100

Transportation Research Record: Journal of the Transportation Research Board, No. 2315, Transportation Research Board of the National Academies, Washington, D.C., 2012, pp. 100–109.DOI: 10.3141/2315-11

Université Paris-Est, Institut Français des Sciences et Technologies des Trans-ports, de l’Aménagement et des Réseaux, Génie des Réseaux de Transports Terrestres et Informatique Avancée, GRETTIA, F-93166 Noisy-le-Grand, France. Corresponding author: N. Farhi, nadir.farhi@ifsttar.fr.

on the basis of the model, and the number of leaders taken into account in anticipation was varied. The effect of the anticipation modeling on transient traffic is thus shown. As expected, it was observed that as the number of leaders taken into account in anticipa­tion increases, the car trajectories are smoothed; the distant followers are slowed down in unstable traffic phases, but they retrieve their nonanticipative trajectories once the traffic has stabilized.

Second, a parameter identification approach for the multi­anticipative model is proposed, and it is shown in a basic example how this method is applied. NGSIM data of vehicle trajectories on a segment of US­101 are used. The purpose here is to present the process of parameter identification, since the data used here are not exhaustive to draw conclusions, and the model is likely to be improved. However, some interesting observations are made on the data sample considered. For example, it was observed that the scatter plot for space headway and average velocities appears to be easy to approximate by a piecewise­linear curve in the case in which the space headway is computed with respect to two leaders as compared with the case in which the space headway is computed with respect to only one leader.

The remainder of this introduction gives a short review of car­following modeling and fixes notations. The notations used are t for time (discrete or continuous), x for distance (car positions), and n for the number of cars. The cars are numbered such that the first car (Car 1) is the leader. These variables are considered:

• x(n, t): cumulative traveled distance of car n from time zero to time t,• y(n, t): intervehicular distance x(n − 1, t) − x(n, t), and• v(n, t): velocity of car n at time t.

Car­following models are often based on a behavioral law Ve (equilibrium speed spacing function) that gives, for equilibrium traffic, the velocity v of a car n as a function of the intervehicular distance y between cars n and n − 1. It is then assumed that the law Ve also holds for transient traffic. A kind of general form of first­order car­following models can then be derived as was done for macroscopic first­order modeling [Lighthill–Whitham–Richards models (13, 14)]:

�v n t V y n t v n te, , , ( )( ) = ′ ( )( ) ( )∆ 1

where Δv(n, t) = v(n − 1, t) − v(n, t).The simplest form for the equilibrium speed–spacing function

Ve(y) is the linear one Ve(y) = αy + β, where α and β are parameters. In this case, the linear car­following model (1, 4) is obtained:

v n t n t, , ( )( ) = ( ) +α β 2

Multianticipative Piecewise-Linear Car-Following Model

Nadir Farhi, Habib Haj-Salem, and Jean-Patrick Lebacque

An extension of the piecewise-linear car-following model to multianticipa-tive driving is proposed. As in the one-car-anticipative model, the stability and the stationary regimes are characterized through a variational for-mulation of the car dynamics. The homogeneous driving case is studied here, and it is shown that with the stationary regime the multianticipative model guarantees the same macroscopic behavior as that for the one-car anticipative model. Nevertheless, in transient traffic, the variance in car velocities and accelerations is mitigated by the multianticipative driving, and the car trajectories are smoothed. A parameter identification of the model is made on the basis of NGSIM data and using a piecewise-linear regression approach.

A multianticipative car­following traffic model is presented, in which drivers control their velocities by taking into account the posi­tions and the velocities of many cars ahead. In basic car­following models (1–4), the car dynamics is described by stimulus–response equations that express the control process of drivers. Each driver accelerates or decelerates depending on his or her speed and on the relative speed and the intervehicular distance with respect to the driver of the car ahead.

Multianticipative car­following models are often extensions of existing one­car anticipative models. In 1968, Bexelius (5) extended the model of Chandler et al. (1) to the multianticipative case. Lenz et al. (6) extended the model of Bando et al. (7). More recently, Hoogendoorn et al. (8) extended the model of Helly (9, 10).

The model presented here is an extension of the piecewise­linear car­following model (11, 12). It is a first­order discrete­time model in which the car velocities are given as a function of the intervehicular distances. It is shown here that the variational formulation made by Farhi holds also for the proposed multianticipative extension (11, 12). That is, the car dynamics is again interpreted as dynamic program­ming equations associated with stochastic optimal control problems of Markov chains, as in the work by Farhi (11, 12). Because of that formulation, it was possible to characterize the stability of the car dynamics and to calculate the stationary regimes.

