## Abstract

This study addresses the uncertainties in hybrid-electric powertrain technology for a 19-passenger commuter aircraft, focusing on two future Entry-Into-Service timeframes: 2030 and 2040. The methodology is split into a preliminary optimization of aircraft design based on nominal technology scenarios followed by Monte Carlo simulations to investigate the impact of diverse technology projections and distribution types. Advanced surrogate modeling techniques, leveraging deep neural networks (DNN) trained on a dataset from an aircraft design framework, are employed. Key outcomes from this work reveal a marked increase in computational efficiency, with a speed-up factor of approximately 500 times when utilizing surrogate models. The results indicate that the 2040 entry-into-service (EIS) scenario could achieve larger reductions in fuel and total energy consumption—20.4% and 15.8% respectively—relative to the 2030 scenario, but with higher uncertainty. Across all scenarios examined, the hybrid-electric model showcased superior performance compared to its conventional counterpart. The battery-specific energy density is proved to be a critical parameter of the aircraft’s performance across both timeframes. The findings emphasize the importance of continuous innovation in battery and motor technologies to target toward greater system-level efficiency and reduced environmental impact.

## 1 Introduction

The electrification of aviation has been in the forefront of research activities for the last decades. Current estimations show that aviation is responsible for 2.4% of all human-induced carbon emissions [1]. In parallel, this industry is considered an economical anchor for transferring goods and people. The worldwide revenue passenger kilometers have increased by over 19 times from 1970 to 2019 [2]. In addition, it is expected that it will considerably grow over the next decades with current trends showing a growth of 5% in passenger numbers every year [3]. Therefore, even if its contribution is less that other energy-related carbon emitting human activities, it is crucial to achieve more sustainable operations. Regulatory bodies from across the globe have set a series of ambitious environmental goals to drive the efforts toward sustainability. Flightpath 2050 [4] summarized the goals set by the European Union both in terms of environmental impact reduction and aircraft operation. On the other hand, Mangerlsdorf et al. [5] discussed the environmental goals presented by the National Aeronautics and Space Administration, while highlighting the different timeframes (denoted as N+) and, finally, different initiatives driven by National Aeronautics and Space Administration to achieve those goals.

Numerous studies focusing on this topic are currently available in open literature. These range in terms of aircraft size, with the majority focusing on large aircraft due to their higher share on the sector’s impact [6]. However, regional air mobility (RAM)—150–800 km and 5–50-passenger aircraft—is an emerging transportation model. Due to the increased urbanization and strain of traditional transportation infrastructure, phenomena such as traffic congestion and longer commute times are observed globally. RAM can offer a transformative solution as it can enable point-to-point connectivity between cities and rural areas.

In addition, electrified regional and commuter aircraft can be realized in shorter timeframes according to a study presented by Schäfer et al. [7]. According to Antcliff et al. [8], RAM can be enabled in the United States by taking advantage of the existing small airport infrastructure and utilizing future aircraft concepts. Justin et al. [9] provided a summary of the reasons behind the low share of operations for smaller regional aircraft. However, the authors highlighted the potential increase of smaller aircraft operations due to the emerging realization of electrified aircraft. Optimization of operations and fleet assignments early in the process can mitigate the environmental impact due to the new operations.

The studies found in open literature also span in terms of level of electrification and consider various powertrain architectures [10]. The hybrid-electric architecture is highlighted as the most promising due to the achievable battery energy density of 500 Wh/kg in a timeframe of 15 years [11]. Its advantage lies in the lower added weight required to fly the aircraft compared to all-electric architectures where the resulted weight is prohibitive for both long and short range missions. Different design approaches are currently developed to model this architecture [12].

As the current trend is to transition from separate subsystem designs to vehicle-centric frameworks considering different disciplines, studies employing design space exploration and optimization utilizing low-fidelity models have gained attention over the years. Such frameworks are essential in understanding the benefits and challenges. However, they are based on first principles rather than detailed design to accomplish rapid evaluations. The underlying powertrain assumptions can considerably affect optimal solutions. These are based on technology forecasts rather than developed technologies. Given the uncertainty in forecasts, it is essential to quantify the impact on the system during the initial design steps and assess the inherent technological risks.

Uzodinma et al. [13] were the first to integrate the technological risk induced during aircraft conceptual design focusing on component-level uncertainty and its impact on vehicle-level metrics. A 150-passenger parallel hybrid-electric aircraft was examined and an uncertainty analysis using a surrogate model of the propulsion framework [14] was employed. Their study emphasized the vital role of uncertainty analysis in future technology development programs and underscored the significance of battery specific energy density.

Gautier et al. [15] built upon the previous work by proposing two methodologies to address limitations. First, they introduced an aircraft design optimization method that considers technological uncertainty, generating robust designs. However, they noted that achieving optimal design comes at the cost of reduced performance, as it must accommodate diverse technological scenarios. Secondly, the authors investigated the impact of uncertainty on aircraft design performance optimized for known technology parameters, revealing potential up to a 1% average performance difference in relative block fuel results compared to the robust designs obtained from the first methodology.

At the time of writing this publication, based on the authors’ knowledge, there is a notable gap in the field of assessing the effect of technological risks on the performance of hybrid-electric architecture for RAM. Previous studies discussed in the previous paragraphs [13,15] are centered around larger aircraft featuring a parallel hybrid-electric architecture. The preference for the parallel architecture arises from its relatively more feasible implementation, but not without substantial challenges, eliminating the need for extensive redesign. Additionally, the cell-level battery energy density is assumed to be in the range 359–795 $Wh/kg$ [13,15], but there is significant uncertainty around this parameter. While there is a substantial body of research on hybrid-electric aircraft in general, there is a distinct absence of research that focuses on uncertainty quantification of powertrain technology on the performance of smaller aircraft.

