Optimizing the battery charging and swapping infrastructure for electric short-haul aircraft—The case of electric flight in Norway

Recent advances in battery technology have opened the possibility for short-haul electric flight. This is particularly attractive for commuter airlines that operate in remote regions such as archipelagos or Nordic fjords where the geography impedes other means of transportation. In this paper we address the question of how to optimize the charging infrastructure (charging power, spare batteries) for an airline when considering a battery swapping system. Our analysis considers the expenditures needed for (i) the significant charging power requirements, (ii) spare aircraft batteries, (iii) the used electricity


Introduction
To limit anthropogenic emissions, the aviation industry targets climate-neutrality by 2050 (UNFCCC, 2021).Opposed to the automotive sector, where electric vehicles are now fully rolled out across the globe by major manufacturers and hold a 10% share of total sales worldwide (Paoli et al., 2022) with up to 80% in Norway (Klesty, 2022), the low energy density of batteries has prevented a similar transition in the aviation sector.Emission reductions in aviation have for now been focused on e.g.improving operations by electric taxiing (Soltani et al., 2020;Guo et al., 2014) or electric urban air mobility (Garrow et al., 2021;Shao et al., 2021).Nevertheless, battery technologies have greatly developed over the last ten years: gravimetric energy density has almost tripled (Balogun et al., 2018), volumetric energy density has increased eightfold (Muralidharan et al., 2022) and prices per kWh have decreased tenfold (Ziegler and Trancik, 2021).As such, battery technology is now at the level where it can be used for small electric aircraft (Holland, 2021;Taraldsen, 2021;Taylor-Marriott, 2022).
Electric aircraft (e-AC) are becoming an alternative for high frequency commuter airlines operating short-haul flights in remote areas with low passenger volumes (Justin et al., 2022).By using electric motors, which use less expensive energy and require less • We propose a recourse model that determines an optimal battery charging infrastructure for a network of airports, assuming a battery swap system.This recourse model is necessary to account for the variability of the flight schedule throughout the year when optimizing the charging infrastructure.This is in contrast to existing studies, which optimize the infrastructure considering a single (representative or peak) day of operations.• Our proposed model minimizes the sum of the infrastructure acquisition costs (charging capacity and spare batteries), and the operational costs (electricity costs and flight delays due to battery charging).• We illustrate our model for a network of 7 hub and 36 regional airports in Norway, using an existing flight schedule from a year of operations.We also consider an already existing configuration of an electric aircraft (Eviation Alice) designed for short-range flights.
The remainder of the paper is structured as follows.In Section 2, we discuss literature on scheduling operations at-and sizing of-battery swapping stations, as well as e-AC charging infrastructure.In Section 3, we describe the problem of e-AC charging infrastructure management.In Section 4, the recourse model for e-AC charging infrastructure management is introduced.In Section 5, we illustrate the performance and results of our model in a case study for a regional carrier operating electric aircraft in Norway.In Section 6 we quantify the advantage gained by our recourse model which accounts for an entire year of operations, versus an optimization model which considers only a single day (peak-traffic or median-traffic day).Concluding remarks and future research directions are given in Section 7.

