A Sensitivity Analysis on the Range Equation of Hybrid-electric aircraft

This paper carries out a sensitivity analysis on the recently proposed hybrid-electric range equation [1]. The proposed hybrid-electric range equation is based on an efficiency-based definition of the degree of hybridization and the efficiencies of respective drivetrains on the range estimation of the hybrid-electric aircraft. The ATR 72 turbo-prop aircraft is chosen as the case study for the sensitivity analysis. The sensitivity analysis done in this paper shows the effects of parameters such as lift to drag ratio, efficiencies, energy densities, payload weight, etc. on the aircraft range. It was observed that variation in aircraft range due to each parameter was distinct and unique. The analysis also depicted the implications of the changes in the above-mentioned parameters from an aircraft designer’s viewpoint. The changes were carried out using the predictions for the year 2050. The sensitivity analysis performed in this work successfully narrowed down the parameters that had the maximum impact on the aircraft range.


Introduction
As part of the efforts towards the abatement of climate change, European Union has set aggressive targets to limit the impact the effect of aviation on the environment.Its Flightpath 2050 [2] strategy aims to reduce CO2, NOx and noise pollution substantially by the year 2050 while maintaining its social and economic benefits.Several green technologies are being researched so as to reduce emission levels.Hydrogen powered aircraft [3] and electric aircraft [4] are some of them.Each and every technology have their own advantages and drawbacks.Hydrogen powered aircraft would reduce the CO2 emissions substantially.However, a lot of infrastructure development has to be done for the production of green hydrogen and its storage.On the other hand, electric aircraft leaves the dependency on fuel behind and would produce no inflight emissions.However, the battery energy density is currently not high enough to make an electric aircraft feasible.
Hybrid-electric aircraft [5] based on a combination of battery and fuel power offers a mid-term solution until the other technologies are sufficiently advanced sufficiently.The hybrid-electric aircraft theory is still in its nascent stage and there are many research gaps to be filled.In the previous work [1] the authors attempted to fill a gap by proposing an efficiency-based hybrid-electric aircraft range equation.This paper produces a sensitivity analysis on this equation, with the aim to identify parameters that would benefit the aircraft range.To achieve this, the paper is organized in the following manner: Section 2 gives a brief overview of the derived efficiency-based hybrid-electric aircraft range equation.Section 3 introduces the aircraft being used as a baseline (ATR 72 [6]), on which this case study will be performed.Section 4 presents the sensitivity analysis using partial differentials identifying key parameters having impact on the range of the aircraft.Section 5 gives the sensitivity analysis using carpet plots on the key parameters.It also proposes a hybrid-electric ATR 72 aircraft based on forecasted values of these parameters for the year 2050.Finally, Section 6 provides a Discussion and Conclusion, highlighting the inferences and implications of the sensitivity analysis.

Hybrid electric range equations and its configurations
The hybrid-electric aircraft propulsion system [5] can be generally divided into -series and parallel configurations.Series configuration is the one in which the components are arranged back to back.The total power goes on to the exit component -the propeller.The power output is affected by the reduction due to efficiencies of the existing in between components.While a parallel configuration has separated mechanical and electrical paths and the combined power of both runs the propeller.The junction in both cases signifies the degree of hybridization (φ). Figure 1 denotes a simplified hybridelectric configuration which can be converted into series and parallel configurations.Table 1 provides the generalized efficiencies and their corresponding efficiencies according to the configurations.Table 1.Relation between efficiencies for series and parallel hybrid-electric configuration [1].
The authors propose a modified and improved aircraft range equation (1) which has been derived in [1] and takes efficiency of different components into consideration.
Where:   is the empty weight of the aircraft   is the energy density of the battery   is the payload weight of the aircraft   is the coefficient of lift   is the weight of battery   is the coefficient of drag   is the weight of the fuel g is the acceleration due to gravity   is the energy density of the fuel  , is the total energy contained in fuel and batteries It was found from our earlier work and from the research that a parallel hybrid-electric configuration is better than a series hybrid-electric one in the way that its more efficient and has less weight.Thus, in this paper the parallel configuration will be adopted for performing the sensitivity analysis.Using the relation of efficiencies from the table 1, we get the definition for a hybrid-electric aircraft in parallel configuration as follows: The paper will also consider the ATR 72 as the base aircraft on which the case study is implemented.

