Traffic Density Estimation with the Cell Transmission Model 1

Transcription

1 Traffic Density Estimation ith the Ce Transmission Mode Laura Muñoz, Xiaotian Sun, Roberto Horoitz 2, Luis Avarez 3 Department of Mechanica Engineering University of Caifornia at Berkeey Berkeey, CA Abstract A macroscopic traffic fo mode, caed the sitchingmode mode (SMM), has been derived from the ce transmission mode (CTM) and then appied to the estimation of traffic densities at unmonitored ocations aong a highay. The SMM is a hybrid system that sitches among different sets of inear difference equations, or modes, depending on the mainine boundary data and the congestion status of the ces in a highay section. Using standard inear systems techniques, the observabiity and controabiity properties of the SMM modes have been determined. Both the SMM and a density-based version of the CTM have been simuated over a section of I-21 West in Southern Caifornia, using severa days of oop detector data coected during the morning rush-hour period. The simuation resuts sho that the SMM and CTM produce density estimates that are both simiar to one another and in good agreement ith measured densities on I-21. The mean percentage error averaged over a the test days as approximatey 13% for both modes. 1 Introduction Freeay traffic data is often avaiabe in the form of occupancy and voume measurements coected from singe or doube oop detectors embedded in the pavement [1]. In conjunction ith effective vehice ength data, these measurements can be converted into macroscopic quantities such as traffic density and speed. Loop detector data sets are often incompete or contain bad sampes. For instance, from [1] it can be seen that approximatey 3% of the possibe oop sampes in Caifornia s District 7, hich contains over 3 freeays, ere missing, on average, over the period from March 22 to February 23. Hoever, on-ramp metering contro strategies, such as ALINEA [2], require accurate oca traffic density information in order to effectivey reguate on-ramp infos to the freeay. It is thus essentia to have a means of reconstructing missing traffic density measurements. To address these concerns, an open-oop density estimator, based on the ce transmission mode (CTM) [3, 4], has been designed, and has been shon to perform e hen tested 1 Research supported by UCB-ITS PATH grant TO Professor, horoitz@me.berkeey.edu; author for correspondence. 3 Professor, Universidad Naciona Autónoma de México /3/$ IEEE 375 ith data from Interstate 21 in Southern Caifornia. We refer to this estimator as the sitching-mode mode (SMM). The sitching-mode mode is a inear time-varying mode, derived from a modified ce transmission mode that uses density instead of occupancy as its state variabe. The ce transmission mode, a macroscopic traffic mode, as seected for this research due to its anaytica simpicity and abiity to reproduce congestion ave propagation dynamics. The CTM has previousy been vaidated for a singe freeay ink (ith no on-ramps or off-ramps) using data from I-88 in Caifornia [5]. The modified CTM, from hich the SMM is derived, is simiar to that of [3, 4], except that it (1) uses ce densities as state variabes instead of ce occupancies, (2) accepts nonuniform ce engths, and (3) aos congested conditions to be maintained at the donstream boundary of a modeed freeay section. Using ce densities instead of ce occupancies permits the CTM to to incude uneven ce engths, hich eads to greater fexibiity in partitioning the highay. Nonuniform ce engths aso enabe us to use a smaer number of ces to describe a given highay segment, thus reducing the size of the state vector [ρ 1...ρ N ] T, here ρ i is the density of the i th ce. Whie it is expected that partitioning a segment into a arge number of ces can improve numerica accuracy, our interest here is to test our methods using a smaer state vector and to simpify the design of estimators and controers. Aoing congested fo rates at donstream boundaries is necessary to enabe the mode to ork ith rea highay data. In the modified CTM, a highay is partitioned into a series of ces. The density of ce i evoves according to conservation of vehices. For the case of a inear highay segment ith no on- or off-ramps, vehice conservation can be ritten as ρ i (k +1)=ρ i (k)+ Ts i (q i (k) q i+1 (k)). (1) Here, k is the time index, T s is the discrete time interva, i is the ength of ce i, and q i (k) is the fo rate, in vehices per unit time, into ce i during the interva [k, k +1).As described in [4], q i (k) is determined by taking the minimum of to quantities: q i (k) =min(s i 1 (k),r i (k)), (2) here S i 1 (k) = min(vρ i 1 (k),q M,i 1 ), is the maxi- Denver, Coorado June 4-6, 23