Prior work and contributions
The concept of battery swaps systems (BSSs) as an alternative to plug-in charging has been proposed as a means of reducing recharge times, to protect the electricity grid against high and unpredictable demand, and to limit battery degradation (Sarker et al., 2015;Sultana et al., 2018).BSSs have mainly been studied in the context of electric cars, electrified public transport busses and delivery drones (see e.g.Schneider et al. (2018), Ayad et al. (2021), and Kwizera and Nurre (2018)).Two research themes regarding BSSs are relevant for this study: the sizing of the BSS infrastructure, and the scheduling of charging of the batteries.
In the past years, several studies have addressed the operations' scheduling of BSSs, which considers determining when to recharge the batteries at the BSS to ensure a sufficient stock.Worley and Klabjan (2011) propose a model to optimize the charging schedule at a BSS for day-ahead planning, while considering a fixed and predetermined demand for full batteries.They formulate the problem as a mixed-integer linear program with the objective of maximizing the revenue by supplying full batteries to customers while minimizing the electricity costs.This model is expended by Nurre et al. (2014), who add Vehicle-to-grid (V2G) charging as a possibility for increasing BSS revenue, and by Park et al. (2017), who study the minimization of waiting time for charged batteries.All of these studies, however, assumed a known demand for batteries before optimization.Sarker et al. (2015) were the first to recognize the uncertainty in the demand of a BSS, proposing a robust optimization approach to ensure a sufficient supply of fully charged batteries.However, this study was limited to using fixed charging times, as opposed to a state-of-charge dependent one.Additional studies have focused on minimizing battery degradation (Wu et al., 2017;Asadi and Nurre Pinkley, 2021;Kwizera and Nurre, 2018).In this paper, we assume that the flight schedule is known and consider deterministic planning like Worley and Klabjan (2011) and Park et al. (2017).We expand upon the previous work done by considering partial charging, and our objective also considers the minimization of the waiting time for a full battery, apart from the electricity costs.Inextricably linked with the optimization of the battery recharge schedule is the problem of recharge infrastructure sizing: determining how many batteries can be charged simultaneously and how many spare batteries to keep in stock at the station.Worley and Klabjan (2011) combine a day-ahead scheduling of a BSS with determining the required spare battery inventory.However, batteries cannot always be acquired a single day ahead, and are hence not a flexible asset.As such, there is a trade-off between making long-term investments in infrastructure (spare batteries and chargers), and short-term operational expenses related to the charging schedule (electricity costs, battery degradation, etc.).This is the reason why the two problems are not solved simultaneously in more recent studies.This two-phase approach to the problem was first proposed by Schneider et al. (2018), who minimize the sum of the investment costs for batteries and chargers and the costs of operating the BSS (customers served against electricity cost).Customer arrival times are assumed to be exponentially distributed.Using a Monte-Carlo Dynamic Programming approach, a near-optimal policy for charge scheduling was developed.Sun et al. (2018) consider a similar problem, but add a minimum level-of-service for the customers, while only considering fixed charging times.Later, Sun et al. (2019) expand the problem to a three-phase approach where the number of required chargers and batteries is determines in two phases.Other studies focus on combining the sizing of BSSs with finding an optimal location (see Mak et al. (2013) or Liu et al. (2016)) or simultaneously routing the vehicles which use the BSS (see Ayad et al. (2021)).For an extensive literature overview of optimization models for BSSs, we refer to Zhan et al. (2022).In this study, we consider a two-phase approach (sizing the charging capacity and spare batteries in the first phase, scheduling the recharging operations in the second), similar to Schneider et al. (2018), Sun et al. (2018).We extend these approaches in by considering a state-of-charge dependent time of charge for the batteries.We also allow for preemptive battery charging during the second phase.
There are a number of studies where optimization of battery charging operations is performed in the context of electric aircraft (e-AC).Justin et al. (2020) consider the minimization of the required investment costs of the infrastructure (chargers and spare batteries) in order to support a BSS for small e-AC.In their analysis, flight schedules and battery recharge times are assumed to be known beforehand.They apply their models for Mukuele Airlines (Hawaii) and Cape Air (New England).Salucci et al. (2019) perform a similar study, considering variable electricity prices throughout the day, but assume identical flight duration and battery recharge times (similar to Nurre et al. (2014)).This work was further expanded by Trainelli et al. (2021), who also considers the procurement and routing of a fleet of e-AC from a hub airport.They minimize the sum of the acquisition costs of the electric aircraft, charging points and spare batteries, and the electricity costs.A similar problem is studied by Mitici et al. (2022), where a fixed electricity price is assumed.A hierarchical cost function is assumed to solve the problem more efficiently (first for aircraft fleet sizing and routing, then for charging infrastructure sizing and scheduling).Guo et al. (2020) also consider the availability of electricity based on renewable energy sources.Finally, the sizing of aircraft charging infrastructure at the gates/apron has been studied by Doctor et al. (2022).
None of these studies have separated the infrastructure sizing (number of aircraft, chargers, spare batteries) from the scheduling of operations.As such, these solutions are optimized for one day of operations, instead of for the (predicted) traffic throughout the entire year.A two-phase approach, such as in Schneider et al. (2018) or Sun et al. (2018), which takes demand seasonality into account has been shown to outperform such a single-phase approach.We introduce a tactical-phase battery recharge scheduling model, an extension of the work of Justin et al. (2020), But opposed to the previous literature on charging infrastructure for electric aircraft, we assume that the infrastructure size is predetermined and cannot be optimized simultaneously.For this, we introduce an infrastructure sizing model which accounts for demand variation throughout the year.This inclusion is the main contribution of our paper.

