calculatingprojectace

OptimizingAircraftTrajectorieswithMultipleCruiseAltitudesinthePresenceofWindsHokK.
Ng1UniversityofCalifornia,SantaCruz,MoffettField,CA94035BanavarSridhar2andShonGrabbe3NASAAmesResearchCenter,MoffettField,CA94035Thisstudydevelopsatrajectoryoptimizationalgorithmforapproximatelyminimizingaircrafttraveltimeandfuelburnbycombiningamethodforcomputingminimum-timeroutesinwindsonmultiplehorizontalplanes,andanaircraftfuelburnmodelforgeneratingfuel-optimalverticalprofiles.
Itisappliedtoassessthepotentialbenefitsofflyinguser-preferredroutesforcommercialcargoflightsoperatingbetweenAnchorage,AlaskaandmajorairportsinAsiaandthecontiguousUnitedStates.
Flyingwindoptimaltrajectorieswithafuel-optimalverticalprofilereducesaveragefuelburnofinternationalflightscruisingatasinglealtitudeby1-3%.
Thepotentialfuelsavingsofperformingen-routestepclimbsarenotsignificantformanyshorterdomesticcargoflightsthathaveonlyonestepclimb.
Wind-optimaltrajectoriesreducefuelburnandtraveltimerelativetotheflightplanroutebyupto3%forthedomesticcargoflights.
However,fortrans-oceanictraffic,thefuelburnsavingscouldbeasmuchas10%.
Theactualsavingsinoperationswillvaryfromthesimulationresultsduetodifferencesintheaircraftmodelsanduserdefinedcostindices.
Ingeneral,thesavingsareproportionaltotriplength,anddependontheen-routewindconditionsandaircrafttypes.
I.
IntroductiondvancedairtrafficmanagementsystemsaredesignedtooptimizethethroughputandefficiencyofthenationalairspacesystemintheUnitedStatestoaccommodategrowthinairtraffic.
Developingoptimalaircrafttrajectoriesnotonlyenhancesairtrafficflowbutalsohelpstheaviationindustrycopewithincreasingfuelcostsand1ResearchScientist,UniversityAffiliatedResearchCenter,MailStop210-8,MemberAIAA.
2SeniorScientistforAirTransportationSystems,AviationSystemsDivision,FellowAIAA.
2SeniorScientistforAirTransportationSystems,AviationSystemsDivision,FellowAIAA.
3ResearchScientist,SystemsModelingandOptimizationBranch,MailStop210-10,AssociateFellowAIAA.
Ahelpstoreduceaviation-inducedclimatechange.
Classicalaircrafttrajectoryoptimizationsaresolvedbyapplyingcalculusofvariationstodeterminetheoptimalityconditions,requiringthesolutionofnon-linearTwo-PointBoundaryValueProblems(TPBVPs)[1].
Inordertoreducethemodelingorderandhencethecomplexity,theenergystateapproximationhasbeensuccessfullyappliedtogenerateanoptimalverticalflightprofile[2,3].
Alternatively,amoregeneralsolutiontoaircrafttrajectoryoptimizationcanbeobtainedbysingularperturbationtheorywhichapproximatessolutionsofhighorderproblemsbythesolutionofaseriesoflowerordersystemswiththesystemdynamicsseparatedintolowandfastmodes[4,5].
However,findingtheglobaloptimalsolutionstotheTPBVPsremainsadauntingtask.
Withthetremendousadvancementinnumericalcomputingpower,TPBVPscanbeconvertedtononlinearprogrammingproblemsthataresolvableevenforproblemswithmanyvariablesandconstraintsusingnumericalalgorithmssuchasdirectcollocationmethods.
Neglectingaircraftdynamicsandapplyingshortestpathalgorithmsingraphtheory,anoptimaltrajectorycanbeapproximatedbythepaththatminimizesthetotallinkcostconnectingtheoriginanddestinationinapre-definednetwork.
Thegraphmethodsoftenrequirelargecomputationtimeandmemoryspacebutguaranteeglobaloptimalsolutions.
Thesenumericalmethodsarewidelyappliedforproblemsinconstrainedairspacesuchasminimumcost-to-climbandobstacle-avoidance.
However,themajorityoftheaforementionedtrajectoryoptimizationsneglecttheeffectofwindintheproblemformulation.
Thefollowingstudiesincludewindeffects.
In1931ErnstZermelo[6]firstproposedtheminimum-timepathproblemforaboatmovingthrougharegionofstrongcurrents.
BrysonandHoapproachedtheproblembydeterminingtheanalyticalequationgoverningtheoptimalheadingdynamicsthatsatisfiesPontryagin'sMinimumPrinciple[1].
Thisresultsinasolutiontotheminimum-timepathforaircrafttravelataconstantaltitudeandspeedinstrongwindsgivencorrectchoiceoftheinitialheading.
Generally,theinitialheadinganglecanbeobtainedbyaniterativeprocessbutconvergenceproblemsmayoccurduetothepresenceofnonlinearitiessuchasjetstreams.
Thesearchcouldconvergetoalocalminimumdependingontheinitialchoice.
Recentstudiesproposeselectingtheinitialheadingbycombiningcalculusofvariationsandgraphtheory,andclaimthatthesemethodsalwaysgenerateglobaloptimalsolutionswithmoderatecomputationaleffort[7,8].
Thesestudiesarefocusedoncomputingtheminimum-timepathatasingleflightlevelandhavenotconsideredtheminimum-fuelroutethatisdividedintosegmentsatseveralflightlevels,identifiedbyincorporatingaircraftweightandfuelburnmodelingintotheoptimizationprocedures.
Thisstudydevelopsapracticaltrajectoryoptimizationalgorithmforaircraftthatapproximatelyminimizescostoftimeandfuelburnbycombiningthemethodforcomputingminimum-timeroutesinwindsonmultiplehorizontalplanesandtheaircraftfuelburnmodelforgeneratingfuel-optimalverticalprofiles.
Itisappliedtoevaluatethepotentialbenefitsofflyingwind-optimalroutesinaseamlessairspaceforcommercialcargoflightsoperatingbetweenAnchorage,AKandmajorairportsinAsiaandthecontiguousUnitedStates.
