Incorporation of Lagrangian measurements in freeway traffic state estimation

Transcription

1 Incorporation of Lagrangian measurements in freeway traffic state estimation Juan C. Herrera Departamento de Ingeniería de Transporte y Logística Pontificia Universidad Católica de Cile Alexandre M. Bayen Systems Engineering, Department of Civil and Environmental Engineering University of California, Berkeley. Abstract Cell pones equipped wit a Global Positioning System (GPS) provide new opportunities for location based services and traffic estimation. Wen traveling on board veicles, tese pones are able to accurately provide position and velocity of te veicle, and can be used as probe traffic sensors. Tis article presents a new tecnique to incorporate mobile probe measurements into igway traffic flow models, and compares it to a Kalman filtering approac. Tese two tecniques are bot used to reconstruct traffic density. Te first tecnique modifies te Ligtill Witam Ricards partial differential equation (PDE) to incorporate a correction term wic reduces te discrepancy between te measurements (from te probe veicles) and te estimated state (from te model). Tis tecnique, called Newtonian relaxation, nudges te model to te measurements. Te second tecnique is based on Kalman filtering and te framework of ybrid systems, wic implements an observer equation into a linearized flow model. Bot tecniques assume te knowledge of te fundamental diagram and te conditions at bot boundaries of te section of interest. Te tecniques are designed in a way in wic does not require te knowledge of on and off ramp detector counts, wic in practice are rarely available. Te differences between bot tecniques are assessed in te context of te Next Generation Simulation program (NGSIM), wic is used as a bencmark data set to compare bot metods. Tey are finally demonstrated wit data from te Mobile Century experiment obtained from undred Nokia N95 mobile pones on I880 in California on February 8, Te results are promising, sowing tat te proposed metods successfully incorporate te GPS data in te estimation of traffic. Corresponding autor: Assistant Professor. Casilla 306, Código 105, Santiago 22, Cile; jc@ing.puc.cl; Tel ; Fax: P.D. student at U.C. Berkeley during te researc presented in tis article. Assistant Professor. 711 Davis Hall, Berkeley, CA 94720; bayen@berkeley.edu; Tel: ; Fax:

2 1. Introduction Traffic congestion in te US alone causes a 78 billion drain on its economy annually. In total, drivers lose 4.2 billion ours and waste 2.9 billion of fuel gallons per year because of traffic congestion (Scrank and Lomax, 2007). In order to improve te performance of te system, several traffic control strategies ave been proposed and implemented over time. Coordination of traffic ligts, ramp metering, dynamic speed limits, and information display troug cangeable message signs (CMS) are examples of suc strategies. Most of te igway flow control strategies require an accurate knowledge of te state of te system (for instance, veicles accumulation or speed distribution on a igway segment). Traffic state estimation is a crucial part of any active control sceme on freeways, and usually requires te deployment of a significant monitoring infrastructure. Te most common way to monitor traffic on freeways is te use of inductive loop detectors. Tese sensors are embedded in te pavement and collect data from veicles as tey pass over tem. However, teir ig installation and maintenance cost prevents massive deployments, especially in developing countries. Alternative ways to monitor traffic make use of information provided by individual veicles. Radio frequency identification (RFID) transponders used to pay tolls (suc as Fastrak in California or EZpass on te East Cost) are used to compute travel time between consecutive locations, wic requires te installation of post readers along te road. Tis tecnology, owever, is not universal and only provides point to point information. Probes veicles reporting traffic conditions are also used for traffic monitoring purposes, but te penetration is not ig enoug to provide sufficient spatio temporal coverage of te transportation network. In te era of convergence of multimedia, communication and sensing platforms, GPS equipped smart pones are becoming an essential contributor of location based services. Given te ig popularity of smart pones, tis new sensing tecnology as te potential of providing a better coverage of te transportation network tan current sensor tecnology does (and witout public infrastructure costs associated wit monitoring). Tis new source of traffic data (speed measurements gatered from GPS equipped pones onboard veicles) creates a scientific callenge for traffic state estimation. Privacy issues and tecnology limitations (pone energy consumption and bandwidt usage) prevent te systematic collection of all data generated by te GPS. Tis means tat te amount of data wic can be used for traffic flow reconstruction is a subset of all data generated by eac GPS. Tis article presents a new algoritm to integrate GPS data in flow models, and compares it to an application of Kalman filtering to te specific problem of interest. Bot metods assume te knowledge of te fundamental diagram and te conditions at bot boundaries of te section of interest. Te metods can work even wen data is not available for te on and off ramps, wic is almost always te case in practice. For tis, te algoritms use data available at intermediate locations on te freeway, transmitted by te mobile devices. Te rest of te article is organized as follows. Section 2 provides some background, summarizing relevant work for traffic state estimation and introducing te metod cosen to address te problem of interest. Te metods are fully described in Section 3. Two data sets are used to test te metods. Section 4 describes bot datasets and presents te main results. Finally, Section 5 states te main conclusions of te present study and proposes some future work directions. 2

3 2. Background Traffic state estimation requires data, wic may differ depending on te type of sensors used to collect it. Numerous freeways around te world, in particular in te US, are equipped wit loop detector stations embedded in te pavement to collect traffic data. For eac lane, tese detectors aggregate information at a given sample time (usually 20 or 30 seconds). Veicle counts, occupancy and speed are among te information tat a detector can collect. Tese measurements provided by static sensors are traditionally referred to as Eulerian measurements, wic means tat te detector measures flow troug a fixed control volume. On te oter and, Lagrangian sensors collect measurements of te system along a particle trajectory (motion of a car). RFID transponders, smart pones and GPS devices onboard veicles providing position and/or velocity are examples of mobile (or Lagrangian) sensors. Data can be used wit different modeling approaces for traffic reconstruction. One class of models does not make use of a traffic flow pysics, and te estimation is based on statistics (current and istorical data). Real time traffic reports are sometimes based on tis type of statistical models. Tese models ave also been a common practice in studies tat use smart pones as traffic sensors, in wic one of te main goals as been speed or travel time estimation on a stretc of road (Sanwal and Walrand, 1995; Westerman et al., 1996; Ygnace et al., 2000; Bar Gera, 2007; Krause et al., 2008). Note tat te first four aforementioned studies use cell tower signal to obtain cell pone position, wic is less accurate tan GPS positioning. In Krause et al. (2008) te autors ave investigated te use of macine learning tecniques to reconstruct travel times on graps based on sparse measurements collected from GPS devices embedded in smart pone and automobiles. Treiber and Helbing (2002) propose a metod tat filters Eulerian data collected from loop detectors (density or velocity) to reconstruct traffic flow. In practice, density can be extracted from occupancy; terefore, density based approaces are possible. Te filter is suc tat in free flow, traffic information propagates downstream, wile in congestion it travels upstream. Even toug tis metod takes into account te way in wic information propagates in traffic streams, it does not make use of any flow model. Anoter class of models is based on traffic flow pysics. Ligtill and Witam (1955), and Ricards (1956) independently proposed a first order partial differential equation (referred to as te LWR PDE) to describe traffic evolution over time and space. Te LWR PDE is a scalar yperbolic conservation law, wic relates canges in density over time to canges in flow over space. Tis law is based on te conservation of veicles principle. Extensions of tis model include a second equation accounting for te fact tat veicles do not accelerate/decelerate instantaneously, wic are known as second order models, suc as in Zang (1998) and Aw and Rascle (2000). Numerical scemes, suc as te Godunov sceme (Godunov, 1959), can be used to discretize tese continuous models (bot first and second order models). Wile te discussions wit respect to first order and second order models led to specific conclusions on bot sides, te metod proposed applies to models of any order. In te present case, we apply it to a first order model (te LWR model); extensions to second order models are straigtforward, owever, tey are outside te scope of tis article. Daganzo (1994, 1995) proposed te cell transmission model (CTM), wic is a discretization of te LWR PDE 1. Te CTM divides te igway in cells of lengt Δx and computes te state of te system (veicle accumulation or density) every Δt units of time in eac cell according to te conservation of veicles principle. Te CTM transforms te nonlinear flux function into a nonlinear discrete operator. Modifications to te CTM can be found in te literature. Munoz et al. (2003) use a ybrid system framework to develop te switcing mode model (SMM), wic 1 Te CTM is a special case of te Godunov sceme wen te fundamental relation between flow and density is assumed to be triangular. 3