Electric aircraft operations with battery swapping
We consider an airline operating a fleet of short-haul electric aircraft, each equipped with one battery.Let  and  denote the set of regional and hub airports, respectively, where these electric aircraft fly between.Aircraft can recharge their batteries at all airports during turnaround, but this is too short to fully recharge it.In order to quickly replenish the battery, the aircraft can swap it for a fully charged one during turnaround.We assume that only the hub airports have the infrastructure needed to swap the batteries of the electric aircraft.We also assume that a battery is swapped with a new, fully-charged battery every time an aircraft visits a hub airport.Fig. 1 shows an example of operations of one electric aircraft.At the start of the day, the aircraft flies from hub H1 to H2 via regional airports R1 and R2.At R2, the battery is partially recharged.This aircraft swaps its battery at H2. From H2, the aircraft flies to H3, where it swaps its battery again.

Electric aircraft flight schedules
Let  denote a set of days of an entire year during which the airline operates.A flight schedule for day  ∈  consists of a list of arrival and departure times for an entire day of operations.Let   denote the time interval during one day when the aircraft fly to and from a set of airports (hubs and regional airports

Fig. 2.
Overview of the battery swapping and recharging process at an airport.The battery charging capacity at the airport is  ℎ = 4 ⋅   (at most four batteries can be charged at max power), and there at 12 spare batteries (three currently being charged, 13 waiting at the depot, minus four which should depart on the four parked aircraft).

Battery swaps
We assume that each hub airport is equipped with one charging station.Upon arrival at a hub airport, the used battery of an electric aircraft is swapped with a new fully charged battery.Let  ℎ  denote the set of batteries which arrive with flights  ℎ  .For each flight  ∈  ℎ  , let   ∈  ℎ  be the battery with which it arrives.Let   denote the time it takes to bring the used battery to the charging station.The same amount of time   is assumed to be required to bring a fully charged battery from the charging station to an aircraft.

Battery charging station
Let  ℎ denote the charging capacity at hub airport ℎ, i.e., the total power with which batteries can be charged simultaneously at the airport.Let  ℎ ⋅   ∈ R + denote the daily cost to provide this charging power at hub airport ℎ.Let   ℎ ∈ N denote the number of spare batteries available at a hub airport ℎ.Let   ∈ R + be the daily cost of having one spare battery at a hub airport.
We assume that batteries can be charged preemptively.We also assume that the price of electricity varies throughout the day and is given by a function   ∶   → R + of time.
Fig. 2 shows an example of a hub airport and its charging infrastructure, i.e., with charging power  ℎ = 4 ⋅   and   ℎ = 12 batteries available at this airport for swap, out of which 3 batteries are charging and 13 batteries are idle in the inventory, from which there are 4 which have to depart with an aircraft currently parked at the airport.Each aircraft needs to depart from the airport with a fully charged battery.

Battery charging station sizing
Given a hub airport ℎ ∈  and set of flight schedules during days , the main objective of our study is to determine a suitable battery charging infrastructure size ( ℎ ,   ℎ ).This problem is solved once for each hub airport before the start of , to allow time for charging station construction.The infrastructure is optimized to minimize the sum of the capital expenditures ( ℎ ⋅   +   ℎ ⋅   ) and average charging schedule operating costs over  are as low as possible.The value of the latter for each  ∈  is given by the battery recharge schedule optimization (Section 3.2), which is used as a subroutine.

