Flow Patterns in Cellular Automata and OptimalVelocity Traffic Models at Highway Bottlenecks

1
Flow Patterns in Cellular Automata and
Optimal-Velocity Traffic Models at Highway
Bottlenecks

• Justin FindlaySupervisor Dr. Peter Berg

2
Introduction

• Three types of Traffic Models
• Microscopic Cellular Automata (CA)
• Microscopic Car-Following (CF)
• Macroscopic Continuum (CT)
• Link between CF and CT established 2000.
• Link between CF and CA not well established.

3
Highway Bottleneck

• Purpose is to study flow patterns near highway
  bottlenecks in the CA and CF models.
• Bottlenecks caused by congestion due to car
  wrecks, construction and on-ramps.
• Modeled using a reduction in the maximum speed
  inside the bottleneck.
• For a vehicle positions
• 0 lt x lt LB, the maximum
• speed in the bottleneck
• applies,
• Outside the bottleneck,
• applies.

Figure 1. Representation of the road with a
bottleneck and open road.
4
Cellular Automata Model

• Discrete in space and time.
• Fluid-dynamical approach based on
• 1D array of sites occupied by a
• vehicle or empty.
• Periodic boundary conditions.
• Updated based on rules at each time step.
• Velocity determined by headway.
• Flux determined by q v(1/?)?
• where v(h) v(1/?)

Figure 2. Velocity as a function of the headway
between cars.
5
Optimal-Velocity CF Model

• Continuous in space and time.
• Equation of motion.
• Optimal-velocity determined in
• the same way as CA model
• in Figure 2.
• Resulting in similar Fundamental
• Diagram (FD).

Figure 3. Fundamental Diagram for the CA model.
OV model uses similar FD with different values.
6
Wave Solutions

• From FD, there are 6
• possible wave solutions
• indicated by chords.

Figure 4. Possible wave solutions from the FD.
Cases (a) and (b) occur in the simulations and
correspond to chord 1 and 2 respectively. Case
(c) (e) correspond to chords 3-5 and do not
occur in the simulations.
7
Wave Solutions from Simulations

• All three solutions here are found in both the CA
  and OV models.
• The remaining possible solutions can be ruled out
  using Continuum theory.

Figure 5. Wave solutions found during the
simulations. (a) Two shocks emerge, one at the
downstream bottleneck and one classical shock on
the open road. (b) one shock at each boundary
of the bottleneck (c) Homogeneous flow.
8
Analyzing Known Wave Solutions

• From the conservation of cars,
• And equating the fluxes of the bottleneck and
  open road.
• The wave solution of Figure 5b is found.
• Once the density is increased beyond this value,
  the wave solution of Figure 5a is found and the
  length of the second plateau can be determined,

9
Traveling-Wave Phase Plane Analysis

• The known solutions were found by analyzing the
  conservation of cars and the flux on the previous
  slide.
• The missing solutions are analyzed through
  stability analysis using second-order continuum
  theory.
• Continuum Model
• From these equations the stability of the
  possible connections in the FD are shown not to
  exist.

Conservation of Cars
Dynamical equation with optimal-velocity
approximation (Lee et al, 2000)
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Conclusion and Future Work

• The CA and OV models can be directly compared
  through wave connections.
• Future work will concentrate on unstable CA and
  CF models.

11
References

• M. Bando, et al. Dynamical model of traffic
  congestion and numerical simulation. Physical
  Review E, 51(2)1035-1042, 1995.
• P. Berg and J. Findlay. Linking cellular
  automata and optimal-velocity models through wave
  selections at bottlenecks. Traffic and Granular
  Flow 05, 2007.
• P. Berg, et al. Continuum approach to
  car-following models. Physical Review E,
  61(2)1056-1066, 2000.
• H. K. Lee, et. al. Macroscopic traffic models
  from microscopic car-following models. Physical
  Review E, 64, 2001.
• K. Nagel, et al. A cellular automaton model for
  freeway traffic. Journal de Physique I,
  22221-2229, 1992.