2 Q(ρ) Q M v ρ J Figure 1: Fo as a function of density mum fo that can be suppied by ce i 1 under freefo conditions, over the interva [k, k +1), and R i (k) = min(q M,i,(ρ J ρ i (k))), is the maximum fo that can be received by ce i under congested conditions, over the same time interva. Eqs. (1) and (2) are the density-based equivaents of those described in [3]. The modified CTM aso uses density-based versions of the merge and diverge as of [4] to incorporate on-ramp and off-ramp fos. The CTM parameters are depicted in the fundamenta diagram of Fig. 1. They can be vaid for a ces or aoed to vary for each ce. The free-fo speed v is the average speed at hich vehices trave don the highay under uncongested (o density) conditions. is the average speed at hich congestion aves propagate upstream through the highay under fuy congested conditions. Q M is the maximum fo that occurs at critica density ρ c, and ρ J is the jam density. It is hepfu to revie here some of the terminoogy and naming conventions that i be used throughout this paper. The congestion status of ce i is determined by comparing the ce density ith the critica density: if ρ i <ρ c,i, the ce has free-fo status, otherise ρ i ρ c,i and the ce has congested status. The SMM sitches beteen severa sets of inear difference equations depending on the vaues of the mainine boundary inputs and on the congestion status of the ces in a section. Each set of inear equations is referred to as a mode of the SMM. The SMM estimates the movement of congestion ave fronts through a highay section. Here, a ave front is understood to be a status transition, upstream of hich nearby ces have one status (e.g. free-fo), and donstream of hich nearby ces have the opposite status. 2 Sitching-Mode Mode ρ ρ In the sitching-mode mode, e describe the ce transmission mode as a hybrid system that sitches beteen 5 sets of inear difference equations, depending on the congestion status of the ces and the vaues of the mainine boundary data. Assuming our state variabe is the ce densities, ρ =[ρ 1...ρ N ] T, the key difference beteen the CTM and the SMM is that, ith respect to density, the former is noninear, hereas each mode of the atter is inear. The SMM can be extracted from the modified CTM by riting each inter-ceuar fo, q i, as either an expicit function of ce density, vρ i 1 (k) or (ρ J ρ i (k)), or as a constant, Q M. This expicit density dependence is achieved by suppying a set of ogica rues that determine the congestion status of each ce, at every time step, based on measurements at the segment boundaries. For simpicity, the fooing assumptions are made: 1. The densities and fos at the upstream and donstream segment boundaries, as e as fos on a the on-ramps and off-ramps, are measured. 2. There is at most one status transition (or ave front) in the highay section. If both the upstream and donstream mainine boundaries are of the same status, i.e., both free-fo or both congested, e assume that a the mainine ces, 1 through N, have the same status; hie if the to boundaries are of different status, there exists a singe ave front in the segment, upstream of hich a the ces have congested (free-fo) status, and donstream of hich a ces have free-fo (congested) status. The singe-ave front assumption is an approximation that is expected to be acceptabe for short highay segments ith ony one on-ramp and off-ramp, such as the exampe ater in this section. To more accuratey dea ith onger sections ith many on- and off-ramps, the sitching ogic can be modified to ao mutipe ave fronts ithin a segment. Since an SMM-modeed section contains at most one congestion ave front, the modes of the SMM can be