Optimization models and methods for equilibrium traffic assignment 9783030341015, 9783030341022

Citation preview

Springer Tracts on Transportation and Traffic

Alexander Krylatov Victor Zakharov Tero Tuovinen

Optimization Models and Methods for Equilibrium Traffic Assignment

Springer Tracts on Transportation and Traffic Volume 15

Series Editor Roger P. Roess, New York University Polytechnic School of Engineering, Brooklyn, NY, USA

About this Series The book series “Springer Tracts on Transportation and Traffic” (STTT) publishes current and historical insights and new developments in the fields of Transportation and Traffic research. The intent is to cover all the technical contents, applications, and multidisciplinary aspects of Transportation and Traffic, as well as the methodologies behind them. The objective of the book series is to publish monographs, handbooks, selected contributions from specialized conferences and workshops, and textbooks, rapidly and informally but with a high quality. The STTT book series is intended to cover both the state-of-the-art and recent developments, hence leading to deeper insight and understanding in Transportation and Traffic Engineering. The series provides valuable references for researchers, engineering practitioners, graduate students and communicates new findings to a large interdisciplinary audience. ** Indexing: The books of this series are submitted to SCOPUS and Springerlink **

More information about this series at http://www.springer.com/series/11059

Alexander Krylatov Victor Zakharov Tero Tuovinen •

•

Optimization Models and Methods for Equilibrium Traffic Assignment

123

Alexander Krylatov Institute of Transport Problems Russian Academy of Sciences Saint Petersburg, Russia

Victor Zakharov Faculty of Applied Mathematics and Control Processes Saint Petersburg State University Saint Petersburg, Russia

Faculty of Applied Mathematics and Control Processes Saint Petersburg State University Saint Petersburg, Russia Tero Tuovinen Faculty of Information Technology University of Jyväskylä Jyväskylä, Finland

ISSN 2194-8119 ISSN 2194-8127 (electronic) Springer Tracts on Transportation and Traffic ISBN 978-3-030-34101-5 ISBN 978-3-030-34102-2 (eBook) https://doi.org/10.1007/978-3-030-34102-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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 efficient solution algorithms for the problem are based on its structural features and practical meaning rather than standard nonlinear optimization techniques or approaches. Therefore, careful consideration on the meaning basis of a traffic assignment problem for efficient algorithm development seems to be of high importance. Nevertheless, the vast majority of books on this topic provide common traffic equilibrium models as well as general description of respective methods as a rule. This book focuses on discussion of the traffic assignment problem as well as the mathematical and practical meaning of variables, functions, and basic principles in 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 constructive criticism and suggestions. The results obtained here are expected to be desired both for practice and theory. The first author is sincerely grateful for the support of this work by a grant of the 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 Saint Petersburg, Russia Jyväskylä, Finland

Alexander Krylatov Victor Zakharov Tero Tuovinen

vii

Contents

Part I

Traffic Flow Issues

1 Introduction . . . . . . . . . . . . . . . . . . . . . 1.1 Brief Traffic Theory Background . . . 1.2 Current Remarkable Research Issues 1.3 Promising Relevant Expansion . . . . References . . . . . . . . . . . . . . . . . . . . . . . Part II

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

3 3 5 8 10

......

17

......

17

. . . .

. . . .

24 30 36 43

.......

45

.......

45

.......

49

.......

55

....... .......

59 69

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Optimization Traffic Assignment Models

2 Principles of Wardrop for Traffic Assignment in a Road Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 User Equilibrium and System Optimum of Wardrop in a Road Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dual Traffic Assignment Problem in a Large Scale Road Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Route-Flow Traffic Assignment as a Fixed Point Problem . 2.4 Link-Flow Traffic Assignment as a Fixed Point Problem . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nash Equilibrium in a Road Network with Many Groups of Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Competitive Traffic Assignment in Case of Many Users’ Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Relationships of Wardrop’s Principles and Nash Equilibrium in Case of Many Users’ Groups . . . . . . . . . 3.3 Nash Equilibrium in Noncooperative Game of Users’ Groups on Routes of a Road Network . . . . . . . . . . . . . . 3.4 Behavioral Model of Competitive Traffic Assignment on a Road Network . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

ix

x

Part III

Contents

Optimization Traffic Assignment Methods

4 Methods for Traffic Flow Assignment in Road Networks . . . 4.1 Gradient Descent for User-Equilibrium Search in Road Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Projection Approach for Route-Flow Traffic Assignment . . 4.3 Projection Approach for Link-Flow Traffic Assignment . . 4.4 Route-Flow Assignment in a Linear Network as a System of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

......

73

...... ...... ......

73 77 86

...... ......

94 99

5 Parallel Decomposition of a Road Network . . . . . . . . . . . . . . . 5.1 Decomposition of a Road Network into Parallelized Subnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Route-Flow Traffic Assignment in a Network with One Pair of Source and Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Route-Flow Traffic Assignment in a General Road Network . 5.4 Link-Flow Traffic Assignment in a General Road Network . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part IV

. . . . 101 . . . . 101 . . . .

. . . .

. . . .

. . . .

106 109 113 118

Optimization Models and Methods for Network Design

6 Topology Optimization of Road Networks . . . . . . . . . . . . . . . . . 6.1 Bi-level Mathematical Programming for the Optimization of a Road Network Topology . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Optimal Capacity Allocation for General Road Network . . . . . 6.3 Optimal Capacity Allocation for Corridor-Type Road Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Optimal Capacity Allocation for Corridor-Type Road Network Under Multi-modal Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Optimal Transit Network Design . . . . . . . . . . . . . . . . . . . . 7.1 Optimality Criteria for a Transit Network Design . . . . . . 7.2 Traffic Assignment in Road Networks with Transit Subnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Optimality Criteria for a Transit Network Design Under Competitive Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Traffic Assignment in Road Networks with Transit Subnetworks Under Competitive Routing . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 121 . . . 121 . . . 124 . . . 127 . . . 133 . . . 139

. . . . . . . 141 . . . . . . . 141 . . . . . . . 149 . . . . . . . 156 . . . . . . . 162 . . . . . . . 176

Contents

Part V

xi

Networking Issues

8 Transportation Processes Modelling in Congested Road Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Signal Control in a Congested Urban Area . . . . . . . . . . . 8.2 OD-Matrix Reconstruction and Estimation Based on a Dual Formulation of Traffic Assignment Problem . . 8.3 Emission Reduction Due to Traffic Reassignment . . . . . . 8.4 Time-Dependent Vehicle Routing in a Congested Urban Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . 179 . . . . . . . 179 . . . . . . . 185 . . . . . . . 190 . . . . . . . 197 . . . . . . . 202

9 Load Flow Estimation in a Transmission Network . . . . . . . . . 9.1 Multi-supplier and Multi-consumer Power Grid System . . . 9.2 Competition of Consumers in Smart Grid Systems . . . . . . . 9.3 Integrated Smart Energy System . . . . . . . . . . . . . . . . . . . . 9.4 Pricing Mechanisms in Multi-generator and Multi-consumer Power Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

205 205 212 217

. . . . . 220 . . . . . 228

Part I

Traffic Flow Issues

The development of models and methods for a traffic assignment problem, on one hand, produces immediate practical results, while on the other, leads to new techniques and approaches for coping with different nonlinear optimization problems. Therefore, the development of this scientific field contributes both in theory (mathematical modelling, operations research, constrained nonlinear optimization, bi-level optimization etc.) and practice (decision-making support in transportation, in-vehicle routing guide systems, transportation planning systems, etc.). Thus, the issues raised in this book definitely deserve wide comprehensive investigation.

Chapter 1

Introduction

Abstract A road network is a large scale system with huge amounts of interdependent elements such as streets, junctions, traffic lights, traffic flows, and the like. Deep understanding of the nature of existing dependencies between the elements of the road network could offer decision makers and managers of different levels significant support in the transportation sphere. Thus, the Introduction chapter is devoted to a brief review on traffic theory development, and it illuminates important research directions in the field to be taken under additional careful investigation. The major attention is paid to problems concerning traffic assignment. The discussion tends to cover both practical and theoretical aspects. Essentially, the Introduction is committed to specify a general line of the book in a short way so as possible.

1.1 Brief Traffic Theory Background Over the past one hundred years the traffic flow theory has been permanently developing. This is primarily due to the fact that the private car takes a more and more significant place in the daily life of each person. People use cars to go to work, to the city, to the theater, and finally just to the store. Therefore, it is natural that such a widespread use of the car by a large number of people leads to significant loads on the elements of road networks of the cities. This is especially noticeable in large cities, where it is typical that a large number of people are moving from one area to another at the same time of the day. Thus, it is reasonable to talk about the existence of regular time-traffic flows between origin-destination areas or OD-matrices. Such flows influence most significantly on the traffic congestions in the road networks. In this regard, it is appropriate to raise the question about the methods for traffic management in conditions of a natural limitation of road networks. Traffic flow theory began to develop 100 years ago. Its mathematical foundations were made in 1912 by professor G.D. Dubelir [1]. That time, the capacity analysis of highways and intersections was under investigation. and later mathematical models based on the probability theory appeared. Such models, as a rule, described the behavior of traffic flows on local streets or intersections, and their implementation © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_1

3

4

1 Introduction

on the entire network had no practical result. In 1934 B.D. Greenshield established the leader-follower model [2]. Such a model is now called microscopic, since solely two vehicles of the whole traffic flow are under consideration at each moment. In 1955 the first macroscopic model appeared in the paper of M.J. Ligthill and F.R.S. Whitham, in which traffic flow was considered as the flow of fluid [3]. In the beginning of 1960s, F. Haight wrote a book which collected all the results obtained up to that time [4]. This book is believed to be the first step on the path of traffic flow theory appearing as a branch of applied mathematics. In 1952 J.G. Wardrop stated two principles of traffic assignment [5]. According to the first principle, “the journey times in all routes actually used are equal and less than those that would be experienced by a single vehicle on any unused route”, while according the second principle “the average journey time is at a minimum”. The first mathematical formulation of these two principles was obtained by M.J. Beckmann, etc. [6]. Subsequently, this mathematical model has become a classic and today it is one of the key positions in traffic flow theory [7–9]. Herewith, it should be mentioned that the traffic assignment problem has a form of an optimization problem if and only if the performance time of a link is dependent on the flow of this link solely. In the 1960 and 1970s the previously obtained results were extended. Note that the traffic assignment model based on the first principle of Wardrop belongs to the class of so-called behavior models [10]. However, approaches exist for traffic flow estimation in a road network, which are not based on the route choice concept. For the first time a set of heuristic rules for traffic flow estimation was proposed by G.V. Sheleykhovskiy in 1936 [11]. The proof of convergence of the Sheleykhovskiy’s method for the problem with transportation constraints appeared in 1967 [12]. In 1970 Wilson defined the entropy approach for traffic flow estimation [13]. Actually, Sheleykhovskiy’s method is an entropy-type method. Fundamentally new formulations of a network equilibrium problem in a form of variational inequalities and the complementarity problem were proposed in the early 1980 [14–16]. The applicability of the network equilibrium model was expanded significantly, since new formulations allowed the use of arbitrary performance function. Therefore, from that moment, traffic flow estimation could be done in case of an arbitrary link performance function, even in case of its complex dependence on the flow of all others links. On one hand, researchers were faced with non-potential mapping, but the models became more applicable. Moreover, variational inequalities or the complementarity problem could not be solved by ordinary optimization tools, but there are special approaches. New ideas in the traffic flow theory appeared in the end of 1990s due to B.S. Kerner, and they were eventually formed in 2009 [17–21]. Kerner identified three phases of traffic flow. His results are based on ideas of microscopic and macroscopic models, while the theory of three phases is rather a description of flow stages than new approach for modeling. Nevertheless, the theory of three phases is useful from a meaningful perspective. Moreover, from the articles written in the recent years we should mention, first of all, the papers of I.V. Konnov, devoted to network equilibrium [22, 23]. Konnov developed a fundamentally new interpretation of the network equilibrium as an

1.1 Brief Traffic Theory Background

5

auction model [22]. He showed that network flow equilibrium problems are particular cases of a general auction market model with divisible commodities and price functions of participants.

1.2 Current Remarkable Research Issues Transportation networks have been an important point of attention for researchers all around the world since the beginning of the 20th century. Such interest is caused by two aspects: (1) the practical significance of the research; (2) rich theoretical contributions. The main directions of research in this field at the moment are • • • •

traffic evaluation, OD-matrix estimation, urban road network design, transit network design.

It is emphasized that the present book is devoted to optimization models and methods for traffic assignment. In other words, it is assumed that travel time on any link solely depends on its flow. Advantages of such models are widely discussed in [9]. One of the main challenges for decision-makers at different levels of management in modern worldwide large cities is coping with enormous traffic jams on their road networks. Authorities facing such complicated problems are forced to implement various quite expensive arrangements. On one hand they tend to put new infrastructure facilities into operation, and on the other—try to reorganize road traffic. As a result, the continuously growing travel demand encourages the development of advanced methodological tools and technological innovations to meet the newly emerging requirements. The need for innovations is felt especially in the area of traffic signal control as soon as numerous signalized junctions of the road networks contribute most significantly by alternating traffic lights. Extra complexity is added by the intricacy of large scale transportation networks and their inner nontrivial coherence. Therefore several researchers focus on the optimization of signal control settings with certain and uncertain travel demand [24]. However, motorization seems to be difficult to limit, thus authorities are forced to design optimal road networks [25]. Investments in this direction will be the more effective, the more accurate is the prediction of demand in the long term. To predict the dynamics of traffic flows, approaches for OD-matrix estimation and traffic assignment are used [26]. The main idea in the sphere of traffic assignment is based on two principles of Wardrop [5]. Indeed, these two principles are actively used in such IT-systems as PTV Vision and TransCad. Despite the fact that the principles of Wardrop are well investigated, new extensions and clarifications appear [27]. However, in-vehicle routing guide systems significantly influence the behavior of drivers today. Therefore, new game-theoretical approaches for traffic flow assignment have appeared [28].

6

1 Introduction

Remarkable that in-vehicle routing guide systems are presented by different business companies then it leads to competitive behavior of many groups of network’s users. Meanwhile, a steadily growing number of customers using such systems leads to an increasing influence on groups’ behavior on eventual traffic assignment. Indeed, navigation providers suggest routes for their customers and directly influence traffic flows assignment online [73]. Nevertheless, there is a lack of methodological tools for modeling the competitive behavior of users’ groups. Traffic assignment models based on game theory principles were investigated by Altman et al. [29–31]. Included in this book results concerning game of navigation providers have no analogues and they contribute in developing traffic assignment models [32]. As a rule, navigation providers choose routes for their customers based only on available information about a current road situation regardless of the actions of other navigation providers. However, the travel time on a route selected by a navigation provider depends not only on its flow, but also on the flows which are guided through this route by other navigation providers. In this case, a reasonable model of competitive routing is a non-cooperative game of navigation providers with the Nash equilibrium as the optimality principle. A Nash equilibrium profile is a set of players strategies such that no player benefits by a unilateral deviation from its strategy. As it is well-known, whenever a Nash equilibrium profile appears unique, players do not need a third party for their action coordination for its implementation. We refer to this situation as the competitive routing problem: several navigation providers operate on a network and each of them seeks the best traffic flows assignment for its customers (e.g. by suggesting them the fastest routes) [29, 30, 33]. Of course, the correlation of Nash and Wardrop equilibrium strategies attracts the interest of researchers. Other powerful tools to impact the traffic flows assignment belong to metropolitan administrations. Indeed, administrative traffic control can be implemented indirectly via network design or transportation planning. Herewith, during transportation planning one has to take into consideration the whole road network, but not try to solve local problems. The division of transportation planning on a system approach and a local approach can be found in different sources [34]. The methodology for traffic flow management and network design is quite well developed today. Indeed, a special class of problems already exists called the Urban Transportation Network Design Problem (UTNDP) [35]. The UTNDP consists of strategic, tactical and operational decisions [36]. Moreover, traffic assignment is deeply dependent on the topology of a road network. Thus, the search for the optimal topology of a road network is an important problem [34]. Therefore the UTNDP, being very multifaceted, can be associated with the following list of issues [35]: 1. Construction of roads or increasing capacity of the streets [37]. 2. Optimal public facilities allocation in a road network [25]. 3. Organization of decision-making hierarchy in transportation planning sphere including strategic, tactical and operational decisions [36]. Such problems are often formulated as bi-level optimization programs [28]. Commonly, the upper level reflects the decision-making of a city authority, which is able to influence the capacity of the network (reconstruction, modernization building of

1.2 Current Remarkable Research Issues

7

roads etc.). Authority tends to minimize the average travel time in a network. The lower level reflects the behavior of drivers, who react to the capacity changing. Each driver seeks to minimize their own travel time from origin to destination. Thus, there is competition between drivers. Herewith, the goal function of the upper level corresponds to the second principle of Wardrop, while the goal function of the lower level corresponds to the second principle of Wardrop. Special attention should be paid to the fact that the implementation of wide tools for the mathematical modeling of traffic processes deeply depends on accurate and full information about traffic demand. All models include the volume of traffic demand as input and are highly sensitive to its changes. Thus, the efficient implementation of mathematical models requires accurate information about ODmatrices. OD-matrix estimation, in turn, is a hard mathematical problem that has been under investigation of researchers for the past 40 years [38–41]. A number of methods and approaches exist for OD-matrix estimation. One of the most efficient methods is based on information from plate scanning sensors [38] or RFID scanning sensors. Indeed, the information obtained from such sensors can be used to reconstruct the routes that vehicles use [42, 43]. In addition, authorities of large modern cities are occasionally faced with challenges in the transportation sphere concerning the creation of special conditions for a certain type of vehicles. For instance, the authorities are interested in encouraging the use of environmentally friendly vehicles on transportation networks because green vehicles decrease total greenhouse gas emissions. Thus, appropriate arrangements should be performed to motivate drivers to use green vehicles instead of gasolinepowered vehicles. To achieve this goal, transit networks designed for green-vehicle routing could prove to be an effective method. However, the question is how to offer the green vehicles attractive trip conditions. Another example is a transit network with toll prices. In this case the special type of vehicles is vehicles with drivers who are ready to pay the toll price. The general question is how to offer the certain type of vehicles the most attractive trip conditions. Because information regarding the amount of these certain vehicles currently on the road is believed to be available, the question could be reformulated quantitatively as to how many routes should be available for only these vehicles to use. We call such routes specialized routes in contrast to common routes. Considering this network design, if the specialized routes are not fully loaded while the common routes are overloaded, the transportation network is unbalanced [44]. Conversely, if the specialized routes are overloaded by the current amount of special vehicles on the road, using special cars will not provide a significant advantage for their drivers. Thus, it is necessary to determine the conditions that guarantee a well-balanced allocation of specialized and common routes in a given transportation network. The above wide reasoning shows us with evidence that the development and proof of theoretical principles for mathematical methods and algorithms which could be used for solving important social and economic problems in a sphere of traffic control and road network design in large cities seem to be of high importance. Most attention should be paid to traffic flow assignment, special methods for nonlinear constrained optimization, duality, and bi-level optimization.

8

1 Introduction

1.3 Promising Relevant Expansion In 1847 G.R. Kirchhoff gave a topological-type proof that the currents in the wires are uniquely determined for wires obeying linear Ohm’s law (the current and potential drop are proportional) [45]. By virtue of this result, a stable state of currents in a network must satisfy Kirchhoff’s laws, which are simply the conservation of electricity and energy. Two concise expressions of these laws were discovered by Maxwell: the junction equation and the mesh equation [46]. Duffin established that a network of nonlinear conductors has a stable state of currents, and this state is unique [47]. These results were obtained through the mathematical programming formulations of the problem of finding currents and voltage in electrical networks. In fact, formulation of a mechanical analog to electric networks was developed. Further mechanical analogies of an electrical network were investigated by [48, 49]. The rapid development of the energy market encouraged the appearance of many suppliers and the electric power transmission systems were faced with great demands throughout the world. Such circumstances have led to the stimulation of several researchers to cope with the corresponding challenges. Due to their simplicity, the first optimization models were addressing an economic dispatch problem. This problem is to define the optimal energy consumption from several suppliers, and is associated with the optimal power flow problem that was first formulated in 1979 [50]. Then the fundamentals of power generation, control and operation were gathered, widely discussed, and developed [51]. However, the physical and engineering properties of a transmission network could contribute to total costs significantly, while the network topology influences the optimal production and use of electricity. The first investigation of the pricing pattern depending on both production and transmission of electricity was made in [52]. The classical model of a “peak load pricing” problem was extended in four ways to reflect the transmission and distribution costs [53]. This investigation intended to promote the economic efficiency in the use of a whole electric power system. Since electrical networks could not be managed directly by imposing the route choices on the currents, researchers concentrated on investigations of the optimal management of power generation, physical network characteristics, generation and transmission pricing. Principles of short-run transmission pricing developed in [52] have got the further profound development with respects to certain properties of power transmission systems [54]. Under transmission constraints the economic dispatch was defined as the maximization of the benefits less the costs subject to the availability of electric plants and characteristics of the transmission network [55, 56]. Due to this definition and the corresponding mathematical formulations the new opportunities for power market organization were clearly recognized by researchers. Authors of [57] drew attention to the fact that the dispatching of transmission is deeply associated with market competition in the electric power industry. An owner of some part of a transmission network is able to bid capacity that connects consumers with the energy supplier. Thus, a consumer has a set of alternative suppliers which are switched in the network by virtue of transmission owners and, hence, the transmission

1.3 Promising Relevant Expansion

9

expansion relies on market forces [58]. Indeed, the market can decide what combination of financial rights (options, obligations and futures) is the most useful to power generation, consumption, and transmission [59]. Dispatchable transmission under the rules of the Regional Transmission Organization was considered in [59]. The dispatch method for the electricity markets consisting of wholesale markets and retail markets was proposed in [60]. Eventually, it seems reasonable to divide the power market into electricity and capacity markets: the electricity market deals with variable production costs, while the capacity market attracts capital investments [61]. In recent decades, the structure of power supply has been changing drastically. The development of communication and computation technologies, ever-growing energy consumption, as well as increasing penetration of alternative energy sources lead the world to a new vision of power grid architecture. These modern grids should incorporate new facilities, such as renewable or local energy generators, energy storage systems, electric vehicles etc., while maintaining and improving the stability of the network. One of the crucial problems in power grid management is the reduction of imbalance between energy demand and supply. The consumption of energy changes over the course of a day, and conventional energy generators cannot respond to these changes instantly. This forces us to either waste energy in the case of overproduction, or to use expensive auxiliary energy generators, such as gas turbines or diesel generators. The increasing use of renewable energy sources makes this problem even more challenging due to the uncontrollable, weather-dependent nature of these types of generation methods [62]. Hence, the power supply system should turn to production-oriented consumption, which leads us to the idea of demand side management (DSM). Much work has been done in the field of demand side management and, more specifically, demand response. Basic pricing mechanisms for systems with a single supplier and multiple consumers are studied in [63, 64]. The distributed generation and storage are considered in [65, 66]. Coalition formation for local networks with the use of methods of cooperative game theory is examined in various scenarios in [67, 68]. Generalized equilibria in dynamic multi-leader-follower games are studied in [69–71]. However, the questions of the power grid’s topology and congestion in transmission lines are not taken into account in these works. The maintenance of the transmission system is crucial to avoid overloads and blackouts in the grid. Therefore, the study of demand side management should also consider the distribution of energy flows in the network. It is important to mention that routing methods from the transportation theory cannot be applied directly to power grids because of the special nature of electricity [72]. Electrical flows are distributed according to the Kirchhoff’s rules, and one cannot route these flows arbitrarily. Nevertheless, the classical power grid model can be presented as a model of single-commodity non-atomic routing [9]. In our case, consumers have their specific energy demands and pursue the goal of cost minimization. Therefore, we can formulate a competitive game of consumers, and this network setting accepts the implementation of the non-atomic routing methods with a single source and a single destination. The corresponding models were studied in [29–31] and have been developed in [73–75].

10

1 Introduction

In the last chapter of this book we merge different ideas and techniques of demand response, game theory and transportation theory, and develop a new gametheoretic model of energy consumption by multiple agents. A non-cooperative game of consumers is formulated under energy transmission constraints. These constraints strongly depend on the topology of a network. Thus, we formulate a corresponding flow assignment problem, while taking into account similarities between the current distribution in power grids and non-atomic routing in transportation networks. The obtained results seem to be promising.

References 1. Semenov VV (2004) Matematicheskoe modelirovanie dinamiki transportnykh potokov megapolisa [Mathematical modelling of traffic flow dynamics in a megalopolis area]. Keldysh Institute of Applied Mathematics, Moscow 2. Greenshields BD (1934) A study of traffic capacity. In: Proceedings (US) highway research board, vol 14, pp 448–494 3. Ligthill MJ, Whitham FRS (1995) On kinetic waves II. A theory of traffic flow on crowded roads. Proc. Royal Soc Ser A. 229(1178):317–345 4. Haight FA (1963) Mathematical theories of traffic flow. Academic Press Inc., New York 5. Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc Inst Civil Eng 2:325–378 6. Beckmann MJ, McGuire CB, Winsten CB (1956) Studies in the economics of transportation. Yale University Press, New Haven 7. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc., Englewood Cliffs, NJ 8. Cascetta E (2009) Transportation systems analysis: models and applications. Springer, New York 9. Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc., New York 10. Di X, Liu HX (2016) Boundedly rational route choice behavior: a review of models and methodologies. Transp Res Part B: Methodol 85:142–179 11. Sheleikhovskii GV (1936) Transportnye osnovaniya kompozitsii gorodskogo plana [Transportation basis of urban plan composition]. Giprogor, Leningrad 12. Bregman LM (1967) Dokazatel’stvo skhodimosti metoda G.V. Sheleikhovskogo dlya zadachi s transportnymi ogranicheniyami [The proof of Sheleikhovskii method convergence for the problem with transportation constraints]. Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki 7(1):147–156 13. Wilson AG (1970) Entropy in Urban and regional modelling. Pion, London 14. Bershchanskii YM, Meerov MV (1983) Teoriya i metodi resheniya zadach dopolnitel’nosti [Theory and methods for solving complementary problems]. Avtomatika i telemekhanika 6:5– 31 15. Dafermos S (1980) Traffic equilibria and variational inequalities. Transp Sci 14:42–54 16. Smith MJ (1979) The existence, uniqueness and stability of traffic equilibria. Transp Res Part B 13:295–304 17. Kerner BS (1998) Experimental features of self-organization in traffic flow. Phys Rev Lett 81(20):3797–3800 18. Kerner BS (2009) Introduction to modern traffic flow theory and control: the long road to three-phase traffic theory. Springer, Berlin 19. Kerner BS, Rehborn H (1996) Experimental features and characteristics of traffic jams. Phys Rev E 53(2):1297–1300

References

11

20. Kerner BS, Rehborn H (1996) Experimental properties of complexity in traffic flow. Phys Rev E 53(5):4275–4278 21. Kerner BS, Rehborn H (1997) Experimental properties of phase transitions in traffic flow. Phys Rev Lett 79(20):4030–4033 22. Konnov IV (2015) On auction equilibrium models with network applications. Netnomics 16:107–125 23. Konnov IV (2013) Vector network equilibrium problems with elastic demands. J Global Optim 57:521–531 24. Chiou S-W (2014) Optimization of robust area traffic control with equilibrium flow under demand uncertainty. Comput Oper Res 41:399–411 25. Friesz TL (1985) Transportation network equilibrium, design and aggregation: key developments and research opportunities. Transp Res Part A 19:413–427 26. Krylatov AY, Shirokolobova AP, Zakharov VV (2016) OD-matrix estimation based on a dual formulation of traffic assignment problem. Informatica (Slovenia) 40(4):393–398 27. Boyce D (2007) Future research on urban transportation network modeling. Reg Sci Urban Econ 37:427–481 28. Hollander Y, Prashker JN (2006) The applicability of non-cooperative game theory in transport analysis. Transportation 33:481–496 29. Altman E, Basar T, Jimenez T, Shimkin N (2002) Competitive routing in networks with polynomial costs. IEEE Trans Autom Control 47(1):92–96 30. Altman E, Combes R, Altman Z, Sorin S (2011) Routing games in the many players regime. In: Proceedings of the 5th international ICST conference on performance evaluation methodologies and tools, pp 525–527 31. Altman E, Kameda H (2001) Equilibria for multiclass routing in multi-agent networks. In: Proceedings of the IEEE conference on Decision and Control, vol 1, pp 604–609 32. Krylatov AY, Zakharov VV, Malygin IG (2016) Competitive traffic assignment in road networks. Transp Telecommun 17(3):212–221 33. Orda A, Rom R, Shimkin N (1993) Competitive routing in multiuser communication networks. IEEE/ACM Trans Netw 1(5):510–521 34. Xie F, Levinson D (2009) Modeling the growth of transportation networks: a comprehensive review. Netw Spat Econ 9:291–307 35. Farahani RZ, Miandoabchi E, Szeto WY, Rashidi H (2013) A review of urban transportation network design problems. Eur J Oper Res 229:281–302 36. Magnanti TL, Wong RT (1984) Network design and transportation planning: models and algorithms. Transp Sci 18(1):1–55 37. Dantzig GB, Harvey RP, Lansdowne ZF, Robinson DW, Maier SF (1979) Formulating and solving the network design problem by decomposition. Transp Res Part B 13(1):5–17 38. Castillo E, Menedez JM, Jimenez P (2008) Trip matrix and path flow reconstruction and estimation based on plate scanning and link observations. Transp Res Part B 42:455–481 39. Shen W, Wynter L (2012) A new one-level convex optimization approach for estimating origindestination demand. Transp Res Part B 46:1535–1555 40. Simonelli F, Marzano V, Papola A, Vitiello I (2012) A network sensor location procedure accounting for o-d matrix estimate variability. Transp Res Part B 46:1624–1638 41. Tong CO, Wong SC (2000) A predictive dynamic traffic assignment model in congested capacity-constrained road networks. Transp Res Part B 34:625–644 42. Castillo E, Menendez JM, Sanchez-Cambronero S (2008) Traffic estimation and optimal counting location without path enumeration using Bayesian networks. Comput Aided Civil Infrastruct Eng 23(3):189–207 43. Li X, Ouyang Y (2011) Reliable sensor deployment for network traffic surveillance. Transp Res Part B 45:218–231 44. Beltran B, Carrese S, Cipriani E, Petrelli M (2009) Transit network design with allocation of green vehicles: a genetic algorithm approach. Transp Res Part C 17:475–483 45. Kirchhoff G (1847) Ueber die aufloesung der gleichungen, auf welcheman bei der untersuchung der linearen vertheilung galvanischer stroeme gefuehrt wird. Annalen der Physik und Chemie 72:497–508

12

1 Introduction

46. 47. 48. 49.

Maxwell JC (1873) A treatise on electricity and magnetism. The Clarendon Press, Oxford Duffin R (1947) Nonlinear networks. IIa. Bull Am Math Soc 53:963–971 Saishu K, Moriwaki Y (1972) Optimal assignment of traffic flows. Electr Eng Jpn 92:113–120 Sasaki T, Inouye H (1974) Traffic assignment by analogy to electric circuits. In: Transportation and traffic theory, Proceedings of the 6th international symposium on transportation and traffic theory, Sydney, pp 495–518 Carpentier J (1979) Optimal power flows. Int J Electr Power Energy Syst 1(1):3–15 Wood AJ, Wollenberg BF (1984) Power generation, control, and operation. Wiley, New York Bohn RE, Caramanis MC, Schweppe FC (1984) Optimal pricing in electrical networks over space and time. Rand J Econ 15(3):360–376 Crew MA, Kleindorfer PR (1979) Public utility economics. St. Martins Press, New York Schweppe FC, Caramanis MC, Tabors RD, Bohn RE (1988) Spot pricing of electricity. Kluwer Academic Publishers, Norwell Chowdhury BH, Rahman S (1990) A review of recent advances in economic dispatch. IEEE Trans Power Syst 5(4):1248–1259 Hogan WW (1992) Contract networks for electric power transmission. J Regul Econ 4:211–242 Chao H-P, Peck S (1996) A market mechanism for electric power transmission. J Regul Econ 10(1):25–59 Hogan WW (1999) Market-based transmission investments and competitive electricity markets. In: John F. Kennedy (ed) School of government, Harvard University, Cambridge, Massachusetts 02138 (Technical report) O’Neill RP, Baldick R, Helman U, Rothkopf MH, Stewart WR (2005) Dispatchable transmission in RTO markets. IEEE Trans Power Syst 20(1):171–179 Yang J, Zhang G, Ma K (2016) Hierarchical dispatch using two-stage optimisation for electricity markets in smart grid. In J Syst Sci 47(15):3529–3536 Dolmatova M (2015) Optimal nodal capacity procurement. Adv Intell Sys Comput 360:415– 423 Ziser CJ, Dong Z, Wong K (2012) Incorporating weather uncertainty in demand forecasts for electricity market planning. Int J Syst Sci 43(7):1336–1346 Mohsenian-Rad A-H, Wong VW, Jatskevich J, Schober R, Leon-Garcia A (2010) Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid. IEEE Trans Smart Grid 1(3):320–331 Veit A, Xu Y, Zheng R, Chakraborty N, Sycara KP (2013) Multiagent coordination for energy consumption scheduling in consumer cooperatives. In: 27th AAAI conference on artifficial intelligence, pp 1362–1368 (2013) Atzeni I, Ordóñez LG, Scutari G, Palomar DP, Fonollosa JR (2013) Demand-side management via distributed energy generation and storage optimization. IEEE Trans Smart Grid 4(2):866– 876 Atzeni I, Ordóñez LG, Scutari G, Palomar DP, Fonollosa JR (2013) Noncooperative and cooperative optimization of distributed energy generation and storage in the demand-side of the smart grid. IEEE Trans Signal Process 61(10):2454–2472 Alam M, Ramchurn SD, Rogers A (2013) Cooperative energy exchange for the efficient use of energy and resources in remote communities. In: Proceedings of the 2013 international conference on autonomous agents and multi-agent systems, pp 731–738 Mihailescu R-C, Vasirani M, Ossowski S (2011) Dynamic coalition adaptation for efficient agent based virtual power plants. In: Multiagent system technologies, pp 101–112 Nie P-Y (2007) Discrete time dynamic multi-leader-follower games with feedback perfect information. Int J Syst Sci 38(3):247–255 Nie P-Y (2011) Dynamic discrete-time multi-leader-follower games with leaders in turn. Comput Math Appl 61(8):2039–2043 Pang J-S, Fukushima M (2005) Quasi-variational inequalities, generalized Nash equilibria, and multileader-follower games. Comput Manag Sci 2(1):21–56 Kok J (2013) The power matcher: smart coordination for the smart electricity grid. Vrije Universiteit Press, Amsterdam

50. 51. 52. 53. 54. 55. 56. 57. 58.

59. 60. 61. 62. 63.

64.

65.

66.

67.

68. 69. 70. 71. 72.

References

13

73. Zakharov VV, Krylatov AY (2016) Competitive routing of traffic flows by navigation providers. Autom Remote Control 77(1):179–189 74. Zakharov V, Krylatov A (2014) Equilibrium assignments in competitive and cooperative traffic flow routing. IFIP Adv Inf Commun Techn 434:613–620 75. Zakharov V, Krylatov A, Ivanov D (2013) Equilibrium traffic flow assignment in case of two navigation providers. IFIP Adv Inf Commun Tech 408:156–163

Part II

Optimization Traffic Assignment Models

Equilibrium flow assignment models offer an efficient tool for describing a wide range of networking processes. The development of such models indeed contributes significantly to the mathematical modeling of real networks. Current available results in the field allow us to describe the selfish behavior of drivers in road networks but not limited. Raised models are turned out to be applicable for different economical assessments, such as the estimation of marginal costs deviation or evaluation of price of anarchy in competitive multi-agents systems. Nevertheless, this field still includes practically important but unresolved issues.

Chapter 2

Principles of Wardrop for Traffic Assignment in a Road Network

Abstract In this chapter is devoted to user equilibrium and system optimum of Wardrop. Discussion on the mathematical formulation of traffic assignment problems with regard to their meaning is available in the Sect. 2.1. The specification of necessary basic statements completes this discussion further. The dual traffic assignment problem with travel times between all origins and destinations as dual variables is considered in the Sect. 2.2. The practical significance of such dual formulation is shown to become evident due to the wide spread of online traffic services. The route-flow assignment problem and link-flow assignment problem are reduced to fixed-point problems with explicit operators in the Sect. 2.3 and Sect. 2.4 respectively. Proofs of corresponding theorems are fully given.

2.1 User Equilibrium and System Optimum of Wardrop in a Road Network J.G. Wardrop formulated two principles of traffic assignment in 1952 [1]. According to the first principle, the journey times on all of the actually used routes are equal, and less than those which would be experienced by a single vehicle on any unused route between each OD-pair. Therefore, Wardrop’s first principle describes an equilibrium traffic assignment pattern, called user equilibrium. According to the second principle the average journey time is a minimum in a network at all. Thus, Wardrop’s second principle describes the optimal traffic assignment pattern, called the system optimum. Nowadays, the traffic assignment problem is a highly urgent research topic from both practical and theoretical perspectives. An increasing scale of networks (transportation, telecommunication etc.) require breakthrough methodological tools for their analysis and design. That is why numerous researchers all around the world are dealing with a traffic assignment problem [2, 3]. Consider a network, presented by a connected directed graph G = (V, E) consisting of sequentially numbered vertices and sequentially numbered edges [4]. Let us introduce: V is the set of the sequentially numbered vertices of G; E is the set of the sequentially numbered edges of G; W is the set of vertices pairs of G (an origin and a destination), w ∈ W ; R w is the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_2

17

18

2 Principles of Wardrop for Traffic Assignment in a Road Network

set of routes between an origin-destination pair w ∈ W ; xe is the traffic flow through the edge e ∈ E, x = (. . . , xe , . . .); xew is the traffic flow through the edge e ∈ E between an OD-pair w, x w = (. . . , xew , . . .); ce is the capacity of the edge e ∈ E, c = (. . . , ce , . . .); frw is the traffic flow through the route r ∈ R w ; F w is the traffic demand between a given OD-pair w ∈ W ; te (xe ) is a smooth nondecreasing function that models the travel time (delay) of the flow xe through the congested edge e ∈ E; w is an indicator: δe,r  w δe,r

=

1, if the edge e ∈ Elies along the route r ∈ R w ; 0, otherwise.

User equilibrium in a road network is such an assignment of the traffic demand F w between available routes R w that the journey times on all the actually used routes are equal and less than those which would be experienced by a single vehicle on any unused route for each OD-pair w ∈ W [1]: 

 te (xe ) ·

w δe,r

e∈E

= tw , frw > 0, ≥ tw , frw = 0,

∀ r ∈ Rw ,

(2.1)

where tw is travel time through any actually used route between OD-pair w ∈ W , when the traffic flow is assigned according to the first principle of Wardrop. The traffic assignment problem was formulated for the first time in [5]. The user equilibrium estimation for a general network in the form of a constrained nonlinear optimization problem was proposed in [6, 7]: min x

subject to



 e∈E

xe

te (u)du,

(2.2)

0

frw = F w ∀w ∈ W,

(2.3)

frw ≥ 0 ∀r ∈ R w , w ∈ W

(2.4)

r ∈R w

with definitional constraints xe =

  w∈W

w frw δe,r ∀e ∈ E.

(2.5)

r ∈R w

Theorem 2.1 ([4, 8]) Solution x ∗ to the optimization problem (2.2)–(2.5) is the user equilibrium of Wardrop in the network G.

2.1 User Equilibrium and System Optimum of Wardrop in a Road Network

19

Proof Lagrangian for the problem (2.2)–(2.5) is L=

 e∈E

xe

 te (u)du + t

w

0

F − w



 frw

r ∈R w

+

  − frw ηrw , r ∈R w

where tw and ηrw ≥ 0, r ∈ R w , w ∈ W are multipliers of Lagrange. According to Kuhn-Tucker, the partial derivatives of the Lagrangian with respect to x in the point x ∗ must be equal to zero. If we substitute (2.5) with (2.2), then route-flows ( frw ) could be considered as variables. Therefore, the partial derivatives of the Lagrangian with respect to frw , r ∈ R w , w ∈ W , in the point x ∗ must be equal to zero as well:  ∂ xe ∂L = te (xe ) · w − tw − frw = 0 ∀r ∈ R w , w ∈ W. w ∂ fr ∂ fr e∈E

(2.6)

Taking into account (2.5) we obtain: ∂ xe w = δe,r ∀e ∈ E, r ∈ R w , w ∈ W, ∂ frw that leads (2.6) to  ∂L w = te (xe ) · δe,r − tw − frw = 0 ∀r ∈ R w , w ∈ W. ∂ frw e∈E

(2.7)

The complementary slackness condition requires that the equalities frw · ηrw = 0 be true. In this case, if frw > 0, then ηrw = 0, while if frw = 0, then ηrw ≥ 0. Consequently  e∈E

 w te (xe ) · δe,r

= tw , if frw > 0, ≥ tw , if frw = 0,

∀ r ∈ Rw .

