TTENG 422 Traffic Operations Module 5 S2024.pdf

Traffic Operations and Management: Traffic flow theory

2. Car Following Models

\ £ Car following theories and models dictates the distance
between vehicles and estimation of traffic density in the
traffic stream.

3. | Distance Headway Charac

Distance headway is defined as the distance

from a selected point on the lead vehicle to the

same point on the following vehicle.

| Point
Ans) = Li + Ent)

dust) = distance headway of vehicle n + 1 at time + (feet)
physical length of vehicle n (feet)
gap length between vehicle n and n + 1 at time z (feet),

Distance

Distance gap is defined as the gap length
between the rear edge of the lead vehicle and
the front edge of the following vehicle.

teristics

4

4. 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:

Anzı = Inga na
where,
dur Alístance headway of vehicle (n+1) (m)
Inf time headway of vehicle (n+1) (sec.)

X= speed of vehicle (n+1) during time period h,. , (m/sec)

Traffic Density is estimated based distance headway as following:

1000
a

Where
k= density (veh./km/lane) , 1000 = 1 km , d = average distance headway in (m/veh.)

5. 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

An the 1960s, three parallel efforts:

/ a- Kometani & Sasaki in Japan

b- Forbs at Michigan State University
c- General Motors R&D team

6. | Car Following Theories:
Notations

o +1 (1+ At)
n= lead vehicle

n+l = following vehicle A] EAN

La = length of lead vehicle(m)
ind Veh 7
Ln+1 = length of following vehicle (m) TED YY
“position of lead vehicle m) 7 DR

position of following vehicle (m) = | |

peed of lead vehicle (m/sec) [hn Fr]
speed of following vehicle (m/sec) 1
cceleration of lead vehicle (m/sec?) Xp 1 1) —————__»

Xn41 = acceleration of following vehicle (m/sec?)

t= attimet x(t) sl

t+At = Atafter timet

|
alt) 10 $a

7. Car Following Theories:
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.
VA e
A ns)
du = Fo an 1a Bh li | the

Assuming a vehicle length of 20 feet, equation (6.5) can be expressed as follows:
4
dun = 1.36 lin 1@)| + 20

8. Car Following Theories:
Pipes’ Theory(2)
Based on Pipes theory, the minimum safe headway can be calculated as follows:
han = 1.36 + —<

Xn +10)

Field results

Minimum safe time headway

f L N
0 10 20 20 40 50 60 0 147 23 #4 587 733 88
Speed (miles) Speed, x, (0) Iisec)

9. | Car Following Theories:
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.
Ya

Y hu = At +

in

©)
Minimum time gaps varied between 1-3 (based on field results), assuming
reaction time 1.5 sec and a vehicle length 20 ft:

hymn = 1.50 + dns = 1.50 fég()] + 20

5,(t)

10. Car Following Theories:
Forbes’ Theory(2)

Forbes' Minim Safe Distance headway and safe time headway

T T T

Field
results

|
T

A 3
+ =
5 = 20
E =
3 &
E E 10
& =

0 10 20 30 40 50 60 0 147 283 m 587 733 88
Speed (miles/he} ‘Speed, x, (1) (tue)

11. | Car Following Theories:
General Motors’ Theory

The Research team at GM developed five generations of car-following
models, all f which took the form:

Response = Function ( Sensitivity , Stimuli)
Response was always represented by the acceleration or deceleration
of thefollowing vehicle

Stimuli was always represented by the relative velocity of the lead
icle and the following vehicle.

e difference in the different generations of the GM model was in the
representation of the sensitivity.

12. | Car Following Theories:
General Motors 15 Model

| Hault +49 = [i ~ in +10)

| Where:
y A TABLE 6.1 Parameter Values for First
At= The reaction time GM Model
a = Señsitivity factor Reaction
4 Time, | Sensitivity,
Measured At a
Value || (sec) (sec)
Minimum 1.0 0.17
Average | 155 0,37
Maximum || 2.2 0.74

13. | Car Following Theories:
General Motors 274 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.

= ar
Where Xnai(t + At) = or re) — Xn AC]
E % ‘|
At= Thé reaction time

al &2= Sensitivity factor

roblem:
It was difficult to implement this model and in selecting the appropriate
sensitivity value! So, they developed the 3'4 model!

