TRB Planning Applications Conference - May 19, 2009

Investigation of Speed-Flow Relations and Estimation of

Volume Delay Functions for Travel Demand Models

in Virginia

TRB Planning Applications Conference - May 19, 2009

Presented by:Jaesup Lee, Virginia Department of TransportationDean Munn, The Corradino Group

Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions

for Travel Demand Models in Virginiafor Travel Demand Models in Virginia TRB Planning Applications Conference, May 19, 2009TRB Planning Applications Conference, May 19, 2009

• Introduction

• Traffic Data used in basic analysis and model estimation

• Data Development and Definition

– Free Flow Speed

– Traffic Flow (Uninterrupted vs. Interrupted)

– Link Capacity

• Various Curve Fittings by Functional Class

• Findings and Further Research

Outline



Introduction

Project Goals

• Use empirical data obtained from Virginia facilities to evaluate speed-flow relationships

• Test various volume-delay functional forms for each facility type and determine which provides the best performance

• Calibrate volume delay function parameters for each facility type

• Outcome should be suitable for implementation in Virginia urban travel models



Introduction

Fundamental Elements of Volume-Delay Estimation

• Converting spot speeds to space-mean speed

• Characteristics of free-flow

• Identifying boundary between uninterrupted and interrupted flow

• Using knowledge of this boundary to estimate the maximum sustainable flow rate (Capacity)

• Use empirical observations to fit VDF curve parameters



Traffic Monitoring System (TSM) Data

• 5,848 locations from 17,400 detector locations available• Three locations per classification selected



Traffic Data

• Data records are a summary of each 15 minute period • Speed bins are in 5 mph increments• Data records are organized by lane and vehicle class

LINKID COUNTERNUMLANE STARTDATEVEHICLEB_1 VEHICLEB_2 VEHICLEB_3 INTERVAL SPEEDBINLABIN01 BIN02 BIN03 BIN04 BIN05 BIN06 BIN07 BIN08 BIN09 BIN10 BIN11 BIN12 BIN13 BIN14 BIN15 BIN16 BIN17 BIN18 BIN19 BIN20 BIN21 SPEEDQUALI010148 1 1 9/1/2007 6 1 1 15 65 0 0 0 0 0 0 1 0 1 3 3 7 4 1 0 0 0 0 0 0 0 5010148 1 1 9/1/2007 6 1 2 15 65 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 5010148 1 1 9/1/2007 6 1 3 15 65 0 0 0 0 0 0 2 2 2 0 0 1 3 0 0 0 0 0 0 0 0 5010148 1 2 9/1/2007 6 1 1 15 65 0 0 0 0 0 0 0 0 0 3 3 1 3 2 1 0 0 0 0 0 0 5010148 1 2 9/1/2007 6 1 2 15 65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5010148 1 2 9/1/2007 6 1 3 15 65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5010148 1 1 9/1/2007 6 1 1 15 65 0 0 0 0 0 0 0 1 0 1 6 4 5 6 2 0 0 0 0 0 0 5010148 1 1 9/1/2007 6 1 2 15 65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5010148 1 1 9/1/2007 6 1 3 15 65 0 0 0 0 0 0 1 1 5 3 3 3 0 0 0 0 0 0 0 0 0 5010148 1 2 9/1/2007 6 1 1 15 65 0 0 0 0 0 0 0 0 0 0 2 6 1 5 1 0 0 0 0 0 0 5010148 1 2 9/1/2007 6 1 2 15 65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5

• CUBE/Voyager script simplifies data• Spot speeds are converted to Space Mean Speeds

Filed Description TIME_PER Time Period TOT_AUTOS Total number of Autos TOT_TRUCK Total number of Trucks TOT_VEH Total Vehicles LANES Maximum number of lanes FRATE Flow rate BLANK1 Separator column AUTO1 Harmonic Space mean speed for autos TRUCK1 Harmonic Space mean speed for trucks BOTH1 Harmonic Space mean speed for all vehicles FDEN1 Flow Density for all vehicles with harmonic space mean speed BLANK2 Separator column AUTO2 Space mean speed for autos ( as per the formula from paper) TRUCK2 Space mean speed for trucks ( as per the formula from paper) BOTH2 Space mean speed for all vehicles ( as per the formula from paper) FDEN2 Flow Density for all vehicles for paper space mean speed