4 combines discrete event dynamics estimation (mode identification) wit nonlinear continuous dynamics state estimation (density estimation). Gomes and Horowitz (2006) modify te original merge rule of te CTM and propose te asymmetric CTM (ACTM). For traffic state estimation, dynamic flow models of te system can be combined wit data collected by sensors, a process known as data assimilation. However, tere exist various tecniques to perform data assimilation. Gazis and Knapp (1971), Szeto and Gazis (1972), and Sun et al. (2004) ave used Eulerian measurements to perform data assimilation using Kalman filtering tecniques. Wile Gazis and Knapp (1971) and Szeto and Gazis (1972) used te conservation of veicles as model, Sun et al. (2004) uses te SMM. Cu et al. (2005) and Nantawicit et al. (2003) ave also performed data assimilation using Kalman filtering tecniques, but wit simulated Lagrangian data. Cu et al. (2005) use te conservation of veicles equation and Kalman filtering tecniques to combine point detection data and probe veicle data into te travel time estimation, wile a second order model is used in Nantawicit et al. (2003) to perform data assimilation. Data assimilation using Lagrangian data to estimate te state of a system is common in oter fields suc as meteorology and oceanograpy. In tese fields, data assimilation metods range from simple, suboptimal tecniques suc as direct insertion, statistical correction, and statistical interpolation to more sopisticated, optimal algoritms suc as inverse modeling, variational tecniques and a family of metods based on Kalman filtering (Paniconi et al., 2003). Te extrapolation of tese tecniques to transportation engineering problems appears to be very promising. A known and simple metod used in oceanograpy is Newtonian relaxation (or nudging) metod (Antes, 1974), wic we use for te present study. Te Newtonian relaxation metod relaxes te dynamic model of te system towards te observations. To tis end, a source term proportional to te difference between te predicted and observed state is included in te constitutive equation of te model (in te present case te LWR PDE). Most of te data assimilation metods use boundary conditions at te boundary of te pysical domain of interest. Tey are assumed to be known at some specific locations (usually at te boundaries of te computational domain). Tus, tese metods make use of Eulerian data as well. Tis article presents two metods for traffic state estimation tat can andle bot Eulerian and Lagrangian measurements, using a first order flow model. Te contributions of tis article specifically include: Te development of a first order flow model wic integrates a nudging term (to perform data assimilation) and its corresponding discretization using te Godunov sceme, An extension of te Kalman filtering based metod used to estimate traffic state, following te work of Sun et al. (2003), wic can incorporate Lagrangian observations, A Lagrangian data selection procedure to incorporate te measurement data into te discrete model, An implementation of bot algoritms on an extensive dataset (te NGSIM data) for wic ground trut is known, An implementation of bot algoritms on te data set from te Mobile Century experiment, and a cross validation wit PeMS data 2. 2 California Performance Measurement System: ttp://pems.eecs.berkeley.edu/ 4

5 3. Description of te approaces Te problem of interest for te present study is te incorporation of Lagrangian data into traffic flow models for freeways. Te present section describes te new approac proposed to address tis problem and te extension of te Kalman filtering approac specific to our problem. It first describes te flow model to be used and ten presents te proposed metods. 3.1 Flow model approac Continuous flow model Te continuous flow model proposed in te 1950's by Ligtill and Witman (1955) and Ricards (1956) describes te evolution of traffic on an infinite road. Te LWR PDE relates density on te road and its flow: k( x, t) + t x [ q( k( x, t)) ] = 0 (1) Te function k(x,t) represents te veicle density in veicles per unit lengt and q(k(x,t)) is te flux function in veicles per unit time (or te fundamental diagram, wic is assumed to be triangular and time space invariant in tis work) at location x and time t. For a finite road, te boundary conditions are given at te upstream and downstream ends of te section (x=a and x=b, respectively), and tey could be obtained from sensors at tese locations, suc as loop detectors. Ideally, tey would be directly prescribed as values k a (t) and k b (t) of te density. k(a,t)= k a (t) and k(b,t)= k b (t), wit t (0,T). (2) Te initial condition corresponds to te density along te road at te beginning of te period of analysis: k(x,0) = k 0 (x), wit x (a,b). (3) Given te dynamics of te traffic, te lack of accurate knowledge of initial conditions can be counter balanced by te flus out effect, i.e. for sufficiently large periods of time, te influence of te initial conditions becomes negligible. For a proper caracterization of te solution to equation (1), weak boundary conditions are required and strong boundary conditions in (2) cannot be used as suc. Boundary conditions only apply on te boundary of te section were caracteristics are entering te computational domain. Te weak boundary conditions for te specific case of te LWR PDE can be found in Bardos et al. (1979) for general yperbolic conservation laws. In te present context, tey can be written in simpler form as follows: and (i) k(a,t) = k a (t) and q (k a (t)) 0 or (ii) q (k(a,t)) 0 and q (k a (t)) 0 or (4) (iii) q (k(a,t)) 0 and q (k a (t)) 0 and q(k(a,t)) q(k a (t)) 5