Swapped aircraft batteries recharge scheduling
Given a certain charging infrastructure size (charging capacity  ℎ and number of spare batteries   ℎ ) and a flight schedule during day  ∈  at a hub ℎ ∈ , we aim to determine a charging schedule for batteries such that the charging schedule operating cost (electricity cost and flight delay cost) is minimized.For this, we need to determine which battery departs with which aircraft, and when to charge these batteries during the day.In the case that the infrastructure is not sufficiently large to ensure that flights depart on time, a penalty cost is incurred.This is a convex piecewise linear function of time, with breakpoints at   .Each time the flight delay is larger then some  ∈   , a cost   () ≥ 0 is incurred per unit of delay time.We assume that these delays are small and thus are assumed to be absorbed on route.
An overview of the model, with the interaction between the Battery charging station sizing problem and the Swapped aircraft batteries recharge scheduling problem, can be seen in Fig. 3.The latter is used as a subroutine in order to evaluate the charging schedule operation cost for a given infrastructure size on a single day.

A recourse model to optimize the charging infrastructure for swapped batteries
We propose a novel recourse model which manages the swapping and charging process for an airline operating a fleet of electric aircraft.This model is able to minimize the infrastructure and charging costs not just for a single day, but for an entire year of operations.consisting of the battery charging station sizing problem (long-term planning phase) and the swapped aircraft batteries recharge scheduling problem (medium-term planning phase).First, we propose a MILP which manages the charging schedule of the aircraft batteries, given a known flight schedule and charging infrastructure.This problem can be solved once a flight schedule Charging infrastructure cost at hub ℎ is made known, weeks before the day of operations.Second, we propose a master-problem to determine the most cost-effective charging infrastructure over an entire year of operations.This uses the swapped aircraft batteries recharge scheduling model as a subroutine.This problem is solved years in advance of the day of operations.
An overview of all used notation can be found in Table 1.

Tactical planning: Swapped aircraft batteries recharge scheduling problem
We first propose the swapped aircraft batteries recharging model, which considers a single day  ∈  and hub ℎ ∈ .The algorithm decides when each battery which arrives at the airport is charged, and to which outbound flight it is assigned.If, because of charging infrastructure limitations, batteries cannot be charged on time, outbound flights must wait until a fully charged battery is available.This formulation minimizes the sum of the cost for electricity used to charge the batteries and the cost of flight delays.
Let us first introduce the following notation for a hub ℎ ∈ .First, we discretize the day   in a set of intervals with length .Let    = min(  ) and    = max(  ), the start and end of the day, respectively.As such we obtain Second, we define the cumulative demand for charged batteries at time  ∈  *  as: This gives the total number of batteries which should have been charged by  in order to allow the flights to be performed on schedule.Third, a battery can only be charged once it has arrived at the charging station, and for each time  ∈  *  , the set of these batteries is given by  ℎ  = { ∈  ℎ  ∶  ≤    }.At the start of   , a number of aircraft may already be present at the airport, which stayed there overnight.The batteries with which these arrived are assumed to have been fully recharged by the start of   .Let   denote the number of these batteries.

Decision variables
We consider the decision variable

Mixed-integer linear problem formulation
We consider the following MILP to manage the recharging schedule at hub ℎ, given a charging infrastructure ( ℎ ,   ℎ ) and a flight schedule on day , which minimizes the operational costs  ℎ ( ℎ ,   ℎ ): The operational costs (Eq.( 1a)) are given by the sum of the electricity costs,  ) , for each interval [,  + ] with  ∈  *  .Constraints (1b) ensure that all batteries which arrive at the airport are charged by the end of the day, ensuring a supply of spare batteries on the next day.Constraints (1c) ensure that during each interval, batteries are not charged longer than the length of the interval, whereas Constraints (1d) ensure that the total charging capacity is not exceeded.Whether or not a battery is ready to be used on an outbound flight by  is determined in Constraints (1e), which force   to be 0 as long as battery  is not fully charged.Finally, Constraints (1f) determine the cumulative supply of spare batteries.

