PAPER #07-3016 Freeway Traffc Shocwave Analyss: Explorng the NGSIM Trajectory Data by Xao-Yun Lu* PATH, Insttute of Transportaton Studes Unversty of Calforna, Bereley Rchmond Feld Staton, Bldg 452 1357 S. 46th Street Rchmond, CA 94804-4648 Tel: 1-510-665 3644, Fax: 1-510-665 3537 Emal: xylu@path.bereley.edu Alexander Sabardons Insttute of Transportaton Studes Unversty of Calforna, Bereley 109 McLaughln Hall, Bereley, CA 94720-1720 E-mal: sabardons@ce.bereley.edu For presentaton and publcaton 86 th Annual Meetng Transportaton Research Board Washngton, D.C. January 2007 November 15, 2006 # Words: 3,109 Plus 6 Fgures (1,500) Total: 4,519 *Correspondng author
Lu/Sabardons 1 Abstract The paper presents the development and applcaton of a numercal algorthm to estmate the propagaton speed of shocwaves on freeways based on vehcle trajectory data. The essence of the algorthm s that the shocwave propagaton speed s the travelng speed of local mnma of consecutve vehcle speed trajectores. The applcaton of the algorthm on the NGSIM datasets from the I-80 and US101 freeways under congested condtons shows that all the shocwaves have smlar propagaton speed of about 11.4 mph (18.34 ph), whch s ndependent of the traffc flow speed pror to congeston. The algorthm developed here s generc and can be used for shocwave analyss based on any vehcle-by-vehcle trajectores estmated from any range sensors. The fndngs have been appled to estmate the upper bound for tme delay error n ln travel tme estmaton from pont sensor data. Keywords: traffc flow, shocwaves, NGSIM, vehcle trajectores
Lu/Sabardons 2 1. Introducton Shocwaves can be defned as boundary condtons between states of traffc flow on hghway facltes (e.g., from free-flow to congeston). Understandng the formaton and characterstcs of shocwaves s mportant n studyng congeston patterns and mpacts, developng mproved analyss tools and desgnng traffc management strateges. The study of shocwaves n turn requres data on vehcle movements and nteractons over tme and space. Such data are very lmted; exstng databases nclude vehcle trajectores produced at Oho State Unversty n the 70 s [1], a FHWA study on vehcle nteractons n the early 80 s [2], and more recently sample vehcle trajectores and shocwaves on a short secton of I-680 freeway [3]. Recently, data on ndvdual vehcle trajectores were collected and made avalable under the Next Generaton Smulaton (NGSIM) project [4], a natonal effort amng to develop mproved algorthms and datasets for calbraton and valdaton of traffc smulaton models. The NGSIM data provde a unque opportunty to nvestgate drver behavor, better understand traffc dynamcs and formulate mproved models. The NGSIM freeway database conssts of vehcle trajectores on two test stes [5]. The I-80 (BHL) test secton s a 0.40 mle (0.64 Km) 6-lane freeway weavng secton wth an HOV lane. Processed data nclude 45 mnutes of vehcle trajectores n transton (4:00-4:15 pm) and congeston (5:00-5:30 pm). The US101 ste s a 0.3 mle (0.5 m) weavng secton wth fve lanes. Processed data nclude 45 mnutes of vehcle trajectores n transton (7:50-8:05 am) and congeston (8:05-8:35 am). The data have been extracted from vdeo recordngs usng machne vson algorthms [6]. The paper presents an exploratory analyss of shocwaves on freeways, and the development and applcaton of a numercal algorthm to estmate the shocwave propagaton speed based on the NGSIM vehcle trajectory data. The wor s part of an ongong study on emprcal understandng of traffc dynamcs and formulaton of mproved models. The algorthm developed here s generc and can be used for shocwave analyss based on any vehcle trajectores estmated from any range sensors as long as the speed trajectores are reasonably smooth. Secton 2 of the paper dscusses qualtatvely some observatons from the nspecton of vehcle speed-tme and dstance-tme plots and the characterstcs of shocwaves. Secton 3 descrbes the formulaton of an algorthm to estmate the shocwave propagaton speed. The numercal mplementaton of the algorthm and the results from ts applcaton to the NGSIM data are presented n Secton 4. Secton 5 presents an applcaton of the fndngs. The last secton summarzes the study fndngs and outlnes future research drectons. 2. Data Observatons The NGSIM data format s vehcle ID, lane and poston at 0.1 sec ntervals. We processed the data to produce tme-dstance and tme-speed plots for each freeway lane, and also speeddstance contour plots. A total of 11,779 vehcles were processed. Detaled descrpton of the data processng and all the plots are ncluded elsewhere [7]. Fgures 1 and 2 show sample speed-tme and dstance-tme plots from the two test stes. Fgure 1 also llustrates dfferent flow regmes and transtons; from free flow to congeston on-set and
