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Unformatted text preview: Introduction to Transportation Engineering
Traffic Flow Models
Dr. Antonio A. Trani
Professor of Civil and Environmental Engineering
Virginia Polytechnic Institute and State University Blacksburg, Virginia
Fall 2009 Virginia Tech 1 Topics for Discussion
• Why modeling trafﬁc?
• Approaches to model trafﬁc
• Parameters connected with trafﬁc ﬂow
• What role do vehicle dynamic/kinematic equations play?
• Examples Virginia Tech 2 Why Trafﬁc Modeling?
Required to estimate capacity of any transportation
facility
Highway capacity  how many cars per hour?
Railway capacity  how many rail cars per hour
Airport capacity  how many aircraft can land per hour? Required to estimate level of service of transportation
facilities
a) Level of service is connected with delays imposed by the
system on vehicles and people To study impacts of our own actions (building
infrastructure) Virginia Tech 3 A Difﬁcult Problem to Understand
• Trafﬁc phenomena is complex
• Trafﬁc phenomena is usually a stochastic process
(described by random variables)
• Los Alamos (New Mexico) statement:
“Modeling trafﬁc phenomena has proven to be more difﬁcult
than predicting and modeling subatomic level reactions
inside the atom  for nuclear warhead simulations” • Statement after four years of work developing the newest
trafﬁc and transportation planning software package
called TRANSIMS Virginia Tech 4 Approaches to Modeling Trafﬁc
Microscopic
Attempts to look into individual driving behaviors
Vehiclefollowing models Macroscopic
Looks at the trafﬁc as a ﬂuidﬂow or heattransfer
phenomena
Vehicles are not identiﬁed individually but as a group of
entities moving on the system Technically, both microscopic and macroscopic models
consider the human in their solutions Virginia Tech 5 Parameters Connected with Trafﬁc Models
• Speed
• Volume and Rate of Flow
• Density or Concentration
• Spacing and Headway
• Clearance and Gap Virginia Tech 6 How Do We Collect Trafﬁc Data?
• External devices
Road trafﬁc counters (loop detectors)
Trafﬁc data collectors
Radar guns
WeightinMotion • Internal devices (invehicle technology)
GPS data collection devices
Car chip collectors
Speed transducer collectors Virginia Tech 7 Loop Detectors
• Hardware/software application
• Measures trafﬁc volume, time stamp, speed, gap source: Sensource
source: Jamar Technologies
Virginia Tech 8 Trafﬁc Data Collectors
• Measure trafﬁc data at intersections (turning movements),
vehicle delays, queue lengths, saturation ﬂows
• Can be connected to software to expedite the analysis TDC12
source: Jamar Technologies
Virginia Tech 9 RADAR/LIDAR Technology
• Measures spot speed (instantaneous speed)
• Used in lawenforcement and also in trafﬁc studies RADAR System
LIDAR System source: Stalker Radar Systems
Virginia Tech 10 Video Trafﬁc Monitoring
• Used in incident detection (hardware/software)
• Can measure realtime (or stored) trafﬁc data; including
volume, occupancy, speed and vehicle class over time source: Autoscope Systems
Virginia Tech 11 Vehicle Collection (Car Chip)
• Measures up to ﬁve vehicle parameters
• Good to monitor driver behavior (or trafﬁc analysis)
• Downloads data to a PC source: http://www.thecarchip.net/ Virginia Tech 12 WeightinMotion (WIM) Devices
• Measure vehicle weight
• Good to measure road infrastructure use and deterioration source: Virginia Tech
