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New Method for Probabilistic Traffic Demand Predictions for En Route Sectors Based on Uncertain Predictions of Individual Flight Events

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Nnth USA/Europe Ar Traffc Management Research and Development Semnar (ATM2011) Berln, Germany, June 2011 New Method for Probablstc Traffc Demand Predctons for En Route Sectors Based on Uncertan Predctons
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Nnth USA/Europe Ar Traffc Management Research and Development Semnar (ATM2011) Berln, Germany, June 2011 New Method for Probablstc Traffc Demand Predctons for En Route Sectors Based on Uncertan Predctons of Indvdual Flght Events Eugene P. Glbo and Scott B. Smth J. A. Volpe Natonal Transportaton Systems Center 55 Broadway, Kendall Square, Cambrdge, MA 02142, U.S.A. {eugene.glbo, Abstract Federal Avaton Admnstraton (FAA) Ar Traffc Flow Management (TFM) decson-makng s based prmarly on a comparson of predcted traffc demand and capacty (usually called Montor/Alert Parameter, or MAP) at varous elements of Natonal Arspace System (NAS) such as arports, fxes and sectors to dentfy potental congeston. The current FAA Traffc Flow Management System (TFMS) and ts decsonsupport tools operate wth determnstc predctons and do not consder the stochastc nature of the predctons. Sector demand predctons appear to be less accurate and stable than predctons for arports and fxes. The major reason s that, unlke arports and fxes where flghts are aggregated n 15-mnute ntervals, TFMS predcts sector demand by aggregatng flghts for each mnute and usng the one-mnute peak demand as a measure for sector demand for entre 15-mnute nterval. Ths paper presents a novel analytcal approach to and technques for translatng characterstcs of uncertanty n predctng sector entry tmes and tmes n sector for ndvdual flghts nto characterstcs of uncertanty n predctng onemnute sector demand counts. The paper shows that expected one-mnute sector demand predctons are determned by a probablstcally weghted average of one-mnute sector entry demand predctons for several consecutve one-mnute ntervals wthn a sldng tme wndow. The wdth of the wndow s determned dependng on probablty dstrbutons of errors n flghts sector entry tme predctons. Expected one-mnute sector demands along wth standard devatons of demand counts are expressed va probablstc averagng of seres of one-mnute determnstc predctons of number of flghts enterng a sector. The results of the paper contrbute to probablstc predctons of congeston n arspace. These analytcal results can also be used to evaluate the mpact of mproved accuracy n flght tmng predctons on reducng uncertanty n traffc demand predctons, hence leadng to better dentfcaton of congeston n arspace. 1. Introducton Federal Avaton Admnstraton (FAA) Ar Traffc Flow Management (TFM) decson-makng s based prmarly on a comparson of traffc demand and capacty predctons at varous Natonal Arspace System (NAS) elements such as arports, fxes and en-route sectors. The current Traffc Flow Management System (TFMS) and ts decsonsupport tools deal wth determnstc predctons and neglect the stochastc nature of the predctons. An mportant part of the Next Generaton Ar Transportaton System (NextGen) program s the transton from determnstc to probablstc TFM that would lead to more realstc and effcent TFM decsons. Research on probablstc TFM and related problems s underway n many organzatons ncludng government, prvate sector and academa. Recent results demonstrate the potental beneft of applyng a stochastc approach to TFM (see, e.g., [1] [13]). A general concept of probablstc TFM for managng congeston n the NAS under uncertan predctons of demand and capacty, as well as an ncremental, probablstc approach to TFM decson-makng can be found n [10] and [11], respectvely. Research results presented n [12] and [13] provde an mportant contrbuton to probablstc TFM descrbng a constructve approach to ncorporatng probablstc weather forecasts nto probablstc TFM, as well as desgn of the modelng tool for evaluatng TFM strateges. The