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Fine Tuning Mathematical Models for Toll Applications

Dr. A. Mekky, P.Eng., A. Tai, M. Khan

Ministry of Transportation, Ontario, Canada

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Presentation Outline Introduction

The GTA Hwy 407 Transportation Tomorrow Survey The GTA Model

The Problem Model Fine-Tuning and Parameter

Testing Results and Conclusions.

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Introduction Transportation models have many

applications in transportation planning One of those applications is the simulation

of toll roads In this kind of application one usually needs

to estimate the expected revenue from a particular toll road investment

The main statistic coming out of the model in this case is the vehicle.km on the toll road

High accuracy of estimation is important.

The Greater Toronto Area

6 Regions

Population 5.4 Million

Employment 2.9 Million

Autos 2.8 Million

Daily Trips 12 Million

Hwy 401

Hwy 401

Hwy 403QEW

Hwy

7

Hwy 407

Gardiner Exp.

Hw

y 40

0

Hwy

10

Hw

y 6

Hwy 6Hwy 7A

Hwy 7

Hw

y 48

Hw

y 40

4

DV

P

Hwy

403

Toronto

YorkDurham

Peel

Halton

Hamilton

Hwy 407

Total Length = 108 km

Average Peak Hr. Trip Length = 19.48 km

Total Peak Hr. Users = 3,300

Total Peak Hr. Vehicle km = 641,500Lake Ontario

Highway 407

East-west north of Toronto

108 km in 4 Regions

Fully Electronic Tollway

Opened in November 1997

Toll about 13c per km

Daily Trips about 300,000.

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Transportation Tomorrow Survey Travel Behaviour Telephone Survey Covers GTA & surrounding areas 1986, 1991, 1996, 2001 Info Contents:

Household Information (locations, etc) Person Information (age, gender, etc) Trip Information (OD, time, modes, etc).

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The GTA Model Since the 1960’s Base Year: 2001 Zones: 1,677 Gateways: 26 Nodes: 12,457 Links: 37,385 Demand: Trip Matrix from TTS (2001) All Trip Purposes, Automobile Trips Only Commercial Vehicles not included explicitly.

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The Problem Original Model Performed

Reasonably well at the Screen-Line Level

However, it didn’t produce good results for observed volumes & the number of users on Hwy 407

This had an effect on adjacent facilities

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The Problem cont’d It was tempting to use tools to

reproduce volumes on links (DEMADJ)

The decision was made to check/ fine-tune main elements of the network first.

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Fine-Tuning (stage 1) Since good part of the network was

“inherited” from the previous “century”, an update exercise was needed: Fine-tuning Free-Flow Speeds

Adjust free flow speed on access roads and parallel facilities to Hwy 407

Adjust free flow speed on Hwy 407 Adjust free flow speed on Hwy ramps

Adjust some centroid connectors

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Fine-Tuning (stage 1) cont’d Adjust Hwy 407 Capacity to 2200

vehicles/ hr/ lane (instead of 1800), as it carries low heavy vehicle traffic

Form of Volume Delay Functions Unchanged

Travel Matrix Unchanged.

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Fine-Tuning (stage 2) Build upon Fine-Tuning of stage 1 Adjust parameters of VDFs (alpha,

beta) Adjust Value of Time for tollway Travel Matrix Unchanged.

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Parameters Testing Alpha for Non-Highways Alpha For Highways Beta for Non-Highways Beta for Highways Value of TimeA macro was written to do several emme/2 run with various combinationsOf the above parameters.

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Hypotheses Tested Base: Original network & original VDF Stage 1 Update: Modified network with

original VDF, no matrix modification Stage 2 Update: Modified network with

modified VDF, no matrix modification DEMADJ on Stage 1 Update DEMADJ on Stage 2 Update.