Therefore, the solution of (2.2)–(2.5) indeed satisfies the requirement (2.1). Hence, the obtained solution is the user equilibrium of Wardrop by definition. For demonstrative purposes consider a network of non-interfering (parallel) routes (Fig. 2.1). Such a network is a particular case of a corridor network that could be called a corridor-type network with n off-ramps [9]. A network of non-interfering (parallel) routes is a directed graph G, which consists of two vertices and n edges (routes). The traffic demand between n source and sink is F, while the traffic assignf i = F. The travel time through any edge is ment pattern is f i ≥ 0, i = 1, n: i=1 a smooth nondecreasing function: ti ∈ C 1 (R + ), ti (x) − ti (y) ≥ 0 when x − y ≥ 0 x, y ∈ R + , i = 1, n, where R + is a non-negative orthant. Moreover, it is believed that ti (x) ≥ 0, x ≥ 0 and ∂ti (x)/∂ x > 0, x > 0, i = 1, n. The user equilibrium for a network of non-interfering routes could be formulated as follows. The user equilibrium in a network of non-interfering routes is such an

20

2 Principles of Wardrop for Traffic Assignment in a Road Network

Fig. 2.1 The network of non-interfering (parallel) routes

assignment of traffic demand F between routes f ∗ = ( f 1∗ , . . . , f n∗ ), that the journey times on all the actually used routes are equal and less than those which would be experienced by a single vehicle on any unused route: ti (

f i∗ )



= t∗ , if f i∗ > 0, ≥ t∗ , if f i∗ = 0,

where t∗ is travel time through any actually used route, when traffic demand is assigned according to the first principle of Wardrop. Considering a network of non-interfering routes allows for getting a clear insight into the first principle of Wardrop. Figure 2.2 demonstrates the part of Saint Petersburg road network near “Park Pobegy” metro station. Common sense dictates that the green and orange routes seem to be equal for moving (quite close time delays), while the red route requires much more time for moving between source and sink. The implementation of the user equilibrium concept (2.1) allows the removal of the red route from investigation as obviously unused for this OD-pair. Therefore, Wardrop’s first principle seems to be an adequate tool for traffic assignment estimation. Note that there is another formulation of the traffic assignment problem (2.2), (2.3) in the form of a nonlinear optimization problem. Because of its form, the problem (2.2), (2.3) is often called link-flow formulation. Alternatively, link-node formulation is   xe te (u)du, (2.8) min x

subject to

 e1 ∈V p

xew1 −

e∈E



0

xew2 = d p , ∀ p ∈ V,

xew ≥ 0, ∀e ∈ E, w ∈ W, with definitional constraints

(2.9)

e2 ∈W p

(2.10)

2.1 User Equilibrium and System Optimum of Wardrop in a Road Network

(a)

21

(b)

Fig. 2.2 Non-interfering routes in the road network of the city. a Two routes. b Three routes

xe =



xew , ∀e ∈ E,

(2.11)

w∈W

where V p is the set of edges terminating at vertix p, W p is the set of edges initiated at vertix p, and ⎧ ⎨ −F w , if vertix p is the source in a pair w ∈ W, w d p = F w , if vertix p is the sink in a pair w ∈ W, ⎩ 0, otherwise.

(2.12)

Remarkable is that the formulations (2.2)–(2.5) and (2.8)–(2.11) are not equal, however, the following theorem establishes the relationships between them. Theorem 2.2 ([10], Flow decomposition theorem) Every route and cycle flow has a unique representation as non-negative link flows. Conversely, every non-negative link flow may be represented as a route and cycle flow (though not necessary uniquely). Since cycles are impossible in formulations (2.2)–(2.5), we could conclude that the feasible set of solutions to (2.8)–(2.11) includes the feasible set of solutions to (2.2)– (2.5), according to Theorem 2.2. A system optimum in a road network is such assignment of traffic demand F w between available routes R w that the average journey time is a minimum [1]. The traffic assignment problem was formulated for the first time in [5]. The system optimum search of a general network in the form of a constrained nonlinear optimization problem was proposed in [6, 7]: x • = arg min x

 e∈E

te (xe )xe ,

(2.13)

22

2 Principles of Wardrop for Traffic Assignment in a Road Network

subject to



frw = F w ∀w ∈ W,

(2.14)

frw ≥ 0 ∀r ∈ R w , w ∈ W

(2.15)

r ∈R w

with definitional constraints xe =

  w∈W

w frw δe,r ∀e ∈ E.

(2.16)

r ∈R w

Theorem 2.3 ([4, 8]) Solution x • to the optimization problems (2.13)–(2.16) is a system optimum in the network G. Proof Lagrangian for the problem (2.13)–(2.16) is L=

 e∈E

 te (xe )xe + m

w

F − w

 r ∈R w

 frw

+

  − frw ηrw , r ∈R w

where mw and ηrw ≥ 0, r ∈ R w , w ∈ W are multipliers of Lagrange. According to Kuhn–Tucker, the conditions partial derivatives of the Lagrangian with respect to x in the point x • must be equal to zero. If we substitute (2.16) with (2.13), then route-flows ( frw ) could be considered as variables. Therefore, the partial derivatives of the Lagrangian with respect to frw , r ∈ R w , w ∈ W , in the point x • must be equal to zero as well:   ∂ xe ∂te (xe ) ∂L te (xe ) + = xe − mw − frw = 0 ∀r ∈ R w , w ∈ W. w ∂ frw ∂ x ∂ f e r e∈E (2.17) Taking into account (2.16) we obtain ∂ xe w = δe,r ∀e ∈ E, r ∈ R w , w ∈ W, ∂ frw that leads (2.17) to   ∂L ∂te (xe ) w te (xe ) + = xe δe,r − mw − frw = 0 ∀r ∈ R w , w ∈ W. (2.18) ∂ frw ∂ x e e∈E The complementary slackness condition requires that the equalities frw · ηrw = 0 be true. In this case, if frw > 0, then ηrw = 0, while if frw = 0, then ηrw ≥ 0. Consequently    ∂te (xe ) = mw , if frw > 0 w te (xe ) + xe δe,r ≥ mw , if frw = 0 ∂ xe e∈E

∀ r ∈ Rw .

(2.19)

2.1 User Equilibrium and System Optimum of Wardrop in a Road Network

23

Therefore, the solution to (2.13)–(2.16) satisfies (2.19). Note that in the left part of (2.19), a mathematical expression of marginal cost is found. The equivalence of marginal costs on all routes between pair w ∈ W means that a minimum average journey time is achieved. Hence, the obtained solution is system optimum by definition. Evidently, in a general case the user equilibrium x ∗ and system optimum x • could not be equal. An assignment that is optimal from the perspective of road network performance could not reflect the route choice strategies of a driver, who actually tries to minimize his/her own travel time. Selfish routing of drivers leads to competition between them in the use of a road network. Therefore, the user equilibrium concept is employed for modeling selfish routing. Note that there is another formulation of the traffic assignment problem (2.13)– (2.16) in the form of a nonlinear optimization problem. Because of its form, the problem (2.13)–(2.16) is often called link-flow formulation. Alternatively, link-node formulation is  te (xe )xe , (2.20) min x

subject to

 e1 ∈V p

e∈E



xew1 −

xew2 = d p , ∀ p ∈ V,

(2.21)

e2 ∈W p

xew ≥ 0, ∀e ∈ E, w ∈ W,

(2.22)

with definitional constraints xe =



xew , ∀e ∈ E,

(2.23)

w∈W

where V p is the set of edges terminating at vertix p, W p is the set of edges initiated at vertix p, and ⎧ ⎨ −F w , if vertix p is source in pair w ∈ W, w d p = F w , if vertix p is sink in pair w ∈ W, ⎩ 0, otherwise.

(2.24)

It should be emphasized in the end that it is the flows on edges that are taken as variables of optimization problems described in this section. A traffic assignment problem with flows on edges as variables is called a link-flow assignment problem. Further, we consider traffic assignment problems with flows on routes as variables. We call such kinds of problems route-flow assignment problems.

24

2 Principles of Wardrop for Traffic Assignment in a Road Network

2.2 Dual Traffic Assignment Problem in a Large Scale Road Network Let us consider a network of general topology presented by directed graph G = (V, E). We use the following notation: W is the set of OD-pairs, w ∈ W , W ⊂ V ; R w is the set of routes connecting an OD-pair w; F w is the traffic demand for T  an w, F = F 1 , . . . , F |W | ; frw is the traffic flow on the route r ∈ R w , OD-pair w w w w w r ∈R w f r = F ; f = { f r }r ∈R w and f = { f }w∈W ; x e is the traffic flow on the w is the arc e ∈ E, x = (. . . , xe , . . .); te (xe ) is the travel cost on the arc e ∈ E; δe,r indicator: 1 if e is included in the route r , 0 otherwise. What follows from Sect. 2.1 is that the user equilibrium in a road network G is reached by such pattern x ∗ that x ∗ = arg min x

subject to



 e∈E

xe

te (u)du,

(2.25)

0

frw = F w , ∀w ∈ W,

(2.26)

r ∈R w

frw ≥ 0, ∀w ∈ W,

(2.27)

with definitional constraints xe =

 

w frw δe,r , ∀e ∈ E.

(2.28)

w∈W r ∈R w

According to the first principle of Wardrop, an equilibrium journey time tw exists for any OD-pair w. Equilibrium journey time tw is the journey time on any actually used route between w under the user equilibrium assignment in a road network. Let us prove that tw are Lagrange multipliers in the optimization program (2.25)–(2.28). Lemma 2.1 tw is the Lagrange multiplier in the optimization program (2.25)–(2.28) corresponding to the constraint (2.26). Proof The Lagrangian of the problem (2.25)–(2.28) is L=

 e∈E

0

xe

te (u)du +

 w

 μw F − w

 r ∈R w

 frw

+



  ηrw − frw ,

w r ∈R w

where μw and ηrw ≥ 0 are Lagrangian multipliers, and differentiation of the Lagrangian yields:  ∂L = te (xe ) − μw − ηrw = 0. ∂ frw e∈r

2.2 Dual Traffic Assignment Problem in a Large Scale Road Network

25

Note that according to complementary slackness ηrw frw = 0. Thus, for frw > 0 the following expression holds 

te (xe ) = μw , ∀r ∈ R w , w ∈ W.

(2.29)

e∈r

Actually, the left part of (2.29) is the journey time on any used route ( frw > 0) between the OD-pair w. In other words μw = tw for any w ∈ W . Eventually, according to the Lemma the following equality is true: tw =



te (xe∗ ) ∀r ∈ R w , w ∈ W.

e∈r

Lemma 2.1 reveals an important meaningful sense of Lagrange multipliers corresponding to (2.26). Efficient applications could be developed due to the dual traffic assignment problem with these Lagrange multipliers as a set of dual variables. For instance, if we know the travel time of a vehicle on any alternative route between an OD-pair, the traffic demand of this OD-pair can be estimated. Indeed, let us describe this idea on a simple case. Let us consider a linear network of parallel routes (Fig. 2.1) to demonstrate the practical significance of a dual traffic assignment problem. We use the following notation: F is the traffic demand between n an OD-pair; f i is the traffic flow on the f i = F; ti ( f i ) = ai + bi f i is the travel route i, i = 1, n, f = ( f 1 , . . . , f n ), i=1 time on the route i, i = 1, n, ai ≥ 0 and bi > 0. As shown earlier, the traffic assignment problem for such a network is f ∗ = arg min f

subject to

n 

n   i=1

fi

ti (u)du,

(2.30)

0

f i = F,

(2.31)

i=1

f i ≥ 0 ∀i = 1, n.

(2.32)

The user equilibrium of Wardrop for a network of parallel routes is reached by such assignment f ∗ = ( f 1∗ , . . . , f n∗ ) of demand F that ti ( f i∗ )



= t∗ > 0 when f i∗ > 0, > t∗ when f i∗ = 0,

i = 1, n.

The following theorem provides an efficient algorithm for traffic demand estimation when equilibrium journey time is known. Moreover, due to the simplicity of the investigated network, deep insight becomes available.

26

2 Principles of Wardrop for Traffic Assignment in a Road Network

Without a loss of generality we assume that routes are numbered as follows: a1 ≤ . . . ≤ an . Theorem 2.4 Traffic demand F for a linear network of parallel routes can be obtained explicitly: F = t∗

k k   1 as − , b b s=1 s s=1 s

(2.33)

where k satisfies a1 ≤ . . . ak < t∗ ≤ ak+1 . . . ≤ an .

(2.34)

Proof According to Lemma 2.1, the equilibrium journey time t∗ is the Lagrangian multiplier that corresponds to the restriction (2.31) of the optimization program (2.30)–(2.32). Since goal function (2.34) is convex, the Kuhn-Tucker conditions are both necessary and sufficient. Let us introduce Lagrange multipliers ηi ≥ 0, i = 1, n for (2.34). The Lagrangian for the optimization problem (2.30)–(2.32) is L=

n   i=1

fi

 ti (u)du + t∗ F −

0

n 

 fi

+

i=1



ηi (− f i ) .

(2.35)

i

The derivative of Lagrangian (2.35) at variable f i has to be equal to zero: t∗ = ti ( f i ) − ηi . The complementary slackness condition states that ηi · f i = 0. This equation holds when at least one of the variables is zero. Thus, if f i > 0, then ηi = 0 and t∗ = ti ( f i ) = ai + bi f i .

(2.36)

However, if f i = 0, then ηi ≥ 0 and t∗ = ti ( f i ) − ηi = ai − ηi . Hence, if ai ≥ t∗ , then f i = 0. On the contrary, if we express f i in terms of t∗ from (2.36) we get fi =

t∗ − ai . bi

2.2 Dual Traffic Assignment Problem in a Large Scale Road Network

27

Therefore, if f i > 0, then t∗ > ai . Thus we are able to define the set of actually used routes (routes with non-zero flows): fi =

 t∗ −ai bi

0,

ai < μ, i = 1, n. ai ≥ μ,

(2.37)

Since routes numbered in such a way that a1 ≤ . . . ≤ an , there exists route k such that a1 ≤ . . . ≤ ak < t∗ ≤ ak+1 ≤ . . . ≤ an . According to (2.31) and (2.37) we obtain n 

fi =

k  μ − as s=1

i=1

bs

=F

and, consequently, F=

k  t∗ − a s s=1

bs

.

Theorem 2.4 states that if we know the travel time of a vehicle on any alternative route between an OD-pair, the appropriate traffic demand can be uniquely reconstructed. Due to such results, the developed approach seems to be promising. The main idea of this method is based on the first principle of Wardrop as follows: if we define the journey time of a vehicle on any actually used route between a certain OD-pair, we believe that the journey time on all other used routes is the same. Note that t∗ could be easily defined between any OD-pair in a real road network. Indeed, online services exist which collect information about the average speed on each arc of a road network. For example, “Yandex.Traffic” based on “Yandex.Maps” reflects the current road situation online (Fig. 2.3). Due to the large scale databases this service is able to make statistical predictions for different time periods and different days of the week. Moreover, the average travel speed through any arc of a road network is in the public domain (Fig. 2.4). Therefore, we can easily determine travel time through the whole route between any pair of origin and destination, using the information about average speed on the arcs. We believe that road traffic is assigned according to the first Wardrop principle. Thereby, if we estimate the travel time through the shortest route between an OD-pair then we obtain t ∗ for this OD-pair. The establishment of an explicit relationship between equilibrium journey time and traffic demand is a nontrivial task for a network of general topology. However, the following theorem allows us to establish at least a formal relationship between them. Introduce T = (. . . , tw , . . .), when w ∈ W .

28

2 Principles of Wardrop for Traffic Assignment in a Road Network

Fig. 2.3 The traffic in Saint-Petersburg city

Fig. 2.4 Data from Yandex.Maps

Theorem 2.5 A dual problem exists for the traffic assignment problem (2.25)–(2.28), which has the following form: max Ξ (T ) where Ξ (T ) is defined by

(2.38)

2.2 Dual Traffic Assignment Problem in a Large Scale Road Network

Ξ (T ) = min



f ≥0

e∈E

xe

te (s)ds +

0



 t

F −

w

w



29

 ,

frw

(2.39)

r ∈R w

w

subject to definitional constraints xe =

 

w frw δe,r , ∀e ∈ E.

(2.40)

r ∈R w

w∈W

Proof We assume that x ∗ is a solution of (2.25)–(2.28). According to Lemma 2.1, tw are Lagrangian multipliers for the constraints (2.26). The Lagrangian for the problem (2.25)–(2.28) is L(x, T ) =

 e∈E

xe

te (s)ds +

0



 t

w

F − w



 frw

+

r ∈R w

w

 w

(− frw )ηrw .

r ∈R w

If L(xe , tw ) has a saddle point (x ∗ , T ∗ ) in an admissible set, then x ∗ is the solution of the problem (2.25)–(2.28), and T ∗ is the solution of the following optimization problem [11]: max L(x ∗ , T ). T

This duality relation holds when the theorem of equivalence is satisfied [11]. Due to the simple form of a linear network of parallel routes, the relationships between equilibrium journey time and traffic demand were defined explicitly (Theorem 2.4). However, it could also be done through dual formulation of the traffic assignment problem: t∗ = arg max ξ(t), t

where ξ(t) is defined as  ξ(t) =

fi

min

f i ≥0, i=1,n

0

 ti (u)du + t

F−

n 

 fi

.

i=1

Note that the dual traffic assignment problem could be efficiently used for ODmatrix estimation. Section 7.2 is devoted to the OD-matrix estimation technique based on insight of tw and the dual traffic assignment problem. The results which were considered here were obtained for the first time by authors in [12].

30

2 Principles of Wardrop for Traffic Assignment in a Road Network

2.3 Route-Flow Traffic Assignment as a Fixed Point Problem The first fixed point formulations of the traffic assignment problem appeared in [13, 14]. However, the obtained results are very general. Special methods to solve fixed point formulations of the traffic assignment problem have not been developed [8]. The contribution of this section is a special technique that reduces the traffic assignment problem on a network of parallel routes to the fixed point problem with a contraction operator. Moreover, in Sect. 3.2 the fixed-point iteration appears, which is proved to converge geometrically with a contractive operator; under some fairly natural conditions the fixed-point iteration is proved to converge quadratically. The technique contributes to the development of parallel decomposition algorithms which is shown in Sects. 4.2 and 4.3 [8]. Such algorithms represent a multi-commodity network as a set of single-commodity subnetworks and, solve the traffic assignment problem on these subnetworks simultaneously [15, 16]. The user equilibrium problem (2.2)–(2.5) for a network of parallel routes has the following form: n  fi  ti (u)du, (2.41) min f

subject to

i=1

n 

0

f i = F,

(2.42)

i=1

f i ≥ 0.

(2.43)

Let us introduce the additional notations: ai (x) = ti (x) −

dti (x) dti (x) x and bi (x) = , i = 1, n. dx dx

Theorem 2.6 ([17]) The optimization problem (2.41)–(2.43) is equivalent to the fixed point problem f = Φ(f), (2.44) where the components f and t ( f ) = (t1 ( f 1 ), . . . , tn ( f n )) are indexed so that a1 ( f 1 ) ≤ · · · ≤ an ( f n ), and the components Φ(f) = (Φ1 ( f ), . . . , Φn ( f ))T look like

(2.45)

2.3 Route-Flow Traffic Assignment as a Fixed Point Problem

 Φi ( f ) =

m s=1 1 F+ m bi ( f i ) s=1 b

as ( f s ) bs ( f s ) 1 s ( fs )

−

ai ( f i ) , bi ( f i )

31

for i ≤ m,

(2.46)

for i > m,

0,

where m is defined from the condition m  am ( f m ) − ai ( f i )

bi ( f i )

i=1

≤F
0 then ηi = 0 and, by (2.51), we have all i = 1, qi j j j [ai + bi f i ]δ Ii = t. On the other hand, if f i = 0 then ηi ≥ 0 and, tˆi ( f i ) = j=1 qi j j j by (2.51), we have tˆi ( f i ) = j=1 [ai + bi f i ]δ Ii ≥ t. These correlations can be rewritten as the condition tˆi ( f i ) =

 qi    = t, if f i > 0, j j j ai + bi f i δ Ii ≥ t, if f i = 0,

i = 1, n.

(2.52)

i = 1, n.

(2.53)

j=1

Express f i in terms of t using (2.52): fi =

⎧ ⎨ qi ⎩ 0,

j=1



j

t−ai j bi



j

δ Ii , if if

qi

j

j

j

j

j=1

ai δ Ii ≤ t,

j=1

ai δ Ii > t,

qi

The functions tˆi ( f i ) for all i = 1, n are convex, and so, the Khun–Tucker conditions are both sufficient and necessary. Therefore, we can say that the assignment f ∗ creates a user equilibrium in the network of parallel routes if and only if there is t∗ such that f ∗ and t∗ satisfy (2.52) and, therefore, (2.53). II. Assume that f ∗ and t∗ are defined. Then each f i∗ , i = 1, n, corresponds to a j∗ j∗ j∗ fixed interval Ii i and the coefficients ai i and bi i correspondingly. Without loss of generality, let us reindex the edges so that j ∗

∗

a11 ≤ · · · ≤ anjn .

(2.54)

j ∗

∗

m+1 . Then, inserting Let m be the index of the edge such that amm ≤ t∗ < am+1 (2.53) into (2.49), we have

n  i=1

fi =

  m j∗  t∗ − ai i j∗

i=1

bi i

j

= F.

(2.55)

2.3 Route-Flow Traffic Assignment as a Fixed Point Problem

33

Therefore, t∗ can be expressed as t∗ =

m j∗ ai F + i=1 m 1 i .

(2.56)

i=1 b ji ∗ i

Put (2.56) into (2.53) to obtain f ∗ explicitly: f i∗

=

⎧ ⎨ ⎩

m j ∗ ai 1 F+ m i=1 1 i j∗ bi i=1 ji ∗ bi

−

j∗

ai

j∗ bi

, if i ≤ m,

i = 1, n.

if i > m,

0,

j ∗

(2.57) ∗

m+1 Find the value of m using (2.55) and the inequality amm ≤ t∗ < am+1 :

 j ∗  m j∗  amm − ai i j∗

bi i

i=1

≤F
m,

0,

where m is defined from the condition m m+1   am+1 − ai am − ai m,

(2.68)

where m is defined by the condition m 

γi ( f i )[ϕm ( f m ) − ϕi ( f i )] ≤ F
ATn−k Ak Gk (xk∗ )ATk

(2.76)

Proof For the ease of explanation, we split the proof into four parts. I. Apply the Khun–Tucker conditions to the optimization problem (2.71)–(2.73) by differentiating the Lagrangian L=

n   i=1

xi

ti (u)du +

0

m  j=1

 ωj bj −

n 

 a ji xi

+

i=1

n 

(−ηi )xi

i=1

with respect to xi and equating the result to zero. The optimal solution of (2.71)– (2.73) x∗ and corresponding Lagrange multipliers ω∗ = (ω1∗ , . . . , ωm∗ )T , ηi∗ ≥ 0, i = 1, n satisfy  ∂ L(xi∗ ) = ti (xi∗ ) − ω∗j a ji − ηi∗ = 0, ∀i = 1, n, ∂ xi j=1 m

or ti (xi∗ ) =

m 

ω∗j a ji + ηi∗ , ∀i = 1, n.

(2.77)

j=1

The complementary slackness condition requires that the equalities ηi∗ xi∗ = 0 n. In this case, if xi∗ > 0 then ηi∗ = 0 and, by (2.77), we have for all i = 1, m ∗ ti (xi ) = j=1 ω∗j a ji . On the other hand, if xi∗ = 0 then ηi∗ ≥ 0 and, by (2.77), we have ti (xi∗ ) ≥ mj=1 ω∗j a ji . Therefore, the correlations between x∗ and ω∗ can be rewritten as the condition:

2.4 Link-Flow Traffic Assignment as a Fixed Point Problem

 ti (xi∗ )

= ≥

m

∗ j=1 ω j a ji ,

if xi∗ > 0,

∗ j=1 ω j a ji ,

if xi∗ = 0,

m

or in a matrix form

39

∀i = 1, n,

(2.78)

tk (xk∗ ) = ATk ω∗ ,

(2.79)

∗ ) ≥ ATn−k ω∗ , tn−k (xn−k

(2.80)

 T where tk (xk∗ ) = t1 (x1∗ ), . . . , tk (xk∗ ) is a set of ti (xi∗ ) such that xi∗ > 0 for i = 1, k, and matrix Ak consists of the columns of matrix A corresponding to  T ∗ ∗ ) = tk+1 (xk+1 ), . . . , tn (xn∗ ) is a set of ti (xi∗ ) such xi∗ . Analogically tn−k (xn−k such that xi∗ = 0 for i = k + 1, n, and An−k consists of the columns of matrix A corresponding to such xi∗ . The functions ti (xi ), i = 1, n, are convex, and so, the Khun–Tucker conditions are both sufficient and necessary. Therefore, we can say that the assignment x∗ is the solution of (2.71)–(2.73) if and only if there is ω∗ such that x∗ and ω∗ satisfy (2.78) and, therefore, (2.79), (2.80). II. Let us approximate the following smooth nondecreasing functions ti (xi ) ∈ C 1 (R1+ ) for xi ≥ 0 by piecewise linear functions: ti (xi ) ≈ tˆi (xi ) =

qi    j j j di + ci xi δ Ii ∀i = 1, n, j=1



where j

δ Ii =

1, if xi ∈ I j , 0, otherwise,

q

and Ii = (Ii1 , . . . , Ii i ) is the set of intervals corresponding  to the linear segments  of the piecewise linear functions tˆi (xi ) with precision ε: ti (xi ) − tˆi (xi ) < ε for all xi ≥ 0, i = 1, n. We now formulate a constrained optimization problem (2.71)–(2.73) when the delay functions are piecewise linear functions ˆ x) = min ˆ x∗ ) = min G(ˆ G(ˆ xˆ

xˆ

n   i=1

0

xˆi

qi    j j j di + ci u δ Ii du,

(2.81)

j=1

subject to Aˆx = b,

(2.82)

xˆ ≥ O.

(2.83)

40

2 Principles of Wardrop for Traffic Assignment in a Road Network

Due to Kuhn–Tucker conditions we have the following correlation between the optimal solution xˆ ∗ = (xˆ1∗ , . . . , xˆn∗ )T of (2.81)–(2.83) and Lagrange multipliers ωˆ ∗ = (ωˆ 1 , . . . , ωˆ m ) corresponding to restriction (2.82) (directly from (2.78)): tˆi (xˆi∗ ) =

qi   j=1

 m ∗  = j=1 ωˆ j a ji , if xˆi∗ > 0, j j j di + ci xˆi∗ δ Ii ≥ mj=1 ωˆ ∗j a ji , if xˆi∗ = 0,

∀i = 1, n.

Express xˆi∗ in terms of ωˆ ∗j , j = 1, m: xˆi∗

=

⎧ ⎨ qi ⎩

m

j=1

j ˆ ∗j a ji −di j=1 ω j ci

 j

δ Ii , if

0,

if

qi

j j j=1 di δ Ii

qi

j

j

≤

j=1 di δ Ii >

m

ˆ ∗j a ji , j=1 ω

m

ˆ ∗j a ji , j=1 ω

(2.84)

i = 1, n. x The functions 0 i tˆi (u)du for any i = 1, n are convex, and so, the Khun–Tucker conditions are both sufficient and necessary. Therefore, we can say that the assignment xˆ ∗ is a solution of (2.81)–(2.83) if and only if there is ωˆ ∗ such that xˆ ∗ and ωˆ ∗ satisfy (2.78) and (2.84). III. Assume that xˆ ∗ and ωˆ ∗ are defined. Then each xˆi∗ , i = 1, n, corresponds to a j∗ j∗ j∗ fixed interval Ii i and the coefficients di i and ci i correspondingly. Introduce the additional notation: ⎛

⎞

⎛ j1 ∗ ⎞ ⎛ jk+1 ∗ ⎞ d1 dk+1 ⎟ ⎜ . . ⎜ .. ⎟ .. ⎟ , Σ = ⎜ .. ⎟ , Σ . . Δk = ⎜ = ⎝ . ⎠ ⎝ . ⎠. k n−k . . ⎠ ⎝ . ∗ 1 j j ∗ 0 · · · jk ∗ dk k dnn c 1 j ∗ c11

··· 0

k

Without loss of generality, assume that the columns of matrix A are renumbered so that (2.85) Σk ≤ ATk ωˆ ∗ , Σn−k > ATn−k ωˆ ∗ ,

(2.86)

then, according to (2.84), we have xˆ k∗ = Δk ATk ωˆ ∗ − Δk Σk , ∗ = O. xˆ n−k

(2.87)

Then, inserting (2.87) into (2.82), we obtain Aˆx∗ = Ak xˆ k∗ = Ak Δk ATk ωˆ ∗ − Ak Δk Σk = b.

(2.88)

2.4 Link-Flow Traffic Assignment as a Fixed Point Problem

41

Δk is a diagonal matrix and rank(Δk ) = k. Therefore, if rank(Ak ) = m, then k ≥ m and a rang of Ak Δk ATk is m (see [20]). Since the order of Ak Δk ATk is m × m, the inverse matrix exists, and from (2.88) we obtain: −1    ωˆ ∗ = Ak Δk ATk b + Ak Δk Σk .

(2.89) j∗

j∗

Inserting (2.89) into (2.87) we obtain correlations between xˆ ∗ and ci , ci j∗ from Ii i :  −1   b + Ak Δk Σk − Δk Σk , xˆ k∗ = Δk ATk Ak Δk ATk

(2.90)

∗ = O, xˆ n−k

under conditions retrieved from (2.85) and (2.86) (when insert (2.89)):  −1   b + Ak Δk Σk , Σk ≤ ATk Ak Δk ATk

(2.91)

 −1   b + Ak Δk Σk . Σn−k > ATn−k Ak Δk ATk

(2.92)

Therefore, the solution of the optimization problem (2.81)–(2.83) with piecewise linear functions can be expressed as (2.90), when (xˆ1∗ , . . . , xˆn∗ ) and the columns of matrix A are renumbered according to (2.91), (2.92).   ˆ  IV. Piecewise linear functions tˆi (xi ), i = 1, n are defined  so that ti (xi ) − ti (xi ) < ε  j  j j for xi ≥ 0 and, consequently, di + ci xi − ti (xi ) < ε for all xi ∈ Ii , j = 1, qi , j

j

j

∀i = 1, n. Moreover, for all i = 1, n, j = 1, qi , at the endpoints of Ii = [x˙i ; x¨i ] the value of the piecewise linear function tˆi (xi ) equals the value of the function ti (xi ) (by the definition of piecewise linear function): 

j

j

j

j

di + ci x˙i = ti (x˙i ) j j j j di + ci x¨i = ti (x¨i )

then

j

j

ci =

j

ti (x¨i ) − ti (x˙i ) j

j

x¨i − x˙i j

j

j

di = ti (x¨i ) −

j x¨i

∀ j = 1, qi , i = 1, n,

(2.93)

j

(2.94)

j

ti (x¨i ) − ti (x˙i )

j

∀ j = 1, qi , i = 1, n,

−

j x˙i

x¨i

∀ j = 1, qi , i = 1, n.

j

As ε → 0, the intervals Ii , j = 1, qi , i = 1, n, collapse to the point x˜i = j j limε→0 [x˙i ; x¨i ], and we have the following limits for (2.93) and (2.94):

42

2 Principles of Wardrop for Traffic Assignment in a Road Network j

j

lim ci =

ε→0

dti (x˜i ) ∀ j = 1, qi , i = 1, n, d xi

(2.95)

j

j

j

lim di = ti (x˜i ) −

ε→0

dti (x˜i ) j x˜ ∀ j = 1, qi , i = 1, n. d xi i

(2.96)

In item (III), we proved that the solution of (2.81)–(2.83) with piecewise linear functions can be expressed as (2.90). However, this statement holds for piecewise linear functions that approximate the initial functions to every degree of accuracy. Therefore, by the smoothness of function ti (xi ), i = 1, n, as ε → 0 and taking into account (2.95) and (2.96), the correlation (2.90) will become (2.74), and (2.91), (2.92) will become (2.75), (2.76) correspondingly. Remark 2.1 As ε → 0 in (2.93) and (2.94), the conditions (2.79) and (2.80), by (2.89), will become:  −1   tk (xk∗ ) = ATk Ak Gk (xk∗ )ATk b + Ak Gk (xk∗ )dk (xk∗ ) ,

(2.97)

 −1   ∗ b + Ak Gk (xk∗ )dk (xk∗ ) . ) ≥ ATn−k Ak Gk (xk∗ )AkT tn−k (xn−k Corollary 2.2 ([19]) Assume that x∗ is a solution of the optimization problem (2.71)– (2.73) when ti (xi ) = di + ci xi , di ≥ 0, ci > 0, i = 1, n. Integer k ≤ n, a set of k components of x and a corresponding set of k columns of A exist (without loss of generality, we assume that these components and columns are numbered in order from 1 to k) so that  −1   b + Ak Gk dk − Gk dk , xk∗ = Gk ATk Ak Gk ATk ∗ = On−k,1 , xn−k

(2.98)

in case rank(Ak ) = m and

where

 −1   dk ≤ ATk Ak Gk ATk b + Ak Gk dk ,

(2.99)

 −1   b + Ak Gk dk , dn−k > ATn−k Ak Gk ATk

(2.100)

⎞ 0 ··· 0 ⎜ 0 1 ··· 0 ⎟ c2 ⎟ ⎜ Gk = ⎜ ⎟ , ⎝ · · · · · · ... · · · ⎠ 0 0 · · · c1k k×k T T   dk = d1 , . . . , dk , dn−k = dk+1 , . . . , dn . ⎛

1 c1

2.4 Link-Flow Traffic Assignment as a Fixed Point Problem

43

In Sect. 3.3 the projection operator is obtained explicitly and the algorithm to solve the link-flow assignment problem in a single-commodity network is developed. In Sect. 4.4 this algorithm is used for coping with the link-flow assignment problem in a multi-commodity network. Stress that given in this section theorem was proved in [19]. Here we give proof of this theorem as a whole because of its important methodological sense.

References 1. Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc Inst Civil Eng 2:325–378 2. Pang J-S, Fukushima M (2005) Quasi-variational inequalities, generalized Nash equilibria, and multileader-follower games. Comput Manag Sci 2(1):21–56 3. Xie J, Yu N, Yang X (2013) Quadratic approximation and convergence of some bush-based algorithms for the traffic assignment problem. Transp Res Part B 56:15–30 4. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc., Englewood Cliffs, NJ 5. Beckmann MJ, McGuire CB, Winsten CB (1956) Studies in the economics of transportation. Yale University Press, New Haven, CT 6. Dafermos SC, Sparrow FT (1969) The traffic assignment problem for a general network. J Res Nat Bureau Stand 73B:91–118 7. Dafermos SC (1968) Traffic assignment and resource allocation in transportation networks. PhD thesis. Johns Hopkins University, Baltimore, MD 8. Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc., New York 9. Shen W, Zhang HM (2009) On the morning commute problem in a corridor network with multiple bottlenecks: its system-optimal traffic flow patterns and the realizing tolling scheme. Transp Res Part B 43:267–284 10. Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall, Englewood Cliffs, NJ 11. Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont, MA 12. Krylatov AY, Shirokolobova AP, Zakharov VV (2016) OD-matrix estimation based on a dual formulation of traffic assignment problem. Informatica (Slovenia) 40(4):393–398 13. Fisk C (1984) A nonlinear equation framework for solving network equilibrium problems. Environ Plan 16A:67–80 14. Fisk C, Nguyen S (1980) A unified approach for the solution of network equilibrium problems. Publication 169. Centre de rechercher sur les transports, Universite de Montreal, Montreal (1980) 15. Chen R-J, Meyer RR (1988) Parallel optimization for traffic assignment. Math Program 42:327– 345 16. Patriksson M (1993) A unified description of iterative algorithms for traffic equilibria. Eur J Oper Res 71:154–176 17. Krylatov AY (2016) Network flow assignment as a fixed point problem. J Appl Ind Math 10(2):243–256 18. Swamy MNS, Thulasiraman K (1981) Graphs, networks, and algorithms. Wiley, New York 19. Krylatov AY (2018) Reduction of a minimization problem for a convex separable function with linear constraints to a fixed point problem. J Appl Ind Math 12(1):98–111 20. Gantmacher F (1959) Theory of matrices. AMS Chelsea Publishing, New York

Chapter 3

Nash Equilibrium in a Road Network with Many Groups of Users

Abstract In this chapter concentrates on the relationships between individual and group behaviour of drivers in a road network. Such relationships are established by comparing the optimal routing of drivers (system optimum of Wardrop), the competitive drivers’ groups routing (Nash equilibrium), and the selfish drivers routing (user equilibrium of Wardrop). Thus, the boundary conditions for traffic assignment in a road network were recently obtained for the first time. Wide analytical discussion on the topic as well as a survey of relevant references are presented. Moreover, a new behavioural model of traffic assignment in case of simultaneous selfish and group behaviour of drivers in a road network is formulated in the last section. An explicit solution to a behavioural traffic assignment problem is offered for a singlecommodity linear network with non-interfering routes.