Strategic planning: battery charging station sizing problem
Second, we propose a recourse model which optimizes the charging infrastructure size, given the flight schedules which reflect the traffic demand variation during an entire year of operations .This functions as the master problem of the recourse model, and is solved only once for the entire year.It determines the expected operational cost of a charging infrastructure by using the optimization model from Section 4.1 as a subroutine.These operational costs are added to the capital expenditures, i.e., the cost of acquiring and maintaining chargers and spare batteries to obtain the total costs, which we aim to minimize.

Decision variables
We consider the decision variables  ℎ ∈ R + and   ℎ ∈ N, the charging capacity and the number of spare batteries at hub airport ℎ, respectively.

Recourse model formulation
We consider the following MILP to determine the suitable charging infrastructure size at a hub, given the distribution of flight schedules , which minimizes the total cost  ℎ : In this formulation,  ℎ is the sum of the daily cost of the charging capacity,    ℎ , the spare batteries     ℎ , and the average operational costs ∑ ∈  ℎ ( ℎ ,   ℎ )∕||, which is obtained from the tactical planning problem of Section 4.1 for each day  ∈ .

Simulated annealing algorithm
The value of ∑ ∈  ℎ ( ℎ ,   ℎ ) cannot be evaluated trivially, hence regular MILP solution algorithms are unsuitable to solve this problem.As ∑ ∈  ℎ ( ℎ ,   ℎ ) is also non-convex in  ℎ and   ℎ , we use Simulated Annealing to find an approximate solution.The approach we use to optimize the airport charging infrastructure is given in Algorithm 1.
The algorithm first obtains an initial solution (lines 2-9).The expected operating costs for each solution are determined.After the initial solution is found, the algorithm iterates towards better solutions by selecting a solution in the neighborhood of the current solution (line 11), evaluating the expected operating costs only for the current solution (line 12).The new solution is accepted if it is an improvement over the current one (line 14), or with a probability less than one if it is not (line 16, to avoid being stuck in local optima).This process continues until a stopping criterion is met.In this section, we apply the model presented in Section 4 to the short-haul flight network of Norwegian Widerøe Airlines, using flight schedule based on the performed flights during October 2021-October 2022.
We use the case of Widerøe Airlines because Norway has a strong position for early implementations of electric aircraft on commercial flights.Due to the challenging terrain, air transport is often the only viable means of travel.This results in a market with a lot of low-passenger short-haul flights: around 77% of all domestic flights in the country are under 400 km (Baerheim et al., 2022).In addition, Norway's electricity is for 98% produced with renewable resources (Olje-og energidepartementet, 2016) such that the electric aircraft can be operated sustainably.
The experiment was performed by implementing the algorithm with Gurobi Optimizer 9.1, on an Intel Core i7-10610U with 8 GB of RAM.

