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Introduction- Fatmah Ebrahim
Traffic Jam FactsThere are 500,000 traffic jams a year.
That’s 10,000 a week.
Or 200-300 a day.
Traffic congestion costs the economy of England £22bn a year1
[1] Eddington Transport Study, Rod Eddington (2006)
Overview• Queuing Theory• Macroscopic Flow Theory• Kinetic Theory• Cellular Automata• Three Phase Theory• Vehicle Following Model• Bifurcations• Computer Model• Mechanical Model• Data Analysis• Conclusions
Queuing Theory- Roger Hackett
Parameters of Queuing Theory
Flow rate: q
Capacity: Q
Intensity: x=q/Q
Pollaczek-Khintchine Formula
d t
2
x
1 x
1 Cs
2
Macroscopic Flow Theory
- Peter Edmunds
Macroscopic Flow TheoryWe need to define three variables:
Spatial density, K: the number of vehicles per
unit length of a given traffic system.
Flow, Q: the number of vehicles per unit time
Speed, v: the time rate of progression
These lead to the fundamental equation of traffic
flow: Q=Kv
We can obtain an equation that resembles the
equation of continuity for fluid flow:
This is based on the assumption that no vehicles
enter or leave the road.
It can be adapted for n traffic lanes and for
inflows or outflows of gΔxΔt.
Modeling traffic flow as a fluid
To solve this, we assume that the flow q is a function of
the density k.
We obtain the equation:
This is solved by the method of characteristics.
Eventually, after introducing a parameter s along the
characteristic curves, the final equation can be derived:
An analytic solution of this equation is usually
impossible and so what is done in practice is to
draw the graph of q=kf(k) against k.
Kinetic Traffic Flow Theory
- Joshua Mann
Developed to explain the macroscopic properties of
gases.
Pressure, temperature and volume are modelled by
considering the motion and molecular composition of
the particles.
Original theory was static repulsion.
Kinetic Theory
Primitive Speed Equation
v
onacceleratidiscrete
discrete
onacceleratidiscrete
discrete
onacceleratismooth
v
pressureconvection
dt
dkVdv
dt
dv
kdt
dv
x
kVar
kx
VV
t
V
21
11
•Convection term: change of the average speed V due to a spatial speed gradient carried with the flow V.
•Pressure term: change of average speed V as a result of individual vehicles that travel at v < V and v > V.
•Smooth acceleration: change of average speed V due to smooth individual accelerations.
•Discrete acceleration 1: change of the average speed V due to events that cause a discrete change in the number of vehicles with expected speed v.
•Discrete acceleration 2: change of the average speed V due to a discrete change in the total number of vehicles.
x
kVar
k
Vdvdt
dv
kW
x
VV
t
V
w
v discrete
w
1
• Where W is a drivers desired speed.
• is the relaxation time.w
Modified Speed Equation
Advantages:
• It provides a realistic representation of multiclass traffic.
• It reproduces phenomena observed in congested traffic.
• It helps to relate traffic flow models to the behaviour of the driver.
Disadvantages:
• The individual behaviour of drivers is still not fully accounted for.
• The model cannot fully describe complex traffic flows in towns.
Advantages and Disadvantages
Cellular Automata Traffic Model- Joshua Mann
An idealization of a physical system.
Physical quantities take a finite set of values and
space and time are discrete.
Traffic flow is modelled using the road traffic rule.
Cellular Automata
Road Traffic Rule Model
Vehicles modelled as point particles moving along a line
of sites.
A vehicle can only move if its destination cell is free.
If the destination cell is freed at the same time as motion
the vehicle does not move until after the cell is vacated
as it cannot observe the other vehicles motion.
110
0
• Traffic light situation.
• Numbers in grid are turn flags and indicate priority.
• Condition allowed is right turn on red light.
Applications
Advantages:
• It enables the study of traffic flow in towns and cities.
• It allows the implication of certain road regulations to be modelled.
Disadvantages:
• It does not account in any way for the behaviour of the driver.
• The individual speed of vehicles is not accounted for.
• The differing sizes of vehicles are not accounted for.
Advantages and Disadvantages
Three Phase Theory- Eóin Davies
The Three Phase Theory of Traffic Flow
Classical Theory (Two Phases):
•Free Flow•Congested
Three Phase (Congested phase split into two):
•Free Flow•Synchronized flow•Wide-moving jam
Fundamental Hypothesis of Three Phase Traffic Theory
Transitions
• Free Flow -> Synchronised Flow
• Synchronised Flow -> Wide-moving Jam
Conclusions• It is qualitative theory.
• It is a description of traffic patterns not an explanation.
• Not widely accepted.
• Based on data from German freeways - there is no reason that the results would match other roads in other countries.
Vehicle Following Model
- Steven Kinghorn
VFM studies the relationship between two successive vehicles.