distinguished by the congestion status of the ces upstream and donstream of the ave front. If there is no ave front in the section, e can use a doubed abe, e.g. Free-fo Free-fo to indicate the absence of any status transition. The five modes are denoted: (1) Free-fo Free-fo (FF), (2) Congestion Congestion (CC), (3) Congestion Free-fo (CF), (4) Free-fo Congestion 1 (FC1), and (5) Free-fo Congestion 2 (FC2). The to modes of Free-fo Congestion are determined by the reative magnitudes of the suppied fo of the ast uncongested ce upstream of the ave front and the receiving fo of the the first congested ce donstream of the ave front. If the former is arger, the SMM is in FC1; hie if the atter is arger, it is in FC2. Respectivey, these to cases are distinguished by hether the congestion ave is traveing backard or forard ithin the segment. Consider the highay segment in Fig. 2, hich is divided into 4 ces. The measured aggregate fos and densities at the upstream and donstream mainine detectors are denoted by q u, ρ u, and q d, ρ d. A five modes of the SMM can be summarized as foos: ρ(k+1) = A s ρ(k)+b s u(k)+b J,s ρ J +B Q,s q M, (3) 3751 Denver, Coorado June 4-6, 23

3 q 1 q u, ρu oop detectors ρ 1 q 2 q 3 q 4 q d, ρd ρ2 ρ3 ρ4 r 2 f 3 Figure 2: Highay segment divided into 4 ces here s =1, 2, 3, 4, 5 indicates the mode (1: FF, 2: CC, 3: CF, 4: FC1, 5: FC2), ρ = [ρ 1...ρ 4 ] T is the state, and u = [q u r 2 f 3 ρ d ] T are the fo and density inputs; specificay, r 2 and f 3 are the measured on-ramp and off-ramp fos entering and eaving the section, subscripted according to their ce of entry or exit. ρ J = [ρ J1 ρ J2 ρ J3 ρ J4 ρ J5 ] T is the vector of jam densities, and q M =[Q M1 Q M2 Q M3 Q M4 ] T is the vector of maximum fo rates. In the FF mode, each ce is abe to accept the fo suppied by its upstream neighboring ce; and ce accepts a of the boundary info q u, hie ce dumps vehices at the free-fo speed v 4. The state matrices are 1 v 1T s A 1 = 1 v 2T s v 2 T s 1 v 3T s v 3 T s T s T B 1 = s Ts v 1 T s q 5 1 v 4T s B J,1 = 4 5, B Q,1 = 4 4. (4) In the CC mode, each ce can ony dump the amount of fo that can be accepted by the donstream neighboring ce. The number of vehices that ce can emit is determined by the boundary density ρ d, hie ce receives vehices up to its capacity. The state matrices are 1 1T s 2 T s A 2 = B J,2 = 1 2T s 1 3T s 3 T s 4 T s 1 4T s T s B 2 = Ts 5 T s 2T s 2 T s 3T s 3 T s 4T s 1 T s 4 T s 5T s B Q,2 = 4 4. (5) In the CF mode, there exists one congestion-to-free-fo transition inside the section. One property of the SMM is that the ave front i aays ie on a ce boundary. Ces 3752 upstream of the ave front behave as congested ces, hie ces donstream of the ave front reease vehices at the free-fo rate. The ave front itsef acts as a botteneck, expeing vehices at maximum aoed rate Q M, and decouping the region upstream of the ave front from the donstream region. For the case here the ave front is ocated in beteen ces 2 and 3, the state matrices are: 1 1T s 2 T s 1 2T s A 3 = 2 1 v 3T s v 3 T s 1 v 4T s T s B 3 = Ts 1 T s 2T s B J,3 = 2 T s 2 B Q,3 = Ts T s. (6) In both FC modes, one free-fo-to-congestion transition exists inside the section. Unike the previous mode, the state matrices change depending on the direction of motion of the ave front. In FC1, the ave front moves donstream. Assuming, for exampe, that the ave front is beteen ces 2 and 3, the state matrices for this mode are: 1 v 1T s v 1 T s A 4 = 2 1 v 2T s vt s 1 4 T s 1 4T s 4 T s T s B 4 = 2 Ts 5 T s B J,4 = 4T s 4 T s 5T s B Q,4 = 4 4. (7) For FC2, the ave moves upstream. Again assuming that the ave front is beteen ces 2 and 3, this mode differs from the previous case in that, due to the dominance of the congested fo rate at the ave front boundary, the tridiagona ro is no the second instead of the third ro, and more terms appear in B J,s : 1 v 1T s 1 A 5 = v 1 T s 1 1 3T s 4 T s 4 T s 1 4T s Denver, Coorado June 4-6, 23