Short-haul network of Widerøe Airlines
Widerøe Airlines is the largest commuter operator in Scandinavia (Baerheim et al., 2022).It operates a fleet of 23 De Havilland Canada Dash 8-100 aircraft (hereafter: Dash 8) for short-haul flights within and to Norway.These have a capacity of 40 passengers or a payload of 4500 kg.These aircraft account for the majority of Widerøe's fleet.We propose a scenario in which this fleet is replaced with electric aircraft, while keeping to the existing flight schedule as much as possible.We have imported the flight data for the Dash 8 between 28 October 2021 and 27 October 2022, using data from Flightradar24 (2022) for each aircraft of the Dash 8 fleet.We have used the actual time of departure, actual landing time, origin and departure of each flight.As such, cancelled flights have not been included in the data.The network spanned by these aircraft consists of 43 airports and 56,000 flights; it is shown in Fig. 4.
The used flight schedules in this case study are based on the analyzed historical flight data.We assume that all itineraries between hubs (possibly via regional airports) which were previously performed by the Dash 8 and are within range of the electric aircraft, are performed by the e-AC.The e-AC start each journey from a hub airport with a fully charged battery.If there are any stopovers at non-hub airports, it will recharge its battery during turnaround.This is done using the existing ground power infrastructure.Last, if there are two subsequent flights for an aircraft, from A to B and from B to C, and the first flight cannot be performed on time (due to battery/charging limitations), then the aircraft directly flies from A to C. Flights A to B and B to C are not included in the schedule.We use these as our (365) flight schedules.
Assuming a range of 610 km (see the next subsection on the e-AC model) approximately 52,000 of the 56,000 flights which we have analyzed can be performed by the electric aircraft.A comparison of the number of flights visiting the 20 busiest airports can be seen in Fig. 5.We propose to use the seven most visited airports in the network as hubs for the electric aircraft fleet, where batteries can be swapped.These airports are in the towns of Bodø (BOO, 15,000 flights annually), Tromsø(TOS, 10,000 flights), Trondheim  (TRD, 10,000 flights), Oslo (OSL, 9000 flights), Bergen (BGO, 5000 flights), Hammerfest (HFT, 5000 flights) and Vadsø(VDS, 4000 flights).Table 2 shows the distribution of the number of daily flights on each of the selected hubs.Even though there is a significant variance in the number of daily flights, there is no seasonal difference for this fleet.This is because the Dash-8 100 aircraft are mainly used for commuting and to deliver cargo, and only to a lesser degree for tourism.

Specifications of the electric aircraft
We propose a scenario where the Dash-8 aircraft are replaced by an e-AC with the specifications of the Eviation Alice (Eviation, 2022).Currently in the certifying phase, the Alice is able to transport ten passengers or a payload of 1150 kg.
The Alice is currently able to fly a distance of 250 nm (approximately 480 km).Assuming a continuation of current battery development (Ziegler and Trancik, 2021), this will increase to 610 km by 2025.It is equipped with a 820 kWh battery which requires slightly over 4 h for a full recharge.We assume that the charging power available at hubs and regional airports is the same.The cruising speed of 260 kts is comparable to the 280 kts of the Dash 8-100, and as such we assume the flight duration is unchanged.We assume that 7% of the battery capacity is required for take-off (Hepperle, 2012).All used characteristics of the  aircraft can be found in Table 3.The electricity cost is derived from historical data (Nordpool Group, 2022), and a resolution of  = 15 min has been used.Flight delay costs are derived from European industry data (Cook and Graham, 2015), which takes flight personnel, maintenance, fuel and passenger costs into account.We use the tactical costs of the ATR-43 as reference and scale them to account for the difference in size (10 passengers instead of 42) and inflation (20%).Using   = {00:00, 00:45, 01:15}, we have found   (0:00) = e9.05,  (0:45) = e10.90, and   (1:15) = e10.95.The original data and this approximation are shown in Fig. 6.
Because of this, the payload is about a quarter of the size of the payload of Dash 8.In this paper we aim to maintain the original flight schedule without optimizing for the difference in the payload.This would required adding more flights to the schedule, and is addressed in Section 5.2.

Results-optimizing the charging infrastructure for widerøe airlines
We apply the optimization framework from Section 4 to the flight network of Widerøe Airlines and the e-AC model to obtain the most cost-effective charging infrastructure at the 7 hub airports.We first discuss the results for the largest hub, Bodø, in detail.After this, we shall present the results for all hubs.Bergen, Bodø, Hammerfest, Oslo, Tromsø, Trondheim and Vadsø Table 5 shows the optimized infrastructure, together with the average operations energy consumption, delays and costs for all seven hubs.The daily costs range from e970 at Hammerfest Lufthavn (HFT) up to e2890 at Bodø Lufthavn, accumulating to a network total of e11,793 per day.It should be kept in mind that this does not account for the decrease in payload caused by using 10-seater aircraft instead of the 37-seater Dash-8.The table shows an approximate marginal cost of e100 per flight, which is independent of the difference in payload, of which the spare battery supply forms the biggest share.Furthermore, the delay costs are not proportional to the number of daily flights.For example, the average delay costs at Vadsøof e142 are larger than the costs at Bodø, of e140, which accommodates three times the number of flights.