ThesectiononPracticalAircraftTrajectoryOptimizationdescribestotaloperatingcost,anddevelopstheverticalprofileandhorizontalmaneuverthatminimizefuelburnandtraveltimeinwindsforcommercialaircraft.
ThesectiononExperimentalSetupappliestheoptimizationalgorithmtocalculatewind-optimaltrajectoriesforcommercialcargoflightsbetweenAnchorage,AKtomajorcitiesintheU.
S.
andAsia.
TheResultssectionpresentspotentialbenefitsofflyingminimum-timetrajectoriesinwindswithafueloptimalverticalprofile.
TheConclusionssectionsummarizestheresults.
II.
PracticalAircraftTrajectoryOptimizationThissectionpresentsthepracticaltrajectoryoptimizationalgorithmthatapproximatestheminimizationofthetotalcostoftraveltimeandfuelconsumptionforaircraftonasphericalsurface.
Optimizingperformanceofasingleaircraftfromtakeofftolandinginthepresenceofairtrafficconstraintsandwindsiscomputationallyintensiveandtimeconsuming.
Futureairtrafficsimulationsystemswillberequiredtooptimizeandsimulatetensofthousandsofaircrafttrajectoriesinsecondsorminutes.
However,itisimpracticalandmaynotbenecessaryforthesesystemstogeneratethousandsofoptimalanddetailedaircrafttrajectories.
Thedetailsofthetrajectoryoptimizationrequireddependontheperformanceindex,theapproachandtheapplication.
Thisstudyadaptsapracticaloptimizationapproachbyassumingatypicalstructureforanaircrafttrajectoryandfocusesonoptimizingdirectoperatingcostduringcruisewhenthetimeandfuelsavingshavethemostimpact.
Atypicalaircrafttrajectoryconsistsofaninitialclimb,asteady-statecruise,andafinaldescent.
Here,aircraftperformanceisoptimizedforthecruisephaseonly.
Typicalaircraftprofilesareappliedforgeneratingtrajectoriesduringinitialclimbandfinaldescentsinceadditionalairtrafficconstraintsareinvolvedinthesestagesandtraveltimeandfuelsavingsaregenerallysmallcomparedtothoseduringcruise.
Thecruisetrajectoryisdividedintosegmentsonseveralaltitudesastheoptimalcruisealtitudeincreasesduetothereductioninaircraftweightasfuelisused.
Thelegalcruisingaltitudesandtheen-routestepclimbtimesareoptimizedbasedonEurocontrol'sBaseofAircraftData,Revision3.
6model[9].
TheaircraftoptimalheadingduringcruiseisthesolutionoftheZermeloproblemderivedonasphericalEarthsurfaceintheabsenceofconstraints.
Thehorizontaltrajectorysegmentsareoptimizedbasedonthecost-to-goassociatedwithextremals(trajectories)generatedbyforwardorbackwardintegratingthedynamicalequationsforoptimalheadingandaircraftmotionfromvariouspointsintheairspace.
Thiscomputationallyefficientalgorithmsearchesforoptimalsolutionsbycombiningcalculusofvariationsanddynamicprogramming.
Thenextsubsectiondefinesaircraftoperatingcostandoutlinesthepracticaloptimizationprocedures.
Theoptimizationproceduresincludetwostages.
ThesubsectiononOptimalVerticalProfiledescribessolvingtheoptimalaircraftcruisealtitudesanden-routestepclimbtimes.
ThesubsectiononHorizontalTrajectoryGenerationpresentstheaircraftmodelandtheproceduresforcalculatingoptimalaircraftheadingsinwindsonmultipleflightlevels.
A.
AircraftOperatingCostThedirectoperatingcostforacruisingaircraftcanbewrittenas:J=[Ct+Cff(m,h,V)]t0tf∫dt,(1)whereCtandCfarethecostcoefficientsoftimeandfuel.
Thefuelflowrate,f,canbeapproximatedbyafunctionofaircraftmass,m,altitude,h,andairspeed,V.
Aggarwal[10]minimizesdirectoperatingcostsforlong-rangetransporttypeaircraftintheabsenceofwinds,andshowsthattheaircraftMachnumberremainsalmostconstantduringcruise.
Thisresultisadoptedheretosimplifythesearchfortheoptimalsolution.
Forachosenconstantairspeedindexedini,Vi,thecostfunctionisrewrittenas:Ji=[Ct+Cff(m,h,Vi)]t0tf∫dt.
(2)TheoptimalVicanbedeterminedthroughaniterativeprocesssuchthatthearrivaltimeconstraintismetandoperatingcostisminimized.
Thisoperatingcostdependsonthetotaltraveltimeandfuelflowrateduringcruise.
Totaltraveltimeisminimizedbydeterminingtheoptimalaircraftheadingthatyieldstheminimum-timetrajectoryinwindsandthefuelflowrateisminimizedbycontrollingcruisealtitudesgivenmandVi.
Thecomputationaleffortishigh,especiallyforlonghaulflights,whenbothwindsandaircraftfuelburnareconsideredintheoptimizationsinceen-routemeteorologicalconditionsvarywithlocation,andaircraftfuelburndependsonaltitude.
Inordertoimprovethecomputationalefficiencyoftheapproach,atwo-stageoptimizationstrategyisproposedfornarrowingdownthesearchscopetofindapproximatesolutiontotheoptimizationproblem.
Thefirststageoptimizesanaircraftverticalprofilealongthetracktodeterminetheoptimalcruisealtitudesanden-routestepclimbtimesgiventheflightlevelsconstraints.
Thesecondstagedevelopswind-optimalhorizontaltrajectorysegmentsbydeterminingoptimalaircraftheadinginwindsbasedonthecruisealtitudesandstepclimbtimesthataresolvedinthefirststage.
B.
OptimalVerticalProfileThissubsectiondevelopsanaircraftverticalprofilebasedonthefuelconsumptionmodelinEurocontrol'sBaseofAircraftDataRevision3.
6(BADA)[9].
First,formulasforcalculatingthecruisealtitudethathastheminimumfuelburnratearederivedbyassumingaconstantcruisespeed.
Then,theyareappliedforoptimizingtheaircraftverticalprofilealongtrackbasedontheaircraftweightandtrueairspeed.
