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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 ,,,,,,,,,,,,