NETWORK CELL ROUTING MODEL FOR CONTROL OF THROUGHPUT AND DELAY OF AIR TRAFFIC

Transcription

1 NETWORK CELL ROUTING MODEL FOR CONTROL OF THROUGHPUT AND DELAY OF AIR TRAFFIC Alex T. Nguyen an John S. Baras, Unversty of Marylan, College Park, MD Abstract Attempts to manage ar traffc by ether ecreasng elay or ncreasng flow can have the reverse effect on the other. In the Natonal Arspace System (NAS), flghts typcally epart at ther chosen tmes, an flow management technques are mplemente n the ar n an effort to maxmze flow, whch coul lea to elay for nvual arcraft. Other areas of the worl allow flghts to epart only at preetermne slot tmes knowng that there s a clear unobstructe path, leang to mnmal elays but possbly unerutlzng arspace. We propose a new approach n leveragng the hghway cell transmsson moel (CTM) to the arspace n the form of a mult-obectve optmzaton that traes between maxmzng throughput an mnmzng elay. The moel s pose as a multcommoty traffc flow nteger program where the constrants are relaxe slghtly from the CTM to examne strateges for achevng optmal throughput an elay. In practce, the moel s envsone to ntally run offlne to etermne a prelmnary soluton to the ntal state of the system. Solutons for subsequent changes n actual state can be etermne by runnng the moel onlne for the ncremental state change. Stochastc events such as convectve weather clearng tmes or capactes can be nclue n the moel to better see the benefts an mpacts of pre-postonng traffc to take avantage of possble future clearng. Introucton In ths paper we present a traffc flow moel that s nspre by the Cell Transmsson Moel (CTM) from hghway traffc. Lghthll-Whtham-Rchars (LWR) ntally propose a set of partal fferental equatons (PDE) that escrbe the ynamcs of hghway traffc flow [1] [2]. Ths was later use by Daganzo [3] [4] to create the Cell Transmsson Moel, n whch a seres of consecutve cells s use to represent ncrements of a hghway. The traffc flow from one cell to the next n the smulaton moel s governe by a scretze approxmaton of the flow concentraton curve from LWR. Ths moel has been wely use n hghway transportaton for a varety of purposes, such as recent work on networke sgnal control. [5] Recently applcatons of ths moel to ar traffc have nclue the CTM(L) moel whch mposes capactes on a set of cells. [6] The moel uses as a ecson varable the amount of flghts to retan n a cell, an moves other flghts to the aacent cell. The obectve functon mnmzes the usage n the network, an the network conssts of fxe preetermne paths between each orgn-estnaton par. We propose a moel that ynamcally routes traffc through congeste areas an optmzes for not only mnmal elay through the system, but also traes wth maxmzng throughput. The moel s formulate as a multobectve nteger program wth smlar structure to a multcommoty traffc flow program. The moel leverages the framework of the Cell Transmsson Moel, but wth routng capabltes an two obectves. Ar traffc flow moels typcally establsh a noe at the ntersecton of each lnk, an entfy the amount of flow nto an out of that noe wth the assumpton that the flow s steay between the noes. In realty, a bottleneck at a certan pont on a lnk ue to weather, heavy traffc, or other reasons can cause backups an ncrease mles n tral restrctons on arcraft heang to the area. The ynamcs of arcraft movement from cell to cell woul be benefcal for tactcal actons. Ths coul be use n combnaton wth hgher-level flow moels that n combnaton woul prove for strategc plannng of traffc flows aroun congeste areas. Whle the specfc ynamcs of nvual vehcles on a hghway are not all the same as those for arcraft n the ar, we can examne the applcaton of the overall framework of the CTM to the ar wthout moelng nvual arcraft /09/$ IEEE. 2.B.2-1 Authorze lcense use lmte to: Unversty of Marylan College Park. Downloae on July 20,2010 at 20:21:06 UTC from IEEE Xplore. Restrctons apply.

