Friction Factors and Trip Lengths;

BPR Curves and Speeds

presented by

Michelle Arnold, AICP, EI

Transportation Planner

AECOM

September 3, 2015

Parts and Format

• Part A: Friction Factors and Trip Lengths

• Part B: BPR Curves and Speeds

• Format

– What are they?

– How are they generated?

– Where are they used in the model?

2

What are Friction Factors

• Friction factors are parameters used in the

gravity model to account for travel time

separation between zones.

• An impedance factor.

• The gravity model distributes trips to the traffic

analysis zones based on impedance between the

traffic analysis zones and their attractions.

3

Gravity Equation

� = �����

��

�= the force between masses

�= gravitation constant

��= the first mass

��= the second mass

�= the distance between the centers of the masses

m1m2

r

4

Gravity Model Equation

��� = ��

�� ���� ��

∑ �� ���� ��"�#�

���= Trips from zone i to zone j

��= Total trips sent from zone i

��= Total trips received by zone j

����= Travel time factor for time between zone i and zone j ((((friction factorfriction factorfriction factorfriction factor))))

��= Socioeconomic adjustment factor for interchanges

.= Number of analysis zones5

Gravity Model

Zone 3

602 Work Trip

Productions

76 Work Trip

Attractions

Zone 4

47 Work Trip

Attractions

Zone 5

82 Work Trip

Attractions

Zone 2

531 Work Trip

Attractions

Zone 1

1080 Work

Trip

Attractions

10

min

ute

s

20 minutes

6

Why are Friction Factors Used

• Obtaining calibrated friction factors is the

principal operation of gravity model calibration.

• The product of the attractions and the friction

factor represents each zone’s relative

attractiveness and accessibility.

• Used to replicate observed trip length frequency

distributions.

7

How Are Friction Factors Generated

Friction factors are created for each trip purpose.

• Household travel surveys; OD surveys.

• Borrowed from another study area or model.

• Census Journey-to-Work for work-based trips.

• Roadside surveys for internal-external trips.

• Formulas.

8

Friction Factors Formulas

• Power function: ��� = 01

where a common value of the exponent a is 2

• Exponential function: ��� = 234�

where m represents the mean travel time

• Gamma function: ��� = a0526�

where t is travel impedance (time in minutes)

a, b, and c are model parameters9

Gamma Function

• Lookup tables are widely used in FSUTMS, although more recently some models use a gamma function.

• Whichever model’s parameters are chosen, they should be checked against observed trip length frequencies for each trip purpose.

• The parameters should be adjusted as needed to obtain the most reasonable model for the region.

10

Friction Factor Quiz

Friction factors attempt to show the effect of

travel time or impedance on trip making.

True or False. A way to estimate the initial

friction factors is to use the factors from a

previous study in a similar area. True

11

Initial Friction Factors Development

• To create initial friction factors, the productions and attractions trip table and a network skims matrix are needed.

• The survey data from an OD matrix provide the trip lengths.

• The trip length frequencies from the initial friction factors are compared to observed trip frequencies from the survey for reasonability.

12

Friction Factors in Calibration

Calibrating a gravity model involves adjusting the

friction factors until the model adequately

reproduces the productions and attractions trip

table and matches the observed average trip

length and frequencies from the survey.

13

Calibration Process

Initial Friction Factors, Travel Time Tables and

Trip Tables

Apply Gravity Model and Distribute Trips

Compare Attractions

Compare Trip Time Distribution and

Average Trip Time

Calibrated Friction Factors

Adjust

Friction

Factors

Adjust

Attraction

Factors

Not Balanced

Balanced?

Not Good

Good Comparison?

14

Calibrated Friction Factors

• Calibrated friction factors

for home-based work.

• Friction factors are

inversely proportional to

the travel time.

• Friction factors have no

units.

15

Friction Factors in the Model

Gravity model done manually.

Gravity model as a function.

