m_{\mathit{fb}}$. Each point of the chart represents a configuration designed using the methodology outlined in Section 2.1. The whole group of the configurations designed by combining all the possible design parameters is represented in Figure 3; it can be seen that the results are scattered widely, and hence it is useful to highlight and discuss the effects of the individual parameters.

The results focusing on the parametric variation of $H_P$ are shown in Figure 4; each figure highlights the points that belong to the investigated interval, indicated in the legend, whereas the excluded points are shaded. It can be noted that the designed configurations with similar $H_P$ exhibit a large variability in terms of block fuel. The great variability of the solutions is due to the fact that just the installed power splitting cannot guarantee the achievement of low fuel consumption if a proper power supply management strategy is not provided.

The results relating to the thermal power supply in cruise ${\Phi }_{t,2}$ are reported in Figure 5; lower values of ${\Phi }_{t,2}$(Figure 5a) correspond to a higher power supplied by the electric chain in cruise, which is the most energy-demanding phase of the mission; this implies a higher request for electrical energy, so there is a battery mass ($m_b$) on board, and this rapidly affects the increases in the aircraft’s MTOW; on the other hand, this corresponds to configurations with a lower $m_{\mathit{fb}}$. As the largest amount of fuel consumption occurs during the cruise phase, the same trends are not detectable by only considering the variations of ${\Phi }_{t,1}$ (climb). Accordingly, it is possible to state that the power supply split strategy during the cruise phase represents the key driver to steer the design optimization toward fuel consumption minimization for hybrid-electric aircraft.

4.2. Optimization Results: Three-Stage Mission

This section provides the results of optimizations performed according to the problem defined in Section 2.2. In this case, we focus on the discretization of the mission into k = 3 stages, namely climb, cruise and descent. Considering the mathematical formulation of the problem in Equation (7), the optimization is carried out considering ${H_P}_{\mathit{max}}$ to be equal to 0.7, ${\phi }_t$ to be equal to 0.56 and ${\phi }_t$ to be equal to 1. Four different sets of optimizations are performed by varying the constraint on the MTOW, considering $\mathit{MTO}{W}_{\mathit{max}}$ = [23, 30, 35, 40]× 10³ $\mathsf{k}{\mathrm{g}}_{\mathsf{f}}$. Each airborne phase of the mission, namely, climb, cruise and descent, is prescribed to have a constant value of ${\Phi }_t$, and the FoM to be minimized is block fuel.

Figure 6 shows the results of the optima design variables, in terms of the statistical analysis of the 10 local optimizations performed for each $\mathit{MTO}{W}_{\mathit{max}}$; these results highlight the effects of relaxing the constraint on the MTOW. The $H_P$ optima are more spread out in the case of configurations with a higher take-off mass, reaching maximum values around 0.64 for configurations with MTOW = 40 × 10³ $\mathrm{k}{\mathrm{g}}_{\mathrm{f}}$. This variability in the distribution of the installed power split has an impact on the power supply management, which also offers multiple different solutions for each optimization case. In general, however, considering the fraction of thermal power supplied in cruise, ${\Phi }_{t,2}$, the effect of relaxing the $\mathit{MTO}{W}_{\mathit{max}}$ constraint is evident, as increasing the MTOW makes gradually lower values of ${\Phi }_{t,2}$ possible, and hence, for there to be more room to carry on board batteries and provide electric energy to accomplish cruise. Regarding the fraction of thermal power supplied in climb, ${\Phi }_{t,1}$, and descent, ${\Phi }_{t,3}$, there is no evident trend; again, according to the preliminary discussion of Section 4.1, this result is expected.

By enabling the optimizer to design aircraft with a higher MTOW, aiming at minimizing $m_{\mathit{fb}}$, it is possible to find solutions with the highest possible utilization of electrical energy in the cruise phase. The battery packs are in fact very heavy and require this relaxation in the MTOW in order to be installed on board. This is evident in Figure 7, which shows the results related to the optimum configuration of each dataset, i.e., for each value of $\mathit{MTO}{W}_{\mathit{max}}$. Specifically, the power supplied by the electric motors and thermal engines of optima configurations has an opposite trend in cruise when increasing MTOW. The key role of the power supply split during the cruise phase emerges again from these optimization results. It is worth noting that the electric motor is always greater than zero, which means that the battery pack supplies power to the electric motors throughout the entire mission, and no recharge of the battery pack occurs.

