TTEng 422 s2021 module 5 Introduction to Traffic Flow Theory

Module 5:
Traffic Flow Theory
& Shockwave
Analysis
TTE 422 Traffic Operations - Copyright © 2021 Wael ElDessouki Spring 2021
1.Car following Model
2.Macroscopic Flow
Models
3.Shockwaves
Applications
229

Car Following Models
Car following theories and models dictates the distance
between vehicles and estimation of traffic density in the
traffic stream.
TTE 422 Traffic Operations - Copyright © 2021 Wael ElDessouki
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Spring 2021

Distance Headway Characteristics
 Distance headway is defined as the distance
from a selected point on the lead vehicle to the
same point on the following vehicle.
 Distance gap is defined as the gap length
between the rear edge of the lead vehicle and
the front edge of the following vehicle.
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Distance Headway Characteristics
 Time headway rather than distance headway is more encountered because
of the greater ease of measuring time headway. Distance headway can be
obtained only photographically, but usually obtained by calculation based
on time headway as follows:
where,
 dn+1= distance headway of vehicle (n+1) (m)
 hn+1= time headway of vehicle (n+1) (sec.)
 = speed of vehicle (n+1) during time periodhn+1 (m/sec)
 Traffic Density is estimated based distance headway as following:
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= ℎ ∗ ̇
̇
=
Where
k = density (veh./km/lane) , 1000 = 1 km , ̅ = average distance headway in (m/veh.)

Car Following Theories:
Background
 Theories describing how one vehicle following another vehicle were
developed in the early 1950s & 1960s
 Pipes was one of the pioneers in developing car-following theories in
the early 1950
 In the 1960s, three parallel efforts:
a- Kometani & Sasaki in Japan
b- Forbs at Michigan State University
c- General Motors R&D team
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Notations
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n = lead vehicle
n+1 = following vehicle
= length of lead vehicle(m)
= length of following vehicle (m)
=position of lead vehicle (m)
=position of following vehicle (m)
̇ =speed of lead vehicle (m/sec)
̇ =speed of following vehicle (m/sec)
̈̇ =acceleration of lead vehicle (m/sec2)
̈ = acceleration of following vehicle (m/sec2)
=
+ ∆ = ∆

Pipes’ Theory (1)
 Pipes theory was based on the following concept:
A good rule for following another vehicle at safe distance is to allow
yourself at least the length of a car between your vehicle and the lead
vehicle for every ten miles per hour of speed at which you are
traveling.
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Pipes’ Theory(2)
 Based on Pipes theory, the minimum safe headway can be calculated as follows:
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Forbes’ Theory(1)
 Forbes approached car-following behavior by considering the reaction
time needed for the following vehicle to perceive the need to
decelerate and apply the brakes.
That is, the time gap between the rear of the lead vehicle and the front
of following vehicle should always be equal to or greater than the
reaction time.
 Minimum time gaps varied between 1-3 (based on field results), assuming
reaction time 1.5 sec and a vehicle length 20 ft:
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Forbes’ Theory(2)
 Forbes’ Minim Safe Distance headway and safe time headway
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General Motors’ Theory
 The Research team at GM developed five generations of car-following
models, all f which took the form:
 Response was always represented by the acceleration or deceleration
of the following vehicle
 Stimuli was always represented by the relative velocity of the lead
vehicle and the following vehicle.
 The difference in the different generations of the GM model was in the
representation of the sensitivity.
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Response = Function ( Sensitivity , Stimuli)

General Motors 1st Model
Where:
 ∆t = The reaction time
 a = Sensitivity factor
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General Motors 2nd Model
 The significant range for the sensitivity value (0.17-0.74) alerted the
investigator that spacing between vehicle should be introduced into the
sensitivity term.
Where:
 a1 & a 2= Sensitivity factor
 Problem:
It was difficult to implement this model and in selecting the appropriate
sensitivity value! So, they developed the 3rd model!
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General Motors 3rd Model
 The relationship between sensitivity and spacing between vehicles:
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General Motors 3rd Model
Where:
 ao = Sensitivity factor
 Later work bridged between this model and the
Greenberg macroscopic model.
 The values in the table were estimated at
different facilities
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General Motors 4th Model
Where:
 a‘ = Sensitivity factor
 Here, the speed was added to the sensitivity term
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General Motors 5th Model
Where:
 al,m = Sensitivity factor
 l,m = power parameters for speed and distance
headway
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Advanced Traffic Flow &
Shock Wave Analysis
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Macroscopic Traffic Flow Models (Review)
Speed & Density Relationship
Density
k
Speed
v
FlowRate
Q
Where
k
Q
v
s
s