4 B 5 = B 4, B J,5 = 3T s 3 T s 4T s 4 T s 5T s B Q,5 = 4 4. (8) At each time step, the SMM determines its mode based on the measured mainine boundary data and the congestion status of the ces in the section. If both ρ u and ρ d have free-fo status, the FF mode is seected, and if both of these densities are congested, the CC mode is seected. If ρ u and ρ d are of opposite status, then the SMM performs a search over the ρ i to determine hether there is a status transition inside the section. This ave front search consists of searching through the ces, in order, ooking for the first status transition beteen adjacent ces. It is expected that some error i be induced in the ave front ocation predicted by the SMM, since the search for a status transition is performed on the states estimated by the SMM, and not the actua, unmeasured state. A stochastic estimation method, based on the SMM, that uses output feedback to correct both estimated densities and predicted ave front ocations, has been deveoped and i be documented in an upcoming PATH report. 2.1 Genera resuts on observabiity Tabe 1 summarizes the observabiity for each SMM mode. The observabiity resuts ere derived using standard inear systems techniques. On the eft side, upstream ces and donstream ces give the status of ces both upstream and donstream of the congestion ave front. If there is no such ave front, both sets of ces have the same status. The right side indicates hich of the to mainine boundary measurements, if either, can be used to make the SMM observabe. To reate the measurements to the states, in Fig. 2, it is assumed that ρ u is a measurement of ρ 1 and ρ d is a measurement of ρ 4. These resuts can be obtained by computing the observabiity matrices for the A s of Eq. (3) ith the output matrices C u =[1]and C d =[1], or ith the combined output matrix C = [ ] Cu T Cd T T. For exampe, for the FF mode, it can be shon that (A 1,C u ) is not observabe, hereas (A 1,C d ) is. From the tabe, it can be seen, as a genera resut, that if a ces have free-fo status, the states are observabe using a donstream measurement, hie in congested mode, they are observabe using an upstream measurement. If there is no donstream measurement avaiabe hen ces are in free-fo mode, or there is no upstream measurement hen ces are congested, as in the ast to cases isted in Tabe 1, the system is unobservabe. This is reated to the ave (information) propagation directions on a highay in different congestion modes. When a highay section is in free-fo mode, the informa- Tabe 1: Observabiity for different SMM modes Upstr. Ces Donstr. Ces Observabe ith Free-fo Free-fo Donstr. Meas. Congested Congested Upstr. Meas. Congested Free-fo Up. and Don. Meas. Free-fo Congested 1 Unobservabe Free-fo Congested 2 Unobservabe Tabe 2: Controabiity for different SMM modes Upstr. Ces Donstr. Ces Controabe from Free-fo Free-fo Upstr. On-Ramp Congested Congested Donstr. On-Ramp Congested Free-fo Not Controabe Free-fo Congested 1 Up. and Don. O.R. Free-fo Congested 2 Up. and Don. O.R. tion propagates donstream at speed v, hich is the vehice traveing speed. Therefore, in order to be abe to estimate the ce densities, the donstream density measurement is needed. When the highay is in congestion, the information propagates upstream at speed, hich is the backard congestion ave traveing speed, and an upstream measurement is needed to estimate densities. 2.2 Genera resuts on controabiity Controabiity resuts are summarized in Tabe 2. These resuts can be derived in a simiar manner as the observabiity resuts; e.g., for the FF mode, if A 1 is compared ith B 1,r2 =[ Ts ] T, the on-ramp-dependent coumn of B 1, it can be shon that the ce densities donstream of on-ramp info r 2 (ρ 2 through ρ 4 ) are controabe from r 2, hereas the upstream density (ρ 1 ) cannot be controed from r 2. When the section is fuy congested, the situation is reversed: ρ 1 is controabe from r 2, but the donstream states ρ 2 through ρ 4 are not controabe from r 2. Generay, a section in free-fo mode is controabe from an on-ramp at its upstream end, hereas a congested section can be controed from an on-ramp at its donstream end. If a section is in CF mode, it cannot be controed by an onramp at either end of the section, hie the opposite is true for the FC modes. 3 Resuts Fig. 3 is a schematic diagram of the freeay section used to test both the modified ce transmission mode and sitching-mode mode. It is a subsection of I- 21 West, approximatey 2 mies in ength, ith four Myrte ML 34.5 Huntington ML 33.5 Santa Anita ML 32.2 ( q u, ρ u) ( q m, ρ m) ( q d, ρ d) q 1 ρ1 ρ2 ρ3 4 ρ 5 ρ6 ρ7 ρ8 ρ q 9 r 2 f 3 r 6 f 7 oop detectors Figure 3: A segment of I-21W divided into ces 3753 Denver, Coorado June 4-6, 23