This work draws its main scope from the studies discussed previously. However, the goal is to perform uncertainty analysis for a hybrid-electric commuter aircraft. The main focus is to comprehend and quantify the effect of uncertain powertrain technology assumptions on the system-level performance. The aircraft conceptual design framework utilized is based on Refs. [16,17] and has been developed on top of the OpenConcept library [18]. The conceptual design framework computational scheme along with a design space exploration study are presented in Ref. [16] while the optimization schemes and operational analysis are discussed in Ref. [17]. The series/parallel partial hybrid-electric architecture is considered and low-fidelity models are incorporated. Due to the high number of simulations required for uncertainty analysis, surrogate models are used to speed up the simulations. Three powertrain technology scenarios (conservative, nominal, and aggressive) are determined for two entry-into-service (EIS) timeframes—2030 and 2040—based on the work presented by Uzodinma et al. [13].

## 2 Methodology

### 2.1 Aircraft Configuration.

The aircraft configuration explored in this work is presented in Fig. 1. The configuration is characterized by a 19-passenger commuter aircraft featuring the series/parallel partial hybrid-electric powertrain architecture. The powertrain system comprises one turboprop engine and one electrically driven propeller (e-propeller) per wing. The e-propeller can be driven either by the battery or by fuel in turbo-electric mode through the engine power off-take. The turboprop engines are coupled with generators that can deliver power to the powertrain system.

### 2.2 Technology Scenarios.

As discussed before, the main focus of this study is to capture the impact of electrical component technology on the aircraft-level performance. Two EIS timeframes are examined (2030 and 2040) by introducing informed ranges of technological performance metrics for all included electrical powertrain components. These are characterized by their efficiency and specific power, with the exemption of the battery that also is characterized by its specific energy density.

The projected technological values are dependent on the anticipated intensity of technology development. As a result, it is crucial to encompass various scenarios. The outlined technology scenarios for each electrical component within both EIS timeframes are outlined in Table 1. These projections are based on the comprehensive technology analysis conducted by Uzodinma et al. [13].

2030 | 2040 | |||||||
---|---|---|---|---|---|---|---|---|

Conservative | Nominal | Aggressive | Conservative | Nominal | Aggressive | |||

Battery | Cell specific energy | $Wh/kg$ | 359 | 489 | 584 | 459 | 638 | 795 |

Specific power | $kW/kg$ | 0.7 | 0.75 | 0.8 | 0.9 | 0.95 | 1.0 | |

Efficiency | % | 80 | 85 | 90 | 85 | 90 | 95 | |

Motor | Specific power | $kW/kg$ | 9.2 | 13.2 | 16.1 | 10.8 | 20.4 | 33.0 |

Efficiency | % | 96.3 | 96.8 | 97.4 | 96.7 | 97.5 | 98.3 | |

Generator | Specific power | $kW/kg$ | 9.2 | 13.2 | 16.1 | 10.8 | 20.4 | 33.0 |

Efficiency | % | 96.3 | 96.8 | 97.4 | 96.7 | 97.5 | 98.3 | |

PMAD | Efficiency | % | 98.2 | 98.5 | 98.8 | 98.5 | 98.9 | 99.4 |

2030 | 2040 | |||||||
---|---|---|---|---|---|---|---|---|

Conservative | Nominal | Aggressive | Conservative | Nominal | Aggressive | |||

Battery | Cell specific energy | $Wh/kg$ | 359 | 489 | 584 | 459 | 638 | 795 |

Specific power | $kW/kg$ | 0.7 | 0.75 | 0.8 | 0.9 | 0.95 | 1.0 | |

Efficiency | % | 80 | 85 | 90 | 85 | 90 | 95 | |

Motor | Specific power | $kW/kg$ | 9.2 | 13.2 | 16.1 | 10.8 | 20.4 | 33.0 |

Efficiency | % | 96.3 | 96.8 | 97.4 | 96.7 | 97.5 | 98.3 | |

Generator | Specific power | $kW/kg$ | 9.2 | 13.2 | 16.1 | 10.8 | 20.4 | 33.0 |

Efficiency | % | 96.3 | 96.8 | 97.4 | 96.7 | 97.5 | 98.3 | |

PMAD | Efficiency | % | 98.2 | 98.5 | 98.8 | 98.5 | 98.9 | 99.4 |

The cell level-specific energy density scenario is based on an S-curve projection fitted on existing historical data for different battery chemistries as presented by Tiede et al. [19]. In parallel, three scenario projections from Ref. [20] were utilized for the specific power of electrical machines. As historical efficiency data were not sufficient for fitting, the efficiency projections were derived by applying annual reduction in losses on the current state-of-the-art values.

### 2.3 Technological Risks: Uncertainty Propagation.

Within this subsection, the framework developed to address the uncertainty in technological risk propagation is introduced and discussed. The scenarios outlined in Table 1 map potential outcomes of electrical component technologies for two EIS timeframes. In the context of optimizing the design of the proposed hybrid-electric aircraft on the nominal technology scenarios, it becomes pivotal to explore the following questions:

How does the hybrid-electric aircraft’s system-level performance evolve during operation under various technological scenarios, after the initial design optimization?

What is the influence of diverse technological scenarios on the hybrid-electric aircraft’s system performance across two EIS timeframes?