16

Trip Distribution Trip Length Frequency

Florida Statewide Model—TLF Distance

17

Trip Distribution Trip Length Frequency

Florida Statewide Model-TLF Minute

18

Trip Distribution Chart

FLSWM Trip Distribution Trip Length Frequency

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Pe

rso

n T

rip

s

Distance (Miles)

HBW HBSH HBSR HBO NHB

19

Trip Distribution Chart

FLSWM Trip Distribution Trip Length Frequency

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

4,000,000

4,500,000

5,000,000

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125

Pe

rso

n T

rip

s

Distance (Minutes)

HBW HBSH HBSR HBO NHB

20

Calibration and Validation Standards

21

Calibration and Validation Standards

Model HBW HBSH HBSR HBO NHB

Statistic

(minutes)

12-35 9-19 11-19 8-20 6-19

FLSWM 19 17 16 17 14

FSUTMS-Cube Framework Phase II: Model Validation and Calibration Standards

October 2, 2008

Average Trip Length and Frequencies by Purpose

22

Friction Factor Quiz

• The output of the gravity model calibration is a

set of calibrated friction factors.

• If the output attractions do not balance with

the input attractions, the attractions are

adjusted and the model is rerun.

• True or False. If the trip time frequency

distribution and average trip time inputs and

outputs do compare satisfactorily, the friction

factors need to be changed. False.23

Calibration with Friction Factors

Matching average observed trip lengths or even

complete trip length frequency distributions is

not sufficient to say that trip distribution model

is validated. The orientation of trips must be

geographically correct or reasonable.

24

What are BPR Curves

• Bureau of Public Roads (BPR) curves are a type

of volume-delay functions used to describe the

speed-flow relationships in a travel demand

model network based on the available link

capacity.

25

BPR Formula

0� = 00� 1 + :;�

<�

=

0�= Congestion flow travel time on link i;

00�= Free-flow travel time on link i;

;�= Volume of traffic on link i per unit of time (flow

attempting to use link i);

<�= Capacity of link i per unit of time;

:= Alpha coefficient, which was assigned a value of 0.15 in

the original BPR curve; and

K= Beta coefficient, the exponent of the power function,

which was assigned a value of 4 in the original BPR curve

26

BPR Formula Meaning

• The formula basically shows that as volume

increases or flow increases relative to capacity,

the speed decreases and travel time increases.

• In the original BPR function, the capacity, ci, used

LOS C, but common practice now is to use LOS E

from the Highway Capacity Manual.

27

Why are BPR Curves Used

• FSUTMS uses equilibrium highway assignment

which means that no trip can be made by an

alternate path without increasing the total

travel time in the network.

• Equilibrium accounts for congestion in traffic

assignment. The process is iterative and

requires adjustments.

28

Why are BPR Curves Used

• BPR curves are used in a capacity restraint assignment to introduce congested time/speed in the route choice.

• For example, if Road X is within your shortest route and it is at free-flow speed, and everyone will take it; but Road X reaches capacity or overcapacity. The BPR curve increases the time for travelers on that road (congested time) so that in the next assignment iteration, Traveler Z remains on Road X and you use Road Y.

29

BPR Curve Criticisms

• Some modelers feel that the standard

BPR curve overestimates speeds for

volume-to-capacity (v/c) ratios

greater than or equal to 1.0 and

underestimates speeds for v/c ratios

less than 1.0.

• In the real world, v/c greater than 1.0

does not exist. There are queues and

stop-and-go traffic flows.

30

Updated BPR Parameters

31

BPR In the Model

32

Standard BPR Curve Example 1

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

Co

ng

est

ed

/Un

con

ge

ste

d S

pe

ed

Ra

tio

Volume-Delay

BPRorg

Interstate

Div Arterial

Undiv Art.

