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1.College of Air Traffic Management，Civil Aviation University of China，Tianjin 300300，P.R.China；2.Air Traffic Management Research Institute，Nanyang Technological University，637460，Singapore；3.School of Mechanical and Aerospace Engineering，Nanyang Technological University，639798，Singapore；4.Aerospace Division，Department of Physics，Technical University of Catalonia?Barcelona Tech，Catalonia 08860，Spain

Abstract: A method for formation flight trajectory optimization was established. This method aims at minimizing fuel consumption of a two-aircraft formation flight，without changing the original trajectory of the leader. Candidate flight pairs were selected from all international flights arriving at or departing from China in one day according to the requirement of the proposed method. Aircraft performance database Base of Aircraft Data（BADA）was employed in the trajectory computation. By assuming different fuel-saving percentages for the following aircraft，pre-flight plan trajectories of formation flight were optimized. The fuel consumption optimization effect under the influence of different trajectory optimization parameters was also analyzed. The results showed that the higher the fuel savings percentage，the longer the flight distance of formation flight，but the smaller the number of formation combinations that can be realized，which is limited by the aircraft performance. The following aircraft flying along the approximate actual flight trajectory can be benefited as well，and the optimal fuel-saving efficiency is related to the expected fuelsaving efficiency of formation flight.

Key words：formation flight；trajectory optimization；fuel-saving；Base of Aircraft Data（BADA）

0 Introduction

Birds take advantage of formation flight to save energy，which has been known for many years.Studies have been carried out applying formation flight to commercial flight aiming at improving trans?portation capacity and reducing greenhouse gas emissions. Simulation and experimental tests have showed that formation flight can reduce fuel con?sumption by up to 20%［1-5］. Similar results were ob?tained in flight tests as well［6-10］.

Research efforts have also been carried out ap?plying formation flight to trajectory optimization.There have been several efforts related to aircraft trajectory optimization to reach a numerical solu?tion，using heuristic optimization techniques［11］，or to develop an analytical solution［12-13］. In very rare cases several flights were sharing similar flight plans including departure and arrival airport，and depar?ture times［14］. Therefore，route optimization was an?alyzed with the same departure or arrival airport［15］.Based on previous results，a completely decentral?ized approach was established［16］，which suppressed the occurrence of large detours in the assembly of flight formations. Recently，an optimization frame?work was also developed that relies on optimal con?trol theory to solve the multiple-phase optimization problem［17］.

Formation flight trajectory optimization has been a hot topic in the last decade. However，most of the studies，optimizing the total fuel consump?tion，were based on the assumption that both the leading aircraft（leader）and the following aircraft（follower）agree to change their original trajectories.In these methods，only the follower’s fuel consump?tion is reduced and the leader’s fuel consumption is increased because of the detour required to match both trajectories. This is unrealistic even when the two flights are operated by the same airline，because the pilot’s performance evaluation is related to fuel consumption. This shortcoming makes it difficult to implement these algorithms into real-world opera?tions. It is more practical if the leader’s operation is not impeded and only the follower’s trajectory is replanned to form the formation. Another factor con?straining the application of current studies is that free-routing airspace is assumed while the airspace structure is not taken into consideration.

In order to address these two problems，a nov?el optimization method of formation flight for mini?mizing fuel consumption is proposed in this paper.We assume that only the follower’s trajectory is re?vised to follow the leader while the leader’s trajecto?ry remains unimpeded. Under this assumption，the leader’s trajectory follows the existing airspace structured，so that the availability of the path of the new formation is guaranteed. Therefore，the pro?posed method could potentially be achievable in the near future.

In the development of this method，this paper focuses on two key questions：Could the follower still be benefited by formation flight，when it fol?lows the actual trajectory of the leader；and how much savings could be achieved？ To answer these questions，we set constraints to candidate flight pairs selection and optimized the rendezvous and separation points for the follower. Then，the forma?tion of flight trajectory and fuel consumption for the follower is optimized. Such trajectory optimization is not geometric optimization，but based on real tra?jectory. Consumption is estimated according to air?craft data. The numerical study results are closer to the actual fuel saving after the formation flight is put into effect.

In this study，historical trajectory data of the whole world on one day are taken as a sample datas?et. Candidate flights are filtered for formation flight，and fuel consumption is estimated by each case.This paper is organized as follows：Section 1 de?scribes the heuristics that acts as filters on the com?binatorial set of all possible candidates. Section 2 de?scribes trajectory computation methodology，includ?ing models for aircraft and trajectory constraints.Section 3 describes the methodology of trajectory optimization. Results analysis for China internation?al flights are provided in section 4. Discussion and conclusions are given in sections 5.