What is the relative importance of each technology in achieving optimal system outcomes?

This work aims to gather information that contribute toward addressing the aforementioned questions.

#### 2.3.1 Uncertainty Propagation on Nominal Optimal Design.

The developed methodology is outlined in Fig. 2. During Phase I, a full design aircraft optimization is performed using the aircraft conceptual model and considering the nominal technology scenario for all components. The full design aircraft optimization is outlined in Table 2 and will be further discussed in Sec. 2.4.

Variable | Quantity | |
---|---|---|

Objective (f) | Block energy | 1 |

Design variables ($x$) | MTOW | 1 |

$Sref$ | 1 | |

$Wbat$ | 1 | |

$Nprop$ | 1 | |

$Neprop$ | 1 | |

$Dprop$ | 1 | |

$Deprop$ | 1 | |

$PGT*$ | 1 | |

$PEM*$ | 1 | |

$PGEN*$ | 1 | |

$\Phi $ | 3 | |

$\varphi $ | 7 | |

Constrains ($c$) | $0\u2264RMTOW\u22645$ | 1 |

$0.2\u2264SOCloiter$ | 1 | |

$0.1\u2264\xi EM$ | 7 | |

$0.2\u2264\xi GT$ | 7 | |

$W/S=345$$kg/m2$ | 1 | |

$Vtip=200$$m/s$ | 1 | |

$P/D2=200$$kW/m2$ | 1 |

Variable | Quantity | |
---|---|---|

Objective (f) | Block energy | 1 |

Design variables ($x$) | MTOW | 1 |

$Sref$ | 1 | |

$Wbat$ | 1 | |

$Nprop$ | 1 | |

$Neprop$ | 1 | |

$Dprop$ | 1 | |

$Deprop$ | 1 | |

$PGT*$ | 1 | |

$PEM*$ | 1 | |

$PGEN*$ | 1 | |

$\Phi $ | 3 | |

$\varphi $ | 7 | |

Constrains ($c$) | $0\u2264RMTOW\u22645$ | 1 |

$0.2\u2264SOCloiter$ | 1 | |

$0.1\u2264\xi EM$ | 7 | |

$0.2\u2264\xi GT$ | 7 | |

$W/S=345$$kg/m2$ | 1 | |

$Vtip=200$$m/s$ | 1 | |

$P/D2=200$$kW/m2$ | 1 |

During Phase II, Monte Carlo sampling considering the different technological scenarios is employed to generate the required sampling space. The distributions of the electrical component assumptions are defined by utilizing the three scenarios (conservative, nominal and aggressive) presented in Table 1.

For each sample, a reduced optimization problem, which focuses on finding the optimal aircraft operation for minimal block total energy consumption, is executed. The reduced optimization problem definitions are summarized in Table 3. It is worth noting that during the operation optimization, a surrogate model of the aircraft conceptual framework is utilized in order to improve computational efficiency and handle the high number of samples required during Monte Carlo simulations. More information on the aircraft surrogate model creation can be found in Sec. 2.5. The optimal points are derived according to Table 3. Therefore, system-level performance parameter distributions can be derived and investigated.

Variable | Quantity | |
---|---|---|

Objective (f) | Block energy | 1 |

Design variables ($x$) | TOW | 1 |

$\Phi $ | 3 | |

$\varphi $ | 7 | |

Constrains ($c$) | $0\u2264RTOW\u22645$ | 1 |

$0.2\u2264SOCloiter$ | 1 | |

$0.1\u2264\xi EM$ | 7 | |

$0.2\u2264\xi GT$ | 7 |

Variable | Quantity | |
---|---|---|

Objective (f) | Block energy | 1 |

Design variables ($x$) | TOW | 1 |

$\Phi $ | 3 | |

$\varphi $ | 7 | |

Constrains ($c$) | $0\u2264RTOW\u22645$ | 1 |

$0.2\u2264SOCloiter$ | 1 | |

$0.1\u2264\xi EM$ | 7 | |

$0.2\u2264\xi GT$ | 7 |

#### 2.3.2 Sensitivity Analysis of Powertrain Technology.

where $E[Y|Xi]$ is the expected value of the quantity of interest *Y* given the variation of the input $Xi$ and $Var(Y)$ is the total variance of *Y*. Through the application of Sobol indices, facilitated by the library SALib [22], it is possible to quantify the contribution of each input, specifically different electrical component technologies, to the total variance.

### 2.4 Aircraft Conceptual Design Framework.

The aircraft conceptual design framework is based on previous work by the authors [16,17] and employs the OpenConcept library [18]. The framework consists of a series of low-fidelity models to describe the airframe and the powertrain system. Models are setup individually and linked up using the OpenMDAO [23] capabilities, which is employed as the base library for the framework development. Energy and power flows between components are defined in order to ensure for consistent calculations. In parallel, first principle aerodynamic calculations [24] and weight estimations [25] based on empirical correlations are employed to estimate the aircraft characteristics, while powertrain components are modeled using constant performance characteristics as described in Table 1. The engine component performance is modeled using Specific Fuel Consumption estimations from the Pratt and Whitney PT6A-67D unit [26], while the propeller performance is based on an empirical efficiency map for a constant speed turboprop obtained by Ref. [27].

The supply power ratio $\Phi $ definition is based on Refs. [28,29]. This parameter controls the power management and distribution (PMAD) component, shown in Fig. 1 and defines the ratio between the battery supply compared to the whole power supplied to the PMAD. The PMAD is a lumped equivalent of the power electronics present in the system, modeled with a constant efficiency. This component also includes power flow checks to ensure consistent calculations. A $\Phi <0$ will indicate a battery charging mode, however this is not considered within this work.