An analysis regarding the solid rationale of Equation (8) has been carried out. Indeed, batteries are limited by both power and energy constraints [61]. The constraint related to the power supplied is generally true in the case of short-time high-power-demand applications, e.g., in the case where the battery pack supplies power only for the take-off. However, in the design scenarios proposed in this paper, the battery pack is utilized throughout the entire mission to minimize fuel consumption. As a result, the battery pack must store a substantial amount of energy, leading to a significant weight increase. The substantial weight of the battery pack means that a large amount of power can be supplied, fulfilling the high-power demand requested in some mission phases such as the take-off. To further support the previous sentence, we can consider Figure 8, which depicts the maximum power supplied by the battery pack at the take-off phase $P_{b,\mathit{max}}$ vs. the installed electric motor power $P_e^i$ for configurations with MTOW = 30 × 10³ kg_f (represented by circles). The maximum power supplied by the battery pack is calculated by multiplying the battery mass with the battery power density, which is assumed to be equal to 1 kW/kg (according to ref. [13]), and taking into account the loss due to battery efficiency (equal to 0.95). The region below the dotted line indicates that the maximum power supplied by the battery pack is not sufficient to guarantee the power requested by the electric motor during the take-off; the opposite occurs for the region over the dotted line. The result clearly shows that all the designed configurations can supply the requested power during the take-off phase, and therefore the energy constraint is the most restrictive one in this case.

The main data relating to the optima configurations for each MTOW are given in

Table 4. Main data of hybrid-electric aircraft optimizations (range 600 nm).

Variable	Value
$\mathit{MTOW}$ [$\mathsf{k}{\mathrm{g}}_{\mathsf{f}}$]	23,000	30,000	35,013	40,049
$H_P$	0.402	0.525	0.438	0.413
${\Phi }_{t,1}$	0.183	0.169	0.151	0.128
${\Phi }_{t,2}$	0.459	0.385	0.243	0.180
${\Phi }_{t,3}$	0.173	0.184	0.120	0.100
$P_t^i$ [MW]	3.593	3.704	5.105	6.102
$P_e^i$ [MW]	2.489	4.221	4.108	4.418
$m_b$ [kg]	4291	8196	10,962	13,698
$m_{\mathit{fb}}$ [kg]	872	764	688	620

, and these reveal that the reductions in thermal power supply in cruise (and therefore the corresponding increases in electrical power supply) result in a swap between the energy sources in the corresponding ratio of their respective gravimetric energy densities. The comparison with the full-thermal reference configuration ($m_{\mathit{fb}}$ = 103 kg), developed by using the methodology outlined in Section 2.1 according to the TLARs reported in Section 3, shows that the block fuel consumption reduces up to 44% by relaxing the constraint on MTOW.

Finally, the previous analyses have been extended to a design range of 400 nm and 800 nm. Specifically, several hybrid-electric aircraft have been designed (at the same range) and compared with the reference thermal tube-and-wing aircraft. The related optimization results are depicted in Figure 9, showing that the in case of a range of 400 nm, the introduction of hybrid-electric propulsion can reduce fuel consumption up to zero in the case of configurations with MTOW = 35 × 10³ $\mathsf{k}{\mathrm{g}}_{\mathsf{f}}$; whereas, in the case of a range of 800 nm, there is very limited benefit in terms of fuel consumption gains.

The results depicted in Figure 9 highlight some interesting aspects related to the design of hybrid-electric regional aircraft: (i) the current projections of powertrain electric components limit the design range; specifically, the most penalizing factor is the low battery energy density which introduces severe weight increases for longer distances and (ii) to introduce noteworthy reductions in fuel consumption, it is necessary to increase aircraft MTOW with respect to the typical values of full-thermal benchmarks. This latter aspect can affect other performance metrics: the direct operating costs of aircraft are expected to increase, as already proposed by [41,57], and then also suggested by [62]. Furthermore, aircraft compatibility with airport infrastructure can be a further issue, since a high MTOW is associated with a large wingspan, which can introduce compliance issues with standardized apron size.

4.3. Optimization Results: Parametric Analysis of the Number of Stages Dividing the Mission

The role of the cruise power supply profiles, both thermal and electric, resulted in being the main drivers to optimizing hybrid-electric performance in terms of block fuel consumption. In this section, a parametric analysis of the number of stages dividing the cruise is proposed; this allows us to split the cruise into shorter intervals, and hence to increase the resolution of the power supply time profiles. This enables an understanding of which degree of power profile resolution is necessary to achieve a more refined outcome by applying the proposed simulative model. This study is divided into two steps: (i) four different cases where the number of stages has been varied up to eight, the MTOW limited to 23 × 10³ kg_f and the range equal to 600 nm, and (ii) an additional case where the number of steps has been increased up to 35 and the MTOW and range up to 40 × 10³ kg_f and 800 nm, respectively. Regarding the first point, the four different cases considered are as follows: “Case 0”, which represents the case of one stage at cruise (cf. Section 4.2) and “Case 1”, “Case 2” and “Case 3” that consider two, four and eight cruise stages, respectively, as reported in

Table 5. Description of the analyzed cases.