In transient traffic, some qualitative results are obtained from the model. First, the car dynamics on a one­lane road of about 10,000 m was simulated, in which the trajectories of a given number of leaders were imposed and the trajectories of all the followers were simulated

Farhi, Haj-Salem, and Lebacque 101

The model in Equation 2 is, of course, not satisfactory. Another well­known car­following model is that of Gazis et al. (2):

�v n t T av n t Tv n t v n t

x n t

p, ,

, ,

,+( ) = +( ) −( ) − ( )

−( )1

1 −− ( )( )x n tl

,( )3

where a reaction time T is considered, and p and l are parameters. For example, if p = 1 and l = 2, the model can simply be obtained by taking Ve(y) = b exp − a/y in Equation 1, satisfying V ′e(y) = aVe(y)/y2.

Other car­following models exist that do not necessarily match the form in Equation 1, such as the models of Bando et al. (7) (opti­mal velocity model), Helly (9, 10), Treiber et al. (15) (intelligent driver model), and others. Bando et al. (7) proposed the optimal velocity model:

�v n t T V y n t v n t Te, , , ( )+( ) = ( )( ) − +( )( )λ 4

This model has been much studied recently for being easily analyzed with mathematical tools.

Helly (9, 10) considered the linear model:

�v n t T v n t y n t Sn, , , ( )+( ) = ( ) + ( ) −( )α β∆ 5

where α and β are parameters, and Sn is the desired distance, which can be linear Sn = S0 + Tvn, with S0 the minimum gross distance between two cars.

The base here is the piecewise­linear car­following model pro­posed by Farhi (12), where the behavioral law Ve is approximated with a minimum–maximum (min­max)­piecewise­linear curve:

V y ye u U w W uw uw( ) = +{ }min max ( )� � α β 6

and where a one­car­anticipative discrete­time car dynamics has been obtained:

x n t x n tx n t x n

u U w W

uw

, , min max, ,

+( ) = ( ) +−( ) −

11

� �

α tt uw( )( ) +{ }β ( )7

where αuw and βuw, for (u, w) ∈ U × W, are parameters, and U and W are two finite sets of indices. The system (Equation 7) is also written, for traffic of v cars 1, 2, . . . , v, as follows:

x t T M x t c nn u U w Wuw

n nuw+( ) = ( ) +{ } ≤ ≤min max� � 1 vv ( )8

where Muw and cuw, for (u, w) ∈ U × W, are matrices and column vectors, respectively.

Two cases were distinguished by Farhi (12):

1. The v cars move on a ring road. In this case, Muw and cuw are given by

Muw

uw uw

uw uw

uw uw

=

−−

−

1 01 0

0 0 1

α αα α

α α

�

� � � �

and

cv

duw t uw

uw uw uw= +

α β β β, , . . . ,

The dynamics (Equation 8) is stable under the condition αuw ∈ [0,1], and the behavior law is realized at the stationary regime:

v yu U w W uw uw= +{ }∈ ∈min max ( )α β 9

where v denotes the asymptotic car velocity (the same for all cars), and y denotes the average intervehicular distance in the ring road (y = 1/d).

2. The v cars move on an “open” road, where the velocity v1(t) of the first car (the leader) varies over time but is stationary. In this case, Muw and cuw are given by

Muw uw uw

uw uw

= −

−

1 0 01 0

0 0 1

�

� � � �α α

α α

and

c t v tuw tuw uw( ) = ( )( )1 , ,β β. . . ,

Again, the dynamics (Equation 8) is stable under the condition αuw ∈ [0,1] and the inverse behavior law at the stationary regime is obtained as follows:

yv

u U w Wuw

uw

= −∈ ∈max min ( )1 10

βα

where v1 denotes the asymptotic velocity of the first car.

The dynamics in Equation 8 was interpreted by Farhi (12) under the assumption αuw ∈ [0,1], ∀ (u, w) ∈ U × W as a dynamic program­ming equation associated with a stochastic game on a controlled Markov chain; more details can be found elsewhere (12).