14. | Car Following Theories:
General Motors 34 Model

The relationship between sensitivity and spacing between vehicles:

Vehicles
far
| apart

Vehicles
close
together

15. | Car Following Theories:
General Motors 34 Model

| .. _ O% (mi
| X, + At) ZOO Exo) ER)

Where: .
TABLE 6.2 Parameter Values for Third GM
At= The reaction time Model

Zi
ao = Sensitivity factor Reaction | Sensitivity
Time, | Parameter,
At %
Later work bridged between this model and the Location (sec) fusec)
reenberg macroscopic model.
General Motors 15 403
The values in the table were estimated at test track
different facilities Holland Tunnel | 1.4 268
Lincoln Tunnel 12 29.8

16. Car Following Theories:
General Motors 4" Model

.. «line + Ar)]
O OF
| Where:

At= The reaction time

EXO) Kur 0)

7
as = Sensitivity factor

if

Hére, the speed was added to the sensitivity term

17. Car Following Theories:
General Motors 5 Model

Cr Len 1 CE + At)]”
Din) — ay (OV

At = The reaction time
on = Sensitivity factor

/
Lm = power parameters for speed and distance

Ice headway exponent, £

Fifth mode

illum

~ First and second

SS

Deal

o Fourth model
2

ZY

_

|
A 1 |
o +1 72

‘Speed exponent, m

18. | | Advanced Traffic Flow &
Shock Wave Analysis

N

19. Macroscopic Traffic Flow Models (Review)
Speed & Density Relationship

vf Free Flow

Speed/Density Models: Speed 73 5

v (km/hr]
| Greenshield's Model(1934):

v, - free flow speed

y

/ Greenberg's Model(1959):

/ v= evn(#2) A en

À Be). 0 ta k (veh/lane/km)
v,=Q/k Density
Where ,
Underwood's Model(1961)/ © = FlowRate
ou y, - Speed

v, - FreeFlowSpeed k — Density

20. Se

Macroscopic Traffic Flow Models (Review)
Flow 8. Density Relationship

\
Flow Rate\Q (veh/lane/hr)

Capacity

Free Flow

Speed

7

Stable
Flow

O=v,*k
Where

O - FlowRate
v, — Speed

Congested
Flow

k — Density
Density
K(vel/lane/km)

Jan!

Critical |
Density Density (kun)

21. Macroscopic Traffic Flow Models (Review)
Speed & Flow Rate Relationship

Speed Free Flow
| v (km/hr) 4 Speed 0
vy==
s
k
Stable nn
% Flow Where
/ O - FlowRate
/ >
v, — Speed
Congested E

Flow

k — Density

Flow Rate
Q(veh/lane/hr)

Capacity

22. 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.

23. Shockwave Analysis

| Types of shock waves :
Frontal stationery
Backward forming (or moving)

Forward recovery (or moving)

Rear stationery
Backward recovery (or moving}

Forward forming (or moving)

24. Shock Wave Analysis

Examples at Signalized Intersection:

| a

/

| (be Af
IT A à
LA
LI
vr
vr

Distance ——=—

2 y
/ X Backward forming 7 Lu.

Forward forming

25. Shock Wave Analysis

Examples along a highway (behind a slow Truck):

| 1 ff 414 Frontal Moving LD
{ 1 oA 52722 Ela dav
23 sei N

=>
8
8
e
3
a
= / LIN
= IIA E LIS N ,

fg LA yg / 1 / Forward forming moving

26. 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
put flow is less and a new steady state

free-flow condition exists, as noted on the

flow-density diagram as state 8

The flow, density, and speed of state B are
denoted as 98, kB, and vB, 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 nowehicles are destroyednor
created.

Density, k
[6

WU

Y Yj, Y)

77 /

27. 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
Nehicles 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 Nz = gt = (ug — 048) kgt NA, just downstream of

the shock w the shock wave speed,
is (UA - oABNA = Gal = Us — Op) kat 5

Then: a da = Ge _ Ag
AB ka — ke Ak

28. | 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:

4 =80 - .75k, and

2 the flows for states A, B, C, and D are 1440. 960, 960, and 600 vehicles per hour
per lane, respectively. 4

Dersity, k

29. Density, k

u =80 - .75k, and
THE TOWS Tor states A, B, C, and D are 1440, 960, 960,
and 600 vehidles per hour per lane, respectively.

30. 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.

31. £
8
2

32. ~<— sourisiq

Lo

33. Hue

34. | | End of Module 5:
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