5 Traffic Fo Myrte Huntington Santa Anita ML 34.5 ML 33.5 ML oop detectors Upstream boundary (34.49) Midde region (33.49) Donstream boundary (32.199) 1 5 Mainine ce densities (veh/mi/ane) 25Apr21 data SMM CTM Time (hours) Figure 4: Measured and simuated mainine densities for a segment of I-21W on Apri5, 21 mainine anes, three mainine oop detector stations abeed Myrte (ML 34.5), Huntington (ML 33.5), Santa Anita (ML 32.2), and additiona detector stations on each ramp. ML stands for mainine, and the numbers, e.g. 34.5, are the absoute postmie indices of the detector stations (postmies are a measurement of distance, in mies, aong the highay). Fig. 3 shos the mainine segment partitioned into eight ces. The on-ramp fo into ce i is r i, and f j is the off-ramp fo exiting ce j. The mainine ce engths chosen for the segment ere [ ] mi. We make severa assumptions in order to reate the measured quantities (q u,ρ u,q m,ρ m,q d,ρ d, and fos measured at each on- and off-ramp) to fos and densities used by the mode: (1) ρ u is a measurement of the density in the first ce, i.e. ρ u = ρ 1 ; (2) simiary, ρ d = ρ 8 ; (3) the midde density ρ m is a measurement of ρ 5, since the midde ML station (Huntington) ies ithin ce 5; (4) r i (or f j ) is equa to the measured on-ramp (or off-ramp) fo at the corresponding on-ramp (or off-ramp) station. The oop detector data used in this study as obtained from the Performance Measurement System (PeMS) [1]. Each oop detector provides measurements of voume (veh/timestep) and percent occupancy every 3 sec. In the case of the ML detectors, densities (veh/mi) can be computed for each ane using density = occupancy g-factor, here the g- factor is the effective vehice ength, in mies, for that detector. For singe oop-detector freeays such as I-21, PeMS provides g-factors cacuated according to the PeMS agorithm, described in [6]. A necessary condition for numerica stabiity is that vehices traveing at the maximum speed may not cross mutipe ces in one time step, that is, vt s i, i =1, 2,...,N. This, combined ith the aforementioned ce engths prohibits a simuation time step as arge as 3 seconds, thus a zeroth-order interpoation as appied to the PeMS data to yied data ith T s = 5 sec. To counteract noise in the PeMS 3-sec data, a 1st-order Butterorth opass fiter Hz as appied to the data using a zero-phase forard-and-reverse fitering technique. One difficuty in seecting a test section is that it is rare for a the oop detectors in a section to be functioning propery at the same time. In the cases here detectors ere not functiona, the data as corrected using information from neighboring sensors. The interpoated, fitered, and corrected data sets ere used as simuation inputs. ith cutoff frequency.1t 1 s Severa of the ce parameters used in these simuations (v = 63 mph, Q M = 8 veh/hr, ρ J = 688 veh/mi) ere estimated through a hand-tuning procedure, herein Eq. (2) as evauated over the 5AM 12PM time range using measured mainine densities in pace of ce densities, ith nomina vaues for v,, and ρ J. v, Q M and ρ J ere subsequenty adjusted to improve the agreement beteen the empirica evauation of Eq. (2) and the measured mainine fos. = mph and ρ c = 127 veh/mi ere then computed as functions of the estimated v, Q M, and ρ J, assuming that a the parameters must satisfy the trianguar fundamenta diagram shape of Fig. 1. Since a fo-density hysteresis oop as often observed in the empirica fo vs. density pots, an approximate fo hysteresis as induced in the modes by reducing from mph to 12.5 mph at 9AM. Spatiay uniform parameters are a reasonabe assumption for this freeay segment, hich contains no abrupt varia Denver, Coorado June 4-6, 23