Thefuelburnforaircraftduringcruise,F,iscalculatedas:F=tf,(3)wheretiselapsedtime.
Thefuelburnrate,f,forjetsandturbopropsisdeterminedbythespecificfuelconsumption,SFC,andthrust,Th,andcanbeexpressedinthefollowingform:f=Cfcr1000SFCTh,SFC=Cf1(1+VCf2).
(4)Cfcr,Cf1,Cf2arethethrustspecificfuelconsumptioncoefficientsandVTASisthetrueairspeed.
Duringcruise,thrustequalsaerodynamicdragforce,D,thatisafunctionofaerodynamicdragcoefficients,CD,airdensity,ρ,aircraftwingreferencearea,S,andtrueairspeed,andisexpressedas:D=12CDρSV2.
(5)Thedragcoefficient,undernormalconditionsandexceptduringapproachandlanding,isspecifiedasafunctionofthedragcoefficientparameters,CD0,CD2,andtheliftcoefficient,CL,where:CD=CD0+CD2CL2,CL=2mgρSV2.
(6)Notethattheaerodynamicdragisafunctionofairdensitythatdependsonaltitude.
Theformulasforcalculatingtheoptimalcruisealtitudethathastheminimumfuelflowratearederivedbytakingthefirstordervariationofthefuelflowratewithrespecttothecruisealtitudetozero.
UndertheInternationalStandardAtmosphere(ISA)conditions,thetropopauseisat11,000metersaltitude;andtheoptimalaircraftcruisealtitude,hopt,atorbelowthetropopausecanbecalculatedbythefollowingformula:hopt=[1ef(m,V)KTRgas2(g+KTRgas)ρ0_ISA2](1000T0_ISA6.
5).
(7)Abovethetropopause,itis:hopt=f(m,V)RgasTtrop_ISA2gρtrop_ISA2+11000,(8)wheref(m,V)=ln(4m2g2CD2S2V4CD0).
Rgasistherealgasconstantforair.
Thetemperaturegradient,KT,thesealevelairdensity,ρ0_ISAandthesealeveltemperature,T0_ISA,areallconstantunderISA.
Theairdensity,ρtrop_ISA,andtemperature,Ttrop_ISA,arealsoconstantsatthetropopause.
Theappendixshowsthederivationoftheseformulas.
OptimalcruisealtitudesarecomputedfromEqs.
(7,8)basedontheatmosphericconstantsandaerodynamicdragcoefficientsthatareaircrafttypedependent.
Theyalsovarywithaircraftmassandairspeed.
Thethrustmarginisalsocheckedtoensurethatitispositivewhendeterminingtheoptimalaltitudesforeachselectionofmassandspeed.
Thethrustmarginisdefinedasthemaximumcruisethrustminusthethrustrequiredtomaintain100ft/minuterateofclimbattheselectedairspeed,attheoptimalcruisealtitude[11].
Figure1showstheoptimalcruisealtitudesforvariouscombinationsofmassandspeedfortheMcDonnellDouglasMD-11aircraft.
Ingeneral,optimalcruisealtitudesarelowerforhigherweightsandincreaseasairspeedincreases.
Notethatoptimalcruisealtitudedecreasesatveryhighairspeedswhenthethrustmarginsbecomenegative.
Inthesecases,optimalaltitudesarerecomputedandloweredtomeetthethrustmarginrequirement.
3300035000350003500035000370003700037000370003900039000390004100041000AircraftMass,1000kgTrueAirspeed,knots450460470480490500200210220230240250260270Fig.
1OptimalcruisealtitudesforMD110100200300343536373839Time,minutesAltitude,1000feetOptimalOperationalFig.
2VerticalprofileduringcruiseforMD11.
Duringaflight,theoptimalcruisealtitudeincreasesduetothecontinuousreductioninaircraftweightasfuelisused.
Anaircraftweightprofileduringcruise,foraconstantairspeed,isgeneratedusingtheinitialaircraftweightatthestartofcruiseandthetimehistoryofthefuelburnrate.
Figure2plotstheverticalprofileforoptimalcruisealtitudesinbluebasedontheweightprofileintheabsenceofwinds.
Duetocurrentairtrafficregulation,however,aircraftarenotpermittedtofollowthisoptimalclimbprofile,andcaninsteadonlycruiseoncertainflightlevelsdependingonthedirectionofflights.
ThisstudyadaptsthestandardinReducedVerticalSeparationMinima(RVSM)thateastboundaircraftflyoddthousandsoffeetwhilewestboundaircraftflyeventhousandsoffeet.
Theoptimallegalcruisealtitudes,hoptLegal,aredeterminedgiventhesetofflyablealtitudes,hLegal,usingthefollowingequation:hoptLegal(t)=minhLegalhopt(t)hLegal{}.
(9)TheverticalprofileforaneastboundflightisplottedinmagentainFig.
2.
Thisverticalprofileisgeneratedwithoutwindsandassumesaninstantaneousstepclimb.
Itisanapproximationfortheverticalaircraftprofilefordeterminingtheen-routelegalcruisealtitudesandthetotaltraveltimeateachaltitude.
C.
HorizontalTrajectoryGenerationThesecondstageoftheproposedoptimizationstrategyoptimizesaircrafthorizontaltrajectorysegmentsinthepresenceofwindsbasedontheverticalprofiledevelopedinthefirststage.
Minimizingtotalaircrafttraveltimealsominimizestotalfuelburnforagivenverticalprofilesincethefuelburnratesarespecifiedalongthetrack.
Theminimum-timetrajectoryconsistsofthehorizontaltrajectorysegmentsonthepre-determinedlegalcruisingaltitudes.
Theyareoptimizedbasedonthecost-to-goassociatedwitheachextremalgeneratedbyforwardorbackwardintegratingthedynamicalequationsforoptimalheadingandaircraftmotionfromvariouspointsintheairspace.
Thissubsectionpresentsthedynamicalequationforoptimalaircraftheadingandthehorizontaltrajectoryoptimizationalgorithm.