The ATR 72 aircraft as a case study
The ATR 72 is a regional turboprop aircraft manufactured by ATR aircraft company.Regional aircraft perform over 40% of all commercial flights worldwide.Around 3020 more regional aircraft [8] would be required in the future.Henceforth, electrifying this type of aircraft first would be a step in the right direction.Figure 2 gives a schematic of ATR 72 aircraft and Table 2  The succeeding section discusses the sensitivity analysis of equation ( 1) carried out by partial differentials.This helps in shortlisting parameters which have a great impact on the aircraft range.

Sensitivity Analysis using partial differentials
Sensitivity analysis can be defined as the method to find the uncertainty in the output of a mathematical model due to the uncertainty in different existing input parameters [8].The first method of carrying out the sensitivity analysis is using the partial differentials.The range equation ( 1) was differentiated with respect to a particular parameter.This was then multiplied with the change in the parameter value.This gives the change in the aircraft range due to the change in the parameter.This is shown in the equation ( 3).An example is also provided (equation ( 4)) which describes the procedure explained above.(4) A sensitivity analysis using partial differentials procedure was carried for various parameters such as lift to drag ratio, propulsive efficiency, energy density of battery, payload weight, energy density of fuel and gas turbine efficiency.The corresponding change in the range for various parameters in stated in the table 3. It was observed that the lift to drag ratio, propulsive efficiency and energy density of battery has the most impact on the aircraft range as compared to other.The aircraft range is most sensitive to these parameters.Therefore, these three parameters were shortlisted and the further analysis was carried out on these parameters.The next section gives an insight into the sensitivity analysis using carpet plots.The parameters narrowed down in this section are worked upon further and the analysis is carried out.A process to design a hybrid ATR 72 using the predictions of these parameters for the year 2050 is also carried out.

Sensitivity Analysis using carpet plots 5.1. Variation of lift to drag ratio
The sensitivity analysis was carried out using carpet plots.Lift to drag ratio for a body is defined as lift generated by a body moving through air divided by aerodynamic drag [10].For a powered aircraft, it is generally specified for a straight and level flight.This ratio is determined using computational fluid dynamics, computer simulations or experimentally using a wind tunnel.Historical data [10] was used to project lift to drag ratio values for upcoming decades which is shown in a plot (figure 3) between lift to drag ratio and years.The futuristic Boeing truss-braced wing [11] for X-66 experimental aircraft is expected to have long, thin wings stabilised by trusses, thus improving lift to drag ratio.Figure 4 gives resulting aircraft range obtained from hybrid-electric range equation after varying lift to drag ratio with degree of hybridization.The first change is from base model (point O) to point A. This will change φ from 0 to 0.5 but will impact payload capacity, emissions, etc. Points A and B are initial & final hybrid ATR 72 aircraft configuration (for φ = 0.5) with respect to prediction for year 2050.The change in the configuration from the point A to B is also depicted schematically (figure 5).The impact on the aircraft range due to the change in lift to drag ratio is also shown.The aircraft is modified to extends its range by 84 kms due to the increased lift to drag ratio from 17.352 to 18.813.The alteration in the aircraft layout is also described here.

Variation of propulsive efficiency
Propulsive efficiency [12] defines the effectiveness of an aircraft engine when it converts the energy stored in fuel into kinetic energy to accelerate the aircraft.The propulsive efficiency has been improved over the years by improving the bypass ratio (BPR), thereby increasing the size of an aircraft engine.However, there is a limit to which the size of an aircraft engine can be increased.There has been extensive research going on in improving the propulsive efficiency.The historical data [13] was used to make a prediction of propulsive efficiency values for the upcoming decades.This is shown in a plot (figure 6) between propulsive efficiency and the years.Figure 7 gives the resulting aircraft range obtained from the hybrid-electric range equation due to the variation of propulsive efficiency with degree of hybridization.The points C and D are the final hybrid ATR 72 aircraft configuration from the iteration in the previous section and the changed configuration with respect to the prediction for the year 2050 of the propulsive efficiency respectively.The change in the configuration from the point C to D is also depicted schematically (figure 8).This schematic continues from the final point B of the iteration in the previous section.The impact on the aircraft range due to the change in propulsive efficiency is also shown in figure 8.The aircraft range increases by 86 kms due to the improve propulsion efficiency from 0.8071 to 0.8701.