The shaft power ratio $\varphi $ is defined as the ratio of the shaft power delivered to the e-propellers $Pes$ and the total power delivered to propulsive devices ($Pes+Ps$). Both parameters can be varied by an external driver—the user or an optimizer—for each mission phase. The parameters remain at the same level for the duration of the corresponding phase.

Within this framework, a nonlinear Newton solver is employed to find the appropriate throttle levels of each powertrain component. The main goal is to satisfy the thrust generation requirements for the whole mission considered. The aircraft top-level requirements, defined by the aircraft class and the CS-23 certification [30], are summarized in Table 4. The hybrid-electric aircraft model rests upon the Beechcraft 1900D aircraft design. Utilizing publicly available data [31], a conventional model is initially developed [16] to serve as the benchmark for comparing the outcomes of the hybrid-electric counterpart.

Parameter | Value |
---|---|

MTOW | $\u2264$8618 kg |

Number of passenger | 19 |

Payload | 1881 (kg) |

19 × (87 + 12) | |

Main range | 600 (NM) |

Diversion range | 100 (NM) |

Loiter duration | 30 (min) |

Cruise altitude | 10,000 (ft) |

Loiter altitude | 1500 (ft) |

Cruise Mach | 0.35 (-) |

Rate of climb | $\u2265$6.35 (m/s) |

(MTOW, SL, ISA) | |

Approach speed | $\u2264$62 (m/s) |

Parameter | Value |
---|---|

MTOW | $\u2264$8618 kg |

Number of passenger | 19 |

Payload | 1881 (kg) |

19 × (87 + 12) | |

Main range | 600 (NM) |

Diversion range | 100 (NM) |

Loiter duration | 30 (min) |

Cruise altitude | 10,000 (ft) |

Loiter altitude | 1500 (ft) |

Cruise Mach | 0.35 (-) |

Rate of climb | $\u2265$6.35 (m/s) |

(MTOW, SL, ISA) | |

Approach speed | $\u2264$62 (m/s) |

A mission profile is predefined externally by describing the aircraft speed and altitude. The mission is split between main mission, diversion mission, and loiter phase with each phase consisting of intermediate points. The main and diversion missions are comprised of the climb, cruise, and descent phases. As shown in Table 4, the loiter phase is a constant altitude flight at 1500 ft for 30 min using one energy source as defined in CS-23 certification [30]. Within this work, the focus is to examine the energy consumption during the main mission and therefore, it is selected that the loiter phase will be powered by fuel. Similarly, the diversion mission is powered by fuel and the hybrid operation optimization focuses on the main mission. A detailed discussion on the mission segmentation as well as the calculation schemes that take place during the mission analysis can be found in Ref. [16] for the interested readers.

The development of this framework is focused toward the versatility of the computations. Parameters and residuals can be either handled by the internal Newton solver or can be exposed as framework outputs and handled by the user or optimizer. The workflow in terms of an XDSM (eXtended Design Structure Matrix) diagram [32] for the aircraft design optimization is illustrated in Fig. 3. The operations corresponding to each block in the diagram are as follows:

Initial design variable levels and design parameters are passed to the optimizer and analysis blocks respectively.

Initialization of the Newton solver.

Atmosphere: the mission profile is used to calculate the speed and density for each intermediate point.

Aerodynamics: inputs are used to calculate the aircraft drag for each mission point.

Mission Analysis: the required thrust and mission point duration are evaluated.

Powertrain: the powertrain component throttles are selected in order to satisfy the thrust requirements.

Weights: the aircraft weight is calculated and updated for each intermediate point by using the fuel mass flow rate and mission point duration.

Check Newton solver convergence. If convergence is not achieved, return to 2; otherwise, continue.

Compute objective

*f*, constrains $c$.Compute new set of design variables. If convergence is not achieved, return to 1; otherwise, return optimal solution.

The aforementioned workflow refers to the optimization problem outlined in Table 2. It is important to note that for other optimization problems, the input–output relationships depicted in the XDSM diagram are subject to updates. Despite these variations, the main structure of the aircraft conceptual design remains consistent as the one shown in Fig. 3. In addition, the objective function *f* selected for the optimization in this work is the Block Total Energy consumption, as illustrated in Fig. 3. This parameter is calculated from the summation of the chemical (fuel) and electrical energy. The fuel and electrical energy consumption are integrated from fuel flow and power delivered by the battery over the mission segments. The term block refers to the energy consumption during the main mission. For interested readers looking for a deeper dive into the design optimization process, extensive details can be found in Ref. [17].

### 2.5 Aircraft Surrogate Model.

The conceptual design framework presented in Sec. 2.4 allows for rapid design evaluation while optimizing the aircraft system. However, due to the focus of the present work, which is the uncertainty analysis of the powertrain technology and its effect on the system-level performance, a large number of optimizations and simulations are required. A rule-of-thumb for Monte Carlo simulations is to simulate a few thousands samples to ensure that the law of large numbers [33] is in place. With the current scheme and optimization, the computational time required for this analysis is prohibitive. Therefore, the aforementioned aircraft design framework will not be employed directly but will be used to train a surrogate model. The faster evaluation of the surrogate model is of major importance due to the high number of evaluations required for this analysis.