Lu/Sabardons 3 buld up (regon a), congested statc state (regon b), and recovery (regon c). The duraton of congeston (or shocwave propagaton) s also shown as tme d over dstance e n ths Fgure. Also, we can observe the correspondence between the speed drops n the mult-vehcle speed envelop and the shocwave n the dstance-tme plot. The speed envelop drop s approxmately proportonal to the shocwave propagaton dstance. Further observatons of these plots show that each wave leads to a valley for the speed curves. The relatve depth of the valley represents speed reducton or the effects of shocwave. Thus, such depth could be used to classfy the shocwave: long term congeston or short term fluctuaton. Each valley corresponds to at last one local mnmum. There could be several local mnmum f the speed remans constant n the valley (e.g., stopped vehcles), but there s a unque mnmum whch corresponds to the earlest tme. The propagaton of those unque mnmum values determnes the shocwave propagaton speed. Fgure 3 shows vehcle trajectores for lanes 1 and 2 of the US101 test ste under congested condtons. Shocwaves are shown as straght lnes propagatng bacwards n tme and space. The speed of the shocwave s the slope of the lne, estmated at about 11 mph. Fgure 4 shows the vehcle trajectores and shocwaves for lanes 2 and 3 of the I-80 test ste under congeston (lane 1 s the HOV lane, operatng under free-flow condtons). These Fgures show that the shocwave characterstcs and speeds are very smlar on both stes. Fgure 5 shows the speed and dstance trajectores of selected vehcles related to sngle shocwave. It s clear that the speed-tme curve s of 2 nd order parabolc type, and the dstancetme curve s a 3 rd order curve. Vehcle trajectores related to multple waves are shown n Fgure 6. The speed-tme and space-tme trajectores for mult waves correspond to combnatons of 2 nd order and 3 rd order curves respectvely. Based on the above qualtatve observatons, we developed a numercal algorthm to estmate the shocwave speed from the vehcle trajectory data. The algorthm searches for those unque local mnmum values for each speed-tme trajectores. It then determnes how those local mnma for each trajectory propagate n space and tme, whch provdes the shocwave propagaton speed. 3. Proposed Algorthm for Estmatng the Shocwave Propagaton Speed The followng notaton s used n ths secton: j vehcle or trajectory ndex (each vehcle corresponds to a trajectory) n a consecutve order; two vehcles are consecutve when (a) they are n the same lane, and (b) vehcle j +1 s behnd vehcle j; the ndex of shocwave: There may be more than one shocwave appeared n one data set; and each vehcle may be nvolved n more than one shoc wave; t - dscrete tme ponts for = 0,1, 2,... K; synchronzed for all the vehcles; t = t t s a constant of 0.1 sec data sample perod; + 1 t - the ndex of tme step;
Lu/Sabardons 4 t - tme ponts correspondng to the local mnmum of the speed-tme trajectory for vehcle j; ( j) yj( t ) - dstance of vehcle j at tme pont t ; all the vehcle trajectores startng from the same pont; v ( t ) - speed of vehcle j at tme pont t ; j For each vehcle j, suppose a tme and dstance coordnate (n a global nerta coordnate system) ( j) ( j) for the unque local mnmum pont correspondng to a speed drop s (, j( )) t y t. The tme ponts for mnma of dfferent consecutve trajectores correspondng to the same shocwave are not necessarly equally dstrbuted,.e., ( j+ 1) ( j t ) t s not necessarly constant for all the vehcles nvolved n the same shocwave. However, there should hold the followng relatonshp for any two consecutve vehcles j and j+1: t ( j+ 1) ( j) > t y t < y t ( ( j+ 1) ) ( ( j) ) j+ 1 j (3.1) whch s a necessary condton for a shocwave appearng at the onset of congeston. Ths s because the vehcle behnd begns to decelerate at shorter dstance relatve a common startng pont, and t s mpossble that t j+ 1 > t j y t y t ( ( j+ 1) ) ( ( j) ) j+ 1 j (3.2) The shocwave propagaton speed () V j for consecutve vehcles s estmated as: V ( j+ 1) ( j) y () 1( ) ( ) j+ t y j t j ( j+ 1) ( j) t t = < 0 (3.3) () The negatve sgn of V j ndcates that the shocwave s bac-propagatng. The mplementaton of the algorthm requres not only the vehcle movng dstance but also the start tracng pont to get the poston of the vehcle wth respect to the nertal coordnate system at any tme. Suppose there are J vehcles nvolved n the -th shocwave. Then the average shocwave propagaton speed s defned as: V 1 (3.4) J 1 () () = Vj J 1 j= 1