practcal value of decson-support tools used for probablstc TFM wll sgnfcantly depend on qualty of data used n the tools to represent uncertanty n the avaton system, n partcular, uncertanty n predctng traffc demand and capacty. Therefore thorough data analyss along wth analytcal tools are needed to examne the sources of uncertanty, to characterze uncertanty and to apply advanced statstcal methods for reducng uncertanty n predctons. Research on uncertanty, whch s the core ssue for probablstc TFM, s conducted n two major drectons: uncertanty n traffc demand predctons and uncertanty n predctng the capacty of NAS elements. Each of those drectons addresses two major elements of the NAS: arports and arspace. Although arports and arspace are equally mportant n the NAS and are generally subject to smlar sources of uncertanty n traffc demand and capacty predctons, there are substantal dfferences n measurng both demand and capacty n arports and arspace. For example, both demand and capacty are better defned and measured for arports than for en route sectors. Moreover, because of the dfferences n measurng traffc demand n arports and n sectors n the current TFMS, the TFM specalsts notced that demand predctons n sectors are more uncertan, more volatle and less relable than those for arports. The TFMS measures traffc demand for arports and sectors for each 15-mnute nterval of the tme perod of nterest. The prncpal dfferences n measurng traffc demands for sectors and arports are as follows: 1. Traffc demand n sectors s based on onemnute aggregate counts vs. 15-mnute arrval or departure counts at arports. 2. Current TFMS determnes traffc demand n sectors on a 15-mnute bass and consders the maxmum one-mnute count wthn a 15-mnute nterval as the traffc demand for the sector for entre 15-mnute nterval, whle a 15-mnute traffc demand for arports ncludes all flghts wth ETAs wthn the 15-mnute nterval. 3. Sgnfcant fractons of one-mnute demand counts n adjacent one-mnute ntervals n a sector mght contan the same flghts whle, at arports, adjacent 15-mnute ntervals contan dfferent flghts. Our research has been focused on developng a methodology that allows for a quanttatve representaton of uncertanty n ar traffc demand predctons for NAS elements [1] - [6]. Our prevous research, reported n [1] [3], was focused on the accuracy of TFMS 15-mnute aggregate traffc demand predctons wthout consderng ndvdual flghts that comprse those aggregate counts. We also developed a regresson model amed at mprovng the accuracy and stablty of those aggregate predctons. The regresson model ncluded demand predctons at three consecutve 15-mnute ntervals wth the nterval of nterest n the mddle of the 45-mnute tme wndow. Includng demand predctons n adjacent ntervals mplctly takes nto account the effect of uncertanty n predctons of arrval tmes for ndvdual flghts. The regresson mproved both the accuracy of demand predctons and the stablty and accuracy of TFMS Montor/Alert. The next step n our research, reported n [4] and [5], was focused on analyzng uncertanty n predctons of arport arrval tmes for ndvdual flghts and developng a methodology for translatng the uncertanty n estmated tme of arrval (ETA) for ndvdual flghts nto uncertanty n aggregate 15-mnute traffc demand predctons for arrval arports. The result was a methodology for probablstc traffc demand predctons at arports wth quanttatve characterstcs of uncertanty n the predctons. The motvaton for usng uncertanty n predctng tmes for ndvdual flghts n the characterzaton of uncertanty n predctng aggregate traffc demand s as follows. Current TFMS provdes determnstc demand predctons by aggregatng flghts whose estmated tmes of arrval or departure (ETAs or ETDs) fall wthn a tme nterval of nterest wthout consderng uncertanty (random errors) n flght s ETA or ETD. Those errors are the major contrbutors nto uncertanty n aggregate demand count predctons. The physcal mechansm that causes the errors n aggregate demand s mgraton of some flghts ETAs from one tme nterval to another durng flght updates due to random errors, so that a flght that counted