#### 2.5.1 Data Creation.

In order to effectively train the surrogate model, representative labeled data are required. The data are created by employing a Latin hypercube sampling approach considering all the technology parameters presented in Table 1. The samples, once created, are passed to the aircraft model. Subsequently, an optimization process, as detailed in Table 3, is carried out for each of these samples. After the optimizations are finalized, the optimal take-off weight (TOW), Block Fuel, Electrical and Total Energy, which are the quantities of interest (QoI), are extracted from each optimization and used as the outputs of the surrogate model.

It is important to note that for the purposes of this work, two surrogate models are trained: one for 2030 and one for 2040. The aforementioned sampling approach is repeated two times with different ranges of values as depicted in Table 1. A total of 5000 samples are created for each EIS. The data are divided into two categories: 80% is allocated for training, and the remaining 20% is reserved for testing. Within the training data, a further 90% is utilized for training the model, while the remaining 10% is utilized to validate the model. This division is crucial to enhance the robustness of the surrogate model and improve its generability.

#### 2.5.2 Surrogate Model Training.

The architecture of the surrogate model is constructed utilizing a deep neural network (DNN) framework, which is implemented through the TensorFlow library [34]. This particular model is tailored for a regression task, characterized by its configuration of eight inputs and four outputs. The model inputs are the uncertain technological parameters presented in Table 1, while the model outputs include the four QoIs mentioned in the Data Creation section. For training, the model employs a mean squared error (MSE) loss function, aiming to minimize the discrepancy between the predicted and actual values. To efficiently navigate through the vast space of possible hyperparameter combinations, a random search optimization is carried out using Keras Tuner [35]. The hyperparameters selected along their range are summarized in Table 5.

Hyperparameter | Selection | Value |
---|---|---|

Layers | Integer | $2\u221210$ |

Units per layer | Integer | $64\u2212512,32$ |

Optimizer | Choice | Adam |

RMSprop | ||

Batch size | Integer | $16\u2212128,16$ |

Activation function | Choice | relu, tanh |

Learning rate | Choice | $10\u22122\u221210\u22125,10\u22121$ |

Regularizer | Float | $10\u22125\u221210\u22121$ |

Hyperparameter | Selection | Value |
---|---|---|

Layers | Integer | $2\u221210$ |

Units per layer | Integer | $64\u2212512,32$ |

Optimizer | Choice | Adam |

RMSprop | ||

Batch size | Integer | $16\u2212128,16$ |

Activation function | Choice | relu, tanh |

Learning rate | Choice | $10\u22122\u221210\u22125,10\u22121$ |

Regularizer | Float | $10\u22125\u221210\u22121$ |

The learning rate is a critical hyperparameter that controls the step size for updating the network’s weights. An optimal learning rate ensures efficient convergence, avoiding overshooting or getting stuck in local minima. On the other hand, optimizers like Adam [36] and RMSprop [37], complement the learning rate by adaptively adjusting it during training. Adam is favored for its rapid convergence, while RMSprop adjusts the learning rate based on recent gradients, maintaining consistent updates. In this study both Adam and RMSprop were examined separately in the hyperparameter search to evaluate which optimizer suites the current dataset.

The batch size is another important factor; it not only influences memory utilization and computational speed but also affects the model’s generalization capabilities. Fine-tuning of the batch size ensures that the gradient estimates are stable and informative, enabling a smoother and more reliable training process.

Furthermore, L2 regularization is incorporated during the training process to ensure the model is not overfitted. L2 regularization encourages the model to develop a less complex and more generalizable internal representation, which can perform better on the validation and test datasets. Finally, a careful consideration is given to the network’s architecture. This includes configuring the number and dimensions of hidden layers and selecting appropriate activation functions.

During the training phase, key callbacks were implemented to enhance model performance and efficiency. These callbacks cover tasks such as preserving optimal weights, stopping training when validation loss increases, and dynamically adjusting learning rates during plateaus. Further information on these implementations is available in the associated code repository ACSurrogateTune.

## 3 Results and Discussion

This section presents and discusses the results obtained by applying the methodology discussed above. First, the surrogate models’ performance is discussed followed by an investigation of the required sample size for Monte Carlo estimates and the uncertainty propagation due to different distribution types and EIS timeframes. Finally, a sensitivity analysis of technological parameters on the aircraft performance is discussed.

### 3.1 Surrogate Model Performance.

After performing the random search in Keras Tuner to tune the models’ hyperparameters, the optimal models were selected. The performance of the surrogate models is based on the MSE validation loss. The different parameter levels tested for the surrogate models were summarized previously in Table 5. The best performing model configurations for both EIS timeframes are summarized in Table 6. Regarding the activation function, the selection is presented for each layer. The selected models present the best performance in terms of both MSE validation and train loss.

Parameter | 2030 | 2040 |
---|---|---|

Number of layers | 2 | 3 |

Optimizer | Adam | RMSprop |

Learning rate | 1.0 × 10^{−3} | 1.0 × 10^{−5} |

Batch size | 96 | 16 |

Activation function | relu/relu | relu/tanh/tanh |

Regularizer | 3.11 × 10^{−5} | 8.77 × 10^{−3} |

Hidden layer 1 | 192 units | 96 units |

Hidden layer 2 | 96 units | 384 units |

Hidden layer 3 | — | 416 units |

Parameter | 2030 | 2040 |
---|---|---|

Number of layers | 2 | 3 |

Optimizer | Adam | RMSprop |

Learning rate | 1.0 × 10^{−3} | 1.0 × 10^{−5} |

Batch size | 96 | 16 |

Activation function | relu/relu | relu/tanh/tanh |

Regularizer | 3.11 × 10^{−5} | 8.77 × 10^{−3} |

Hidden layer 1 | 192 units | 96 units |

Hidden layer 2 | 96 units | 384 units |

Hidden layer 3 | — | 416 units |

In parallel, the MSE train and validation loss history were examined, shown in Fig. 4. This is required to ensure model convergence as well as avoid any over- or underfitting. In Fig. 4, the convergence from 10 selected trials for EIS 2030 is presented in gray, whereas the best performing model is highlighted in blue and black colors. The results show that the selected model is converged as well as not overfitting the data since the train $LMSE$ is lower that the validation $LMSE$.