1 Candidate Search for Formation Flight

As fuel consumption estimation for the whole trajectory is very time-consuming. Aiming at tens of thousands of trajectories，it is impractical to esti?mate every trajectory. Hence，candidates selection is carried out only by the positions of the departure airport（AD）and arrival airport（AA），subscripts let?ters“l(fā)”and“f”are for leader and follower，respec?tively. As the leader’s trajectory has not been changed，we assume the leader trajectory fromADltoAAlis the great circle route（GCR）for filter can?didates. And the spherical perpendicular fromADfandAAfto the leader’s GCR is made. As shown in Fig.1，the red line is the trajectory of the leader；the upper green line is the trajectory of the follower for case 1；the lower green line is the trajectory of the follower for case 2.

Fig.1 Geometric construction of two cases of two-aircraft formation mission

We take the two cases shown in Fig. 1 as sam?ples. In both cases，theADfandAAfhave footpoint（F）on leader GCR or its extension line. As rendez?vous point（PR）and separation point（PS）have not been set，we assume that the follower departs fromADfand passesFfor departure（FD）andFfor arriv?al（FA） sequentially，then arrives atAAffinally.Therefore，F(xiàn)DisPR，andFAisPS. In case 1，bothADf1andAAf1haveFon the leader’s GCR asFD1andFA1，respectively. And in case 2，bothADf2andAAf2haveFon the extension line of leader GCR.During filtering，we takeADlandAAlasFD2andFA2，which means the follower flies along the dotdash line in Fig.1.BetweenFDandFAis the overlap?ping trajectory，which is the part of the trajectory where formation flight will take place.

After the route is set，we require the flight di?rection of both aircraft are identical. Longitude is chosen as the indicator of direction. The east and the west are presented by positive and negative lon?gitude，respectively，and International Date Change Line is simplified to be ±180°. To make sure two aircraft have the same flight direction，Eq.（1）needs to be met，where lng refers to longitude

To ensure that the benefit of formation exceeds the cost of detouring，κdis introduced as the mini?mum ratio of overlapping. We divide the formation flight trajectory of the follower into three phases，dDAF，dAAF，anddFF，wheredDAFis the distance be?tweenADfandFD，dAAFis the distance betweenAAFandFA，dFFis the distance betweenFDandFA，or the distance of overlap trajectory，as shown in Fig.2.

Fig.2 Geometric construction of overlap ratio estimation

As presented by Eq.（2），the ratio ofdFFto the sum ofdDAF，dAAFanddFFmust be no less than κd，to ensure that the follower has sufficient formation flight distance. Obviously，the larger the overlap ra?tio，the more efficiency the follower can obtain.

Formation flight requires the follower to unite with the leader. To achieve this，normally，the fol?lowers need to adjust their departure time. A thresh?old must be set to the shift of departure time. AsFDis obtained already，we assume that the leader and the follower arrive atPRsimultaneously fromFDandADF，respectively. From the actual trajectory of the leader，we can find the tracking point closest toFD，and take the time when the leader passed it astl，and take the takeoff time of the follower astf.We de?fine the time threshold astlagand therefore we im?pose the following constraint

Besides trajectory information，aircraft types are also contained in the dataset. The aerodynamic performances of aircraft are taken into consideration.The follower is not a match for the leader，if its up?per limit of airspeed and flight altitude is lower than the corresponding value of the leader during its cruise phase.

In this paper，worldwide historical trajectory data of 20 January，2017 are employed，including 140 000 flights of ADS-B record. Fourteen thou?sand flights that arrived at or departed from China are extracted as a sample dataset. Candidates for for?mation flight are selected from the dataset. Based on the above-mentioned requirements，we obtain the sensitiveness of the number of candidates toκdandtlapas shown in Table 1.

Table 1 Number of formation flight candidates

The largertlagor the smallerκd，the more for?mation flight candidates.Smallerκdmakes the candi?dates be excessive，and is time-consuming for fur?ther trajectory optimization. However，largerκdlim?it the number of candidates.tlagdepends on opera?tional constraints，but longer than 1 h is probably unbearable. For trajectory optimization in this pa?per，we filter candidates by settingκd= 0.8，andtlag= 3 600 s. Thus 1 233 candidate flight pairs met the requirements.