Variation of battery energy density
The battery energy density [14] is defined as the energy stored by batteries per kg of its weight.The current battery energy density is around 200-230 Wh/kg and there has been constant research going on in this field in order to enhance these values.New chemical combinations for batteries are being tested regularly which can improve the battery energy density further so that it can be used extensively in aircraft.The historical data [15] was used to make a projection of battery energy density values for the upcoming decades.This is shown in a plot (figure 9) between battery energy density and the years.Figure 10 gives the resulting aircraft range obtained from the hybrid-electric range equation due to the variation of battery energy density with degree of hybridization.The points E and F are the final hybrid ATR 72 aircraft configuration from the iteration in the previous section and the changed configuration with respect to the prediction for the year 2050 of the battery energy density respectively.The change in the configuration from the point E to F is also depicted schematically (figure 11).This schematic continues from the final point D of the iteration in the previous section.The impact on the aircraft range due to the change in battery energy density is also shown.The aircraft range changes by 368.2 kms due to the increase of the battery density from 228Wh/kg to 570Wh/kg predicted for the year 2050.The change in aircraft layout (addition of more batteries) is also shown.density.Initial thought was always that as a hybrid electric aircraft will require batteries, these would come at the cost of payload (its weight and space).However, in the analysis carried out in this paper it was shown that other parameters such as lift to drag ratio, propulsive efficiency and battery energy density would be more effective in increasing aircraft range and reaching very close to existing ATR 72 aircraft value.

Discussion and Conclusion
A sensitivity analysis was carried out on modified hybrid-electric range equation which was derived in a previous paper.The range equation contains various parameters such as efficiencies, lift to drag ratio, different weights (empty and payload), total input energy, energy densities of fuel and battery, etc.The analysis was carried out for the ATR 72 case study.By using partial differentials, it was found that lift to drag ratio, propulsive efficiency and battery energy density have the most impact on the aircraft range.Predictions of the variation of these parameters was made for the next 3-4 decades using the extrapolation of existing data.These predictions were then used to carry out the sensitivity analysis using carpet plots.A hybrid ATR 72 aircraft was also proposed with the changes using the above-mentioned predictions in order to achieve the range of the existing fully-fuelled ATR 72 aircraft.The new design was reached after adjusting parameters such as lift to drag ratio, propulsive efficiency and battery energy density and their impacts on aircraft range was observed.The changes were also depicted thematically.This shows that with careful adjustment of the parameters, the range of current ATR 72 aircraft can be achieved with the parameters extrapolated to the year 2050.This would also lead to saving in the aviation fuel consumption and reduction in emissions.

Figure 3 .
Figure 3. Prediction in variation of lift to drag ratio.

Figure 4 .
Figure 4. Sensitivity analysis using carpet plots for lift to drag ratio.

Figure 5 .
Figure 5. Proposed hybrid-electric ATR 72 aircraft based on predicted values of lift to drag ratio.

Figure 6 .
Figure 6.Prediction in variation of propulsive efficiency.

Figure 7 .
Figure 7. Sensitivity analysis using carpet plots for propulsive efficiency.

Figure 8 .
Figure 8. Proposed hybrid-electric ATR 72 aircraft based on predicted values of propulsive efficiency.

Figure 9 .
Figure 9. Prediction in variation of battery energy density.

Figure 10 .
Figure 10.Sensitivity analysis using carpet plots for battery energy density.

Figure 11 .
Figure 11.Proposed hybrid-electric ATR 72 aircraft based on predicted values of battery energydensity.Initial thought was always that as a hybrid electric aircraft will require batteries, these would come at the cost of payload (its weight and space).However, in the analysis carried out in this paper it was shown that other parameters such as lift to drag ratio, propulsive efficiency and battery energy density would be more effective in increasing aircraft range and reaching very close to existing ATR 72 aircraft value. 

Table 2 .
provides its basic specifications.Specifications of ATR 72 aircraft.

Table 3 .
Change in the aircraft range due to the change in different parameters using partial differentials.