The models’ performance is further examined by testing them on unseen data, namely, the test data (20% of the original dataset). The root mean squared error (RMSE) and the coefficient of determination ($R2$) for each model output are summarized in Table 7. The values presented in the matrix show that the model can adequately predict the QoI.

2030 | 2040 | ||||
---|---|---|---|---|---|

Parameter | Units | RMSE | $R2$ | RMSE | $R2$ |

MTOW | kg | 0.19 | 0.999 | 1.07 | 0.998 |

Block fuel | kg | 0.13 | 0.999 | 0.36 | 0.999 |

Block total | kW × h | 1.16 | 0.999 | 3.79 | 0.999 |

Energy | |||||

Electrical | kW × h | 0.55 | 0.999 | 2.13 | 0.999 |

Energy |

2030 | 2040 | ||||
---|---|---|---|---|---|

Parameter | Units | RMSE | $R2$ | RMSE | $R2$ |

MTOW | kg | 0.19 | 0.999 | 1.07 | 0.998 |

Block fuel | kg | 0.13 | 0.999 | 0.36 | 0.999 |

Block total | kW × h | 1.16 | 0.999 | 3.79 | 0.999 |

Energy | |||||

Electrical | kW × h | 0.55 | 0.999 | 2.13 | 0.999 |

Energy |

Finally, the regression charts for all output parameters and models were examined. Here, due to space constrains, the block total energy for EIS 2030 is presented in Fig. 5. The desired performance of the model is verified for the whole design space.

Going beyond the aspects of model selection and performance, it is crucial to examine the achieved speed-up factor for the surrogate model. While the framework is able to ran in the order of seconds for one iteration, an optimization requires several iterations to converge as well as additional iterations to compute gradients. The optimization time depends on the complexity of the optimization (e.g., Tables 2 and 3) with the operation optimization being faster due to less design variables and constrains. In addition, it is important to note that this time depends on the computational device that the framework is deployed. Nevertheless, the optimization times are in the order of magnitude of mins to an hour. Due to the fact that Monte Carlo simulations and sensitivity analysis requires thousands of samples (optimizations) as well as numerous iterations to select the required sample size, this study would not be possible to achieve without the development of surrogate models. In order to showcase the importance of the surrogate model, the speed-up factor is calculated. This metric quantifies the time efficiency gained when opting for the surrogate model and is determined by comparing the time taken for the original aircraft model and the surrogate model, including the training time, to produce the required number of samples for this work. The training time consists of the time to perform optimizations for training data creation as well as the hyperparameter tuning process to fully train the model. The prediction time for the DNN models are in order of ms. Therefore, the calculated speed-up factor for the presented surrogate model is in the order of $5\xd7102$, highlighting a substantial reduction in modeling time.

### 3.2 Selection of Required Sample Size.

In order to be able to adequately describe the Monte Carlo estimates, a study on the number of required samples is important. Within this work, the Max to Sum plots—or M/S plots—approach is employed to achieve this goal. An M/S plot refers to a graphical representation of the ratio of the maximum value to the cumulative sum of the values. M/S plots of the four moments (mean, variance, skewness and kurtosis) for the system-level QoI distributions are typically used to identify the number of samples required to describe a distribution through Monte Carlo simulations. Within this work, this procedure has been repeated for all QoI, distribution types and for the two EIS timeframes examined. In Fig. 6, an indicative example is presented, showcasing the M/S plot for Block Total Energy consumption within the EIS 2030 timeframe, utilizing a triangular distribution type for all technological parameters.

Figure 6 is split into two subfigures to present the convergence of the Monte Carlo estimates for the left and right tails. As observed in Fig. 6, the distribution characteristics are converging with an increasing number of samples. In this case the 5000 samples would not be sufficient to describe the distribution. Therefore, after examining the graph, 55,000 samples were selected to extract the distribution information presented in the following Figures. The 55,000 samples were sufficient for all other parameters examined within this study. The surrogate model is proved as crucial here to efficiently handle a large sample size for accurate distribution characterization and ensuring computational efficiency.

### 3.3 Uncertainty Propagation—Distribution Types.

The distributions for the TOW, Fuel, Total Energy consumption, and the degree of hybridization (DoH) during the main mission are presented herein. The results are presented relativized ($\Delta $) to the conventional Beechcraft 1900D aircraft except for the DoH since the conventional aircraft does not utilize electrical energy during its operation. For the total energy consumption, the hybrid-electric aircraft calculation considers both fuel and electrical energy consumption, whereas for the conventional one, the fuel consumption is converted to energy. This conversion for both aircraft is made through multiplying the amount of fuel consumed with the lower heating value of the Jet-A fuel, which is assumed to be 43 kJ/kg.

In this analysis, three distribution types are utilized uniform, triangular, and truncated normal—to represent different assumptions about the studied technology parameters. These distributions signify diverse levels of knowledge regarding the expected values of each parameter. The different types of distributions for the battery energy density are shown in Fig. 7. However, similar distributions are derived from the scenarios in Table 1 and applied to all parameters presented in this table.