2 Applcaton of the CTM woul prove more ntuton nto the ynamcs of the congeston along each segment whch woul not typcally be obvous when usng a typcal flow moel that only nclues noes at ntersectons. Knowlege of where an when arcraft woul be mpacte by the congeston ownstream, how many woul be affecte, an ther spee reuctons or vectors are mportant n the management of the traffc. CTM Overvew In the general CTM moel, each arcraft s on an nepenent path p (treate as a sngle commoty) from an orgn arport to a estnaton arport. Lnks are broken nto cells, numbere from 1 to gong n the ownstream recton (recton of travel). The length of each cell can be set to be such that wthout congeston, vehcles woul travel from one cell to the next when the tme goes from t to t +1. There s only one-way travel on each secton of cells. Ths s acceptable snce n the Natonal Arspace System (NAS) the routes are one-way for partcular alttue levels. The state varable s n () t, the number of vehcles contane n cell at tme t. The flow s controlle by () y t, the number of arcraft that s to flow nto cell n the tme nterval (t, t +1). The state transton equaton can be wrtten smply as n ( t+ 1) = n ( t) + y ( t) y ( t) + 1 In the CTM, the relatonshp between the flow rate an traffc ensty s erve from a scretze verson of the PDEs from the LWR moel. [4] The flow rate s approxmate by a trapezoal functon of the traffc ensty. When there s free flow, the total flow s proportonal to the number of vehcles n the cell. When the traffc ensty ncreases, the flow levels off an s cappe at a certan maxmal flow amount. As traffc further ncreases beyon the amount that yels the maxmum flow, the flow begns to ecrease. Ar traffc s not as elastc a hghway traffc n the way that the flow cannot reach to zero as traffc levels ncrease. However, there are small spee controls that can be mplemente to slghtly lower arcraft spee urng areas of hgh congeston. Alternatvely, effectve spee reucton or holng wthn a cell can be vewe as arcraft vectorng to reuce the spee component along the recton of travel. The nflow y () t s thus expresse as { } y () t = mn n (), t Q(), t N () t n () t 1 where the parameters use are: N () t max number of vehcles that can be present n cell at tme t Q () t max number of vehcles that can flow nto cell when the clock avances from t to t+1 The two equatons above escrbe the ensty of flghts n the cells an how flghts flow from one cell to the next. Applcaton to Ar Traffc Routng Here we apply the CTM to an optmzaton form for use wth arcraft n the ar. The ntent s to moel the arcraft flow rather than optmal groun holng strateges. However, results to mprove network flow wll nherently lea to certan amount of flghts beng hel on the groun n orer to optmze elay an throughput n the ar. The flow equaton from the CTM has a mn{ } operator that requres the flow to be no greater than any of the three terms, an also equal to one of the three terms. In orer to optmze the elay an throughput, we relax the mn{ } operator to allow y () t to take on values less than all of the terms. In ong so, flghts can take a cautous approach an not avance as quckly f there s a possbly of a ecrease n capacty ownstream. Flghts can even reman out of the arborne network an reman on the groun wthout ncurrng hgh costs. Ths mnmzes the arborne elay by not havng planes take off an wat n the ar at bottleneck ponts. To construct the network for the moel, the man flows of traffc n the arspace can frst be etermne va a clusterng metho. These paths are then converte nto a network of cells whch can moel the movement of arcraft along each route an propagate elays ue to weather or other flow 2.B.2-2 Authorze lcense use lmte to: Unversty of Marylan College Park. Downloae on July 20,2010 at 20:21:06 UTC from IEEE Xplore. Restrctons apply.

3 constrane areas to etermne the effects fferent control methos have on throughput an elay. The network s set up such that each arcraft s estne for a partcular estnaton arport, an may take fferent paths to reach the estnaton to avo congeston an elays. Fgure 1 shows a sample network wth two orgns an two estnatons. A bottleneck exsts n cell 3, an causes the maorty of traffc from Orgn 1 to Destnaton 1 that woul ornarly travel along cells to nstea travel along the path Source 01 n = 8 Source 07 n = 8 y = Scheule epartures y = Scheule epartures Org 1 1 Org N = Dest 1 6 Dest 2 10 Fgure 1. Sample Network Setup Repostory cells are nclue n front of the orgn arports to prove a source of arcraft. As flghts are scheule to epart, they are sent from the repostory cells to the epartng arport by settng y, () t to be the number of flghts scheule to epart for estnaton urng the tme pero t to t+1. At the orgn arport, they coul be elaye on the groun before takng off f congeston exsts n the network. The orgn arport s assume to have nfnte capacty for arcraft to ncur elay pror to eparture. Smlarly, a snk cell can be ae after each estnaton for the flghts to move away from the estnaton once they arrve. For the problem wth multple commotes an routng, the state an flow varables are reefne as: n () t number of vehcles contane n cell at tme t gong to estnaton y, () t flow from cell to cell n the tme nterval (t,t+1) that are estne for. The parameters are: N () t max number of vehcles that can be present n cell at tme t Q () t max number of vehcles that can flow nto cell when the clock avances from t to t +1 l ( ) Inex of the cell at estnaton. G s Set of snks (the cells that are mmeately after the estnaton cells). R Set of cells aacent to cell from whch flghts can flow nto cell. S Set of cells aacent to cell that flghts can move to from cell. A Total no. of arrvals for estnaton. a, Incates whether there s a val path from cell to an aacent cell. The capacty values for arcraft ensty an flow are allowe to vary by locaton an tme n orer to moel changes n weather patterns, specal use arspace, or other congeston. 2.B.2-3 Authorze lcense use lmte to: Unversty of Marylan College Park. Downloae on July 20,2010 at 20:21:06 UTC from IEEE Xplore. Restrctons apply.