TheaircraftequationsofmotionataconstantaltitudeabovethesphericalEarth'ssurfaceare:˙φ=Vcosψ+u(φ,θ,h)Rcosθ,(10)˙θ=Vsinψ+v(φ,θ,h)R,(11)˙m=f,(12)subjecttotheconditionsthatthrustequalsdrag,flightpathangleiszero,andtheboundaryconstraintsaremet.
φislongitudeandθislatitude,ψisheadingangle,andRistheEarth'sradius.
Theeast-componentofthewindvelocityisu(φ,θ,h),andthenorth-componentofthewindvelocityisv(φ,θ,h).
ItisassumedthattheEarthisasphereandR>>h.
Thepreviousstudy[12]appliesPontryagin'sMinimumPrincipletodeterminetheheadinganglethatisassumedtobethecontrolavailableforaircraftduringcruiseforminimizingthecostoftraveltimeandclimateimpact.
Whenneglectingtheclimateimpact,thehorizontaltrajectoryisoptimizedbydeterminingtheheadinganglethatminimizestraveltimeinthepresenceofwinds.
Thedynamicalequationfortheoptimalaircraftheading[12]is:˙ψ=Fwind(ψ,φ,θ,u,v,V)Rcosθ,(13)whereFwind(ψ,φ,θ,u,v,V)isaircraftheadingdynamicsinresponsetowindsandisexpressedas:Fwind(ψ,φ,θ,u,v,V)=[sinψcosψu(φ,θ,h)φ+cos2ψsinθu(φ,θ,h)+cos2ψcosθu(φ,θ,h)θv(φ,θ,h)φ+sinψcosψsinθv(φ,θ,h)+cosψsinψcosθv(φ,θ,h)θ+Vcosψsinθ+cos2ψv(φ,θ,h)φ].
(14)Arecentstudy[8]developedthesamedynamicalequationforcomputingminimum-timepath.
Thesestudies[8,12]reducedthetrajectoryoptimizationproblemfromatwo-pointboundaryvalueproblemtoaninitialvalueproblem.
Numericalalgorithmssuchascollocationmethods,usedin[12],orinterpolationtechniques,asusedin[8],canbeappliedtodeterminetheoptimalinitialaircraftheading.
Inthecasethatanaircraftcruisesatasinglealtitude,theminimum-timetrajectoryiscompletelyspecifiedbyintegratingEqs.
(10,11,13)simultaneouslyfromtheorigintothedestinationusingtheoptimalinitialaircraftheading.
Ingeneral,anaircraftcruisesatmultiplealtitudestoaccommodateweightreduction,therebyminimizingfuelburn,asdescribedintheprevioussubsection.
Theminimum-timetrajectoryisthecombinationofwind-optimalextremalsonseveraldifferentaltitudes,eachsolvedusingwindconditionsonthataltitude.
Thisstudyemploystheconceptofdynamicprogrammingforconstructingtheminimum-timetrajectorywithwind-optimalextremalsatdifferentcruisealtitudes.
Theoptimalverticalprofileprovidestheinitialandsubsequentoptimalcruisealtitudesaswellasthetransitiontimesbetweenthealtitudes.
Thefollowingfivestepscomputethehorizontalminimum-timetrajectory.
1.
Usingarangeofdifferentinitialheadinganglesatthestartoftheinitialcruisesegment,acollectionofwind-optimalextremalsisgeneratedbyforwardintegratingEqs.
(10,11,and13)untilthefirststepclimbtime.
Thisstepisillustratedbyplottingthewind-optimalextremalsinblueinFig.
3foraflightfromAnchoragetoHongKong.
Fig.
3Wind-optimalextremalsfromAnchorage,AKandHongKong2.
Usingarangeofdifferentfinalheadinganglesatthedestination,anothercollectionofwind-optimalextremalsisgeneratedbybackwardintegratingEqs.
(10,11,and13)atthesecondcruisealtitude.
Thisisdoneforafixedperiodoftime,estimatedbasedonthedifferencebetweenthetotaltraveltimeandthefirststepclimbtime.
TheMagentacurvesinFig.
3showstheseextremalsatthesecondcruisealtitude.
Theseextremalsprovidetheminimumtime-to-gotothedestinationfromanypointsalongthem.
Delaunaytriangulationandinterpolationtechniquescanbeappliedtoestimatethetime-to-gofromanypointinthecoveredairspaceregionusingthepointsontheextremals.
3.
Identifytheextremalfromthecollectiongeneratedinstep1abovethathastheminimumtime-to-goatthefirststepclimbtime(i.
e.
theendoftheextremals).
Thisstepdeterminesthewind-optimaltrajectorysegmentatthefirstcruisealtitudeandtheaircraftpositionatthefirststepclimbtime.
ThetrajectorysegmentandthestepclimbpositionareshownbythebluedashedlineandthebluecrossinFig.
3.
Notethattheminimumtime-to-gocalculatedbasedontheextremalsinstep2isonlyanapproximationwhentheaircraftcruisesatmorethantwoaltitudes.
4.
Afterclimbingtothenewcruisealtitude,repeatstep1andstep2atthenewstartingpositionandaltitude.
ThisisdemonstratedinFig.
4,whichplotsthenewstartingpositionasabluecrossandthenextstepclimbpositionasamagentacross.
Theextremalsandtheminimum-timetrajectorysegmentatthenewaltitudeareplottedassolidanddashedmagentalines.
Thisstepisrepeateduntilthelastcruisealtitudeisreached.
Fig.
4Wind-optimalextremalson2altitudes5.
Whenthelastcruisealtitudeisreached,theminimum-timetrajectoryatthelastaltitudeiscalculatedbyforwardintegratingEqs.
(10,11,and13)usingthecorrectheadingangleatthestartofthelastcruisealtitude.
Theheadingangleiscalculatedbyinterpolatingtheaircraftheadingsfromthebackwardextremalsatthefinalcruisealtitude.
Thehorizontalwind-optimaltrajectoryforaflightfromAnchorage,AKtoHongKongonAugust1,2010isshowninFig.
5.
Thewindvectorsatflightlevels360,380and400areplottedinblue,magentaandgreen,respectively.
Aflightlevelisastandardaltitudeofanaircraftinhundredsoffeet.
Eachtrajectorysegmentisoptimizedwithrespecttothewindsattheassociatedflightlevel.