Analyzed Case	Description
Case 0	One stage at cruise phase
Case 1	Two stages at cruise phase
Case 2	Four stages at cruise phase
Case 3	Eight stages at cruise phase

.

Figure 10 shows that by increasing the dividing stages, the optimizer finds a thermal power supply profile that decreases during cruise; compared to the single-stage cruise case, the solutions steer towards a profile with a higher ${\Phi }_t$ at the beginning of the cruise phase and a lower ${\Phi }_t$ at the end, as shown in Figure 11 for the optima configurations; an opposite trend is reflected for the power supplied by the electric motors.

The powertrain efficiency $\eta$, defined by Equation (13), increases in the last stages of the cruise phase, as seen in Figure 12, in cases having more than one cruise dividing stage. Increases in the powertrain efficiency occur in the case where less power is demanded to the thermal chain, as reported in [11], because the electric power chains exhibit higher individual component efficiencies. Hence, the optimizer with a widened design space, i.e., a larger number of dividing stages, searches for solutions with a higher $\eta$. This causes the energy supplied by the battery pack $E_b$, calculated by means of Equation (14), to slightly increase, potentially enabling related fuel saving.

$$\eta =P/(P_b+P_F)$$

(14)

$$E_b= {\int }_0^t{P}_b\mathit{dt}$$

(15)

However, for the considered application, $E_b$ is 2084 kWh, 2086 kWh, 2085 kWh and 2090 kWh for the four cases analyzed, respectively, highlighting that the actual impact of improving the power supply profile to increase $\eta$ is only theoretically impactful on the performance. Indeed, the related fuel consumption in the case of eight cruise divisions is 870 kg, only 2 kg less than the optimization based on a single cruise stage. Due to the conceptual nature of this approach, the difference is clearly negligible. It is evident that the increase in the number of stages does not generate any benefits in terms of the improvement of the power supply profile. In fact, the amount of electric and thermal energy for the optimized aircraft is very similar for all of the investigated configurations, and using a simplified mission discretization, i.e., made by the main three phases, seems to be enough to properly understand and optimize the overall aircraft performance.

In the second step, the number of dividing stages has been increased by considering three additional cases with 19, 27 and 35 cruise stages, to explore if an even finer resolution of the cruise can effectively impact the optimized power profile. The results for the design range of 600 nm- are reported in Figure 13, showing that the optima configurations are close to the low border of the configurations designed in the parametric study proposed in Section 4.1; it clearly appears that the trade-off border identified by the very simplified parametric exploration cannot be exceeded by improving the resolution of the problem. To check if the finer splitting of the cruise is instead effective for a longer range, the case of 800 nm has been simulated with the same approach. However, as depicted in Figure 14, the same general outcomes have been achieved. Increasing the design range more is not useful, as the hybrid-electric propulsion would not lead to any performance benefit in this scenario.

5. Discussion

The results proposed in Section 4 lead to interrelated conclusions that are of significant importance for the conceptual design of regional hybrid-electric aircraft equipped with a parallel powertrain, which can be summarized as follows: (i) optimizing the power supply split between the electric and thermal chains is of key relevance to achieving specific performance benefits, such as reduced fuel consumption, and (ii) given the limited route length, optimizing the power supply split in the three main airborne phases (climb, cruise, descent) is sufficient, whereas the tighter temporal discretization of the power profile would increase the computational cost without providing any additional engineering insights. This enables exploratory studies of such technologies to be performed with rather simple models, thereby avoiding the formalization, the implementation, and the computational cost of much more complex simulative models, such as those discussed in [45]. Focusing more on the technical data proposed in this manuscript, it is worth underlining that the results are also related to the modeling of the powertrain adopted in this work, namely that (i) the efficiency of the components of the electric chain is constant and does not depend on the supplied power and (ii) the brake-specific fuel consumption (BSFC) is constant and does not depend on the power supplied by the thermal engine. These two hypotheses are reasonable in the conceptual design phase since they allow for the initialization of the design process and the obtaining of preliminary results, even though some detailed information is not available (e.g., engine performance map). This study showed that, when dealing with the conceptual design of hybrid-electric aircraft and assuming that the thermal power fraction is constant in each airborne phase of the mission, a complex power management strategy is not worth considering, since it increases complexity without any significant advantage on fuel consumption. Therefore, a constant thermal power fraction in each airborne phase is sufficiently accurate to achieve useful information for the subsequent preliminary design process. However, it is worth noting that neglecting the dependency between the BSFC and the power supplied by the thermal engine, and the dependency between the electric motor, battery and power electronic efficiency and the power supplied, may lead to an incorrect assessment of aircraft fuel consumption and related emissions. This aspect, indeed, has already been investigated and emphasized in ref. [63], highlighting that more refined models to assess this dependency should be considered.