Lu/Sabardons 5 4. Numercal Implementaton of Shocwave Speed Estmaton Algorthm The mplementaton of the above descrbed shocwave speed estmaton algorthm conssts of the followng steps: (a) flterng to smooth the speed-tme trajectores, (b) searchng local mnmum of the smoothed trajectores, (c) clusterng the tme and dstance ponts correspondng to those local mnma, and (d) calculatng the average shocwave propagaton speed from the dstance-tme ponts correspondng to the clustered local mnma. Vehcle trajectory flterng The algorthm requres fndng the mnmum values of the speed trajectores and ther correspondng tme and dstance ponts. The trajectores from the NGSIM data used n the study have dsturbances due to measurement nose and estmaton error, whch may cause problems n the search for the true mnmum values of speed trajectores. We apply a low pass flter and a rate lmt flter to smooth the trajectory data to elmnate those rrelevant local mnma due to nose. The followng low pass (Butterworth) lnear flter s used for flterng purpose: x1( ) = 0.4320 x1( 1) 0.3474 x2( 1) + 0.1210 vn( ) x2( ) = 0.3474 x1( 1) + 0.9157 x2( 1) + 0.0294 vn( ) vout ( ) = 0.4984 x1( ) + 2.7482 x2( ) + 0.0421 vn ( ) = 1,2,... Where x 1 (), x 2 () are the flter state varables, v n s the nput speed sgnal and v out s the output fltered speed. The ntal condton s chosen as ( x1(0), x 2(0)) = (0,0). Fgures 5 and 6 show smoothed speed-tme and dstance-tme trajectores usng the above methods. Search for local mnma of speed trajectores Several methods exst for search local mnmum such as Newton gradent method, steepest decent method for contnuous functons and genetc algorthm for dscontnuous functons [8,9]. However, these methods are not sutable for the purpose of ths analyss, because we need to account for the followng: (a) Although lnear flter has been used to smooth the speed-tme trajectores, the speed obtaned from dstance measurement by numercal dfferentaton stll may stll have estmaton nose, whch may cause traps n the local mnmum search. Thus the algorthm should be able to move out of the traps due to nose n speed trajectory. (b) It s suffcent to locate only the approprate mnma,.e., speed drops exceedng some threshold. Small speed drops that are not due to a shocwave should be fltered out. (c) The algorthm should smple to facltate mplementaton on large data sets The proposed algorthm s called movng wndow mnmum searchng. A tme wndow of wdth L s specfed (the number of tme ponts to be searched), and the algorthm searches progressvely (after each progressve search, moves one step forward). The algorthm searches the local mnmum of the speed trajectory for any vehcle j whch may be related to some shocwave. If the speed s constant for a tme nterval (e.g., the speed s zero for stopped
Lu/Sabardons 6 vehcles) there are uncountable number of mnmum values. In ths case, the frst mnmum s regstered, whch s the unque mnmum ( j v ) for each speed valley, and corresponds to a unque ( ) ( j) ( j) tme dstance pont, j( ) t y t. The algorthm can be descrbed n generc semantcs: For j = 1: J Whle t < T =1 If < L v = mn v t, v t,..., v t { ( 1) ( 2) ( )} t s the frst tme pont among {,..., } j j j j If Then s regster t s as 1 =+1 Else v = mn v t, v t,..., v t ( j ) ( j) t and also y( t ) t t ( s { ( ) ( + 1) ( + )} t + s the frst tme pont among {,..., L} ( ) j j j j L If s v = v t +, then j j s 1 regster t + s as 1 =+1 If Loop End =+1 Whle Loop End j=j+1 For Loop End ( j ) ( j) t and also y( t ) 1 1 ) such that v v ( t ) t t + ( s L) such that j j s 1 =, Note that for short tme wndow L, more local mnmum other than the shocwave related mnmum could be regstered. Also f L s specfed too long, some true local mnmum values related to shocwaves may be mssed. In ths applcaton, L was set as 20 seconds, or 200 tme steps,.e., shocwaves lastng less than 20 seconds are gnored. Clusterng of local mnmum values The local mnma obtaned from the algorthm descrbed n the prevous secton belong to each trajectory. It s necessary to cluster consecutve local mnma to form a chan of mnma for a shocwave. The mnma should satsfy the followng crtera to be ncluded n a cluster: Two consecutve mnma and ther tme ponts should have the same sequental order, or: ( j+ 1) ( j) t t > δ
