Validation and verification methodology to calibrate the ...
Traffic signal design-I · 2019-01-02 · Greenshield's model • Example –Calibrate Greenshields...
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Traffic stream models Tom Mathew
Transcript of Traffic signal design-I · 2019-01-02 · Greenshield's model • Example –Calibrate Greenshields...
Traffic stream models
Tom Mathew
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Traffic stream models
• Macroscopic
– Expression of the average behavior of the
vehicles at the specific location and time
• Mesoscopic
– Small group of traffic entities with activities
and interactions
• Microscopic
– Space-time behavior of the systems’ entities
(i.e. vehicle and drivers)
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Traffic stream models
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Traffic stream models
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Traffic stream models
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Traffic stream models
• Macroscopic Stream Models
– Greenshield's linear
– Greenberg's logarithmic
– Underwood's exponential
– Pipe's generalized
– Multi regime models
• Two and Three regime
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Greenshield's model
• Linear speed-density relationship
Relation between speed and density
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Greenshield's model
• Description
– v = mean speed
– k = density
– 𝒗𝒇 = free flow speed
– 𝒌𝒋= jam density
– When density becomes zero, speed
approaches free flow speed
Two parameter model
𝒗 = 𝑣𝑓 −𝑣𝑓
𝑘𝑗𝒌
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10
20
30
40
50
60
70
80
90
0 50 100 150 200 250
Sp
ee
d (
v)
Density (k)
vf=80, kj=200
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Greenshield's model
• Relation between flow and density
𝒗 = 𝑣𝑓 −𝑣𝑓
𝑘𝑗𝒌
𝒒 = 𝒌 𝒗
𝒒 = 𝒌 𝑣𝑓 −𝑣𝑓
𝑘𝑗𝒌
𝒒 = 𝑣𝑓𝒌 −𝑣𝑓
𝑘𝑗𝒌2
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500
1000
1500
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3500
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4500
0 50 100 150 200
Flo
w (
q)
Density (k)
vf=80, kj=200
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Greenshield's model
• Boundary conditions
– Maximum flow qmax
– Density corresponding to max. flow ko
– Speed corresponding to max. flow vo
• Model parameters
– Jam density kj
– Free flow speed vf
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Greenshield's model
• Derivation of ko
– We have
– Differentiating
𝒒 = 𝑣𝑓𝒌 −𝑣𝑓
𝑘𝑗𝒌2
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Greenshield's model
• Derivation of qmax
• Derivation of vo
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Greenshield's model
• Relation between speed and flow
𝒗 = 𝑣𝑓 −𝑣𝑓
𝑘𝑗𝒌
𝒒 = 𝒌 𝒗
𝒌 = 𝑘𝑗 −𝑘𝑗
𝑣𝑓𝒗
𝒒 = 𝑘𝑗𝒗 −𝑘𝑗
𝑣𝑓𝒗2
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0 1000 2000 3000 4000 5000
Speed (v
)
Flow (q)
vf=80, kj=200
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Greenshield's model
• Calibration
– Determination of model parameters
– Free flow speed (vf)
– Jam density (kj)
where x is density and y denotes speed
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Greenshield's model
• Calibration
– Using linear regression method
OR
a is
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Greenshield's model
• Example
– Calibrate Greenshields model
using the data give in the
table
– Find the maximum flow
– Find the density
corresponding to a speed of
30 km/hr
No K v
1 171 5
2 129 15
3 20 40
4 70 25
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Greenshield's model
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Model Comparison
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Model Comparison
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Greenberg's model
• Logarithmic relation
– Advantage
• Analytical derivation
• Good at congestion
– Drawbacks
• Infinite speed
• Poor at low densities
𝒗 = 𝑣𝑜𝑙𝑛𝑘𝑗
𝒌
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Underwood's model
• Exponential Model
– Advantage
• Good at low speed
– Drawbacks
• Speed is zero only at
infinity density
• Poor at high densities
𝒗 = 𝑣𝑓𝒆−𝒌𝑘𝑜
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Pipes' model
• Generalized Model
– When n is 1 Pipe’s model resembles
Greenshield’s model
𝒗 = 𝒗𝒇 1 −𝒌
𝒌𝒋
𝒏
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Comparison of Models
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0 50 100 150 200 250
Sp
eed
V(k
mp
h)
Density k (veh/km)
k = 200, vf = 90, ko = 100, v0 = 45
Liner n = 1
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Comparison of Models
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30
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60
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0 50 100 150 200 250
Sp
eed
V(k
mp
h)
Density k (veh/km)
k = 200, vf = 90, ko = 100, v0 = 45
Liner n = 1
Log
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Comparison of Models
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0 50 100 150 200 250
Sp
eed
V(k
mp
h)
Density k (veh/km)
k = 200, vf = 90, ko = 100, v0 = 45Liner n = 1
Log
Exp
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Comparison of Models
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80
90
100
0 50 100 150 200 250
Sp
eed
V(k
mp
h)
Density k (veh/km)
k = 200, vf = 90, ko = 100, v0 = 45 Gen n = 0.5
Liner n = 1
Gen n = 2
Log
Exp
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Model Comparison
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Model Comparison
Model Comparison
• Limitations
– Assumes uniform behaviour
– Poor predictability
– Generalization
• Need for multi-regime models
– Traffic states divided into various regimes
– Behaviour depends on the regime
– Separate models for each regime
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Multiregime model
• Eddie’s Two Regime Model
– Based on field data (Chicago)
Regime 1:
Exponential
Regime 2:
Logarithmic
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Multiregime model
• Eddie’s Two Regime Model
– Based on field data (Chicago)
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Multiregime model
• Eddie’s Two Regime Model
Greenshields
Model
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Multiregime model
• Three Regime Model
– Free flow
– Normal
– Congested
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Multiregime model
Free flow
CongestedNormal
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Multiregime model
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Multi-regime models
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Speed –Flow: Effect of location
• Simultaneous treatment of q k v
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Conclusion
• Concerns
– The current status of mathematical models for speed-
flow concentration relationships is in a state of flux
– The models that dominated for nearly 30 years are
incompatible with the data currently being obtained
– but no replacement models have yet been developed
Lieu 1999, Traffic-Flow Theory
US DOT, Federal Highway Administration
http://www.tfhrc.gov/pubrds/janfeb99/traffic.htm
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Conclusion
• Trends
– Despite those words of caution, it is important
to note that there have been significant
advances in understanding traffic stream
behavior since 1980’s leading to a better
understanding of traffic operation
– Efforts to implement ITS will provide
challenges for applying this improvement
– Equally important, ITS will likely provide the
opportunity for acquiring more and better data
to advance understanding of traffic operations
Thank You
Questions?