2 Trajectory Computation Method?ology

When computing the aircraft trajectory，sever?al considerations must be taken into account. First，the aircraft performance model is needed，with the definition of a mathematical model representing air?craft dynamics and its performance. Second，in or?der to obtain operationally sound trajectories，flight envelope and air traffic management（ATM）con?straints must also be specified，as detailed in“Tra?jectory constraints modeling”［18］. Finally，atmo?spheric variables，such as wind，temperature，and pressure，are also needed，since they have a signifi?cant influence on the trajectory.

A point-mass dynamic model，Eurocontrol’s Base of Aircraft Data（BADA），and the Interna?tional Standard Atmosphere（ISA）have been con?sidered in this paper. More mathematical details on the formulation of this computation methodology can be found in Ref.［19］.

2.1 Aircraft model

Accurate estimation of fuel consumption in for?mation flight needs to be in accord with reality. To obtain realistic results，an aircraft performance mod?el（APM）is required to accurately represent aircraft behavior. BADA is an APM based on the kinetic ap?proach developed and maintained by Eurocontrol，with the active cooperation of aircraft manufacturers and operating airlines. BADA was designed for tra?jectory prediction and simulation for purposes of ATM research and operations，and has a high repu?tation within the academic and research world. A multi-platform library，pyBADA designed for a rap?id，easy and transparent integration in Python of the BADA APM for ATM research purposes，is used in this study. The applications of pyBada include air?craft performance modeling，trajectory prediction and optimization，and visualization［19］.

pyBADA assumes a nonlinear point-mass rep?resentation of the aircraft，where forces are applied at its center of gravity. It is reduced to what is com?monly called a gamma-command model，where con?tinuous vertical equilibrium is assumed. Gammacommand point-mass models have been reported to provide sufficient fidelity for ATM purposes. Air?craft dynamics are described in the air reference frame assuming flat nonrotating earth and neglecting wind components，yielding to the following set of differential equations

wherex=[v，s，h，m] is the state vector，in whichvis the true airspeed，sthe along path distance，hthe altitude andmthe mass of the aircraft；Tthe to?tal thrust；Dthe aerodynamic drag；gthe gravity ac?celeration，assumed to be constant；γthe aerody?namic flight path angle andFFthe fuel flow. The control vector considered isu=[γ，π]，whereπis the throttle setting.

All aerodynamic and engine parameters are rep?resented by continuous polynomials that use manu?facturer performance data encoded in BADA. The ISA has also been considered in this model.

2.2 Trajectory constraints modeling

Besides the dynamics of Eq.（4），other con?straints must be specified to model certain operation?al aspects or limits. In pyBADA，the initial and fi?nal conditions of the problem are taken. At the mo?ment the slats are retracted after the take-off or ex?tended before the landing. The remaining parts of the take-off and approach are not considered because the trajectory is heavily constrained by operational procedures. For the initial point of the trajectory，the mass of the aircraft is not fixed. All the remain?ing state variables are fixed to typical operational values.

Generic bounding constraints on certain vari?ables are specified as follows

wherevCASis the calibrated airspeed（CAS）and γmin，γmax，vCASmin，Mamin（minimum Mach number），vmax（velocity maximum in operations）andMamax（maximum Mach in operations）are aircraft depen?dent scalars.

3 Pre?flight Plan Trajectory Opti?mization

Horizontal and vertical pre-flight plan trajecto?ry optimization was carried out based on the whole trajectory of the follower，including formation flight part and trajectories where the follower flied alone.Fuel flow was calculated by aircraft model，which is described in Section 2. As only one location forPRorPSof a minimum of fuel consumption was exist on the trajectory，we used a bubble sort algorithm to realize the optimization.

3.1 Horizontal route optimization

In this part，PRandPSlocation on the trajecto?ry of the leader were optimized. The new trajectory of the follower from the initial position toPRand fromPSto terminal position need to be built，if the leader and the follower do not have the sameADorAA. As shown in Fig.3，the blue line is the trajecto?ry of the follower for formation flight.

In case 1，the trajectory of the follower was shorter than that of the leader. The assumption was made that the trajectory below FL100 was governed by terminal airspace structures. So that the trajecto?ry of the follower remained unchanged below FL100，presenting the taking off or landing phase.And we adjusted the time of those trajectories to en?sure they could match the formation flight. The ini?tial position was where the follower reached FL100，and the terminal position was where the fol?lower descent under FL100. The trajectories that needed to be built were from initial position toPRand fromPSto terminal position. When the follower passedPR，it had an identical position with the lead?er，both horizontally and vertically. And they also should have performed at the same airspeed. A simi?lar condition should be achieved when the follower leaved the leader atPS. To allow the follower to unite with the leader，the follower and the leader passedPRorPSat the same time. Further descrip?tion is expressed in vertical route optimization.

