AbstractTravel time estimation is a critical ingredient for

transportation management and traveler information-both infrastructure-based and in-vehicle. Focusing on freeway travel time estimation for display on roadside variable message signs, this paper describes a concept developed from principles of traffic flow for establishing optimal sensor density.

The methods are based on computing the magnitude of under- and over-prediction of travel time during shock passages. The midpoint method and Coifman methods in four situations are calculated during three types of shock waves considering representative traffic dynamics situations. Vehicle hours traveled (VHT) is used to evaluate travel time estimation errors. Relationships between travel time estimation errors and sensor spacing are established.

Optimal sensor spacing expressions are calculated considering the trade-off between cost of VHT estimation error and the sensor construction cost.

Comparison of optimal sensor spacing is performed among different travel time estimation methods in each type of shock waves. Sensitivity analysis is also performed, and a summary provided about the relationships between actual VHT, predicted VHT, VHT errors, total cost, optimal sensor spacing and variables speed and flow in different traffic states, segment length and sensor spacing.

ResultsOptimal sensor spacing depends only on speed

and flow values and cost coefficients.In Shock wave AC+CD/CE, when the ratio of

Cu/Cd is less than 0.4/0.25, the optimal sensor spacing increases sharply with the ratio decreases; when larger than 0.4/0.25, slowly.

Comparisons of optimal sensor spacing indicate the longest optimal sensor spacing in AC+CD is Coifman method 3, in AC+CE is Coifman method 2.

Add Error, Abs Error and Penalty Error are found to be inversely proportional to the sensor density; Under and Over Error are also found to be inversely proportional but combined with constants.

AcknowledgmentsGalen McGill of ODOT posed the sensor density question

and supported this research. Dr. Robert Bertini and David Lovell established fundamentals of this research.

www.its.pdx.edu

Impact of Sensor Spacing on Freeway Travel Time Estimation for Traveler Information

Wei Feng and Dr. Robert Bertini, Portland State University

Sensor Spacing Optimization

Midpoint AC

vAC

s=l/4

n

n(n-1)/2

n(n-1)/2

vms

0 0.2 0.4 0.6 0.8 1-5%

0%

5%

10%

15%

20%

11.50

11.55

11.60

11.65

11.70

11.75

11.80

11.85

11.90AC+CD

Error (Additive)Error (Absolute Value)Error (Penalty)Predicted VHTActual VHT

Sensor Spacing (miles)

VHT

Erro

r

VHT

/ m

ile

T2T10 t, dt T2

l=s

vf

vc

vAC

vms

Coifman 1 AC

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5AC+CD

Coifman4=Coifman3

Coifman2

Coifman1

Midpoint

Cu/Cd

S*

(Mil

es

)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

2

4

6 AC+CECoifman4=Coifman3

Coifman2

Coifman1

Midpoint

Cu/Cd

S*

(Mil

es

)

Frontal stationary

Backwardrecovery

Forwardrecovery

Forwardforming

Backwardforming

Rear stationary

x

tA. Types of Transitions

q

k

qC

vf vc

vAC

qAA

B C

D

vCD

EqE vCE

B. Assumed Traffic Flow Relation

A

B

C

D

A

x

t

bn

tdeact

vCDvAC

vfDA

AC

CD

vmsA

B

C

E

x

t

bn

tdeact

vCEvAC

vf

AC CEvms

Optimal Sensor spacing S*

Travel Time Estimations

0 0.2 0.4 0.6 0.8 10%

5%

10%

15%

20%

13.4113.4213.4313.4413.4513.4613.4713.4813.4913.5013.51

AC+CE

Error (Additive)Error (Absolute Value)Error (Penalty)Predicted VHTActual VHT

Sensor Spacing (miles)

VHT

Erro

r

VHT

/ m

ile

upred

uactu VHTVHTCCMin

s

lCVHTVHTC d

oact

opredo

ou CC , : cost coefficients to convert under or over estimated VHT error into $, [ $ / hr ]

l s: segment length, (mile) : sensor spacing, (mile)

C: total cost, ($)

Fundamentals and Assumptions

Additive error, Absolute error and Penalty error are linearly related to sensor spacing;

Predicted VHT linearly approaches actual VHT as sensor spacing decreases;

dC : sensor cost, ($ per sensor)

Midpoint AC with n sensors

VHT Errors vs. Sensor Spacing

ttftc

vf

vc

vAC

vms

sl

under

Congestion signal

over

t

x j1 j2j3

vAC

vf

l=s vc

vms

Coifman 2 AC

l=s

vf

vc

vAC

vms

s/2Coifman 1

Coifman 2

Coifman 3 AC

In backward forming shock wave AC and backward recovery shock wave CD, midpoint method both under and over predict VHT, while Coifman 1 and 2 methods only under predict VHT;

In forward recovery shock wave CE, midpoint both under and over predict VHT, while Coifman 1 and 2 methods only over predict VHT

CECDACcfCBAopred

upred

oact

uact vvvvvqqqslfVHTVHTVHTVHT ,,,,,,,,,,,,