3.1 Competitive Traffic Assignment in Case of Many Users’ Groups The rapid development of information technologies in the past three decades leads to the emergence of different specialized telecommunication systems. Nowadays these systems are introduced in almost every field of human activity, including transportation. Without a shadow of doubt the influence of these systems on decision-making increases significantly. One could observe the impact of in-vehicle route guidance and information (IVRGI) systems on route choices in his/her daily trips. Moreover, the permanent innovative development of such systems is noticeably related to the creation of intelligent vehicles. Indeed, from a consumer perspective one of the main attributes of any intelligent vehicle is an automatic drive regime that is associated with an automatic in-vehicle route guidance system. Therefore, guidance systems seem to be an integral part of the concept of an intelligent vehicle. Actually, an automatic guidance system is a great advantage of intelligent vehicles, and not only from a consumer perspective. The traffic flow of intelligent vehicles could be automatically assigned by the central guidance system in such a way that the overall travel time of all road network users could be minimized. In other words, © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_3

45

46

3 Nash Equilibrium in a Road Network with Many Groups of Users

the system optimum could be reached in the network by imposing the optimal route choices on the users. Such an assignment of the traffic flow is often called an involuntary system optimum, unlike a voluntary system optimum [1]. In a voluntary system optimum case, after paying special charging tolls the users reached the system optimum, although initially they were tending to the user equilibrium assignment [2–4]. Here, it should be mentioned that the user equilibrium assignment is supposed to take place when all drivers try to minimize their own travel time without the support of any guidance system. As a result, the overall travel time spent by all users assigned according to the interest of each atomic user is more than the overall travel time in a system optimum case. Therefore, the central guidance system is capable of reducing congestions by imposing system optimum assignment on the intelligent vehicles. Moreover, according to [5] there is considerable government interest in the development of in-vehicle guidance systems. This interest reflects a belief that such systems could produce benefits in four ways [5]: • to improve peoples’ knowledge of the network and assist them in finding efficient routes; • to reduce unnecessary mileage, traffic volumes, and hence congestion; • to link in-vehicle guidance systems with traffic control and, perhaps, with road pricing systems; • to obtain globally more efficient routing patterns (system optimum). Therefore, governments feel possible positive effects from the implementation of in-vehicle guidance systems and try to formalize them. From the mathematical perspective, all these advantages could be expressed by the system optimum principle. Hence, all the abovementioned ways of producing benefits to the traffic systems are already discussed in the previous paragraph in a short form of the “optimizational vocabulary”. Despite the interest of governments, as a rule the major contribution in the development of guidance systems is produced by different private business companies. By virtue of the competitive structure of economics, these companies are forced to compete with each other in offering their own users better service. First of all, “better service” means less travel time from origin to destination. Thus, each company seeks to route the flows of its own users in a way that minimizes their average travel time. At the same time, others try to minimize the average travel time of their users’ routing in the same road network. Due to the described circumstances the competitive traffic assignment problem appears. The non-cooperative nature of the relations between the companies leads to a set of such optimization programs that the unknown variables of any of these programs are independent parameters in all the others. Therefore, the competitive traffic assignment problem should be formulated in a game theoretic form with Nash equilibrium search [6]. The system optimum assignment obtained by in-vehicle guidance systems with one provider is assumed to differ from the assignment imposed by the Nash equilibrium strategies of the competitive companies offering route guidance to their consumers. Thus, investigating the relationships between Wardrop’s system optimum associated with the traffic assignment problem and Nash equilibrium associated with

3.1 Competitive Traffic Assignment in Case of Many Users’ Groups

47

the competitive traffic assignment problem seems quite important. Indeed, when the flows of intelligent vehicles are large enough, the competitive guidance systems could deviate the traffic assignment from system optimum significantly. This fact should be taken into consideration by traffic engineers, transportation planners, network designers and the like in transportation modeling. Note that a huge amount of researches are already devoted to Wardrop’s principles of traffic assignment. An extensive review of existing models and methods implemented for traffic assignment evaluation was conducted by M. Patriksson in 1994 [7]. This outstanding book was republished in the beginning of 2015 without loss of its relevance [2]. In spite of the various results already obtained in this branch of applied mathematics, new investigations are still appearing to contribute to namely practically expanding numerical techniques. For instance, in the past decade much attention has been paid to a class of bush-based algorithms [8]. According to [9], it is the ability of such algorithms to solve large-scale traffic assignment problems at a high level of precision that attracts numerous researchers. On the other hand, a relatively weak development of theoretical principles concerning the traffic assignment problem has occurred since the early 1950s. Indeed, the first principle of Wardrop associated with the user equilibrium assignment states that: • The journey time on all actually used routes is equal, and less than that which would be experienced by a single vehicle on any unused route. This principle, cited from Wardrop’s article [4] and published in 1952, is still relevant and useful for the evaluation of the traffic flow assignment. In other words, traffic engineers and decision-makers at different levels of management use this principle to identify used routes and their flows in a network. Another principle of Wardrop, associated with the system optimum and offered in the same article, states that: • The average journey time is a minimum. This principle could not be used for the evaluation of the traffic flow assignment since not one driver seeks to minimize the average time. Any driver tries to minimize their own travel time and, hence, the first principle of Wardrop is more appropriate for the purpose of such evaluation. Nevertheless, the relationships between the user equilibrium and system optimum define important economical conclusions in the sphere of transportation about toll pricing on the links of a network [1]. Eventually, due to Wardrop’s second principle it is possible to evaluate the toll prices that could be charged from the users of a road network. Therefore, both of Wardrop’s principles are useful from practical perspective. However, the drawback is that they are not applicable in cases where, instead of atomic drivers, the behavior of groups of drivers (with a common group interest) should be taken into consideration. Thus, by virtue of the rapid development of guidance systems, the need for a new assignment principle is expected to increase significantly in the upcoming years. If the term “group of users” (UG) is understood as a set of all drivers following the advice of the same guidance system, then a new principle could be formulated roughly as follows:

48

3 Nash Equilibrium in a Road Network with Many Groups of Users

• Under competitive conditions the average journey time of each group of users is a minimum. Note that the explicit mention of the competitive environments in this principle is important, since without the competitive behavior of companies creating guidance systems the second principle of Wardrop is sufficient for the purpose of appropriate modeling. Further, we will associate this competitive traffic assignment principle with the competitive traffic assignment problem. The first attempt to define traffic equilibria in forms of network games was made in the late 1950s by Charnes and Cooper [10]. They described the user equilibrium flow as a non-cooperative Nash equilibrium in a game where the players are pairs of origin– destination (OD pairs) competing to minimize the travel time of their respective commodity flows. Further discussion along this line is developed by Dafermos in [11, 12]. However, these first investigations of the relationships between a Wardrop equilibrium and network games have not driven to any formal expressions. Rosenthal studied a discrete version of the user equilibrium traffic assignment problem in 1973 [13]. It should be stressed that the players are defined as individual travelers, with strategy spaces equal to their respective sets of routes available. Travelers are seeking to minimize their individual travel time, i.e. their payoff functions. The game is shown to be equivalent to a non-cooperative, pure-strategy Nash game in the traffic network. Therefore, Rosenthal was the first one who formulated the special case of a competitive traffic assignment problem as we defined it above. This case is special since each UG solely consists of one user. Devarajan extended the discrete version to the continuous case in 1981. However, like Charnes and Cooper, Devarajan also defines OD pairs as the players and considers the payoff functions [14]: ϕw (y) =

 e∈E w

ye

te (s)ds, ∀w ∈ W,

0

where W is a set of UGs, w ∈ W ; E w is a set of links included in the routes between an OD pair w; ye is the traffic flow on a congested link e; te is the travel time through a congested link e. Hence, his formulation does not correspond to the competitive traffic assignment principle. Nevertheless, it was proved that the Nash game thus defined is equivalent to a Wardrop equilibrium. In the mid-1980s, more general game formulations of traffic equilibria were offered by Fisk [15] as well as Haurie and Marcotte [16]. In the 1990s the development of computer networks motivated researchers to begin the investigation of competitive routing in multiuser communication networks [17]. According to Orda et al., a single administrative domain was no longer a valid assumption in networking. Then the communication networks shared by selfish users with their own given flow demands were considered and modeled as noncooperative games by several research groups [18–21]. Due to these researches, different properties of such systems are established, and the conditions of existence and uniqueness of the Nash equilibrium in multiuser communication networks are widely studied.

3.1 Competitive Traffic Assignment in Case of Many Users’ Groups

49

During the 2000s Altman et al. have extended the results obtained for multiuser communication networks, and then implemented them to road networks [18, 22–24]. Unlike Haurie and Marcotte, Altman et al. established the convergence of the Nash equilibrium in network games to the Wardrop equilibrium as the number of players grows under weaker convexity assumptions [22]. Therefore, they raised the question of the relationships between the Nash equilibrium in noncooperative n-person network games and the Wardrop equilibrium in the traffic assignment problem. In 2015 I.V. Konnov proposed a fundamentally new interpretation of the network flow equilibrium problem as an auction model [25]. Konnov showed that an equilibrium model deeply depends on equilibration mechanisms and information frameworks, taken as a basis. Therefore, an auction equilibrium model of Konnov differs from the Nash equilibrium model since Konnov utilizes another behavioral pattern. Therefore, the differences between Wardrop and Nash equilibria are totally reflected in the behavioral models of a participant that are used [25]. Generally, one driver could not influence a road network at all. That is why one driver does not use the information about the route choices of all others, but instead uses some generalized indicators like average traffic jams at the moment. The Nash equilibrium search in a network game with a selfish driver as a player leads us to a noncooperative game. Naturally, Wardrop’s equilibrium in such a case is not equal to Nash equilibrium [13, 14, 25].

3.2 The Relationships of Wardrop’s Principles and Nash Equilibrium in Case of Many Users’ Groups Consider a transportation network presented by directed graph G = (V, E). We assume that there is a set of OD pairs W and sets of routes R w between each OD pair w ∈ W . Moreover, we introduce the following notation: F w is the demand between w; frw is the flow of UG j through route r ∈ R w ; xe is the flow through the arc e ∈ E, x = (. . . , xe , . . .); te (x) = te (xe ) is the travel time of flow xe through a congested w is an indicator: arc e ∈ E; δe,r  w = δe,r

1, if route r ∈ R w ”includes” arc e ∈ E, 0, otherwise.

According to [2, 3, 26], the equal travel time on all actually used routes, that is less than the travel time on any unused route, could be reached by an assignment strategy obtained from the following optimization program: x

ue

= arg min x

 e∈E

0

xe

te (u)du,

(3.1)

50

3 Nash Equilibrium in a Road Network with Many Groups of Users

subject to



frw = F w ∀w ∈ W,

(3.2)

frw ≥ 0 ∀r ∈ R w , w ∈ W,

(3.3)

r ∈R w

with definitional constraints xe =

  w∈W

w frw δe,r ∀e ∈ E.

(3.4)

r ∈R w

In turn, the minimum average travel time could be reached by an assignment strategy obtained from the following optimization program: xso = arg min T (x) = min x

subject to



x



te (xe )xe ,

(3.5)

e∈E

frw = F w ∀w ∈ W,

(3.6)

frw ≥ 0 ∀r ∈ R w , w ∈ W,

(3.7)

r ∈R w

with definitional constraints xe =

 

w frw δe,r ∀e ∈ E.

(3.8)

w∈W r ∈R w

Now, consider the same transportation network presented by directed graph G = (V, E), the set of OD pairs W and the corresponding sets of routes R w , w ∈ W . According to the competitive traffic assignment principle, each group tries to assign its users on the available routes from origin to destination in such a way that their average travel time is minimum. We introduce the following notation: M = {1, . . . , m} is users’ groups (UG); F jw > 0 is the demand of UG j between w, F j = the set of j j jw F ; xe is the flow of UG j through the arc e ∈ E, x j = (. . . , xe , . . .), w∈W jw x − j = (x 1 , . . . , x j−1 , x j+1 , . . . , x m ) and xe = (xe1 , . . . , xem ); fr is the flow of UG jw jw  w jw j through route r ∈ R ; f = ( f 1 , . . . , f |R w | ) is the assignment of the flow F jw through possible routes R w ; f j = (. . . , f j,w , . . .) is the strategy of UG j, and f − j = ( f 1 , . . . , f j−1 , f j+1 , . . . , f m ); f = ( f 1 , . . . , f m ) is the set of all strategies w is an indicator: of all UGs; δe,r  w = δe,r

1, if route r ∈ R w ”includes” edge e ∈ E, 0, otherwise.

3.2 The Relationships of Wardrop’s Principles and Nash Equilibrium …

51

Each UG tries to minimize the average travel time of its own users. Therefore, the following optimization programs could be formulated for all j = 1, m [28]: ∗

x j = arg min Tmj (x) = min xj

subject to



xj



te (xe )xej ,

(3.9)

e∈E

frjw = F jw ∀w ∈ W,

(3.10)

frjw ≥ 0 ∀r ∈ R w , w ∈ W,

(3.11)

r ∈R w

with definitional constraints xej =

 

w frjw δe,r ∀e ∈ E,

(3.12)

r ∈R w

w∈W

xe =

m 

xej ∀e ∈ E.

(3.13)

j=1

Note that for each j ∈ M the set x − j is not fixed, but induced by the assignment decisions of other UGs. Therefore, we obtain a competitive traffic assignment problem that could be reformulated in the form of a noncooperative   netj j j work game with penalty functions Tm , j = 1, m: Γm M, {Fm } j∈M , {Tm } j∈M , where  j jw jw Fm = { f jw | fr ≥ 0 ∀r ∈ R w , = F jw , ∀w ∈ W }, where r ∈R w f r xej =

 

w frjw δe,r

and

w∈W r ∈R w

xe =

m 

xej .

j=1

The consideration of the competitive traffic assignment problem in a game theoretic form leads us to the Nash equilibrium search. The Nash equilibrium in the game Γm ∗ is realized by strategies xmne = (x1 , . . . , xm ∗ ) such that    ∗ ∀ j ∈ M. Tmj xmne ≤ Tmj x j , x− j

(3.14)

There are certain relationships between optimization programs (3.1)–(3.4), (3.5)– (3.8) and (3.9)–(3.13). These relationships are defined by the following Lemma: Lemma 3.1 ([27]) Ceteris paribus it is true, that • the system optimum goal function is equal to the sum of the goal functions of all UGs:  T (x) = Tmj (x) ∀x, (3.15) j∈M

52

3 Nash Equilibrium in a Road Network with Many Groups of Users

• the competitive atomic and group behavior of vehicles deviate the transportation system from the optimal state:  T (x ) so

≤ T (xue ) in an atomic case, ≤ T (xmne ) in a group case,

(3.16)

particulary, if ta (xa ) = const, there are strict inequalities:  T (xso )

< T (xue ) in an atomic case, < T (xmne ) in a group case.

(3.17)

Proof (1) Consider the sum of goal functions of all UGs: m  j=1

Tmj (x) =

m  

te (xe )xej =

j=1 e∈E



⎛ te (xe ) ⎝

e∈E

m  j=1

⎞ xej ⎠ =



te (xe )xe = T (x).

e∈E

(2) Since xso is the global minimum of T (x), then T (xso ) ≤ T (x) for any feasible x, and, particularly for xue and xmne . (3a) It is well known that the system optimum is equal to user equilibrium only if travel times te (xe ) do not depend on the value of flow [2]. Indeed, xue satisfies an equality:  w te (xue )δe,r = tw ∀r ∈ R w , w ∈ W, (3.18) e∈E

where tw is an optimal travel time through any route between w in an user equilibrium state. On the other hand, xso satisfies another equality:  e∈E

te (xso ) +

 ∂te (xso ) w δe,r = mw , ∂ xe

(3.19)

that is true for all r ∈ R w , w ∈ W , and where mw is an optimal marginal cost of any route between w in a system optimum state. Expression (3.18) could be equal to (3.19) only if  ∂te (xso ) w δe,r = 0 ∀r ∈ R w , w ∈ W, ∂ x e e∈E that means te (xe ) = const for all e ∈ E. (3b) To prove that T (xso ) < T (xne ) when te (xe ) = const, let us consider the Lagrangian of the optimization program (3.9)–(3.13) associated with any UG j = 1, m:      j j j,w jw jw L = te (xe )xe + mm fr (− frjw )ηrjw , F − + e∈E

r ∈R w

r ∈R w

3.2 The Relationships of Wardrop’s Principles and Nash Equilibrium … j,w

53

jw

where mm and ηr ≥ 0 are Lagrange multipliers. According to Kuhn–Tucker conjw ditions (for optimal fr > 0) we obtain   ∂te (xe ) j w te (xe ) + = xe − mmj,w δe,r =0 j j ∂ xe ∂ x e e∈E

∂L j

or equivalently

  ∂te (xe ) j w j,w te (xe ) + x e δe,r = mm . j ∂ xe e∈E

Moreover, since xe =

m

j

j=1

(3.20)

j

xe then ∂ xe /∂ xe = 1 and, hence,

∂te (xe )

=

j ∂ xe

∂te (xe ) ∂ xe ∂te (xe ) · = . j ∂ xe ∂ xe ∂ xe

Therefore, (3.20) could be rewritten as   ∂te (xne m ) j∗ ne w te (xm ) + xe δe,r = mmj,w ∀ j = 1, m. ∂ x e e∈E

(3.21)

Eventually, the expression (3.18) could be equal to (3.19) only if  ∂te (xne ) m

e∈E

∂ xe

w δe,r = 0 ∀r ∈ R w , w ∈ W,

that means te (xe ) = const for all e ∈ E. Since the travel time in real road networks significantly depends on congestion, one could believe that the competitive atomic and group behavior of vehicles strictly increases the average travel time. Unfortunately, Lemma 3.1 does not give any answer to the question about the ratio of xue xne . The only fact we obtained is that T (xue ) and T (xne ) are both larger than T (xso ), i.e. the atomic and group behavior of drivers increases the average travel time in a road network. Theorem 3.1 ([27]) The following inequalities hold: T (xso ) ≤ T (xmne ) ≤ T (xue ).

(3.22)

Proof The left part of (3.22) is proved by previous Lemma 3.1. Let us concentrate on the proof of the right part of (3.22). First of all, we should compare network games  j Γm and Γm+1 . According to Lemma 3.1, the equality T (x) = j∈M Tm (x) and this is true for any amount of UGs. Therefore,

54

3 Nash Equilibrium in a Road Network with Many Groups of Users

T (x) =

m 

Tmj (x) =

j=1

m+1 

j

Tm+1 (x) ∀x.

(3.23)

j=1 j

The sets of constraints of games Γm and Γm+1 satisfy the inclusion ∪mj=1 Fm ⊆ j

∪m+1 j=1 Fm+1 . Subsequently, since the minimum of the function on the we have the inequality (3.24) T (xmne ) ≤ T (xne m+1 ). Note that in a real road network |F| (the absolute amount of all vehicles) is a maximum amount of UGs. In such a case each atomic vehicle tries to minimize its own travel time. Formally, in this situation m and F j could be expressed as m = |F| and F j = 1 for all j = 1, |F|. Hence, due to (3.24) we obtain ne ) ∀m = 1, |F|. T (xmne ) ≤ T (x|F|

(3.25)

On the other hand, the set of strategies in the network game Γ|F| is a Nash equilibrium if no vehicle can do better by unilaterally changing their route. However, this is a Wardrop equilibrium by definition. Therefore, the Nash equilibrium in the network game Γ|F| and Wardrop equilibrium in an appropriate network should be equal. Nevertheless, a certain clarification should be made. Consider the optimization program (3.1)–(3.4) associated with the user equilibrium of Wardrop. The solution of this program satisfies (3.18) and, without a doubt, it is generally not integer. Therefore, if we find the Nash equilibrium in the network game Γ|F| in pure strategies, then the Nash equilibrium is not equal to the ne ) ≤ T (xue ). Nevertheless, if we require x to be inteWardrop equilibrium: T (x|F| ger in (3.1)–(3.4) and call the solution the integer user equilibrium of Wardrop, then the Nash equilibrium in pure strategies is equal to the integer Wardrop equilibne ) = T (xue ). Eventually, if we find the Nash equilibrium in Γ|F| in mixed rium: T (x|F| strategies, then the Nash equilibrium in mixed strategies can be equal to the Wardrop ne ) ≤ T (xue ). Therefore, depending on the additional conditions: equilibrium: T (x|F|    ne     ne  = T xue or T x|F| < T xue . T x|F| Corollary 3.1 ([27]) If te (xe ) = const, the following inequalities hold T (xso ) = T (x1ne ) < T (x2ne ) < . . . < T (xmne ) < ne ne < T (xm+1 ) < . . . < T (x|F| ) ≤ T (xue ).

(3.26)

Proof By virtue of (3.21): • for m UGs the following m equations hold: j = 1, m:  e∈E

te (xne m)

+

∂te (xne m) j

∂ xe

∗ xej

 w δe,k = mmj,w ,

(3.27)

3.2 The Relationships of Wardrop’s Principles and Nash Equilibrium …

55

• for m + 1 UGs the following m + 1 equations hold: j = 1, m + 1:   ∂te (xne j,w m+1 ) j ∗ ne w te (xm+1 ) + xe δe,k = mm+1 . j ∂ xe e∈E

(3.28)

ne Therefore, xmne and xm+1 could be equal only if te (xe ) = const, then due to (3.24) we obtain (3.26).

The obtained results testify that, in the case of competing UGs on the transportation network, the average travel time of traffic flows between origin and destination areas possibly increases in comparison to the case of only one UG. In other words, the competitive using of a network leads to the appearance of some increment in the average travel time. Theorem 3.1 allows us to establish some rules that characterize the border conversion of the competitive assignment (Nash equilibrium) into a noncompetitive assignment (user equilibrium and system optimum). Here they are: • if j = 1, then the Nash-equilibrium is converted into the system optimum of Wardrop, • if F j = 1 for all j = 1, m, then the Nash equilibrium in pure strategies can be converted into integer the user equilibrium of Wardrop, • if F j = 1 for all j = 1, m, then the Nash equilibrium in mixed strategies can be converted into the user equilibrium of Wardrop, • if F j → 0 for all j = 1, m, or equivalently m → ∞, then the Nash equilibrium is converted into the user equilibrium of Wardrop. Moreover, due to Corollary 3.1, one could state three important conclusions: 1. The competitive in-vehicle route guidance systems decrease the average travel time in an urban traffic area in comparison to an atomic vehicle guidance. 2. The smaller the amount of competitive in-vehicle guidance systems, the shorter the average travel time in an urban traffic area. 3. The centralized guidance system guarantees the least travel time in an urban traffic area. Therefore, since in modern worldwide cities drivers chose routes by their own even competitive guidance systems could decrease the average travel time. Consequently, from this perspective, the development of intelligent vehicles could significantly improve traffic conditions in urban areas.

3.3 Nash Equilibrium in Noncooperative Game of Users’ Groups on Routes of a Road Network The solution of (3.9)–(3.13) is a Nash equilibrium link-flow traffic assignment. Generally, it is impossible to reconstruct Nash equilibrium route flows [2]. This section is devoted to the methodology of route-flow Nash equilibrium search. The route flow

56

3 Nash Equilibrium in a Road Network with Many Groups of Users

competitive traffic assignment problem could be reduced to a nonlinear system of equations. For simplicity, let us consider the implementation of this methodology to a network of parallel routes. Consider the road network presented by digraph G, consisting of two nodes (source and sink) and n noninterfering (parallel) routes. The traffic demand between source and sink is F, the assignment of F among the routes is f i ≥ 0, i = 1, n: n f = F, f = ( f 1 , . . . , f n )T . The travel time through a route is ti ∈ C 1 (R + ), i i=1 ti (x) − ti (y) ≥ 0 when x − y ≥ 0 ∀x, y ∈ R + , i = 1, n, where R + is a set of nonnegative numbers. Moreover, we believe that ti (x) ≥ 0 ∀x ≥ 0 and ∂ti (x)/∂ x > 0 ∀x > 0, i = 1, n. Assume that F is presented by m ≥ 2 groups of the users of a network:  F = mj=1 F j . Each user group j seeks to minimize travel the time of F j from  j j source to sink: f j = f 1 , . . . , f n . Thereby, there is competition between UGs in using common routes since  the jassignment of any group depends on the assignment of all the others: f i = mj=1 f i . The competition of groups leads to a competitive equilibrium search. The competitive traffic assignment under the competition of users’ groups is ∗ j∗ j∗ such an assignment of F j , j = 1, m, through routes f j = ( f 1 , . . . , f n ), that the average travel time of any UG is minimum [27, 28]. The traffic assignment problem in case of group competition is proved to be formulated as a set of optimization problems [27]. For a network of parallel routes, one can find the following set of problems: j = 1, m: n  ∗  j T j f j = min ti ( f i ) f i , (3.29) f

subject to

n 

j

i=1

j

fi = F j ,

(3.30)

i=1 j

f i ≥ 0, i = 1, n.

(3.31)

Therefore, are faced with game   we Nash equilibrium  search in noncooperative   j n j j j j j Γm M, F j∈M , T j∈M , where F = f | f i ≥ 0 ∀i = 1, n, i=1 f i  = F j ∀ j ∈ M, and T j is the penalty function of UG j ∈ M. Theorem 3.2 The Nash equilibrium search in game Γm is equivalent to the following set of nonlinear equations

3.3

Nash Equilibrium in Noncooperative Game of Users’ Groups ...

⎧ ⎪ ⎪ t ( f ) + t1 ( f 1 ) f 11 − ⎪ ⎪ ⎨ 1 1 ... ⎪ ⎪ ⎪

1 ⎪ ⎩ tn ( f n ) + t n ( f n ) f n − .. . ⎧ ⎪ ⎪ t ( f ) + t1 ( f 1 ) f 1m − ⎪ ⎪ ⎨ 1 1 ... ⎪ ⎪ ⎪ ⎪ ⎩ tn ( f n ) + tn ( f n ) f nm −

57

 F 1 + ns=1 tts (( ffs )) s s n 1

=0

 F 1 + ns=1 tts (( ffs )) s s n 1

=0

s=1 t ( f s ) s

s=1 t ( f s ) s

(3.32)

 F m + ns=1 tts (( ffs )) s s n 1

=0

 F m + ns=1 tts (( ffs )) s s n 1

=0

s=1 t ( f s ) s

s=1 t ( f s ) s

Proof Apply Kuhn–Tucker conditions to optimization problem (3.29) and (3.31) for any j ∈ M. The first-order optimality conditions for the Lagrangian L = j

n 

 j

ti ( f i ) f i + ω

F −

j

j

i=1

n 

 fi

j

+

i=1

n 

j

j

ηi (− f i )

i=1

j

with respect to f i yield: ∂L ∂ti ( f i ) j j = ti ( f i ) + f i − ω j − ηi = 0, j ∂ fi ∂ fi j

where ω j and ηi ≥ 0, i = 1, n are Lagrange multipliers. Therefore, we have: ∂ti ( f i )

ti ( f i ) + or, by

∂ fi

∂ti ( f i ) ∂ fi

we obtain ti ( f i ) +

j

=

j

j

j

f i = ω j + ηi , i = 1, n,

∂ti ( f i ) ∂ f i ∂ti ( f i ) · = , j ∂ fi ∂ fi ∂ fi

∂ti ( f i ) j j f i = ω j + ηi , i = 1, n. ∂ fi j

(3.33)

j

The complementary slackness condition requires ηi f i = 0 for all i = 1, n. Thus, j j j if f i > 0 then ηi = 0 and hence, according to (3.33), ti ( f i ) + ∂ti ( fji ) f i = ω j . If j

j

f i = 0 then ηi ≥ 0 and, according to (3.33), ti ( f i ) + correlations could be written as follows

∂ti ( f i ) j fi j ∂ fi

∂ fi

≥ ω j . The obtained

58

3 Nash Equilibrium in a Road Network with Many Groups of Users

ti ( f i ) +

∂ti ( f i ) ∂ fi

j

 fi

j

j

= ω j , if f i > 0, j

≥ ω j , if f i = 0,

i = 1, n.

(3.34)

j

The functions ti ( f i ) f i for all i = 1, n are convex, and so, the Khun-Tucker conditions are both sufficient and necessary. Therefore, we can say that the assignment ∗ f j , j ∈ M creates a Nash equilibrium in the network of parallel routes if and only ∗ ∗ ∗ if there is ω j such that f j and ω j satisfy (3.34). j Express f i > 0 from (3.34) for any route i = 1, n ti ( f i ) ωj −

ti ( f i ) ti ( f i )

j

fi = and insert into (3.30) ωj

n  i=1

then we obtain

(3.35)

 ti ( f i ) 1 − = Fj

ti ( f i ) i=1 ti ( f i ) n

 F j + ns=1 tts (( ffss )) s . ω = n 1 j

(3.36)

s=1 ts ( f s )

Insert (3.36) into (3.35) and we obtain for all i = 1, n and j = 1, m: ti ( f i ) +

 F j + ns=1 tts (( ffss )) s − = 0. n 1

j ti ( f i ) f i

s=1 ts ( f s )

Therefore, the Nash equilibrium search in a network of parallel routes could be reduced to a system of nonlinear equations. The important Corollary of this result is the explicit Nash equilibrium assignment in a linear network of parallel routes. Note that the explicit assignment is obtained solely due to the solution of the appropriate system of linear equations. Corollary 3.2 Assume that ti ( f i ) = ai + bi f i , i = 1, n. Without loss of generality, routes are indexed as follows: a1 ≤ . . . ≤ an . The Nash equilibrium in the game Γm is reached by the following strategies fi

j∗

1  q f , m + 1 q=1 i m

j

= fi −

3.3

Nash Equilibrium in Noncooperative Game of Users’ Groups ...

59

for all j = 1, m, where j fi

k j

j 1 F +F+ = k j bi

as s=1 bs

1 s=1 br

−

ai , bi

for all i = 1, n, j = 1, m, when 1  an − ai , m + 1 s=1 bi n

Fj > for all j = 1, m.

Proof For a linear network of parallel routes the system (3.32) is [28]: ⎛

2 ⎜1 ⎜ ⎜ .. ⎜. ⎜ ⎝1 1 thus, by

 i=1

⎞⎛ 1 ⎜ 1⎟ ⎟⎜ .. ⎟ ⎜ ⎜ .⎟ ⎟⎜ ⎠ 1 ··· 2 1 ⎝ 1 ··· 1 2

1 2 .. .

··· ··· .. .

1 1 .. .

f i1 f i2 .. .

⎞

⎛

m1 −ai bi m2 −ai bi

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎟=⎜ . ⎜ ⎟ ⎜ −ai f im−1 ⎠ ⎝ mm−1 bi m m −a m i fi b

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

i

j

f i = F j , we obtain: Fj + F + mj = k j

k j

1 s=1 br

as s=1 bs

∀ j = 1, m,

that allows us to find a Nash equilibrium explicitly. Theorem 3.2 allows us to reduce the route-flow Nash equilibrium search to a system of nonlinear equations. Note that in decomposition algorithms (see chap. 5) a road network is represented as a set of parallelized subnetworks. Therefore, due to theorem 3.2, the route-flow Nash equilibrium traffic assignment in a general topology network could be reduced to a system of nonlinear equations.

3.4 Behavioral Model of Competitive Traffic Assignment on a Road Network A behavioral model is a model that describes the behavior of a crowd of people. In particular, the first principle of Wardrop creates a behavioral model of traffic flows. However, it is assumed that each individual driver tries to minimize their own travel time, nevertheless many drivers use in-vehicle guide systems for routing in their

60

3 Nash Equilibrium in a Road Network with Many Groups of Users

daily trips. Thus, groups of users appear in a road network (each group uses its own guide system). Each guide system seeks to minimize the travel time of all its customers. Therefore, the behavioral model of a traffic flow assignment has to take into consideration both the atomic (individual) users and the groups of users. This section is devoted to the development of such a model. We assume that there are groups of users (in-vehicle guide systems, logistics companies, etc.) as well as independent (individual) users [29]. Each group of users tends to minimize the average travel time of its users, while each individual user tends to minimize their own travel time. The mathematical formalization of such a model is a noncooperative game. There are m players associated with m groups of users, and player m + 1 associated with all individuals users. Each ”player” tends to minimize their own penalty function under the influence of all the others. Therefore, there is Nash equilibrium search in a noncooperative game. As the transportation network model, consider a directed graph G = (V, E). m ≥ 2 navigators assign the traffic flows of their customers on the network G. Besides groups’ competition there are individual users who tend to minimize their own travel time. Introduce the following notation: R w is the set of routes between w ∈ W ; xe is the traffic flow on arc e ∈ E, x = (. . . , xe , . . .); te is the travel time (delay) on the edge e ∈ E; M = {1, . . . , m} is the set of UGs; F j,w > 0 is the traffic demand of UG  j between w ∈ W ; F m+1,w is the traffic demand of all indij w j,w is the overall traffic demand between w ∈ W ; xe vidual users; F = m+1 j=1 F j

j,w

is the traffic flow of UG j through edge e ∈ E, x j = (. . . , xe , . . .); fr is the traffic flow of UG j through route r ∈ R w ; frm+1,w is the traffic flow of individual users through route r ∈ R w ; frw = ( fr1,w , . . . , frm,w , frm+1,w ), r ∈ R w , and − j,w j−1,w j+1,w w fr = ( fr1,w , . . . , fr , fr , . . . , frm,w , frm+1,w ); δe,r is the indicator:  w = δe,r

1, if route r ∈ R w ”includes” edge e ∈ E, 0, otherwise.

  Let us analyze the noncooperative game G M, {F j } j∈M , {T j } j∈M , where M =  j,w j,w {1, . . . , m, m + 1} and F j = { f j,w | f k ≥ 0 ∀r ∈ R w , = F j,w } for r ∈R w f r any j ∈ M, and penalty functions T j (x) =

   te xe1 , . . . , xem , xem+1 xej , ∀ j = 1, m, e∈E

T

m+1

(x) =

 e∈E

xem+1 0

  te xe1 , . . . , xem , u du,

3.4

Behavioral Model of Competitive Traffic Assignment on a Road Network

when xej =

 

61

j,w frj,w δe,r ,

w∈W r ∈R w

xe =

m+1 

xej .

j=1

Let us prove that the Nash equilibrium in game G is such an assignment of flows in which the individual users minimize their own travel time, while the UGs minimize the average travel time of their users. Theorem 3.3 The Nash equilibrium x ∗ in game G is such a traffic assignment where the individual users minimize their own travel time while the UGs minimize the average travel time of their users simultaneously. Proof Each player of G, associated with a group of users, is represented by an optimization problem with the penalty function Tj subject to F j , j = 1, m. The Lagrangian of the problem for j = 1, m is L = j



 te

xe1 , . . . , xem , xem+1



 j xe

+m

j,w

F

j,w

−





fr

j,w

+

r ∈R w

e∈E



− fr

j,w



j,w

ηr

r ∈R w

and the Lagrangian of the problem for j = m + 1 is L m+1 =

 e∈E

xem+1 0







te xe1 , . . . , xem , u du + tm+1,w F m+1,w −



 frm+1,w +

r ∈R w

+

  − frm+1,w ηrm+1,w . r ∈R w

Differentiate the Lagrangians with respect to fr j

∂ xe ∂ fr

j,w

j,w

for j = 1, m + 1, by

q

w = δe,r

and

∂ xe ∂ fr

j,w

= 0 when q = j,

we obtain ∂L j ∂ fr

j,w

     ∂te xe1 , . . . , xem , xem+1 j 1 m m+1 te x e , . . . , x e , x e + = xe − j ∂ xe e∈E  j,w j,w w δe,r −m − ηr = 0.

62

3 Nash Equilibrium in a Road Network with Many Groups of Users

and ∂ L m+1 ∂ frm+1,w

=

    w te xe1 , . . . , xem , xem+1 − tm+1,w − ηrm+1,w δe,r = 0. e∈E

   te xe1 , . . . , xem , xem+1 +

We obtain

e∈E

  ∂te xe1 , . . . , xem , xem+1





j,w

= m j,w , fr > 0, j,w ≥ m j,w , fr = 0,

(3.37)

    w = tm+1,w , frm+1,w > 0, te xe1 , . . . , xem , xem+1 · δe,r ≥ tm+1,w , frm+1,w = 0.

(3.38)

+

j ∂ xe

xej

·

w δe,r

for j = 1, m and

e∈E

Therefore, the game theoretic approach allows us to formulate a behavior model considering selfish and group competition on a road network. The game G is a complex computational problem. Furthermore, a series of publications demonstrated that for a higher efficiency, a transportation network should be represented by a set of independent subnetworks, each consisting of two nodes (origin and destination areas) and parallel routes [20, 30, 31]. In this case, we formulate the competitive routing problem for any pair of origin and destination areas, with subsequent transfer of the derived results to any other pair of such areas. Consider a graph composed of two origin and destination areas connected by n parallel arcs (routes). m UGs assign the traffic flows of their customers on a given network which is also used by individual users (so-called player (m + 1)). Introduce notation: N = {1, . . . , n} is the set of routes; M = {1,. . . , m, m + 1} is the set of j players; F j > 0 is the traffic demand of player j; F = m+1 j=1 F is the overall traffic  j j demand; f i ≥ 0 is the traffic flow of player j through route i; f i = m+1 j=1 f i is the traffic flow through route i; ti ( f i ) is the travel time of flow f i through route i. Define j j strategies of player j as a vector f j = ( f 1 , . . . , f n )T such that n 

j

fi = F j .

i=1

The travel time of the traffic flow of player j through route i depends not only on this flow, but also on the traffic flows guided by the rest of the players through the route. In other words, the travel time of the traffic flow of player j through route i

3.4

Behavioral Model of Competitive Traffic Assignment on a Road Network

63

equals the travel time of the traffic flow f i through route i. We will estimate the travel time of the traffic flow f i through route i using the linear delay function: ti = ai + bi f i , ai ≥ 0, bi > 0, i = 1, n. Note that this linear delay function could be interpreted as a linear BPR delay function. In the general case, BPR delay functions have the term ( f i /ci ) raised to a power β. The latter value is defined via real–time estimation of traffic flows on road network segments. Value f i is the sum of all transport vehicles moving from an origin area to a destination route i. Therefore, our game of (m + 1) players corresponds to  through j f and hence the travel time of the traffic flow f i through route i acquires f i = m+1 j=1 i the form m+1  j ti = ai + bi · fi . j=1

Therefore, in a road network of parallel routes a set of strategies n inj the noncoopj j erative game G has to satisfy F j = { f j | f i ≥ 0 ∀i = 1, n, i=1 f i = F } for all j ∈ M, when penalty functions are Tj =

n    j ti f i1 , . . . , f im , f im+1 f i ,

j = 1, m

(3.39)

i=1

Tm+1 =

n   i=1

0

f im+1

  ti f i1 , . . . , f im , u du.

(3.40)

By the minimization of T j for j = 1, m we actuallyminimize the average travel time j−1 j+1 fˆi1 , . . . , fˆi , fˆi , . . . , fˆim , of the traffic flow F j for j = 1, m: for any fixed n we obtain the“local” system optimum for the users of UG j [3]. fˆim+1 i=1 Consequently, minimization of the penalty functions of the players in the game G 1, m. causes the minimization of the average travel  time of users of UGs j when j =  n j−1 j+1 m+1 m 1 In this case, for each j ∈ M the quantity fˆi , . . . , fˆi , fˆi , . . . , fˆi , fˆi i=1

is not fixed, but induced by the strategies of the other players. This leads to the competitive routing problem and a Nash equilibrium search in the game G. Lemma 3.2 f ∗ is the Nash equilibrium in game G if and only if m j for j = 1, m and tm+1 (Lagrange multipliers) exist, such that ⎛ ⎞ j−1 j∗ m+1 j   > 0, ∗ q j q ⎠ = m , for f i ⎝ fi + 2 fi + fi ai + bi j∗ j ≥ m , for f i = 0, q=1 q= j+1

(3.41)

64

3 Nash Equilibrium in a Road Network with Many Groups of Users

for all i = 1, n, j = 1, m and ai + bi ·

m+1 

 q fi

q=1

∗

= tm+1 , for f im+1 > 0,

(3.42)

∗

≥ tm+1 , for f im+1 = 0,

for all i = 1, n. Proof Let us employ the Kuhn–Tucker conditions. The functionals (3.39) and (3.40), and the feasible solution domains F j for all j = 1, m + 1; hence, the Kuhn–Tucker conditions play the role of necessary and sufficient optimality conditions here. Construct the Lagrangian for the minimization problems (3.39) and (3.40) subject to F j for all j = 1, m + 1: n    j ti f i1 , . . . , f im , f im+1 f i + m j L =

 F −

j

j

i=1

n 

 fi

j

+

i=1

n 

 j j ηi − f i

i=1

for UG j = 1, m and L m+1 =

n   i=1

f im+1

0







ti f i1 , . . . , f im , u du + tm+1 F m+1 −

n 

 f im+1 +

i=1

+

n 

  ηim+1 − f im+1 .

i=1 j

The first-order optimality conditions for the Lagrangian with respect to f i yield ⎛ m j = ai + bi · ⎝

j−1 

q

fi + fi

j∗

q=1

for UG j = 1, m and tm+1 = ai + bi ·

+

m 

⎞ ∗

q j j f i ⎠ + bi f i − ηi

q= j+1

m+1 

q

j

f i − ηi

q=1

for individual users (player m + 1), for all routes i = 1, n. The complementary slackj j∗ ness condition ηi f i = 0 for all i = 1, n and j = 1, m + 1 dictates that, whenever ∗ j j f i > 0, the result is ηi = 0 and we arrive at the first conditions in (3.41) and (3.42). j∗ j In the case of f i = 0, the constraints ηi ≥ 0 on the Lagrange multipliers brings to the second conditions in (3.41) and (3.42).

3.4

Behavioral Model of Competitive Traffic Assignment on a Road Network

65

Corollary 3.3 f ∗ is the Nash equilibrium in a game G if and only if m j and tm+1 (Lagrange multipliers) exist, such that f i1

+ · · · + 2 fi



j∗

+ ··· +

f im+1

=

m j −ai bi

0,

, for ai < m j , for ai ≥ m j ,

(3.43)

for all i = 1, n, j = 1, m and f i1

+ · · · + fi

j∗

 + ··· +

Proof According to Lemma 3.2 for f i f i1 + · · · + 2 f i and for all i = 1, n:

m+1 

=

f im+1

j∗

j∗

tm+1 −ai bi

0,

j

j=1

(3.44)

> 0 for all i = 1, n and j = 1, m, then

+ · · · + f im =

fi =

, for ai < tm+1 , for ai ≥ tm+1 .

m j − ai >0 bi

tm+1 − ai > 0, bi

that means ai < m j and ai < t j . For f i

j∗

= 0 for all i = 1, n and j = 1, m:

m j = ai + bi

m+1 

q

fi

q=1,q= j

and for all i = 1, n tm+1 = ai + bi

m 

q

fi

q=1

that means m j ≤ ai for j = 1, m and tm+1 ≤ ai , since traffic flows could not be negative.  Corollary 3.4 In the game G, the Nash equilibrium is of the form f ∗ m1 , . . . , mm ,  tm+1 for some m j , j = 1, m and tm+1 . Proof Introduce the following notation

66

3 Nash Equilibrium in a Road Network with Many Groups of Users

⎛

2 ⎜1 ⎜ ⎜ = ⎜ ... ⎜ ⎝1 1

A = Am+1×m+1

⎞ 1 ··· 1 1 2 ··· 1 1⎟ ⎟ .. . . .. .. ⎟ , . . . .⎟ ⎟ 1 ··· 2 1⎠ 1 · · · 1 1 m+1×m+1

T  and fi = fi1 , . . . , fim , fim+1 , where  j fi

=

m j −ai bi

0,

, for ai < m j , for ai ≥ m j ,

for all i = 1, n, j = 1, m and  fim+1

=

tm+1 −ai bi

0,

, for ai < mm+1 , for ai ≥ tm+1 ,

for all i = 1, n. Then (3.43) and (3.44) could be rewritten in a matrix form as follows  ∗ ∗ T A f i1 , . . . , f im ∗ , f im+1 = fi , ∀i = 1, n.

(3.45)

This matrix equation has a unique solution, since all rows of the square matrix A(m+1)×(m+1) are linearly independent. In other words, the game G possesses a unique positive Nash equilibrium. The following matrix is inverse to A: ⎛

A

−1

0 ··· ⎜ 1 ··· ⎜ ⎜ .. . . =⎜ . . ⎜ ⎝ 0 0 ··· −1 −1 · · · 1 0 .. .

We have fi

j∗

0 0 .. .

−1 −1 .. .

