On The Generation Mechanisms of Stop-Start Waves in Traffic Flow

H. Michael Zhang, ProfessorDepartment of Civil and Environmental Engineering

University of CaliforniaDavis, CA 95616

Distinguished ProfessorSchool of Transportation Engineering

Tongji University Shanghai, China

The Sixth International Conference on Nonlinear MechanicsAugust 12-15, 2013, Shanghai, China

Outline of Presentation

•Features of congested traffic•Conventional wisdom about stop-start waves•An alternative explanation of stop-start waves•Discussions and conclusion

Traffic congestion is everywhere

from Los Angeles

to Beijing

Features of congested traffic

• Phase transitions

• Nonlinear waves

• Stop-and-Go Waves (periodic motion)

Phase transitions

Nonlinear waves

Vehicle platoon traveling throughtwo shock waves

flow-density phase plot

Stop-and-Go Waves (Oscillations)

Scatter in the phase diagram is closely related to stop-and-go wave motion

Conventional Wisdom

• Phase transitions: nonlinearity and randomness in driving behavior

• Nonlinear waves: nonlinear, anisotropic driving behavior

• Stop-and-Go waves: stochasticity +– H1: instability in CF (ODE) or Flow (PDE)– H2: Lane change

Models and Evidences• Microscopic

– Modified Pipes’ model – Newell’ Model– Bando’ model

• Macroscopic continuum – LWR model– Payne-Whitham model– Aw-Rascle, Zhang

model

min , /n f nx v s t l

( ) 1 exp ( ) /n f n fx t v s t l v

*( ) , 1/n n nx t a u s x t a

* 0t xq

0,t xv 2

*0t xx

v vcv vv

* * * *1/ , , ,s u s v q v q v

0,t xv *

t x

v vv v c v

'*c v

Illustration: cluster solutions in the Bando model with a non-concave FD

L=6,000 m, l=6m, T=600s, dt=0.1s, j=167 veh/km, N=300 veh, average gap=14 m, Avg. occ is 0.3 .Vehicles randomly placed on circular road with 0 speed

Evidence I: cluster solutions in the Bando model with a non-concave FD

Evidence II: cluster solutions in PW model with a non-concave FD

L=22.4km, T=0.7 hr=5s,

From Kerner 1998

Evidence II: cluster solutions in PW model with a non-concave FD

location

Time=500

Difficulties with CF/Flow Models

• Due to instability, cluster solutions are sensitive to initial conditions and the resolution of the difference scheme

• Wave magnitudes and periods are hard to predict and often not in the same order of magnitudes with observed values

An Alternative Explanation

• Main cause: lane changing at merge bottlenecks

• Mechanism: “route” and lane-change location choice produces interacting waves

• Model: network LWR model

The Model

, 1 1 1( ) min{ ( ), , [ ( )]}, 1, , fi i i a i i ay t x t C N x t i

( ) ( ) ( ) ( )( ) (1 )min{ ( ), , }, ( ) min{ ( ), , }

1 1t st s t s

u u

S t S t S t S tf t r D t f t r D t

r r r r

( ) mid{ ( ), ( ) ( ), (1 ) ( )}t t d s df t D t S t D t p S t

( ) mid{ ( ), ( ) ( ), ( )}s s d t df t D t S t D t pS t

Link flow

Diverge flow

Merge flow

The Mechanism

Numerical Results

Discussions• Solutions can be obtained analytically if FD is

triangular• Wave periods are controlled by free-flow speed,

jam wave speed, and distance between diverge and merge “points”

• Only under sufficiently high demands stop-start waves occur

• Stop-start waves travel at the speed of jam wave• Stop-start waves do not grow in magnitude

Conclusions

• Stop-start waves can arise from ‘routing’ and lane change choices

• A network LWR model can produce stop-start waves with right periods and magnitudes

• But waves do not grow, need to introduce instability/stochastic elements