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CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

AN ADAPTIVE TIME GAP CAR-FOLLOWING MODEL

MACROSCOPIC STABILITY OF A FLOW STUDY BY SIMULATION

Sylvain Lassarre∗ Michel Roussignol‡ Antoine Tordeux∗‡

∗INRETS, Groupe d’Analyse du RIsque et de sa Gouvernance, 23 rue Alfred Nobel, 77420

Champs-sur-Marne, France.

‡Université Paris-Est Marne-la-Vallée, Laboratoire d’Analyse et de Mathématiques Appliquées,

5 boulevard Descartes, 77454 Marne-la-Vallée, France

TRB 88 th Annual Meeting — SESSION # 779

TITLE

Effects of Driver Behavior on Traffic Flow Characteristics

SPONSORED BY

Traffic Flow Theory and Characteristics

PRESIDING OFFICER

Michael J. Cassidy, University of California, Berkeley 15/01/2009 — Washington DC

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Partners

GARIG

• Groupe d’Analyse du RIsque et de sa Gouvernance.
• One of the 19 research unities of INRETS (The French

National Institute for Transport and Safety Research). LAMA

• Laboratoire d’Analyse et de Mathématiques Appliquées.
• Research unity of Paris-Est University and CNRS.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Our objective

• Define a microscopic longitudinal traffic model

◮ at a local scale (≈ from 200 m to 10 km) ; ◮ heterogeneous : vehicles distinguished according to their type and characteristic ; ◮ in an uni-directionnal context.

• Evaluate the impact of microscopic driver behavior on the

macroscopic state of a flow.

• Evaluate the impact of some traffic exploitation strategies
• n the performances and safety of the drivers.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Why multi-agent microscopic modelling ?

(and not a macroscopic one) A traffic flow, composed of numerous vehicles, can be apprehended as a macroscopic phenomenon. ◮ In a microscopic approach of modelling, a macroscopic phenomenon emerges from the interactions of a collective of agents. ◮ Traffic flow characteristics, complex and hardly visible, are explained by the behavior of the drivers, better known, who compose it.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Presentation main lines

• Context of the microscopic traffic flow modelling.
• Definition of an original car-following model.
• Macroscopic stability study by simulation.

Table of contents : CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Fondamental variables – Notes used

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Distinction of free / interactive case

• FIG. 1: Fondamental diagramm of mean speed / density and

flow / density. Results obtained on real data.

• ρj

critical density threshold ;

• ρ < ρj

free trafic : vehicles traveled at desired speed ;

• ρ > ρj

interactive trafic : speed regulation.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Existence of a microscopic safety state

• FIG. 2: Distance gap and mean distance gap according to the
• speed. Results obtained on real data.
• Existence of safety distance that is an increasing fonction
• f speed ;
• The consequence of a reaction time ?

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Models of the 1950s

• Defined by a fonction of acceleration or speed.
• Discrete definition with a time step δt equal to the reaction

time T r.

• Two fondamental variables institute a duality :

◮ the difference of speed with the predecessor vi+1 − vi ; ◮ the difference of position with the predecessor xi+1 − xi.

• Macroscopic equilibrium function of mean speed or flow vs

density associated by integration.

• No explicit definition of a microscopic safety state.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Models of the 1950s

• Some famous authors : Greenshields - Chandler -

Kometani and Sasaki - Greenberg - Edie - Underwood - Newell - Helly - Pipes - Rothery- Bexelius.

• A general model define by Gazis, Herman and Rothery

(1961) : ai(t + T r) = λc vi(t + T r)l (xi+1(t) − xi(t))m (vi+1(t) − vi(t)) with λ (distinguished between acceleration or deceleration phase), l and m some parameters.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Models from the 1990s to the present day

• Defined by a fonction of acceleration or speed.
• Continuous definition (implementation of a discretization

scheme in simulation, distinction δt / T r).

• Explicite definition of a microscopic safety state modeled

by a function of targeted safety speed, distance or time.

• Relaxation process applied to the speed, the distance gap
• r a function of these variables and the targeted safety

state function.

• Anticipation strategies incorporating several predecessors.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Models from the 1990s to the present day

• Some famous authors : Gipps - Bando and Hasebe -

Jiang, Zu and Zhu - Helbing and Tilch - Treiber and Hennecke - Wagner, Lubashevsky and Mahnke - Lenz - Addision and Low.