⎞

⎟ ⎟ ⎟ . ⎟ ⎟ 1 −1 ⎠ −1 m + 1 m+1×m+1 j

= fi − fim+1 ,

(3.46)

for all j = 1, m and i = 1, n, ∗

f im+1 = (m + 1)fim+1 −

m  q=1

q

fi ,

(3.47)

3.4

Behavioral Model of Competitive Traffic Assignment on a Road Network

67

 j j  for all routes i = 1, n. As far as fi = fi m j for all i = 1, n, j = 1, m and fim+1 =    j∗ j∗  fim+1 tm+1 , we have that f i = f i m1 , . . . , mm , tm+1 for all i = 1, n and j = 1, m + 1. Lemma 3.3 Let f ∗ be the Nash equilibrium in a game G. If F j > F q , then ω j > ωq , for all j, q = 1, m. Proof Consider the equality Fj =

n 

fi

j∗

i=1

=

n 

j

fi −

i=1

n 

fim+1 ,

(3.48)

i=1

that holds for all j = m. Therefore, compare F j and F q , for all j, q = 1, m, n to 1, j q n fi and i=1 fi . we should compare i=1 Imagine that F j > F q , but m j < mq . In this case, n  i=1

j

fi =



 {i: ai 0 if t > ai for all routes i = 1, n. The condition that mi , j = 1, m m+1 are larger than all ai , i = 1, n is equivalent to (3.52) and (3.53). and ti

3.4 Behavioral Model of Competitive Traffic Assignment on a Road Network

69

Note that (3.55) is the equilibrium travel time on the route i, i = 1, n. Moreover, (3.55) is equivalent to the equilibrium travel time of F individual users [32]. Corollary 3.5 Assume that inequalities (3.52) and (3.53) hold. The overall travel time of F both individual and group users is equal to the overall travel time of F individual users in a network of parallel routes. Proof The overall travel time of F through routes i, i = 1, n, is T =

n 

ti ( f i ) f i .

(3.56)

i=1

According to (3.55) and [32], the travel time of flows on alternative actually used routes is the same:  F + ns=1 abss ∗ m+1 t =t = n 1 . s=1 bs

Therefore, in case of both individual and group users or in case of individual users only, expression (3.56) is T =

n  i=1

ti ( f i ) f i = t∗

n 

f i = t∗ · F.

i=1

Inequalities (3.52) and (3.53) hold in congested road networks. In other words, if (3.52) and (3.53) hold, then all n routes are used by F individual and group users.

References 1. Gartner NH (1980) Optimal traffic assignment with elastic demands: a review part I: analysis framework. Transp Sci 14(2):174–191 2. Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc, Mineola, N.Y. 3. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc, Englewood Cliffs, N.J. 4. Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc Inst Civ Engineers 2:325–378 5. Bonsall P (1992) The influence of route guidance advice on route choice in urban networks. Transportation 19:1–23 6. Nash J (1951) Non-cooperative games. Ann Mathematics 54:286–295 7. Patriksson M (1994) The traffic assignment problem: models and methods. VSP, Utrecht, The Netherlands 8. Zheng H, Peeta S (2014) Cost scaling based successive approximation algorithm for the traffic assignment problem. Transp Res Part B 68:17–30 9. Xie J, Yu N, Yang X (2013) Quadratic approximation and convergence of some bush-based algorithms for the traffic assignment problem. Transp Res Part B 56:15–30

70

3 Nash Equilibrium in a Road Network with Many Groups of Users

10. Charnes A, Cooper WW (1958) Extremal principles for simulating traffic flow in a network. Proc Natl Acad Sci U S A 44:201–204 11. Dafermos SC (1971) An extended traffic assignment model with applications to two-way traffic. Transp Science 5:366–389 12. Dafermos SC, Sparrow FT (1969) The traffic assignment problem for a general network. J Res Natl Bur Standards 73B:91–118 13. Rosenthal RW (1973) The network equilibrium problem in integers. Networks 3:53–59 14. Devarajan S (1981) A note on network equilibrium and noncooperative games. Transp Research 15B:421–426 15. Fisk CS (1984) Game theory and transportation systems modelling. Transp Research 18B:301– 313 16. Haurie A, Marcotte P (1985) On the relationship between nash-cournot and wardrop equilibria. Networks 15:295–308 17. Orda A, Rom R, Shimkin N (1993) Competitive routing in multiuser communication networks. IEEE/ACM Trans Networking 1(5):510–521 18. Altman E, Basar T, Jimenez T, Shimkin N (2002) Competitive routing in networks with polynomial costs. IEEE Trans Autom Control 47(1):92–96 19. Korilis YA, Lazar AA (1995) On the existence of equilibria in noncooperative optimal flow control. J Assoc Comput Machinery 42(3):584–613 20. Korilis YA, Lazar AA, Orda A (1995) Architecting noncooperative networks. IEEE J Sel Areas Commun 13(7):1241–1251 21. La RJ, Anantharam V (1997) Optimal routing control: game theoretic approach. In: Proceedings of the 36th IEEE conference on decision and control. pp 2910–2915 22. Altman E, Combes R, Altman Z, Sorin S (2011) Routing games in the many players regime. In: Proceedings of the 5th international ICST conference on performance evaluation methodologies and tools. 525–527 23. Altman E, Kameda H (2001) Equilibria for multiclass routing in multi-agent networks. In: Proceedings of the 40th IEEE Conference on 1. pp 604–609 24. Altman E, Wynter L (2004) Eguilibrium, games, and pricing in transportation and telecommunication networks. Netw Spat Economics 4:7–21 25. Konnov IV (2015) On auction equilibrium models with network applications. Netnomics 16:107–125 26. Beckmann MJ, McGuire CB, Winsten CB (1956) Studies in the economics of transportation. Yale University Press, New Haven, CT 27. Krylatov AYu, Zakharov VV, Malygin IG (2016) Competitive traffic assignment in road networks. Transp Telecommunication 17(3):212–221 28. Zakharov VV, Krylatov AY (2016) Competitive routing of traffic flows by navigation providers. Automation and Remote Control. 77(1):179–189 29. Roughgarden T (2005) Selfish routing and the price of anarchy. MIT Press 30. Daganzo CF (1994) The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transpn Res B 28:269–287 31. Korilis YA, Lazar AA, Orda A (1999) Avoiding the braess paradox in non-cooperative networks. J Appl Prob 36:211–222 32. Krylatov AYu (2014) Optimal strategies for traffic management in the network of parallel routes. Vestn St-Peterbg Univ Prikl Mat Inform Protsessy Upr 2:121–130

Part III

Optimization Traffic Assignment Methods

Equilibrium traffic assignment methods are actually nonlinear optimization methods. However, as a rule, the most efficient of them are based on the problem content and thereby provide additional unexpected insight on the topic. Thus, the development of equilibrium traffic assignment methods contributes both to nonlinear optimization and comprehensive understanding of the practical problems under investigation.

Chapter 4

Methods for Traffic Flow Assignment in Road Networks

Abstract In this chapter is devoted to approaches for solving traffic flow assignment problems. The most popular gradient descent method for solving traffic assignment problems is discussed in the first section. New projection algorithms based on the obtained, explicitly fixed-point operators for the route-flow assignment problem and link-route assignment problem are presented in the third and fourth sections respectively. Obtained operators is proved to be contractive that leads to the linear convergence of provided algorithms. Moreover, under some fairly natural conditions the algorithms converge quadratically. The technique for representing a linear route-flow assignment problem in the form of a system of linear equations is presented in the fourth section. A simple example demonstrates the evident usability of the developed technique for its implementation and further extensions.

4.1 Gradient Descent for User-Equilibrium Search in Road Networks The most common algorithm for traffic flow assignment is the Frank–Wolfe algorithm and its extensions. In fact, the Frank–Wolfe algorithm is a gradient descent method. Initially, M. Frank and P. Wolfe developed their algorithm to cope with problems of quadratic programming [1]. However, this algorithm is able to solve more general problems when the goal functions are pseudo-convex and smooth, and a feasible set of solutions is nonempty, compact and convex. Generally, the Frank–Wolfe algorithm for (2.2)–(2.5) is the following sequence of steps. Step 0 (Initialization) Let x 0 be a feasible solution of the traffic assignment problem (2.2)–(2.5), L B D = 0, ε > 0 and q = 0. Step 1 (Search direction generation) Let us define L(x) =



te (xeq )xe .

(4.1)

e∈E

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_4

73

74

4 Methods for Traffic Flow Assignment in Road Networks

and solve the following linear program min L(x), x

subject to



(4.2)

frw = F w ∀w ∈ W,

(4.3)

frw ≥ 0 ∀r ∈ R w , w ∈ W,

(4.4)

r ∈R w

with definitional constraints  

xe =

w∈W

w frw δe,r ∀e ∈ E.

(4.5)

r ∈R w

If y q is a solution of the linear program (4.1)–(4.5), then pq = y q − x q is the search direction at this step. Step 2 (Convergence check) L B D := max{L B D, L(y q )}. If  xeq

 e∈E

0

te (u)du − L B D < ε, LBD

(4.6)

then the algorithm is converged to x q . Otherwise, proceed. Step 3 (Line search) Line search lq is the solution of an one-dimensional problem:  lq = arg min l

 e∈E

0

q

q

xe +lpe

   te (u)du  0 ≤ l ≤ 1 .

(4.7)

Step 4 (Update) x q+1 = x q + lq pq . Step 5 (Convergence check) If  xeq+1

 e∈E

0

te (u)du − L B D < ε, LBD

then the algorithm is converged to x q+1 . Otherwise, q := q + 1 and go to Step 1. As an initial solution x 0 “all-or-nothing” assignment could be taken. In other words, the trip demand could be assigned through the shortest path only. Since the constraints of the traffic assignment problem are linear (as well as separable), the linear subproblems (4.2)–(4.5) could be decomposed to |W | independent problems. Each independent problem is a shortest path problem  x one origin for destination pair, when the travel time on any arc is fixed ∂/∂ xe e∈E 0 e te (u)du = q te (xe ) where xe = xe . What is remarkable is that in such a case set of all possible routes should not be pre-defined.

4.1 Gradient Descent for User-Equilibrium Search in Road Networks

75

The traffic assignment problem is a particular case of a nonlinear constrained optimization problem (4.8) min Ψ (x), x∈X

where Ψ is convex and continuous differentiable of the polyhedral set X . Since y q is the solution of the linearized problem and Ψ is the convex, then L(y q ) =







te (xeq ) yeq − xeq ≤ te (xeq ) xe∗ − xeq ≤ Ψ ∗ ,

e∈E

e∈E

where x ∗ is a solution of (4.8), i.e. L(y q ) is a lower bound on the optimal value of  xq  (4.8). Moreover, if L(y q ) = e∈E 0 e te (u)du for some q, then x q is a solution of (4.8), and the convergence criterion (4.6) seems to be reasonable.  e∈E  xeHowever, traffic engineers avoid utilizing (artificial) goal function 0 te (u)du in the convergence criterion, since this function does not possess any physical or economic sense. A common alternative is the comparison of travel times on alternative used routes between an origin–destination pair (they should be equal)  max w∈W

max

r,r∈R w

trw ( frw )

−

trw ( f rw )



  w w  fr , f r > 0

at each iteration of the algorithm. Such a criterion requires gathering information about the used routes, which is not needed for an ordinary Frank–Wolfe algorithm. Nevertheless, sometimes the route-flow values could be used for equilibrium solution analysis [2–4]. The lower bound of the equilibrium solution provides a pure estimation of current solution [5]. Linearization is replacement initial goal functions with tangents. Indeed, in a congested road network the difference between en equilibrium assignment and en “all-or-nothing” assignment is significant. Numerous convergence tests under different convergence criterion are available in [6]. The first papers devoted to the Frank–Wolfe algorithm as a method for traffic flow assignment are [7–11]. The first applications of the Frank–Wolfe algorithm to flow assignment in telecommunication networks were made in [12–14]. According to interpretation, the Frank–Wolfe algorithm is known as [15]. • (Linearization) This interpretation of the Frank–Wolfe algorithm follows directly from a form of an approximation scheme, when only one tangent plane is used for approximating a goal function. • (Conditional gradient) The Frank–Wolfe algorithm is actually gradient descent: each step aims to find the direction of the most negative directional derivative of a goal function among all feasible directions. • (Flow deviation) Such an interpretation is based on behavioral terms, since at each iteration some drivers choose a less loaded route. • (Convex combination) Step 3 of the algorithm could be interpreted as the optimal convex combination forming of a linearized subproblem solution and current initial problem solution.

76

4 Methods for Traffic Flow Assignment in Road Networks

Note that the Frank–Wolfe algorithm became famous due to papers [3, 9, 16]. Nowadays, it is the most commonly used method for traffic assignment in the sphere of transportation planning. The vast majority of computations in the Frank–Wolfe algorithm are associated with finding the shortest path at Step 1 (up to 90 % in large-scale applications). In fact, (4.2)–(4.5) is a problem of finding the shortest path between each originq

destination pair in a directed graph with positive travel delays on arcs te (xe ) . Due to the non-negativity of travel delays on arcs, the Dijkstra algorithm could be used q [17]. Herewith, travel delays te (xe ) change from iteration to iteration. Therefore, it is quite natural to use the solution of subproblem (4.2)–(4.5), obtained in the q-th iteration, as an initial solution to the (q + 1)-th iteration. This is a way of improving any traffic assignment algorithm which requires finding the shortest path [18–23]. The convergence rate of the Frank–Wolfe algorithm for nonlinear constrained optimization was widely investigated [24–28]. Link-flow assignment is not unique and the Frank–Wolfe algorithm converges to the set of optimal solutions. Moreover, if travel delays are differentiable functions, the derivative of the goal function is Lipschitz continuous on a set of feasible solutions, and exact line search in Step 3 may be replaced by inexact line searches, such as the Armijo and Goldstein step length rules. The unsatisfactory performance of the Frank–Wolfe algorithm was mentioned enough early. The convergence rate of the algorithm was widely investigated via its application to different nonlinear problems, for instance [24, 29]. Eventually, linear convergence of the algorithm was established by researchers. The earliest examples of poor Frank–Wolfe algorithm performance were given in [30, 31]. The main reason for such a poor performance is the manner of the direction search (Step 1). The optimal solution and search direction primarily depend rather on a set of feasible solutions than on a goal function. Therefore, when the optimal solution is reached, a search direction tends to become orthogonal to the steepest descent direction [30]. Deep investigation of the Frank–Wolfe algorithm for the traffic assignment problem is available in [32]. While gradient descent for non-constrained optimization gives 90◦ between sequential search directions, the respective angle in the Frank–Wolfe algorithm is 120◦ . The reason of such a phenomenon is the rapidly decreasing step length in both algorithms, which explains the term “zig-zagging” that is used for describing the behavior of the Frank–Wolfe algorithm. A remarkable study was conducted in [33]. According to the obtained results, the Frank–Wolfe algorithm could also generate cycle routes as a part of the solution (especially at the first iteration). Herewith, cycle flows will be unlikely excluded from the next iterations that impacts negatively on the efficiency of the algorithm. Such empiric results seem to be natural since all generated routes for (4.2)–(4.5) will have some flow in any newly generated assignment. Eventually, any solution generated in a finite number of iterations will include actually unused routes with non-zero flow. Konnov suggested an adaptive version of the partial linearization method which makes selective component-wise steps satisfying some descent condition and utilizes a sequence of control parameters [34]. This method can solve optimization problems

4.1 Gradient Descent for User-Equilibrium Search in Road Networks

77

where the goal function is the sum of a smooth function and a convex non-necessary smooth function. The good convergence rate of the method was demonstrated via its application to the network equilibrium problem.

4.2 Projection Approach for Route-Flow Traffic Assignment Theorem 2.6 states that the user equilibrium problem formulated as the optimization problem (2.41)–(2.43) is equivalent to the fixed point problem (2.44) for the network of parallel routes. The representation of the user equilibrium problem as (2.44) allows us to reformulate the problem of finding the equilibrium assignment in the network of parallel routes in the form of a simple iterative process f q+1 = Φ(f q ),

(4.9)

where the components f q are indexed so that q

a1 ( f 1 ) ≤ · · · ≤ an ( f nq ),

(4.10)

while the components Φ(f q ) = (Φ1 ( f q ), . . . , Φn ( f q ))T are as follows: Φi ( f ) = q

⎧ ⎨ ⎩

1 q bi ( f i )

 F+ ks=1 k

q as ( f s ) q bs ( f s ) 1 s=1 b ( f q ) s s

0

−

q

ai ( f i ) q bi ( f i )

for i ≤ k,

(4.11)

for i > k,

when k is defined by k q q  ak ( f ) − ai ( f ) k

i

q

i=1

bi ( f i )

≤F
0 for all i = 1, k, and so, k  q ts ( fsq )  q F −

ti ( f ) q s=1 f s − ts ( f sq ) q+1 fi − f i∗ = f i − f i∗ −  iq + = q  ti ( f i ) k ti ( f i ) q  s=1 ts ( f s )

=



q fi

−

k s=1

−

f i∗



q ti ( f i ) − ti ( f i∗ ) + t ∗ − − q ti ( f i )

  q (t ( f )−t ( f ∗ ))+t ∗ q ( f s − f s∗ ) − s s t  ( sf q )s s s = m k ti ( fik ) s=1 ts ( f sk )

= k −

s=1



q fi

−

f i∗



q ti ( f i ) − ti ( f i∗ ) t∗ − − q q − ti ( f i ) ti ( f i )

  q k (t ( f )−t ( f ∗ )) q ( f s − f s∗ ) − s st  ( f qs) s s=1 t∗ s s +  q k k ti ( fiq ) ti ( f i ) s=1 q  s=1 ts ( f s )



q

ti ( = f i − f i∗ − Thus

q f i ) − ti ( f i∗ ) q ti ( f i )

1 q ts ( f s ) 1 q ts ( f s )

  q (ts ( f s )−ts ( f s∗ )) q ∗ ( f − f ) − q s s s=1 ts ( f s ) . k ti ( fiq )

k −

=

s=1 ts ( f sq )

4.2 Projection Approach for Route-Flow Traffic Assignment

79

− f i∗ = 

k ( f q − f ∗ )− (ts ( fsq )−ts ( fs∗ ))  q ∗ s q q

s s=1 t ( f )−t ( f ) ts ( f s ) i i i i = f i − f i∗ − − . q k t  ( f q ) t( f ) q+1

fi



i

i

(4.15)

i i s=1 t  ( f q ) s s

For the ease of explanation, let us introduce the notations ζi (x) = (x −

f i∗ )

k  ti (x) − ti ( f i∗ ) ti (x) − and ςi (x) = q , i = 1, k,  ti (x) t( f ) s=1 s s

in this case, (4.15) becomes q+1 fi

−

f i∗

k = ζi (

q fi )

q s=1 ζs ( f s ) . q ς ( fi )

−

(4.16)

Note that by the Lagrange mean value theorem, for all i = 1, k we have ti ( f i ) − ti ( f i∗ ) = ti (θi )( f i − f i∗ ), q

q

q

where θi belongs to the interval between f i and f i∗ so that we have q

q

 q 

t  (θ ) q q f i − f i∗ , i = 1, k. ζi ( f i ) = 1 − i iq ti ( f i )

(4.17)

 q  t  (θ ) q q q

Define Ui ( f i ) = 1 − t i ( fiq ) , i = 1, k, U ( f q ) = U1 ( f 1 ), . . . , Uk ( f k ) and subi i stitute (4.17) with (4.16) q+1 fi

−

f i∗

= Ui (

q q fi ) fi

−

f i∗



q q s=1 Us ( f s ) f s q ς ( fi )

k −

− f s∗

,

we have      q+1  q   q − f i∗  ≤ Ui ( f i ) ·  f i − f i∗  +  fi

k      1 Us ( f q ) ·  f q − f ∗  . q · s s s ς ( f i ) s=1 (4.18) Sum up the left parts of (4.18) over i = 1, k and note that k  i=1

Then

 1 1 1 q = q · k  ς ( fi ) t ( f ) s=1 i=1 i i k

1 q ts ( f s )

= 1.

80

4 Methods for Traffic Flow Assignment in Road Networks k  k k              q+1  Ui ( f q ) ·  f q − f ∗  + Us ( f q ) ·  f q − f ∗  , − f i∗  ≤  fi i s s s i i i=1

s=1

i=1

and therefore, k  k         q+1 ∗ 2 Ui ( f q ) ·  f q − f ∗  . − fi  ≤  fi i i i i=1

(4.19)

i=1

We have ||U ( f q )|| → 0 as f q → f ∗ , i.e.   ∀ε : 0 < ε < 1 ∃ρ : 2 U ( f q ) ≤ ε < 1 for f q ∈ Sρ ( f ∗ ), where Sρ ( f ∗ ) = { f : || f − f ∗ || ≤ ρ}. Therefore, since f q ∈ Sρ ( f ∗ ) and due to (4.19), we have k  k     q   q+1   f − f ∗  ≤ ερ < ρ, − f i∗  ≤ ε  fi i i i=1

i=1

and so, f q+1 ∈ Sρ ( f ∗ ). Thus, if f 0 ∈ Sρ ( f ∗ ) then { f q } ∈ Sρ ( f ∗ ) and k k    q   0   f − f ∗  ≤ εq  f − f ∗ , i i i i i=1

i=1

whence it follows that { f q } converges geometrically. Note that Theorem 4.1 holds for the delay functions that are smooth on the edges of the network of parallel routes. However, if the delay functions are twice continuously differentiable then we have Theorem 4.2 ([35]) Let in some neighborhood of f ∗ the functions ti (·) ∈ C 2 , i = 1, n be such that     t (x) ≥ αi > 0 and t  (x) ≤ βi < ∞. (4.20) i i Then there is a δ-neighborhood of f ∗ such that for an arbitrary choice of the initial approximation f 0 from this neighborhood, the series {f q } of the iterative process (4.9) remains within the neighborhood and converges quadratically to f ∗ . Proof If ti (·) ∈ C 2 (R ≥0 ) for all i = 1, n, then by the Taylor expansion formula, we have 1 q q q

q q 2 ti ( f i∗ ) = ti ( f i ) + ti ( f i ) f i∗ − f i + ti (Θi ) f i∗ − f i , i = 1, k, 2

(4.21)

where Θi is some point between f i and f i∗ . Since ti ( f i∗ ) = t ∗ , i = 1, k, let us rewrite (4.21) as q

q

4.2 Projection Approach for Route-Flow Traffic Assignment

81

1 q q q

q q 2 t ∗ = ti ( f i ) + ti ( f i ) f i∗ − f i + ti (Θi ) f i∗ − f i , i = 1, k. 2

(4.22)

Rewrite (4.14) as follows: 

ti ( f i ) + ti ( f i ) f i q

q

q+1

q

− fi



=

F−



k s=1

q

fs −

k

q

ts ( f s ) q ts ( f s )



ti ( f i ) s=1 ts ( f sq ) q

and add this expression to (4.22):  1  q q+1 q q 2 − f i∗ = ti (Θi ) f i∗ − f i + Ω, i = 1, k, t ∗ + ti ( f i ) f i 2 where Ω= Since F =

k i=1

F−



k s=1

k

q

fs −

q

ts ( f s ) q ts ( f s )

(4.23)

 .

1 s=1 ts ( f sq )

f i∗ , we have 

k Ω=

q

f s∗ − f s + k 1

s=1



q

ts ( f s ) q ts ( f s )

.

(4.24)

s=1 ts ( f sq )

q

Find ti ( f i ), i = 1, k from (4.22) and insert it into (4.24): 

k Ω=

s=1

q

f s∗ − f s +

t∗ q ts ( f s )

q

− f s∗ − f s − k 1

 1 ts (Θs ) 2 ts ( f sq ) q



q 2

f s∗ − f s

 ,

s=1 ts ( f sq )

whence it follows that 1 Ω = t∗ − 2

ts (Θs ) ∗ s=1 ts ( f sq ) f s − k 1 s=1 ts ( f sq )

k

q

q 2

fs

.

(4.25)

Inserting (4.25) into (4.23), we have q ti ( f i )



q+1 fi

−

f i∗



1 1 q q 2 = ti (Θi ) f i∗ − f i − 2 2

ts (Θs ) ∗ s=1 ts ( f sq ) f s − k 1 s=1 ts ( f sq )

k

q

q 2

fs

,

i = 1, k (4.26)

82

4 Methods for Traffic Flow Assignment in Road Networks

then, 1 t  (Θ ) ∗ 1 q+1 q 2 fi − fi − − f i∗ = i  qi fi 2 ti ( f i ) 2 q

and so,

k

ts (Θs ) s=1 ts ( f sq ) q



k

q 2

f s∗ − f s

, i = 1, k,

ti ( f i ) s=1 ts ( f sq ) q

  1  t  (Θ q )     q+1 ∗  i qi  ·  f ∗ − f q 2 + f − f  i i ≤ i i    2 ti ( f i ) 1 1 + · q ·  2 ti ( f i )

  q    ts (Θs )   ∗ q 2 s=1  ts ( f sq )  · f s − f s , k 1 s=1 |ts ( f sq )|

k

which, after the summation over i = 1, k, looks like  k  k       ti (Θiq )   ∗  q+1  q 2    − f i∗  ≤  fi  t  ( f q )  · fi − fi . i i i=1 i=1 Since (4.20) holds in some neighborhood of f ∗ , we have k  k    βi  ∗  q+1  q 2 − f i∗  ≤ · fi − fi  .  fi α i=1 i=1 i

(4.27)

Therefore, by definition it follows from (4.27) that the series {f q } of the iterative process (4.9) quadratically converges for an arbitrary choice of f 0 from some δneighborhood of f ∗ . Remark 4.1 Explicitly expressed, the iterative process (4.9) looks like

q+1

fi

q

= fi −

q ti ( f i ) q  ti ( f i )

+

F−



k s=1

k

q

fs −

ti ( f i ) s=1 ts ( f sq ) q

q

ts ( f s ) q ts ( f s )

 .

Note that at each new iterative step the optimization problem (2.41)–(2.43) is solved for the case of linear delay functions that correspond to the tangents of the initial delay functions at the points obtained at the previous step, at which an optimal assignment of flows is achieved. Theorems 4.1 and 4.2 suggest that the iterative process (4.9) for f 0 from some δ-neighborhood of f ∗ converges to an user equilibrium point for the flow assignment F in the network of parallel routes; moreover, when the delay functions meet some reasonable conditions on the links, we have quadratic convergence. As may be seen from the proofs of both theorems, one of the key requirements to the neighborhood

4.2 Projection Approach for Route-Flow Traffic Assignment

83

is the equality k = k ∗ . In other words, as soon as the iterative process (4.9) finds an optimal number of routes, the quadratic convergence is achieved. On the other hand, we can give the algorithm a hint about which routes are optimal for example by having implemented the optimal route search procedure described in [36]. Moreover, it should be noted that in a loaded network all possible routes are used; thus, we can set f0 to be f0 = (F/n, . . . , F/n)n . Due to projection operator Φ (2.44) defined explicitly via (2.45)–(2.47) the corresponding projection algorithm was developed. At each iteration, the algorithm performed the following three steps. (k + 1) iteration: 1. To index m k components f k and t ( f k ) so that a1 ( f 1k ) ≤ a2 ( f 2k ) ≤ . . . ≤ am k ( f mk k ). 2. To find m k+1 ≤ m k (the amount of non-zero components f k+1 ) from the condition  am k+1 ( f mk k+1 ) − ai ( f ik )

m k+1

bi ( f ik )

i=1

 am k+1 +1 ( f mk k+1 +1 ) − ai ( f ik )

m k+1

≤F
0.1, i = 1, n} is the set of used routes. 4. If     ti ( f ik+1 ) − t j ( f jk+1 ) < ε, i, j∈R k+1

then terminate, with f k+1 as the approximate solution. Otherwise, let k := k + 1, and go to Step 1. The amount of iterations required by these two algorithms is available in Table 4.2. An infinite amount of iterations (∞) means the zig-zagging behavior of the algorithm. What is highly remarkable is that the simple topology of the network allows us to draw revealing insights. Indeed, the solutions of the network’s flow assignment problem corresponding to pattern 1, 2 and 3 have no zero components, i.e. there are no unused routes. However, the solutions of the network’s flow assignment problem

4.2 Projection Approach for Route-Flow Traffic Assignment

85

Table 4.2 The projection algorithm versus the Frank–Wolfe algorithm: amount of iterations FW-algorithm Projection algorithm Pattern 1 Pattern 2 Pattern 3 Pattern 4 Pattern 5

2 2 3 ∞ ∞

2 2 3 7 3

corresponding to pattern 4 and 5 have zero components, i.e. there are unused routes. Actually, route 3 in pattern 4, and routes 3 and 5 in pattern 5 are obviously too “expensive” to use. Thus, in advance it is quite clear that f i∗ = 0 for i = 3 in pattern 4 and f i∗ = 0 for i ∈ {3, 5} in pattern 5. According to Table 4.2, the Frank–Wolfe algorithm demonstrates zig-zagging behavior when there are zero components in an equilibrium solution of a network’s flow assignment problem. Nevertheless, the projection algorithm does not pay much attention to such a “trouble” (zero components in an equilibrium solution) and demonstrates high convergence rate. The projection algorithm demonstrates such a performance primarily due to the third step of each iteration. Indeed, each iteration clarifies zero components in an equilibrium solution and excludes them from consideration (4.28)-I I . Eventually, the projection algorithm seeks the solution in the space of non-zero components (red routes on Fig. 4.1 correspond to zero components). ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

(a)

⎛ q⎞ f1 ⎜ q f2 ⎟ ⎜ ⎜ q⎟ f3 ⎟ ⎟→⎜ ⎜ .. ⎟ ⎜ . ⎠ ⎝ q fn I

⎞ q+1 f1 q+1 ⎟ f2 ⎟ q+1 ⎟ f3 ⎟ ⎟ .. ⎟ . ⎠ q+1 fn

⎛ ⎜ ⎜ ⎜ ⎝

q ⎞ ⎛ q+1 ⎞ f1 f1 q ⎟ f2 ⎟ ⎜ .. ⎟ .. ⎟ → ⎝ . ⎠ . ⎠ q+1 q f kq+1 f kq

II

(b)

Fig. 4.1 The network of parallel routes. a FW-algorithm, b Projection algorithm

(4.28)

86

4 Methods for Traffic Flow Assignment in Road Networks

In turn, the Frank–Wolfe algorithm operates in the space of all components (4.28)-I (Fig. 4.1), and experiences zig-zagging behavior in a neighborhood of zero components. The larger the difference between the alternative routes, the higher the probability to experience zig-zagging behavior. The results obtained here show an obvious advantage of the projection approach for a network’s flow assignment problem. The efficiency of the developed projection algorithm is primarily caused by the exclusion of zero components during solving. Therefore, it is highly promising to develop these projection algorithms. Generally, it is quite a complicated task to find projection operator [40].

4.3 Projection Approach for Link-Flow Traffic Assignment Theorem 2.8 states that the user equilibrium problem formulated as the optimization problem (2.71)–(2.73) is equivalent to the fixed point problem (2.74) for the network of parallel routes. The representation of the user equilibrium problem as (2.74) allows us to reformulate the problem of finding the equilibrium assignment in the network of parallel routes in the form of a simple iterative process q+1

xk

 −1   q q q q b + Ak Gk (xk )Dk (xk ) − = Gk (xk )ATk Ak Gk (xk )ATk q q −Gk (xk )Dk (xk ), q+1 xn−k = O,

(4.29)

where the components xq+1 and the columns A are indexed so that  −1   q q q q b + Ak Gk (xk )Dk (xk ) , Dk (xk ) ≤ ATk Ak Gk (xk )ATk  −1   q q q q b + Ak Gk (xk )Dk (xk ) . Dn−k (xn−k ) > ATn−k Ak Gk (xk )ATk Theorem 4.3 ([41]) For any x0 from the feasible set (2.72), (2.73), the series {xq } of (4.29) remains within the feasible set and converges geometrically to x∗ .

Proof Let x∗ be an optimal solution of (2.71)–(2.73). Consider xq+1 − x∗ . Let us assume that xq belongs to the δ-neighborhood of x∗ and that k = k ∗ . For the ease of explanation, let us introduce the notation  −1   q q q q q q b + Ak Gk (xk )Dk (xk ) , Yk (xk ) = Gk (xk )ATk Ak Gk (xk )ATk   q+1 in this case, xk − xk∗ becomes q+1

xk

− xk∗ = Yk (xk ) − Gk (xk )Dk (xk ) − xk∗ . q

q

q

q

(4.30)

4.3 Projection Approach for Link-Flow Traffic Assignment

Note that

q

q

q

87

q

q

Gk (xk )Dk (xk ) = Gk (xk )tk (xk ) − xk , and, consequently, (4.30) is q+1

xk

  q q q q q − xk∗ = xk − xk∗ − Gk (xk )tk (xk ) + Yk (xk ). q

(4.31)

q

Consider the product Gk (xk )tk (xk ):   q q q q q Gk (xk )tk (xk ) = Gk (xk ) tk (xk ) − tk (xk∗ ) + Gk (xk )gk (xk∗ ), where

(4.32)

⎛ t1 (x q )−t1 (x ∗ ) ⎞ 1

t  (x ) q

1

⎜ q1 1 ∗ ⎟   ⎜ t2 (x2 )−tq2 (x2 ) ⎟ ⎜ t2 (x2 ) ⎟ q q Gk (xk ) tk (xk ) − tk (xk∗ ) = ⎜ ⎟. .. ⎟ ⎜ ⎠ ⎝ q .

(4.33)

tk (xk )−tk (xk∗ ) q tk (xk )

By the Lagrange mean value theorem let us rewrite (4.33): ⎛ t  (θ q ) 1

1 q

t1 (x1 ) q t2 (θ2 ) q  t2 (x2 )

x1 − x1∗ q

⎞

⎟ q x2 − x2∗ ⎟ ⎟ ⎟= .. ⎟ . ⎠

 q tk (θk ) q ∗ x − x q  k k   −1tk (xk ) q q q ∗ = Gk (xk ) Gk (Θk ) xk − xk ,

⎜   ⎜ ⎜ q q Gk (xk ) tk (xk ) − tk (xk∗ ) = ⎜ ⎜ ⎝



(4.34)

where θi belongs to the interval between xi and xi∗ , i = 1, k, Θk = (θ1 , . . . , θk )T . In the view of (4.34), we substitute (4.32) with (4.31): q

q+1 xk

−

xk∗

q

= Ek −

q Gk (xk )

q

q

q

  −1 !  q q q q q xk − xk∗ − Gk (xk )tk (xk∗ ) + Yk (xk ), Gk (Θk ) (4.35)

where Ek is the identity matrix of size k. Note that b + Ak Gk (xk )Dk (xk ) = Ak xk∗ + Ak Gk (xk )tk (xk ) − Ak xk = q

q

q

q

q

    q q q q = −Ak xk − xk∗ + Ak Gk (xk ) tk (xk ) − tk (xk∗ ) + Ak Gk (xk )tk (xk∗ ) =

88

4 Methods for Traffic Flow Assignment in Road Networks

  −1 !  q q q q xk − xk∗ + Ak Gk (xk )tk (xk∗ ). = −Ak Ek − Gk (xk ) Gk (Θk ) In this case, we obtain

q

q

Yk (xk ) =

(4.36)

 −1   −1 !  q q q q q xk − xk∗ + Ak Ek − Gk (xk ) Gk (Θk ) = −Gk (xk )ATk Ak Gk (xk )ATk  −1 q q q +Gk (xk )ATk Ak Gk (xk )ATk Ak Gk (xk )tk (xk∗ ). Consider the second summand from right part of (4.36) separately, putting (2.97) instead of tk (xk∗ ):  −1 q q q Ak Gk (xk )tk (xk∗ ) = Gk (xk )ATk Ak Gk (xk )ATk  −1  −1  q q q b+ = Gk (xk )ATk Ak Gk (xk )ATk Ak Gk (xk )ATk Ak Gk (xk∗ )ATk  +Ak Gk (xk∗ )Dk (xk∗ ) = q

= Gk (xk )ATk

  −1 −1  q q Ak Gk (xk )ATk b+ Ak Gk (xk )ATk Ak Gk (xk∗ )ATk  +Ak Gk (xk∗ )Dk (xk∗ ) =

  −1   q b + Ak Gk (xk∗ )Dk (xk∗ ) = = Gk (xk ) ATk Ak Gk (xk∗ )ATk = Gk (xk )tk (xk∗ ), q

q

q

thus Yk (xk ) is as follows

q

q

Yk (xk ) =

 −1   −1 !  q q q q q xk − xk∗ + Ak Ek − Gk (xk ) Gk (Θk ) = −Gk (xk )ATk Ak Gk (xk )ATk +Gk (xk )tk (xk∗ ). q

Eventually, we obtain from (4.35):

4.3 Projection Approach for Link-Flow Traffic Assignment q+1

xk q −Gk (xk )ATk

89

  −1 !  q q q xk − xk∗ − − xk∗ = Ek − Gk (xk ) Gk (Θk )

 −1   −1 !  q q q q T xk − xk∗ + Ak Gk (xk )Ak Ak Ek − Gk (xk ) Gk (Θk ) +Gk (xk )tk (xk∗ ) − Gk (xk )tk (xk∗ ) q

or q+1

xk

q

  −1 !  q q q xk − xk∗ − − xk∗ = Ek − Gk (xk ) Gk (Θk )

 −1   −1 !  q q q q q − Gk (xk )ATk Ak Gk (xk )ATk xk − xk∗ . Ak Ek − Gk (xk ) Gk (Θk ) (4.37)   −1  q

q q , Uk xk = Ek − Gk (xk ) Gk (Θk )

Denote

and

 −1 q

q q Γk xk = Gk (xk )ATk Ak Gk (xk )ATk Ak ,

thus (4.37) could be rewritten as q+1

xk

  q  q q q  q − xk∗ = Uk xk xk − xk∗ − Γk xk Uk xk xk − xk∗ .

Thus, we have componentwise inequalities        q  q  q    q+1 q q xk − xk∗  ≤ Uk xk  xk − xk∗  + Γk xk Uk xk xk − xk∗ , and, hence,

Since

    q   q   q   q+1  xk − xk∗  ≤ Ek + Γk xk  Uk xk  xk − xk∗ .

(4.38)

q

Ak Γk xk = Ak

(4.39)

   q q  for any xk from the feasible set, a matrix Mk exists such that Γk xk  ≤ Mk . Thus, the inequality (4.38) is as follows           q+1 q q xk − xk∗  ≤ Ek + Mk Uk xk  xk − xk∗ ,   where Ek + Mk is a matrix with finite positive fixed elements. Then

90

4 Methods for Traffic Flow Assignment in Road Networks

     q   q   q+1  xk − xk∗  ≤ Ek + Mk  · Uk xk  · xk − xk∗ .