n one nterval can be counted n another (earler or later) nterval after updatng ts ETA. The queston s how to formalze the translaton of characterstcs of uncertanty n ndvdual flght tmng predctons nto characterstcs of uncertanty n aggregate traffc demand predctons. A method that allowed for ths translaton was frst proposed n [7]. The method provded the analytcal means for obtanng a probablty dstrbuton of aggregate demand at a specfc tme nterval of nterest through probablty dstrbutons of errors n tmes of arrvals of ndvdual flghts predcted to arrve at ths specfc nterval of nterest. The method, however, dd not consder an extended set of flghts that also ncludes the flghts predcted to arrve n several adjacent ntervals. Consderng the probabltes for those flghts to arrve wthn the nterval of nterest wll affect the Probablty Densty probablty dstrbuton of aggregate demand for the nterval of nterest. Another approach was proposed n [8] for probablstc predcton of aggregate traffc demand n en route sectors based on probablstc characterstcs of uncertanty for ndvdual flghts tmes to be n a sector. In partcular, the paper focused on consderng the probablty dstrbutons of flght departure tmes from orgn arports for estmatng the expected number of flghts n a sector. The paper presents a methodology and analytcal results that demonstrate how characterstcs of uncertanty n predcton of tmes for ndvdual flghts translates nto characterstcs of uncertanty n predcton of aggregate one-mnute traffc demand counts n sectors. Lke our prevous work, t s based on a statstcal analyss of current TFMS data. The translaton of the characterstcs of uncertanty n TFMS predctons of tmes for ndvdual flghts nto characterstcs of uncertanty n predctons for aggregate demand counts s a challengng problem. However, t s much more complcated for sectors than for arports because of the dfferences n measurng traffc demands for these NAS elements. The paper s organzed as follows. Secton 2 presents characterstcs of uncertanty n TFMS predctons of tmes for ndvdual flghts to cross sector boundary and be n a sector Secton 3 presents the methodology for recalculatng characterstcs of uncertanty n ndvdual flghts tmng predctons nto characterstcs of uncertanty n aggregate onemnute demand counts for both enterng a sector and beng n a sector. Conclusons are gven n Secton 4. The Appendx contans a more detaled dervaton of mathematcal expressons necessary for probablstc sector demand predctons. 2. Characterzaton of uncertanty n TFMS flghts sector entry and sector occupancy tme predctons TFMS estmates and perodcally updates sector entry tmes for ndvdual flghts. To estmate the accuracy of those predctons, the TFMS data was collected durng the days of moderate demand when there were no TFM ntatves. Wthout such nterference by TFM control, the errors n predctons can be measured by the dfference between predcted and actual tmes. The analyss was conducted on the TFMS data for sxteen en route sectors: ZBW02, ZBW09, ZBW17, ZBW20, ZBW46, ZID82, ZID83, ZID86, ZLC06, ZLC16, ZMP20, ZOB57, ZOB67, ZOB77, ZSE14, and ZTL43. The data ncluded repeated updates for flght sector entry and sector ext tmes. Altogether, durng Aprl 10 through Aprl 16 of 2009, there were approxmately 834,000 tme predctons for 39,000 flghts analyzed. The look-ahead tmes (LAT) for predctons vared from 0 (for actual tmes) to 3 hours. The data analyss was performed on the above mentoned data and on addtonal data set collected n Aprl and June of 2007 and reported n [4]. The detaled results of analyss of accuracy n sector entry tme predctons can be found n the Volpe report [6]. The dstrbuton of predcton errors and ther parameters (average and standard devaton) were estmated separately for actve (arborne) flghts and for proposed (not yet departed) flghts. Fgure 2-1 llustrates the typcal probablty densty functons of predcton errors n sector entry tmes for actve and proposed flghts. 