Transportation Institute Virginia Tech 13 GPS Data Collectors
• Measure vehicle performance parameters
• Good for trafﬁc behavioral studies
• Used in vehicle tracking (ﬂeet applications) source: cybergraphy Virginia Tech 14 Sample GPS Data
• Sample data collected in Phoenix, AZ
• Using Global Positioning System technology
• Smoothed speed is sometimes necessary to remove data
outliers Time (s)
0
2
4
6
8
10
12
14
16
18 Smoothed
Speed (km/h) Speed (km/h)
41.43
41.43
41.43
41.43
52.11
45.17
55.60
48.82
58.92
52.35
60.76
55.30
61.86
57.59
66.28
60.63
65.73
62.42
65.91
63.64 Acceleration
(m/s2)
0.00
0.00
0.52
0.51
0.49
0.41
0.32
0.42
0.25
0.17 Virginia Tech Fuel (l/s)
HC (mg/s)
CO (mg/s) NO
0.00000
0.00000
0.00000
0.00268
2.36032
37.23742
0.00344
3.12970
56.08933
0.00363
3.40516
63.20015
0.00381
3.69682
70.53934
0.00386
3.83389
73.50874
0.00385
3.89856
74.39976
0.00421
4.46872
88.83619
0.00401
4.29287
82.74030
0.00395
4.27326
81.33448 15 Sample GPS Car Data
Data collected in Phoenix driving on arterial roads
100
Raw Speed 90 Smoothed Speed 80 Speed (km/h) 70
60
50
40
30
20
10
0
0 100 200 300 400 500 600 700 800 Time (s) Virginia Tech 16 Phoenix Car Data (Detail)
Data collected in Phoenix driving on arterial roads
100
Raw Speed
Smoothed Speed 90
80 Speed (km/h) 70
60
50
40
30
20
10
0
100 150 200 250 Time (s) Virginia Tech 17 Sample GPS Car Data
Data collected in Phoenix driving on arterial roads
100
Raw Speed 90 Smoothed Speed 80 Speed (km/h) 70
60
50
40
30
20
10
0
0 100 200 300 400 500 600 700 800 Time (s) Virginia Tech 18 Sample Trafﬁc Data
Data collected at various locations
Holland (Beltway)
Germany (Autobahn)
U.S. (I4) These plots demonstrate how speed (u), density (k) and
ﬂow (q) are related Virginia Tech 19 Highway 401 Data (U.S.)
The data shows the basic relationship between speed (u)
and ﬂow (q) Data courtesy of Dr. H. Rakha (VTTI)
Virginia Tech 20 Highway 401 Data (U.S.)
The plot shows the basic form of the densityspeed
relationship Data courtesy of Dr. H. Rakha (VTTI)
Virginia Tech 21 Highway 401 Data (U.S.)
• Basic Density vs. Flow Relationship Data
Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 22 Autobahn Data (Germany)
The data shows the basic relationship between speed (u)
and ﬂow (q) Data
Data courtesy of Dr. H. Rakha (VTTI)
Virginia Tech 23 Autobahn Data (Germany)
The plot shows the basic form of the densityspeed
relationship Data courtesy of Dr. H. Rakha (VTTI)
Virginia Tech 24 Autobahn Data (Germany)
• Basic density (k) vs. ﬂow (q) relationship Data
Data courtesy of Dr. H. Rakha (VTTI) Virginia Tech 25 Macroscopic Trafﬁc Flow Models
We cover two basic models:
• Greenshield
• Greenberg
In all trafﬁc ﬂow models, the following fundamental
trafﬁc ﬂow equation applies, q = u⋅k (1) where: q is the trafﬁc ﬂow (vehicles/hr per lane), u is the
ﬂow speed (km/hr) and k is the ﬂow density (vehicles per
lanekm) Virginia Tech 26 Greenshield’s Model (circa 1936)
Assumes a linear relationship between ﬂow speed (u) and
ﬂow density (k)
u
Speed[km/h] Freeﬂow speed uf k
u = u f ⋅ 1 –  k j
Jam density kj k
Density [veh/kmlane] k
q = u ⋅ k = u f ⋅ 1 –  ⋅ k k j
Virginia Tech (2) 27 Greenshield’s Model
2 k
q = u f ⋅ k – 
kj (3) Maximum ﬂow Flow[veh/h] k2 k –  q = uf ⋅ kj qm Jam density
Density [veh/km]
0 km kj k where: k m is the density for maximum ﬂow and q m is the
maximum ﬂow.