Lu/Sabardons 7 Two consecutve mnma should come from dfferent but also consecutve speed trajectores; ( 1) ( ) Correspondng dstance pont should satsfy the condton (1),.e. ( j+ ) ( j y ) j+ 1 t < y j t The followng condtons are satsfed: max y t y t ( j+ 1) ( j) j+ 1( ) j( ) V t ( j+ 1) ( j) t V mn (0.1) where δ > 0 s pre-specfed tme threshold; V mn, V max are possble mnmum and maxmum shocwave propagaton speed respectvely. In the algorthm mplementaton, we used the followng values: δ = 0.1 sec V mn = 3 ft/sec (1 m/sec) V max = 30 ft/sec (10 m/sec) The algorthm for buldng the clusters can be descrbed as follows: ( j) ( j) Step 0: Set j = 1, =1 and Intalze cluster 1 wth ( t, yj( t )) Step 1: Set j = j+1 ( j) ( j) ( j) ( j) Step 2: Tae a tentatve pont ( t, y ( )) 0 j t from trajectory j and chec f ( t, ( )) 0 y 0 j t 0 and ( ( j 1) ( j t, 1) y 1( )) j t satsfy the condton n (1.5) for all the prevous clusters set up so far; ( j) ( j) If YES: regster ( t, yj( t )) as the follower of ( ( j 1) ( j t, 1) yj 1( t )) n the same cluster; 0 0 If NO: Intalze a new cluster wth the frst mnmum pont of the second trajectory: Set =+1; ( j) ( ) (, ( j ) j ) t y t ; Notce that the subscrpt of has been ncreased by 1; ( j) ( j) Step 3: Remove the tentatve pont (, j( )) Step 4: Repeat Step 2 t y t from the set of mnma for trajectory j Ths process wll stop after all the mnma for all the trajectores have been put n some cluster. Estmaton of the shocwave speed Once the clusterng has been acheved, the average shocwave propagaton speed can be calculated from the clustered mnma usng Equatons (3.3) and (3.4). The algorthm was appled to all NGSIM data sets. The estmated shocwave speed was found to be 11.4 mph (18.3 ph).
Lu/Sabardons 8 5. Applcatons The fndngs on shocwave characterstcs and propagaton speeds can be used to dentfy the spatal and temporal mpacts of congeston, and to develop and calbrate traffc flow models. In ths secton, we llustrate how the results can be used to estmate the upper bound for tme delay error n the ln travel tme estmaton from pont sensors, such as loop detectors. Suppose the detecton method needs nformaton from both upstream and downstream loop detectors. In ths worst case, let τ c denote the upper bound for the shocwave to reach the upstream detector and V shoc s the shocwave propagaton speed. The followng relatonshp holds: L= V τ (5.1) shoc c where L s the detector spacng. Or equvalently L τ c = V (5.2) The upper bound for tme delay error estmaton can be expressed as: shoc L τ = τ + τ = + τ (5.3) max c 0 0 Vshoc Where the tme delay τ 0 s caused by data processng delays such as aggregaton over tme, and/or by persstence checng to reduce false alarm n congeston onset detecton. For example, f the data update rate s 0.1s, the data processng delay s 0.5s and persstence checng s 10 steps, then τ 0 = 1.5s. The estmated as above shocwave speed V shoc s 5.1 m/sec (16.72 ft/sec). Equaton (5.3) becomes L τ max = + τ 0 (5.4) 5.1 whch provdes the upper bound for tme delay from the dstance between the two detector statons and the nstant pont speed at the downstream staton. Equaton (5.4) can be used by traffc engneers to determne how dense the loop statons should be nstalled on hghways to acheve the acceptable estmaton error threshold τ max. Dscusson The estmated shocwave propagaton speed of 11.4 mph (18.3 ph) s the same for both the test stes for the tme perods studed. The estmated speed from the algorthm s the same wth the drect measurement from the Fgures 3 & 4. Also, ths estmate s also n agreement wth the estmated shocwave speed of 12.5 mph reported usng the I-680 data [3]. It s also demonstrated how the results can be used for the estmaton of upper bound for tme delay error n the ln travel tme estmaton based on the detecton at two termnal statons such as loops,
Lu/Sabardons 9 whch mmedately leads to a relatonshp between sensor densty and tme delay error threshold. The tme perods analyzed were all n congeston and transton. We wll nvestgate subject to the avalablty of data, the shocwave speed and characterstcs under dfferent traffc flow condtons. Also, we wll nvestgate the effects of the congeston causes on shocwave characterstcs. The algorthm s beng mplemented as part of a software tool to automate the data processng and analyss of large data sets nvolvng vehcle trajectores and other mcroscopc data. The fndngs on shocwave characterstcs and propagaton speeds are used n the development and calbraton of traffc analyss models for oversaturated freeway flow. The results are also used n assessng the effectveness of alternatve methods and survellance technologes to dentfy the congeston onset[7].