Processed Data

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 20.00 40.00 60.00 80.00 100.00 120.00

Density (Veh/ Mi/ Ln)

Speed vs Density

0 5 10 15 20 250

100

200

300

400

500

600

0 5 10 15 20 2510

20

30

40

50

60

70

80

0 5 10 15 20 250

10

20

30

40

50

0 5 10 15 20 250.1

0.2

0.3

0.4

0.5

0.6

Time of Day

Lane1

Lane2

Lane3

Lane1

Lane2

Lane3

Lane1

Lane2

Lane3

Lane1

Lane2

Lane3

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Flow (Veh/ Hr/ Ln)

Speed vs Flow

Speed by Time of Day



Estimating Free Flow Speed

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 20.00 40.00 60.00 80.00 100.00 120.00


1. HCM recommends using mean value for low volume conditions2. Standard practice also includes using 85th Percentile speed

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 20.00 40.00 60.00 80.00 100.00 120.00

71.1 mph

73.7 mph

Min Max Min Max

55.5 75.2 71.1 1.8 73.0 5.3 76.0 71.1 6.5 73.7

Alternative MethodAll Flow Conditions

Observed Speed Range

Mean85th

PercentileStandard Deviation

Low Volume ConditionsHCM Method

Observed Speed Range

Mean85th

PercentileStandard Deviation



0

500

1000

1500

2000

2500

3000

0.00 20.00 40.00 60.00 80.00 100.00 120.00

Density (Veh/Mi/Ln)

Flo

w (

Veh

/Hr/

Ln

)

Defining Interrupted Flow

1. Plots of flow vs density and speed vs density show two flow states

2. Others have defined the transition point as the maximum flow or the density at maximum speed, but this is not representative of typical conditions

3. Statistical techniques can define the transition between the two states

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 20.00 40.00 60.00 80.00 100.00 120.00


Flow vs. Density

Speed vs. Density

Interrupted Flow Interrupted Flow




We define interrupted flow as: Any speed below the threshold where there is 0.0001% probability that it is the same as freeflow.

Speed Distribution

0

500

1000

1500

2000

2500

3000

3500

4000

50 55 60 65 70 75 80

Observed Speed

Fre

qu

ency

Observed

Freeflow Normal Distribution

For our Rural Freeway example:The computed threshold is 62.58 MPH



0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 20.00 40.00 60.00 80.00 100.00 120.00

Density

Sp

eed Uninterrupted

Interrupted


Rural Freeway example, with flow states identified



Interrupted Flow

Histogram – Percent with Interrupted Flow vs. Flow Density

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

2.50

7.50

12.5

017

.50

22.5

027

.50

32.5

037

.50

42.5

047

.50

52.5

057

.50

62.5

067

.50

72.5

077

.50

82.5

087

.50

92.5

097

.50

102.

50

Flow Density

Pro

bab

ility

of

Inte

rru

pte

d F

low

Per

cen

t w

ith

Inte

rru

pte

d F

low



Estimating Capacity1. Our data shows a classic logistic distribution2. We estimated parameters (using density as the only variable) to

create a probability function that best fits the data

3. Capacity corresponds to flow density with a 50% probability of

being interrupted

PI = 1/[1 + e(1D+

0)] , where D = Density (veh/mi)

Probability Function

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

2.50

7.50

12.50

17.50

22.50

27.50

32.50

37.50

42.50

47.50

52.50

57.50

62.50

67.50

72.50

77.50

82.50

87.50

92.50

97.50

102.5

0

Flow Density

Pro

bab

ilit

y

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Fre

qen

cy

Observed Frequency

Predicted byProbability Function



0

500

1000

1500

2000

2500

3000

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

Flow Density

Mea

n F

low

(O

bse

rved

)

99.5% Probability of Interrupted Flow



Capacity Estimates

This example gives a 39.7 pc/mi Density Threshold or a 2384 pc/hr Max Flow Rate



Fitting Volume Delay Functions

Volume-Delay Functions - Using the computed capacity, the following volume delay functions were estimated based on speeds during uninterrupted flow

BPR: R = R0[1 + (V/C)^]

Akcelik: R=R0+D0+0.25T[(V/C-1)+sqrt{(V/C-1)2+(16J(V/C)L2)/T2}]