ThewindmagnitudesanddirectionsaretakenfromtheGlobalForecastSystem(GFS).
GFSisaglobalnumericalweatherpredictioncomputermodelrunbytheNationalOceanic&AtmosphericAdministration(NOAA)fourtimesaday.
Itproducesforecastsupto16days,andproducesaforecastforevery3rdhourforthefirst180hours,andafterthat,every12hours.
Thehorizontalresolutionisroughlyequivalentto0.
5*0.
5degreelatitude/longitude.
GFSdatahas64unequally-spacedverticalisobaricpressurelevelsrangingbetween0.
25-1000mb,withenhancedresolutionatlowandhighaltitude.
Fig.
5Thewind-optimaltrajectoryfromAnchorage,AKtoHongKongThecomputationtimeforsolvingthewind-optimaltrajectoryon3flightlevelsisabout8secondswhenrunningaMatlabR2012aprogramonaMacProwithdual2.
66GHz6-coreprocessorsand64GBmemory.
Thecomputationtimeforsolvingawind-optimaltrajectoryatasingleflightlevelisabout6secondsonthesameplatform.
Thisalgorithmcanbeimplementedconvenientlyinothercomputerlanguagessinceitdoesnotrequirecomplexnumericalsolvers.
Notethatgeneratingthewind-optimalextremalsinparallelcanfurtherreducesthecomputationtime.
Theperformancesofthetwowind-optimaltrajectoriesareevaluatedbysimulatingthecompleteaircrafttrajectoriesfromtakeofftolandingforaBoeing777-200withatakeoffweightof258,300kgandacruisespeedof486knotsusingthewind-optimalflightpaths.
Thetypicalaircraftprofilesduringtakeoff,stepclimbs,andlandingarebasedonBADA[9].
Thehorizontaltrajectoryduringcruiseissimulatedbasedonthewind-optimalflightpathatasingleflightlevelandthewind-optimalflightpathonmultipleflightlevelswithen-routestepclimbs,respectively.
Thetotalfuelconsumptionandtraveltimeforthewind-optimalflightonasingleflightlevelis64.
4tonnesand569minutes,respectively.
Thewind-optimalflightthatcruiseson3flightlevelsburns62.
4tonnesoffuelandtravelsfor562minutes.
Flyingwind-optimalwithen-routestepclimbssavesabout3.
2%offueland1.
2%oftraveltimewhencomparedtothatonasingleflightlevel.
Thispracticalapproachintegratesthetechniqueofaircraftwind-optimalheadingandtheconceptofDynamicProgramming(DP)toapproximatelyoptimizetheperformanceofaircrafttrajectoriesinthepresenceofwinds.
Thedetailsofderivingthedynamicalequationforaircraftwind-optimalheadingcanbefoundinthepaststudies[8,12].
TheapplicationofDPfordeterminingthesubsequentstepclimblocationsisdiscussedintheAppendix.
III.
ExperimentalSetupsThepracticaltrajectoryoptimizationalgorithmisappliedtoassessthepotentialbenefitsofflyingwind-optimaltrajectorieswithanoptimalverticalprofileforcommercialcargoflightsoperatingatAnchorage,AK.
ThetrajectorycomputationsuseairtrafficandglobalwinddatafromOctober2010,obtainedfromtheGFS.
Theperformanceofthewind-optimaltrajectoriesisevaluatedbysimulatingthecompleteaircrafttrajectoriesfromtakeofftolandingusingthewind-optimalflightpaths.
Thetypicalaircraftprofilesduringtakeoff,stepclimbs,andlandingarebasedonBADA.
TedStevensAnchorageInternationalAirport(ANC)isamajorhubairportforcargoflightsbetweentheU.
S.
andAsia,beinglessthan9.
5hoursfrom90%oftheindustrializedworld.
Forthispaperweanalyzenearly12,500FederalExpress(FedEx)andUnitedParcelService(UPS)cargoflightsfrom2010.
Basedonthesedata,Fig.
6showsthe10mostpopularoriginairportsforflightstoANC.
Almost80%ofinboundFedExandUPSflightsdepartfromthese10originairports.
Over90%ofoutboundFedExandUPSflightsfromANCflytothe10destinationairportsshowninFig.
7.
Table1liststheairportnamesandcodesforthetoporiginsanddestinations.
RJAAandKINDisthetoporiginanddestination,respectively,inadditionaltotheninecommonairports.
About99%oftheaircraftinthedatasetbelongtothefiveaircrafttypesshowninFig.
8.
ThesearetheMcDonnell-DouglasMD-11(MD11),Boeing767-300(B763),Boeing747-400(B744),Boeing777-200LR(B77L),andMcDonnell-DouglasDC-10(DC10).
Table1TopairportsLouisvilleInternational(KSDF)MemphisInternational(KMEM)HongKongInternational(VHHH)ShanghaiPudongInternational(ZSPD)SeoulIncheonInternational(RKSI)NewarkLibertyInternational(KEWR)TaiwanTaoyuanInternational(RCTP)OsakaKansaiInternational(RJBB)OaklandInternational(KOAK)NaritaInternational(RJAA)IndianapolisInternational(KIND)AircrafttrajectoriesareoptimizedbasedontheaerodynamicparametersandaircraftperformancedataprovidedbyBADAforthefiveaircrafttypesdescribedabove.
BADA[9]doesnotincludetheB77L,sotheaerodynamicparametersandaircraftperformancedatafortheBoeing777-200(B772)areusedinstead.
Hightakeoffmassisassumedforeachaircrafttypesincecommercialcargoflightsareusuallyheavilyloaded.
Typicalaircraftclimbanddescentprofilesduringinitialclimb,en-routestepclimbsandfinaldescentareobtainedforeachaircrafttypefromBADA[9]performancefilesforgeneratingthecompleteaircrafttrajectories.
Thesearedefinedbasedonaltitude,airspeedandmassforeachphase.
Theoptimalspeeddependsonthecostperformanceindexthatisgenerallyspecifiedbytheairlinesandchosenaccordingtotheirpriorities.
Thecurrentoptimizationproceduresolvesfortheoptimalspeedbyiteratingarangeofcruisespeedsthatrequiresamoredetailedaerodynamicmodel.