(4.40)

 q  q We have Uk xk  → 0 as xk → xk∗ , i.e.   q   q ∀ε : 0 < ε < 1 ∃ρ : Ek + Mk  · Uk xk  ≤ ε < 1 when xk ∈ Sρ (xk∗ ),     q where Sρ (xk∗ ) = xk : xk − xk∗  ≤ ρ . Therefore, since xk ∈ Sρ (xk∗ ) and due to (4.40), we have    q   q+1  xk − xk∗  ≤ ε xk − xk∗  ≤ ερ < ρ, q+1

and so, xk



q ∈ Sρ xk∗ . Thus, if xk0 ∈ Sρ (xk∗ ), then {xk } ∈ Sρ (xk∗ ) and     q x − x∗  ≤ εq x0 − x∗  , k k k k q

whence it follows that {xk } converges geometrically. Note that Theorem 4.3 holds for the delay functions that are smooth on the edges of the network. However, if the delay functions are twice continuously differentiable, we have Theorem 4.4 ([41]) Let the functions ti (·) ∈ C 2 (R1+ ), i = 1, n on the feasible set of solutions ((2.72) and (2.73)) be such that     t (x) ≥ αi > 0 and t  (x) ≤ βi < ∞. i i

(4.41)

Thus for any x0 from the feasible set (2.72), (2.73), the series {xq } of (4.29) remains within the feasible set and converges quadratically to x∗ . Proof For the ease of explanation, we split the proof the three parts. I. If for all i = 1, n the functions ti (·) ∈ C 2 (R1+ ), then by the Taylor expansion formula we have 1 q q q

q q 2 ti (xi∗ ) = ti (xi ) + ti (xi ) xi∗ − xi + ti (θi ) xi∗ − xi , i = 1, k, 2 where θi is some point between xi and xi∗ or in a matrix form q

q

 1 2  −1  −1  q q q q q Gk (Θk ) xk∗ − xk + xk∗ − xk , tk (xk∗ ) = tk (xk ) + Gk (xk ) 2

(4.42)

∗ q 2 where, for the ease presentation, xk − xk is a componentwise exponentiation

of q of the 2 power of xk∗ − xk . q Express tk (xk ) from (4.31) as follows    −1  −1 q q q+1 q q q q xk − xk = Gk (xk ) Yk (xk ) tk (xk ) + Gk (xk )

4.3 Projection Approach for Link-Flow Traffic Assignment

91

and add it to (4.42):   −1  q q+1 tk (xk∗ ) + Gk (xk ) xk − xk∗ = 2  −1 1   q −1  ∗ q q q q Gk (Θk ) xk − xk . = Gk (xk ) Yk (xk ) + 2

(4.43)

q

II. Express tk (xk ) from (4.42)  1 2  −1  −1  q q q q q tk (xk ) = tk (xk∗ ) − Gk (xk ) Gk (Θk ) xk∗ − xk − xk∗ − xk 2 and, in the view of (2.97), we obtain:  −1   q tk (xk ) = ATk Ak Gk (xk∗ )ATk b + Ak Gk (xk∗ )Dk (xk∗ ) −  1 2 −1  −1   q q q q Gk (Θk ) xk∗ − xk − xk∗ − xk . − Gk (xk ) 2 q

q

q

Substitute tk (xk ) with Yk (xk ):  −1   q q q q q q q Ak xk∗ + Ak Gk (xk )tk (xk ) − Ak xk , Yk (xk ) = Gk (xk )ATk Ak Gk (xk )ATk then

 −1  q q q q Ak xk∗ + Yk (xk ) = Gk (xk )ATk Ak Gk (xk )ATk −1    q b + Ak Gk (xk∗ )Dk (xk∗ ) − +Ak Gk (xk ) ATk Ak Gk (xk∗ )ATk  1 2  −1  −1   q q q q q Gk (Θk ) xk∗ − xk − xk∗ − xk − Ak xk , − Gk (xk ) 2

or

 −1    q q q q q Ak xk∗ − xk + Yk (xk ) = Gk (xk )ATk Ak Gk (xk )ATk  −1   q +Ak Gk (xk )ATk Ak Gk (xk∗ )ATk b + Ak Gk (xk∗ )Dk (xk∗ ) −   1 2   −1  q q q q xk∗ − xk , −Ak xk∗ − xk − Ak Gk (xk ) Gk (Θk ) 2

thus we have

 −1 q q q q Yk (xk ) = Gk (xk )ATk Ak Gk (xk )ATk

92

4 Methods for Traffic Flow Assignment in Road Networks

  −1   q Ak Gk (xk )ATk Ak Gk (xk∗ )ATk b + Ak Gk (xk∗ )Dk (xk∗ ) − 2   −1  1 q q q xk∗ − xk , − Ak Gk (xk ) Gk (Θk ) 2 or

 −1   q q q b + Ak Gk (xk∗ )Dk (xk∗ ) − Yk (xk ) = Gk (xk )ATk Ak Gk (xk∗ )ATk  −1 2  −1  1 q q q q q xk∗ − xk , Ak Gk (xk ) Gk (Θk ) − Gk (xk )ATk Ak Gk (xk )ATk 2

in the view of (2.97), we eventually have Yk (xk ) = Gk (xk )tk (xk∗ )− q

q

q

(4.44)

 −1 2  −1  1 q q q q q − Gk (xk )ATk Ak Gk (xk )ATk xk∗ − xk . Ak Gk (xk ) Gk (Θk ) 2 III. Placing (4.44) into (4.43), we have   −1  q q+1 tk (xk∗ ) + Gk (xk ) xk − xk∗ = = tk (xk∗ ) +

2 1   q −1  ∗ q Gk (Θk ) xk − xk − 2

−1 2  −1  1  q q q q xk∗ − xk , − ATk Ak Gk (xk )ATk Ak Gk (xk ) Gk (Θk ) 2 or

  1 2  −1  q+1 q q q xk − xk∗ = Gk (xk ) Gk (Θk ) xk∗ − xk − 2

(4.45)

 −1 2  −1  1 q q q q q xk∗ − xk . − Gk (xk )ATk Ak Gk (xk )ATk Ak Gk (xk ) Gk (Θk ) 2 Therefore, due to (4.45), we have   1  2  −1     q+1 q q q ∗   xk − xk  ≤ Gk (xk ) Gk (Θk )  xk∗ − xk  + 2  2  −1   1   q   q q q  + Γk (xk ) Gk (xk ) Gk (Θk )  xk∗ − xk  2 or

 1 2     −1    q+1 q q ∗    x∗ − xq  . G ≤ E − x + M (x ) G (Θ ) xk k k  k k k k k k k  2

(4.46)

4.3 Projection Approach for Link-Flow Traffic Assignment

93

   q   −1    g (x )  q q Note that Gk (xk ) Gk (Θk )  is a matrix with the fractions  gi (Θiq )  on the diagonal, i

i

and zeros as other elements. Therefore, in the view of (4.41), we have from (4.46):   1  2   q+1   ∗ q Ek + Mk Ωk Δ−1 xk − xk∗  ≤ k xk − xk  , 2

(4.47)

where Δk and Ωk are square matrices of size k with αi and βi on the diagonals respectively, and zeros as all other elements. Therefore, by definition it follows from (4.47) that the series {xq } of the iterative process (4.29) quadratically converges to x∗ for an arbitrary choice of x0 from the feasible set of solutions ((2.72) and (2.73)). q

q

q

q

q

q

Remark 4.2 Since b = Ak xk and Gk (xk )Dk (xk ) = = Gk (xk )tk (xk ) − xk , the iterative process (4.29) could be rewritten as follows q+1

xk

 −1   q q q q q = xk − Gk (xk ) Ek − ATk Ak Gk (xk )ATk Ak Gk (xk ) tk (xk ), q+1

xn−k = O.

(4.48)

Note that the expression in the square brackets is a generalized projection matrix. Therefore, the projection operator for the problem (2.71)–(2.73) is obtained explicitly. Remark 4.3 Explicitly expressed, the iterative process (4.29) looks like q+1

xk

q

q

q

q

q

= xk − Gk (xk )tk (xk ) + Yk (xk )

or componentwise q

q+1

xi

q

= xi −

gi (xi ) q q q q + yi (x 1 , . . . , x k ). gi (xi )

Therefore, at each new iterative step the optimization problem (2.71)–(2.73) is solved for the case of linear delay functions that correspond to the tangents of the initial delay functions at the points obtained at the previous step, at which an optimal assignment of flows is achieved. Theorems 4.3 and 4.4 suggest that the iterative process (4.29) for x0 from the δneighborhood of x∗ converges to an user equilibrium point for the flow assignment F in the network with one origin-destination pair; moreover, when the delay functions meet some reasonable conditions on the links, we have quadratic convergence. As may be seen from the proofs of both theorems, one of the key requirements to the neighborhood is the presence of information about actually used edges of the road network. However, the explicitly obtained projection operator (4.48) could be utilized efficiently to cope with respective variational inequalities. The traffic assignment

94

4 Methods for Traffic Flow Assignment in Road Networks

problem is proved to be formulated in terms of variational inequalities as well [15]. Therefore the obtained results also contribute to projection methods for variational inequalities.

4.4 Route-Flow Assignment in a Linear Network as a System of Linear Equations Consider the road network of parallel routes. We assume that each alternative (parallel) route consists of edges with different characteristics. We call such a network the road network of parallel heterogeneous routes. Introduce notations: N = {1, . . . , n} is the set of parallel (non-interfering) routes between an origin-destination pair; L i = {1, . . . , li } is the set of consequently numbered edges of the route i, i = 1, n; F is the trip demand between an norigin-destination pair; f i is the flow through route f i = F; h il is a free traffic flow on the edge l of i, i = 1, n, f = ( f 1 , . . . , f n ), i=1 route i, h i = (h i1 , . . . , h ili ), i = 1, n; til ( f i , h il ) is the travel time on the congested edge l of route i, l = 1, li , i = 1, n. Let us formulate the traffic assignment problem in the network of parallel routes in the form of an optimization program [42]: min z( f ) = min f

f

li  n   i=1 l=1

fi

til (u, h il )du,

(4.49)

0

subject to n 

f i = F,

(4.50)

i=1

f i ≥ 0 ∀i = 1, n.

(4.51)

Note that (4.49) has ( f i , h il ) as arguments in the til (.) for all l = 1, li , i = 1, n, in spite of the fact that the control variables vector is f = ( f 1 , . . . , f n ), while free flows h il for all l = 1, li , i = 1, n, are some fixed parameters. Such a formalization is made intentionally, since h = (h 1 , . . . , h n ) carries a very important meaning. Let us introduce additional notations: ai =

li  j=1

ai j , bi =

li  j=1

bi j , h i =

li 

bi j h i j , i = 1, n.

(4.52)

j=1

Lemma 4.1 Without loss of generality, we assume that routes are numbered as follows

4.4 Route-Flow Assignment in a Linear Network as a System of Linear Equations

a1 + h 1 ≤ . . . ≤ an + h n .

95

(4.53)

Thus the user-equilibrium in the network of parallel heterogeneous routes is achieved by the following traffic assignment fi =

F+ bi

r

as +h s bs 1 s=1 bs

s=1

r

−

ai + h i , i ≤ r, bi

(4.54)

f i = 0, r < i ≤ n,

(4.55)

r r   ar + h r − ai − h i ar +1 + h r +1 − ai − h i ≤F< . bi bi i=1 i=1

(4.56)

where r is such that

Proof The Lagrangian for the problem (4.49) with constraints (4.50), (4.51) is L=

n   i=1

fi

" (ai + bi u + h i )du + t

∗

F−

0

n 

# fi

i=1

+

n 

ηi (− f i ).

i=1

We differentiate The Lagrangian: ∂L = (ai + bi f i + h i ) − t∗ − ηi ∀i = 1, n. ∂ fi

(4.57)

According to the complementarity condition ηi f i ∗ = 0 for all i = 1, n. Therefore, in the view of (4.57), we obtain:  ai + bi f i + h i

= t∗ , for f i ∗ > 0, ≥ t∗ , for f i ∗ = 0,

∀i = 1, n.

, for ai + h i < t∗ , for ai + h i ≥ t∗ ,

∀i = 1, n.

whence ∗

fi = If we take r such that

then

 t∗ −ai −h i bi

0,

ar + h r ≤ t∗ < ar +1 + h r +1 , r  i=1

fi =

r  t∗ − ai − h i = F, bi i=1

(4.58)

96

4 Methods for Traffic Flow Assignment in Road Networks

whence we express

 s F + rs=1 as b+h t = r 1 s . ∗

s=1 bs

Eventually, to define r we can use (4.58). Theorem 4.5 The optimization problem (2.2)–(2.5) with linear delay functions is equivalent to a system of linear equations and linear inequalities for variables f . Proof Consider an arbitrary route r ∈ R w between the OD-pair w ∈ W . The route r consists of edges r j (up to the notation), j = 1, lrw . The flow frw ≥ 0 is assigned through the route r ∈ R w between an OD-pair w ∈ W , hence, each edge r j, j = 1, lrw , is loaded by the flow frw . Moreover, now we suppose that each edge r j, j = 1, lrw , could also be loaded by flows from some other routes f pν : for each edge r j, j = 1, lrw , the sum of such flows is a stationary flow h rwj on the edge r j of route r ∈ R w of the OD-pair w ∈ W . In this case, for each pair w ∈ W , the following optimization problem could be formulated lrw  f w  r tr j (u, h rwj )du, min w f

r ∈R w j=1

subject to



0

frw = F w ,

r ∈R w

frw ≥ 0, with definitional constraints   f pν δ νj, p , ∀ j ∈ r, r ∈ R w , w ∈ W, h rwj = ν∈W p∈R ν \r

i.e. the sum of all route flows crossing edge j of route r , besides frw . According to lemma, the user equilibrium for any such subnetwork is frw

=

r w

asw +h ws s=1 bsw w brw rs=1 b1w s

Fw +

−

arw + h rw , r ≤ r w, brw

frw = 0, r w < r ≤ n,

(4.59) (4.60)

where r is such that r r   arww +1 + h rww +1 − arw − h rw arww + h rww − arw − h rw w ≤ F < , brw brw r =1 r =1 w

w

(4.61)

4.4 Route-Flow Assignment in a Linear Network as a System of Linear Equations

97

when a1w + h w1 ≤ . . . ≤ anw + h wn ,

(4.62)

where w

arw

=

lr 

w

arwj ,

j=1

brw

=

lr  j=1

w

brwj ,

h rw

=

lr 

brwj h rwj , ∀r ∈ R w , w ∈ W.

j=1

Let us show that the simultaneous solving of systems (4.59), (4.60) for all OD-pairs w ∈ W will lead to user equilibrium in the network G. Assume that  w f = . . . , f , . . . is a system consisting of (4.59), (4.60) for all OD-pairs w ∈ W , w

but it is not the user equilibrium. That means there is at least one f which is not w the user-equilibrium for the OD-pair w. However, if f satisfies the system of linear w equations (4.59), (4.60) for all w ∈ W , then f satisfies any subsystem (4.59), (4.60) w for concrete w. However, if f satisfies (4.59), (4.60) for some w, then due to the w lemma, f is the user equilibrium for OD-pair w. Thus, we obtain contradiction. Therefore, the simultaneous solving of systems (4.59), (4.60) for all w ∈ W indeed leads to user equilibrium in the network G. Example 4.4.1 Consider the network presented in Fig. 4.2. The graph is assumed to be directed and consists of two origins (circles) and two destinations (squares). Each source–sink pair is connected by two routes. The first pair (the red one) is connected by routes 1 and 2, while the second pair (the green one) is connected by routes 3 and 4. Route 1 consists of one edge; route 2 consists of edges 2.1, 2.2 and 2.3; route 3 consists of edges 3.1, 3.2 and 3.3; route 4 consists of one edge. Note that 2.2 and 3.2 are the same edge. Nevertheless, it is necessary to give any edge of a route its unique number. Therefore, there are two traffic assignment problems in two subnetworks of parallel routes: the red one and the green one (Fig. 4.3). Moreover F1 = 20 and F2 = 30. The characteristics of all four routes are given in table 4.3.

Fig. 4.2 Road network

98

4 Methods for Traffic Flow Assignment in Road Networks

Fig. 4.3 Road network Table 4.3 The characteristics of the routes Route 1 Route 2 i ai bi a2i b2i 1 2 3

4

0.5

2 1.5 1.5

0.2 0.15 0.2

Route 3 a3i b3i

Route 4 ai bi

1.5 1.5 1

5

Table 4.4 The characteristics of the routes Route 1 Route 2 i =1 i =2 ai bi hi

4 0.5 0

5 0.55 0.15 · f 3

0.15 0.15 0.2

0.4

Route 3 i =3

Route 4 i =4

4 0.5 0.15 · f 2

5 0.4 0

Then, due to (4.52), we obtain the integral characteristics of each route (the table 4.4). Due to (4.54), we have f1 =

f2 =

f3 =

20 + 0.5

20 +

+

5+0.15 f 3 0.55

1 + 0.5 0.55

4

0.51

+

−

4 0.5

5+0.15 f 3 0.55

1 1 + 0.5 0.55

−

5 + 0.15 f 3 0.55

4+0.15 f 2 + 5 10.5 1 0.4 0.5 0.5 + 0.4

−

4 + 0.15 f 2 0.5

0.55

4 0.5



30 +

f4 =

30 +

4+0.15 f 2 + 5 10.5 1 0.4 0.4 0.5 + 0.4

−

5 0.4

4.4 Route-Flow Assignment in a Linear Network as a System of Linear Equations

or

99

⎛

⎞⎛ ⎞ ⎛ ⎞ 1 0 −0.14 0 11.42 f1 ⎜0 1 ⎜ ⎟ ⎜ ⎟ 0.14 0 ⎟ ⎜ ⎟ ⎜ f 2 ⎟ = ⎜ 8.57 ⎟ , ⎝ 0 0.17 ⎠ ⎝ ⎠ ⎝ f3 1 0 14.44 ⎠ f4 0 −0.17 0 1 15.55

whence we obtain the route-flow assignment as a solution of the system of linear equations: ⎞ ⎛ ⎞ ⎛ 13.3 f1 ⎜ f 2 ⎟ ⎜ 6.7 ⎟ ⎟ ⎜ ⎟=⎜ ⎝ f 3 ⎠ ⎝ 13.3 ⎠ . f4 16.7 Eventually, we can substitute the obtained route-flows with the delay functions of the corresponding routes and establish t1 ( f 1 ) = t2 ( f 2 ) = 10.7, t3 ( f 3 ) = t4 ( f 4 ) = 11.7. Therefore, the user equilibrium assignment is actually achieved.

References 1. Frank M, Wolfe P (1956) An algorithm for quadratic programming. Naval Res Logist Quart 3:95–110 2. Gartner NH (1980) Optimal traffic assignment with elastic demands: a review Part II: algorithmic approaches. Transp Sci 14(2):192–208 3. Van Vliet D, Dow PDC (1979) Capacity-restrained road assignment. Traffic Eng Control 20:296–305 4. Van Vliet D (1976) Road assignment – I: principles and parameters of model formulation. Transp Res 10:137–143 5. Larsson T, Patriksson M (1992) Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transp Sci 26:4–17 6. Rose G, Daskin MS, Koppelman FS (1988) An examination of convergence error in equilibrium traffic assignment models. Transp Res 22B:261–274 7. Golden BL (1975) A minimum-cost multicommodity network flow problem concerning imports and exports. Networks 5:331–356 8. LeBlanc LJ, Morlok EK, Pierskalla WP (1974) An accurate and efficient approach to equilibrium traffic assignment on congested networks. Transp Res Record 491:12–23 9. LeBlanc LJ, Morlok EK, Pierskalla WP (1975) An efficient approach to solving the road network equilibrium traffic assignment problem. Transp Res 9:309–318 10. Nguyen S (1973) A mathematical programming approach to equilibrium methods of traffic assignment with fixed demands. Publication 138. Départment d’Informatique et de Reserche Opérationelle. Université de Montréal, Montréal 11. Steenbrink PA (1974) Optimization of transport networks. Wiley, London 12. Fratta L, Gerla M, Kleinrock L (1973) The flow-deviation method: an approach to store-andforward computer communication network design. Networks 3:97–133 13. Klessig RW (1974) An algorithm for nonlinear multicommodity flow problems. Networks 4:343–355 14. Yaged B Jr (1971) Minimum cost routing for static network models. Networks 1:139–172 15. Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc., Mineola, N.Y.

100

4 Methods for Traffic Flow Assignment in Road Networks

16. Florian M, Nguyen S (1976) An application and validation of equilibrium trip assignment methods. Transp Sci 10:374–390 17. Dijkstra EW (1959) A note on two problems in connexion with graphs. Numerische Mathematik 1:269–271 18. Gallo G, Pallottino S (1982) A new algorithm to find the shortest paths between all pairs of nodes. Discrete Appl Math 4:23–35 19. Gallo G, Pallottino S (1986) Shortest path methods: a unifying approach. Math Progr Study 26:38–64 20. Halder AK (1970) The method of competing links. Transp Sci 4:36–51 21. Loubal PS (1967) A network evaluation procedure. Highway Res Record 205:96–109 22. Rodionov VV (1968) The parametric problem of shortest distances. U.S.S.R. Comput Math Math Phys 8:336–343 23. Wollmer R (1964) Removing arcs from a network. Oper Res 12:934–940 24. Canon MD, Cullum CD (1968) A tight upper bound on the rate of convergence of the FrankWolfe algorithm. SIAM J Control 6:509–516 25. Dem’janov VF, Rubinov AM (1965) On the problem of minimization of a smooth functional with convex constraints. Soviet Math Doklady 6:9–11 26. Dunn JC (1979) Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals. SIAM J Control Optim 17:187–211 27. Dunn JC (1980) Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM J Control Optim 18:473–487 28. Levitin ES, Polyak BT (1966) Constrained minimization methods. USSR Compu Math Math Phys 6:1–50 29. Zangwill WI (1969) Nonlinear programming: a unified approach. Prentice-Hall Inc, Englewood Clifs, N.J 30. Wolfe P (1970) Convergence theory in nonlinear programming. In: Abadie J (ed) Integer and Nonlinear Programming. North-Holland, Amsterdam (1970) 31. Zangwill WI (1969) Convergence conditions for nonlinear programming algorithms. Manag Sci 16:1–13 32. Lupi M (1986) Convergence of the Frank-Wolfe algorithm in transportation networks. Civil Eng Syst 3:7–15 33. Janson BN, Zozaya-Gorostiza C (1987) The problem of cyclic flows in traffic assignment. Transp Res 21B:299–310 34. Konnov IV (2017) An adaptive partial linearization method for optimization problems on product sets. J Optim Theory Appl 175(2):478–501 35. Krylatov AY (2016) Network flow assignment as a fixed point problem. J Appl Ind Math 10(2):243–256 36. Marcotte P (1986) Network design problem with congestion effects: a case of bilevel programming. Math Progr 34(2):142–162 37. Greenshields BD (1934) A study of traffic capacity. Proc (US) Highway Res Board 14:448–494 38. Long J, Gao Z, Zhang H, Szeto WY (2010) A turning restriction design problem in urban road networks. Eur J Oper Res 206(3):569–578 39. Yang H, Yagar S (1995) Traffic assignment and signal control in saturated road networks. Transp Res Part A 29(2):125–139 40. Krylatov AY, Shirokolobova AP (2017) Projection approach versus gradient descent for networks flows assignment problem. Lect Notes Comput Sci 10556:345–350 41. Krylatov AY (2018) Reduction of a minimization problem for a convex separable function with linear constraints to a fixed point problem. J Appl Ind Math 12(1):98–111 42. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc, Englewood Cliffs, N.J

Chapter 5

Parallel Decomposition of a Road Network

Abstract In this chapter a new road network decomposition approach is proposed. Necessary statements are proved in the first section, and the formal description of the approach as well as a brief review on relevant references are presented. The offered approach is completely based on the very features of the equilibrium traffic assignment and, consequently, forms the methodological basis for direct solving algorithms. The implementation of the developed approach to a single-commodity network can be found in the second section. A new decomposition technique for route-flow assignment in a general topology network is presented in the third section. The fourth section is reserved for a decomposition technique for link-flow assignment in a general topology network. Moreover, capabilities for route-flows aggregation and link-flows disaggregation which appeared by virtue of the new approach are discussed.

5.1 Decomposition of a Road Network into Parallelized Subnetworks Parallel decomposition algorithms are a special type of algorithms for traffic assignment. Mainly, parallel decomposition algorithms are utilized for traffic assignment in a multi-commodity network. Such a network could be decomposed into a set of subnetworks and the traffic assignment problem could then be parallelly solved for all subnetworks. Parallel decomposition algorithms work especially efficiently under the presence of many processors. Since the problem could be large-scale, the processors should be powerful enough. The synchronization mechanism significantly affects the performance efficiency of parallel decomposition algorithms [1–4]. Indeed, theoretically, parallel decomposition could increase the convergence rate of any algorithm by virtue of parallel, non-sequential data processing. However in fact, some processors may be more powerful than others and, hence, they are forced to wait for the least powerful processor. Moreover, the efficiency of parallel computations could be decreased because of memory conflict or slow communication between the processors [1, 5]. Since the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_5

101

102

5 Parallel Decomposition of a Road Network

speed of computing and communication between the processors may be very different, the productivity of the whole system could decrease. Asynchronous computing accelerates the convergence of iterative methods [6, 7]. Note the principle of partial asynchronization. According to this principle, the upper bound for delays of data exchange between the processors and the upper bound for the difference between frequencies of parallel processors are determined. For instance, the principle of partial asynchronization was first used for algorithms of partial linearization [8–10]. A block Jacobi algorithm is a naturally parallel decomposition algorithm which allows the saving of the problem structure but ignores the mutual influence of flows between different OD-pairs. Actually, |W | traffic assignment problems are solved at each iteration q: 

x (q + 1) = arg min w w

x

e∈E

 j xew + j=w xe (q)

te (s)ds,

0

with constrained x w for all w ∈ W . The Jacobi-type algorithm is developed for the link-node formulation of the traffic assignment problem [11]. The piecewise linear approximation copes with the subproblems. Computational experiments demonstrate the increasing of the parallel algorithm’s efficiency when increasing the scale of the problem. Such a behavior seems to be natural, since the time costs for repeating a part of algorithm decrease. The algorithm from [12] looks like a Jacobi-type algorithm. The method is based on a weighted separable approximation of goal function at each iteration q: x (q + 1) = arg min w w

x

 e∈E

 j σ xew + j=w xe (q)

te (s)ds, σ > 0,

(5.1)

0

i.e. this is an extension of the standard approach via the introduction of weight ω. The subproblems are solved by an algorithm close to those from [11]. The parallel and sequential versions of the algorithm and their combinations are considered. Time costs and delays in communication between the processors are compared in the different versions of the algorithm. The algorithm developed in the paper [13] utilizes x w (q + 1) = arg min w x

 e∈E

σ xew

te (s)ds,

(5.2)

0

and each subproblem is solved by a dual approach. What is remarkable is that a goal function of any subproblem does not depend on iteration. Therefore, the algorithm is expected to be less efficient than a Jacobi approach one. The algorithms above are based on the solutions of subproblems with the same goal functions as in the original problem. Actually, this is their advantage, since the amount of iterations required for achieving a precise solution are few. On the other hand, the

5.1 Decomposition of a Road Network into Parallelized Subnetworks

103

nonlinear nature of goal function increases the complexity to the problem. Moreover, the complexity of the subproblems corresponding to different processors may differ significantly and, consequently, it could lead to the decreasing of efficiency of the parallel decomposition algorithm. Asynchronous computing is one of the possible solutions here. Another possible solution is to change the subproblems via making their complexity closer. The projection method for solving the decomposed traffic assignment problem was proposed in [14]. Application of this method was made in [15]. Higher efficiency of the developed algorithm in comparison to the Frank-Wolfe algorithm is established for a low-scale network. However, this algorithm is too expansive for large-scale networks, since the projection operation does not utilize the network structure. The original method of network parallel decomposition is proposed in this study. The network is decomposed into a set of independent subnetworks, and the traffic assignment problem is solved for each subnetwork independently. The most important feature of the developed algorithms is that route-flows instead of link-flows are taken as variables. Therefore, the obtained methods contribute to the development of route-flow assignment algorithms. Consider the road network presented by directed graph G = (V, E). Each OD-pair w ∈ W has its set of routes R w . Each route r ∈ R w consists of a sequence of edges er ∈ Er ⊂ E. Let h wer be the stationary flow on the edge er of route r when   r ∈ R w and h rw = {h wer }er ∈Er for all r ∈ R w . Introduce the function trw frw , h rw which defines the travel time of flow frw through the route r between OD-pair w. The travel time of the flow through the route naturally depends on the stationary flows on all edges along this route. Consider the following optimization problem: [PSPw ] min w f

subject to

 r ∈R w



0

frw

  trw u, h rw du,

frw = F w ,

(5.3)

(5.4)

r ∈R w

frw ≥ 0, ∀r ∈ R w ,

(5.5)

for any OD-pair w ∈ W . We call the problem (5.3)–(5.5) or PSPw as the parallelized subproblem of the traffic assignment problem in the network G for OD-pair w ∈ W . We call the subnetwork corresponding to subproblem PSPw as the parallelized subgraph PSGw of graph G for OD-pair w ∈ W . The assumption to be especially emphasized is that when we solve subproblem PSPw for any w, the route-flows of all other OD-pairs are believed to be stationary. Theorem 5.1 The solution of the traffic assignment problem (1.2)–(1.5) in network G satisfies PSPw for all w ∈ W simultaneously.

104

5 Parallel Decomposition of a Road Network

Proof The Lagrangian of the problem (5.3)–(5.5) is L = w

 r ∈R w

frw 0

trw

      w w w w u, h r du + t F − fr + (− frw )ηrw , r ∈R w

r ∈R w

where tw and ηrw ≥ 0 are the multipliers of Lagrange. Differentiate L w with respect to frw :   ∂ Lw = trw frw , h rw − tw − ηrw = 0 w ∂ fr or

  trw frw , h rw = tw + ηrw .

(5.6)

The complementarity slackness requires that frw · ηrw = 0 for all r ∈ R w , w ∈ W . Thus, if frw > 0, then ηrw = 0. If frw = 0, and then ηrw ≥ 0. From (5.6) we obtain   = tw , for frw > 0, trw frw , h rw ≥ tw , for frw = 0,

(5.7)

Since the goal function of the problem is convex, Kuhn-Tucker conditions are necessary and sufficient. Thus f w is a solution of (5.3)–(5.5) if and only if (5.7) holds. Herewith, (5.7) guarantees that the travel time on all actually used routes is equal and less than the free travel time on unused routes. Thus, (5.7) guarantees that a solution of (5.3)–(5.5) satisfies the first principle of Wardrop. The simultaneous holding of (5.7) for all w ∈ W leads to such an assignment that the travel time on all used routes is equal and less than the free travel time on any unused route between all OD-pairs. Therefore, the user equilibrium of Wardrop is achieved. Theorem 5.1 gives formal proof of the fact that the simultaneous independent solving of traffic assignment problems in the parallelized subnetworks PSGw leads to the user equilibrium assignment of Wardrop. However, the following meaningful interpretation of simultaneous independent solving of PSPw , w ∈ W could be given. Consider the network game when each OD-pair is associated with a player. Each player w ∈ W seeks to minimize their penalty function (5.3)subject to (5.4) and (5.5),  that can be written in the form of a non-cooperative game Ω W, {Fw }w∈W , {Pw }w∈W ,  w w where Fw = { f w | frw ≥ 0 ∀r ∈ R w , r ∈R w f r = F } for all w ∈ W , and Pw =

 r ∈R w

0

frw

  trw u, h rw du ∀w ∈ W.

Trip flows between different OD-pairs use a common network and, consequently, we have competition between the OD-pairs for the resources (capacities) of the network.

5.1 Decomposition of a Road Network into Parallelized Subnetworks

105

The competition lead us to Nash equilibrium search in the network game Ω. The Nash equilibrium in Ω is achieved by such f ∗ that    ∗ ∀w ∈ W, Pw f ∗ ≤ Pw f w , f −w

(5.8)

where f −w is a vector of the route-flows between all OD-pairs from W , besides the concrete pair w. Players decide how to assign flows simultaneously, which means that (5.9) h rw = h rw ( f −w ) ∀ r ∈ R w , w ∈ W. Theorem 5.2 ([16]) The Nash equilibrium in network game Ω is the user equilibrium of Wardrop in network G. Proof The Nash equilibrium in the game Ω is achieved by f w∗ satisfying (5.8) for each player w ∈ W . Let us show that if for any two routes p ∈ R w and r ∈ R w such that  w  w fp

0

fr

t pw (u, h wp )du >

trw (u, h rw )du,

0

(5.10)

a part of flow Δf from route p moves to route r , then the value of Pw will decrease. Indeed, if (5.10) holds, then 

f pw 0

 t pw (u, h wp )du −

frw

0

trw (u, h rw )du = δ > 0.

Since integrals with variable upper limits from (5.10) are continuous functions, positive Δf 1 and Δf 2 exists such that 

f pw −Δf 1

0



frw +Δf 2

0

 t pw (u, h wp )du >

f pw 0

 trw (u, h rw )du
0, r = 1, 2 fr = br 1 br fr =

1 ∗ ar + h r ·t − , for fr > 0, r = 3, 4 br 2 br

We have a linear system f 1 = 44 + 0.2 f 1 − 0.2 f 2 − 0.2 f 3 f 2 = 56 − 0.2 f 1 + 0.2 f 2 + 0.2 f 3 f 3 = 27 − 0.3 f 1 f 4 = 123 + 0.3 f 1 which has the following solution ⎛ ⎜ ⎜ ⎜ ⎝

with corresponding time delays

⎞ ⎛ ⎞ f 1∗ 35 ⎜ ⎟ f 2∗ ⎟ ⎟ ⎜ 65 ⎟ = ⎟ ⎜ ⎟, f 3∗ ⎠ ⎝ 17 ⎠ 133 f 4∗

(5.17)

5.3 Route-Flow Traffic Assignment in a General Road Network

113

⎛

⎞ ⎛ ⎞ t1 ( f 1∗ ) 52 ⎜ t2 ( f ∗ ) ⎟ ⎜ 52 ⎟ 2 ⎜ ⎟ ⎜ ⎟ ⎝ t3 ( f 3∗ ) ⎠ = ⎝ 33 ⎠ . t4 ( f 4∗ ) 33 Therefore, (5.17) is the route-flow traffic assignment in the network G 2 corresponding to the user-equilibrium of Wardrop. As mentioned above, route-flow assignment could easily convert to link-flow assignment, but never vice versa. For the network G 2 we have ⎞ ⎞ ⎛ ⎛ ∗⎞ ⎛ f 1∗ 35 x1 ⎟ ⎜ ⎜ x∗ ⎟ ⎜ f 2∗ ⎟ ⎟ ⎜ 65 ⎟ ⎜ 2∗ ⎟ ⎜ ∗ ⎟ ⎜ 17 ⎟ ⎜x ⎟ ⎜ f 3 ⎟ ⎟ ⎜ ⎜ 3∗ ⎟ ⎜ ∗ ⎟ ⎟ ⎜ ⎜x ⎟ = ⎜ (5.18) ⎜ 4∗ ⎟ ⎜ ∗ f 4 ∗ ⎟ = ⎜ 133 ⎟ . ⎜ x ⎟ ⎜ f + f ⎟ ⎜ 52 ⎟ ⎟ ⎜ 5∗ ⎟ ⎜ 1 ∗ 3 ⎟ ⎜ ⎠ ⎝ 17 ⎠ ⎝ x6 ⎠ ⎝ f3 100 x7∗ f 1∗ + f 2∗ Thus, link-flow assignment (5.18) in the network G 2 corresponds to the userequilibrium of Wardrop. For the ease of explanation we considered the linear network. However, the described technique could also be applied to non-linear network. Indeed, Sects. 2.3 and 4.4 give us full theoretical basis to convert the traffic assignment problem in the non-linear network to a system of non-linear equations.

5.4 Link-Flow Traffic Assignment in a General Road Network Consider a network with two OD-pairs (|W | = 2) presented by directed graph G 2 . Let circles be sources, while squares are sinks. The characteristics of the edges of the network G 2 are available in Table 5.7 (Fig. 5.6). Graph G 2 could be represented as two subnetworks SG1 and SG2 (each subnetwork has only one OD-pair). The network with the green OD-pair is assumed to be the first one (Fig. 5.7). The network with the red OD-pair is assumed to be the second one (Fig. 5.8).

Table 5.7 Characteristics of edges of G 2 e 1 2 3 ae be

4 0, 2

4,5 0,2

4,5 0,5

4

5

6

7

8

4 0,3

5 0,15

4,5 0,15

5,5 0,2

6 0,3

114

5 Parallel Decomposition of a Road Network

Fig. 5.6 The network with two OD-pairs

Fig. 5.7 The first subnetwork

Fig. 5.8 The second subnetwork

Let us describe the sequence of steps for decomposing the link-flow traffic assignment in the general network. Example 5.4.1 For the ease of explanation we believe that the travel time is directly proportional to the flow on an edge. Assume that trip demands F 1 = 100 and F 2 = 150 should be assigned using edges in the network G 2 (Fig. 5.3). Let us represent G 2 as two subnetworks SG1 and SG2 (Figs. 5.7 and 5.8). Thus, edges 4, 5 and 6 are

5.4 Link-Flow Traffic Assignment in a General Road Network

115

common edges for both subnetworks. Introduce notations: xew is the flow on the edge e between the OD-pair w, e = 1, 8 and w = 1, 2. Evidently, for the first OD-pair x12 = 0 and x82 = 0, while for the second OD-pair x21 = 0, x31 = 0 and x71 = 0. Therefore, we have two optimization problems. The optimization problem for the first subnetwork is  x4   x5  x6 min (4 + 0, 3u) du + (5 + 0, 15u) du + (4, 5 + 0, 15u) du , x1

0

0

0

subject to x41 + x51 = 100 x41 − x61 = 0 x41 ≥ 0, x51 ≥ 0, x61 ≥ 0 The optimization problem for the second subnetwork is  min x2



x4

+

x2



0

(4, 5 + 0, 5u) du+

0



x5

(4 + 0, 3u) du +

0

x3

(4, 5 + 0, 2u) du +

 (5 + 0, 15u) du +

0

x6

 (4, 5 + 0, 15u) du ,

0

subject to x22 + x32 = 150 x22 − x52 = 0 x32 − x62 = 0 x2 ≥ 0, x3 ≥ 0, x52 ≥ 0, x61 ≥ 0 Therefore, for the first subnetwork we have



 11 0 100 , b1 = , A1 = 1 0 −1 0 ⎛

⎞ ⎛ ⎞ 3, 33 0 0 4 G1 = ⎝ 0 6, 67 0 ⎠ , D1 = ⎝ 5 + 0, 15x52 ⎠ . 0 0 6, 67 4, 5 + 0, 15x62

116

5 Parallel Decomposition of a Road Network

And for the second subnetwork we have ⎛ ⎞ ⎛ ⎞ 11 0 0 150 A2 = ⎝ 1 0 −1 0 ⎠ , b2 = ⎝ 0 ⎠ , 0 1 0 −1 0 ⎛

5 ⎜0 2 G =⎜ ⎝0 0

⎞ ⎛ ⎞ 0 0 0 4, 5 ⎟ ⎜ 2 0 0 ⎟ 4, 5 ⎟ ⎟ , D2 = ⎜ 1 ⎠. ⎝ ⎠ 0 6, 67 0 5 + 0, 15x5 4, 5 + 0, 15x61 0 0 6, 67

For each subproblem we have the following expressions (Sect. 2.4 in the first chapter): x1 = G1 A1

T

   T −1  1 A1 G1 A1 b + A1 G1 D1 − G1 D1 ,

x2 = G2 A2

T

   T −1  2 A2 G2 A2 b + A2 G2 D2 − G2 D2 .

The right-side operators are contraction operators (Sect. 4.3 in the third section). Thus, fixed-point iterations could be run and converge to the unique solutions. Let us start from the following initial feasible solutions: ⎛ x1 =

⎞

x41 ⎝ x51 ⎠ x61

⎞ ⎛ ⎞ 75 x2 50 ⎟ ⎜ ⎜ ⎟ x 3⎟ ⎜ 75 ⎟ = ⎝ 50 ⎠ , x2 = ⎜ ⎝ x52 ⎠ = ⎝ 75 ⎠ , 50 x62 75 ⎛

⎞

⎛

then the iterative processes converge quadratically to ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ ⎞ 90 x2 x41 27 ⎜ x3 ⎟ ⎜ 60 ⎟ 1 2 1 ⎟ ⎜ ⎟ x = ⎝ x5 ⎠ = ⎝ 73 ⎠ , x = ⎜ ⎝ x52 ⎠ = ⎝ 90 ⎠ . 27 x61 x62 60 ⎛

Thus, we obtain solutions for traffic assignment problems for SG1 and SG2 when there are two actually used routes between the first OD-pair (links 1, 4, 6, 8 and links 1, 5, 8) and two actually used routes between the second OD-pair (links 2, 5, 7 and links 3, 6, 7). Indeed, compute travel times through the corresponding routes: t1 (x1 ) + t4 (x4 ) + t6 (x6 ) + t8 (x8 ) = 89.5, t1 (x1 ) + t5 (x5 ) + t8 (x8 ) = 89.5, and

5.4 Link-Flow Traffic Assignment in a General Road Network

117

t2 (x2 ) + t5 (x5 ) + t7 (x7 ) = 87.5, t3 (x3 ) + t6 (x6 ) + t7 (x7 ) = 87.5. However, one more possible route exists for the second OD-pair, consisting of links 2, 4, 6, 7. Let us check if this route is actually used or not. For such a purpose we are able to exploit the criterion on used routes from Sect. 2.4 of the first chapter: D24

A24

compare with

T



A2 G2 A2

 T −1

 2  b + A2 G2 D2 − G2 D2 ,

(5.19)

where D24 = (t4 (x4 ) − t4 (x4 )x41 ) = 12 and ⎛

⎞ 0 A24 = ⎝ −1 ⎠ . 1 Compute right-side of (5.19): A24

T

   T −1  2 A2 G2 A2 b + A2 G2 D2 − G2 D2 = 12.

Since D24 = A24

T



A2 G2 A2

 T −1

 2  b + A2 G2 D2 − G2 D2 ,

then x42 = 0 for the second subproblem, so the route consisting of links 2, 4, 6, 7 between the second OD-pair is unused (see Sect. 2.4 of the first chapter). Thus, eventual solutions for the subproblems are ⎛ ∗⎞ ⎛ ⎞ ⎛ ⎞ x2 90 ∗⎞ x41 27 ∗⎟ ⎜ ⎜ ⎟ x ∗ ∗ 3 2 ⎟ ⎜ 60 ⎟ . = ⎝ x51 ⎠ = ⎝ 73 ⎠ , x = ⎜ 2∗ ⎠ = ⎝ ⎝ x5 90 ⎠ ∗ 27 x61 2∗ 60 x6 ⎛

x1

∗

Since we find equilibrium link-flow traffic assignments for both subnetworks, we can aggregate flows to obtain equilibrium link-flow traffic assignment for the whole network: ⎞ ⎛ ∗⎞ ⎛ 100 x1 ⎜ x2 ∗ ⎟ ⎜ 90 ⎟ ⎟ ⎜ ∗⎟ ⎜ ⎜ x3 ⎟ ⎜ 60 ⎟ ⎟ ⎜ ∗⎟ ⎜ ⎜ x ⎟ ⎜ 27 ⎟ ∗ 4 ⎟. ⎜ ⎟ ⎜ x =⎜ ∗ ⎟=⎜ ⎟ ⎜ x5∗ ⎟ ⎜ 163 ⎟ ⎜ x ⎟ ⎜ 87 ⎟ ⎟ ⎜ 6∗ ⎟ ⎜ ⎝ x7 ⎠ ⎝ 150 ⎠ 100 x8∗

118

5 Parallel Decomposition of a Road Network

Therefore, the described approach gives link-flow assignments for each OD-pair of the network. In other words, this approach allows us to get disaggregated linkflows in the road network. We would like to stress once again that for the ease of presentation we considered a linear network. However, the same technique could be applied to a non-linear network. Theoretical basis for such an application is given in Sects. 2.4 and 4.3.