6 (i) k(b,t) = k b (t) and q (k b (t)) 0 or (ii) q (k(b,t)) 0 and q (k b (t)) 0 or (5) (iii) q (k(b,t)) 0 and q (k b (t)) 0 and q(k(b,t)) q(k b (t)) dq In equations (4) and (5), q (k) is te slope of te flux function q(k), defined as q '( k) =. Te interpretation of te dk matematical formulae above, wic are required for te problem (1) (2) (3) to be matematically well posed, i.e. were (2) as been replaced by (4) (5), is not straigtforward. Tese conditions state tat te upstream boundary condition is relevant (and tus can be applied in te strong sense) only wen a free flow condition is observed at tat point. Oterwise, te boundary condition is irrelevant and conditions are dictated by downstream traffic. Te opposite is true for te downstream boundary, in wic te boundary condition is relevant only if congestion is observed at tat point. Note tat tese conditions were derived empirically by Daganzo (1995) from pysical principles. Te interpretation of te conditions in (4) is as follows (interpretation for condition (5) is similar). One of te tree alternative is possible, based on te state of te system directly downstream x=a (i.e. at x=a + ε, were ε is small). Eiter (i). Tis means tat it is possible to impose te desired density k a (t) upstream from te section of interest, terefore k(a,t) = k a (t). For example, a low flow situation in wic te upstream inflow is dictated by te demand falls into tis category. Or (ii). Bot te desired inflow and realized inflow are congested flows, given by te negative slope of q( ). In tis case, te solution, i.e. te boundary condition, is driven by te state, i.e. te inflow allowed by te current state of congestion. Or (iii). Te current flow is congested, but te desired inflow (demand) is not, owever, its flow value is greater tan te realized flow value on te igway. Wile te pysical reasoning between tese tree cases could probably be organized wit a different Boolean logic, see in particular (Daganzo, 1995) for a pysical analysis of tis situation, formula (4) above is matematically required for te problem to be well posed. In particular, it ensures existence and uniqueness of an entropy solution to tis equation (Oleinik, 1957; Bardos et al., 1979) for a bounded domain, from wic follows convergence of te Godunov discretization sceme, wic is key to ensure te proper numerical solution to te problem. Te first known instantiation of tis Boolean type condition is due to LeFloc (1988), and more recently adapted for igway specific flux functions by Strub and Bayen (2006). In practice, tese boundary conditions are implemented using gost cells. Tese cells correspond to te input and output cells as proposed by Daganzo (1994), and are presented in subsequent sections Discretization metod For implementation purposes, te LWR PDE needs to be discretized. To tis end, te freeway section is divided into I cells (eac one of lengt Δx distance units, and indexed by i). Time is divided into H time steps (eac one of lengt Δt time units, and indexed by ). In order to meet te Courant Friedrics Lewy (CFL) stability condition (LeVeque 2002), wic states tat a veicle traveling at te free flow speed v f cannot traverse more tan one cell in one time step, te condition Δt v f Δx sould be met. At every time step, te model estimates te density in eac cell according to te following expression: 6

7 ( q q ) i = 1, 2... I and = 0,1... H ki = ki r i+ 1 i (6) Te parameter r is te inverse of te speed needed to travel one cell in exactly one time step (i.e. r = Δt/Δx), wile k i is te discrete approximation of te density in cell i at time step. Te variable q i is te flow into cell i between time and +1, and depends nonlinearly on te density of cells i 1 and i. It can be computed using te Godunov sceme as follows: q( ki ) q( kc) qi = q( ki 1) min i 1 { q( k ), q( k )} i if kc < ki < ki 1 if ki < kc < ki 1 if ki < ki 1 < kc if ki 1 ki (7) In equation (7), k c is te critical density, wic corresponds to te point at wic te flux reaces its only maximum (te flux function is concave). As was explained earlier, weak boundary conditions are required for a proper caracterization of te solution of te LWR PDE in (1). In te implementation, one gost cell is inserted at eac boundary of te section. Gost cells are outside of te pysical domain, and are virtual cells on wic te Godunov update equations (6) and (7) automatically apply weak boundary conditions (4) and (5), by construction of te Godunov sceme (LeVeque, 2002; Strub and Bayen, 2006). Te gost cells contain te boundary conditions and allow te first and last cells of te computational domain to be updated depending on te existing traffic conditions (free flow, congested, or a combination of te two). Tis approac was successfully implemented and tested in earlier work (Strub and Bayen, 2006) wit traffic data collected from loop detectors. Te equations for te gost cells (i=0 and i=i+1) and for te initial conditions (=0) are given by equation (8) and (9), respectively: Δt Δt k = ( ) 1 = ( ) = 0, Δ k t dt and k ( 1) I + Δ Δ k t dt H t t ( 1) Δt b t a 0 1 xi ki = k ( x) dx i 1,2... I x x 0 = Δ i 1 (8) (9) 3.2 Proposed metods In order to perform traffic state estimation (veicle accumulation or density, in tis case) along te section of interest using Lagrangian data, specific data assimilation metods need to be developed. If accurate traffic data from all locations was available at all times, traffic state estimation would not be needed since te state can be directly inferred from te data. In practice, owever, data is not always available from all te locations and is, in general, very sparse. As explained in Section 2, different approaces can be used to obtain estimates for locations witout measurements. Te approaces adopted ere use traffic flow models, based on veicle accumulation (or density). Two metods are proposed and discussed in te following sections, wic rely on te following assumptions: 7

8 GPS enabled mobile pones traveling on te section of interest are sampled in time. Tey report teir position and velocity at specific time intervals T. Boundary conditions are known. Tat is, te density at bot ends of te section of interest is available. Tis data can be provided by loop detectors. However, as we sall see later, te availability of loop detector data at te boundaries is not a critical requirement. Te fundamental diagram is assumed to be triangular and known. Information from intermediate ramps is not required. In fact, tis constitutes one of te main features of te metods proposed since, in reality, information from ramps is rarely available Definitions: state, estimated, and observed variables Te state variable caracterizes te state of a dynamical system. In te present case, it corresponds to veicle density, and it is denoted by k(x,t). By definition, it satisfies te pysical model, wic is te LWR PDE in (1) in te present case. Te state estimate (or estimated variable) is te result of te state estimation process, and it is distinguised from te model variable by te use of a at. We denote k ˆ ( x, t) te state estimate. To perform te estimation, it is assumed tat some variable is measured. In te present case, te available data consists of position and velocity measurements from GPS equipped veicles at different times. Terefore, a relationsip between te velocity and te density is needed in order to relate te measured or observed variable and te estimated variable. Tis issue is addressed in te next paragraps. Te observed position and velocity of an equipped veicle labeled j at time t are denoted by s j o (t) and v j o (t), respectively (superscript o denotes observation). It is assumed tat te observed velocity at te corresponding location and time is provided by te individual probe velocity measurement. Using v(x,t) to denote te average velocity field on te freeway, we defined te observed velocity at time t and location s j o (t) by: v o (s j o (t),t) = v j o (t) (10) Tis assumption may not be appropriate wen dealing wit multi lane freeways, were different lanes migt ave different speeds. In tese cases, tis problem requires a specific treatment, wic sould use more sopisticated models, and are out of te scope of tis work 3. As a first approac, we treat te freeway as a single traffic stream. Te fundamental diagram relates te flow, te density and te velocity of te flow on a section of road. In particular, tis relationsip can be used to infer te density from te velocity. Tat is, using te fundamental diagram, v o (s j o (t),t) can be converted into te observed density at te point of measurement, denoted by k o (s j o (t),t). In reality, k o (s j o (t),t) is not te observed density but an estimate based on te speed measurement and te fundamental diagram. Since te fundamental diagram is only a model, and in reality flow density points (or traffic states) do not necessarily lay on a line, tis conversion is expected to introduce error in te observed density. 3 In multi lane freeways, more tan one veicle could send a measurement from te same location s j o (t) at te same time t. In tis case, te observed velocity is te average of all te measurements from location s j o (t) at time t. 8