AnticipAtion Modeling

Presented in this section is an extension of the model (Equation 7) to multianticipative traffic, in which each car chooses its velocity depending on the intervehicular distance with respect to a given number m of cars ahead of the considered car (multiple leaders). In order to situate this model with respect to the existing multi­anticipative models and to explain the extension, a short review of multianticipative car­following models is given.

A straightforward multiple­leader extension of the model of Chandler et al. (1) is the Bexelius model (5):

�v n t T v n tjj

j

m

, , ( )+( ) = ( )( )

=∑α ∆

1

11

where

v· = acceleration; αj, j = 1, 2, . . . , m = sensitivity parameters with respect to

jth car ahead; andΔv(j)(n, t) = v(n − j, t) − v(n, t).

102 Transportation Research Record 2315

This model is very simple but permits some mathematical analysis.Hoogendoorn et al. noted the inconvenience of the additive form

of the Bexelius model (Equation 11) and proposed the following modification (8):

�v n t T v n tj m jj, min , ( )+( ) = ∆ ( )≤ ≤

( )1 12α

Hoogendoorn et al. (8) also proposed a multianticipative generalization for the Helly model (Equation 5):

�v n t T v n t x n t Sjj

j

m

jj, , ,+( ) = ∆ ( ) + ∆ ( ) −( )

=

( )∑α β1

1jj

j

m

n( ) =

∑1

2

13( )

where Δx(j)(n, t) = x(n − j, t) − x(n, t). Lenz et al. (6) generalized the Bando model (Equation 4) as follows:

�v n t Vx n t

jv n tj e

j

,,

,( ) =∆ ( )

− ( )

( )κ

=∑j

m

1

14( )

where κj expresses the sensitivity with respect to the jth leader.A multiple­leader extension is proposed here for the piecewise­

linear car­following model (Equation 7). A minimum form is used as in Equation 12 (rather than an additive form as in Equation 11). Moreover, a uniform form for the sensitivity with respect to the intervehicular distance is used as in Equation 14 (the intervehicular distance with respect to the jth leader is divided by j). The dynamics is considered:

x t x tn n j m

j

u U w W+( ) = ( ) + +( )≤ ≤−

∈ ∈1 11

1min min maxλ

αuuwn j n

uw

x t x t

j− ( ) − ( )

+

β ( )15

where m is the number of leaders taken into account in anticipation, and λ ≥ 0 is a discount parameter with respect to the leader index. The dynamics in Equation 15 can be written simply as follows:

x t x t

x t

n n j m u U w W

juwn j

+( ) = ( ) + ≤ ≤ ∈ ∈

−

1 1min min max

α(( ) − ( )

+

x t

jn

juwβ ( )16

where ∀(u, w) ∈ U × W, (αjuw)j, 1 ≤ j ≤ m are increasing nonnegative sequences, and (βjuw)j, 1 ≤ j ≤ m are increasing sequences.

The interpretation of the minimum operator with respect to the jth leader in Equation 16 is that a car n maximizes its velocity under the following constraints:

x t x t

x t x t

n n u U w W

juwn j n

+( ) − ( ) ≤

( ) − ( )∈ ∈

−

1 min max

αjj

j mjuw

+

≤ ≤β 1

One consequence of anticipation in driving is that the informa­tion that a car i, for i = n − 1, n − 2, . . . , max(1, n − m), deceler­ates at time t is immediately transmitted to car n, which reacts at time t + 1 instead of t + n − i. The discounting with respect to the leader indices, made by introducing the multiplicative term (1 + λ)j−1, permits favoring of closer leaders over distant ones. If

λ = 0, the cars respond equally to the stimulus of all the leaders j, with j = 1, 2, . . . , m.

In the following section, the stability of the car dynamics (Equation 15) is studied, and the existence of stationary regimes is characterized. Two cases are distinguished: traffic on a ring road and traffic on an open road. In both cases, the asymptotic car positions are given when stationary regimes exist. The transient traffic for the car dynamics (Equation 15) is treated in the section on transient traffic.