References 1. Bertsekas DP, Tsitsiklis JN (1989) Parallel and distributed computation: numerical methods. Prentice-Hall, London 2. Greenbaum A (1989) Synchronization costs on multiprocessors. Parallel Comput 10:3–14 3. Hockney RW, Jesshope CR (1988) Parallel computers 2: architecture, programming and algorithms. Adam Hilger, Bristol 4. Hwang K, Briggs FA (1985) Computer architecture and parallel processing. McGraw-Hill, Singapore 5. Kung HT (1976) Synchronized and asynchronous parallel algorithms for multiprocessors. In: Traub JF (ed) Algorithms and complexity: new directions and recent results. Academic Press, NY. 1, pp 53–200 (1976) 6. Bertsekas DP, Castañon DA (1991) Parallel synchronous and asynchronous implementations of the auction algorithm. Parallel Comput 17:707–732 7. Chajakis ED, Zenios SA (1991) Synchronous and asynchronous implementations of relaxation algorithms for nonlinear network optimization. Parallel Comput 17:873–894 8. Tseng P (1992) On the rate of convergence of a partially asynchronous gradient projection algorithm. SIAM J Optim 1:603–619 9. Tsitsiklis JN, Bertsekas DP, Athans M (1986) Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans Autom Control AC-31:803–812 10. Tsitsiklis JN, Bertsekas DP (1986) Distributed asynchronous optimal routing in data networks. IEEE Trans Autom Control AC-31:325–332 11. Feijoo B, Meyer RR (1988) Piecewise-linear approximation methods for nonseparable convex optimization. Manag Sci 34:411–419 12. Chen R-J, Meyer RR (1988) Parallel optimization for traffic assignment. Math Progr 42:327– 345 13. Larsson T, Migdalas A, Patriksson M (1993) A partial linearization method for the traffic assignment problem. Optimization 28:47–61 14. Rosen JB (1960) The gradient projection method for nonlinear programming, Part I: linear constraints. J Soc Ind Appl Math 8:181–271 15. Schwartz M, Cheung CK (1976) The gradient projection algorithm for multiple routing in message-switched networks. IEEE Trans Commun COM-24:449–456 16. Devarajan S (1981) A note on network equilibrium and noncooperative games. Transp Res Part B 15B(6):421–426

Part IV

Optimization Models and Methods for Network Design

Nowadays, more than ever, such class of problems as network design seems to be of extreme importance from the practical perspective. However, despite its evident practical importance the network design is a collection of disparate approaches frequently far from the achievements of advanced mathematics. Meanwhile, the optimal network design is most often a bi-level nonlinear optimization problem that does not have any precise efficient algorithm today to be implemented in a general case. Comprehensive analytical researches no less than others could contribute to eliminating such a shortcoming.

Chapter 6

Topology Optimization of Road Networks

Abstract In this chapter is mainly devoted to the capacity allocation problem as one of the most significant for road network topology optimization. A brief review on problems concerning network design and relevant fields is given in the Sect. 6.1. Capacity allocation control for a general topology network in the form of a mathematical problem is formulated in the Sect. 6.2. The Sect. 6.3 is devoted to solving the capacity allocation problem for a single-commodity linear network of non-interfering routes. The solution is obtained explicitly that allows to make practically substantial conclusions. The Sect. 6.4 addresses the problem of optimal capacity allocation control under multi-modal traffic flows. The multi-modality influence on optimal control strategy for capacity allocation is also discussed.

6.1 Bi-level Mathematical Programming for the Optimization of a Road Network Topology Well-formulated models of traffic processes, both from theoretical and practical perspectives, consider the route-choice behavior of drivers. Meanwhile, decisions concerning management, control and investment should be made in the interests of society. Therefore, transportation planning is a typical example of a hierarchical process when the society decides what to improve in the road network at the upper level, while drivers decide how to move at the lower level. In other words, the behavior of drivers can be estimated (predicted) but never dictated. For instance, the society chooses the streets for which the capacity is to be increased, but it is drivers who choose the routes most appropriate for them. Mathematical models describing such kinds of bi-level relationships are problems of bi-level mathematical programming or bi-level optimization [1]. A bi-level optimization problem describes a hierarchical system consisting of two levels of decision-making. The upper level, called a leader, controls variables y ∈ Y , while the lower level, called a follower, controls variables x ∈ X . The relationship between these two decision-making levels is established via their goal functions φ(x, y) and f (x, y). Eventually, there is a game between a leader and a follower, called Stackelberg competition [2, 3]. According to Stackelberg competition, the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_6

121

122

6 Topology Optimization of Road Networks

first player, leader, chooses y ∈ Y minimizing his/her goal function φ(x, y), and the second player, follower, reacts on the leader’s choice by choosing such x ∈ X that minimizes his/her goal function f (x, y). Therefore, the decision of a follower depends on the decision of the leader, i.e. x = x(y). Mathematically it could be written as follows: If mapping x : Y → X exists such that for any fixed y ∈ Y f (x(y), y) ≤ f (x, y), ∀x ∈ X, and if there exists y ∗ ∈ Y such that   φ x(y ∗ ), y ∗ ≤ φ (x(y), y) , ∀y ∈ Y, then the pair (x ∗ , y ∗ ), where x ∗ = x(y ∗ ), is called Stackelberg equilibrium with the first player as the leader and the second player as the follower. This is the most natural description of bi-level programs [1]. Theorem 6.1 ([4]) If Y and X are compact in R m and R n respectively, and if φ and f are real continuous functions on X × Y , then Stackelberg equilibrium exists (with any player as a leader). Consider the following problem of bi-level optimization: min φ (x(y), y)

(6.1)

x(y) = arg min f (x, y) .

(6.2)

y∈Y

where x∈X

We assume that X ⊆ R n and Y ⊆ R m are convex and compact, functions f and φ are continuous on X × Y and (6.2) has unique solution x(y) ∈ X for any y ∈ Y . In this case, according to Theorem 6.1, Stackelberg equilibrium exists in (6.1), (6.2). Thus, the problem (6.1), (6.2) is a general bi-level model in the sphere of transportation planning when y are the variables of the upper level (decisions of an authority/society), x are the variables of the lower level (estimation of drivers’ reaction). A road network topology is not only roads and intersections but the set of all road facilities (traffic lights, pedestrian crossings, etc.). However, commonly, when the road network is presented by a graph, edges are the roads and nodes are the intersections. In such a case, constraints caused by different road facilities could be taken into account via goal function (see Sect. 8.1). Thus, the optimal road network topology design is the design of nodes and edges as well as signal control, parking prices, toll pricing, etc. Note particularly that signal control significantly influences the optimal node-edge combinations in the graph [5–9].

6.1 Bi-level Mathematical Programming for the Optimization of a Road …

123

Depending on the form of the optimization problem there are different network design problems [10]: 1. Discrete network design problem considers discrete changes such as building of new roads, addition of new lanes, defining of one-way streets, or left turns restriction. 2. Continuous network design problem considers continuous changes such as capacity allocation, defining of green light cycle, toll pricing. 3. Mixed network design problem considers discrete and continuous changes simultaneously. Almost all research on the topic concerns the improvement of an already existing road network. Indeed, nowadays there is no need to build new cities. The challenge is to improve the quality of life in the already built cities. Note that researchers very rarely take the multi-modality of traffic flows into consideration. However, different modes of transport utilize real road networks. The following approaches exist for road network design under multi-modality of traffic [10]: • Different modes of transport do not cross. In this case, each mode of transport has its own road network. Such an approach could be efficient, for instance, when buses have separate lanes. • Different modes of traffic flows do not cross. For instance, when buses and private cars have common lanes. • Different modes of traffic flows interact. The majority of network design problems under multi-modality do not reflect the mutual influence of decisions made by different modes. The optimization problem of the lower-level is always a behavioral model of traffic flows in a road network. Most often the lower level is the user-equilibrium of Wardrop (see Sect. 2.1). Nevertheless, other behavioral models exist, such as the stochastic user equilibrium [11]. Drivers are assumed not to possess any information about exact travel time. Therefore, the following input for network design problem exists: (1) topological connectivity of nodes and edges; (2) trip demands between each OD-pair in the form of a matrix or vector; (3) capacity, amount of lanes, free travel time and parameters of delay functions; (4) constraints concerning physical and economical conditions; (5) possible set of projects for feasible network improvement; (6) costs of any possible project. Depending on the concrete network design problem, precise mathematical algorithms, heuristic or meta-heuristic algorithms could be used to cope with the problem. Precise algorithms are based on the mathematical properties of the problem. However, precise methods become unable for solving large scale problems. Heuristic and meta-heuristic methods are more efficient here, despite the fact that they cannot

124

6 Topology Optimization of Road Networks

achieve global optimum. These kinds of algorithms do not require any information about the mathematical properties of the problem. Herewith, the computational speed of such algorithms is visibly higher than the computational speed of precise algorithms. Precise algorithms were applied to the network design problem in [12–15]. However, the largest road network in that paper has 40 nodes and 99 edges. Sometimes the bi-level optimization problem of network design could be reduced to a one-level optimization problem [16–18]. For instance, one can define the equivalence of travel time on actually used routes as constraints [19–21]. Heuristic algorithms are the most useful class of methods for network design from practical perspectives. There are heuristic algorithms based on special representation of the lower level [22, 23] as well as algorithms coping with the bi-level optimization problem directly [6, 24, 25]. Meta-heuristic algorithms were applied to the network design problem in [26–28].

6.2 Optimal Capacity Allocation for General Road Network Consider the road network presented by directed graph G = (V, E). Let us introduce notations: V is the set of nodes of G; E is the set of edges of G; W is the set of OD pairs of G, w ∈ W ; R w is the set of routes between the OD-pair w ∈ W ; xe is the traffic flow on the edge e ∈ E, x = (. . . , xe , . . .); ce is the capacity of the edge e ∈ E, c = (. . . , ce , . . .); frw is the traffic flow through the route r ∈ R w ; F w is the trip demand between the OD-pair w ∈ W ; te (xe , ce ) is the travel delay of xe on the w is the indicator: congested edge e ∈ E with capacity ce ; δe,r  w = δe,r

1, if edge e ∈ Eis “included” in the route r ∈ R w ; 0, otherwise.

The capacity allocation problem for the network G can be formulated as the following bi-level optimization program: T (x ∗ , c∗ ) = min c

subject to





te (xe , ce )xe ,

(6.3)

e∈E

ce ≤ C,

(6.4)

e∈E

ce ≥ ce ∀e ∈ E,

(6.5)

when traffic is believed to be assigned according to the first principle of Wardrop:

6.2 Optimal Capacity Allocation for General Road Network

x ∗ = arg min x



with constraints

 e∈E

xe

te (u, ce )du,

125

(6.6)

0

frw = F w ∀w ∈ W,

(6.7)

frw ≥ 0 ∀r ∈ R w , w ∈ W

(6.8)

r ∈R w

and definitional constraints xe =

  w∈W

w frw δe,r ∀e ∈ E.

(6.9)

r ∈R w

Consider the optimization problem of the upper level. According to (6.5), the capacity of any edge could be increased from ce , e ∈ E. Constraint (6.4) states that the increasing of the overall capacity of the network is limited by C. This constraint could be caused by the budget for road network modernization. Goal function (6.3) minimizes the average travel time in the road network. Herewith, minimization of the upper level is possible only with respect to c since the authority can only control capacity. The reaction of drivers on capacity changes is described by the optimization problem (6.6)–(6.9). Mathematically, (6.6)–(6.9) is actually some mapping Φ: c → x. The solution x ∗ for (6.6)–(6.9) is proved to be user-equilibrium of Wardrop [29–31]. Unfortunately, generally x may not be expressed as a function depending on c. Moreover, as mentioned above, no concrete properties of Φ could be established. Therefore, the analysis of (6.3) depending on c could not be done because of the absence of any properties of the mapping. In other words, it is impossible to establish the direction of changing c depending on changing x. However, in a particular case such an analysis could be done. The particular case is the linear network of parallel (non-interfering) routes. We consider this class of networks in details in Sects. 6.3 and 6.4. It is almost impossible to solve the bi-level optimization problem of network design precisely since the problem is NP-hard. Even a simple bi-level optimization problem with both linear upper and lower levels is NP-hard [32]. Another trouble is the non-convexity of an optimization problem. Even in the case when both levels are convex, the convexity of the bi-level problem could not be guaranteed [33]. Moreover, (6.3)–(6.9) is not a unique formalization of the network design problem. Indeed, the lower level problem (6.6)–(6.9) could be rewritten in a node-link

form (2.8)–(2.11). In this case we obtain an alternative form of the network design problem:  te (xe , ce )xe , (6.10) T (x ∗ , c∗ ) = min c

e∈E

126

6 Topology Optimization of Road Networks

subject to



ce ≤ C,

(6.11)

e∈E

ce ≥ ce ∀e ∈ E,

(6.12)

when traffic is believed to be assigned according to the first principle of Wardrop: min x

with constraints

 e1 ∈V p

 e∈E



xew1 −

xe

te (u)du,

(6.13)

0

xew2 = d p , ∀ p ∈ V,

(6.14)

e2 ∈W p

xew ≥ 0, ∀e ∈ E, w ∈ W,

(6.15)

and definitional constraints xe =



xew , ∀e ∈ E,

(6.16)

w∈W

where V p is the set of edges terminating at node p, nd W p is the set of edges initiated at node p, and ⎧ ⎨ −F w , if node p is the source in pair w ∈ W, w d p = F w , if node p is the sink in pair w ∈ W, ⎩ 0, otherwise.

(6.17)

Theorem 6.2 ([31]) Solutions for the traffic assignment problem in both forms (6.6)–(6.9) and (6.13)–(6.16) exist. Theorem 6.2 guarantees the existence of a solution to the lower level problem. However, formulation (6.6)–(6.9) is more demonstrative because of its difficult applicability [31]. Moreover, the uniqueness of the solution in both formulations could be guaranteed only under a row of assumptions. Theorem 6.3 ([31]) Let te (·) be a strictly increasing function for any e ∈ E, then the optimization problem (6.6)–(6.9) (as well as (6.13)–(6.16)) has the unique solution. Theorem 6.3 guarantees the uniqueness of the solution to the traffic assignment problem when delay functions are strictly increasing. However, direct use of (6.13)– (6.16) in computational procedures leads to the necessity of processing huge amount of variables even for medium-scale networks. Efficient algorithms are based on the properties of the traffic assignment problem and the user-equilibrium principle [34].

6.2 Optimal Capacity Allocation for General Road Network

127

Eventually, it is computational procedures that lead to different assignments with the same value of the goal function [31]. Thus, additional troubles appear during the investigation of mapping Φ. However, according to Corollary 2.1, equilibrium traffic assignment could be obtained explicitly for the network of parallel routes. The explicit solution to the lower level problem allows us to represent the initial bi-level optimization problem as a one-level problem. Moreover, we use linear BPR-delay functions: ti ( f i ) =

ti0

 fi 1+ , ∀i = 1, n. ci

First of all, such a delay function has the concrete sense. Indeed, ci could be interpreted as the capacity of route i, i = 1, n. Secondly, as it is shown in Sect. 6.3, there are a number of reasons associated with the convenience.

6.3 Optimal Capacity Allocation for Corridor-Type Road Network Consider a network presented by two nodes and n directed edges (non-interfering routes). We call such a network a network of parallel n routes. F is the trip demand, f i = F, f =( f 1 , . . . , f n ). while f i ≥ 0 is the flow through the edge i = 1, n: i=1 The time delay on the edge i is the function ti ( f i , ci ) = ti0 1 +

fi ci

, where ti0 is

the free travel time on the edge i, and ci is the capacity of the edge i, i = 1, n, c = (c1 , . . . , cn ). Network design problem (6.3)–(6.9) for the network of parallel routes is T ( f ∗ , c∗ ) = min c

subject to

n 

n 

ti ( f i , ci ) f i

(6.18)

i=1

(ci − ci ) ≤ C,

(6.19)

ci ≥ ci ∀i = 1, n,

(6.20)

i=1

when traffic is believed to be assigned according to the first principle of Wardrop: f ∗ = arg min f

n   i=1

0

fi

ti (u, ci )du

(6.21)

128

6 Topology Optimization of Road Networks

with constraints

n 

f i = F,

(6.22)

i=1

f i ≥ 0 ∀i = 1, n,

(6.23)

where ci is the initial capacity of the edge i, i = 1, n, and C is the maximum overall network capacity changing to be achieved. Theorem 6.4 ([35]) Consider a congested linear network of parallel routes. Assume that

 n  max j=1,n t 0j − 1 . (6.24) F≥ (ci + C) ti0 i=1 Let I0 and I be subsets of indexes of routes such that I0 ∩ I = ∅, I0 ∪ I = {1, . . . , n} and ti01 = ti02 ∀ i 1 , i 2 ∈ I0 , ti01 < ti02 ∀ i 1 ∈ I0 , i 2 ∈ I. The solution to the network design problem (6.18)–(6.23) is the following pair of vectors ( f ∗ , c∗ ):  ∗ > ci , if i ∈ I0 , ci = ci , if i ∈ I,  (ci∗ − ci ) = C,

and

(6.25)

i∈I0

while f ∗ is the solution of (6.21)–(6.23) when c∗ is substituted. Proof According to [36], when (6.24) holds

for the  network of parallel routes with fi 0 linear BPR-delay functions ti ( f i , ci ) = ti 1 + ci , i = 1, n, then the following traffic assignment is guaranteed: f i∗

ci F + ns=1 cs = 0 n cs − ci , i = 1, n. ti s=1 t 0

(6.26)

s

In other words, the network is congested since all n routes are used. Moreover, according to (6.24), the network is still congested even in the case when all routes are increased with C. For the linear BPR-delay function (6.18) takes the following expression T ( f, c) =

n  i=1

ti0

fi 1+ ci

 fi .

(6.27)

6.3 Optimal Capacity Allocation for Corridor-Type Road Network

129

Substitute (6.26) into (6.27):   n  F + ns=1 cs ci F + ns=1 cs n cs · 0 n cs − ci , T ( f, c) = T ( f (c), c) = ti s=1 t 0 s=1 t 0 i=1 s

s

  n F + ns=1 cs  ci F + ns=1 cs n cs · − ci , T ( f (c), c) = n cs ti0 s=1 t 0 s=1 t 0 i=1

or

s

s

then we obtain n n F + ns=1 cs  F + ns=1 cs  ci F + ns=1 cs · · − · ci T ( f (c), c) = n cs n n cs cs t0 s=1 t 0 s=1 t 0 s=1 t 0 i=1 i i=1 s

s

s

and   n n  F + ns=1 cs F + ns=1 cs  T ( f (c), c) = n cs · F+ cs − n cs · ci , s=1 ts0

or

s=1 ts0

s=1

i=1



2 n F + ns=1 cs F + n cs  n cs − n s=1 · ci . T ( f (c), c) = cs s=1 ts0

s=1 ts0

i=1

Thus T ( f (c), c) =

F2 + 2 · F ·

2 2  n  n n n ci + F · i=1 s=1 cs + s=1 cs s=1 cs n cs n cs − , s=1 ts0

s=1 ts0

that eventually leads to F2 + F · n T ( f (c), c) =

n

s=1 cs cs s=1 ts0

,

or in a compact form: T ( f (c), c) = F ·

F + ns=1 cs n cs . s=1 ts0

Therefore, we have the following optimization problem 

∗

T f (c ), c

∗



F + ns=1 cs = min F · n cs c

s=1 ts0

(6.28)

130

6 Topology Optimization of Road Networks

subject to

n 

(ci − ci ) ≤ C,

(6.29)

ci ≥ ci ∀i = 1, n.

(6.30)

i=1

Lagrangian of the problem (6.28)–(6.30) is  n  n   F + ns=1 cs +ω (ci − ci ) − C + ηi (ci − ci ), L = F · n cs s=1 ts0

i=1

i=1

where ω ≥ 0 and ηi ≥ 0, i = 1, n, are the multipliers of Lagrange. Let us differentiate Lagrangian with respect to ci ∂L ∂ =F· ∂ci ∂ci



 F + ns=1 cs n cs + ω − ηi s=1 ts0

and equate it to zero: n

cs s=1 ts0

F·

  − F + ns=1 cs ·  2 n cs s=1 ts0

1 ti0

+ ω − ηi = 0.

According to the complementary slackness (ci − ci ) · ηi = 0 for all i = 1, n. Thus, if ηi > 0, then ci = ci . If ci > ci , then ηi = 0. We have:  F·

F+

n

s=1 cs



 n

·

1 ti0

cs s=1 ts0

− 2

n

cs s=1 ts0



= ω for ci > ci , ≤ ω for ci = ci ,

∀i = 1, n.

(6.31)

 n  Analogically, according to the complementary slackness ω · i=1 (ci − ci ) − C n = 0. Therefore, we have two cases. If i=1 (ci − ci ) < C, then ω = 0, that leads (6.31) to  F·

F+

n

s=1 cs

 n



·

1 ti0

cs s=1 ts0

− 2

n

cs s=1 ts0



= 0 for ci > ci , ≤ 0 for ci = ci ,

∀i = 1, n.

(6.32)

Assume that there are such indexes i that ci > ci . Without loss of generality we believe that this is true for the first r indexes {1, . . . , n}. Thus we have ci > ci for i = 1, r , which leads (6.32) to the system

6.3 Optimal Capacity Allocation for Corridor-Type Road Network

⎧ ⎪ ⎪ [F + c1 + . . . + cr ] · ⎨ .. . ⎪ ⎪ ⎩ [F + c1 + . . . + cr ] ·

131

1 t10

=

c1 t10

+ ... +

cr , tr0

1 tr0

=

c1 t10

+ ... +

cr , tr0

which has a solution if and only if t10 = . . . = tr0 . However, if t10 = . . . = tr0 , then F is zero. Therefore, the solution c∗ should be searched on the border of noptimal ∗ feasible set (6.29), i.e. i=1 (ci − ci ) = C and ω∗ > 0. Consider (6.31) when ω > 0. Assume that indexes i exist such that ci > ci . Without loss of generality we believe that the first r indexes among {1, . . . , n}. In this case ci > ci for i = 1, r . In the view of (6.32), we have the system ⎧ ⎪ [F + c1 + . . . + cr ] · ⎪ ⎪ ⎨

1 t10

=

⎪ ⎪ ⎪ ⎩ [F + c + . . . + c ] · 1 r

1 tr0

=

ω F

·

ω F

·





c1 t10

+ ... +

cr tr0

c1 t10

+ ... +

cr tr0

.. .

2

2



+



+

c1 t10

+ ... +

cr tr0

c1 t10

+ ... +

cr tr0





,

,

which has a solution if and only if t10 = . . . = tr0 . If t10 = . . . = tr0 , then F= and since

r

i=1 ci

=C+

ω · [c1 + . . . + cr ]2 , t10 · F

r

i=1 ci ,

then

ω = t10

C+

2

F r

i=1 ci

.

According to (6.31), routes i = 1, r with ci > ci have the property ti0 < t 0j , where j = r + 1, n. Indeed, for any route j = r + 1, n, in the view of (6.31), we have  F·

F+

n

s=1 cs

 n



·

1 ti0

cs s=1 ts0

− 2



n

cs s=1 ts0

=ω>F·

F+

n

s=1 cs

 n



·

1 t 0j

cs s=1 ts0

− 2

n

cs s=1 ts0

,

that is true if and only if ti0 < t 0j . Corollary 6.1 ([35]) If |I0 | = 1 then the solution of the bi-level optimization problem (6.18)–(6.23) is unique. If |I0 | > 1 then there are infinite alternatives of c∗ that lead to the same f ∗ . Herewith, any pair of such ( f ∗ , c∗ ) leads to the minimum of (6.18) subject to (6.19), (6.20).

132

6 Topology Optimization of Road Networks

Proof According to (6.26), the following non-zero flows correspond to the optimal c∗ : c∗ F + n cs∗ − ci∗ , i = 1, n. (6.33) f i∗ = 0i n s=1 cs∗ ti s=1 t 0 s

Moreover, according to the theorem (6.33) transforms f i∗

ci∗ F + s∈{1,...,n}\I0 cs∗ + s∈I0 cs∗ = 0 − ci∗ , cs∗ cs∗ ti + s∈{1,...,n}\I t 0 s∈I t 0 0

0

s

s

and in the view of (6.25) we obtain ci∗ F + s∈{1,...,n}\I0 cs∗ + C + s∈I0 cs ∗ − ci∗ , fi = 0 C+ s∈I0 cs cs∗ ti + s∈{1,...,n}\I0 ts0

t 0j

where j ∈ I0 . Therefore, changes of ci∗ for i ∈ I0 in the view of (6.25) do not lead to changes of f ∗ . We can see that according to Theorem 6.4 the shortest routes should be extended in the first place to decrease the average travel time in the road network. Building of ring or radian roads could be an efficient measure only if there is not any possibility for increasing the capacity of already existing roads. A good real example of

Fig. 6.1 The road network of Saint Petersburg city

6.3 Optimal Capacity Allocation for Corridor-Type Road Network

133

such a conclusion is West High Speed Diameter in Saint Petersburg city. The Diameter creates additional the shortest route between the south and the north of Saint Petersburg. The obtained theoretical results state that the most efficient investment in the capacity of a road network is increasing the capacity of the shortest routes between origins and destinations. Thus, one of the most real measures for increasing the road network capacity is the prohibition to park vehicles along the shortest routes between all origins and destinations. In other words, traffic engineers should define the shortest routes between all origins and destinations and prohibit the parking of vehicles along these routes. Figure 6.1 illustrates this approach for the road network of Saint Petersburg city. There are 7 experimental OD-pairs (from peripheral areas to the center): (1, 8), (2, 8), (3, 8), (4, 8), (5, 8), (6, 8), (7, 8). The shortest routes between OD-pairs are bold. From the perspectives of the obtained results, it is these routes that should be extended by traffic engineers in the first place.

6.4 Optimal Capacity Allocation for Corridor-Type Road Network Under Multi-modal Traffic In this section, under multi-modal traffic flows, we understand the presence of different groups of transport. We assume that there are groups of users of a road network who seek to minimize the average journey time of their groups, and there are individual users of the network who seek to minimize their own journey time. Consider traffic assignment in the network of parallel routes under such kind of multi-modality. F is n f i = F, the trip demand, while f i ≥ 0 is the flow through the edge i = 1, n: i=1 f = ( f 1 , . . . , f n ). Moreover, the trip demand of UG j is F j , j = 1, m and the trip demand of all individuals is F m+1 . The time delay on the edge i is the function  ti ( f i , ci ) = ti0 1 + cfii , where ti0 is the free travel time on the edge i, ci is the capacity of the edge i, i = 1, n, c = (c1 , . . . , cn ). The network design problem (6.3)–(6.9) for the network of parallel routes is T ( f ∗ , c∗ ) = min c

subject to

n 

n 

ti ( f i , ci ) f i

(6.34)

i=1

(ci − ci ) ≤ C,

(6.35)

ci ≥ ci ∀i = 1, n,

(6.36)

i=1

when traffic is believed to be assigned according to both selfish drivers and group:

134

6 Topology Optimization of Road Networks

f i∗ =

m 

fi

j∗

∗

+ f im+1 , i = 1, n,

(6.37)

j=1

for all UGs j, j = 1, m, we have f

j∗

= arg min

n 

j

f

ti ( f i1 , . . . , f im , f im+1 , ci ) f i

j

(6.38)

i=1

with constraints

n 

j

fi = F j ,

(6.39)

i=1 j

f i ≥ 0 ∀i = 1, n,

(6.40)

and for individuals ∗

f m+1 = arg min f m+1

with constraints

n   0

i=1

n 

f im+1

ti ( f i1 , . . . , f im , u, ci )du

f im+1 = F m+1 ,

(6.41)

(6.42)

i=1

f im+1 ≥ 0 ∀i = 1, n,

(6.43)

where ci is the initial capacity of the edge i, i = 1, n, and C is the maximum overall network capacity changing to be achieved. Theorem 6.5 Consider a congested linear network of parallel routes. Assume that F≥

n 

(ci + C)

max j=1,n t 0j ti0

i=1

and Fj + F ≥

n  i=1

(ci + C)

max j=1,n t 0j ti0

 −1

 − 1 ∀ j = 1, m.

(6.44)

(6.45)

Let I0 and I be subsets of indexes of routes such that I0 ∩ I = ∅, I0 ∪ I = {1, . . . , n} and ti01 = ti02 ∀ i 1 , i 2 ∈ I0 , ti01 < ti02 ∀ i 1 ∈ I0 , i 2 ∈ I.

6.4 Optimal Capacity Allocation for Corridor-Type Road Network …

135

The solution to the network design problem (6.34)–(6.43) is the following pair of vectors ( f ∗ , c∗ ):  ∗ > ci , if i ∈ I0 , ci = ci , if i ∈ I,  (ci∗ − ci ) = C,

and

(6.46)

i∈I0

while f ∗ is the solution of (6.38) and (6.41) when c∗ is substituted. Proof In Sect. 3.4 it is shown that when (6.44) and (6.45) hold for the network of parallel routes with linear BPR-delay functions ti ( f i , ci ) = ti0 1 + cfii , then the following traffic assignment is guaranteed: fi

j∗

=

ci Fj · n 0 ti s=1

(6.47)

cs ts0

for all j = 1, m, i = 1, n, and ∗

f im+1 =

ci F m+1 + ns=1 cs · − ci n cs ti0 s=1 t 0

(6.48)

s

In other words, (6.44) and (6.45) guarantee that the network is congested since all n routes are used. Moreover, according to (6.44) and (6.45), the network is still congested even in the case when all routes are increased with C. For the linear BPR-delay function, (6.34) takes the following expression T ( f, c) =

 fi fi . ti0 1 + ci i=1

n 

(6.49)

Substitute (6.47) and (6.48) with (6.49):   n  F + ns=1 cs ci F + ns=1 cs n cs · 0 n cs − ci , T ( f, c) = T ( f (c), c) = ti s=1 t 0 s=1 t 0 i=1 s

or

s

  n F + ns=1 cs  ci F + ns=1 cs n cs · − ci , T ( f (c), c) = n cs ti0 s=1 t 0 s=1 t 0 i=1 s

then we obtain

s

136

6 Topology Optimization of Road Networks

n n F + ns=1 cs  ci F + ns=1 cs F + ns=1 cs  T ( f (c), c) = n cs · · − · ci n n cs cs t0 s=1 t 0 s=1 t 0 s=1 t 0 i=1 i i=1 s

s

s

and   n n  F + ns=1 cs F + ns=1 cs  T ( f (c), c) = n cs · F+ cs − n cs · ci , s=1 ts0

or

s=1 ts0

s=1

i=1

2 n F + ns=1 cs F + ns=1 cs  n cs T ( f (c), c) = − n cs · ci . 

s=1 ts0

s=1 ts0

i=1

Thus T ( f (c), c) =

F2 + 2 · F ·

2 2  n  n n n ci + F · i=1 s=1 cs + s=1 cs s=1 cs n cs n cs − , s=1 ts0

s=1 ts0

that eventually leads to F2 + F · n T ( f (c), c) =

n

s=1 cs cs s=1 ts0

,

or in a compact form: T ( f (c), c) = F ·

F + ns=1 cs n cs . s=1 ts0

Therefore, we have the following optimization problem 

∗

T f (c ), c subject to

∗



F + ns=1 cs = min F · n cs

n 

c

(6.50)

s=1 ts0

(ci − ci ) ≤ C,

(6.51)

ci ≥ ci ∀i = 1, n.

(6.52)

i=1

Lagrangian of the problem (6.50)–(6.52) is  n  n   F + ns=1 cs +ω (ci − ci ) − C + ηi (ci − ci ), L=F· n cs s=1 ts0

i=1

i=1

6.4 Optimal Capacity Allocation for Corridor-Type Road Network …

137

where ω ≥ 0 and ηi ≥ 0, i = 1, n, are the multipliers of Lagrange. Let us differentiate Lagrangian with respect to ci ∂L ∂ =F· ∂ci ∂ci



 F + ns=1 cs n cs + ω − ηi s=1 ts0

and equate it to zero: n

cs s=1 ts0

F·

  − F + ns=1 cs ·  2

1 ti0

n cs s=1 ts0

+ ω − ηi = 0.

According to the complementary slackness (ci − ci ) · ηi = 0 for all i = 1, n. Thus, if ηi > 0, then ci = ci . If ci > ci , then ηi = 0. We have:  F·

F+

n

s=1 cs



 n

·

1 ti0

cs s=1 ts0

− 2

n

cs s=1 ts0



= ω for ci > ci , ≤ ω for ci = ci ,

∀i = 1, n.

(6.53)

 n  Analogically, according to the complementary slackness ω · i=1 (ci − ci ) − C n = 0. Therefore, we have two cases. If i=1 (ci − ci ) < C, then ω = 0, that leads (6.53) to  F·

F+

n

s=1 cs

 n



·

1 ti0

cs s=1 ts0

− 2

n

cs s=1 ts0



= 0 for ci > ci , ≤ 0 for ci = ci ,

∀i = 1, n.

(6.54)

Assume that there are such indexes i that ci > ci . Without loss of generality we believe that this is true for the first r indexes {1, . . . , n}. Thus we have ci > ci for i = 1, r , that leads (6.54) to the system ⎧ [F + c1 + . . . + cr ] · ⎪ ⎪ ⎨ .. . ⎪ ⎪ ⎩ [F + c1 + . . . + cr ] ·

1 t10

=

c1 t10

+ ... +

cr , tr0

1 tr0

=

c1 t10

+ ... +

cr , tr0

which has a solution if and only if t10 = . . . = tr0 . However, if t10 = . . . = tr0 , then F is zero. Therefore, the solution c∗ should be searched on the border of noptimal ∗ feasible set (6.51), i.e. i=1 (ci − ci ) = C and ω∗ > 0. Consider (6.53) when ω > 0. Assume that indexes i exist such that ci > ci . Without loss of generality we believe that the first r indexes among {1, . . . , n}. In this case ci > ci for i = 1, r . In the view of (6.54), we have the system

138

6 Topology Optimization of Road Networks

⎧ ⎪ [F + c1 + . . . + cr ] · ⎪ ⎪ ⎨

1 t10

=

⎪ ⎪ ⎪ ⎩ [F + c + . . . + c ] · 1 r

1 tr0

=

ω F

·

ω F

·





c1 t10

+ ... +

cr tr0

c1 t10

+ ... +

cr tr0

.. .

2

2



+



+

c1 t10

+ ... +

cr tr0

c1 t10

+ ... +

cr tr0





,

,

which has a solution if and only if t10 = . . . = tr0 . If t10 = . . . = tr0 , then F= and since

r

i=1 ci

=C+

ω · [c1 + . . . + cr ]2 , t10 · F

r

i=1 ci ,

then

ω = t10

C+

2

F r

.

i=1 ci

According to (6.53), routes i = 1, r with ci > ci have the property ti0 < t 0j , where j = r + 1, n. Indeed, for any route j = r + 1, n, in the view of (6.53), we have  F·

F+

n

s=1 cs

 n



·

1 ti0

cs s=1 ts0

− 2



n

cs s=1 ts0

=ω>F·

F+

n

s=1 cs

 n



·

1 t 0j

cs s=1 ts0

− 2

n

cs s=1 ts0

,

that is true if and only if ti0 < t 0j . Corollary 6.2 If |I0 | = 1 then the solution of the bi-level optimization problem (6.34)–(6.43) is unique. If |I0 | > 1 then there are infinite alternatives of c∗ that lead to the same f ∗ . Herewith, any pair of such ( f ∗ , c∗ ) leads to the minimum of (6.34) subject to (6.35), (6.36). Proof According to (6.47) and (6.48), the following non-zero flows correspond to the optimal c∗ : c∗ F + n cs∗ f i∗ = 0i n s=1 − ci∗ , i = 1, n. (6.55) cs∗ ti s=1 t 0 s

Moreover, according to the theorem, (6.55) transforms f i∗

ci∗ F + s∈{1,...,n}\I0 cs∗ + s∈I0 cs∗ = 0 − ci∗ , cs∗ cs∗ ti s∈{1,...,n}\I t 0 + s∈I t 0 0

0

s

s

and in the view of (6.46) we obtain ci∗ F + s∈{1,...,n}\I0 cs∗ + C + s∈I0 cs ∗ − ci∗ , fi = 0 C+ s∈I0 cs cs∗ ti + s∈{1,...,n}\I0 ts0

t 0j

6.4 Optimal Capacity Allocation for Corridor-Type Road Network …

139

where j ∈ I0 . Therefore, changes of ci∗ for i ∈ I0 in the view of (6.25) do not lead to changes of f ∗ . Therefore, optimal changing of the network under multi-modality is equal to optimal changing of the network with solely individual users. Certainly, the obtained results are true under all taken assumptions. However, they should be taken into consideration when planning investments in a road network. Indeed, the obtained results contribute in the development of methodological tools for decision-making support in the sphere of transportation planning.

References 1. Migdalas A (1995) Bilevel programming in traffic planning: models, methods and challenge. J Glob Optim 7(4):381–405 2. Stackelberg H (1952) The theory of the market economy. Oxford University Press, London 3. Mazalov VV (2014) Mathematical game theory and applications. Wiley 4. Simaan M, Cruz JB Jr (1973) On the Stackelberg strategy in nonzero-sum games. J Optim Theory Appl 11:533–555 5. Cascetta E, Gallo M, Montella B (2006) Models and algorithms for the optimization of signal settings onurban networks with stochastic assignment. Ann Oper Res 144(1):301–328 6. Chiou S (2008) A hybrid approach for optimal design of signalized road network. Appl Math Model 32(2):195–207 7. Meneguzzer C (1995) An equilibrium route choice model with explicit treatment of the effect of intersections. Transp Res Part B 29(5):329–356 8. Wey WM (2000) Model formulation and solution algorithm of traffic signal control in an Urban network. Comput Environ Urban Syst 24(4):355–377 9. Wong SC, Yang C (1999) An iterative group-based signal optimization scheme for traffic equilibrium networks. J Adv Transp 33(2):201–217 10. Farahani RZ, Miandoabchi E, Szeto WY, Rashidi H (2013) A review of Urban transportation network design problems. Eur J Oper Res 229:281–302 11. Daganzo CF, Sheffi Y (1977) On stochastic models of traffic assignment. Transp Sci 11(3):253– 274 12. Chen M, Alfa AS (1991) A network design algorithm using astochastic incremental traffic assignment approach. Transp Sci 25(3):215–224 13. Drezner Z, Wesolowsky GO (1997) Selecting an optimum configuration of one-way and twoway routes. Transp Sci 31(4):386–394 14. LeBlanc LJ, Boyce DE (1986) A bilevel programming algorithm for exact solution of the network design problem with user-optimal flows. Transp Res Part B 20(3):259–265 15. Long J, Gao Z, Zhang H, Szeto WY (2010) A turning restriction design problem in Urban road networks. Eur J Oper Res 206(3):569–578 16. Gao Z, Sun H, Zhang H (2007) A globally convergent algorithm for transportation continuous network design problem. Optim Eng 8(3):241–257 17. Gao Z, Wu J, Sun H (2005) Solution algorithm for the bi-level discrete network design problem. Transp Res Part B 39(6):479–495 18. Zhang H, Gao Z (2009) Bilevel programming model and solution method for mixed transportation network design problem. J Syst Sci Complex 22:446–459 19. Lo HK, Szeto WY (2009) Time-dependent transport network design under cost-recovery. Transp Res Part B 43(1):142–158 20. Szeto WY, Jaber X, O’Mahony M (2010) Time-dependent discrete network design frameworks considering land use. Comput-Aided Civil Infrastruct Eng 25(6):411–426

140

6 Topology Optimization of Road Networks

21. Szeto WY, Lo HK (2008) Time-dependent transport network improvement and tolling strategies. Transp Res Part A 42(2):376–391 22. Marcotte P, Marquis G (1992) Efficient implementation of heuristics for the continuous network design problem. Ann Oper Res 34(1):163–176 23. Marcotte P (1986) Network design problem with congestion effects: a case of bilevel programming. Math Program 34(2):142–162 24. Suh S, Kim T (1992) Solving nonlinear bilevel programming models of the equilibrium network design problem: a comparative review. Ann Oper Res 34(1):203–218 25. Ziyou G, Yifan S (2002) A reserve capacity model of optimal signal control with userequilibrium route choice. Transp Res Part B 36(4):313–323 26. Mathew TV, Sharma S (2009) Capacity expansion problem for large Urban transportation networks. J Transp Eng 135(7):406–415 27. Miandoabchi E, Farahani RZ (2010) Optimizing reserve capacity of Urban road networks in a discrete network design problem. Adv Eng Softw 42(12):1041–1050 28. Poorzahedy H, Rouhani OM (2007) Hybrid meta-heuristic algorithms for solving network design problem. Eur J Oper Res 182(2):578–596 29. Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc Inst Civil Eng 2:325–378 30. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc., Englewood Cliffs, NJ 31. Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc., Mineola, NY 32. Ben-Ayed O, Boyce DE, Blair CE III (1988) Ageneral bilevel linear programming formulation of the network design problem. Transp Res Part B 22(4):311–318 33. Luo Z, Pang J, Ralph DC (1996) Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge 34. LeBlanc LJ, Morlok EK, Pierskalla WP (1975) An efficient approach to solving the road network equilibrium traffic assignment problem. Transp Res 9:309–318 35. Krylatov AY (2017) Optimal strategies for road network’s capacity allocation. Vestnik SanktPeterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya. 13(2):182–192 36. Krylatov AY (2014) Optimal strategies for traffic management in the network of parallel routes. Vestnik Sankt-Peterburgskogo Universiteta, Prikladnaya Matematika, Informatika, Protsessy Upravleniya. 2:121–130

Chapter 7

Optimal Transit Network Design

Abstract In this chapter, the optimal transit network design is under investigation. A special kind of transport is assumed to be given the best traffic conditions in a road network (the smallest amount of travel time between origins and destinations). Optimality criteria for transit network design in case of selfish routing are defined in the first section. The Sect. 7.2 is devoted to the estimation of selfish traffic assignment in a network with a transit subnetwork. Optimality criteria for transit network design in case of competitive drivers’ groups routing are defined in the Sect. 7.3. The Sect. 7.4 is devoted to traffic assignment in case of competitive drivers’ groups routing in a network with a transit subnetwork. Explicit illustrative solutions are presented for a single-commodity road network with non-interfering routes.