9 If a triangular fundamental diagram is used, a problem arises wen te speed v j o (t) is ig. In fact, if te measured speed v j o (t) v f, were v f is te free flow speed, different combinations of flow and density ave te same free flow speed. Indeed, under free flow conditions it is not possible to observe te local density troug speed measurements in te way described in te previous paragrap. For tese cases, free flow conditions will be assumed and a free flow density value (denoted k FF ) will be used. Te value for k FF can cange in time and space, and can be obtained from istorical data. Tis seems reasonable considering tat our main interest is to obtain accurate density estimates specifically wen congestion arises. Terefore, te observed density is given by: w kj o o if v ( s j ( t), t) < v f o o o o k ( s j ( t), t) = v ( s j ( t), t) + w (11) FF k oterwise Parameters k J and w are te jam density and te slope of te rigt branc of te triangular fundamental diagram, respectively. Te observed velocity v o (s j o (t),t) comes from equation (10) Newtonian Relaxation metod Te Newtonian relaxation (or nudging) metod is a simple euristic metod tat as been used for data assimilation in te field of environmental fluid mecanics (Antes 1974). In oceanograpy, GPS equipped drifters are used to estimate te velocity field of rivers, using sallow waters models. Te extension of tis tecnique to transportation engineering problems appears to be very promising, since GPS equipped veicles are similar to te drifters, and we aim to estimate te state of te freeway in terms of te veicle density. Te Newtonian relaxation metod relaxes te dynamic model of te system towards te observations. To tis end, a source term called te nudging term, proportional to te difference between te estimated and observed state, is included in te constitutive equation of te model, wic in te present case is te LWR PDE in (1): J kˆ q( kˆ) o p p + = ( x s j ( t j ), t t j ) t x j= 1 p t t j Ω j o p p o p p [ kˆ( s j ( t ), t ) k ( s j ( t ), t )] λ (12) j j Te summation over te index j in te RHS of equation (12) accounts for te J different veicles equipped wit Lagrangian sensors, wile te second summation includes all te observations sent by eac veicle j before te current time t. Te expression in equation (12) assumes tat veicle j sends observations at times j j t Ω j, were Ω j t represents te set of times until t at wic measurements from veicle j are performed and used for data assimilation (note tat necessarily t j p < t since only observations from te past are available at current time t). Te nudging factor λ(δ x,δ t ) represents te weigt of eac observation to be applied to te solution. Tis weigt is expected to become negligible (and eventually to become zero) away from te measurement location and after te measurement time. For tis reason, we ave adopted ere an expression for te nudging factor λ(δ x,δ t ) tat takes tis into account, and can be found in Isikawa et al. (1996): p t j 9

10 2 1 δ exp x δ exp < t if δ = x X nudge and0 δt Td λ ( δ x, δt ) Ta X nudge Td (13) 0 oterwise Te factor dies out on a space and time scale of X nudge and T d, respectively. Terefore, close to were and wen te observation is made, λ(δ x,δ t ) nudges te solution towards te observations. Te parameter T a as units of time and determines te strengt of te nudging factor. A numerical example to sow te effect of te nudging factor is provided in Appendix A. In te present context, te nudging term adds or removes veicles from te state of te flow model depending on weter te model underestimates or overestimates te number of veicles on te freeway. In Section we will discuss te implication of adding or removing veicles on te conservation of veicles principle. Numerical implementation: discrete model Te discretization of te nudging term in te RHS of equation (12) needs to be added to equation (6). Te final expression for te discretized model is as follows: ( q q ) J ˆ + 1 ˆ o p p ( ( ), ) ˆm jp o, m ki = ki r i+ 1 i Δt λ x i s j t j Δt t j k k c jp c jp j= 1 p Δt t Ω j j i = 1,2... I jp and = 0,1... H 1 (14) Te notation introduced in equation (14) is explained below: x i : location of te beginning of cell i, x i = x 0 +(i 1) Δx, for i=1,2,,i, were x 0 is te beginning of te section of interest, c jp : cell index corresponding to location s j o (t j p ), wic is te location were te veicle j is at time t j p wen o p s its p t observation is sent, j ( t j ) x0 c jp =, were s o j (t p j ) > x 0, Δx m jp : time step corresponding to location t j p, wic is te time (in time steps units) wen te p t p t j observation from veicle j occurs, m jp =. Δt Te last term in te RHS of equation (14) is te discretization of te nudging factor times te difference between estimated and measured density (terms inside te square brackets). Te nudging factor is given by te expression in (13) Kalman filtering based metod Kalman filtering is a recursive metod used to estimate te state of a discrete process governed by a linear stocastic dynamical system (Bar Salom and Li 1993) in te presence of noisy measurements. Te metod 10

11 assumes tat te way in wic te state of te system (density in tis case) evolves is linear and known, and is referred to as te dynamics or state equation (wic includes a process noise). Noisy measurements of te output of te system (i.e. observed density) are available, and te measurement or observation equation relates te output and te state of te system. Knowing te covariance of bot te process and te measurement error, te metod obtains te best estimate of te state of te system in te sense of te least squares. Kalman filtering tecniques ave been proposed to perform traffic state estimation in te presence of loop detector data (Sun et al and 2004). Given te nonlinearity of te model in (6), conventional Kalman filtering can only be applied to linear subsets of tis dynamics. Tis tecnique can be extended to cases in wic data is provided by mobile sensors, wic is described next. State space representation Te state space representation of te system consists of two equations: te dynamics (or state) equation and te measurement (or observation) equation. Te dynamics equation describes ow te state of te process (density) evolves over time and space. Since te constitutive equation in (6) is nonlinear, it needs to be linearized first. Te ybrid system framework used in Munoz et al. (2003) is adopted for tis purpose. Discrete event dynamics estimation is performed to identify te traffic condition or mode of te section of interest. Tat is, te mode of eac cell (i.e. free flow or congested) needs to be determined at te beginning of eac time interval. Previous studies (Munoz et al. 2003, Sun et al. 2004) ave proposed different ways to identify te mode on a sort section of igway. For longer sections, te number of possible modes increases, adding complexity to te mode identification. In te present study, state estimates (and indirectly Lagrangian measurements) are used to identify te mode. At te end of time interval, density estimates for every cell are available. Tese estimates consider all te observations collected until time step (inclusive) and are referred to as te a posteriori estimates at time step. Te a posteriori estimate at cell i at time step is denoted by k ˆ+ i, ˆ+, i. Te mode cosen for eac cell will depend on te value of. In oter words, if k > k, cell i is congested; oterwise, cell i is in free flow (k c is te critical c density). Note tat te mode is being identified by using te state estimates and not by measuring te actual state of te system. Once te mode for eac cell as been identified, te flow into cell i between time step and +1, q i, becomes linear in te densities in cells i 1 and i. Terefore, equation (6) can be written as follows: J Q + 1 = A k + B u + B kj + B qmax w (15) k + Bold letters represent matrix or vector notation. Te scalars k J and q max are te jam density and te maximum flow, respectively. Te vector u is te input vector at time, wic includes te density at te boundaries of te domain, and w is te process error (caused for instance by te fact tat not all te entry/exit counts are T J available). Equation (5) is linear in te state k, wic is defined as k = [ k 1 k2... ki ]. Te matrices A, B, B and B Q depend on te traffic conditions or mode of te section, wic sows te importance of te mode identification step. Te algebraic expression of tese matrices for specific cases can be found in (Munoz et al 2003). k ˆ+ i, 11