Virginia Tech 28 Greenshield’s Model
The condition for maximum ﬂow (qm) is achieved when,
km um = kj
2 = uf
 (4) 2 Then qm = uf kj
4 (5) Prove this relationship using calculus. Virginia Tech 29 Greenberg’s Model (circa 1959)
Assumes a nonlinear relationship between ﬂow speed (u)
and ﬂow density (k)
k j
u = c ⋅ ln k (6) kj
q = u ⋅ k = c k ⋅ ln  k (7) where: k j is the jam density, c is a model constant (later
to be proven the speed for maximum ﬂow), u is the
space mean speed (just like in other models) and k is the
ﬂow density.
(8) Virginia Tech 30 Greenberg’s Model
Use calculus to prove that, km = kj
e (9) is the density for maximum ﬂow.
You can also prove that the speed for maximum ﬂow
occurs at, um = c (10) Virginia Tech 31 Greenberg’s Model
Since the relationship q = u ⋅ k for all trafﬁc ﬂow
conditions, the condition for maximum ﬂow (qm) is,
ck j
qm =  (11) e Prove this relationship using calculus. Virginia Tech 32 Example 1
The Blacksburg Middle school board hires you as a
transportation engineer to ease complaints from parents
driving vehicles and making a left turn to the school
entrance during the peak hour in the morning (see Figure
1).
The road is divided and has a left turn queueing island
allowing cars to stop before making the turn.
Measurements at the road by the town engineer indicate
that trafﬁc ﬂow in this section has a jam density of 70
veh/kmlane and the free ﬂow speed of 50 km/hr
(restricted by the speed limit). Assume Greenshield’s
model trafﬁc ﬂow conditions hold true. Virginia Tech 33 School
Entrance To Blacksburg R = 12 m. Spacing = S Car Traffic
Counters Car
To Radford FIGURE 1. Blacksburg Middle School Trafﬁc Situation. The typical acceleration model for a car is known to be:
a = 4.0 – 0.1 V where: a is the acceleration of the car (in m/s2) and V is
the vehicle speed in m/s. During the morning peak period,
trafﬁc counters at the site measure an average of 20
Virginia Tech 34 vehicles per kilometer per lane traveling from Radford to
Blacksburg (see Figure 1).
a) Find the typical spacing (S) and the average headway
(h) between vehicles traveling from Radford to
Blacksburg during the peak morning period.
b) Find if the average headway (h) allows a typical driver
to make a left turn if the driver has a perception/reaction
time of 0.5 seconds. The radius of the curve to make a left
turn is 12 meters. According to AASHTO standards, the
critical vehicle length is 5.8 meters. Virginia Tech 35 Solution to Part (a)
Find the Spacing (Sp) between vehicles. Since the density
of the trafﬁc ﬂow is known to be 20 veh/kmla we
compute the spacing as the reciprocal of the density 1
1
S p =  =  = 0.05 kilometers
k
20
Sp = 50 meters
To ﬁnd the headway we need to ﬁgure out how fast the
cars are traveling on the road. We use Greenshield’s
model to estimate the speed when k = 20 veh/kmla. Virginia Tech 36 Space Mean Speed vs. Density Diagram
u (km/hr) uf = 50 km/hr (13.89 m/s)
kj = 70 veh/kmla 50 37.5 0
70 20 k (veh/kmla) u
50
u = u f – f k = 50 –  ( 20 ) = 35.71 km/hr
kj
70
Virginia Tech 37 Solution to Part (a)
Traveling at 35.71 km/hr (9.92 m/s) the headway (h)
between successive cars is, 50
h =  = 5.04 seconds
9.92 Virginia Tech 38 Solution to Part (b)
To check if the turning vehicle can make a safe maneuver,
check the time to turn against the headway (h) calculated
in part (a). Account for the reaction time of the turning
vehicle.
