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Introduction to Transportation Engineering

Traffic Flow Models

Dr. Antonio A. TraniAssociate Professor of Civil and Environmental Engineering

Virginia Polytechnic Institute and State University

Blacksburg, VirginiaFall, Spring 2005

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Topics for Discussion

• Why modeling traffic?

• Approaches to model traffic

• Parameters connected with traffic flow

• What role do vehicle dynamic/kinematic equations play?

• Examples

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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)

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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 sub-atomic 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

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Approaches to Modeling Traffic

Microscopic

Attempts to look into individual driving behaviors

Vehicle-following models

Macroscopic

Looks at the traffic as a fluid-flow or heat-transfer 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

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Parameters Connected with Traffic Models

• Speed

• Volume and Rate of Flow

• Density or Concentration

• Spacing and Headway

• Clearance and Gap

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How Do We Collect Traffic Data?

• External devices

Road traffic counters (loop detectors)

Traffic data collectors

Radar guns

Weight-in-Motion

• Internal devices (in-vehicle technology)

GPS data collection devices

Car chip collectors

Speed transducer collectors

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Loop Detectors

• Hardware/software application

• Measures traffic volume, time stamp, speed, gap

source: Sensource

source: Jamar Technologies

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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

source: Jamar TechnologiesTDC-12

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RADAR/LIDAR Technology

• Measures spot speed (instantaneous speed)

• Used in law-enforcement and also in traffic studies

source: Stalker Radar SystemsLIDAR System

RADAR System

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Video Traffic Monitoring

• Used in incident detection (hardware/software)

• Can measure real-time (or stored) traffic data; including volume, occupancy, speed and vehicle class over time

source: Autoscope Systems

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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/

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Weight-in-Motion (WIM) Devices

• Measure vehicle weight

• Good to measure road infrastructure use and deterioration

source: Virginia TechTransportation Institute

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GPS Data Collectors

• Measure vehicle performance parameters

• Good for traffic behavioral studies

• Used in vehicle tracking (fleet applications)

source: cybergraphy

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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) Speed (km/h)

Smoothed

Speed (km/h)

Acceleration

(m/s2) Fuel (l/s) HC (mg/s) CO (mg/s) NO

0 41.43 41.43 0.00 0.00000 0.00000 0.00000

2 41.43 41.43 0.00 0.00268 2.36032 37.23742

4 52.11 45.17 0.52 0.00344 3.12970 56.08933

6 55.60 48.82 0.51 0.00363 3.40516 63.20015

8 58.92 52.35 0.49 0.00381 3.69682 70.53934

10 60.76 55.30 0.41 0.00386 3.83389 73.50874

12 61.86 57.59 0.32 0.00385 3.89856 74.39976

14 66.28 60.63 0.42 0.00421 4.46872 88.83619

16 65.73 62.42 0.25 0.00401 4.29287 82.74030

18 65.91 63.64 0.17 0.00395 4.27326 81.33448

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Sample GPS Car Data

Data collected in Phoenix driving on arterial roads

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800

Time (s)

Sp

ee

d (

km

/h)

Raw Speed

Smoothed Speed

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Phoenix Car Data (Detail)

Data collected in Phoenix driving on arterial roads

0

10

20

30

40

50

60

70

80

90

100

100 150 200 250

Time (s)

Sp

ee

d (

km

/h)

Raw Speed

Smoothed Speed

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Sample GPS Car Data

Data collected in Phoenix driving on arterial roads

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800

Time (s)

Sp

ee

d (

km

/h)

Raw Speed

Smoothed Speed

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Sample Traffic Data

Data collected at various locations

Holland (Beltway)

Germany (Autobahn)

U.S. (I-4)

These plots demonstrate how speed (u), density (k) and flow (q) are related

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Highway 401 Data (U.S.)

The data shows the basic relationship between speed (u) and flow (q)

Data courtesy of Dr. H. Data courtesy of Dr. H. Rakha Rakha (VTTI)(VTTI)

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Highway 401 Data (U.S.)

The plot shows the basic form of the density-speed relationship

Data courtesy of Dr. H. Data courtesy of Dr. H. Rakha Rakha (VTTI)(VTTI)

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Highway 401 Data (U.S.)

• Basic Density vs. Flow Relationship

Data courtesy of Dr. H. Data courtesy of Dr. H. Rakha Rakha (VTTI)(VTTI)

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Autobahn Data (Germany)

The data shows the basic relationship between speed (u) and flow (q)

Data courtesy of Dr. H. Data courtesy of Dr. H. Rakha Rakha (VTTI)(VTTI)

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Autobahn Data (Germany)

The plot shows the basic form of the density-speed relationship

Data courtesy of Dr. H. Data courtesy of Dr. H. Rakha Rakha (VTTI)(VTTI)

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Autobahn Data (Germany)

• Basic density (k) vs. flow (q) relationship

Data courtesy of Dr. H. Data courtesy of Dr. H. Rakha Rakha (VTTI)(VTTI)

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Macroscopic Traffic Flow Models

We cover two basic models:

• Greenshield

• Greenberg

In all traffic flow models, the following fundamental traffic flow equation applies,

(1)

where: is the traffic flow (vehicles/hr per lane), is the

flow speed (km/hr) and is the flow density (vehicles per lane-km)

q u k⋅=

q uk

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Greenshield’s Model (circa 1936)