12 Te measurement or observation equation projects te state vector into te measurements provided by Lagrangian sensors, y, wit te one predicted by te model: y = C k + v (16) Te vector v represents te measurement noise. Te matrix C is time dependent, and its size and elements depend on te location of te measurements. It only contains zeros and ones, wose position in te matrix C depend on te locations at wic te measurements are taken. Te time dependency of C is a major callenge and is directly linked to te Lagrangian aspect of te measurements. In summary, te dynamics or state equation and te measurement or observation equation of te system are given by equation (15) and (16), respectively. Kalman filtering Te following notation is used: T kˆ : a priori state estimate of k, were kˆ = [ k ˆ kˆ... kˆ 1 2 I ], + kˆ ˆ kˆ +, [ kˆ + : a posteriori state estimate of k k + =, T, were k ˆ+, I ], P : a priori estimate error covariance, were e = k kˆ is te a priori estimate error, + P : a posteriori estimate error covariance, were e + = k + kˆ is te a posteriori estimate error. Te difference between te a priori and a posteriori estimates at time step is te fact tat te a priori estimates do not take into account te observations collected at time step, wile te a posteriori estimates do. Tat is, te a posteriori estimate is an updated version of te a priori estimate. Kalman filtering provides a set of recursive equations to estimate te vector state. Te equations are as follows (Bar Salom and Li 1993): kˆ + 1 = A kˆ B u P = A P A + Q F kˆ P + 1 T + 1 = P + 1 C + 1 C+ 1 ˆ = k+ 1 + F ( ) + = I F C P+ 1 T T [ P ] C R ( y C k ) +1ˆ +1 (17) (18) (19) (20) (21) Initial conditions ˆk 0 and P 0 are assumed to be known. Q and R are te covariance matrices of te process and measurement error, respectively. F is known as te Kalman gain at time step. Note tat tere is a striking similarity between te two metods: te second term in equation (20) represents a non pysical source (correction) term, wic modifies te a priori value of te state of te system, similar to te nudging term. Implementation Te state vector contains te density in eac cell. At te beginning of time step +1, a posteriori estimates at time step are available. Te traffic conditions on te network at te end of time step need to be identified in order to determine wic set of matrices A, B, B J and B Q to use. Te mode will be identified wit te process 12

13 outlined previously, wic indirectly uses te Lagrangian observations collected. Note, owever, tat te mode identification is te most callenging task in te implementation of tis metod. Once te mode as been identified at te beginning of time step +1, equation (17) and (18) are used to obtain te a priori density estimate and its covariance, respectively. At tis point, Lagrangian data becomes available to te model, i.e. te observed local density at time +1 will be known for some cells (te quantity and position of te Lagrangian sensors at +1 will determine ow many and for wic cells te density is observed). Wit tis information, te observed vector y +1 and matrix C +1 can be constructed. Ten, te Kalman gain is computed using equation (19). Finally, te a posteriori density estimate and its covariance are obtained using equation (20) and (21), respectively. In te event tat no Lagrangian observation is available at time +1, te matrix C +1 is set equal to zero, wic implies tat te Kalman gain is also zero. In tis case, te a priori and a posteriori density estimates are te same. 3.3 Comments on te proposed metods Note first tat in bot metods, te observed density is computed using equation (11), wic includes at least tree error sources. Te first source of error as to do wit equation (10), wen te velocity of an individual veicle is assumed to correspond to te velocity at tat given location. Tis error may be iger for low velocities, wen te variability in speed among veicles is ig. Te second source of error is related to te fact tat te fundamental diagram is not exact. Tus, te velocity to density conversion, as expressed in equation (11), introduces error. Tis error may not be negligible for free flow because of te approximation made for tese cases 4. Te tird source of error corresponds to te measurement error in te velocity v j o (t), wic is expected to be small given te accuracy of GPS. Bot metods are conceptually similar. Tey add or remove veicles depending on te difference between an estimated density and te observed density computed using GPS data. For tis, tey used a so called observer equation, wic is derived from te flow model, but includes modifications wic integrate te measurements. Te metods differ, owever, in te way tis difference between measurement and estimate is used. Te Kalman filtering based metod assumes tat an observation obtained from cell i at time step only corrects or updates te density on te corresponding cell and time step. Te flow model propagates te effect in time and space. On te oter and, te Newtonian relaxation metod uses te observation to directly affect te density of neigboring cells and for future times. Terefore, te effect of eac observation is propagated in time and space directly troug te nudging term, but also troug te flow model. Tis may be useful wen te number of observations is low, because if cell i is congested at time step (according to te observation), neigboring locations are expected to also be congested for a certain period of time (congestion takes time to vanis). Some tradeoffs between te two metods include te following: Modeling complexity. Te Newtonian relaxation metod is trivial to develop on almost any flow model, since it only consists in adding a source term weiged properly by te nudging factor. For Kalman filtering, te difficulty consists in finding te proper linearization, wic in te present case cannot be obtained by linearizing te dynamics directly, but by identifying te proper modes (ence te ybrid system approac), wic is a researc topic in itself. 4 Note tat if density estimates are used to obtain travel times (by computing te velocity), te value used for k FF would not affect te travel time estimates significantly. Terefore, travel times estimates migt be more accurate tan density estimates under free flow conditions. 13