StAbility AnAlySiS And StAtionAry regiMeS

As in the work by Farhi (11, 12), v cars moving on a one­lane road without passing is considered, first when the cars move on a ring road and then on an open road.

traffic on ring road

The cars moving on a ring road, indices n to j in the dynamics (Equation 15), are cyclic in the set {1, 2, . . . , v}. The idea is that the two minimum operators in Equation 15 can be summarized in only one minimum operator and then the one­car anticipative form for the dynamics can be retrieved. The set of all pairs of indices (j, u) is denoted Z with 1 ≤ j ≤ m and u ∈ U:

Z z j u j m u U= = ( ) ≤ ≤ ∈{ }, ,1

The dynamics (Equation 15) is then written as follows:

x t M x t c nn z Z w WzW

n nzW= +( ) ( ) +{ } ≤ ≤∈ ∈1 1min max vv ( )17

where the matrices M zw = M juw and the column vectors czw = c juw are given as follows:

M

j j

j j

juw

juw juw

juw juw

=

−

−

1 0 0 0

0 1 0

α α

α α

α

�

� �

� � � � � jjuw

juw

juw

juw juw

j

j

j

j j

α

α

α α

� � � � �

� � �

�

0 0

0 0 0 1−

and

cj

v

djuw

juw

juw juw juw=

+

α

β β β, , . . . ,

t

The dynamics in Equation 18 has the same form as in Equation 8. It is then interpreted as a dynamic programming equation associated with a stochastic game on a controlled Markov chain. The stability is guaranteed under the condition αjuw ∈ [0,1], ∀( j, u, w) ∈ {1, 2, . . . , m} × U × W; more details can be found elsewhere (11, 12). The stationary regime is characterized by the additive eigenvalue problem:

v x M x cn j m u U w Wjuw

n njuw+ = +{ }≤ ≤ ∈ ∈min min max1 11

18

≤ ≤n v

( )

Farhi, Haj-Salem, and Lebacque 103

where v is the asymptotic car velocity, the same for all cars, and the vector x is the asymptotic car positions, given up to an additive constant. The following result gives a solution for the system shown later in Equation 19.

Theorem 1. If ∀ ( j, u, w) ∈ {1, 2, . . . , m} × U × W, αjuw ∈ [0, 1], the system in Equation 19 admits a solution (v, x) given by

v y

x v y v y

u U w Wuw uw

t

= +{ }= −( ) −( )

∈ ∈min max

, ,

α β1 1

1 2 .. . . , y , 0( )

where y = 1/d is the average intervehicular distance on the ring road.Proof. Following the same approach as in the work by Farhi

(11, 12), the following solution is obtained for the system in Equation 19:

v yz Z w W

zw zwj m u U w

= +{ } =∈ ∈ ≤ ≤ ∈

min max min min maxα β1 ∈∈

+{ }= −( ) −( )( )

Wjuw juw

t

y

x v y v y y

α β

1 2 0, , ,. . . ,

Then, since (αjuw)j and (βjuw)j are increasing sequences (with respect to j) and y ≥ 0,

min min max min max1≤ ≤ ∈ ∈ ∈

+{ } =j m u U w W

juw juwu U w

yα β∈∈

+{ }W

uw uwyα β1 1

■

A particular case is important here, which is the nondiscounting case where λ = 0 in Equation 15. In this case, the matrices Mjuw and the vectors cjuw still depend on j, whereas the parameters αjuw and βjuw are independent of j for all j ∈ {1, 2, . . . , m}. Thus the average car speed v coincides with the average car speed obtained in the one­car anticipative model (Equation 7):

v yu U w W

uw uw= +{ }∈ ∈

min max α β

where αuw = αjuw and βuw = βjuw ∀ j ∈ {1, 2, . . . , m}.

traffic on open road

The speed v1(t) of the first car over time is given, since the cars move on an open road. In addition, in order to analyze the stability and the stationary regime of the car dynamics, it is assumed that the velocity of the first car approaches a constant value v1. That is to say, limt→+∞ v1(t) = v1. Moreover, the number of anticipation cars for the first m cars cannot be m (it is less than m). More precisely, the number of anticipation cars for a car numbered n is min(n − 1, m).

The dynamics (Equation 15) is

x t M x t cn j m u U w Wjuw

n+( ) = ( ) +≤ ≤ ∈ ∈1 1min min max nn

juw

n v

{ }≤ ≤1 19( )

where the matrices Mjuw and the vectors cjuw are as follows (see Equation Box 1).

The entries (Mjuw)ik and (c juw)i for i, k ≤ j, do not play any role in the car dynamics since (cjuw)i = +∞ ∀i ≤ j. Those entries cor­respond to anticipation of a car i with respect to its jth leader that does not exist since i ≤ j.