4 We esre to mnmze the elay whle also maxmzng the throughput. A measure of elay can be expresse as the total amount of tme the arcraft spen n the system. Throughput can be etermne by the number of arcraft that ext the system. Several optons exst for optmzng several obectve functons. In ths moel we use the weghtng factor wf for the throughput functon an combne the two functons nto the sngle obectve. The total usage s mnmze mnus a weghte factor of the total throughput. We coul nstea maxmze throughput mnus the usage, but the values for usage ten to be large an coul result n negatve obectve values. The moel s formulate as follows: mn c n ( t) w y ( t) Subect to: f l( ), l( ) + 1 Gs t t R n ( t+ 1) = n ( t) + y ( t) y ( t) k,, k R S t,, (1) y, () t n() t t,, (2) S n () t N() t t, (3) y, () t Q() t t, (4) R n (), t y () t, A cost factor c can be nclue n the elay functon for each cell f certan routes are esre more than others. In aton, a hgher cost can be apple to all cells except the orgn cells on the groun to encourage flghts reman on the groun to reuce ar elay. The flow conservaton equaton for each estnaton an cell s capture n Constrant 1. Ths shows that the number of arcraft n the next tme ncrement s equal to the number currently n the cell, plus the amount comng n from varous cells, mnus the amount extng. Constrants 2-4 essentally correspon to the terms n the mn{} functon n the CTM moel. Constrant 2 ensures that the amount of flow leavng a cell cannot be greater than that currently n the cell. Up to now, each estnaton, or commoty, s hanle nepenently. Constrant 3 proves a capacty for the flghts n a partcular cell, across all estnatons. An smlar capacty constrant (4) can be nclue to lmt the amount of flow nto a partcular cell for all commotes. Atonal constrants nee to be ae to ensure that flghts only go to ther ntene estnaton. Ths coul be one by forcng the total number of arrvals at each estnaton over all tme to be equal to the epartures as follows: () nl( ) t = A (5a) t Ths constrant requres that there be enough tme for all flghts to reach ther estnaton, an woul fx the throughput. Alternatvely, ths coul be one by restrctng flghts from arrvng at estnatons that are not ntene for them. h n () 0 l ( ) t = (5b) t h Ths constrant oes not restrct all flghts to actually reach the estnaton n the tme allotte, thus allowng for varatons n throughput base on the amount of tme avalable n the moel. The constrant, though, may cause flghts to waner aroun takng the path of least resstance n any recton f t can t reach the estnaton wthn the moelng tme, but the mnmal usage / elay obectve shoul lmt that. In aton, another constrant nees to be ae to force arrvals to move to the aacent snk at each estnaton so they on t lnger at the estnaton over tme to satsfy arrval constrant. y + () t = n () t t, (6) l( ), l( ) 1 l( ) For executon n a solver program, the moel may be mofe wth atonal parameters an mofcatons n the set of nces for the varous terms as follows: n ( t+ 1) = n ( t) + a y ( t) a y ( t) k, k,,, k t,, (7) 2.B.2-4 Authorze lcense use lmte to: Unversty of Marylan College Park. Downloae on July 20,2010 at 20:21:06 UTC from IEEE Xplore. Restrctons apply.