Inaddition,currentBADAparameters,suchasthedragcoefficients,mayonlybevalidatthespecifiednominalspeeds.
Duetotheselimitations,thisstudydoesnotoptimizecruisespeedforeachflight.
Inthispaper,thetrueairspeedduringcruiseiscalculatedfromthetypicalMachnumbersuggestedbyBADA.
Fig.
6Top10originsin2010Fig.
7Top10destinationsin2010Fig.
8Top5aircrafttypesin2010IV.
ResultsThissectionpresentsthepotentialbenefitsforcommercialcargoflightsofflyingwind-optimaltrajectorieswithanoptimalverticalprofile.
TheresultsarebasedonairtrafficandglobalwinddatafromOctober2010.
Paststudiesfocusondevelopingminimum-timetrajectoriesinwindsatasinglealtitudewithoutconsideringpotentialfuelsavingsfromen-routestepclimbs.
ThesubsectiononComparingWindOptimalTrajectories,below,compareswindoptimaltrajectorieswithanoptimalverticalprofileandonmultipleflightlevels(WOMFL)towindoptimaltrajectoriesatasingleflightlevel(WO1FL).
ThetrajectoriesarecomparedforcargoflightsbetweenANCandthetop10originanddestinationairportsforflightstoandfromANC.
ThesubsectiononComparingFlightPlanandWindOptimalTrajectoriesassessestheperformanceofthewindoptimaltrajectorieswithanoptimalverticalprofile(WOMFL)andtheflightplantrajectories(FP)alongthesameverticalprofile.
KSDFKMEMVHHHZSPDRKSIKEWRRCTPRJBBKOAKRJAA0.
500.
511.
522.
533.
5OriginAirportsSavings,%MD11B763B744B772DC10KSDFKMEMZSPDKOAKRCTPVHHHKINDKEWRRJBBRKSI0.
500.
511.
522.
5DestinationAirportsSavings,%MD11B763B744B772DC10Fig.
9PotentialfuelbenefitsforstepclimbsA.
ComparingWindOptimalTrajectoriesThissubsectionanalyzestheWOMFLandWO1FLtrajectoriesforcargoflightstoandfromANC.
TheperformanceoftheWOMFLtrajectoriesisevaluatedbyassessingandcomparingaveragefuelconsumptionandtraveltimeofthesetrajectoriestothatoftheWO1FLtrajectories.
Figure9showstheaveragefuelsavingsinbargraphsforeachorigin,destinationandaircrafttype,forallcargoflightsinOctober,2010toandfromANC.
FlyingtheWOtrajectorieshasapositivefuelsavingforalloftheoriginsanddestinationsexceptKOAKandKIND.
Fuelsavingsarebetween0.
1%and3%dependingontheairportpairs.
Theyaredirectlyproportionaltothelengthoftheroutes,whichdeterminesthenumberofen-routestepclimbs.
Theaveragetraveltime,fuelburn,andnumberofstepclimbsforthetop10originsareshowninTable2.
Theaveragetimeandfuelsavingsareshowninparentheses.
Thepotentialfuelsavingsareverysmallforalldomesticflightssincemostofthemhaveonlyonestepclimb.
FlightsbetweenANCandKOAKorKINDhavetheshortesttraveltime.
Fortheseroutes,performinganen-routestepclimbactuallyincreasesthetotalfuelburn.
Notethattheadditionalfuelburnrequiredforeachstepclimbisnotconsideredwhendevelopingtheaircraftverticalprofile,whichassumesinstantaneousclimbforsimplicityintheoptimization.
Mostinternationalflightssavemorethan1%fueldependingontheaircrafttypes.
TheB744andB772havethelargestamountoffuelsavingswhiletheMD11andB763havesmallerandsimilaramountoffuelsavings.
TheWOMFLtrajectoriesforallthedomesticflightshaveequalorlongertraveltimesthanthecorrespondingWO1FLtrajectories.
Theadditionaltimeinvolvedinclimbinganddescendingisnotnegligiblefortheserelativelyshorttrips.
TheWOMFLtrajectoriesforallinternationalflightsexceptRJAA,whichistheshortestoftheinternationalroutes,havepositivetimesavings.
Thetimesavingsnotonlydependonthetriplengthbutalsoontheflightdirection.
Thetraveltimeforeachflightisaffectedbyen-routewindmagnitudeanddirection.
Therefore,long-distancewind-optimaltrajectoriesnotonlygainfuelsavingsfromen-routestepclimbsbutmayalsobenefitfromhigh-speedwindsathigheraltitudes.
Table2Averagewind-optimalperformanceOriginACTypeTime,[minute]Fuel,[tonne]Steps[no.
]KSDFMD11368(0)52.
5(0.
03)1B763384(-1)33.
0(0.
01)1B744363(-1)66.
1(0.
15)1KMEMMD11369(0)52.
6(0.
03)1B772368(1)47.
7(0.
20)1DC10374(1)51.
9(0.
21)2VHHHMD11545(4)74.
0(1.
32)3B744534(5)93.
1(2.
25)2B772542(4)67.
4(2.
06)3ZSPDMD11462(2)64.
1(0.
76)2B763478(3)40.
0(0.
75)2B744453(3)80.
6(1.
41)2B772459(3)58.
3(1.
40)2RKSIMD11411(2)57.
9(0.
63)2B763426(3)36.
2(0.
60)2OriginACTypeTime,[minute]Fuel,[tonne]Steps[no.
]B744404(3)72.
9(1.
12)2KEWRMD11386(0)54.
8(0.
06)1B763402(-1)34.
3(0.
04)1B772385(-1)49.
7(1.
13)1RCTPMD11498(1)68.
5(0.
75)2B763516(1)42.
8(0.
71)2B744489(2)86.
2(1.
48)2RJBBMD11389(1)55.
1(0.
45)2B763402(2)34.
4(0.
46)2B772386(4)49.
9(1.
15)2KOAKMD11225(0)33.
3(0)0DC10228(-1)33.
3(-0.
07)1RJAAMD11362(-1)51.
6(0.
13)2B763374(-1)32.
3(0.
14)1DC10365(1)50.
7(0.
29)1B.