6 Tabe 3: Mean percentage errors of ρ 5 estimates for severa different days Date CTM SMM Mar. 15, Mar. 27, Apr. 2, Apr. 1, Apr. 25, mean std. dev tions in geometry. Both the sitching mode and modified CTM ere simuated for the section of Fig. 3 using data coected from I-21 West for severa eekdays, over the interva 5AM-12PM, during hich the morning rush-hour congestion normay occurs. It as assumed that the upstream and donstream mainine data (q u,ρ u,q d,ρ d ), as e as the ramp fo data, ere knon, hereas the midde density, ρ m, as considered to be missing, hence in need of estimation. The purpose of the test as to determine hether the modes coud accuratey reproduce ρ m. Fig. 4 shos each of the three measured densities compared ith its corresponding simuated density, for the ces nearest the ML stations, for a particuar morning (4/25/1, 5AM 12PM). In the top graph, the measured upstream density, ρ u, is potted aong ith the simuated ce density, ρ 1, for both the sitching and modified ce transmission modes. The ce 8 density, ρ 8, is compared ith ρ d in the bottom pot. Note that the simuated ρ 1 and ρ 8 are not identica to the nearby measured densities; this discrepancy beteen the mode outputs and the knon measurements ρ u and ρ d can be eiminated using an appropriate cosed-oop estimation scheme. In the midde graph, ρ m is potted against the ce 5 density ρ 5 =ˆρ m. A the densities dispayed in Fig. 4 ere divided by the number of ML anes. Tabe 3 shos the mean-percentage error, defined as E MPE = 1 M M k=1 ρm(k) ˆρm(k) ρ m(k), of each of the estimates for five different days in 21. The mean error over the five days is approximatey 13%. The resuts indicate that both the SMM and modified CTM provide a good estimate of ρ m. As seen in Fig. 4 and Tabe 3, the performance of the to modes is quite simiar. 4 Concusions and Future Work The CTM-derived sitching-mode mode can be used as a freeay traffic density estimator. It is usefu for determining the controabiity and observabiity properties of the highay, hich are of fundamenta importance in the design of data estimators and ramp-metering contro systems. We are currenty orking to extend the SMM-based data estimation methods to the remaining portions of I-21, and to perform more extensive testing to determine the best sets of parameters for the ce transmission and sitching-mode modes. Since off-ramp fo data for I-21 is generay incompete or unavaiabe, e are currenty deveoping a method, based on the sitching-mode mode, for estimating off-ramp fos. Additionay, e are investigating the sitching-mode mode as a basis for designing and testing ne oca ramp metering strategies. A hybrid system mode cosey reated to the SMM has aready been used to anayze the stabiity of oca traffic responsive ramp metering controers [7]. In addition to the design of controers and estimators, faut detection and faut handing agorithms can be deveoped based on the SMM. This is important since data avaiabiity and data integrity are of great concern hen impementing ramp metering contro agorithms in the fied. Acknoedgments The authors oud ike to thank Gabrie Gomes for his assistance in providing geometrica and time-series data for I-21. References [1] Freeay Performance Measurement Project. http: //pems.eecs.berkeey.edu/. [2] Markos Papageorgiou, H.S. Habib, and J.M. Bossevie. ALINEA: A Loca Feedback Contro La for Onramp Metering. Transportation Research Record, 132:58 64, [3] Caros F. Daganzo. The Ce Transmission Mode: A Dynamic Representation of Highay Traffic Consistent ith the Hydrodynamic Theory. Transportation Research - B, 28(4): , [4] Caros F. Daganzo. The Ce Transmission Mode, Part II: Netork Traffic. Transportation Research - B, 29(2):79 93, [5] Wei-Hua Lin and Dike Ahanotu. Vaidating the Basic Ce Transmission Mode on a Singe Freeay Link. PATH Technica Note 95-3, Institute of Transportation Studies, University of Caifornia at Berkeey, [6] Zhanfeng Jia, Chao Chen, Ben Coifman, and Pravin Varaiya. The PeMS Agorithms for Accurate, Rea-Time Estimates of g-factors and Speeds from Singe-Loop Detectors. In 21 IEEE Inteigent Transportation Systems Conference Proceedings, pages , Oakand, CA, August 21. [7] Gabrie Gomes and Roberto Horoitz. A Study of To Onramp Metering Schemes for Congested Freeays. In 23 American Contro Conference Proceedings, Denver, CO, June to appear Denver, Coorado June 4-6, 23