Sensitivity analysis: battery capacity and daily number of flights
We perform a sensitivity analysis to assess the impact of the technological and economical assumptions we have made on the results.This is done for the development of battery energy density and the daily number of flights.We evaluate the impact of these on the optimal infrastructure, as well as the computational requirements to solve the problems.We show the results for Bodø Airport.
Table 6 shows the impact of the aircraft battery performance.We assume that the energy density of the batteries increases by an annual rate of 8% (Ziegler and Trancik, 2021), resulting in a 610 km flight range.We compare this rate with a pessimistic scenario of 0% (range of 480 km) annual increase of the energy density, 4% (range of 540 km), 12% (range of 685 km) and 16% annual increase (range of 775 km).Table 6 shows the required infrastructure, investment, as well as the total optimization time and average optimization time per day subroutine.The results show that above 540 km, range no longer is a limiting factor on the number of flights, stabilizing at around 29.6-29.8.Above this number, the energy requirements increase linearly, affecting the total costs.Only in the most pessimistic scenario (range of 480 km), the range does impede the flight schedule, reducing the number of daily flights by a quarter.In this case, the entire infrastructure is reduced in size and the costs decrease by a third.Finally, in all cases, the computational time is limited to under 30 min for the entire algorithm, and under half a second for each day of operations.
Table 7 shows the impact of the number of electric flights performed on the infrastructure and costs.We assumed that the original schedule is adhered as much as possible, but this does decrease the total payload which can be carried to about 25% of the original schedule.We compare this with 50%, 75% and 100% by doubling, tripling, and quadrupling the number of times each flight is performed.Table 7 shows the average number of flights at Bodø, the infrastructure requirements and costs, and the optimization time.
The results show a sub-linear increase of the costs as a function of number of flights.The infrastructure does approximately linear with the number of flights, but the delays remain fairly limited.The computational time increased from 26 to 201 min for the infrastructure optimization and from 0.46 to 3.5 s for each recharge scheduling optimization.
6. Quantifying the benefit of recourse optimization considering air traffic for an entire year vs. infrastructure optimization considering only one day of traffic

Infrastructure optimization considering only a peak-day or a median-day of air traffic
In order to assess the performance of the recourse model proposed in Section 4.2, we consider two alternative battery charging infrastructure optimization approaches.These approaches optimize the infrastructure using only the flight schedule from a single day of operations.For this, we use either the peak traffic day (PD) or a median traffic day (MD), according to the total number of flights.Using a peak-day flight schedule corresponds to the desire to obtain an infrastructure which is able to cope with the most congested days, although this expensive infrastructure may be overly large during less busy days.Using a median day as reference leads to obtaining an infrastructure which performs well on most days of operations but may be insufficient during congested days.
The two alternative infrastructure optimization approaches use a version of the MILP from Section 4.1 modified in two respects.First, the charging capacity  ℎ and the number of spare batteries   ℎ are model variables instead of input parameters.Second, their associated costs (  and   ) are included in the objective function.As such, the following model is considered: For our analysis,  ( ) = e2890, compared to   = e4730 (+58%) for the () solution and  ( ) = e3140 (+6%) for the ( ) solution.Table 8 shows a breakdown of the costs, split up into charging capacity, spare battery, electricity and delay costs.Compared with the ( ) solution, the ( ) solution trades charging capacity for an extra battery and a significantly larger infrastructure investment, but reduces the expected delays and is able to profit from slightly lower electricity costs.On the other hand, the () solution consists of a relatively small charging capacity, but causes significant flight delays (on average a total of 2:36 h daily).
Last, Fig. 9 shows the cost of the three infrastructure solutions depending on the number of flights on the days of operation.Each point represents one of the 356 days of operations, which are sorted by number of flights.Generally speaking, the daily costs increase with the number of flights.Additionally, the () and ( ) infrastructures are stressed more for a large number of flights then the ( ) infrastructure.
Note that for a small number of flights, not the () but the ( ) solution has the lowest total costs, aided by the flexibility of a larger charging capacity.For busier days, the ( ) solution starts to outperform the other two, with the costs of the ( ) and () solutions increasing to e5000 and e24,000 (not on chart), respectively.

