C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY A N ADAPTIVE TIME GAP CAR - FOLLOWING MODEL MACROSCOPIC STABILITY OF A FLOW STUDY BY SIMULATION S ylvain L assarre ∗ M ichel R oussignol ‡ A ntoine T ordeux ∗‡ ∗ 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 Effects of Driver Behavior on Traffic Flow Characteristics TITLE Traffic Flow Theory and Characteristics SPONSORED BY Michael J. Cassidy, University of California, Berkeley PRESIDING OFFICER 15/01/2009 — Washington DC
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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 on the performances and safety of the drivers.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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 : C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY Fondamental variables – Notes used
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY Distinction of free / interactive case F IG . 1: Fondamental diagramm of mean speed / density and flow / density. Results obtained on real data. • ρ j critical density threshold ; • ρ < ρ j free trafic : vehicles traveled at desired speed ; • ρ > ρ j interactive trafic : speed regulation.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY Existence of a microscopic safety state F IG . 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 of speed ; • The consequence of a reaction time ?
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY Models of the 1950s • Defined by a fonction of acceleration or speed. • Discrete definition with a time step δ t equal to the reaction time T r . • Two fondamental variables institute a duality : ◮ the difference of speed with the predecessor v i + 1 − v i ; ◮ the difference of position with the predecessor x i + 1 − x i . • Macroscopic equilibrium function of mean speed or flow vs density associated by integration. • No explicit definition of a microscopic safety state.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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) : v i ( t + T r ) l a i ( t + T r ) = λ c ( x i + 1 ( t ) − x i ( t )) m ( v i + 1 ( t ) − v i ( t )) with λ (distinguished between acceleration or deceleration phase), l and m some parameters.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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 δ t / 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 or a function of these variables and the targeted safety state function. • Anticipation strategies incorporating several predecessors.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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) : � � � � dv i ( t ) v i + 1 ( t ) − v i ( t ) ( x i + 1 ( t ) − x i ( t )) γ + 1 + A 1 ℓ = C V e − v i ( t ) dt τ x i + 1 ( t ) − x i ( t ) with V e the equilibrium targeted safety speed function, C , A , γ and τ some parameters.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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).
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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 T i ( t ) = ∆ i ( t ) v i ( t ) , v i ( t ) � = 0 . towards a safety time gap T s ( v i ( t )) = f ( v i ( t )) that is a function of v i ( t ) speed ( f ( . ) is the function of safety distance gap).
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY 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 r i ) 2 a min i + 1 a min if a min < a min i i + 1 2 v a min i + 1 − a min i i a min i + 1 a min and v > T r i T min ( v ) = i a min i + 1 − a min i i a min i + 1 − a min i − v T r i otherwise . 2 a min i + 1 a min i with a min < 0 the capacity of deceleration of a vehicle.
C ONTEXT D EFINITION OF A MODEL S IMULATION RESULTS C ONCLUSION B IBLIOGRAPHY Normative approach to a microscopic safety state : Illustrative example F IG . 3: Distance and time gap minimum allowing to avoid collision according to the speed and the capacity of deceleration of the i + 1 = − 5 m / s 2 et T r = 1 . 5 s . considered vehicle. a min minimum safety distance gap, m 100 minimum safety time gap, s 2.5 80 2.0 60 1.5 40 1.0 20 0.5 0 0 10 20 30 40 50 0 10 20 30 40 50 speed, m/s speed, m/s
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