FlightPlanandWindOptimalTrajectoriesInthefuture,withimprovementsinairtrafficmanagement,routestructuresarelikelytobecomemoreflexible.
Withthesemoreflexibleroutestructures,futureairtrafficwillbeabletochoosetoflytheirpreferredroutestoagreaterextentthanisthecasetoday.
ThissubsectionassessesthepotentialbenefitsfortheFedExandUPScargoflightswhentheyflyWOMFLtrajectoriesinsteadofFProutes.
BoththeWOMFLandFPtrajectoriesfollowtheoptimalverticalprofiles.
Table3presentstheaveragesavingsperflight,indescendingorder,forthetoporiginsanddestinationstoandfromANC.
ThepotentialsavingsfortheinboundflightstoANCareshowninthebluecellsandthedatafortheoutboundflightsfromANCareinthegreencells.
Theabbreviations"n/a"indicatewheredataareunavailableforsomeflightdirections.
Figure10showstheaveragepercentagesavingsforfuelinsolidbargraphsforeachorigin,destinationandaircrafttype.
Table3Potentialbenefitsforwind-optimalAirportACTypeTimeSavings[minute]FuelSavings[tonne]VHHHMD1113541.
496.
10AirportACTypeTimeSavings[minute]FuelSavings[tonne]B763n/a45n/a3.
14B74417522.
637.
60B7726500.
695.
35RKSIMD1115471.
815.
71B76336382.
672.
83B7444460.
567.
26B772n/a23n/a2.
61RCTPMD1114351.
604.
10B76325331.
762.
32B74416n/a2.
40n/aZSPDMD1120262.
463.
13B76325101.
840.
68B7442634.
090.
48B7725n/a0.
55n/aRJBBMD118160.
982.
01B76310170.
771.
27B744n/a15n/a2.
40B77220112.
281.
28KSDFMD11870.
950.
94B7634100.
330.
75B744460.
690.
96KEWRMD11630.
820.
41B7635n/a0.
38n/aB772750.
780.
60KMEMMD11560.
590.
84B772560.
630.
70DC10560.
580.
70RJAAMD110n/a0.
04n/aB7638n/a0.
65n/aDC101n/a0.
12n/aKINDMD11n/a3n/a0.
34AirportACTypeTimeSavings[minute]FuelSavings[tonne]B772n/a4n/a0.
54KOAKMD11200.
270DC10100.
140Theaveragefuelsavingsarehighlycorrelatedtothetimesavingsforeachorigin,destinationandaircrafttype.
ThesavingsforeachflightareproportionaltothetotaltraveltimedifferencebetweentheWOMFLandFPtrajectoriessincebothfollowthesameverticalprofile.
Themagnitudesoftimesavingsareaffectedbyen-routewindconditionsandthestructureoftheoriginalflightplans.
KSDFKMEMVHHHZSPDRKSIKEWRRCTPRJBBKOAKRJAA012345678OriginAirportsSavings,%MD11B763B744B772DC10KSDFKMEMZSPDKOAKRCTPVHHHKINDKEWRRJBBRKSI0246810DestinationAirportsSavings,%MD11B763B744B772DC10Fig.
10Potentialfuelbenefitsforflyingwind-optimalFlyingtheWOMFLtrajectorieshasapositivesavinginfuelandtimeforalltheoriginsanddestinationswhencomparedtotheFPtrajectories.
Thedomesticflightshaveupto3%savingsthatareequivalenttoabout5-10minutesintimeand0.
4-1.
0tonnesoffuelforthetopthreedomesticairports:KSDF,KEWRandKMEM.
Theinternationalflightscanpotentiallysavemorethan10%byflyingwindoptimaltrajectories.
FortheoutboundflightstoVHHH,RKSIandZSPD,traveltimeisreducedby23-54minutesandfuelburnby2.
3-7.
6tonnes.
Ingeneral,thesavingsareproportionaltothetriplengths.
Theyalsodependonthedirectionofflightandaircrafttypes.
Long-distanceinternationalflightsgainlargetimeandfuelsavingsforallaircrafttypes.
DomesticflightswithMD11,B744,orB772havelargerfuelsavingsthanwithB763andDC10.
ThedomesticflightsfromANCtoKOAKandtheinternationalflightsfromRJAAtoANChavethesmallestsavings,respectively.
Notethattheaccuracyoftheassessedbenefitsisgreatlyaffectedbythequalityoftheflightplansinthedatabase.
FPtrajectoriesthataresimulatedusingflightplanswithanincompletelistofwaypointsleadtoanunderestimationoftherequiredtraveltimeandfuelburn.
Thisisbecauseflightsareassumedtoflygreatcircleroutesbetweenwaypoints.
Ifaflightplanhasfewwaypoints,theFPtrajectoryissimilartothegreatcircletrajectoryforthatflight.
Subjecttothewindconditions,greatcircletrajectoriescanbeverysimilartowind-optimaltrajectories,especiallyfortheshorttrips.
Hence,thebenefitofflyingawindoptimaltrajectoryisunderestimatedfortheseflights.
Severalotherfactorsalsoaffecttheaccuracyoftheresults.
TheactualverticalflightpathofeachFPtrajectoryisapproximatedbythoseofWOMFLtrajectorysincetheactualaircrafttakeoffweightandairspeedprofileforeachflightarenotavailableinthecurrentdatabase.
EachsimulatedFPtrajectoryfollowstheverticalpaththatassumesatypicalaircraftweightandanairspeedprofilefromBADAinsteadoftheoriginalverticalprofile.
ThisapproximationmayresultsintheperformanceunderestimationforFPtrajectories.
Inaddition,thepotentialbenefitsforWOMFLtrajectoriesneedtobediscountedsincetheyneglecttheadditionalcostsinvolvedinoperatinginsideforeignairspace.
V.
ConclusionThisstudydevelopedatrajectoryoptimizationalgorithmthatminimizesthecostoftimeandfuelburnbyintegratingamethodforcomputingminimum-timeroutesinwindsonmultiplehorizontalplanesandanaircraftfuelburnmodelforgeneratingfuel-optimalverticalprofiles.