Conclusions
This paper proposes a two-phase recourse model optimization framework for battery swap and recharge operations for an airline operating a fleet of short-haul electric aircraft.This framework integrates the scheduling of swapped battery recharges with the sizing of the recharge infrastructure, without merging them completely to account for the fact that the infrastructure size cannot be altered daily.The objective for the former is to minimize the charging station operational costs, comprised of the electricity and the caused delay cost.For the latter, the objective is to minimize the sum of the infrastructure acquisition cost and the average operational cost throughout the year.The scheduling problem, posed as a mixed-integer linear program, is used as a subroutine for the infrastructure sizing problem, which is solved using a simulated annealing algorithm.
Our optimization framework is applied in a case study for Norway's Widerøe Airlines, where the airline's fleet of DHC Dash 8-100 aircraft is replaced with a fleet of electric aircraft based on Eviation Alice's specifications.We use flight data from 28 October 2021 and 27 October 2022 and keep to the original schedule as much as possible.It is assumed that the aircraft will be able to swap their batteries at the seven most visited airports in the network, such that approximately 52,000 out of the 56,000 analyzed flights can be performed.The results show that a combined power supply consists of 4412 kW and that 25 spare batteries are required, such

Fig. 1 .
Fig. 1.Example of a day of operations for an electric aircraft.

Fig. 3 .
Fig. 3. Recourse model for sizing the charging infrastructure and recharge scheduling for e-AC battery swaps.Sizing the charging infrastructure is a strategic planning problem, whereas creating a recharge schedule is a tactical planning problem.

Fig. 4 .
Fig. 4. Network of the Dash 8-100 of Widerøe airlines, flown between the 28th of October 2021 and the 27 of October 2022.

Fig. 5 .
Fig. 5. Histogram of number of annual flights at the twenty most visited hubs of Widerøe.Flight data between October 28, 2021 and October 27, 2022 is used.

Fig. 6 .
Fig. 6.Actual and piece-wise linear approximation of the flight delay costs.The actual costs are based on European reference values for the ATR-43(Cook and Graham, 2015).

Fig. 7 .
Fig. 7. Optimal recharging schedule of the swapped batteries at Bodø Lufthavn on the peak traffic day (August 25, 2022), using  ℎ = 900 kW and   ℎ = 6.Each bar represents a battery, which starts and ends when it arrives at and departs from the charging station, respectively.The color of the bar indicates the current battery state-of-charge.Each arrow shows the scheduled and actual departure time of the flights to which the batteries are assigned.
).Let  ℎ  and  ℎ  denote the set of flights operated by electric aircraft that arrive and depart at hub airport ℎ ∈  on day  ∈ , respectively.Let    ∈   denote the arrival time of flight  ∈  ℎ  and let ∈   denote the departure time of flight  ∈  ℎ  .S. van Oosterom and M. Mitici

Table 1
Overview of used nomenclature.

Table 2
Daily flight statistics at the considered hubs of Widerøe.The average, median, standard deviation and maximum of the number of daily flights is given.Flight data between October 28, 2021 and October 27, 2022 is used.

Table 4
Infrastructure (charging capacity [ ℎ ] and spare batteries [  ℎ ]), average operational characteristics and costs (electricity and caused hours of delay) at Bodø Lufthavn.

Table 5
Infrastructure and expected operational cost for the considered hubs in the network.The first column lists the airport codes and average daily number of flights.Here,   ℎ and   ℎ give the average daily electricity and delay cost.

Table 6
Optimal charging infrastructure at Bodø Airport given various battery energy densities.Both the computational time of the infrastructure optimization and the average per daily recharge scheduling optimization are given.The green case represents the nominal case considered in Section 5.1.1.

Table 7
Optimal charging infrastructure at Bodø Airport given various average number of flights per day.Both the computational time of the infrastructure optimization and the average per daily recharge scheduling optimization are given.The green case represents the nominal case considered in Section 5.1.1. min