It is assumed that the velocity v1(t) of the first car reaches a fixed value v1 at the stationary regime. The stationary regime is thus characterized as follows:

v x M x cn j m u U w Wjuw

n njuw+ = +{ }≤ ≤ ∈ ∈min min max1

11 20≤ ≤n v ( )

Mj j

juw

juw juw

=−

1 0 0 0

0 1 0 0 0

0 0 1 0 0 0

0 0 1α α 0

0 0 0 1

0 0 0

0 0 0 0 1

α α

α α

juw juw

juw

j j

j

−

− jjuw j

j

j

j

+

1

2

1

++

=

( )+∞

+∞

2

1

v

c

v t

juw

juw

juw

juw

ββ

β

++

12

12

jjj

v

EQUATION BOX 1

104 Transportation Research Record 2315

The following result gives a solution for the system in Equation 21.Theorem 2. For all y ∈ R satisfying minu∈U maxw∈W(α1uwy + β1uw)

= v1, the couple (v, x) is a solution for the system in Equation 21, where v = v1 and x is given up to an additive constant by

x v y v y yt= −( ) −( )( )1 2 0 21, , , ( ). . . ,

Proof. The proof is similar to that of Farhi’s Theorem 3 (12). Let y ∈ R satisfying

min maxu U w W

uw uwy v∈ ∈

+{ } =α β1 1 1

Let x be given later by Equation 22. Then ∀n ∈ {1, 2, . . . , v} and

min min max min1 1≤ ≤ ∈ ∈ ≤

+ =j m u U w W

juw

n njuw

jM x c

≤≤ ∈ ∈

∈ ∈

+( ) +

=

m u U w Wjuw juw n

u U w W

y xmin max

min max

α β

αα β1 1

1

uw uw n

n

y x

v x

+( ) +

= +

Moreover, the optimal strategy at the stationary regime is (j, u, w) = (1, u, w) such that α1u w y + β1u w = v1. Indeed,

M x c y xuw

n nuw

uw uw n

u U w

1 11 1 + = +( ) +

=∈

α β

min max∈∈

≤ ≤ ∈ ∈

+( ) +

=

Wuw uw n

j m u U w Wjuw

y xα β

α

1 1

1min min max yy x

M x

juw n

j m u U w W

juw

n

+( ) +

= ≤ ≤ ∈ ∈

β

min min max1

++ cnjuw

■

Theorem 2 gives the car velocity at the stationary regime with one stationary configuration of cars (the uniform configuration) and gives the optimal strategy for drivers in that regime. The car velocity obtained is the same as the one obtained for the one­car anticipative model (Equation 7). Moreover, the optimal strategy of driving at the stationary regime in the case of the multianticipative model is to drive by taking into account only one leader: ( j

–, u, w) =

(1, u, w). That is, in the stationary regime, once the traffic has sta­bilized, it is not necessary for drivers to take more than one leader into consideration.

Moreover, the stationary configurations of the cars in the two cases of one­car anticipation and multianticipation models may coincide. This case is interpreted as follows. Even though the cars reduce their approach in the multianticipative dynamics (because of the minimum operator over the leader indices) compared with their movement under the one­car anticipative dynamics, as long as the cars approach the stationary regime, in which the traffic is stabilized, they retrieve what they have lost in the transient regime. Therefore, by introducing the minimum on the multianticipative dynamics, the traffic becomes smoother without a decrease in the stationary car speed.

trAnSient trAffic

The same example as that presented by Farhi (12) is used, which is adapted to the multianticipative case in order to make a comparison. The car dynamics (Equation 15) is simulated. The time unit is taken

as half a second (1/2 s), and as the distance unit, 1 m. The parameters of the model are the same as those in Example 1 of Farhi (12) (the parameters were determined by approximating a given behavior law). More precisely, the behavior law considered here is approximated by the following piecewise­linear curve of six segments:

�V y y y y y y( ) = +{ + + +max min , , ,α β α β α β α β α1 1 2 2 3 3 4 4 5 ++{+ }}

βα β

5

6 6y

where the parameters αi and βi for i = 1, 2, . . . , 6 follow:

for parameter αi:

1 = 0, 2 = 0.54, 3 = 0.32, 4 = 0.13, 5 = 0.34, and 6 = 0.

for parameter βi:

1 = 0, 2 = −8.1, 3 = −1.47, 4 = 6.11, 5 = 10.6, and 6 = 14.