14 Implementation difficulty. Te benefit of te Newtonian relaxation metod is tat tat any solver used for forward simulations (of te model) can be directly used and modified to include te nudging term, a feature wic as been extensively used in oter fields, in particular in oceanograpy. For Kalman filtering, additional update equations (outlined earlier) need to be added. Parameter tuning. Te nudging term requires tuning tree parameters, wic are usually cosen based on pysical considerations, wile te Kalman filtering approac requires assumptions on te covariance matrices. Computational cost. Bot metods ave rougly te same cost (i.e. te cost of a forward computation), wit additional computations required for te Kalman filter because of te observer equation. Convergence of te estimator. Wile te convergence of te Kalman filter can be assessed teoretically (Bar Salom and Li 1993), Newtonian relaxation is a euristic tecnique for wic no convergence results is known to te autors. Convergence of te model. By definition of te Godunov sceme used, te discrete model used in Newtonian relaxation witout te nudging term can be sown to converge to te solution of te LWR PDE. An extension of tis property can be used to develop a convergence result for a PDE wic includes a continuous nudging term. No suc counterpart is known to te autors for te Kalman filter. As was briefly explained before, mass (veicle) conservation is not satisfied by te metods proposed in tis article. Tis is a common feature of estimation tecniques, so te next paragraps explain te considerations wic ave to be taken into account wen doing estimation, and ow our metods fit in tese considerations. Traditionally, estimation makes te assumption tat te pysics of a penomenon are governed by a constitutive model (Bar Salom and Li, 1993). Te equations for suc models can be of various nature, in particular partial differential equations like equation (1) or difference equations like in equation (6). Because of modeling inaccuracy (models are never perfect), and because of measurement noise (measurements are never noise free), empirically measured data almost never satisfies a model perfectly, i.e. plugging te data in a model will violate te model. In te present case, te model is represented by equation (1) wit corresponding continuous state variable k(x,t), and te corresponding discrete model by equation (6), wit discrete state variable k i. Wit empirically measured data, k(x,t) would never satisfy equation (1) exactly, and k i would never satisfy equation (6) exactly. Te field of estimation as produced a large number of tecniques wic are based on so called observer equations, wic are traditionally denoted by ats (terefore ˆ k( x, t) and kˆi ), wic integrate measurements into constitutive equations, but usually do not satisfy modeling assumptions anymore (i.e. in te present case, k ˆ ( x, t) and kˆi will not satisfy mass or veicle conservation). Te field of estimation and data assimilation is based on constructing evolution equations for tese estimator states ( ˆ k( x, t) and k ˆ in te present case), wic are sown to minimize discrepancy error between measurements and model estimates). For example, Kalman Filtering equations add a corrective term to te model equations (terefore violate te model), and provide a least square estimator of te state of te system. In te present case, te Newtonian relaxation metod adds or removes veicles from a cell depending on te relative values of te observed and estimated density. If te counts were known exactly at every entry and exit point of te network, tis addition/removal of veicles would violate te conservation of veicles. In practice i 14

15 owever, on and off ramps are rarely equipped wit loop detectors (in addition to te fact tat specifically to te case of te igway, te observed quantities are never perfect because of measurement errors). In particular, loss or gain of veicles in between loop detectors is a well known problem in traffic engineering 5. Te present metod can tus be used to incorporate Lagrangian data in place of tis missing loop detector data. Tis metod tus conveniently bypasses te modeling of networks: for mainlines wit on and off ramps only (no major intersection between igways), it replaces te merge diverge junctions by Lagrangian data incorporation. Wen no loop detector data is available at ramps, numerical simulations are simply underdetermined, because of te lack of inflow and outflow information. Wen data from ramps is available, because of measurement errors, te data migt be inconsistent wit te model, in particular veicle conservation. Te Newtonian relaxation metod steers te state of te model locally (in x and t) towards te Lagrangian measurements, wic is a way to reestablis a value of te state closer to te actual state of te system werever suc measurements are available. Terefore, it compensates for te lack of inflow and outflow information or te corresponding inaccuracy. Because of te considerations discussed above, te conservation of veicles tat incorporates te on and offramps is replaced by te nudging term, wic performs te required local adjustments based on measurements. 4. Assessment of te metods Te Newtonian relaxation metod (NR) and te Kalman filtering metod (KF) are implemented on two datasets. In te first one, we ave full knowledge of all positions and speeds of all veicles during te entire experiment, wic enables an extensive validation of te metod against ground trut. In te second one, we use GPS data obtained from smart pones from a subset of drivers and tus demonstrate te applicability of te metods to tis novel way of gatering traffic data. 4.1 Implementation wit NGSIM data Traffic data from te Next Generation Simulation (NGSIM) project 6 is used to evaluate te proposed approac. NGSIM data as been extracted from video, wic provides ground trut trajectories for all veicles. Te data consist of te trajectories of all veicles entering a stretc of US Higway 101S in Los Angeles, CA, during 45 minutes (from 7:50am to 8:35am on June 15, 2005). Te site is approximately 0.4 miles in lengt, wit 5 mainline lanes (see Figure 1). An auxiliary lane exists between te on ramp and te off ramp. Transition from free flow to congestion appens during te first minutes of te data set (i.e. part of te section was in te free flow mode wile te rest was in te congested mode). For implementation purposes, te five mainline lanes were considered. Te full section (1920 ft) as been divided into I=16 cells of Δx = 120 ft eac, and te total simulation time (2610 sec) as been divided into H = 2175 time steps of Δt = 1.2 seconds eac. Parameters of te fundamental diagram maximum flow (q max ), jam density (k j ), critical density (k c ), free flow speed (v f ) and wave speed (w) were extracted from te PeMS system 7. 5 Tere are algoritms to correct for te measurement errors. For instance, miscounts arising during congestion can be corrected once free flow conditions are restored. See te appendix in (Cung et al. 2007) for more details. 6 ttp://ngsim.camsys.com/. 7 Assuming a triangular sape of te fundamental diagram: q max =2040 vppl, k j =205 vpmpl, k c =30 vpmpl, v f =68 mp, and w = 11.7 mp. 15

16 ~ 0.4 miles loop detector No measurement assumed to be known at ramps Figure 1. Higway US101 S used for te NGSIM dataset. Te data is processed in order to emulate loop detector data at bot boundaries of te section. Since all veicle trajectories are known, tis can easily be done witout usual measurement error associated wit loop detectors. Tese emulated detectors will provide boundary conditions as defined in Section 3.2. Ramps counts could be inferred in te same manner, but tey are assumed to be unknown, a realistic assumption given te lack of available infrastructure for most freeways in Nort America. Scenarios investigated Penetration rate and sampling strategies. Te implementation of te algoritms selects a subset of te NGSIM data and treats it as Lagrangian data used for te assimilation. Twelve different scenarios were investigated to account for different penetration rates and sampling strategies. Te penetration rate P is te proportion of trajectories tat are cosen (randomly) as equipped veicles. Tese equipped veicles report teir position and speed every time interval T. Te reported speed is te average speed over te last τ seconds (Figure 2 sketces tis sampling strategy). Te values cosen for P, T, and τ determine te total number of Lagrangian measurements created for eac case investigated. Five different values of P and two values of T were investigated (τ was assumed 6 seconds in all scenarios). Table 1 sows te scenarios investigated and te average number of Lagrangian measurements per mile lane per minute for eac one of tem. Distance i-t report n i-t tracking begins (i-1)-t report τ T Time Figure 2. Scematic of te sampling strategy on equipped veicle n. 16