Collector

Centroid

Toll

33

Standard BPR Curve Example 2

0.000

0.200

0.400

0.600

0.800

1.000

1.200

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

Co

ng

est

ed

/Un

con

ge

ste

d S

pe

ed

Ra

tio

Volume-Delay

BPRorg

FT10

FT20

FT30

FT40

FT50

FT60

FT65

FT70

FT80

FT90

34

Other Volume Delay Functions

• BPR Curve (Modified, Fitted, Updated, etc.)

• Davidson and Modified Davidson Function

• Akçelik Function

• Conical Delay (Spiess) Function

• Linear

• Logarithmic

• Exponential

• Power

35

Comparison of VDF Curves

36

Calibration and Validation Standards

• Adjustments to BPR factors should be based on

more sophisticated calculations of capacity-

related attributes and not arbitrarily modified

solely to improve model validity for selected

facility types.

• One should not manipulate speeds and/or

capacities to fit validation statistics.

37

FSUTMS Recommended VFACTORS

38

FSUTMS Recommended VFACTORS

39

Speeds in FSUTMS Models

• Use a lookup table of area types, facility types, and lanes that assign speeds and capacities.

• Use posted speeds instead a lookup table for speed values. Posted speeds are readily available from FDOT’s Roadway Characteristics Inventory (RCI).

• The speeds in the FSUTMS lookup tables are based on very limited data.

40

Updating SPDCAP and VDFs

• Development of Speed Models for Improving Travel Forecasting and Highway Performance Evaluation (December 2013)

• It recommended traffic assignment improvement through changes in speed values and volume-delay function factors.

• The research noted that different volume-delay functions corresponded well to certain facility types.

41

SPDCAP File

42

FSUTMS Default Model Parameters

43

Speed Data

• Volume-delay functions require free flow

speeds.

• Speed data collection may not be feasible due to

time or budget constraints.

• FDOT has HERE and INRIX data.

44

HERE Data

• 30 million IDs available in Florida.

• Each ID corresponds to a NAVTEQ network

segment.

• Each record has a timestamp with a 5-minute

interval for a total of 288 intervals per day.

• Uses the Traffic Message Channel (TMC) system.

• Updated monthly.

• Speed must be computed.

45

HERE Data Example

46

HERE Data User Limits

• HERE data is only available for license to

DOTs, MPOs, FHWA and its federal partners.

• Selected government users may allow access

to their contractors to support their work

only.

• Unless you have an agreement with a

designated user, you must purchase the data.

47

INRIX Data

• Over 711 million records in Florida from July 1, 2010 to June 30, 2011.

• Average speeds in 5-minute intervals for 24 hours per day.

• Average speed computed by INRIX.

• Uses the Traffic Message Channel (TMC) system.

• FDOT purchased the data to support speed research. No new data has been purchased.

• Restricted to FDOT and MPO for their projects only.

48

HERE and INRIX Data

• Mainly gauge congestion.

• Used for other purposes, e.g., volume-delay functions.

• The quality of data varies depending on the facility type.

• User limited to FDOT, MPOs and other government agencies.

• Model Task Force May 2015 contain a presentation that compared HERE, Bluetooth, and INRIX speed data.

49

References

• Development of Speed Models for Improving Travel Forecasting and

Highway Performance Evaluation (Moses, Ren et al, 2013).

• Greater Treasure Coast Regional Planning Model Version 3.3

(GTCRPM3.3) Model Enhancements and Application Draft Report

(Corradino Group, 2010).

• An Introduction to Urban Travel Demand Forecasting—A Self

Instructional Text (FHWA, 1977).

• NCHRP Report 716: Travel Demand Forecasting: Parameters and

Techniques (TRB, 2012).

• FSUTMS-Cube Framework Phase I: Default Model Parameters (FDOT

Systems Planning, 2007)

• FSUTMS-Cube Framework Phase II: Model Calibration and Validation

Standards (FDOT Systems Planning, 2008)

50

Questions

51

Contact Information

Michelle Arnold, AICP, EI

Transportation Planner

AECOM

(850) 402-6329

michelle.arnold@aecom.com

52