The results for EIS 2030, as depicted in Fig. 8, showcase distinct characteristics of the distributions. The normal distribution showcases a higher confidence in the nominal scenario followed by the triangular distribution. The uniform distribution, in contrast, assigns equal probability to all scenarios. This can be further exemplified in Fig. 7, where the normal and triangular distributions exhibit similarity, indicating a higher confidence in the nominal scenario, while the uniform distribution implies an equal likelihood across all scenarios. These distributions, reflected in the responses, manifest varying degrees of variance in potential outcomes.

The nominal design shown in Fig. 8 corresponds to the maximum take-off weight (MTOW) for this aircraft. Through optimization (Table 3), the TOW is adjusted to ensure consistency in calculations. Although the TOW of the hybrid-electric aircraft shows an increase compared to conventional one, its variance in relation to the technology parameters is minimal. This is due to the fact that changes in component specific powers and battery specific energy alter the available power and energy but not the weight. Essentially, as specific power and energy density vary, the power output and energy capacity change, but the weight of these components remains constant. On the other hand, component efficiencies and battery energy density, which are crucial for electrical energy usage during flight, directly impact the Block Fuel consumption and, consequently, the TOW. These factors influence the aircraft’s overall efficiency, affecting fuel requirements and thus the TOW calculation.

Figure 8 also displays the $\Delta $ Block Fuel consumption. The 2030 nominal design achieves a 17% reduction in block fuel, highlighting the benefits of hybrid-electric propulsion. However, the Block Fuel major variation shown in the graph is primarily due to the technology parameters that define the electrical efficiency of the powertrain system, as previously explained. Given that Block Fuel constitutes about 6% of the MTOW in the nominal design, its variation leads to a relatively minor fluctuation in the TOW.

A smaller 13.6% decrease is observed in the Block Total Energy consumption for the nominal aircraft design. This study assumes that the battery is fully charged at the start of the mission. However, in practical scenarios and depending on the energy source, charging the battery might lead to emissions. Thus, it becomes crucial to assess the system-level efficiency, rather than solely focusing on the Block Fuel consumption. The Block Total Energy consumption also exhibits significant variation, as anticipated. This is largely attributed to the variation in the DoH. It is observed that a higher DoH results in reduced Block Fuel and Total Energy consumption.

However, it is important to note that for both Block Fuel and Total Energy consumption, the worst-case scenarios do not exceed the corresponding values of the conventional model, indicating a reduction across the entire sampling space. This is a positive outcome, as even in conservative scenarios, the performance does not deteriorate compared to the conventional aircraft. However, the extent of the benefit is considerably smaller. Therefore, it is safe to conclude that inherit technology risk discussed previously is of major importance and should be considered when investigating such concepts.

### 3.4 Uncertainty Propagation—Technology Projections.

Within this subsection, the results between the two EIS timeframes (2030 and 2040) are presented and compared. The technology parameters’ distributions are assumed to be triangular for this set of results. The results are presented in Fig. 9 where the QoI distributions are shown in the form of violin graphs, visualizing both the median and standard deviation of the distributions as well as their probability density.

It is observed that, for the EIS 2040, a higher MTOW is required to minimize the Block Total Energy. As anticipated due to the advanced technology scenarios, the hybrid-electric aircraft for EIS 2040 demonstrates a capacity to achieve reduced Block Fuel and Block Total Energy consumption when compared to conventional aircraft and the EIS 2030 counterpart. This is attributed to the increased utilization of electrical energy, as evidenced by the higher DoH levels illustrated in Fig. 9.

Nevertheless, a comparative analysis of the two EIS timeframes reveals that EIS 2040 exhibits a higher level of uncertainty. This increased uncertainty arises from a greater variance in projected performance. Such variability is anticipated, considering the extended timeframe for the projection of technologies involved. This confirms that projecting performance over longer timeframes introduces inherent technology risks. While the distributions of Block Fuel, Total Energy, and DoH for EIS 2040 are skewed toward better performance, it is important to highlight the potential for instances where they could present worse or equal performance compared to EIS 2030.

### 3.5 Sensitivity Analysis.

The significance of each electrical component technology is illustrated in Fig. 10 through the first-order Sobol indices across both EIS timeframes. Notably, the specific energy density of the battery emerges as the key parameter influencing the operation of the hybrid-electric aircraft in both EIS scenarios. However, there is a slight decrease in its importance for EIS 2040, as shown in Fig. 10. In the rank of components, the battery holds the top position, followed by the PMAD and motor components.

Regarding the rest battery parameters, it is observed that its specific power has a minor impact, as the specific energy density dominates the battery sizing and, consequently, operation. Similarly, the battery efficiency demonstrates minimal influence on the block total energy consumption compared to other parameters.

When analyzing the results, it is crucial to consider the relative variations among different technology scenarios and components. In EIS 2030, efficiencies display marginal fluctuations across different scenarios, while the specific power of components demonstrates a more notable range of variation. Therefore, for this timeframe, the motors’ specific power is identified as the second most influential factor. On the contrary, in the subsequent timeframe (2040), there is a remarkable improvement in the motors’ specific power, with component efficiencies taking over as the primary variable affecting energy consumption.

As for generators, their usage during the main mission is limited. This choice is driven by the optimization that prioritizes battery energy consumption during the main mission. Generators come into play during diversion mission, where their purpose is to power the e-propellers. However, while generators do impact the total fuel consumption and, consequently, the MTOW, their influence is small compared to other key parameters in the system.