17 Table 1. Twelve scenarios investigated in te NGSIM data set. Case P (%) T (sec) # of Lagrangian Measurements per mile lane per minute Te travel time for te section of interest is around 2 minutes under congested conditions. Tus, scenarios wit T = 150s assume tat eac equipped veicle sends only one report wile it is traveling te section. Almost continuous tracking for equipped veicles is assumed for scenarios 7 to 12 (T =10s and τ =6s). Because of privacy issues, it is not clear if suc a sampling strategy would be socially acceptable for a commercial product, a problem addressed in (Ho et al., 2008). Tis issue is still open and generates ongoing debates. Tey were investigated to evaluate te traffic reconstruction potential of te proposed metod. Parameters selection. Te nudging factor λ(x,t) depends on tree parameters. Nudging parameters are typically set euristically based on pysical considerations. Parameters X nudge and T d determines ow far in space and time, respectively, an observation influence te solution. For instance, if congestion is observed at location x at time t, it is expected tat congestion also exists at locations in [x X nudge, x+ X nudge ] between times [t, t+ T d ]. For te present case, te values for X nudge and T d are 180 ft and 15 seconds, respectively. Te parameter T a can be seen as a gain tat determines te strengt of te nudging term. Terefore, it as to be tuned as a control parameter of te model. For te present study, values of 10, 20 and 30 seconds ave provided good results for te scenarios investigated. Results Te twelve scenarios presented in Table 1 were implemented following te metods described in Section 3. For comparison purposes, we use a scenario wic only incorporates information from boundary detectors, wic we will refer to as Eulerian data only (EDO). Tis scenario was implemented according to te numerical sceme presented earlier, and does not make use of te ramp counts for te estimation. 17

18 Te Root Mean Square Error 8 (RMSE) of te accumulation of veicles (i.e. number of veicles per cell per time step) is used as a metric for accuracy of te metod. Te true accumulation of veicles can easily be computed since all veicle trajectories are known. Te RMSE for te EDO case is 2.6. Figure 3 sows te ground trut and estimated evolution over time (orizontal axis) and space (vertical axis) of te accumulation of veicles for te EDO case. Colors close to red correspond to ig accumulation of veicle (i.e. congestion). Figure 4 sows te same information for scenarios 1, 3, and 9 using te NR (left) and te KF (rigt). It can be seen tat all te estimates are able to capture te main sockwaves traversing te section of interest. Te intensity of te sockwave, owever, is not equally captured by different scenarios. Figure 3. Veicle accumulation per cell for a) ground trut, and b) EDO case. Figures 3 and 4 suggest tat te main difference among te estimations occurs during te first minutes (before 8:00am), wen some waves emanate from intermediate locations. In fact, te fundamental difference between te EDO scenario and scenarios wit Lagrangian measurements in terms of te RMSE appens during tis period of time. Te added value of Lagrangian measurements is tus clear, as it enables te metods to capture penomena oterwise not detectable wit loop detectors only. For te remaining period of time (until 8:30am), te scenarios wit Lagrangian measurements do not sow a significant improvement wen compared wit te EDO. Figure 5 sows te evolution of te actual total accumulation of veicles on te entire section over time and its corresponding estimate wit EDO and scenario 5 for NR and KF. Tis grap confirms tat te main difference occurs during te first minutes. It is expected tat for longer sections between detectors, wit more intermediate ramps and probably more waves emanating from intermediate locations, te difference between EDO and scenarios wit Lagrangian measurements would be larger. Table 2 sows te RMSE for eac scenario investigated and its corresponding Percentage of Improvement (PoI) RMSEEDO RMSEi wen compared wit te EDO case. Te PoI for scenario i is computed as PoI i = 100. Note RMSE EDO ( ) 8 Te RMSE is defined as: RMSE = zˆ z 2, were ẑ H and z are te estimator and its actual value at time step, respectively, and H is te total number of observations. 18

19 tat te RMSE for eac scenario corresponds to te average over 20 different realizations 9. Figure 6a sows te same information as a function of te total number of Lagrangian measurements used. Figure 4. Veicle accumulation (veicles per cell) estimated using Newtonian relaxation metod (left) and Kalman filtering tecniques (rigt) for scenarios 1 (top), 3 (middle), and 9 (bottom). Figure 6a suggests an increasing performance of te algoritms as te number of observations increases. Te performance reaces a saturation level for large number of observations (more tan 40 observations per mile lane per minute in tis case) at about 22% for te NR and 31% for te KF. Te solid line sows te performance for scenarios 1 to 6 (one observation per equipped veicle), wile te dotted line corresponds to scenarios 7 to 12 (almost continuous tracking). For te NR, te lines in Figure 6a also suggest tat for similar numbers of observations (scenarios 4 and 7), aving more veicles sending observations less frequently seems to produce more accurate results tan aving few veicles being continuously tracked. Tis gap is not observed in te KF implementation. 9 For te same penetration rate P and sampling strategy (P,τ), different realizations consider different veicles as equipped veicles and different times wen measurements are sent. 19

20 Veicle accumulation in te section (ve) Groud trut EDO 80 Scenario 5 (NM) Scenario 5 (KF) 60 7:50 8:05 8:20 8:35 Time Figure 5. Total veicle accumulation on te entire section. Table 2. RMSE for eac scenario and its improvement wit respect to te EDO case. Newtonian relaxation Kalman filtering Scenario RMSE Improvement (%) RMSE Improvement (%) Ideally, te observed density computed using te observed velocity and te fundamental diagram (Section ) sould matc te ground trut density. As expected, it is not te case in practice. Figure 7 sows te true density computed from veicle trajectories versus te observed density computed for scenario 4 (te trend is very similar for te oter scenarios). For a given value of true density, several values of observed density are obtained. In Figure 7, te circle is te mean of tese values, and te range covers one standard deviation. Tis figure reveals tat for te same actual density, different speed reports are sent, suggesting tat (i) te speed reported is not representative of te actual state on te cell, and/or (ii) in congestion, a given density can be acieved at different velocities (or same velocity yields different densities). We believe tat te main source of error is related wit (ii). Indeed, it is well known tat te congested branc of te fundamental diagram is not a line of points but a cloud of points for non stationary traffic (Cassidy, 1998). 20

21 Figure 6. Improvement in te RMSE as te number of Lagrangian measurements canges (Sce: scenario): (a) computing observed density using te fundamental diagram (Section 3.2.1), (b) using te actual density computed from veicle trajectories as te observed one. Note tat bot graps are at different scales. 900 a) 900 b) Observed density Observed density True density True density Figure 7. True (computed from veicle trajectories) versus Observed density (computed using te fundamental diagram as described in Section 3.2.1) for scenario 3 (left) and 9 (rigt). 21