The car dynamics on a one­lane road of about 10,000 m is simu­lated. The number of leaders and the velocity of the first car (over time) are varied to show the effect of multianticipation on transient traffic. The trajectories are smoothed by anticipation, as shown by the notation lists for parameters α and β above. The trajectory of the first car is the same for all views of Figure 1. Although the number of leaders taken into account by drivers cannot exceed five in prac­tice, the car dynamics is simulated here with anticipation with up to 100 leaders. This procedure was done out of curiosity, but it can be interesting in the case, for example, where one would like to study the traffic of communicating cars or automatic ones.

pArAMeter identificAtion

A parameter identification method is proposed here based on (piece­wise) linear regression. With measured data on the car positions and velocities for a given section, the method permits determination of the optimal parameters that match the model with the measured data. Since the dynamics of the model is simply the car veloci­ties given as functions of the intervehicular distances, the optimal parameters would be the ones that approximate the scatter plot of instantaneous intervehicular distances and velocities.

The variable y(m,λ) is denoted as follows:

�y n tx n j t x n t

mj m

j

, , min, ,

λ

λ( ) ≤ ≤

−

( ) =+( ) −( ) − (

1

11 ))( )

j

For fixed values of m and λ and with measured velocities v(n, t) and intervehicular distances x(n − j, t) − x(n, t) for every car n, the scatter plot V(y) is approximated by a piecewise­linear curve:

V y yu U w W uw uw� �( ) = +( )∈ ∈min max ( )α β 22

Farhi, Haj-Salem, and Lebacque 105

(a)

(c)

(e)

(b)

(d)

(f)

FIGURE 1 Traffic on one-lane road: x-axis, time; y-axis, car position (number of cars taken into account in anticipation 5 1, 5, 10, 20, 50, and 100; road length 5 10,000 m; total simulation time 5 500 s).

The min­max piecewise­linear approximation (Equation 23) is based on a piecewise­linear regression approach in which the number of segments as well as the points where they intersect are determined optimally by deterministic dynamic programming. The optimization with respect to the parameters m and λ is done numerically by varying the two parameters in convenient intervals and then determining the optimal parameters.

For fixed values of parameters m and λ, the scatter plot V(y(m,λ)) is approximated by linear regression on separated intervals. The inter­vals are determined by a dynamic programming approach. More precisely, the axis of intervehicular distances y is divided into unity intervals (yi, yi+1). Starting from the first interval, a linear regression is made on that interval, and then for the second interval it is decided whether to make only one linear regression for the two intervals or

106 Transportation Research Record 2315

to approximate the scatter plot on the second interval with another linear segment. Then the same procedure is done for the third unity interval, and so on. The optimal control problem associated with the decision process is solved here. The intervehicular distance plays the role of time in this decision process.

The decision at the unity interval yi, denoted here by r(yi), is in {0, 1}:

• r(yi) = 0 if it is decided to make one linear regression for the interval yi together with the intervals before it and• r(yi) = 1 if it is decided to start a new linear regression from the

unity interval yi.

Costs k(y, N, r) are defined to minimize regression errors and penal­ize large segmentation (limit the number of segments used in the approximation): k(yi, N, r) = the error of regression at the unit inter­val (yi, yi+1), when the interval (yi−N, yi) is wholly approximated by one segment, and when the decision r is taken at the stage yi.Then the costs k(y, N, r) are given by

• k(y, N, r) = linear regression error in (yi−N, yi+1) if r = 0 and• k(y, N, r) = linear regression error in (yi, yi+1) + φ if r = 1, where

φ is a penalty for starting a new linear regression. Then the following optimal control problem is solved:

min , , ( )max

λ∈=

( )∑Γ k y N ry

y

0

23

where Γ is the set of boolean strategies on {0, 1, . . . , ymax}. That is, γ : {0, 1, . . . , ymax} y → r {0, 1}. The value function associated with Equation 24 is

G y N k z N rz y

y

, , , ( )max

( ) = ( )=

∑ 24

G satisfies the dynamic programming equation:

G y

G y N k y N G y N ki i i

max

, min , , , ,

( ) =

( ) = ( ) + +( )+

0

0 11 yy N G yi i, , ,

( )

1 1

25

1( ) + ( ){ }+

The parameter identification method is thus summarized as fol­lows. For each pair of parameters (m, λ), y(m,λ) is calculated. Then the curves V( y(m,λ)) are approximated by using the piecewise­ linear regression approach explained earlier [that is, by solving the dynamic programming equation (Equation 26)]. A total regression error is obtained for each approximation (m, λ). Finally, the best pair (m, λ) is determined that gives the minimal total error.