## 4 Conclusions

This work explores the complex dynamics between technological advancements and the operational capabilities of hybrid-electric aircraft, providing key insights pivotal to the aviation industry’s shift toward electrification. By analyzing a 19-passenger commuter aircraft equipped with a series/parallel partial hybrid-electric powertrain, the research evaluates the influence of technological uncertainties on aircraft performance across two projected EIS scenarios—2030 and 2040.

The impact of inherit performance uncertainty due to the technology projections required to design electrified aircraft is analyzed. A novel two-phase methodology is introduced, beginning with an aircraft design optimization to freeze the aircraft configuration based on a nominal technology scenario. This initial step is succeeded by Monte Carlo simulations, which are applied to the optimized aircraft design to further assess its operational viability. To navigate the complexities of large-scale simulations more efficiently, surrogate models are employed, utilizing deep neural networks. These surrogate models are pivotal in overcoming the computational hurdles associated with selecting iterative sample sizes during uncertainty and sensitivity analysis. This ensures a thorough and efficient exploration of the potential impacts of technological advancements on aircraft performance.

The study achieves remarkable precision in forecasting key performance metrics, with the maximum RMSE recorded at 3.79 kWh for the Block Total Energy consumption, the objective of the optimization. Furthermore, the surrogate model outputs demonstrate good accuracy, as evidenced by a coefficient of determination ($R2$) exceeding 0.99 across all outputs. This high level of accuracy facilitates the efficient execution of an extensive number of simulations, resulting in a significant speed-up factor of $5\xd7102$.

The methodology is applied to the specified 19-passenger commuter aircraft, utilizing different distribution types (uniform, triangular, and truncated normal) for technological parameters, which unveil varying degrees of uncertainty. These uncertainties are a reflection of the current state of technology and its anticipated progression, emphasizing the importance of evaluating how different assumptions can influence the results of the study. The findings indicate that the uniform distribution presents the greatest uncertainty regarding the aircraft’s fuel and energy consumption.

Upon examining two projected EIS timeframes, the analysis shows a slightly better performance in energy consumption reduction for EIS 2040 (15.7%) compared to EIS 2030 (13.6%), but with increased uncertainty due to higher variances in the distributions. Nonetheless, the hybrid-electric aircraft consistently demonstrated energy consumption reductions in all scenarios when compared to conventional aircraft. Even under worst-case conditions, the performance of the hybrid-electric model remained superior to that of the conventional aircraft. Notably, the benefits were somewhat diminished, with reductions in energy consumption being 8.7% for EIS 2030 and 9.5% for EIS 2040. These findings confirm the advantages of electrified propulsion across the range of technologies investigated.

In both EIS timeframes analyzed, the specific energy density of the battery emerged as the most significant factor influencing outcomes. The findings highlight the pivotal importance of advancements in battery and motor technologies. This emphasizes the need for ongoing refinement in estimating these technological parameters and their probability distributions. Enhancing the accuracy of these estimates is vital for reducing uncertainties in predictive models, thereby enabling more accurate projections of aircraft performance and its environmental impacts.

In conclusion, this work significantly enriches the comprehension of the relationship between technological advancements and hybrid-electric aircraft performance. It emphasizes the crucial contributions of Monte Carlo simulations and surrogate modeling in obtaining these insights. By addressing and quantifying the associated impacts, this work makes a valuable contribution to the ongoing efforts toward aviation electrification. The detailed methodology discussed provides a framework for navigating technological risks, ultimately contributing in the aviation sector’s path toward sustainability.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

- AC =
aircraft

- DOE =
design of experiments

- DNN =
deep neural network

- EIS =
entry-into-service

- M/S =
maximum to sum ratio

- MSE =
mean squared error

- MTOW =
maximum take-off weight

- PMAD =
power management and distribution

- QoIs =
quantities of interest

- RAM =
regional air mobility

- RMSE =
root mean squared error

- TOW =
take-off weight

- $Deprop$ =
E-propeller diameter (m)

- $Dprop$ =
propeller diameter (m)

- $Eel$ =
electrical block energy (kWh)

- $Etot$ =
total block energy (kWh)

- $FN$ =
thrust (kN)

- $LMSE$ =
mean squared error loss

- $mAC$ =
aircraft mass (kg)

- $Neprop$ =
propeller speed (rpm)

- $Nprop$ =
E-propeller speed (rpm)

- $Pbat$ =
power delivered by battery (kW)

- $Pes$ =
power delivered to e-propeller (kW)

- $PGT,off$ =
engines’ power off-take (kW)

- $Ps$ =
total power delivered to propulsive devices (kW)

- $P/D2$ =
propeller loading ($kW/m2$)

- $PEM*$ =
motor rated power (kW)

- $PGEN*$ =
generator rated power (kW)

- $PGT*$ =
engine rated power (kW)

- $RMTOW$ =
maximum take-off weight residual (kg)

- $RTOW$ =
take-off weight residual (kg)

- $R2$ =
coefficient of determination

- $SOCloiter$ =
state of charge - end of loiter

- $Sref$ =
wing area ($m2$)

- $Vtip$ =
propeller tip speed ($m/s$)

- $Wbat$ =
battery weight (kg)

- $\Delta $ =
relative to conventional aircraft

- $\xi EM$ =
motor throttle

- $\xi GT$ =
engine throttle

- $\varphi $ =
shaft power ratio

- $\Phi $ =
supply power ratio