22 To determine ow te error in te observed density affects te performance of te metods, all scenarios were investigated assuming tat te density computed from te Lagrangian observations was te actual one. Tat is, te observations are perfect (error free). Te improvement of te RMSE wit respect to te EDO case is sown in Figure 6b. As expected, te accuracy of te estimates improves, acieving up to 40% (NR) and 50% (KF) of total improvement compared wit te EDO case. Te gap between te solid and te dotted line observed in Figure 6a for te NR is not observed in part b of te figure, suggesting tat wen te Lagrangian observations report te correct density, te sampling strategy does not really matter. Statistical approaces suc as te ones mentioned in Section 2 can be used to compute te velocity field. Using te fundamental diagram, te velocity field can be converted into te density field. Tis process is useful to compare te results obtained wit te Newtonian relaxation metod and te results obtained wit a simple statistical approac in tis case, te metod based on te approac proposed by Sanwal and Walrand (1995). In terms of te RMSE, only scenarios 11 and 12 sow an improvement compared to te EDO case (3.4% and 5.7%, respectively). Te rest of te scenarios yield a RMSE iger tan te one obtained for te EDO case. Tat is, for tis case, a metod based on a traffic flow model produce more accurate estimates of te veicle accumulation tan a simple statistical metod. Tis is not surprising since te statistical metod is meant to estimate velocity and not veicle accumulation. Te main difference between te estimates from te EDO scenario and from scenarios using Lagrangian measurements was observed during te periods in wic sockwaves emanate from intermediate points. Tis result is insigtful regarding te benefits of using Lagrangian measurements. By collecting data from individual veicles at different times and locations, we were able to capture te sockwave generated in te middle of te section (between detectors). Terefore, in te presence of Lagrangian data, inter detector spacing could be increased. Unfortunately, te NGSIM section is too sort to test tis statement. Te next section describes and provides result of a field deployment conducted to obtain Lagrangian measurements from GPS enabled smart pones on a longer section of igway. 4.2 Field experiment: Mobile Century Nicknamed te Mobile Century experiment, a field experiment was carried out on February 8, 2008, involving 100 veicles carrying GPS enabled Nokia N95 pones. Tese veicles drove repeated loops of six to ten miles in lengt continuously for nine ours on freeway I 880 near Union City, CA (see Figure 8) in order to acieve up to a 5% penetration rate of te total volume of traffic on te igway during te experiment. 165 UC Berkeley drivers were ired for tis experiment 10. Tis section of igway was selected specifically for its interesting traffic properties, wic include alternating periods of free flow and congestion trougout te day. In particular, te nortbound direction (NB) presents a recurrent and severe bottleneck between Tennyson Rd. and CA92 during te afternoon. On te day of te experiment tere was an accident during te morning tat activated a non recurrent bottleneck at tis same location. Te section is also well covered wit existing loop detector stations feeding into te PeMS system, wic was used to assess te quality of te estimates. 10 A full description of te experiment is visible in video format at ttp://traffic.berkeley.edu. 22

23 Figure 8. Deployment section on igway I 880, Ca, for te Mobile Century experiment. Te N95 Nokia pones used for te study stored teir position and velocity data every 3 seconds, wic after processing te data enables te reconstruction of every equipped veicle trajectory. For operational reasons, lanes 1 to 4 (lane 1 being te leftmost lane) on te section between Decoto Rd. and Winton Ave. in te NB direction were considered. Te period of time analyzed starts at 10am and ends at 6pm. Te 6.5 miles were divided into I = 44 cells of Δx = 780 ft eac, and te 8 ours period of meaningful data was divided into H = 3600 time steps of Δt = 8 seconds eac. As in te NGSIM case, parameters of te fundamental diagram were extracted from PeMS 11. Different scenarios Te N95 Nokia pones stored position and velocity logs every 3 seconds. Te penetration rate P is given by te number of smart pones on te road. During te day, tis rate ranged between 2% and 5% of te total flow. Using tis data, we canged te values of T to construct different scenarios following te metod outlined in te previous section (τ = 2 for all tree scenarios, were τ corresponds to te number of logs over wic te average speed is computed). Table 3 sows te scenarios investigated in tis case and te total number of Lagrangian measurements per lane mile per our tat eac one produces. Table 3. Scenarios investigated using data from field experiment. Case T (sec) # of Lagrangian measurements per mile lane per minute Te parameter coice for X nudge and T d is te same as for te NGSIM case (180 ft and 15s, respectively). Te values of T a cosen for tis experiment are 10s for te first two scenarios and 30s for te tird one. 11 Assuming a triangular sape (and using te same notation as before): q max =2275 vppl, k j =152 vpmpl, k c =35 vpmpl, v f =65 mp, and w = 19.4 mp. 23

24 Results Unlike for te NGSIM data, in tis case we only ave access to trajectories of te equipped subset of veicles. Terefore, trajectories and veicle accumulation ground trut are not known. To evaluate te results we use information collected by 17 loop detector stations installed along te section of interest using te PeMS system. Figure 9 sows te density field produced by PeMS. Te density field collected by PeMS captures te morning accident at postmile 26 (between Tennyson Rd and CA92). It also sows te spatial and temporal extent of te queue tat was formed upstream. Te situation at tis location does not fully recover, and te bottleneck remains active until te evening. A sort wave propagates upstream for about 2 miles at 12:30pm. At 2pm congestion starts, and by 3pm all te section between postmile 21 (Decoto Rd) and 26 exibit some serious level of congestion. Note tat te severity of congestion around postmile 24 (between Wipple Rd. and Industrial Blvd.) is iger tan for some oter locations, suggesting te presence of a second active bottleneck in series at tis postmile. Figure 9. Density field (in vpm) using 17 loop detector stations deployed in te section of interest (obtained troug PeMS). Figure 10 sows estimated density for eac scenario in Table 3 using NR (left) and KF (rigt). Recall tat eac scenario only makes use of two detector stations (at bot boundaries of te section) in addition to te Lagrangian data provided by te cell pones. Te main congested pattern observed from detectors is well captured in te tree scenarios investigated and for te two metods. All te scenarios captured te accident at postmile 26 at 10:30am, even toug not all te equipped veicles were on te igway by ten (veicles were released between 10 10:30am). Te location, intensity and duration of te incident are well replicated in eac scenario. Te sort wave at 12:30pm is also observed on te tree scenarios, but wit different duration and intensity. Te main congestion tat starts at 2pm can be seen in te tree scenarios as well. Te more severe congestion observed during te afternoon from detectors around postmile 24 is also observed from Lagrangian data. In summary, tere is a good agreement between te density field provided by PeMS and te reconstruction using Lagrangian data. As expected, te agreement wit PeMS data improves wit te number of Lagrangian observations. 24

25 Figure 10. Density field (in vpm) using te Newtonian relaxation metod (left) and te Kalman filtering tecniques (rigt) for scenario 1 (top), 2 (middle), and 3 (bottom). For eac scenario, te boundary data is provided by loop detectors. Te value cosen for k FF in equation (11) was arbitrarily set to 25 vpmpl. Different values of k FF cange te estimates but not significantly. Te main difference observed for different values of k FF occurs during te transition between free flow and congested periods. Cases wit iger values of k FF tend to congestion more easily tan lower values. Note tat te estimates sown in Figure 10 make use of only two detector stations located at bot boundaries of te section. However, in te absence of detector data at te boundaries of te section, te results are still very accurate. For instance, we can assume tat te Eulerian boundary data is also provided by equipped veicles. To tis end, te concept of virtual trip lines (VTL) introduced in Ho et al. (2008) is useful. A VTL can be tougt of a virtual loop detector tat records speed of equipped veicles as tey cross it. In our case, every 5 minutes a VTL at 25