9% 8% 7% 6% 5% 4% 3% 2% 1% 0% Predcted - Actual Sector Entry Tme (mnutes) Actve Proposed Fgure 2-1 Probablty densty of errors n sector entry tme predctons The results of analyss show and Fgure 2-1 llustrates that For actve flghts, the dstrbutons of predcton errors are somewhat asymmetrc wth a heaver rght-hand tal, ndcatng a tendency for flghts to enter sectors earler than predcted For the proposed flghts, the dstrbutons of predcton errors are asymmetrc wth heavy left-hand tals, ndcatng a tendency for flghts on the ground to, on average, enter sectors later than predcted. The standard devaton of error s substantally lower for actve than for proposed flghts. It was n the 4 to 12 mnute range for actve flghts, and n the 15 to 22 mnute range for proposed flghts. Sectors vary n sze and n the manner flghts traverse them. As a result, there are sgnfcant dfferences among the sectors n tme-n-sector for a flght. As for the accuracy of tme-n-sector predctons, t s much better than for flght s sector entry tmes and s not much dfferent for actve and proposed flghts and for dfferent LAT. For example, the standard devaton of tme-n-sector error was 4 mnutes or less for both actve and proposed flghts [6]. The probablty dstrbutons of flght tmng predctons are fundamental for aggregatng flghts n probablstc one-mnute sector demand predctons. 3. Probablstc predctons of sector demand counts The number of flghts n a sector durng a specfc one-mnute nterval ncludes the flghts that entered the sector durng ths nterval and the flghts that entered the sector earler and are stll n a sector. As standard devatons of errors n sector entry tme predctons are sgnfcantly greater than one mnute, there are sgnfcant probabltes that a flght would enter a sector earler or later than the flght s ETA. Therefore, a probablstc demand predcton at a specfc one-mnute nterval would need to consder TFMS one-mnute demand predctons for several consecutve one-mnute ntervals surroundng the nterval of nterest. The probablstc predctons of sector one-mnute demand counts requre the followng steps: 1. Translate a flght s tme predctons and assocated predcton errors nto the probablty for the flght to enter a sector durng a partcular mnute. 2. Develop a probablstc characterzaton for the number of flghts enterng a sector, based on the probabltes from step Develop probablstc count predctons for the number of flghts present n a sector durng a partcular mnute (a one-mnute traffc demand counts for a sector). Each step s consdered below. 3.1 Probablty for a flght to enter a sector durng a one-mnute nterval. Our report [6] presents a detaled analytcal approach for determnng probabltes for the flghts to enter a sector durng a one-mnute nterval dependng on the flghts estmated sector entry tmes (ETAs) and probablty dstrbutons of errors n predctng sector entry tmes. The status of ndvdual flghts (actve or proposed) s taken nto account by usng dfferences n accuracy of predctng sector entry tme for actve and proposed flghts. Ths secton presents an example of the flght probablty to enter a sector durng a onemnute nterval. Followng the notaton of [6], s a one-mnute nterval of nterest that s between the begnnng of mnute and the begnnng of mnute (+1), k s a one-mnute nterval for a flght s estmated sector entry tme (ETA), F(x) s a cumulatve dstrbuton functon (CDF) of predcton error for sector entry tme, P, s a probablty for a flght determnstcally k predcted to enter a sector durng a one-mnute nterval k to enter a sector durng a one-mnute nterval, P, 0.5 [F( k + 1) F( k 1)]. k It should be noted that, snce the CDFs are dfferent for actve and proposed flghts, separate calculatons of probabltes are performed for actve and proposed flghts. These formulas permt the calculaton of probabltes for varous ntervals of nterest, ncludng a seres of consecutve ntervals, e.g.,, +1, +2, etc. For the sake of smplcty, n the examples presented below, we assume that the predcton errors are symmetrcally dstrbuted. Table 3-1 shows several values of probabltes for dfferent k, surroundng the nterval of nterest. Ths table llustrates the probabltes for a flght to cross a sector boundary at an nterval of nterest f t determnstcally predcted to enter a sector earler (k ) or later (k ) or on tme (k = ). It also llustrates