• A general model define by Aw, Klar, Materne and Rascle

(2002) :

dvi(t) dt

= C

vi+1(t)−vi(t) (xi+1(t)−xi(t))γ+1 + A 1 τ

• Ve
• ℓ

xi+1(t)−xi(t)

• − vi(t)
• with Ve the equilibrium targeted safety speed function, C,

A, γ and τ some parameters.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Meta-models

• The four Wiedemann’s model driving situation :
• 1. Free driving (distance gap large, desired speed ϑ)
• 2. Regulation (slower predecessor)
• 3. Stable following (safety distance, Action Point model)
• 4. Braking (emergency braking).
• The Treiber Human Driver Meta-Model elements :
• 1. Finite reaction time (involving a delay)
• 2. Modelling of appreciation errors (noise function of

environnemental conditions)

• 3. Temporal anticipation (to paliate to the reaction time)
• 4. Spatial anticipation (several predecessors).

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Fondamental concept retained

• 1. Existence of a strictly positive reaction time ;
• 2. Explicit definition of a certain microscopic safety state that

depends on speed ;

• 3. Explicit definition of a certain regulation strategy in order to

reach the safety state ;

• 4. Definition of an anticipation strategy allowing to paliate to

the reaction time ;

• 5. Implementation of asymmetric longitudinal behavior where

acceleration and deceleration phases are distinguished ;

• 6. Introduction of a stochastic noise.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Key parameter retained

• Reaction time seems to be an essential parameter of the

car-following context (Leutzbach).

• Reaction time and time gap seem to be complementary :

◮ the reaction time defines a physiological delay ; ◮ the time gap defines a physical delay.

→ Model based on the regulation of the time gap Ti(t) = ∆i(t) vi(t) , vi(t) = 0. towards a safety time gap T s(vi(t)) = f(vi(t))

vi(t)

that is a function of speed (f(.) is the function of safety distance gap).

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Normative approach to a microscopic safety state

• The existence of a reaction time and some limited

deceleration capacity can engender collisions. A minimum safety time (or distance) allows to palliate to the collisions

• Approach notably developed by Kometani, Gipps or

Wagner. T min

i

(v) =         

(T r

i )2

2v amin

i+1amin i

amin

i+1−amin i

if amin

i

< amin

i+1

and v > T r

i amin

i+1amin i

amin

i+1−amin i

T r

i − v 2 amin

i+1−amin i

amin

i+1amin i

• therwise.

with amin < 0 the capacity of deceleration of a vehicle.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Normative approach to a microscopic safety state : Illustrative example

• FIG. 3: Distance and time gap minimum allowing to avoid collision

according to the speed and the capacity of deceleration of the considered vehicle. amin

i+1 = −5 m/s2 et T r = 1.5 s.

10 20 30 40 50 0.5 1.0 1.5 2.0 2.5 speed, m/s minimum safety time gap, s 10 20 30 40 50 20 40 60 80 100 speed, m/s minimum safety distance gap, m

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Targeted microscopic safety state and macroscopic performances of a homogeneous flow

Homogeneous flow : ∀i, vi = ϑ (free state) Ti = T s(vi) (interactive state)

• Microscopic → Macroscopic

Mean distance gap : ¯ ∆ = 1/ρ − ¯ ℓ ; Mean speed : Ve(ρ) = min{ϑ, f −1(1/ρ − ¯ ℓ)} ; Flow : Qe(ρ) = ρ × min{ϑ, f −1(1/ρ − ¯ ℓ)} ; Critical density : ρc =

1 f(ϑ)−¯ ℓ.

• Macroscopic → Microscopic

Targeted safety time gap : T s(v) = 1/V−1

e

(v)−¯ ℓ v

; Targeted safety distance gap : f(v) = 1/V−1

e (v) − ¯

ℓ.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Targeted microscopic safety state and macroscopic performances : Illustrative example

• FIG. 4: Targeted safety time as linear function of speed and corresponding

fundamental flow-density plot according to, at left, targeted safety time increasing, and, at rigth, decreasing.