Conical: R=R0[2 + sqrt(2(1-V/C) + 2) –(1-V/C) – ]



• Curve Fitting - non-linear regression

• Goodness of Fit – R-squared – Root Mean Square Error– Non-Parametric tests e.g. Chi-Square

• Other Criteria - suitability for model applications





Urban Interstate

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function

Alpha Beta R-squared RMSEBPR 0.24 3.82 0.947 6.017%Conical 34.5 1.0 0.954 9.489%Akcelik 1.11E-04 1.0 0.957 8.787%

J T




Rural Interstate

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Urban Expressway

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Rural Principal Arterial

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Urban Other Principal Arterials

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Rural Minor Arterial

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Urban Minor Arterial

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Rural Collector

0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Urban Collector


J T

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function




Rural Local

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 0.50 1.00 1.50 2.00 2.50

Density Ratio

Sp

eed

Observed Traffic

BPR Function

Conical Function

Akcelik Function


J T




Summary of calibrated inputs to VDF fitting process

Low High Low High Low High

Rural Interstate 70.3 73.7 1747 2384 28.3 39.9Rural Other Principal Arterial 49.0 51.5 980 1130 23.2 41.3Rural Minor Arterial 41.6 57.8 922 1188 19.1 35.9Rural Major Collector 34.1 47.8 450 589 17.0 25.5Rural Local 28.4 45.7 368 377 11.9 19.3Urban Interstate 63.6 66.0 1442 1710 29.5 32.7Urban Freeways and Expressways 46.7 65.4 810 1981 14.2 36.6Urban Other Principal Arterials 45.2 52.9 888 1291 25.1 29.7Urban Minor Arterial 39.6 51.9 752 1181 18.9 28.4Urban Collector 40.6 51.2 669 853 17.4 28.0

Miles per Hour Hourly PCE/Lane Hourly PCE/Mile

CapacityFlow Rate Flow DensityFreeflow Speed

Functional Classification

VDF Inputs




Summarized results from VDF fitting process

Low High Low High Low High Low High

Rural Interstate 0.15 0.15 4.2 5.0 78.9 178.9 2.5E-05 4.9E-05Rural Other Principal Arterial 0.25 0.69 1.7 3.2 3.1 29.4 1.8E-04 1.6E-03Rural Minor Arterial 0.11 0.14 1.5 1.9 19.2 23.9 2.3E-04 2.7E-04Rural Major Collector 0.35 0.37 1.4 1.9 5.3 7.0 8.3E-04 1.4E-03Rural Local 0.26 0.41 1.0 1.9 5.3 6.0 1.2E-03 1.4E-03Urban Interstate 0.16 0.24 3.3 3.8 29.2 57.6 7.5E-05 1.4E-04Urban Freeways and Expressways 0.10 0.42 1.9 8.7 6.6 89.0 5.2E-05 7.1E-04Urban Other Principal Arterials 0.24 0.28 1.9 2.9 14.8 19.0 2.4E-04 3.6E-04Urban Minor Arterial 0.16 0.29 2.0 4.5 15.6 57.7 1.0E-04 4.0E-04Urban Collector 0.08 0.35 0.4 3.5 12.7 28.1 1.8E-04 5.2E-04

BPRAlpha JAlpha Beta

Functional Classification

ConicalVolume Delay Functions

Akcelik



Initial Findings

• Standard VDF functions are all capable of performing adequately across road classes

• For a given road class, VDF parameters fitted for one location, seem to be transferable to other locations

• Goodness of fit measures do not strongly differentiate between functions

• The Akcelik function, with its more rigorous theoretical underpinnings, seems to work very well



Next Steps

• Additional facility types

• Check model transferability to other facilities

• Compare HCM capacity, planning capacity, and empirical capacity

• Continue to automate analysis process

• Test functions in urban models (assignment convergence, average travel speeds)

• New VDF functional forms and calibrated parameters will become part VDOT modeling standards



Q & A

Thank you !

Contact points

Jaesup Lee: [email protected]

Dean Munn: [email protected]

Jeremy Raw : [email protected]

TRB Planning Applications Conference - May 19, 2009

Documents

Transcript of TRB Planning Applications Conference - May 19, 2009