In case 2，as above mentioned in Section 1，PSandPRwere close toADandAA. The new trajectory of the follower followed the original trajectory be?low FL100 as above mentioned in case 1. In ideal conditions，the follower unites with the leader fromADtoAA. But the follower needs to wait until the leader achieves the cruise altitude，or holds an alti?tude before it arrivesAAf. As the actual flight trajec?tory is not ideal，it is hard to predict the cruising alti?tude of the leader. We assumed the follower needed to achieve 80% max altitude of the leader at first，then maintained the altitude until the follower united with the leader atPR. ThePSmust be above 80%max altitude of the leader also.

The mass of aircraft is a key factor in trajectory modeling. We took the nominal mass in BADA as the mass when the follower passed the initial posi?tion，and we took 90% of the maximum landing mass as the mass when the follower passed the ter?minal position，for every type of aircraft.

For horizontal route optimization，the position ofPRandPSwere optimized depending on assumed fuel-saving percentage（AFS）. AFS is the differ?ence between estimate fuel consumption and origi?nal fuel consumption as a percentage of original fuel consumption. As estimating fuel consumption for each trajectory for optimization is too time-consum?ing，we presented the fuel-saving effect as the reduc?tion of distance during formation flight.dADPRis the distance betweenADfandPR，dPRPSis the distance betweenPRandPS，anddPSAAis the distance be?tweenPSandAAf.dtotalis the total flying distance of the follower. We optimizePRandPSby minimizing thedtotal，and take the position reporting point as the results

Three values of AFS，10%，20% and 30%，were used to analyze the sensitiveness of the posi?tion ofPRandPSto AFS. As the trend of position change can be explained explicitly，other values of AFS are not listed here. Subscript 10，20，30 repre?sent AFS = 10%，20%，30%. Each pair ofPRandPSare shown in Fig.3 as a schematic.PR10andPS10is the closest pair，PR20andPS20is the further one，andPR30andPS30is the farthest. In other words，pair with AFS = 10% has the shortest for?mation flight distance，and AFS = 30% has the longest. With higher AFS，the follower prefers to follow the leader for a longer distance，which also means the follower can obtain more benefit from for?mation flight.

3.2 Vertical profile optimization

The whole flight trajectory has three phases，namely climb，cruise and descend. According to alti?tude，there are four segments in both climb and de?scent，as shown in Fig.4［20］. Formation flight needs precise relative position control. The follower needs to maintain a fixed horizontal position relative to the leader. Getting too close will be risky for the follow?er to be involved in the wake vortex of the leader，but getting too far will make the follower lose the benefit of formation flight. Therefore，formation flight is maintained in the cruise phase mostly，when the leader is in the cruise phase.

As described for the horizontal route optimiza?tion，both cases have the original trajectory that is performed below FL100 by the follower before the formation flight starts，as shown in Fig.5. The blue dashed line is the trajectory which is the same as the original trajectory of the follower. When the follow?er cannot unite with the leader directly，as in case 2，it achieves 80% max altitude of the leader at first，and unite with the leader atPR. AfterPS，which is at 80% max altitude of the leader，the fol?lower maintains the altitude until it starts descend?ing.

Fig.4 Model for the vertical profile

Fig.5 Vertical profile optimization

In both horizontal route optimization and verti?cal profile optimization，the constraint of waypoint and altitude restriction from air traffic control for the climb and descent phase of the follower was not con?sidered，because they are relatively minor parts of the whole flight operation.

4 Numerical Study of International Flights of China

Based on the optimization method described，1 233 pairs of flights were selected from historical trajectory data of the whole world on one day to form a sample dataset.

By setting different fuel-saving percentages，different numbers of pairs can be obtained. Accord?ing to the previous section，the higher the fuel-sav?ing percentage，the closerPRandPSare to depar?ture and arrival airport，the shorter the climb and de?scent distance of the follower. In some cases，due to the shortened climb or descent distance，the follow?er cannot achieve the altitude of the leader within the climb phase or get ground within the descent phase within the maximum performance range. It is shown in Table 2 that a higher fuel efficiency results in a smaller number of candidate flight pairs.

Table 2 Number of formation flight pairs

A pair of flights that was suitable for all param?eters was chosen for demonstration. The leader de?parted from London Heathrow International Air?port，to Shanghai Pudong International Airport，and its type was B777. The follower departed from Helsinki-Vantaa Airport，to Beijing Capital Interna?tional Airport，and its type was A333.