A first application of the proposed identification method is shown in the following paragraphs. The basis here is the NGSIM data of vehicle trajectories on a segment of US­101 (Hollywood Free­way) in Los Angeles, California. The data were collected between 7:50 a.m. and 8:05 a.m. on June 15, 2005. A preliminary analysis of the trajectories showed that according to the multianticipative model presented here, the nth leaders’ positions for n < 3 are redun­dant in the data considered here, even with a null discount parameter (λ = 0). That is,

� �y n t y n t n tm , ,, , ,λ λ λ( ) ( )( ) ( ) ∀ ∀≥ ≥ 0;3

Consequently, no more than three leaders are considered here (m ∈ {1, 2, 3}). Indeed, it can be seen from the dynamics (Equa­tion 15) that for λ = 0, taking into account more than one leader in anticipation does not change anything in the case in which the traffic is accelerating, because the spacing to the jth leader is bigger than j times the spacing to the first leader:

x t x t j x t x t jn j n n n− −( ) − ( ) ( ) − ( )( ) =≥ . 1 1 2, , . . .

Nevertheless, this application is realistic because, in the case of accelerating traffic, anticipation is not significant.

Figure 2 is the scatter plot for y(m,0) with the average car velocities (the average over all cars) for m ∈ {1, 2, 3}. It seems that V(y(2,0)) and V(y(3,0)) (V also denotes the average car velocity over all cars) can easily be approximated with piecewise­linear curves, com­pared with V(y(1,0)). There is no justification for that approximation currently.

The results of the parameter identification are given in Figure 3, which shows the total errors from the piecewise­linear regression obtained for varied values of parameters m ∈ {1, 2, 3} and λ ∈ [0, 5], increased by 0.1. The optimal parameters obtained here are m = 2 and λ = 1.5. As mentioned earlier, it is not worth considering values of m that exceed 3 (for the data considered here). Also, for large values of λ, the same results are retrieved as if it is taken that m = 1. Therefore, the optimization here is significant.

The scatter plot for y(2,1.5) with V is approximated by the following curve of four segments:

V y y y� � �( ) = − +{ }{max , min . . , . . ,0 0 38 1 90 0 11 2 95 10 }} ( )26

The approximation is shown in Figure 4.As shown in Figure 4, it is not easy to identify one behavior for a

large number of drivers. In fact, the parameter identification should be done for each driver. The identification of the behavior law for a randomly selected driver has given the following result. The optimal parameters (m, λ) are m = 2 (anticipation with two leaders) and λ = 0 (no discounting). The curve V(y(2,0)) is approximated as follows:

V y y� �2 0 2 00 0 33 1 71 8, ,max , min . . ,( ) ( )( ) ≈ −( )( )

The approximation is shown in Figure 5, which also shows the approximation of the curve V(y(1,0)):

V y y� �1 0 1 00 0 26 0 9 8, ,max , min . . ,( ) ( )( ) ≈ −( )( )

As obtained by the parameter identification approach, it can be seen in Figure 5 that the curve V(y(2,0)) is well fitted by a piecewise­linear curve compared with the curve V(y(1,0)). The analytical results for the stability of the dynamics and the stationary regimes presented in the section on stability analysis and stationary regimes do not nec­essarily hold for the case of several drivers. Driver heterogeneity will be considered in later research.

concluSion

An extension of the piecewise­linear car­following model to multi­anticipative driving is presented. The minimum form used for taking into account more than one leader, with the discounting parameter used to favor the closest leaders over the distant ones, seems to be

Farhi, Haj-Salem, and Lebacque 107

(a)

00 20 40 60

Inter-vehicular distance

80 100 120 140

123456789

101112131415

Ave

rage

car

-vel

ocity

(b)

00 20 40 60

Inter-vehicular distance

80 100 120 140

123456789

101112131415

Ave

rage

car

-vel

ocity

(c)

00 20 40 60

Inter-vehicular distance

80 100 120 140

123456789

101112131415

Ave

rage

car

-vel

ocity

FIGURE 2 Average car velocity function of intervehicular distance (l 5 0): (a) one leader, (b) two leaders, and (c) three leaders.

FIGURE 3 Identification of parameters m and l.