5 a, y, () t n() t t,, (8) n () t N() t t, (9) a, y, () t Q() t t, (10) Results & Dscusson The moel proves a means to traeoff between reucng elay an maxmzng throughput. By austng the weghtng factor n the obectve functon, numerous solutons can be obtane wth varyng egrees of elay an throughput. Fgure 2 shows a sample output showng the resultng elay (represente as usage n the system) an throughput for varyng values of the weghtng factor w. Usage f Usage/Delay vs Flow Flow Ieal Fgure 2. Delay vs Throughput Traeoff Obectve The usage/elay vs flow curve shows a traeoff between the two obectves, wth the esre recton beng towars greater flow an lower usage. The knee n the curve ncates a tunng n the moel that possbly proves an optmal balance between the two obectves. Decson makes can use ths to talor the moel to the esre traeoff. Ths moel proves the ntal framework for ncluson of atonal features such as stochastcty, possbly n the form of parameters. It woul be eal f an nteger soluton coul be obtane n an effcent manner n orer to hanle atonal complextes later. Moels that have a constrant matrx that s totally unmoular wll have relaxatons that yel nteger solutons an can be solve quckly. Typcal multcommoty flow moels usually o not have constrant matrces that are totally unmoular, wth the excepton of those that have ether two or less sources or snks [7]. The constrant matrx for ths problem has the followng form: N1 ϒ1 A= N2 0 0 ϒ 2 a n A have { } All elements a 1, 0, + 1. Ν 1 an ϒ1 have columns that ether have all 0 s or have exactly one +1 an one -1. Wth ust these two submatrces, the constrant matrx woul be totally unmoular. However, N2 an ϒ2 are also nclue, an correspon to the flow capacty constrants across the multple estnatons. Whle a sngle entty submatrx below the flow constrants woul preserve the total unmoularty of A, these two submatrces have the form[ I I I...]. However, not satsfyng the usual suffcent conton for total unmoularty oes not mean that the problem cannot yel nteger solutons. It s possble that n some cases the lnear program relaxaton may prouce nteger solutons. Ths woul be an eal stuaton for computatonal effcency. Ths moel, whle smlar to a multcommoty flow moel, has more structure that may lea t to yel nteger solutons from the relaxaton. The solutons erve thus far on a small number of samples use seem to ncate some optmsm for nteger solutons for the moel. The actual computatonal effcency for a small sample s shown n Fgure 3. The ncrease n computaton tme relatve to the network sze s shown along wth the respectve number of arports use n the scenaro. The computaton tme appear to yel well wth the network sze an arport sze. However, ths may vary greatly epenng on the specfc network structure. 2.B.2-5 Authorze lcense use lmte to: Unversty of Marylan College Park. Downloae on July 20,2010 at 20:21:06 UTC from IEEE Xplore. Restrctons apply.

6 Computaton Tme Computaton Tme Num Arports Fgure 3. Computaton Tme 10-Cell network 20-Cell network Conclusons & Future Work A moel has been evelope that moels the movement of flghts across a network an proves for a traeoff between mnmal elay an maxmal throughput. Inspre by the Cell Transmsson Moel, ths moel captures the movement of arcraft from cell to cell wthn a network, an etermnes the optmal routng of arcraft through congeste areas. Ongong work on the moel nclue etermnng effcent nteger solutons to the problem, as well as makng the moel stochastc wth ncluson of parameters to moel ranomness n flght eparture tmes an ts effect on elay an throughput. References [1] Lghthll, M.J., J.B. Whtham, 1955, On Knematc Waves. I Flow movement n long rvers. II A theory of Traffc Flow on Long Crowe Roas, Proc. Royal Socety, A 229, pp [2] Rchars, P.I., 1956, Shockwaves on the Hghway, Operatons Research, Vol. 4, No. 1, pp [3] Daganzo, C., The cell transmsson moel: A ynamc representaton of hghway traffc consstent wth the hyroynamc theory, Transportaton Research Part B, Vol. 28B, No.4, pp [4] Daganzo, C., The cell transmsson moel, Part II: Network traffc, Transportaton Research Part B, 29(2). [5] Gabrel G., Roberto Horowtz, Alex Kurzhanskya, Pravn Varaya, Jamyoung Kwon, 2008, Behavor of the cell transmsson moel an effectveness of ramp meterng, Transportaton Research Part C 16, [6] Sun, D., Alexanre Bayen, Multcommoty Euleran-Lagrangan Large-capacty Cell Transmsson Moel for en route traffc, AIAA Journal of Guance, Control an Dynamcs, vol. 31, no. 3, pp , [7] Evans, J. R., J. J. Jarvs an R. A. Duke, "Matros, Unmoularty, an the Multcommoty Transportaton Problem," Presente at the Jont ORSA/TIMS Meetng, Chcago, th Dgtal Avoncs Systems Conference October 25-29, B.2-6 Authorze lcense use lmte to: Unversty of Marylan College Park. Downloae on July 20,2010 at 20:21:06 UTC from IEEE Xplore. Restrctons apply.