The horizontal trajectory is shown in Fig.6.The red line is the trajectory of the leader；the green line is the original trajectory of the follower；and the blue line is the optimized trajectory with formation for the follower. It can be seen from Fig.6 that the trajectories of the leader and the follower basically conformed to the GCR，and the trajectories almost coincided in a relatively long-range. This is because during long-distance flights， international flights need to fly through various waypoints，and the num?ber of routes that can be passed is limited. Most in?ternational routes in the same direction will pass the same waypoints and routes. It is for this reason that international flights are more convenient for forma?tion flight.

Fig.6 Horizontal route sample

The vertical profile is shown in Fig.7. It can be seen that during the cruise phase，the altitude of the follower’s original trajectory was higher than the leader’s. However，to achieve formation flight，the follower maintained the same altitude as the leader fromPR，and descended after it passedPS. The opti?mized trajectory of the follower below FL100 was the same original trajectory. Trajectory optimization was performed only for the trajectory from FL100 to thePR，and from thePSto FL100. After the fol?lower climbed to the altitude ofPS，it maintained the altitude until it passedPSto unite with the lead?er. After the follower passedPS，it maintained the altitude for a while，then descent directly.

Fig.7 Vertical profile sample

Different formation flight routes were built based on different AFS. But AFS is just an as?sumed value. The actual flight may not achieve this fuel-saving percentage. To analyze the sensitiveness of the actual fuel saving percentage to the AFS，four optimized trajectories were achieved with differ?ent AFS，named case 10，case 20，case 30 and case 40，referring to AFS = 10%，20%，30%，40%，respectively.

The original and the four cases fuel consump?tion for followers were estimated with different actu?al fuel saving percentage of formation flight（AFS?FF）. Comparing with original fuel consumption，the actual fuel saving percentage was obtained. For each case，the average actual fuel saving percentag?es（actual percentage）are shown in Table 3.

When AFSFF is 0%，the follower united with the leader and flied longer than the original trajecto?ry，but obtained no benefits from formation flight，which made its actual percentage all below 0. With the increase of AFSFF，the actual percentage is in?creasing，but still lower than AFSFF. This is be?cause the follower could not follow the leader for the whole flight，and we took several constraints in?to account，including the trajectory below FL100 and the altitude changing of the follower. Flying along the original trajectory below FL100 and fol?lowing the leader at nonoptimal altitude reduced its fuel saving. The case with the maximum actual per?centage changed as the AFS change. The case 10 were the best at AFSFF = 0% and 5%，case 20 at AFSFF = 10%，case 30 at AFSFF = 15% and 20%，case 40 at AFSFF = 30% and 40%. The AFSFF of the optimized trajectory was not consis?tent with the actual percentage，which was always lower than the AFSFF. The reason isPRandPSwere optimized only by AFS and distance，as shown in Eq.（6）. The difference between climb，cruise and descent phases were omitted in such method. The higher the AFS，the furtherPRandPS，as shown in Fig.3. In a hypothetical condition，the extra flight segment would lead to lower fuelsaving，But the extra flight segment has a high prob?ability that it exists during the climb or descent phase，which results in the follower saves more fuel if it chooses the trajectory with higher AFS.

5 Conclusions

Recent developments of formation flight were reviewed. Focusing on actual trajectories，we pro?pose a new method for trajectory optimization of for?mation flight. Based on trajectory computation meth?odology and trajectory optimization，the results show that the follower can be benefitted if it does not follow the geometric optimal route，but follows the actual trajectory of the leader. Assumed fuel-sav?ing percentage has an effect on trajectory optimiza?tion and actual fuel-saving efficiency affects the tra?jectory selection. The higher the actual fuel saving percentage，the more benefitted the formation flight trajectory with a higher assumed fuel-saving percent?age will be. Nevertheless，there are still several fu?ture developments that could be expected. Firstly，during the climb and descent phase，the trajectory optimization of the follower does not take the con?straints of waypoint and air control into account.Secondly，the actual fuel saving percentage depends a lot on the relative position of the follower to the leader，which is difficult to be maintained during the formation flight，and further affects the trajectory se?lection.

Finally，human factors are not in the scope of this article. However，formation flight is certainly against current separation standards，which leads to very different operational requirements for pilots and controllers involved. The work introduced in this pa?per is just a preliminary study of a new concept of formation flight operation，and is still far from appli?cation in the civil aviation industry.