,
/
Density
k (veh/lane/km)
Speed
v (km/hr)
0
0
vf Free Flow
Speed
Jam
Density
speed
flow
free
v
k
k
v
v
f
jam
f










 1
*
Greenshield’s Model(1934):
= ∗ ln
− Constant
Greenberg’s Model(1959):
eed
FreeFlowSp
v
e
v
v
f
k
k
f
jam










*
Underwood’s Model(1961):
Speed/Density Models:
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Flow & Density Relationship
Density
k
Speed
v
FlowRate
Q
Where
k
v
Q
s
S




,
*
Flow
Rate
Q
(veh/lane/hr)
Density
k (veh/lane/km)
Jam
Density (kjam)
Critical
Density
0
Capacity
Congested
Flow
Stable
Flow
0
Free Flow
Speed
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Speed & Flow Rate Relationship
Density
k
Speed
v
FlowRate
Q
Where
k
Q
v
s
S




,
Flow Rate
Q (veh/lane/hr)
Speed
v (km/hr)
0
Capacity
Congested
Flow
Stable
Flow
0
Free Flow
Speed
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Shock Wave Analysis
 Flow-speed-density states change over space and time. When these
changes of state occur a boundary is established that demarks the time-
space domain of one flow state from another. This boundary is referred to
as a shock wave.
 In some situations the shock wave can be very mild, like a platoon of high-
speed vehicles catching up to a slightly slower moving vehicle.
 In other situations the shock wave can be a very significant change in flow
states, as when high-speed vehicles approach a queue of stopped
vehicles.
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Shock Wave Analysis
Types of shock waves :
 Frontal stationary
 Backward forming (or moving)
 Forward recovery (or moving)
 Rear stationary
 Backward recovery (or moving)
 Forward forming (or moving)
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Shock Wave Analysis
Examples at Signalized Intersection:
Frontal stationary
Backward forming
Backward Recovery
Forward forming
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Shock Wave Analysis
Examples along a highway (behind a slow Truck):
Frontal Moving
Backward Recovery
Forward forming moving
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SHOCK WAVE EQUATIONS
 Consider an uninterrupted segment of
roadway for which a flow-density
relationship is known.
 For some period of time,. a steady-state
free-flow condition exists, as noted on the
flow-density diagram as state A. The flow,
density, and speed of state A are denoted
as qA, kA, and uA, respectively.
 Then, for the following period of time, the
input flow is less and a new steady state
free-flow condition exists, as noted on the
flow-density diagram as state B.
 The flow, density, and speed of state B are
denoted as qB, kB, and uB, respectively.
Note that in state B, the speed (uB) will be
higher, and these vehicles will catch up
with vehicles in state A over space and
time.
 At the shock wave boundary, the number
of vehicles leaving flow condition B (NB)
must be exactly equal to the number of
vehicles entering flow condition A (NA)
since no vehicles are destroyed nor
created.
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SHOCK WAVE EQUATIONS
 At the shock wave boundary, the number of vehicles
leaving flow condition B (NB) must be exactly equal to the
number of vehicles entering flow condition A (NA) since no
vehicles are destroyed nor created.
 The speed of vehicles in flow condition B just upstream of the
shock wave boundary relative to the shock wave speed is
(uB – wAB)
 The speed of vehicles in flow condition A, just downstream of
the shock wave boundary relative to the shock wave speed,
is (uA - wAB).
 Then:
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SHOCK WAVE Examples 1
 Use the flow-density diagram and combinations of the four flow states (A, B, C, D) in
the shown Figure to draw distance-time diagrams (showing shock wave and
vehicular trajectories) that result in the following types of shock waves: (a) frontal
stationary, (b) backward forming, (c) forward recovery, (d) rear stationary, (e)
backward recovery, and (f) forward forming., Then:
 Repeat the problem with numerical solutions. Assume that the flow-density diagram is
based on a linear Greenshields model, where:
 u =80 - .75 k, and
 the flows for states A, B, C, and D are 1440, 960, 960, and 600 vehicles per hour per
lane, respectively.
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SHOCK WAVE Examples 2
 The individual lanes on a long, tangent, two-lane directional
freeway have identical traffic behavior patterns and each follows
a linear speed-density relationship. It has been observed that the
capacity is 2000 vehicles per hour per lane and occurs at a speed
of 40 km/hr. On one particular day when the input flow rate was
1800 vehicles per hour per lane, an accident occurred on the
opposite side of the median which caused a gapers‘ block and
caused the lane density to increase to 75 vehicles per mile. After
15 minutes the accident was removed and traffic began to return
to normal operations. Draw the distance-time diagram showing
shock waves and selected vehicle trajectories.
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End of Module 5:
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Spring 2021

TTEng 422 s2021 module 5 Introduction to Traffic Flow Theory

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