Lu/Sabardons 10 ACKNOWLEDGEMENTS Ths wor was performed as part of the Calforna PATH Program (TO5320) of the Unversty of Calforna, n cooperaton wth the State of Calforna Busness, Transportaton, and Housng Agency, Department of Transportaton (Caltrans). The contents of ths report reflect the vews of the authors who are responsble for the facts and the accuracy of the data presented heren. The contents do not necessarly reflect the offcal vews or polces of the State of Calforna. Ths report does not consttute a standard, specfcaton, or regulaton.
Lu/Sabardons 11 REFERENCES 1. Trterer, J., Investgaton of Traffc Dynamcs by Aeral Photogrammetry Technques, Transportaton Research Center, Oho State Unversty, Report 278, 1975. 2. Smth, S., Freeway data collecton for studyng vehcle nteractons. Techncal Report. Report No. FHWA/RD-85/108, U.S. Department of Transportaton, Federal Hghway Admnstraton, Offce of Research, Washngton, DC, 1985. 3. Cofman, B., Tme Space Dagrams for Thrteen Shoc Waves, PATH Report No. UCB-ITS-PWP- 97-1, Unversty of Calforna, Bereley, 1997. 4. Alexads, V., Colyar, J., Halas, J., Hranac, R., and McHale, G. The Next Generaton Smulaton Program. ITE Journal, Vol. 74, No. 8, 2004, pp. 22-26. 5. NGSIM Homepage. FHWA. http://ngsm.fhwa.dot.gov. 6. Sabardons A., and V. Alexads, Traffc Data Through the Bereley Hghway Laboratory, Worshop on Traffc Modelng, Sedona, AZ, September 2005 7. Lu, X. Y., A. Sabardons, and et al, Hghway Traffc Parameter Estmaton and Congeston On-Set Detecton, PATH Worng Paper, 2006. 8. M. S. Bazaraa, H. D. Sheral and C. M. Shetty, Nonlnear Programmng Theory and Algorthms, John Wley & Sons, Inc., 1993. 9. Press, W. H., S. A. Teuolsy, W. T. Vetterlng and B. P. Lannery, Numercal Recpes n C, The Art of Scentfc Computng, 2nd Ed., Cambrdge Unversty Press, 1992.
Lu/Sabardons 12 LIST OF FIGURES Fgure 1. Shocwave Characterstcs (NGSIM US101 Dataset, 7:50-8:05am, Lane 1) Fgure 2. Shocwave Characterstcs (NGSIM, I-80 Dataset, 5:00-5:15 pm, Lane 2) Fgure 3. Vehcle Trajectores & Shocwave Speed (NGSIM,US-101 Dataset, 8:20-8:35am) Fgure 4. Vehcle Trajectores & Shocwave Speed (NGSIM, I-80 Dataset, 5:15-5:30 pm) Fgure 5. Vehcle Trajectores: Sngle Wave Propagaton-(NGSIM, US-101 Dataset, Lane 1) Fgure 6. Vehcle Trajectores: Combnaton of Three Waves-(NGSIM, US-101 Dataset, Lane 1)
Lu/Sabardons 13 Fgure 1. Shocwave Characterstcs (NGSIM, US101 Dataset, 7:50-8:05am, Lane 1)
Lu/Sabardons 14 Fgure 2. Shocwave Characterstcs (NGSIM, I-80 Dataset, 5:00-5:15 pm, Lane 2)
Lu/Sabardons 15 Fgure 3. Vehcle Trajectores & Shocwave Speeds (NGSIM, US-101 Dataset, 8:20-8:35 am)
Lu/Sabardons 16 Fgure 4. Vehcle Trajectores and Shocwave Speeds (NGSIM, I-80 Dataset, 5:15-5:30 pm)
Lu/Sabardons 17 Fgure 5. Vehcle Trajectores: Sngle Wave Propagaton (NGSIM, US101 Dataset, Lane 1)
Lu/Sabardons 18 Fgure 6. Vehcle Trajectores: Combnaton of Three Waves (NGSIM, US101 Dataset, Lane 1)