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 Transportation
Dean Munn, The Corradino Group
Outline
• 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
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
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
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
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
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Traffic Monitoring System (TSM) Data
• 5,848 locations from 17,400 detector locations available
• Three locations per classification selected
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
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
010148
010148
010148
010148
010148
010148
010148
010148
010148
010148
010148
COUNTERNUM
LANE
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
1
1
1
2
2
STARTDATE
VEHICLEB_1 VEHICLEB_2 VEHICLEB_3 INTERVAL SPEEDBINLA
BIN01
9/1/2007
6
1
1
15
65
9/1/2007
6
1
2
15
65
9/1/2007
6
1
3
15
65
9/1/2007
6
1
1
15
65
9/1/2007
6
1
2
15
65
9/1/2007
6
1
3
15
65
9/1/2007
6
1
1
15
65
9/1/2007
6
1
2
15
65
9/1/2007
6
1
3
15
65
9/1/2007
6
1
1
15
65
9/1/2007
6
1
2
15
65
BIN02
0
0
0
0
0
0
0
0
0
0
0
BIN03
0
0
0
0
0
0
0
0
0
0
0
• CUBE/Voyager script simplifies data
• Spot speeds are converted to Space Mean Speeds
Filed
TIME_PER
TOT_AUTOS
TOT_TRUCK
TOT_VEH
LANES
FRATE
BLANK1
AUTO1
TRUCK1
BOTH1
FDEN1
BLANK2
AUTO2
TRUCK2
BOTH2
FDEN2
Description
Time Period
Total number of Autos
Total number of Trucks
Total Vehicles
Maximum number of lanes
Flow rate
Separator column
Harmonic Space mean speed for autos
Harmonic Space mean speed for trucks
Harmonic Space mean speed for all vehicles
Flow Density for all vehicles with harmonic space mean speed
Separator column
Space mean speed for autos ( as per the formula from paper)
Space mean speed for trucks ( as per the formula from paper)
Space mean speed for all vehicles ( as per the formula from paper)
Flow Density for all vehicles for paper space mean speed
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
BIN04
0
0
0
0
0
0
0
0
0
0
0
BIN05
0
0
0
0
0
0
0
0
0
0
0
BIN06
0
0
0
0
0
0
0
0
0
0
0
BIN07
0
0
0
0
0
0
0
0
0
0
0
BIN
1
0
2
0
0
0
0
0
1
0
0
Processed Data
Speed vs Density
Speed by Time of Day
600
Lane1
Lane2
Lane3
500
400
300
200
100
0
0
5
10
15
20
25
10
15
20
25
Speed vs Flow
80
70
60
50
40
30
Lane1
Lane2
Lane3
20
10
0
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
5
Estimating Free Flow Speed
1. HCM recommends using mean value for low volume conditions
2. Standard practice also includes using 85th Percentile speed
73.7 mph
71.1 mph
HCM Method
Low Volume Conditions
Observed Speed Range
Standard
Min
Max
Mean
55.5
75.2
71.1
Alternative Method
All Flow Conditions
Deviation
85th
Percentile
1.8
73.0
Observed Speed Range
Min
5.3
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Max
Mean
76.0
71.1
Standard
Deviation
85th
Percentile
6.5
73.7
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
3000
2500
Flow (Veh/Hr/Ln)
2000
1500
Interrupted Flow
Interrupted Flow
1000
500
0
0.00
20.00
40.00
60.00
80.00
100.00
120.00
Density (Veh/Mi/Ln)
Flow vs. Density
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Speed vs. Density
Defining 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
4000
Observed
3500
Freeflow Normal Distribution
Frequency
3000
2500
For our Rural Freeway example:
The computed threshold is 62.58 MPH
2000
1500
1000
500
0
50
55
60
65
Observed Speed
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
70
75
80
Defining Interrupted Flow
Rural Freeway example, with flow states identified
80.0
70.0
60.0
Speed
50.0
Uninterrupted
40.0
Interrupted
30.0
20.0
10.0
0.0
0.00
20.00
40.00
60.00
80.00
Density
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
100.00
120.00
Interrupted Flow
Histogram – Percent with Interrupted Flow vs. Flow Density
100.00%
90.00%
Probability
Interrupted Flow
Flow
withofInterrupted
Percent
80.00%
70.00%
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
2.