7.1 Optimality Criteria for a Transit Network Design Authorities of large cities are typically faced with challenges in the transportation sphere concerning creating of special conditions for certain types of vehicles. For instance, authorities are interested in encouraging the use of environmentally friendly vehicles on transportation networks because green vehicles decrease total greenhouse gas emissions. Thus, appropriate arrangements should be performed to motivate drivers to use green vehicles instead of gasoline-powered vehicles. To achieve this goal, a transit network designed for green-vehicle routing could prove to be an efficient method. However, the question is how to offer attractive trip conditions to green vehicles. Another example is a transit network with toll prices. In this case a special type of vehicles is the vehicles with drivers who are ready to pay the toll price. The general question is how to offer the certain type of vehicles the most attractive trip conditions. Because the information regarding the amount of these certain vehicles currently on the road is believed to be available, the question could be reformulated quantitatively as how many routes should be available for only these vehicles to use. We call such routes the specialized routes in contrast to common routes. Considering this network design, if the specialized routes are not fully loaded while the common routes are overloaded, then the transportation network is unbalanced [1]. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_7

141

142

7 Optimal Transit Network Design

Conversely, if the specialized routes are overloaded by the current amount of special vehicles on the road, then using special cars will not provide a significant advantage for their drivers. Thus, it is necessary to determine the conditions that guarantee a well-balanced allocation of specialized and common routes in a given transportation network. This chapter is devoted to determining such conditions. In this section we intend to establish criteria of optimality for different types of transit networks. First, for ease of presentation, we consider a network of parallel routes including one OD-pair. Then we extend the obtained results to a general topology network. Nevertheless, in the first place it is assumed that there are only parallel routes (i.e., no intersections) from an origin to a destination. One set of these routes is defined as specialized and another as common. The specialized routes are assumed to be used only by special vehicles, while common routes can be used by both special and non-special vehicles. A transportation network represented by a digraph with one origin-destination pair and n parallel links is considered. Each edge is associated with a route from the origin to the destination. We use the following notation: N = {1, . . . , n} is the set of numbers of routes; N1 = {1, . . . , n 1 } is the set of numbers of specialized routes; N2 = {n 1 + 1, . . . , n 2 = n} is the set of numbers of common routes; G is the amount of special vehicles on the network; F is the amount of non-special vehicles on the network; gi is the flow of special vehicles on route i, i = 1, n, g = (g1 , . . . , gn ); f i is the flow of non-special vehicles on route i, i = n 1 + 1, n 2 , f = ( f n 1 +1 , . . . , f n ); ti0 is the free  traveltime on route i, i = 1, n; ci is the capacity of route i, i = 1, n; ti ( f i ) = ti0 1 + cfii is the travel time of flow f i through congested route i, i = 1, n. The travel time is modeled by a BPR-delay function [2]. Conditions must be defined that can guarantee (1) Wardrop user equilibrium on the entire transportation network, (2) the use of specialized routes solely by special vehicles, and (3) the reasonable allocation of specialized and common capacities where all specialized routes are used. The user equilibrium state on the network corresponds to the situation when the travel time of each vehicle between a fixed origin-destination pair is identical [3]. It indicates that special vehicles appear as an attractive alternative for daily trips until the travel time through the specialized routes is less than or equal to the travel time of vehicles through the other flows. Mathematically, this problem could be formulated as the following optimization problem: min z(g, f ) = min g, f

subject to

g, f

n  1  i=1

gi

ti (u)du +

0

n  i=1

 n2 

gi + f i

 ti (u)du

(7.1)

i=n 1 +1 0

gi = G,

(7.2)

7.1 Optimality Criteria for a Transit Network Design n 

143

f i = F,

(7.3)

i=n 1 +1

gi ≥ 0 ∀i = 1, n,

(7.4)

f i ≥ 0 ∀i = n 1 + 1, n 2 .

(7.5)

The unknown variables in (7.1)–(7.5) describe the traffic flows through all possible routes (both specialized and common) for each fixed value of n 1 . Generally, to solve such a problem, one must exploit complex computational procedures; however, this problem could be simplified by evaluating the boundary value of n 1 directly. The value of n 1 is shown to be a boundary when all special vehicles use only specialized routes, and the travel time of each special vehicle is less than or equal to the travel time of vehicles in the other flows that use common routes. The solution of (7.1)–(7.5) could contain routes with zero flows in the Wardrop user-equilibrium state (if ∃i = 1, n : gi = 0 and f i = 0). Such a problem may occur when the initial set of possible routes is poorly defined. Therefore, in this study, we assume that the initial set of possible routes is well balanced or fully loaded. Thus, the conditions of a fully loaded initial set of possible routes must be defined. At the same time, specialized routes must offer special vehicles less travel time between origin-destination nodes. The following is thus introduced: Definition 7.1 n 1 is deemed to be the optimal value if and only if: • The travel time of a special vehicle travelling via specialized routes is less than or equal to the travel time of any vehicle travelling via common routes; • The set of specialized routes is fully loaded when all special vehicles only use the specialized routes. Without a loss of generality, we assume that when n 1 is defined, the routes are numbered as follows: t10 ≤ . . . ≤ tn01 and tn01 +1 ≤ . . . ≤ tn02 .

(7.6)

Lemma 7.1 ([4]) Suppose all special vehicles G use only specialized routes. Then, the set of n 1 specialized routes is fully loaded if and only if G>

n1 

 ci

i=1

tn01 ti0

−1 ,

(7.7)

and the set of |n 2 − n 1 | special routes is fully loaded if and only if F>

n2  i=n 1 +1

 ci

tn02 ti0

−1 .

(7.8)

144

7 Optimal Transit Network Design

Proof If all special vehicles use only specialized routes, then the optimization problems (7.1)–(7.5) could be considered as two independent problems: (1) for special vehicles: min z 1 (g) = min g

g

subject to

n1 

n1   i=1

gi

ti (u)du

(7.9)

0

gi = G,

(7.10)

i=1

gi ≥ 0 ∀i = 1, n 1 ,

(7.11)

gi = 0 ∀i = n 1 + 1, n 2 .

(7.12)

(2) for non-special vehicles:  n2 

min z 2 ( f ) = min f

f

subject to

n2 

fi

ti (u)du

(7.13)

i=n 1 +1 0

f i = F,

(7.14)

i=n 1 +1

f i ≥ 0 ∀i = n 1 + 1, n 2 .

(7.15)

Considering the Lagrangian of the problems (7.9)–(7.12) L = 1

n1   i=1

gi 0

 ti (u)du + t

1

G−

n1 

gi

i=1

+

n1 

ηi gi

i=1

and differentiation yields: ∂ L1 = ti (gi ) − t1 + ηi = 0, ∂gi where t1 ≥ 0 and ηi ≥ 0 for i = 1, n 1 are the Lagrange multipliers. Due to the Kuhn–Tucker conditions

7.1 Optimality Criteria for a Transit Network Design

ti (gi )

145

= t1 when gi > 0, ≤ t1 when gi = 0,

(7.16)

for i = 1, n 1 . The inequality gi > 0 for i = 1, n 1 indicates that each specialized  route from the set of n 1 routes is used. Based on (7.16), if gi > 0, then ti0 1 + gcii = t1  1  and gi = tt 0 − 1 ci > 0 for i = 1, n 1 . Consequently, t1 > ti0 for i = 1, n 1 and, i

based on (7.6), t1 > tn01 . Thus, we obtain G=

n1 

gi =

i=1

 n1  tn01 − 1 ci > − 1 ci . ti0 ti0 i=1

n1  1  t i=1

The problem defined in (7.13)–(7.15) could be investigated similarly, eventually leading to (7.8). Lemma 7.1 defines the rule of determining the optimal n 1 . If inequality (7.7) holds, then all specialized routes are used when all special vehicles drive on only the specialized routes. A simultaneous execution of conditions (7.7), (7.8) indicates that the transportation network is fully loaded (i.e., all routes are in use) even though condition (7.7) does not guarantee that all special vehicles use only the specialized routes. To motivate special vehicles to use only the specialized routes, the decisionmakers must organize traffic in such a way that using specialized routes would be a preferable choice. Theorem 7.1 ([4]) Suppose that the initial set of routes is fully loaded with traffic demands G and F. The flow of specialized routes G uses only the specialized routes if and only if

n 2

n 1 c F + i=n ci G + i=1 1 +1 i

n 1 ci ≤ n 2 . (7.17) ci i=n 1 +1 ti0

i=1 ti0

Proof If all special vehicles use only specialized routes, the optimization problem defined by (7.1)–(7.5) could be considered in a form of two independent problems that are defined by (7.9)–(7.12) and (7.13)–(7.15). Due to (7.16), we obtain

n 1  t1 i=1 ti0 − 1 ci = G and, consequently,

n 1 ci G + i=1 1 t = n 1 ci . i=1 ti0

Based on (7.16): ti (gi ) =

n 1 ci G + i=1

n 1 ci when gi > 0, i=1 ti0

(7.18)

146

7 Optimal Transit Network Design

when n 1 is defined in such a way that all specialized routes are used, Eq. (7.18) is true for i = 1, n 1 . Therefore, (7.18) defines the travel time of any vehicle in the flow of special vehicles. Similarly, we can prove for (7.13)–(7.15) that

n 2 F + i=n c 1 +1 i ti ( f i ) = n 2 when f i > 0, ci

(7.19)

i=n 1 +1 ti0

for i = n 1 + 1, n 2 . Expression (7.19) defines the travel time of any vehicle from the flow of non-special vehicles. Consequently, inequality (7.17) claims that for any special vehicle, it is preferable to travel via specialized routes based on the travel time. Theorem 7.1 defines the second rule of determining the optimal n 1 . Indeed, If an authority provides special vehicles with routes in a way that condition (7.17) holds, then drivers will identify the advantage of using special cars (i.e., less travel time). Now consider a road network of general topology. A transportation network defined by a directed graph G that includes a set of sequentially numbered nodes V and a set of sequentially numbered links E is considered. The following notation is then used: R w is the set of possible routes between an origin-destination pair w; R1w is the set of specialized routes such that R1w ⊂ R w ; R2w is the set of common routes such that R2w ⊂ R w ; R1w ∩ R2w = ∅ and R1w ∪ R2w = R w ; G w and F w are the transportation demands for special and non-special vehicles between a given origin-destination pair w respectively; grw when r ∈ R w is the flow of special vehicles through route r , g = {grw }rw∈R w ; frw when r ∈ R2w is the flow of non-special vehicles through route r , f = { frw }rw∈R w ; xe is the transportation flow 2 through the link e ∈ E, x = (. . . , xe , . . .); ce is the capacity of link e ∈ E; te (xe ) is w w is an indicator such that δe,r =1 the travel time through congested link e ∈ E; δe,r w if the edge e lies along to route r between w, and δe,r = 0 otherwise. In these notations, the problem defined by (7.1)–(7.5) could be reformulated for a network with a general topology:   xe min Z (x) = min te (u)du (7.20) x

x

e∈E

0

subject to 

grw = G w ∀w ∈ W,

(7.21)

frw = F w ∀w ∈ W,

(7.22)

r ∈R w

 r ∈R2w

7.1 Optimality Criteria for a Transit Network Design

147

grw ≥ 0 ∀r ∈ R w , w,

(7.23)

frw ≥ 0 ∀r ∈ R2w , w,

(7.24)

with definitional constraints    w w grw + frw δe,r grw δe,r + . xe = w r ∈R1w

(7.25)

w r ∈R2w

Unknown variables in the optimization problem defined by (7.20)–(7.25) include the traffic flows via the available routes (both specialized and common). The most important aspect in this problem is that its solution depends on the set of specialized routes (i.e., the specialized subnetwork) used. To find the optimal solution to this problem, we consider a set of specialized routes as a parameter that is governed by an authority. Thus, a bi-level control system is considered. Special traffic flow assignments must consider the reactions of special and non-special vehicles based on the concept of Wardrop user equilibrium. In the previous subsection, we developed an approach that supports decisionmaking in a specialized transit network design. The primary criterion for decisionmakers when provided with certain routes that are used by only one type of vehicle (e.g., special vehicles) is the effectiveness of the network capacity allocation. The developed methodology allows us to find such a set of specialized routes that the following points are true: • The travel time of any amount of special vehicles (i.e., from zero flow up to all available special vehicles) using specialized routes is less than or equal to the travel time of the vehicles from the other traffic flows between the same origin-destination pair. • If all special vehicles only use specialized routes, then every specialized route is used, and the specialized subnetwork is not overloaded. The first statement guarantees the absolute advantage (i.e., less travel time) for special vehicles travelling via specialized routes. The second statement claims that the transportation network capacities are allocated effectively. The simultaneous execution of these two criteria can motivate drivers to use special vehicles and also guarantees the effectiveness of the network capacity allocation. Thus, by analogy with Theorem 7.1, we could state the following: Statement 7.1.1 Suppose that all special vehicles use only specialized routes, and the set of specialized routes is fully loaded. The special traffic flow G w ∀w uses only specialized routes if and only if   te (xe ) ≤ te (xe ) ∀r1 ∈ R1w , r2 ∈ R2w , w, e∈Er1

e∈Er2

where Er is the set of links belonging to route r ∈ R w for all w.

148

7 Optimal Transit Network Design

Therefore, the results obtained for a simple transportation network of parallel routes allow us to develop a two-step method: 1. Define the initial set of possible routes as fully loaded. 2. Define the set of specialized routes so that they offer special vehicles shorter travel times compared to the travel times of vehicles in the other traffic flows. This method is applied to a transportation network with a general topology. First, we evaluate the boundaries of the set R1w that could be identified when all special vehicles use only specialized routes and the travel time of any of these vehicles is less than or equal to the travel time of non-special vehicles using common routes. In this case, the problem defined by (7.20)–(7.25) can be divided into two independent problems: (1) for special vehicles: min Z 1 (x) = min x

x

 e∈E

xe

te (u)du

(7.26)

0

subject to 

grw = G w ∀w,

(7.27)

grw ≥ 0 ∀r ∈ R1w , w,

(7.28)

grw = 0 ∀r ∈ R2w , w,

(7.29)

r ∈R1w

with definitional constraints xe =

 w

w grw δe,r ,

(7.30)

r ∈R1w

(2) for non-special vehicles: min Z 2 (x) = min x

x

 e∈E

xe

te (u)du

(7.31)

0

subject to 

frw = F w ∀w,

(7.32)

frw ≥ 0 ∀r ∈ R2w , w,

(7.33)

r ∈R2w

7.1 Optimality Criteria for a Transit Network Design

149

with definitional constraints xe =



w frw δe,r .

(7.34)

w r ∈R2w

The optimization problems defined by (7.26)–(7.30) and (7.31)–(7.34) are ordinary linear problems that identify user equilibria on the transportation network [3]. Unfortunately, for a general network topology, it is impossible to obtain explicit conditions as in a network of parallel routes. However, there are currently many algorithms to solve this type of a problem (e.g., the Frank-Wolf algorithm [5]). Thus, a decision-maker could identify an optimal specialized subnetwork by following these steps: 1. Define the initial set of specialized routes R1w for each origin-destination pair. 2. Solve problems (7.26)–(7.30) and (7.31)–(7.34) using an available computational tool. 3. Determine if the travel times of special vehicles travelling via the specialized subnetwork are less than or equal to the travel times of vehicles travelling via the other traffic flows. 4. Determine if all specialized routes are used when all special vehicles are travelling via the specialized subnetwork. 5. If steps 3 and 4 are not true, then go back to item 1; otherwise, the optimal specialized subnetwork has been constructed. As shown, the design of a specialized transit network with a general network topology is a complex computational problem. To solve the problem defined by (7.26)–(7.30) and (7.31)–(7.34), special information technologies with specific software are required. However, in certain circumstances, a decision maker may desire estimated values with low accuracies. Then, the conditions obtained in the previous subsection could provide a convenient tool to support decision-making in the case of a general network topology.

7.2 Traffic Assignment in Road Networks with Transit Subnetworks In the previous section, we defined rules for the design of a specialized transit network. The conditions of Wardrop user equilibrium on a network with a specialized subnetwork are now determined. Let us begin from the network of parallel routes. Theorem 7.2 ([4]) Suppose the sets of specialized and common routes are fully loaded:   n1 n2   tn01 tn02 G> ci − 1 and F > ci −1 . ti0 ti0 i=1 i=n +1 1

150

7 Optimal Transit Network Design

The traffic flow assignment of special and non-special vehicles on the network of parallel routes achieve Wardrop user equilibrium (g ∗ , f ∗ ) if and only if (1) if n 1 satisfies

n 2

n 1 F + i=n c ci G + i=1 1 +1 i

n 1 ci ≤ n 2 , ci i=n 1 +1 ti0

i=1 ti0

then gi∗

1 cs ci G + ns=1 = 0 n 1 cs − ci for i = 1, n 1 , ti s=1 t 0 s

f i∗ =

n 2

ci F + s=n 1 +1 cs

n 2 − ci for i = n 1 + 1, n 2 ; cs ti0 s=n 1 +1 t 0 s

(2) if n 1 satisfies

n 2

n 1 F + i=n c ci G + i=1 1 +1 i

n 1 ci > n 2 , ci i=n 1 +1 ti0

i=1 ti0

then gi∗ =

1 cs ci G 1 + ns=1

− ci for i = 1, n 1 , n 1 cs ti0 0 s=1 t s

gi∗

+

f i∗

n 2 c ci G 2 + F + s=n 1 +1 s

n 2 = 0 − ci for i = n 1 + 1, n 2 , cs ti s=n 1 +1 t 0 s

where G 1 + G 2 = G and

n 2

n 1 F + G 2 + i=n c ci G 1 + i=1 1 +1 i

n 1 ci

n 2 = . ci i=n 1 +1 ti0

i=1 ti0

Proof The proof of the statement follows directly from Theorem 7.1 and Corollary 2.1. For a general topology network, by analogy with Theorem 7.2, the following can be formulated: Statement 7.2.1 Suppose that all routes are fully loaded: grw > 0 ∀r ∈ R1w and



 grw + frw > 0 ∀r ∈ R2w ∀w ∈ W.

The traffic flow assignment of special and non-special vehicles on a general network topology achieves Wardrop user equilibrium (g ∗ , f ∗ ) if and only if

7.2 Traffic Assignment in Road Networks with Transit Subnetworks

(1) if R1w satisfies 



te (xe ) ≤

e∈Er1

te (xe ) ∀r1 ∈ R1w , r2 ∈ R2w , w,

e∈Er2

then • grw ∗ is defined as the solution to the problem defined by (7.26)–(7.30), • frw ∗ is defined as the solution to the problem defined by (7.31)–(7.34); (2) for R1w satisfies 



te (xe ) >

e∈Er1

te (xe ) ∀r1 ∈ R1w , r2 ∈ R2w , w,

e∈Er2

then • grw ∗ r ∈ R1w is defined as the solution to the problem A   xe min Z1 (x) = min te (u)du x

x

e∈E

0

subject to 

grw = G w1 ∀w,

r ∈R w

grw ≥ 0 ∀r ∈ R1w , w, with definitional constraints xe =



w grw δe,r ,

r ∈R1w

w

  • grw ∗ + frw ∗ is defined as the solution to the problem B   xe min Z2 (x) = min te (u)du x

x

e∈E

0

subject to 

 grw + frw = G w2 + F w ∀w,

r ∈R2w



 grw + frw ≥ 0 ∀r ∈ R w , w,

151

152

7 Optimal Transit Network Design

with definitional constraints xe =



 w grw + frw δe,r ,

w r ∈R2w

where G w1 + G w2 = G w , G w1 and G w2 + F are the traffic flows through the specialized and common routes such that   te (xe ) = te (xe ) ∀r1 ∈ R1w , r2 ∈ R2w , w. e∈E k1

e∈E k2

Therefore, specialized routes are assumed to be used only by special vehicles. Herewith, specialized routes available for special vehicles only should be allocated optimal, i.e., in a user-equilibrium state, the travel time of special vehicles is less than or equal to the travel time of non-special vehicles between the same origin-destination nodes. Moreover, we developed a procedure for defining this set of specialized routes where the optimal allocation takes place. Optimization problems A and B lead to user equilibrium in both network and specialized transit subnetwork when special vehicles use specialized routes only [3]. Clearly, for special vehicles, it is reasonable to drive through specialized routes only if their travel times are less than or equal to the travel times of non-special vehicles. Therefore, the sets of specialized routes R1w for all w ∈ W are optimal if and only if the optimal travel time of special vehicles from program A is less than or equal to the optimal travel time of non-special vehicles from program B. However, the following theorem allows us to avoid considering the problems with the traffic equilibrium assignment on the specialized and common subnetworks separately. Theorem 7.3 ([6]) User-equilibrium assignment of special and non-special vehicles on the general network with specialized routes could be found as a solution to the following optimization program: min Z (x, G 1 , G 2 ) = min

x,G 1 ,G 2

with constraints for all w ∈ W

x,G 1 ,G 2



 e∈E

xe

te (u)du,

(7.35)

0

grw = G w1 ,

(7.36)

frw = G w2 + F w ,

(7.37)

r ∈R1w

 r ∈R2w

G w1 + G w2 = G w ,

(7.38)

grw ≥ 0 ∀r ∈ R1w ,

(7.39)

7.2 Traffic Assignment in Road Networks with Transit Subnetworks

153

frw ≥ 0 ∀r ∈ R2w ,

(7.40)

G w1 ≥ 0 and G w2 ≥ 0,

(7.41)

with definitional constraints xe =



w grw δe,r +



w r ∈R1w

w frw δe,r , ∀e ∈ E.

(7.42)

w r ∈R2w

Proof By definition, user equilibrium corresponds to the situation when the journey times in all actually used routes are equal to and less than those that would be experienced by a single vehicle on any unused route. The Lagrangian of problems (7.35)–(7.42) is L=

 e∈E

xe

te (u)du +

0

+σ

⎡

⎛

⎣tw1 ⎝G w1 −

w∈W

 w



G − w

G w1

−

⎞

⎛

+

 

w∈W

 w



grw ⎠ + tw2 ⎝G w2 + F w −

r ∈R1w

G w2



frw ⎠ +

r ∈R2w

(−grw )ηrw +

r ∈R1w

 w

⎞

  w∈W

 w

+ γ1 − G 1 + γ 2

  − G w2 ,

(− frw )ξrw +

r ∈R2w

where tw1 , tw2 , ηrw ≥ 0 (r ∈ R1w ), ξrw ≥ 0 (r ∈ R2w ), γ1w and γ2w for all w ∈ W are Lagrangian multipliers. Goal functions are convex, and Kuhn–Tucker conditions hold. Differentiate L by all unknown variables and equate it to zero w ∈ W : ∂Z ∂L = w − tw1 − ηrw = 0, r ∈ R1w , w ∂gr ∂gr

(7.43)

∂L ∂Z = − tw2 − ξrw = 0, r ∈ R2w , w ∂ fr ∂ frw

(7.44)

∂L = tw1 − σ w − γ1w = 0, ∂G w1

(7.45)

∂L = tw2 − σ w − γ2w = 0. ∂G w2

(7.46)

Note that for r ∈ R1w for all w ∈ W :  ∂ Z ∂ xe  ∂Z w = = te (xe )δe,r = trw (grw ), w w ∂gr ∂ x ∂g e r e∈E e∈E

154

7 Optimal Transit Network Design

and for r ∈ R2w for all w ∈ W :  ∂ Z ∂ xe  ∂Z w = = te (xe )δe,r = trw ( frw ). w ∂ frw ∂ x ∂ f e r e∈E e∈E Moreover, for conditions (7.39)–(7.41) complementary slackness holds and, consequently, (7.43)–(7.46) could be reformulated for all w ∈ W

(7.47)

r ∈ R2w , ≥ tw2 , for frw = 0,

= σ w , for G w1 > 0, w

(7.48)

trw ( frw )

= tw1 , for grw > 0,

r ∈ R1w ,

trw (grw )

≥ tw1 , for grw = 0, = tw2 , for frw > 0,

t1

tw2

≥ σ w , for G w1 = 0, = σ w , for G w2 > 0, ≥ σ w , for G w2 = 0.

(7.49)

(7.50)

Therefore, conditions (7.47) and (7.48) guarantee the user equilibrium assignment on the specialized and common subnetworks, respectively. Moreover, if G w1 > 0 and G w2 > 0, then tw1 = σ w and tw2 = σ w . Hence, tw1 = tw2 . In this case, trw (grw ) = trw ( f rw ), r ∈ R1w , r ∈ R2w . Therefore, the user equilibrium assignment is on the whole network. Due to the proof of Theorem 7.3, we obtain for all w ∈ W : if G w1 = G w and = 0, then tw1 = σ w tw2 ≥ σ w . Hence, tw1 ≤ tw2 and

G w2

trw (grw ) ≤ trw ( f rw ), r ∈ R1w , r ∈ R2w .

(7.51)

Therefore, instead of using a procedure based on optimization programs A and B, we can use a procedure based on the optimization program from Theorem 7.3. Actually, according to (7.51), the sets of specialized routes R1w for all w ∈ W are optimal if and only if the optimal solution (g ∗ , f ∗ , G ∗1 , G ∗2 ) of program (7.35)–(7.42) is such that G w1 ∗ = G w and G w2 ∗ = 0 for all w ∈ W . Remark 7.2.1 Note that the traffic assignment problem in the network with transit subnetwork in a form of optimization program is important for bi-level optimization. For instance, in Chap. 8 we consider a problem of emission reduction and as we can see, the presence of Theorem 7.3 cannot be overestimated at least in this case. It is remarkable that for a network of parallel routes presented by digraph with one origin-destination pair and n parallel links where each link is associated with a route from origin to destination, we obtained explicit conditions to define an optimal

7.2 Traffic Assignment in Road Networks with Transit Subnetworks

155

specialized route allocation (7.35). However, now the same result could be obtained via solving the following constrained optimization problem:  min

g, f,G 1 ,G 2

z(g, f, G 1 , G 2 ) =

min

g, f,G 1 ,G 2

subject to

n1   i=1

n1 

gi

 n2 

ti (u)du +

0

fi

 ti (u)du ,

i=n 1 +1 0

gi = G 1 ,

i=1 n2 

f i = G 2 + F,

i=n 1 +1

G 1 + G 2 = G, gi ≥ 0 ∀i = 1, n, f i ≥ 0 ∀i = n 1 + 1, n 2 , G 1 ≥ 0 and G 2 ≥ 0, when G ∗1 = G and G ∗2 = 0. Therefore, we have the following corollary for Theorem 7.3. Corollary 7.1 n 1 is optimal if and only if the following two conditions hold: G>

n1  i=1

 ci

tn01 ti0

−1

and

F>

n2 

 ci

i=n 1 +1

tn02 ti0

−1 ,

n 2

n 1 F + i=n c ci G + i=1 1 +1 i

n 1 ci ≤ n 2 , ci i=1 ti0

(7.52)

(7.53)

i=n 1 +1 ti0

when, without loss of generality, the routes are numbered as follows t10 ≤ . . . ≤ tn01 and tn01 +1 ≤ . . . ≤ tn02 .

(7.54)

Note that Theorem 7.2 and Corollary 7.1 give the same result. However, the ways to obtain this result are different. Conditions (7.52)–(7.54) are corollary of a more general statement while conditions from Theorem 7.2 are obtained as an independent result, without any prospect to be generalized.

156

7 Optimal Transit Network Design

7.3 Optimality Criteria for a Transit Network Design Under Competitive Routing In this section we define rules for the design of a specialized transit subnetwork under group routing in a road network. Let us begin from the network of parallel routes. A transportation network that is represented by a directed graph with one origin-destination pair and n parallel routes is considered. It is assumed that there are specialized routes that are used only by special vehicles and common routes that can be used by both special and non-special vehicles. The following notation is used: N = {1, . . . , n} is the set of numbers of routes; N1 = {1, . . . , n 1 } is the set of numbers of specialized routes; N2 = {n 1 + 1, . . . , n 2 = n} is the set of numbers of common routes; M = {1, . . . , m} is the set of user groups (UG) using the transportation j ∈ M; G j is the special-vehicle demand of UG

m network, j j j such that G = j=1 G ; F is non-special-vehicle demand of UG j such that

j F = mj=1 F j ; gi when i = 1, n is the special traffic flow of UG j through route i; j

f i when i = n 1 + 1, n 2 is the non-special traffic flow of UG j through route i; g j = j j j j (g1 , . . . , gn ) and f j = ( f n 1 +1 , . . . , f n 2 ), g = (g 1 , . . . , g m ), f = ( f 1 , . . . , f m ) and g − j = (g 1 , . . . , g j−1 , g j+1 , . . . , g m ), f − j = ( f 1 , . . . , f j−1 , f j+1 , . . . , f m ); ti0 and ci are the freetravel time  and capacity of route i; Fi is the traffic flow through route i; ti (Fi ) = ti0 1 + Fcii is the travel time through congested route i. The travel time is modeled by a BPR-delay function [2]. Every UG seeks to minimize the travel time of its users:  min z (g, f ) = min j

gj, f

j

gj, f

n1 

j

j ti (Fi ) gi

+

n2 

ti (Fi )



j gi

+ fi

j

  (7.55)

i=n 1 +1

i=1

for all j ∈ M, with constraints n 

j

gi = G j ∀ j ∈ M,

(7.56)

i=1 n 

j

f i = F j ∀ j ∈ M,

(7.57)

gi ≥ 0 ∀ j ∈ M, i = 1, n,

(7.58)

f i ≥ 0 ∀ j ∈ M, i = n 1 + 1, n 2 .

(7.59)

i=n 1 +1 j

j

  Thus, we define a non-cooperative game Γm M, {G j , F j } j∈M , {T j } j∈M , where {G j , F j } j∈M is the set of strategies (g j , f j ) satisfying constraints (7.56)–(7.59) and

7.3 Optimality Criteria for a Transit Network Design Under Competitive Routing

157

T j (g, f ) is the penalty function for j ∈ M. The Nash equilibrium in the game Γm is reached by strategies (g ∗ , f ∗ ) such that     ∗ ∗ ∀ j ∈ M. T j g∗, f ∗ ≤ T j g j , f j , g− j , f − j The competitive relationships between the different groups of users lead to mutual influence on their travel times, which addresses the problem of finding the Nash equilibrium. When the behavior of a UG is modeled by the optimization problem defined in (7.55)–(7.59) finding the Nash equilibrium is complex; however, the boundary value of n 1 could be estimated directly and corresponds to the situation when all special vehicles travel only via specialized routes, and their travel times are less than or equal to the travel times of non-special vehicles assigned to common routes. The definition of the optimal n 1 is shown in Definition 7.1. Without loss of generality, it is assumed that when n 1 is defined, the routes are numbered as follows: t10 ≤ . . . ≤ tn01 and tn01 +1 ≤ . . . ≤ tn02 .

(7.60)

Lemma 7.2 ([4]) Suppose all special vehicles G j for all j ∈ M use only specialized routes. The set of n 1 specialized routes is then fully loaded if and only if 1 1  ci Gj > m + 1 i=1

n





tn01

−1

ti0

∀ j ∈ M,

(7.61)

and the set of |n 2 − n 1 | common routes is fully loaded if and only if n2  1 ci F > m + 1 i=n +1 j

1



tn02 ti0

−1

∀ j ∈ M.

(7.62)

Proof If all special vehicles travel only via specialized routes, then the optimization problem defined in (7.55)–(7.59) could be considered as two independent problems: (1) for special vehicles: j

min z 1 (g) = min gj

subject to

gj

n1  i=1

j

n1 

j

ti (Fi )gi

∀j ∈ M

(7.63)

i=1

gi = G j ∀ j ∈ M,

(7.64)

158

7 Optimal Transit Network Design j

gi ≥ 0 ∀ j ∈ M, i = 1, n 1 ,

(7.65)

gi = 0 ∀ j ∈ M, i = n 1 + 1, n 2 ;

(7.66)

j

(2) for non-special vehicles: j min z 2 ( fj

subject to

f ) = min f

n2 

j

n2 

ti (Fi ) f i

j

∀j ∈ M

(7.67)

i=n 1 +1

j

f i = F j ∀ j ∈ M,

(7.68)

i=n 1 +1 j

f i ≥ 0 ∀ j ∈ M, i = n 1 + 1, n 2 .

(7.69)

Based on Corollary 3.2, all routes are used for the problems defined in (7.63)– (7.66) and (7.67)–(7.69) if and only if conditions (7.61) and (7.62) hold, respectively. Lemma 7.2 describes the first rule that defines the optimal amount of specialized routes in the case of competitive routing. If inequality (7.61) holds, then the specialized transit network is fully loaded when all special vehicles travel only via specialized routes. The simultaneous execution of (7.61) and (7.62) indicates that the entire transportation network is fully loaded (i.e., every route is used). At the same time, condition (7.61) does not guarantee that all special vehicles travel only through specialized routes. To motivate special vehicles to travel only via specialized routes, the decision-maker must design a specialized transit network that provides favorable specialized routes. Theorem 7.4 ([4]) Suppose that the initial set of routes is fully loaded by traffic flows G and F. The flow of special vehicles G then uses only specialized routes if and only if   n1   ti0 ci ci j + Ψ (G, m) ≤ Ψ (G , 1) − m+1 m+1 ti0 i=1  0   n2  ti ci ci j ≤ + Φ(F, m) Φ(F , 1) − m+1 m+1 ti0 i=n +1 1

∀ j ∈ M,

(7.70)

7.3 Optimality Criteria for a Transit Network Design Under Competitive Routing

where Ψ (x, y) =

Φ(x, y) =

x+

x+

y

159

n 1

r =1 cr cr r =1 tr0

m+1

n1

,

y n 2 cr m+1

n 2 r =nc1r +1 r =n 1 +1 tr0

(7.71)

.

(7.72)

Proof Based on Corollary 2.3.1, when conditions (7.61) and (7.62) hold, the optimal assignment (g ∗ , f ∗ ) could be expressed explicitly. If one substitutes the explicit g ∗ and f ∗ with (7.63) and (7.67), respectively, then the left and right parts of (7.70) will be obtained, where the left part describes the travel time of the traffic flow of UG j via specialized routes and the right part describes the travel time of the traffic flow via common routes. Theorem 7.4 identifies the second rule regarding the optimal amount of specialized routes when many user groups are present. If the authority defines a set of specialized routes such that condition (7.70) holds, then for any UG, it will be preferable for its special users to travel via specialized routes. Thus, non-special vehicles will identify that special vehicles spend less time travelling between a given origin-destination pair; this will provide motivation for drivers to use special vehicles. Conditions (7.61) and (7.62) thus guarantee full loading of all specialized and common routes. Now consider a general topology network that is represented by a directed graph G with a sequentially numbered set of nodes V and a sequentially numbered set of links E. The following notation is used: M = {1, . . . , m} is the set of UGs, j ∈ M; R w is the set of possible routes between an origin-destination pair w; R1w is the set of specialized routes, R1w ⊂ R w ; R2w is the set of common routes, R2w ⊂ R w ; R1w ∩ R2w = ∅ and R1w ∪ R2w = R w ; G j,w and F j,w are the demands of special and non-special an origin-destination pair w by UG j, respectively,

vehicles between j,w G w = mj=1 G j,w and F w = mj=1 F j,w ; gr when r ∈ R w is the special traffic j,w

flow of UG j through route r ; g j,w = (. . . , gr , . . .), g j = (. . . , g j,w , . . .) and j,w g = (g 1 , . . . , g m ), g − j = (g 1 , . . . , g j−1 , g j+1 , . . . , g m ); fr when r ∈ R2w is the j,w non-special traffic flow of UG j through route r ; f j,w = (. . . , fr , . . .), f j = j,w 1 m −j 1 j−1 j+1 ,f , . . . , f m ); xe (. . . , f , . . .) and f = ( f , . . . , f ), f = ( f , . . . , f j is the traffic flow on link e ∈ E, x = (. . . , xe , . . .); xe is the traffic flow of UG j on j link e ∈ E, x j = (. . . , xe , . . .); ce is the capacity of link e ∈ E; te (xe ) is the travel w w is an indicator: δe,r = 1 if link e “belongs” time through congested link e ∈ E; δe,r w to route r between w, δe,r = 0 otherwise. The problem defined by (7.55)–(7.59) could be reformulated for a general network topology:  te (xe )xej ∀ j ∈ M (7.73) min Z j (x) = min xj

xj

e∈E

160

7 Optimal Transit Network Design

subject to 

grj,w = G j,w ∀ j ∈ M,

(7.74)

frj,w = F j,w ∀ j ∈ M,

(7.75)

grj,w ≥ 0 ∀r ∈ R w , j ∈ M,

(7.76)

frj,w ≥ 0 ∀r ∈ R2w , j ∈ M,

(7.77)

r ∈R w

 r ∈R2w

with definitional constraints    w w grj,w δe,r + , grj,w + frj,w δe,r xej = w r ∈R1w

(7.78)

w r ∈R2w

xa =

m 

xaj .

(7.79)

j=1

  Thus, the non-cooperative game Γ M, {G j , F j } j∈M , {H j } j∈M , where {G j , F j } j∈M is the set of strategies (g, f ) satisfying (7.74)–(7.77) and T j (x (g, f )) is the penalty function for j ∈ M. The Nash equilibrium in the game Γ is reached by strategies (g ∗ , f ∗ ) such that       ∗ ∗ ∀ j ∈ M. T j x g∗, f ∗ ≤ T j x g j , f j , g− j , f − j As in the previous problem, the unknown variables are the traffic flows through both the specialized and common routes. Different allocations of the specialized transit network (i.e., the set of specialized routes) lead to various equilibrium assignments. Therefore, the set of specialized routes could be considered as a parameter for an authority to manage. Thus, a bi-level control system is evident for a general network topology when competitive routing is considered. Applying the procedure developed in the preceding section, we are able to identify the specialized subnetwork defined by the following: • The travel times of any amount of special vehicles (i.e., from zero flow up to all available vehicles) travelling via specialized routes are less than or equal to the travel times of the vehicles in the other flows between given origin-destinations pairs. • If all special vehicles only use specialized routes, then the specialized transit network is fully loaded between each origin-destination pair.

7.3 Optimality Criteria for a Transit Network Design Under Competitive Routing

161

Thus, by analogy with Theorem 7.4 one could formulate the following: Statement 7.3.1 A special traffic flow G w for all w runs through specialized routes only if and only if 

te (xe )xej ≤

e∈Er1



te (xe )xej ∀r1 ∈ R1w , r2 ∈ R2w , w,

e∈Er2

where Er is the set of links belonging to route r ∈ R w for all w. Due to Statement 7.3.1, one could estimate the boundaries of the set R1w , which occur when all specialized routes are used by all of the available special vehicles, and their travel time is less than or equal to the travel times of the non-special vehicles. Thus, the problem defined by (7.73)–(7.79) could be separated as follows: (1) for special vehicles: j

min Z 1 (x) = min xj

xj



te (xe )xej ∀ j ∈ M

(7.80)

e∈E

subject to 

grj,w = G j,w ∀ j, w,

(7.81)

grj,w ≥ 0 ∀r ∈ R1w , j, w,

(7.82)

grj,w = 0 ∀r ∈ R2w , j, w,

(7.83)

r ∈R1w

with definitional constraints xej =



w grj,w δe,r ∀ j ∈ M,

(7.84)

w r ∈R1w

xe =

m 

xej ,

(7.85)

j=1

(2) for non-special vehicles: j

min Z 2 (x) = min xj

subject to

xj

 e∈E

te (xe )xej ∀ j ∈ M

(7.86)

162

7 Optimal Transit Network Design



frj,w = F j,w ∀ j, w,

(7.87)

frj,w ≥ 0 ∀r ∈ R2w , j, w,

(7.88)

r ∈R2w

with definitional constraints xej =



w frj,w δe,r ∀ j ∈ M,

(7.89)

w r ∈R2w

xe =

m 

xej .