Itisappliedtoevaluatethepotentialbenefitsofflyingwind-optimalroutesinaseamlessairspaceforcommercialcargoflightsoperatingbetweenAnchorage,AlaskaandmajorairportsinAsiaandthecontiguousUnitedStates.
Flyingwindoptimaltrajectorieswithfuel-optimalverticalprofilesreducesaveragefuelburnofinternationalflightsflyingatasinglecruisealtitudeby1-3%.
Long-distanceflightsnotonlygainfuelsavingsfromen-routestepclimbsbutmayalsobenefitfromhigh-speedwindsathigheraltitudes.
Thepotentialfuelsavingsofperformingen-routestepclimbsareverysmallforthedomesticcargoflightssincemostofthemhaveonlyonestepclimb.
Theoptimaltrajectoriesreducefuelburnandtraveltimerelativetotheirflightplanroutesforalltheairportpairs.
Domesticflightscanpotentiallysaveupto3%onfuelburnandtraveltime,whichisequivalenttoabout5-10minutesintimeand0.
4-1.
0tonnesoffuelforthetopthreedomesticairports.
Internationalflightscanpotentiallysaveupto10%onfuelandtraveltimebyflyingwindoptimaltrajectories,reducingtraveltimeby23-54minutesandfuelburnby2.
3-7.
6tonnes.
Theactualsavingsinoperationswillvaryfromthesimulationresultsduetodifferencesintheaircraftmodelsanduserdefinedcostindices.
Ingeneral,thesavingsareproportionaltothetriplengths,anddependontheen-routewindconditionsandaircrafttypes.
AppendixA.
OptimalCruiseAltitudesTakingthefirstderivativeoftheaircraftfuelflowratewithrespecttothealtitudetoobtainthefollowingequations,fh=Cfcr1000(DSFCh+SFCDh),(A1)whereSFCh=Cf1Cf2VTAShandDh=12CD0SVTAS2CD24m2g2ρ2SVTAS2&'()*+ρh.
(A2)SFCisaconstantwhenVTASremainsunchangedforvariouscruisealtitudes,i.
e.
,VTASh=0SFCh=0.
Ingeneral,ρh≠0;andsolvingtheequation:Dh=0yieldsCD0SVTAS2=CD24m2g2ρ2SVTAS2.
Thisimpliesρ2=CD24m2g2CD0S2VTAS4.
(A3)Belowthetropopause(altitudeh=11000m),thetemperatureisaconstant.
TheairdensityisρISA=ρtrop_ISA*e(gRgasTtrop_ISA)(h11000).
(A6)TheformulasfortheoptimalcruisealtitudesunderISAareobtainedusingEquations(A3-A6)B.
ApplicationofDynamicProgrammingTheconceptofDPisappliedtodeterminethelocationsforen-routestepclimbsatthestepclimbtimes.
Theoptimalstepclimbslocationsareapproximatedbasedonthetotalcost-to-goassociatedwitheachwind-optimalextremal.
AssumingthatthereareN-1possiblestepclimbsandnpossiblelocationsforeachclimb,aforwarddynamicprogrammingapproachwithN-1stagesandnpossiblestatesinthesetofSNforeachstagecanbeformulatedas:J(SN)=0J(SN1)=minSN1{J(SN)+CN1(SN1)}J(SN2)=minSN2{J(SN1)+CN2(SN2)}MJ(S1)=minS1{J(S2)+C1(S1)}J(S0)=minS0{J(S1)+C0(S0)}(A7)Notethat,thestartingcostattheoriginJ(SN)iszerowithNstagestogo.
Theoptimalcost-to-goattheN-1stage,J(SN-1),isthelinkcost,CN-1(SN-1),fromtheorigintothecurrentstate.
ThelinkcostistheassociateddirectoperatingcostspecifiedbyEq.
(2)alongthewind-optimalflightpath.
Ingeneral,theoptimalcost-to-goatanyarbitrarystagesdependsontheoptimalcost-to-gointhepreviousstageandtheminimumcostfrompreviousstagetothecurrentstage.
TheoptimalcostisrepeatedlysolveduntilthedestinationisreachedatS0.
TheoptimalsolutioncanbefoundinO(Nxnxn)operations.
Eachoperationinvolvecomputationofthelinkcostanddeterminingtheoptimalcost-to-go.
Thecomputationtimeforthisapproachisveryhighsincetheoptimalsolutionateachstatehastobecomputedtoprovidethefinalsolution.
ThealgorithmdevelopedinthisstudyapproximatestheaforementionedapproachbysolvingN-1individualDPproblemsthateachhasoneclimb(N=2)inordertoreducethecomputationaleffort.
Eachofthe1-stageDPproblemcanbewrittenasthefollowing:J(S2)=0J(S1)=minS1{J(S2)+C1(S1)}J(S0)=minS0{J(S1)+C0(S0)}J(S0)=minS1{C1(S1)+C0(S0)}(A8)Thesolutionforeachproblemistheoptimalstepclimblocationthatbecomestheoriginforthesubsequentproblems.
ThecomplexityofthisalgorithmisreducedtoO(Nxn).
Thisalgorithmwhenappliedintherealtimealsoaccommodatesthefrequentlyupdatedweatherforecastanddynamicnatureofatmosphericconditionsbyoptimizingtheaircrafttrajectorysegmentsateachstepclimbusingmostupdatedweatherforecast.
TheDPapproachinEq.
(A8)isequivalenttothatofEq.
(A7)whenaircraftperformsonly1stepclimb.
Thisistruefordomesticcargoflightsoperatingbetweenthe2mostpopularcitypairs.
ThealgorithmcanbeformulatedbyadaptingEq.
(A7)forflightswithmorethan1stepclimbtogainbetteraccuracy.
Thecurrenttwo-stepsoptimizationapproachlimitsaircraften-routestepclimbsatthetimeinstantsandflightlevelsthataredeterminedbytheaircraftverticalprofiletoenhancecomputationalefficiency.
Ingeneral,alargerselectionforpossibleaircraftcruisealtitudesandclimbingtimesthatresultsinlargerNandncanbeselectedfortheDPapproach.
Combiningthehorizontalandverticaloptimizationproceduresprovidesabetterapproximationoftheglobaloptimalsolutionbutsignificantlyincreasesthecomputationalefforts.
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