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Main ResultsNon Hwy Alpha

Hwy Alpha

Value of Time

VKM on Hwy 407

Users On Hwy 407

Avg Trip Length, Km

Actual 641,500 32,920 19.48

BASE 1 1 22 486,300 43,290 11.23

UPDATED S1 1 1 22 596,100 40,720 14.64

NEW-VDF S2 0.70 1.95 29 644,800 41,640 15.49

DJ-UPDATED S1

1 1 22 630,800 46,790 13.48

DJ-NEW-VDF S2

0.70 1.95 29 659,500 45,310 14.55

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Main Results cont’d% Changes Induced by DEMADJ On Stage1

(Actual vs Resultant)O-D Toronto Durham York Peel Halton Hamilton External SumToronto -4.8 6.3 22.2 -19.1 -41.4 -53.4 18.3 -3.1Durham 13.9 1.2 -10.9 26.8 -28.8 -64.6 2.0 3.3York -0.3 11.7 -1.8 50.5 20.6 -69.4 22.1 2.3Peel 1.9 -3.7 59.5 7.9 2.2 -10.8 -8.0 8.4Halton -75.2 -65.0 -33.3 12.6 2.3 17.3 -8.5 -3.2Hamilton -67.6 69.2 -14.6 -1.3 31.7 -0.2 -1.7 2.8External 1.7 -10.0 -13.3 6.9 4.9 -2.6 -0.2 -0.4Sum -4.0 1.3 6.4 6.2 5.4 0.0 -0.1 0.9

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% Changes Induced by DEMADJ On Stage 2 (Actual vs Resultant)

O-D Toronto Durham York Peel Halton Hamilton External SumToronto -6.9 7.2 23.5 -7.7 -34.1 -52.3 60.4 -3.1Durham 31.3 0.4 -38.0 15.5 -20.1 -18.3 -3.3 4.1York 1.9 4.2 -5.4 32.0 -10.8 -50.6 0.5 -0.6Peel -5.0 9.1 53.8 4.5 7.2 5.8 -4.9 4.8Halton -52.8 -70.0 -47.2 1.8 2.5 15.1 -8.0 -3.5Hamilton 16.7 105.9 7.3 -42.4 38.1 -0.4 -2.3 4.0External 39.9 -5.6 -22.8 -2.7 -3.8 -2.5 -0.2 0.1Sum -3.3 0.7 2.3 2.9 6.7 0.0 0.1 0.2

Main Results cont’d

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Screen Line Analysis(Estimated/ Counts Ratio)

ScreenLine 2001 COUNT BASE UPDATED NEW VDF DJ-UPDATED DJ-NEW-VDFToronto Durham 4728 0.95 0.98 0.97 1.01 1.03Durham Toronto 16576 0.75 0.91 0.89 1.00 1.02

Total 21305 0.80 0.93 0.90 1.00 1.03Toronto York 34412 1.03 0.96 0.96 1.23 1.22York Toronto 49300 1.17 1.07 1.06 1.13 1.14

Total 83712 1.11 1.03 1.02 1.17 1.17Toronto Peel 28725 1.13 1.11 1.07 1.04 1.05Peel Toronto 38878 1.03 1.01 1.01 0.99 1.02

Total 67603 1.07 1.05 1.04 1.01 1.03Peel York 10413 1.18 1.27 1.28 1.46 1.38York Peel 7531 0.84 0.89 1.01 1.23 1.22

Total 17944 1.04 1.11 1.17 1.36 1.31York Durham 1746 1.01 0.92 0.91 0.93 0.88Durham York 5012 2.07 1.45 1.54 1.40 1.46

Total 6758 1.80 1.31 1.38 1.28 1.31Peel Halton 11955 1.15 1.22 1.23 1.29 1.34Halton Peel 24115 1.04 1.08 1.07 1.05 1.08

Total 36071 1.08 1.13 1.12 1.13 1.17Halton Hamilton 8919 0.91 0.91 0.91 0.86 0.87Hamilton Halton 16816 0.92 0.93 0.93 1.01 1.07

Total 25735 0.91 0.93 0.93 0.96 1.00Grand Total 259127 1.06 1.04 1.04 1.10 1.12

Hwy 407 counts vs Base Assignment

R-Square = 76%

Hwy 407 counts vs Stage 1 Update

R-Square = 71%

Hwy 407 counts vs Stage 2 Update

R-Square = 72%

R-Square = 94%

Hwy 407 counts vs DJ on Stage 1 Update

Hwy 407 counts vs DJ on Stage 2 Update

R-Square = 89%

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Conclusions Compared to fine-tuning using network and VDF

parameters, adjustment of the matrix with DEMADJ distorted the original matrix and gave larger errors in average trip length, VKM and the number of Hwy 407 users

Fine-tuned model performed well at the screen-line level as well as gave better average trip length, VKM and number of users on Hwy 407

It is advised that models should be validated using info re average trip length, VKM if possible

Heavy usage of DEMADJ without checking other parameters is not recommended.

END