The time available to execute a safe turn is (h)  0.5
seconds to account for reaction time, t available = 5.04 – 0.5 = 4.54 seconds
Technically we should use the gap between two
successive vehicles to estimate the time to turn left. In this
case we have to subtract the time traveled by the
oncoming vehicle to cover its car length at 9.92 m/s
Virginia Tech 39 t gap 5.8
= 5.04 – 0.5 –  = 3.96 seconds
9.92 The distance traveled by a vehicle with a linearlyvarying
acceleration model is,
–k t
–k t
k1
v0
k 1 t S =  –  ( 1 – e ) +  ( 1 – e )
2
k2
k2 k2
2 2 Note that t is either 3.96 or 4.54 seconds (depending on
your assumption on when the stopped vehicle starts the
left turn).
Using values of k 1 , k 2 of 4 and 0.1, respectively, the left
turning vehicle travels 35.6 meters in 4.54 seconds and 28
meters in 3.96 seconds.
Virginia Tech 40 A plot of distance traveled vs. time is shown in the
following diagram. The total distance to be traveled in the
left turn maneuver to reach a safe point is, 2πR
2 π ( 12 )
d =  + L =  + 5.8 = 24.65 meters
4
4
The vehicle can execute the turn safely. Virginia Tech 41 Distance vs. Time Proﬁle (Turning Car)
The car reaches 24.65 m in 3.73 seconds The car travels 35.65 m in 4.54 seconds
Virginia Tech 42 Differentiation of Speeds Used in Trafﬁc Analysis
Two types of speed sused in trafﬁc analyses:
• timemean speed
• spacemean speed
The timemean speed u is deﬁned in the following way:
t N 1
u t =  ⋅
N ∑u (12) i i=1 where u represents recorded speed of the ith vehicle.
i We see that the timemean speed can be calculated by
calculating the arithmetic time mean speed.
Virginia Tech 43 Space Mean Speed
The spacemean speed is the average speed that has been
used in the majority of trafﬁc models. Let us note a section
of the highway whose length equals D. We denote by t
the time needed by the ith vehicle to travel along this
highway section. The spacemean speed u is deﬁned in
the following way:
i s D
u s =  = D
N
t
1
 ⋅
ti
N (13) ∑
i=1 Virginia Tech 44 N 1
The expression  ⋅ ∑ t represents the average travel time t
i N i=1 of the vehicles traveling along the observed highway
section. Virginia Tech 45 Example 2
Measurement points are located at the beginning and at
the end of the highway section whose length equals 1 km
(see ﬁgure below). The recorded speeds and travel times
are shown in the Table.
1 km A Measurement points Virginia Tech B 46 Example 2
Table 1. Recorded speeds and travel times.
Vehicle number Speed at point A [km/h] Travel time between point A
and point B [sec] 1 80 45 2 75 50 3 62 56 4 90 39 5 70 53 Speeds of the ﬁve vehicles are recorded at the beginning
of the section (point A). The vehicle appearance at point A
and point B were also recorded.
a) Calculate the timemean speed and the spacemean
speed.
Virginia Tech 47 Solution to Example 2
The timemean speed u at point A is:
t N 1
u t =  ⋅
N ∑u i i=1 1
1
km
u t =  ⋅ ( 80 + 75 + 62 + 90 + 70 ) =  ⋅ 377 = 75.4 5
5
h Virginia Tech 48 Solution to Example 2
The spacemean speed represents measure of the average
trafﬁc speed along the observed highway section. The
spacemean speed is:
D
u s = N
1
 ⋅
ti
N ∑
i=1 The total travel time for all ﬁve vehicles is:
243
45 + 50 + 56 + 39 + 53 = 243 [ sec onds ] =  [ h ] = 0.0675 [ h ] .
3600 Virginia Tech 49 The spacemean speed is calculated as:
1
km
 km
u s =   = 74.07 1
h
h
 ⋅ ( 0.0675 )
5 Virginia Tech 50 ...
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This note was uploaded on 12/31/2011 for the course CEE 3604 taught by Professor Katz during the Fall '08 term at Virginia Tech.
 Fall '08
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