Despite its promising results, the proposed approach has intrinsic limitations that restrict its applicability to the conceptual phase of aircraft design. The model relies on simplified assumptions, such as constant efficiency of the electrical powertrain components and constant specific fuel consumption for the thermal engine. While these simplifications are justified at this early design stage, they do not fully capture the system’s complexity, neglecting, for instance, dynamic variations in the power profiles or the influence of environmental factors such as altitude and temperature. Another significant limitation lies in the representation of the aircraft as a point mass. While this assumption greatly reduces computational complexity, it excludes the ability to analyze phenomena related to the distribution of aerodynamic forces along the structure, such as the interaction between aerodynamic loads and structural deformations. Furthermore, it does not account for effects such as the variable position of the center of mass during the mission, the trim drag, or the impact of maneuvers on the aircraft flight mechanics. These aspects, crucial in more advanced design stages, are omitted in the simplified model. Additionally, the approach employs step functions to model power transitions between consecutive flight phases, disregarding the gradual changes that occur in reality. This limitation results in an incomplete representation of system dynamics, which is particularly relevant when designing control systems.

The entire methodological framework has been developed to provide a quick and general overview of promising configurations, rather than to address the detailed analysis required in advanced stages of aircraft design. Beyond the conceptual phase, more refined methodologies, such as optimization based on advanced control techniques and detailed thermodynamic and structural modeling, are necessary. These tools can incorporate temporal and spatial variables with greater resolution, offering results that are more representative of actual performance. Even with these limitations, the proposed model is a valuable starting point for early decision-making, thanks to its ability to balance computational cost and effectiveness. However, it is essential to recognize that its approximations must be replaced with more complex and realistic models as the design process advances, requiring greater accuracy and a detailed performance analysis of the aircraft and its systems.

6. Conclusions

This paper aimed to explore the applicability of simplified simulative approaches for performance analyses of regional hybrid-electric aircraft. While the results suggest that these methods can provide useful insights into power split management during flight, further validation through high-fidelity modeling is necessary to confirm their accuracy. In fact, by using a simplified mission simulation approach based on the aircraft point mass model and Euler numerical integration, and dividing the airborne mission into only three phases, i.e., climb, cruise and descent, it is possible to obtain the optima time profiles of the thermal and electrical power supply to minimize fuel consumption. This approach, therefore, allows us to obtain sound information in the conceptual phase, avoiding the formalization and utilization of much more complex and computationally expensive simulation models. The results of the conceptual assessment of a hybrid-electric aircraft belonging to the regional category and equipped with a parallel powertrain show that (i) the power supply split between electric and thermal chains throughout the missions is a key variable to be handled to optimize the aircraft’s performance; (ii) cruise power profiles have the greatest impact on performance, as maximizing the share of electrical power supplied in this phase leads to significant reductions in fuel consumption; (iii) refining the time power profile resolution by increasing the number of stages dividing the mission does not provide any benefit; for the airborne phases, the ideal number of stages is three, covering climb, cruise and descent, respectively; and (iv) hybrid-electric propulsion could provide performance gains at short and very-short range, whereas at a range of 800 nm, the benefits in terms of fuel consumption reduction are very limited. For short ranges, on the other hand, increasing the MTOW enables the transportation of larger battery mass and further reductions in fuel consumption, indicating that a trade-off between different performance metrics should be carefully assessed for hybrid-electric aircraft. Further investigations for this research will consider the impact of battery recharging and the investigation of a different FoM to investigate factors such as emissions. Furthermore, the selection of more accurate and realistic thermal engine maps, when available, could lead to a more accurate fuel consumption prediction for the presented methodology.

This article is a revised and expanded version of the paper entitled “Power Management Supply Optimization for Hybrid-Electric Regional Aircraft”, which was presented at 34th ICAS Conference, Florence, Italy, 9–13 September 2024, ref. [64].