FIGURE 5 Parameter identification of one-driver behavior: (a) one-leader anticipation and (b) two-leader anticipation.

0

1

2

3

4

Vel

ocity

5

6

7

8

0 5 10 15 20 1st leader space headway

25 30 35 40

(a)

(b)

0

1

2

3

4

Vel

ocity

5

6

7

8

0 5 10 15 20 Aggregation of 1st and 2nd leader space headway

25 30 35 40

FIGURE 4 Approximation of law V(y(2,1.5)) with min-max piecewise-linear curve (Number of leaders taken into account m 5 2; discounting parameter l 5 1.5).

0 0 20 40 60 80 100 120

5

10

15

Car

vel

ocity

V(y~

(2,1

.5))

Inter-vehicular distance y~(2,1.5)

Farhi, Haj-Salem, and Lebacque 109

convenient to recapture the main characteristics of anticipative traf­fic. It is shown that the trajectories are smoothed by anticipation in transient traffic without affecting the stationary regimes. That is, the anticipation does not slow down the stationary traffic. The minimum form used for anticipation allows the same variational formulation to be made as was done in one­car anticipative traf­fic, and this formulation allows the characterization of the stability of the car dynamics and calculation of the stationary regimes. The identification test done here is only a first step in analyzing the pro­posed model. In­depth analyses on exhaustive data should be done in the future to improve the modeling approach presented here. In particular, it will be attempted to extend the model in a way that takes into consideration heterogeneity in driving.

referenceS

1. Chandler, R. E., R. Herman, and E. W. Montroll. Traffic Dynam­ics: Studies in Car Following. Operations Research, Vol. 6, 1958, pp. 165–184.

2. Gazis, D. C., R. Herman, and R. W. Rothery. Nonlinear Follow­the­Leader Models of Traffic Flow. Operations Research, Vol. 9, No. 4, 1961, pp. 545–567.

3. Gazis, D. C., R. Herman, and R. B. Potts. Car­Following Theory of Steady­State Traffic Flow. Operations Research, Vol. 7, No. 4, 1959, pp. 499–505.

4. Herman, R., E. W. Montroll, R. B. Potts, and R. W. Rothery. Traf­fic Dynamics: Analysis of Stability in Car Following. Operations Research, Vol. 7, No. 1, 1959, pp. 86–106.

5. Bexelius, S. An Extended Model for Car Following. Transportation Research, Vol. 2, No. 1, 1968, pp. 13–21.

6. Lenz, H., C. K. Wagner, and R. Sollacher. Multianticipative Car­Following Model. European Physical Journal B, Vol. 7, 1999, pp. 331–335.

7. Bando, M., K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama. Dynamical Model of Traffic Congestion and Numerical Simulation. Physical Review E, Vol. 51, No. 2, 1995.

8. Hoogendoorn, S. P., S. Ossen, and M. Schreuder. Empirics of Multi­anticipative Car­Following Behavior. In Transportation Research Record: Journal of the Transportation Research Board, No. 1965, Transportation Research Board of the National Academies, Washington, D.C., 2006, pp. 112–120.

9. Helly, W. Simulation of Bottlenecks in Single Lane Traffic Flow. In Proceedings of the Symposium on the Theory of Traffic Flow, Elsevier, New York, 1959.

10. Helly, W. Simulation of Bottlenecks in Single Lane Traffic Flow. In Theory of Traffic Flow, Elsevier Publishing Co., New York, 1961, pp. 207–238.

11. Farhi, N. Modelisation Minplus et Commande du Trafic de Villes Regu-lières. PhD thesis. University of Paris, Panthon­Sorbonne, 2008.

12. Farhi, N. Piecewise Linear Car­Following Modeling. arXiv:1107.5869v2, 2011.

13. Lighthill, J., and J. B. Whitham. On Kinematic Waves II: A Theory of Traffic Flow on Long, Crowded Roads. Proceedings of the Royal Soci-ety, Series A, Vol. 229, 1955, pp. 281–345.

14. Richards, P. I. Shock Waves on the Highway. Operations Research, Vol. 4, 1956, pp. 42–51.

15. Treiber, M., A. Hennecke, and D. Helbing. Congested Traffic States in Empirical Observations and Microscopic Simulations. Physical Review E, Vol. 62, 2000, p. 1805.

The Traffic Flow Theory and Characteristics Committee peer-reviewed this paper.