50
7.
50
12
.5
0
17
.5
0
22
.5
0
27
.5
0
32
.5
0
37
.5
0
42
.5
0
47
.5
0
52
.5
0
57
.5
0
62
.5
0
67
.5
0
72
.5
0
77
.5
0
82
.5
0
87
.5
0
92
.5
0
97
.5
0
10
2.
50
0.00%
Flow Density
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Estimating Capacity
1. Our data shows a classic logistic distribution
2. We estimated parameters (using density as the only variable) to
create a probability function that best fits the data
PI = 1/[1 + e(b D+ b )] , where D = Density (veh/mi)
1
0
3. Capacity corresponds to flow density with a 50% probability of
being interrupted
Probability Function
120.00%
120.00%
Observed Frequency
80.00%
80.00%
60.00%
60.00%
40.00%
40.00%
20.00%
20.00%
0.00%
0.00%
Flow Density
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Freqency
100.00%
Predicted by
Probability Function
2.
50
7.
5
12 0
.5
17 0
.5
22 0
.5
27 0
.5
32 0
.5
37 0
.5
42 0
.5
47 0
.5
52 0
.5
57 0
.5
62 0
.5
67 0
.5
72 0
.5
77 0
.5
82 0
.5
87 0
.5
92 0
.5
97 0
10 .50
2.
50
Probability
100.00%
Capacity Estimates
0.50% Probability of Interrupted Flow
50.0% Probability of Interrupted Flow
99.5% Probability of Interrupted Flow
3000
Mean Flow (Observed)
2500
2000
1500
1000
500
0
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
Flow Density
This example gives a 39.7 pc/mi Density
Threshold or a 2384 pc/hr Max Flow Rate
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
90.00
100.00
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:
Conical:
Akcelik:
R = R0[1 + a(V/C)^b]
R=R0[2 + sqrt(a2(1-V/C) + b2) –a(1-V/C) – b]
R=R0+D0+0.25T[(V/C-1)+sqrt{(V/C-1)2+(16J(V/C)L2)/T2}]
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Fitting Volume Delay Functions
• 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
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Urban Interstate
Alpha
0.24
34.5
1.11E-04
J
Beta
R-squared
3.82
0.947
1.0
0.954
1.0
0.957
T
RMSE
6.017%
9.489%
8.787%
80.0
Observed Traffic
70.0
BPR Function
Conical Function
60.0
Akcelik Function
Speed
50.0
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Rural Interstate
Alpha
0.15
78.9
4.89E-05
J
Beta
R-squared
4.18
0.790
1.0
0.790
1.0
0.812
T
RMSE
4.491%
8.381%
7.376%
80.0
Observed Traffic
70.0
BPR Function
Conical Function
60.0
Akcelik Function
Speed
50.0
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Urban Expressway
Alpha
0.18
49.1
8.41E-05
J
Beta
R-squared
4.63
0.666
1.0
0.625
1.0
0.646
T
RMSE
4.162%
6.430%
5.770%
80.0
Observed Traffic
70.0
BPR Function
Conical Function
60.0
Akcelik Function
Speed
50.0
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Rural Principal Arterial
Alpha
0.69
3.1
1.63E-03
J
Beta
R-squared
1.68
0.880
1.2
0.862
1.0
0.780
T
RMSE
5.674%
6.442%
10.362%
60.0
Observed Traffic
BPR Function
50.0
Conical Function
Akcelik Function
Speed
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
Urban Other Principal Arterials
BPR
Conical
Akcelik
Alpha
0.24
19.0
2.42E-04
J
Beta
R-squared
2.86
0.874
1.0
0.770
1.0
0.784
T
RMSE
4.602%
14.797%
13.860%
60.0
Observed Traffic
BPR Function
50.0
Conical Function
Akcelik Function
Speed
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Rural Minor Arterial
Alpha
0.14
19.2
2.30E-04
J
Beta
R-squared
1.51
0.248
1.0
0.243
1.0
0.245
T
RMSE
4.752%
5.348%
5.282%
70.0
Observed Traffic
60.0
BPR Function
Conical Function
Speed
50.0
Akcelik Function
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Urban Minor Arterial