the relatve sgnfcance of the probabltes dependng on the tme dfference k. The probabltes have been calculated for the Gaussan dstrbuton F (x) wth zero mean and standard devatons of σ = 4 mnutes (whch mght correspond to the accuracy of predctons for actve flghts), and σ = 15 mnutes (whch mght correspond to the accuracy of predctons for proposed flghts). Table 3-1 Probabltes for Flghts to Enter a Sector durng Interval Probablty σ = 4 σ = 15 P, P,+1 = P, P,+2 = P, P,+3 = P, P,+4 = P, P,+5 = P, P,+6 = P, P,+7 = P, P,+8 = P, P,+9 = P, P,+10 = P, P,+15 = P, P,+20 = P, P,+25 = P, P,+30 = P, P,+35 = P, P,+40 = P, These probabltes tend to be small (less than 0.10), snce they correspond to sngle mnutes. For example, f a flght s determnstcally predcted to enter a sector two mnutes earler (k = -2) or two mnute later (k = +2) than the nterval of nterest, the probablty for the flght to enter a sector durng nterval s smaller: P, -2 = P, For the actve flghts (σ = 4 mnutes) wth ETAs at least nne mnutes from the nterval of nterest (earler or later) the probabltes to enter a sector durng nterval are neglgbly small (less than 0.01). It s mportant to note that when predctons of sector entry tmes are less accurate, the probablty for a flght to enter a sector n a partcular mnute becomes smaller, even n the mnute that t was forecast to enter the sector (approxmately when σ = 15). Another mportant observaton from Table 3-1 s that, snce the standard devatons of predcton errors are much larger than one-mnute, the probabltes for the flghts to enter a sector do not vary much from one mnute to the next. In partcular, for σ = 4 mn, the probabltes for flghts wth ETAs n ntervals, +1 and -1 to arrve to a sector n nterval are, respectvely equal to 0.099, and 0.096,.e., they are nearly dentcal. For σ = 15, the correspondng probabltes a practcally the same for the flghts wth ETAs n thrteen (!) ntervals around the nterval of nterest (ncludng nterval ). 3-2 Number of Flghts Enterng a Sector Durng a One-mnute Interval The results of the prevous secton can be used for characterzaton of uncertanty n predcton of number of flghts enterng a sector. Our report [6] descrbes a detaled analytcal approach to the probablstc predcton of aggregate one-mnute sector entry counts based on probabltes for ndvdual flghts to enter a sector and on determnstc predctons of sector entry counts at varous one-mnute ntervals. In ths analyss, ndependence of errors n predctng tmes among the flghts s assumed. The man result of takng nto consderaton characterstcs of uncertanty n flghts sector entry tme predctons s that the flghts wth ETAs n several adjacent one-mnute ntervals to the nterval of nterest wll be consdered n calculatng the aggregate demand n the nterval of nterest. For example, f d k flghts determnstcally predcted to enter a sector durng one-mnute nterval k, there s the probablty P,k for each of the d k flghts to enter the sector durng the one-mnute nterval of nterest. Assumng that the errors n flght arrval predctons are ndependent, the probablty dstrbuton of number of flghts from d k that can be counted n nterval s a bnomal dstrbuton. The total random number of flghts predcted for a specfc one-mnute nterval s equal to the sum of the random numbers of flghts from several adjacent ntervals k that can be counted n the nterval of nterest. The number of adjacent ntervals β that should be taken nto account depends on relatve values of probabltes P, k. β = max k, where max k s the dstance beyond whch the probabltes small and should be neglected. P, k become too Accordng to the propertes of the bnomal dstrbuton, the expected one-mnute count d and the standard devaton of the one-mnute count are equal to, respectvely [14]: Flghts Enterng n the Mnute Flghts Enterng n the Mnute d = P kd, (3.1), k k σ = P, k ( 1 P, k ) d k, (3.2) k where d k s a determnstc predcton of sector entry counts for a one-mnute nterval k. The probablstc predcton of one-mnute flght counts enterng a sector can be represented by the expected number d and the uncertanty are
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