Optimization models and methods for equilibrium traffic assignment

231 Pages·2020·1.745 MB·English

by  Krylatov A., Zakharov V., Tuovinen T

Springer Tracts on Transportation and Traffic
Alexander Krylatov
Victor Zakharov
Tero Tuovinen
Optimization
Models and
Methods
for Equilibrium
Traffic Assignment
fi
Springer Tracts on Transportation and Traf c
Volume 15
Series Editor
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Brooklyn, NY, USA
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Alexander Krylatov Victor Zakharov
(cid:129) (cid:129)
Tero Tuovinen
Optimization Models
and Methods for Equilibrium
fi
Traf c Assignment
123
Alexander Krylatov Victor Zakharov
Institute of Transport Problems Faculty of AppliedMathematics
Russian Academy of Sciences andControl Processes
Saint Petersburg, Russia Saint PetersburgState University
Saint Petersburg, Russia
Faculty of AppliedMathematics
andControl Processes
Saint PetersburgState University
Saint Petersburg, Russia
Tero Tuovinen
Faculty of Information Technology
University of Jyväskylä
Jyväskylä, Finland
ISSN 2194-8119 ISSN 2194-8127 (electronic)
SpringerTracts onTransportation andTraffic
ISBN978-3-030-34101-5 ISBN978-3-030-34102-2 (eBook)
https://doi.org/10.1007/978-3-030-34102-2
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The sum of the values of excess demands
across all markets must equal zero.
Walras’ law
Preface
Optimization models and methods for an equilibrium traffic assignment is a very
specific research field. It is well known that a traffic assignment problem can be
formulated in the form of a nonlinear optimization program. However, the most
efficientsolutionalgorithmsfortheproblemarebasedonitsstructuralfeaturesand
practicalmeaningratherthanstandardnonlinearoptimizationtechniquesorapproa-
ches. Therefore, careful consideration on the meaning basis of a traffic assignment
problem for efficient algorithm development seems to be of high importance.
Nevertheless,thevastmajorityofbooksonthistopicprovidecommontrafficequi-
libriummodelsaswellasgeneraldescriptionofrespectivemethodsasarule.
Thisbookfocusesondiscussionofthetrafficassignmentproblemaswellasthe
mathematical andpracticalmeaningofvariables, functions, andbasic principlesin
order to obtain its wide analytical comprehension. New approaches, methods, and
algorithms based on the original methodological technique being developed by
authors in their publications for several past years and the corresponding
prospective implementations are compiled here. The book may be of interest to a
wide range of readers, such as students on civil engineering programs, traffic
engineers, developers of traffic assignment algorithms, etc. The authors hope that
the book will be useful to all these readers, and are sincerely waiting for con-
structive criticism and suggestions. The results obtained here are expected to be
desired both for practice and theory.
Thefirstauthorissincerelygratefulforthesupportofthisworkbyagrantofthe
Russian Science Foundation (No. 17-71-10069—Development of methodological
tools for traffic flow assignment and optimization applicable to the creation of
intelligent transportation systems).
Saint Petersburg, Russia Alexander Krylatov
Saint Petersburg, Russia Victor Zakharov
Jyväskylä, Finland Tero Tuovinen
vii
Contents
Part I Traffic Flow Issues
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Brief Traffic Theory Background. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Current Remarkable Research Issues . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Promising Relevant Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Part II Optimization Traffic Assignment Models
2 Principles of Wardrop for Traffic Assignment in a Road
Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 User Equilibrium and System Optimum of Wardrop
in a Road Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Dual Traffic Assignment Problem in a Large Scale Road
Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Route-Flow Traffic Assignment as a Fixed Point Problem. . . . . . . 30
2.4 Link-Flow Traffic Assignment as a Fixed Point Problem. . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Nash Equilibrium in a Road Network with Many Groups
of Users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Competitive Traffic Assignment in Case of Many Users’
Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Relationships of Wardrop’s Principles and Nash
Equilibrium in Case of Many Users’ Groups . . . . . . . . . . . . . . . . 49
3.3 Nash Equilibrium in Noncooperative Game of Users’
Groups on Routes of a Road Network. . . . . . . . . . . . . . . . . . . . . 55
3.4 Behavioral Model of Competitive Traffic Assignment
on a Road Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
ix
x Contents
Part III Optimization Traffic Assignment Methods
4 Methods for Traffic Flow Assignment in Road Networks . . . . . . . . . 73
4.1 Gradient Descent for User-Equilibrium Search in Road
Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Projection Approach for Route-Flow Traffic Assignment. . . . . . . . 77
4.3 Projection Approach for Link-Flow Traffic Assignment . . . . . . . . 86
4.4 Route-Flow Assignment in a Linear Network as a System
of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Parallel Decomposition of a Road Network . . . . . . . . . . . . . . . . . . . 101
5.1 Decomposition of a Road Network into Parallelized
Subnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Route-Flow Traffic Assignment in a Network with One Pair
of Source and Sink. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Route-Flow Traffic Assignment in a General Road Network . . . . . 109
5.4 Link-Flow Traffic Assignment in a General Road Network. . . . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Part IV Optimization Models and Methods for Network Design
6 Topology Optimization of Road Networks . . . . . . . . . . . . . . . . . . . . 121
6.1 Bi-level Mathematical Programming for the Optimization
of a Road Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Optimal Capacity Allocation for General Road Network. . . . . . . . 124
6.3 Optimal Capacity Allocation for Corridor-Type Road
Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4 Optimal Capacity Allocation for Corridor-Type Road Network
Under Multi-modal Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7 Optimal Transit Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.1 Optimality Criteria for a Transit Network Design . . . . . . . . . . . . . 141
7.2 Traffic Assignment in Road Networks with Transit
Subnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3 Optimality Criteria for a Transit Network Design Under
Competitive Routing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.4 Traffic Assignment in Road Networks with Transit
Subnetworks Under Competitive Routing. . . . . . . . . . . . . . . . . . . 162
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Contents xi
Part V Networking Issues
8 Transportation Processes Modelling in Congested Road
Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.1 Signal Control in a Congested Urban Area. . . . . . . . . . . . . . . . . . 179
8.2 OD-Matrix Reconstruction and Estimation Based
on a Dual Formulation of Traffic Assignment Problem . . . . . . . . . 185
8.3 Emission Reduction Due to Traffic Reassignment. . . . . . . . . . . . . 190
8.4 Time-Dependent Vehicle Routing in a Congested Urban
Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9 Load Flow Estimation in a Transmission Network. . . . . . . . . . . . . . 205
9.1 Multi-supplier and Multi-consumer Power Grid System . . . . . . . . 205
9.2 Competition of Consumers in Smart Grid Systems . . . . . . . . . . . . 212
9.3 Integrated Smart Energy System . . . . . . . . . . . . . . . . . . . . . . . . . 217
9.4 Pricing Mechanisms in Multi-generator and Multi-consumer
Power Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228