Alpha
0.16
57.7
1.00E-04
J
Beta
R-squared
4.53
0.475
1.0
0.521
1.0
0.545
T
RMSE
4.270%
8.560%
6.880%
60.0
Observed Traffic
BPR Function
50.0
Conical Function
Akcelik Function
Speed
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Rural Collector
Alpha
0.35
7.0
8.28E-04
J
Beta
R-squared
1.94
0.172
1.1
0.200
1.0
0.203
T
RMSE
9.346%
9.186%
9.196%
60.0
Observed Traffic
BPR Function
50.0
Conical Function
Akcelik Function
Speed
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
Alpha
0.21
28.1
1.79E-04
J
BPR
Conical
Akcelik
Urban Collector
Beta
R-squared
3.06
0.566
1.0
0.549
1.0
0.569
T
RMSE
5.345%
8.234%
7.419%
70.0
Observed Traffic
60.0
BPR Function
Conical Function
Speed
50.0
Akcelik Function
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
BPR
Conical
Akcelik
Rural Local
Alpha
0.26
5.3
1.15E-03
J
Beta
R-squared
1.00
0.050
1.1
0.074
1.0
0.077
T
RMSE
10.611%
11.155%
11.316%
80.0
Observed Traffic
70.0
BPR Function
Conical Function
60.0
Akcelik Function
Speed
50.0
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
Density Ratio
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
2.00
2.50
Fitting Volume Delay Functions
Summary of calibrated inputs to VDF fitting process
Freeflow Speed
Miles per Hour
Functional Classification
Rural Interstate
Rural Other Principal Arterial
Rural Minor Arterial
Rural Major Collector
Rural Local
Urban Interstate
Urban Freeways and Expressways
Urban Other Principal Arterials
Urban Minor Arterial
Urban Collector
Low
70.3
49.0
41.6
34.1
28.4
63.6
46.7
45.2
39.6
40.6
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
High
73.7
51.5
57.8
47.8
45.7
66.0
65.4
52.9
51.9
51.2
VDF Inputs
Capacity
Flow Rate
Flow Density
Hourly PCE/Lane
Hourly PCE/Mile
Low
Low
1747
980
922
450
368
1442
810
888
752
669
High
2384
1130
1188
589
377
1710
1981
1291
1181
853
28.3
23.2
19.1
17.0
11.9
29.5
14.2
25.1
18.9
17.4
High
39.9
41.3
35.9
25.5
19.3
32.7
36.6
29.7
28.4
28.0
Fitting Volume Delay Functions
Summarized results from VDF fitting process
BPR
Functional Classification
Rural Interstate
Rural Other Principal Arterial
Rural Minor Arterial
Rural Major Collector
Rural Local
Urban Interstate
Urban Freeways and Expressways
Urban Other Principal Arterials
Urban Minor Arterial
Urban Collector
Alpha
Low
High
0.15
0.25
0.11
0.35
0.26
0.16
0.10
0.24
0.16
0.08
Volume Delay Functions
Conical
Beta
Alpha
Low
High
Low
High
0.15
0.69
0.14
0.37
0.41
0.24
0.42
0.28
0.29
0.35
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
4.2
1.7
1.5
1.4
1.0
3.3
1.9
1.9
2.0
0.4
5.0
3.2
1.9
1.9
1.9
3.8
8.7
2.9
4.5
3.5
78.9
3.1
19.2
5.3
5.3
29.2
6.6
14.8
15.6
12.7
178.9
29.4
23.9
7.0
6.0
57.6
89.0
19.0
57.7
28.1
Akcelik
J
Low
2.5E-05
1.8E-04
2.3E-04
8.3E-04
1.2E-03
7.5E-05
5.2E-05
2.4E-04
1.0E-04
1.8E-04
High
4.9E-05
1.6E-03
2.7E-04
1.4E-03
1.4E-03
1.4E-04
7.1E-04
3.6E-04
4.0E-04
5.2E-04
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
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
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
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009
Q&A
Thank you !
Contact points
Jaesup Lee: jaesup.lee@VDOT.Virginia.gov
Dean Munn: dmunn@corradino.com
Jeremy Raw : jeremy.raw@VDOT.Virginia.gov
Investigation of Speed-Flow Relations and Estimation of Volume Delay Functions
for Travel Demand Models in Virginia
TRB Planning Applications Conference, May 19, 2009