5 10 20 30 1.0 1.2 1.4 1.6 1.8

Ts(v) increasing

speed, m/s time gap, s 0.00 0.10 0.20 0.0 0.2 0.4 0.6 0.8 density, veh/m flow, veh/s 5 10 20 30 1.0 1.2 1.4 1.6 1.8

Ts(v) decreasing

speed, m/s time gap, s 0.00 0.10 0.20 0.0 0.2 0.4 0.6 0.8 density, veh/m flow, veh/s

simulation

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Model definition

• Stochastic continuous definition by an adaptative

differential system :   

dxi(t) dt

= vi(t) (i)

dTi(t) dt

= λi(t) (T s(vi(t)) − Ti(t)) + Ei(t) (ii)

◮ The deterministic relation (i), extracted from kinematic models, models the motion of the vehicles. ◮ The stochastic relation (ii), models vehicles interaction and has to be calibrated.

• Through relationship T = ∆

v , the system can be rewritten

  

dxi(t) dt

= vi(t)

dvi(t) dt

= vi(t)vi+1(t)−vi(t)

∆i(t)

− λi(t)

• vi(t)T s(vi(t))

Ti(t)

− vi(t)

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Parameter required

• T s(v).

The targeted safety time that defines the security related behavior of a driver confronted to collision risks. Plays an important role as for the macroscopic performances of a flow.

• λ ∈ ]0, 1].

Quantity defining different regulation strategies. A dynamic definition of λ allows to restitute a asymmetric longitudinal behavior.

• E.

Allows to incorporate a variability of driver response.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Discretization scheme

Determinitic case. Implementation of

• an implicit Euler scheme to (i) :

xi(t + δt) = xi(t) + δt vi(t + δt);

• an explicit Euler scheme to (ii) :

Ti(t + δt) = Ti(t) + δtλi(t) (T s(vi(t)) − Ti(t)) . In a cyclic environment, one obtains a scheme for the development of the system through an inverse matrix calculation. The scheme calculates synchronically the speed of vehicles at time t + δt knowing the state of the system at time t.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Illustrative example 1

different regulation strategies according to λ

• FIG. 5: Example of a deceleration situation according to different values of the

parameter λ. Respectively from left to right, acceleration rate, speed and time gap according to the time. Results obtained by simulation.

10 20 30 40 50 −3.0 −1.5 0.0 time, s acceleration, m s2 10 20 30 40 50 5 10 15 20 time, s speed, m/s 10 20 30 40 50 2 4 6 8 time, s time gap, s

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Illustrative example 2

different regulation strategies according to λ

• FIG. 6: Example of a catching-up situation according to different values of the

parameter λ. Respectively from left to right, acceleration rate, speed and time gap according to the time. Results obtained by simulation.

10 20 30 40 50 −2 2 4 time, s acceleration, m s2 10 20 30 40 50 5 10 15 20 time, s speed, m/s 10 20 30 40 50 5 10 15 time, s time gap, s

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Reaction time and anticipation strategy

Ti(t) ˜ Ti(t) = ˜

xi+1(t)−xi(t)−(ℓi+1+ℓi)/2 vi(t)

with ˜ xi+1(t) = vi+1(t − T r) + t

t−T r ˜

vi+1(u) du

• Without anticipation, the predecessor speed is supposed

constant :

✞ ✝ ☎ ✆

˜ xi+1(t) = xi+1(t − T r) + T rvi+1(t − T r)

• With anticipation, the predecessor speed is estimated with

v j

i+1 by supposing the speed of the j th predecessor

constant :

☛ ✡ ✟ ✠

˜ xi+1(t) = xi+1(t − T r) + t

t−T r v j i+1(u) du

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Results observed

Macroscopic stability of a flow according to driver behavior

• A decreasing targeted safety time function allows the easy

emergence of deceleration waves (traffic jams).

• Anticipation strategies (or an increasing targeted safety

time function) are a spatial homogeneity factor.

NETLOGO simulation software modul

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Experience protocol

A rectilinear periodical network of length 1012 m is considered.

• Experience 1 : single trajectory

◮ 1 trajectory of the system ; ◮ 45 vehicles ; ◮ homogeneous initial conditions, one vehicle is perturbated ; ◮ observation time : 500 s.

• Experience 2 : average results

◮ 10 independent trajectories of the system ; ◮ variable number of vehicles (from 5 to 100) ; ◮ random initial conditions ; ◮ observation time : 100 s (above a delay of 900 s).