Assumes a linear relationship between flow speed (u) and flow density (k)

(2)

Density [veh/km-lane]

uf

kj

Speed[km/h]

Jam density

Free-flow speed

k

u

u uf 1 kkj

---– ⋅=

q u k⋅ uf 1 kkj----–

k⋅ ⋅= =

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Greenshield’s Model

(3)

where: is the density for maximum flow and is the maximum flow.

q uf k k2

kj-----–

⋅=

Density [veh/km]

Flow[veh/h]

qm

km kj0

Maximum flow

Jam density

q uf k k2

kj

----– ⋅=

k

km qm

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Greenshield’s Model

The condition for maximum flow (qm) is achieved when,

(4)

Then

(5)

Prove this relationship using calculus.

kmkj

2---=

um

uf2----=

qm

uf kj4

---------=

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Greenberg’s Model (circa 1959)

Assumes a nonlinear relationship between flow speed (u) and flow density (k)

(6)

(7)

where: is the jam density, is a model constant (later

to be proven the speed for maximum flow), is the

space mean speed (just like in other models) and is the flow density.

(8)

u ckjk---- ln⋅=

q u k⋅ ckkjk---- ln⋅= =

kj cu

k

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Greenberg’s Model

Use calculus to prove that,

(9)

is the density for maximum flow.

You can also prove that the speed for maximum flow occurs at,

(10)

km

kje----=

um c=

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Greenberg’s Model

Since the relationship for all traffic flow conditions, the condition for maximum flow (qm) is,

(11)

Prove this relationship using calculus.

q u k⋅=

qm

c kje

------=

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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/km-lane and the free flow speed of 50 km/hr (restricted by the speed limit). Assume Greenshield’s model traffic flow conditions hold true.

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FIGURE 1. Blacksburg Middle School Traffic Situation.

The typical acceleration model for a car is known to be:

where: is the acceleration of the car (in m/s2) and is the vehicle speed in m/s. During the morning peak period, traffic counters at the site measure an average of 20

R = 12 m.

SchoolEntrance Spacing = S

Car

Car

To Radford

To BlacksburgTrafficCounters

a 4.0 0.1V–=

a V

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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.

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Solution to Part (a)

Find the Spacing (Sp) between vehicles. Since the density of the traffic flow is known to be 20 veh/km-la we compute the spacing as the reciprocal of the density

kilometers

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/km-la.

Sp

1k--- 1

20------ 0.05= = =

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Space Mean Speed vs. Density Diagram

km/hr

uf = 50 km/hr (13.89 m/s)u (km/hr)

k (veh/km-la)70

kj = 70 veh/km-la50

020

37.5

u uf

uf

kj

---k– 50 5070------ 20( )– 35.71= = =

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Solution to Part (a)

Traveling at 35.71 km/hr (9.92 m/s) the headway (h) between successive cars is,

secondsh 509.92---------- 5.04= =

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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,

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

tavailable 5.04 0.5– 4.54= =

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seconds

The distance traveled by a vehicle with a linearly-varying acceleration model is,

Note that is either 3.96 or 4.54 seconds (depending on your assumption on when the stopped vehicle starts the left turn).

Using values of , 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.

tgap 5.04 0.5– 5.89.92----------– 3.96= =

S k1tk2

------ k1

k2

2----- 1 e k2t––( )– v0

k2

---- 1 e k2t––( )+=

t

k1 k2

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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,

meters

The vehicle can execute the turn safely.

d 2πR4

---------- L+ 2π 12( )4

----------------- 5.8+ 24.65= = =

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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

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Differentiation of Speeds Used in Traffic Analysis

Two types of speed sused in traffic analyses:

• time mean speed

• space mean speed

The time-mean speed is defined in the following way:

(12)

where represents recorded speed of the i-th vehicle.

We see that the time-mean speed can be calculated by calculating the arithmetic time mean speed.

ut

ut1N---- ui

i 1=

N

∑⋅=

ui

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Space Mean Speed

The space-mean speed is the average speed that has been used in the majority of traffic models. Let us note section of the highway whose length equals D. We denote by the time needed by the i-th vehicle to travel along this highway section. The space-mean speed is defined in the following way:

(13)

ti

us

usD

1N---- ti

i 1=

N

∑⋅-------------------- D

t----= =

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The expression represents the average travel time

of the vehicles traveling along the observed highway section.

1N---- ti

i 1=

N

∑⋅ t

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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.

Measurement points

1 km

A B

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Example 2

Table 1. Recorded speeds and travel times.

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 time-mean speed and the space- mean speed.

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

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Solution to Example 2

The time-mean speed at point A is:ut

ut1N---- ui

i 1=

N

∑⋅=

ut15--- 80 75 62 90 70+ + + +( )⋅ 1

5--- 377⋅ 75.4 km

h-------= = =

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Solution to Example 2

The mean-space speed represents measure of the average traffic speed along the observed highway section. The mean-space speed is:

The total travel time of all five vehicles equals:

.

usD

1N---- ti

i 1=

N

∑⋅--------------------=

45 50 56 39 53+ + + + 243 ondssec[ ] 2433600------------ h[ ] 0.0675 h[ ]= = =

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The mean-space speed equals:

us1

15--- 0.0675( )⋅----------------------------- km

h------- 74.07 km

h-------= =