Each following vehicle responds to the vehicle directly in front.
Following vehicle Leading vehicle
tvn tvn 1
S
Velocity of the following vehicle
Separation distance between two vehicles
Velocity of the leading vehicle tvn
tvn 1
S
Vehicle Following Model (VFM)
Response = Sensitivity Stimulus
General form of model
Response – Braking or accelerating
Sensitivity – Driver reaction time
Stimulus – Change in relative speed
One example of a VFM equation: -
Other VFM’s have different variations in sensitivity. For example, a VFM developed by Gazis, Herman & Potts (1959) has a greater sensitivity for smaller spacing between vehicles: -
Stimulus
nn
ySensitivitsponse
tvtvaAcc 11Re
(1)
(2)
Stimulus
nn
ySensitivit
sponse
tvtvS
aAcc 1
2
Re
(3)
Speed of leading vehicle
Speed of following vehicle
Limitations – following vehicles only react to the vehicles directly in front. However, majority of drivers would look further ahead to gauge traffic conditions.
Computer simulations can be created to introduce many different types of traffic systems (Traffic lights, lanes closer etc)
By applying a vehicle following model, we can study how congestion might be caused and develop ways to reduce it.
VFM in Computer simulation
Bifurcations- Roger Hackett
Bifurcations
This is the reaction time delay vehicle following model.
The Computer Model- Alex Travis
v0: desired velocity ; the velocity the vehicle would drive at in free traffic s*: desired dynamical distances0: minimum spacing; a minimum net distance that is kept even at a complete stand-still in a traffic jam T: desired time headway; the desired time headway to the vehicle in front a: acceleration of vehicleb: comfortable braking decelerationδ is set to 4 as conventions: distance of vehicle aheadv: velocity of vehicle∆v: velocity difference or approaching rate between the vehicle and that of the vehicle directly ahead.
Intelligent Driver Model
Acceleration on free road Deceleration due to car ahead
v0: desired velocity ; the velocity the vehicle would drive at in free traffic s*: desired dynamical distances0: minimum spacing; a minimum net distance that is kept even at a complete stand-still in a traffic jam T: desired time headway; the desired time headway to the vehicle in front a: acceleration of vehicleb: comfortable braking decelerationδ is set to 4 as conventions: distance of vehicle aheadv: velocity of vehicle∆v: velocity difference or approaching rate between the vehicle and that of the vehicle directly ahead.
Graphs Produced for Single Lane Model
The Mechanical Model- Eóin Davies
Mechanical Model
Q=k.v
Q=Flow k=density v=velocity
•Want to confirm this relation.
•Need to measure these variables.
Release balls at a fixed rate.
Density and speed of balls varies when angle of ramp changes
xFigure 1
Mechanical Model
1. Set value of flow by releasing bearings at fixed intervals.
2. Measure speed of balls at certain angle of ramp.
3. Measure Density at different flow rates.
4. Use Q=k.v to calculate flow.
Method
Comparing set flow and flow calculated using K.vRamp angle
(low to High) Calculated flow (k.u) Set Flow (No. Of balls/Sec)
10.790.430.33
1.000.500.33
20.810.480.34
1.000.500.33
30.950.500.33
1.000.500.33
40.960.530.35
1.000.500.33
50.950.530.37
1.000.500.33
Results
Data Analysis- Peter Edmunds
Data AnalysisWe needed to analyze data to investigate which of the theories
already mentioned is the most appropriate for traffic flow.
On the 28th of January our group attempted to take data from the
M1. This was a failure.
Professor Heydecker from the Transport Department at UCL very
kindly allowed us to use his data, taken in conjunction with the
Highways Agency.
Data AnalysisThe M25 data seems to be in agreement with
Greenshield’s original model.
In truth, however, every road is different and will
produce different curves.
In modern traffic data analysis an amalgam of each
theory is used, along with empirical data for the road in
question.
Conclusion- Fatmah Ebrahim
Theory Pros Cons
Queuing Theory
Macroscopic Theory Empirical corroboration Limited applications
Kinetic Theory Multi-class traffic modeling
Cannot model for stop-and-go traffic scenarios
Cellular Automata Can model for stop-and-go traffic scenarios
All components are modeled identically
Vehicle-Following Models
Good for creating computer simulations
Cannot account for unexpected incidents
Three-Phase Theory
Describes complex congestion patterns Not widely tested
Summary
Future Possibilities Automated Highway Systems (AHS)Experiment carried out by National Automated Highway Systems Consortium In 1997
Thanks For Listening
Eóin DaviesFatmah EbrahimPeter EdmundsRoger Hackett
Steven KinghornJoshua MannAlex Travis
with thanks toDr.Stan Zochowski and Dr. BG Heydecker
For more information or a full report please visit our website
http://ucltrafficproject.wordpress.com