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) constant

• T s > T r. With or without anticipation strategies,

perturbations seem to be absorbed.

• T s = T r. Without anticipation, perturbations seem to

spread indefinitely, without getting larger or smaller. With anticipation strategies, perturbations seem to be absorbed.

• T s < T r. Collisions take place. With anticipation strategies,

provided the number of interacting vehicles is suffcient, no collision takes place.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) constant — Experience 1

• FIG. 7: Space / time / density diagram and speed of the disrupted vehicle and the

next 20 vehicles as a function of time. λ = 0.3, j = 1, T r = 1 s. Top : T s = 1.1 s. Bottom : T s = 1 s.

t i m e space d e n s i t y 10 20 30 40 5 10 15 20 time, s speed, m/s time s p a c e d e n s i t y 10 20 30 40 5 10 15 20 time, s speed, m/s

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) constant — Experience 2

• FIG. 8: Presence of collisions according to different values of T s and j.

λ = 0.3, T r = 1 s.

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 target safety time, s

• nb. of predecessors
• presence of collisions

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) linear

T s(v) = T r + av if a > 0 T r + a(v − ϑ)

• therwise
• T s(v) increasing, spatial heterogeneity factor. With or

without anticipation strategies, perturbations seem to be absorbed.

• T s(v) decreasing, spatial homogeneity factor. Without

anticipation, perturbations seem to amplify themself and spread indefinitely (with anticipation strategies, perturbations seem to be absorbed).

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) linear — Experience 2

• FIG. 9: Fundamental diagrams of flow / density (A) and speed variance /

density (B) according to different density and values of a. λ = 0.3, j = 1, T r = 1 s.

a −0.03 −0.02 −0.01 0.00 0.01 0.02 0.03 density, veh/m 0.02 0.04 0.06 0.08 flow, veh/s 0.2 0.4 0.6 0.8

(A)

a −0.03 −0.02 −0.01 0.00 0.01 0.02 0.03 density, veh/m 0.02 0.04 0.06 0.08 speed variance 0.0 0.5 1.0 1.5 2.0 2.5

(B)

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) non-linear

Statistical estimation done a NGSIM sample, shows a targeted safety time function strongly decreasing : T s(v) = γ2 + γ1 ln(v + 1)/v with γ1 ≈ 2.5 et γ2 ≈ 0.75. By simulation, ones observes that this definition of the targeted safety time function is strongly factor of spatial heterogeneity. The system’s convergence towards spatial homogeneity need the implementation of a efficient number of vehicles in interaction.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) non-linear — Experience 1

• FIG. 10: Space / time / density diagram and speed of the disrupted vehicle and

the next 20 vehicles as a function of time. λ = 0.3, T r = 1 s. Top : j = 1. Bottom : j = 5.

t i m e s p a c e d e n s i t y 10 20 30 40 5 10 15 20 time, s speed, m/s t i m e space d e n s i t y 10 20 30 40 5 10 15 20 time, s speed, m/s

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) non-linear — Experience 1

• FIG. 11: Space time diagram. The black trajectory is the perturbated vehicle
• trajectory. λ = 0.3, T r = 1 s, j = 1.

100 200 300 400 500 −400 −200 200 400 time, s space, m

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

T s(v) non-linear — Experience 2

• FIG. 12: Fundamental diagrams of flow / density (A) and speed

variance / density (B) according to different density and values of j. λ = 0.3, T r = 1 s.

nb of predecessors 5 10 15 20 density, veh/m 0.02 0.04 0.06 0.08 flow, veh/s 0.2 0.4 0.6 0.8

(A)

nb of predecessors 5 10 15 20 density, veh/m 0.02 0.04 0.06 0.08 speed variance 1 2 3

(B)

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

Conclusion — Prospect

⋆ Conclusion

• Definition of a car-following model including
• parametrable safety state (function T s(.))
• different regulation strategies (λ)
• reaction time (T r)
• anticipation strategies (j)
• Parameters allow to control macroscopic stability of a flow.
• Targeted safety time function decreasing with speed : an

explication to the observed traffic high level of instability. ⋆ Prospects

• Dynamic definition of λ in order to restituate an asymmetric

longitudinal behavior.

• Evaluate the impact of the noise.

CONTEXT DEFINITION OF A MODEL SIMULATION RESULTS CONCLUSION BIBLIOGRAPHY

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