(7.90)

j=1

The optimization problems defined by (7.80)–(7.85) and (7.86)–(7.90) are complex, particularly when determining the Nash equilibrium of the network [7]. Unfortunately, for a general network topology, it is impossible to obtain explicit expressions for the Nash equilibrium.

7.4 Traffic Assignment in Road Networks with Transit Subnetworks Under Competitive Routing In the preceding section, the rules to obtain the optimal amount of specialized routes are identified. The Nash equilibrium of the network with a specialized transit subnetwork will now be defined. Theorem 7.5 ([4]) Suppose that sets of all routes are fully loaded  n1 0  t 1 n 1 Gj > ci −1 ∀j ∈ M m + 1 i=1 ti0 and

n2  1 F > ci m + 1 i=n +1 j

1



tn02 ti0

−1

∀ j ∈ M.

Then, the traffic flow assignment on the network of parallel routes is the Nash equilibrium (g ∗ , f ∗ ) if and only if (1) if n 1 satisfies

7.4 Traffic Assignment in Road Networks with Transit Subnetworks …

163

  n1   ti0 ci ci j + Ψ (G, m) Ψ (G , 1) − m+1 m+1 ti0 i=1     n 2  ti0 ci ci j ≤ Φ(F , 1) − + Φ(F, m) m+1 m+1 ti0 i=n +1 1

∀ j ∈ M, when Ψ (x, y) and Φ(x, y) are defined in (7.71) and (7.72), then 1  q b , m + 1 q=1 i m

j∗

j

gi = bi − where j

bi =

⎧ ⎨ ci

ti0

n 1 s Gj+ m s=1 G + r =1 cr

n 1 cr

⎩0

r =1 t 0 r

fi

j∗

− ci for i = 1, n 1 , for i = n 1 + 1, n 2 ,

∀ j ∈ M,

1  q − y , m + 1 q=1 i m

=

j yi

where j yi

ci F j + = 0 ti

m

n 2 s r =n 1 +1 cr s=1 F +

n 2 cr r =n 1 +1 tr0

− ci for i = n 1 + 1, n 2 ,

∀ j ∈ M, (2) if n 1 satisfies   n1   ti0 ci ci j + Ψ (G, m) > Ψ (G , 1) − m+1 m+1 ti0 i=1  0   n2  ti ci ci j + Φ(F, m) > Φ(F , 1) − m+1 m+1 ti0 i=n +1 1

∀ j ∈ M, when Ψ (x, y) and Φ(x, y) are defined in (7.71) and (7.72), then j∗

1  q b , m + 1 q=1 i m

j

gi = bi −

164

7 Optimal Transit Network Design

where j

j bi

ci G + = 0 1 ti

m

n 1 s r =1 cr s=1 G 1 +

n 1 cr − ci for i = 1, n 1 , ∀ j ∈ M, r =1 tr0



j∗

gi + f i

j∗



1  q y , m + 1 q=1 i m

j

= yi −

where j

j yi

ci F j + G 2 + = 0 ti

m  s=1

 2 F s + G s2 + rn=n c 1 +1 r

n 2

cr r =n 1 +1 tr0 j

− ci

j

for i = n 1 + 1, n 2 , ∀ j ∈ M, when G j = G 1 + G 2 ∀ j ∈ M and G 1 =

j G 2 = mj=1 G 2 such that

m j=1

j

G1,

  n1   ti0 ci ci j + Ψ (G 1 , m) ≤ Ψ (G 1 , 1) − m+1 m+1 ti0 i=1  0   n2  ti ci ci j j + Φ(F + G 2 , m) ≤ Φ(F + G , 1) − 2 m+1 m+1 ti0 i=n +1 1

∀ j ∈ M, when Ψ (x, y) and Φ(x, y) are defined in (7.71) and (7.72). Proof The statement above directly follows from Theorem 7.4 and Corollary 2.3.1. By analogy with Theorem 7.5, we could formulate the following: Statement 7.4.1 Suppose that the set of all routes is fully loaded: grj,w > 0 ∀r ∈ R1w and



 grj,w + frj,w > 0 ∀r ∈ R2w ,

∀ j ∈ M, w. In such a case, the traffic flow assignment on the network of parallel routes is the Nash equilibrium (g ∗ , f ∗ ) if and only if (1) if R1w satisfies  e∈Er1

te (xe )xej ≤

 e∈Er2

te (xe )xej r1 ∈ R1w , r2 ∈ R2w

7.4 Traffic Assignment in Road Networks with Transit Subnetworks …

165

∀ j ∈ M, w, then ∗

• g j ∀ j ∈ M are defined as the solution of the problem defined by (7.80)–(7.85), ∗ • f j ∀ j ∈ M are defined as the solution of the problem defined by (7.86)– (7.90), (2) if R1w satisfies 

te (xe )xej >

e∈Er1



te (xe )xej ∀r1 ∈ R1w , r2 ∈ R2w ,

e∈Er2

∀ j ∈ M, w, then j,w ∗ w }r ∈R w 1

• {gr

∀ j ∈ M are defined as the solution of:  j te (xe )xej ∀ j ∈ M min Z1 (x) = min xj

xj

(7.91)

e∈E

subject to 

j,w

grj,w = G 1

∀ j, w,

(7.92)

grj,w ≥ 0 ∀r ∈ R1w , j, w,

(7.93)

r ∈R1w

with definitional constraints  w grj,w δe,r ∀ j ∈ M, xej =

(7.94)

w r ∈R1w

xe =

m 

xej ,

(7.95)

j=1 j,w ∗

• {gr

, fr

j,w ∗ w }r ∈R w 2

∀ j ∈ M are defined as the solution of:  j te (xe )xej ∀ j ∈ M min Z2 (x) = min xj

xj

(7.96)

e∈E

subject to 

 j,w grj,w + frj,w = G 2 + F j,w ∀ j, w,

(7.97)

r ∈R2w

grj,w ≥ 0 ∀r ∈ R2w , j, w,

(7.98)

frj,w ≥ 0 ∀r ∈ R2w , j, w,

(7.99)

166

7 Optimal Transit Network Design

with definitional constraints   w grj,w + frj,w δe,r ∀ j ∈ M, xej =

(7.100)

w r ∈R2w

xe =

m 

xej ,

(7.101)

j=1 j,w

j,w

j,w

j,w

where G 1 + G 2 = G j,w , G 1 and G 2 + F j,w ∀ j ∈ M are the traffic flows through the specialized and common routes such that 



te (xe )xej ≤

e∈Er1

te (xe )xej ∀r1 ∈ R1w , r2 ∈ R2w ,

e∈Er2

∀ j ∈ M, w. Therefore, specialized routes are available for special vehicles only and should be allocated optimal, i.e., in a competitive equilibrium (actually the Nash equilibrium) state, where the average travel time of special vehicles is less than or equal to the average travel time of non-special vehicles between the same origin-destination nodes. Moreover, based on Statement 7.4.1 we can formulate two independent optimization programs to find an optimal set of specialized routes: (A) For specialized routes, for all j ∈ M j

min Z 1 = min xj

xj



te (xe )xej ,

e∈E

with constraints for all w ∈ W 

grj = G j,w ,

r ∈R1w

grj ≥ 0, ∀r ∈ R1w , with definitional constraints xej =

  w∈W

w grj δe,r , ∀e ∈ E,

r ∈R1w

xe =

m  j=1

xej ,

7.4 Traffic Assignment in Road Networks with Transit Subnetworks …

167

(B) For non-specialized routes, for all j ∈ M j

min Z 2 = min xj

xj



te (xe )xej ,

e∈E

with constraints for all w ∈ W 

frj = F j,w ,

r ∈R2w

frj ≥ 0, ∀r ∈ R2w , with definitional constraints xej =

  w∈W

w frj δe,r , ∀e ∈ E,

r ∈R2w

xe =

m 

xej .

j=1

Sets of optimization programs A and B could be expressed as non-zero sum games and could lead to a Nash equilibrium assignment on the specialized and common subnetworks, respectively, when special vehicles use only specialized routes. Clearly, it is reasonable to run special vehicles solely through specialized routes if and only if their average travel time is less than or equal to the average travel time of non-special vehicles. Therefore, the sets of specialized routes R1w for all w ∈ W are optimal if and only if the optimal average travel time of special vehicles from the set of programs A is less than or equal to the optimal average travel time of non-special vehicles from the set of programs B. However, the following theorem allows us to avoid solving the two problems and leads to one optimization program. Theorem 7.6 ([8]) The Nash equilibrium assignment of special and non-special vehicles on the general network with specialized routes and several groups of users could be found as a solution for the following set of optimization programs j ∈ M: j

j

min Z j (x j , G 1 , G 2 ) = min

j j x j ,G 1 ,G 2

j j x j ,G 1 ,G 2



te (xe )xej ,

(7.102)

e∈E

with constraints for all w ∈ W  r ∈R1w

j,w

grj = G 1 ,

(7.103)

168

7 Optimal Transit Network Design



j,w

frj = G 2 + F j,w ,

(7.104)

r ∈R2w j,w

j,w

G1 + G2

= G j,w ,

(7.105)

grj ≥ 0 ∀r ∈ R1w ,

(7.106)

frj ≥ 0 ∀r ∈ R2w ,

(7.107)

j,w

G1

j,w

≥ 0 and G 2

≥ 0,

(7.108)

w frj δe,r , ∀e ∈ E.

(7.109)

with definitional constraints xej =





w grj δe,r +

w r ∈R1w

w r ∈R2w

xe =

m 

xej , ∀ j ∈ M.

(7.110)

j=1

Proof The Lagrangian of the problem (7.102)–(7.110) for all j ∈ M is L = j



te (xe )xe +

e∈E



⎡

⎛

⎣m j,w 1

⎝G j,w 1

−



⎞

⎛

j j,w gr ⎠ + m2

⎝G j,w 2

+F

j,w

r ∈R1w

w∈W

−

   j,w j,w j j,w j j,w + σ j,w G j,w − G 1 − G 2 + (−gr )ηr + (− fr )ξr r ∈R1w j,w 

+ γ1 j,w

j,w

j,w 

− G1

j,w



⎞ fr

j⎠

+

r ∈R2w

r ∈R2w

j,w 

+ γ2



j,w 

− G2

,

j,w

j,w

j,w

where m1 , m2 , ηr ≥ 0 (r ∈ R1w ), ξr ≥ 0 (r ∈ R2w ), γ1 and γ2 are Lagrangian multipliers for all j ∈ M. Goal functions are convex, and, hence, Kuhn–Tucker conditions hold. Differenj j tiate L j in all gr and fr , and equate it to zero for all j ∈ M: ∂L j j ∂gr

∂L j ∂ fr

j

= =

∂Z j j

∂gr

∂Z j j

∂gr

j,w

(7.111)

j,w

(7.112)

− m1 − ηrj,w = 0, r ∈ R1w , − m2 − ξrj,w = 0, r ∈ R2w ,

7.4 Traffic Assignment in Road Networks with Transit Subnetworks …

∂L j j,w ∂G 1

∂L j j,w

∂G 2

169

j,w

j,w

= 0,

(7.113)

j,w

j,w

= 0.

(7.114)

= m1 − σ j,w − γ1

= m2 − σ j,w − γ2

Note that for r ∈ R1w for all w ∈ W : ∂Z j j

∂gr

=

 ∂ Z j ∂ xej j

e∈E

j

∂ xe ∂gr

=



te (xe ) +

e∈E

∂te (xe ) j

∂ xe

w xej δe,r = Tr j (grj ),

and for r ∈ R2w for all w ∈ W : ∂Z j ∂ fr

j

=

 ∂ Z j ∂ xej j

e∈E

∂ xe ∂ fr

j

=

 ∂te (xe ) j w j j te (xe ) + x e δe,r = Tr ( f r ). j ∂ xe e∈E

Moreover, for conditions (7.106)–(7.108) complementary slackness holds and, consequently, (7.111)–(7.114) could be reformulated for all j ∈ M, w ∈ W :

Tr j (grj )

j

j

j,w

j

= m2 for fr j,w j ≥ m2 for fr

= σ j,w for j,w m1 ≥ σ j,w for

= σ j,w for j,w m2 ≥ σ j,w for

Tr ( fr ) j

j,w

= m1 for gr > 0, j,w j ≥ m1 for gr = 0, > 0, = 0,

∀r ∈ R1w ,

(7.115)

∀r ∈ R2w ,

(7.116)

j,w

G 1 > 0, j,w G 1 = 0,

(7.117)

j,w

G 2 > 0, j,w G 2 = 0.

(7.118)

Therefore, conditions (7.115) and (7.116) guarantee the system optimum assignj j ment on specialized and common subnetworks, respectively, where Tr (gr ) and j j Tr ( fr ) are the so-called marginal costs (for details one can see [9]). Moreover, j,w j,w j,w j,w j,w j,w if G 1 > 0 and G 2 > 0, then m1 = σ j,w and m2 = σ j,w . Hence, m1 = m2 . j j j j In this case, Tr (gr ) = Tr ( f r ), r ∈ R1w , r ∈ R2w . Therefore, the system optimum assignment is on the whole network. j,w

j,w

Due to the proof of Theorem 7.6, we obtain that if G 1 = G j,w and G 2 = 0, j,w j,w j,w j,w then m1 = σ j,w and m2 ≥ σ j,w . Hence, m1 ≤ m2 and, consequently, for all j ∈ M, w ∈ W : (7.119) Tr j (grj ) ≤ Trj ( f rj ), r ∈ R1w , r ∈ R2w .

170

7 Optimal Transit Network Design

If we assume that there are free flows on the edges of a network, then the following generalization of Theorem 7.5 takes place. Theorem 7.7 ([8]) n 1 is optimal if and only if the following two conditions hold: 1 1  G > m + 1 i=1

n



j

n2  1 F > m + 1 i=n +1



j

  h n1 1+ − 1 ci − h i , cn 1 ti0

tn01

  h n2 1+ − 1 ci − h i , cn 2 ti0

tn02

1

(7.120)

(7.121)

  n1   ti0 ci + h i ci ci + h i j ≤ + Ψ (G, m) Ψ (G , 1) − m + 1 ci m+1 ti0 i=1  0   n2  ti ci + h i ci ci + h i j ≤ + Φ(F, m) Φ(F , 1) − m + 1 ci m+1 ti0 i=n +1 1

∀ j ∈ M, where Ψ (x, y) =

x+

y m+1

(7.122)

n 1

(cs + h s )

n 1s=1cs , s=1 ts0

Φ(x, y) =

x+

y m+1

n 2

(cs + h s )

n 2s=n 1 +1cs , s=n 1 +1 ts0

when, without loss of generality, the routes are numbered as follows   h1 hn t10 1 + ≤ . . . ≤ tn01 1 + 1 , c1 cn 1

(7.123)

  h n +1 hn tn01 +1 1 + 1 ≤ . . . ≤ tn02 1 + 2 . cn 1 +1 cn 2

(7.124)

Proof For convenience, let us divide the proof into four parts. I. For the network of parallel routes, the program (7.102)–(7.110) could be written as follows, for all j ∈ M: min

j j g j , f j ,G 1 ,G 2

j j z j (g j , f j , G 1 , G 2 ) =

min

j j g j , f j ,G 1 ,G 2

⎤ ⎡ n1 n2     j j j ⎦ ⎣ ti (Fi ) gi + ti (Fi ) gi + f i i=1

i=n 1 +1

(7.125) with constraints

7.4 Traffic Assignment in Road Networks with Transit Subnetworks … n 

j

171

j

gi = G 1 ∀ j ∈ M,

(7.126)

j

(7.127)

i=1 n 

j

f i = G 2 + F j ∀ j ∈ M,

i=n 1 +1 j

gi ≥ 0 ∀ j ∈ M, i = 1, n,

(7.128)

f i ≥ 0 ∀ j ∈ M, i = n 1 + 1, n 2 .

(7.129)

j

If all special vehicles run through specialized routes only, then the optimization problem (7.125)–(7.129) could be considered as two independent problems: (1) For specialized routes, for all j ∈ M: j min z 1 (g j ) gj

= min gj

n1 



m

q

n1 

j

gi ,

ci

i=1

subject to

gi + h i

q=1

1+

ti0

(7.130)

j

gi = G j ,

(7.131)

i=1 j

gi ≥ 0, i = 1, n 1 ,

(7.132)

(2) For non-specialized routes, for all j ∈ M: j min z 2 ( fj

f ) = min f

j



n2 

m

1+

ti0

q

q=1

n2 

j

fi ,

ci

i=n 1 +1

with constraints

fi + h i

(7.133)

j

(7.134)

f i ≥ 0, i = n 1 + 1, n 2 .

(7.135)

fi = F j ,

i=n 1 +1 j

II. Consider the optimization program (7.130)–(7.132). Differentiate the Lagrangian of this problem L = j

n1  i=1

 ti0

1+

m q=1

q

gi + h i ci



 j gi

+

j m1

G − j

n1  i=1

j gi

+

n1  i=1

j

j

ηi (−gi ),

172

7 Optimal Transit Network Design j

in gi and equate it to zero 

m

q

q=1

ti0 1 +

j

gi + gi + h i



ci

j

j

− m1 − ηi = 0.

  j j Due to complementary slackness ηi gi = 0 , we obtain for all j ∈ M  ti0

j−1 q=1

1+

q

j

gi + 2gi +

m

q

q= j+1

gi + h i



ci

j

j

= m1 for gi > 0, j j ≥ m1 for gi = 0.

(7.136)

Condition (7.136) for all j ∈ M could be formulated in a matrix form: A · gj = bj,

(7.137)

⎞ 2 1 ··· 1 ⎜ 1 2 ··· 1 ⎟ ⎟ ⎜ =⎜ , ⎟ ⎝ · · · · · · ... · · · ⎠ 1 1 · · · 2 m×m ⎛

where A = Am×m

 j bi

=

j m1 j − 1 ci − h i > 0 for gi > 0. ti0

(7.138)

It is quite clear that A−1 is ⎛

A−1

1 1 ⎞ − m+1 · · · − m+1 1 ⎟ m · · · − m+1 m+1 ⎟ =⎜ . ⎟ . . ⎝ ··· . ··· ⎠ ··· 1 1 m − m+1 · · · m+1 − m+1 m×m m m+1 ⎜− 1 ⎜ m+1

j

Then, if gi > 0, i = 1, n 1 , j ∈ M j∗

gi =

m  m 1 j q bi − b , m+1 m + 1 q=1,q = j i

or j∗

1  q b . m + 1 q=1 i m

j

gi = bi −

(7.139)

7.4 Traffic Assignment in Road Networks with Transit Subnetworks …

173

III. For convex goal functions, Kuhn–Tucker conditions are necessary and sufficient. j Then, from (7.138) we obtain: gi > 0, i = 1, n 1 , j ∈ M 

j m1 − 1 ci − h i > 0, ti0

and, hence, j m1

>

 hi 1+ . ci

ti0

(7.140)

Let us numerate the routes in such a way that (7.123) holds. We also suppose j that gi > 0, i = 1, n 1 , j ∈ M and, hence, (7.140) holds for all i = 1, n 1 , j ∈ M. ∗ j j∗ Now, if we take into consideration that bi = gi1 + . . . + 2gi + . . . + gim ∗ and, consequently, n1 m   j bi = Gs + G j , (7.141) s=1

i=1

it allows us to eventually obtain n1 



i=1

 m j  m1 − 1 c − h Gs + G j . = i i ti0 s=1 j

Therefore, we are ready to express m1 for all j ∈ M:

m j m1

=

s=1

Gs + G j +

n 1

n 1

s=1 (cs

+ hs )

cs s=1 ts0

.

(7.142)

Moreover, due to (7.141), (7.138) and (7.140), the following inequality holds m 

Gs + G j >

s=1

n1 



  h n1 1+ − 1 ci − h i , cn 1 ti0

tn01

i=1

or in matrix form A(G , . . . , G ) > (1, . . . , 1) 1

m T

T

n1 



i=1

tn01



ti0

hn 1+ 1 cn 1







− 1 ci − h i ,

that leads to −1

(G , . . . , G ) > A (1, . . . , 1) 1

m T

T

n1  i=1



  h n1 1+ − 1 ci − h i , cn 1 ti0

tn01

174

7 Optimal Transit Network Design

or

1 1  G > m + 1 i=1

n



j

  h n1 1+ − 1 ci − h i . cn 1 ti0

tn01

Therefore, we obtain the condition (7.120). It is clear that condition (7.121) could be obtained by the same analytical chain. IV. Consider the overall travel time of traffic flow G j : 

m j n1  g + h i j=1 i j T1 (G j ) = ti0 1 + gi . ci i=1 If we put (7.139), (7.142), then we obtain j bi

ci = 0 ti

m s=1

Gs + G j +

n 1

n 1

s=1 (cs

+ hs )

cs s=1 ts0

− ci − h i ,

and m 

j bi

j=1

ci (m + 1) = 0 ti

m s=1

Gs + m

n 1 cs

n 1

s=1 (cs

+ hs )

− m (ci + h i ) .

s=1 ts0 j

Therefore, we are ready to express gi explicitly: j∗

gi =

ci G j + ti0

1 m+1

n 1

n 1 s=1cs

(cs + h s )

−

s=1 ts0

1 (ci + h i ) , m+1

and m 

j∗ gi

j=1

ci = 0 ti

m j=1

Gj +

m n 1 s=1 (cs m+1

n 1 cs s=1 ts0

+ hs )

−

m (ci + h i ) . m+1

Then, the following equality holds:  ti0

m 1+

j j=1 gi

ci

+ hi



 =

ti0 ci + h i + m + 1 ci

m j=1

m G j + m+1

n 1

Introduce the following function: Ψ (x, y) =

x+

y m+1

n 1

(cs + h s )

n 1s=1cs . s=1 ts0

n 1

s=1 (cs

cs s=1 ts0

+ hs )

.

7.4 Traffic Assignment in Road Networks with Transit Subnetworks …

175

Eventually, we can claim that T1 (G j ) =

  n1   ti0 ci + h i ci ci + h i j + Ψ (G, m) Ψ (G , 1) − . m + 1 ci m+1 ti0 i=1

By analogy, it is simple to prove that T2 (F j ) =

 0   n2  ti ci + h i ci ci + h i j , + Φ(F, m) Φ(F , 1) − m + 1 ci m+1 ti0 i=n +1 1

where

x+

Φ(x, y) =

y m+1

n 2

(cs + h s )

n 2s=n 1 +1cs . s=n 1 +1 ts0

Therefore, (7.122) is proved. j

j

If G 1 is the number of special vehicles for UG j using specialized routes and G 2 is the number of special vehicles for UG j using common routes, then due to the Theorem 7.6, we could claim that conditions (7.120)–(7.122) would also appear in the optimal decisions of the following set of programs for all j ∈ M: min z (g , f ) = min j

gj, f

j

j

j

n 1 

j

f

j ti (G i )gi

+

i=n 1 +1

i=1

with constraints

n1 

n2 

j

j

gi = G 1 ,

i=1 n2 

j

j

fi = G 2 + F j ,

i=n 1 +1 j

j

G 1 + G 2 = G j ∀ j ∈ M, j

gi ≥ 0 ∀i = 1, n 1 , j

f i ≥ 0 ∀i = n 1 + 1, n 2 , j

j

G 1 ≥ 0 and G 2 ≥ 0.

 ti (Fi ) f i

j

,

176

7 Optimal Transit Network Design

Because there is a lack of methodological tools based on competitive behavioral models which could support decision-makers with regard to encouraging the use of special vehicles, this paper has investigated the problem of competitive traffic flow assignment in a specialized transit network. Therefore, the obtained results contribute in the development of such methodological tools.

References 1. Beltran B, Carrese S, Cipriani E, Petrelli M (2009) Transit network design with allocation of green vehicles: a genetic algorithm approach. Transp Res Part C 17:475–483 2. U.S. Bureau of Public Roads (ed) (1964) Traffic assignment manual. U.S. Department of Commerce, Washington, DC 3. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall Inc., Englewood Cliffs, NJ 4. Zakharov V, Krylatov A (2015) Competitive green-vehicle assignment on a transportation network. In: Chapter 13: Game theory and applications. Volume 17: Game-theoretic models in mathematical ecology. Nova Science Publishers Inc., pp 360–398 5. Frank M, Wolfe P (1956) An algorithm for quadratic programming. Naval Res Logist Q 3:95–110 6. Zakharov V, Krylatov A, Volf D (2018) Green route allocation in a transportation network. Comput Methods Appl Sci 45:71–86 7. Zakharov VV, Krylatov AY (2016) Competitive routing of traffic flows by navigation providers. Autom Remote Control 77(1):179–189 8. Krylatov AY, Zakharov VV (2016) Competitive traffic assignment in a green transit network. Int Game Theory Rev 18(2):1–14 9. Patriksson M (1994) The traffic assignment problem: models and methods. Dover Publications Inc., New York

Part V

Networking Issues

Theoretical achievements and practical needs are often not isomorphic, and additional effort is required to somehow implement the theory to coping with practical problems. Successful implementation sometimes equals to important discovery. Occasionally, an implementation pattern is even able to change the research direction significantly.

Chapter 8

Transportation Processes Modelling in Congested Road Networks

Abstract In this chapter, the models of different transportation processes in a congested road network are considered. The first section is devoted to a signal control problem formulated as a bi-level optimization program. An analytical solution for a two-commodity linear road network offers a practical and illustrative result to be taken into consideration by decision-makers in this sphere. A new algorithm for OD-matrix estimation based on the dual traffic assignment problem is described in the second section. The third section is devoted to the problem of emission reduction. The approaches presented in this book are shown to be well-implemented for coping with such problems. The time-depended vehicle routing problem in a congested road network is considered in the last section.

8.1 Signal Control in a Congested Urban Area This section is devoted to the development of methodological tools for area traffic signal control under user equilibrium flow pattern with certain demand. Mathematical programming approach is suggested to be applied for achieving this purpose. Note that research exists in which mathematical programming was successfully employed to solve constrained optimization problems of signal control settings [1–3]. Moreover, the bi-level programming technique was also implemented for tackling the problem of optimal signal control setting [4, 5]. However, due to the huge sizes of road networks in corresponding bi-level mathematical programs, a solution of lower level cannot be expressed explicitly [6]. Thereby computationally tractable algorithms for a bi-level network design problems appeared [7]. Consider the network presented by directed graph G that includes a set of consecutively numbered nodes V and a set of consecutively numbered arcs E. We also use the following notation: W is the set of origin-destination pairs, w ∈ W ; J is the set of signalized intersections in the network, J ∈ V ; E j is the set of links terminating to intersection j, j ∈ J ; R w is the set of possible routes between w; F w is the trip demand between w; frw is the traffic flow through route r , r ∈ R w ; xe is the traffic flow on the arc e ∈ E, x = (. . . , xe , . . .); λe is the green time proportion on link e ∈ E (green light timing in the whole traffic lights cycle), λ = (. . . , λe , . . .); te (xe , λe ) is © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_8

179

180

8 Transportation Processes Modelling in Congested Road Networks

w w the travel time through congested signalized link e ∈ E; δe,r is an indicator: δe,r =1 w if link e is a part of route r between w, and δe,r = 0 otherwise. Now we are able to formulate the following bi-level programming problem:

min Z 1 (x, λ) = min x,λ

subject to

x,λ





te (xe , λe )xe ,

(8.1)

e∈E

λe = 1 ∀ j ∈ J,

(8.2)

0 ≤ λe ≤ 1 ∀e ∈ E,

(8.3)

e∈E j

when traffic is believed to be assigned according the first principle of Wardrop: min Z 2 (x) = min x

x



with constraints

 e∈E

xe

te (u, λe )du,

(8.4)

0

frw = F w ∀w ∈ W,

(8.5)

frw ≥ 0 ∀r ∈ R w , w ∈ W,

(8.6)

r ∈R w

with definitional constraints xe =



w frw δe,r .

(8.7)

w r ∈R w

In a similar way a bi-level program was formulated for global optimal signal setting under queuing network equilibrium conditions [5]. For this purpose the queuing delay influencing storage capacity was introduced. Such constraints take into account queue length on congested links before signalized junctions. Eventually queues increasing at the approaches of an intersection were restricted so that upstream intersections were not blocked. Here we do not deal with such constraint because it could lead to an empty set of possible decision in a bi-level program when demand becomes too large. For the ease of presentation, consider the network of n parallel routes. We use the following notation: N = {1, . . . , n} is the set of numbers of all routes; L i is the set of sequentially numbered links of route i, i = 1, n, |L i | = li ; F is the trip demand between OD-pair; f i is the traffic flow through route i, i = 1, n, f = ( f 1 , . . . , f n ); til0 , cil and λil free travel time, capacity and green time proportion of link l belonging  to route i, l = 1, . . . , li , i = 1, n; λi = (λi1 , . . . , λili ), i = 1, n; til ( f i , λil ) = til0 1 +

fi λil cil

is the travel time through congested signalized link l of

8.1 Signal Control in a Congested Urban Area

181

route i, l = 1, li , i = 1, n. We are modeling travel time as modified linear BPR cost function [2, 5, 8]. When signal control pattern is set, the Wardrop user equilibrium assignment of traffic flows could be formulated as follows: min z( f ) = min f

li  n  

f

i=1 l=1

subject to

n 

fi

0

til0

 1+

u λil cil

 du,

(8.8)

f i = F,

(8.9)

i=1

f i ≥ 0 ∀i = 1, n.

(8.10)

Lemma 8.1 ([9]) Assignment f ∗ is the user equilibrium of Wardrop if and only if such t (Lagrange multiplier) exists that li 

til0

 1+

l=1

fi λil cil



= t, if f i > 0, ≥ t, if f i = 0.

(8.11)

Proof To obtain (8.11), we use the Kuhn-Tucher theorem. As soon as the goal function is linear, Kuhn-Tucker conditions are necessary and sufficient. Therefore the Lagrangian of the problem (8.8)–(8.10) is L=

li  n   i=1 l=1

0

fi

til0

 1+

u λil cil





du + t F −

n  i=1

fi

+

n 

ηi (− f i ).

i=1

Let us differentiate L with respect to f i and equate it to zero so that we obtain t=

li  l=1

 til0 1 +

fi λil cil

 − ηi .

(8.12)

According to the Kuhn-Tucker condition of complementary slackness f i ηi = 0 for all i = 1, n it is clear that  = 0, if f i > 0, ηi (8.13) ≥ 0, if f i = 0. Thus due to (8.12) and (8.13) we obtain (8.11). i 0 Introduce the following additional notation: ti0 = ll=1 til (free travel time through   −1 0 t li il the whole route i) and κi (λi ) = for all i = 1, n. l=1 λil cil

182

8 Transportation Processes Modelling in Congested Road Networks

Corollary 8.1 ([9]) Assignment f ∗ is the user equilibrium of Wardrop if and only if such t (Lagrange multiplier) exists that 

t − ti0 κi (λi ), if ti0 < t, fi = 0, if ti0 ≥ t.

(8.14)

Proof If f i > 0 for some i = 1, n then from (8.11)

f i = t − ti0 κi (λi ) > 0 and, consequently, we obtain the first condition from (8.14). If f i = 0 for some i = 1, n then from (8.11) ti0 ≥ t and we obtain the second condition from (8.14). Without loss of generality, we assume that the routes are numbered as follows: t10 ≤ . . . ≤ tn0 .

(8.15)

Theorem 8.1 ([9]) When (8.15) holds, the user equilibrium of Wardrop in the problem (8.8)–(8.10) is reached under the following assignment:  fi =

κi (λi )

 F+ ks=1 ts0 κs (λs ) k s=1 κs (λs )

− ti0 κi (λi ), if i ≤ k, if i > k,

0,

(8.16)

for all i = 1, n, where k is defined from k 

κi (λi )(tk0 − ti0 ) ≤ F




n2 

ci

i=n 1 +1

tn02 ti0

−1 ,

(8.63)

n 2 n 1 F + i=n c ci G + i=1 1 +1 i n 1 ci ≤ n 2 , ci

(8.64)

i=n 1 +1 ti0

i=1 ti0

when routes are numbered as follows: t10 ≤ . . . ≤ tn01

and tn01 +1 ≤ . . . ≤ tn02 ,

(8.65)

1 cs ci G + ns=1  − ci , i = 1, n 1 , gi = 0 n 1 cs ti s=1 t 0

where

s

n 2

fi =

ci F + s=n 1 +1 cs n 2 − ci , i = n 1 + 1, n 2 . cs ti0 s=n 1 +1 t 0 s

Proof Conditions (8.63)–(8.65) are directly obtained from Theorem 7.1. Explicit gi for i = 1, n 1 and f i for i = n 1 + 1, n 2 are obtained from Corollary 2.3.1. According to Theorem 7.4.1, for the network of parallel routes, the optimal condition for n 1 could be expressed explicitly from the program (8.53)–(8.61). Theorem 8.3 The number of green routes n ∗1 guarantees a minimal level of emissions in the network of parallel routes in the competitive case if and only if it is the decision of the following optimization program: min E(g, f ) = min n1

subject to

n1

n 1 

αi · ti (gi ) · e

βi

li ti (gi )

+

n2 

 αi · ti ( f i ) · e

βi

li ti ( f i )

, (8.66)

i=n 1 +1

i=1

1 1  G > m + 1 i=1

n



j

n2  1 F > m + 1 i=n +1 j

1

   h n1 1+ − 1 ci − h i , cn 1 ti0

tn01



   h n2 1+ − 1 ci − h i , cn 2 ti0

tn02

(8.67)

(8.68)

8.3 Emission Reduction Due to Traffic Reassignment

195

  n1   ti0 ci + h i ci ci + h i j ≤ + Ψ (G, m) Ψ (G , 1) − m + 1 ci m+1 ti0 i=1   0  n2  ti ci + h i ci ci + h i j ≤ + Φ(F, m) Φ(F , 1) − m + 1 ci m+1 ti0 i=n +1 1

∀ j ∈ M, where Ψ (x, y) =

x+

(8.69)

n 1

(cs + h s ) n 1s=1cs ,

y m+1

s=1 ts0

Φ(x, y) = and

x+

y m+1

n 2

(cs + h s ) n 2s=n 1 +1cs , s=n 1 +1 ts0

⎡

⎤ m  c + h c i i i ⎦ , i = 1, n 1 , gi = ⎣ 0 Ψ (G j , 1) − m m+1 ti j=1 ⎡

⎤ m  c + h c i i i⎦ fi = ⎣ 0 Φ(F j , 1) − m , i = n 1 + 1, n 2 , m+1 ti j=1 when routes are numbered as follows     h1 hn t10 1 + ≤ . . . ≤ tn01 1 + 1 , c1 cn 1

(8.70)

    h n +1 hn ≤ . . . ≤ tn02 1 + 2 . tn01 +1 1 + 1 cn 1 +1 cn 2

(8.71)

Proof Conditions (8.67)–(8.71) follow directly from Theorem 7.4. Explicit gi for i = 1, n 1 and f i for i = n 1 + 1, n 2 follow from Corollary 2.3.1. Theorems 8.2 and 8.3 are quite important because they define the rule for green route allocation that guarantees two conditions: (1) green routes offer less travel time; and (2) a minimal level of emissions is reached. Indeed, consider Fig. 8.5a and d. As an example, consider the network of ten parallel routes. We assume that Table 8.3 reflects the physical characteristics of this network, and the travel demands are: G = 200 green vehicles, F = 300 non-green vehicles. Let us change the number of green routes from one to nine and check out the obtained result. It is quite evident that increasing the number of green routes leads to the decreasing of the travel time of green vehicles and the increasing of the travel time of non-green vehicles (Fig. 8.5b and d). At the same time, the lower the number

196

8 Transportation Processes Modelling in Congested Road Networks

Table 8.3 Characteristics of the network of ten parallel routes i 1 2 3 4 5 6 7 ti0

(min) 6 ci (veh/h) 52 li (km) 4

6.5 29 4.3

7 18 4.6

7.5 84 5

(a) Total emission son non-green routes

(c) Totale mission son green routes

8 25 5.3

8.5 24 5.6

9 70 6

8

9

10

9.5 91 6.3

10 57 6.6

10.5 73 7

(b) Travel time through non-green routes

(d) Travel time through green routes

Fig. 8.5 The relationship between travel time and the level of emission

of green routes available, the higher the speed (the less travel time) of non-green vehicles. However, as it follows from Fig. 8.5a, the higher the speed that the nongreen vehicles have, the more greenhouse gases they emit. Moreover, it could be verified that four green routes are the best balance between low emissions and an acceptable travel time of non-green vehicles. Actually, the level of emissions for nongreen vehicles is nearly uniform when the number of green routes changes from four to nine (Fig. 8.5a), while the travel time increases monotonically. Simultaneously, the harmful contribution of green vehicles is rather insignificant. Therefore, n 1 = 4 would be kind of a compromise number for a decision-maker. This line of reasoning reflects the path of decision-making that is available due to Theorem 8.2. An analogical line could be considered for Theorem 8.3.

8.3 Emission Reduction Due to Traffic Reassignment

197

Because there is a lack of methodological tools that can be used to support decision-makers in decreasing greenhouse gas emission levels, this section investigated the problem of determining the allocation of available green route capacity. The emission function was introduced, and an analysis of green route allocation that guarantees less travel time and a minimal level of emissions was performed. The approach for designing a green transit network that offers green vehicles shorter travel times between given origins and destinations was discussed.

8.4 Time-Dependent Vehicle Routing in a Congested Urban Area A task of practical importance for companies that offer forwarding service in large transport networks is to optimize costs. Such a task can largely be solved due to efficient vehicle routing. For this reason, the effective algorithms for solving the vehicle routing problem are given much attention by researchers. The vehicle routing problem was formulated for the first time by Dantzig and Ramser [45]. This is a key problem in the field of transportation, goods distribution and logistics. A vehicle routing problem (VRP) is understood as a wide class of well-known combinatorial optimization problems. The main goal in these types of tasks is to find a set of routes for vehicles that serve multiple customers with a given demand, geographically distributed on a network. VRP belongs to the class of NP-hard problems. Thus, the application of precise algorithms to solve such problems in the case of large-scale networks within a reasonable time is difficult [46]. Instead, precise algorithms, various heuristic and meta-heuristic algorithms are used to quickly find a good solution. However, optimality is not guaranteed in this case and, as a rule, even error may not be estimated. Herewith, experiments show that the number of generated solutions may be drastically small. To compare the efficiency of heuristic algorithms for a certain class of problems, publicly available test cases (http:// www.hec.ca/chairedistributique/data) are used. The results are compared with the best found solutions and evaluated according to several criteria. Algorithms are constantly improving and, thus, the list of the best known solutions is updated all the time. The main criterion for evaluating the efficiency of algorithms is accuracy. Since in most cases the optimal solution is unknown, the solution obtained using the investigated algorithm is compared with the best known solutions obtained by other algorithms. The results of many algorithms can strongly depend on the choice of the algorithm parameters. Thus, the same algorithm may show different results on the same set of tasks with different parameters. Moreover, each run of the algorithm can produce different solutions. As a result, when one publishes the results of the evaluation of algorithm efficiency, we can be faced with a different initial basis. Moreover, as mentioned in [47], computational results can be affected by a certain amount of rounding during calculations.

198

8 Transportation Processes Modelling in Congested Road Networks

The second important criterion is the speed of the algorithm. Often speed and accuracy depend from each other. Many heuristic algorithms are iterative, and thus, one of the possible stop criterion is the amount of iteration. The solution found by a quick algorithm is often worse because of low thoroughness of the solution space exploring. Other algorithms spend more time, but find better solutions. Actually, the choice between fast and slow algorithms is dependent on the context. For example, slow algorithms are not suitable or ambulance routing, but if one has enough time for transportation planning, more time for searching good routes may be spent in order to reduce transportation costs. Unfortunately, the complexity theory is not applicable for many meta-heuristic algorithms, so speed in this case directly means computing time. This time depends not only on the design of the algorithm, but also on factors such as the performance of the hardware, the programming language, the quality of software implementation and data structures. Thus, comparing the algorithms according to this criterion is not quite correct. Many researchers do not provide the run time of their algorithm, but everybody discusses the possible spheres of applications. The other two criteria for an algorithm estimation are simplicity and flexibility [47]. In this work, simplicity means the simplicity of software implementation and algorithm settings. The easier the algorithm is to implement, and the less parameters it has, the easier it is to use in real applications. This is the reason for using simple and fast but inefficient heuristic algorithms such as the Clarke-Wright algorithm [48, 49]. Flexibility means the possibility to expand the set of constraints without crucial changes of the model. For instance, it can mean the simplicity to adapt the algorithm developed for the vehicle routing problem without time constraints to the case with such limitations. There is a well-known statement that the traffic reaches an equilibrium after some time [50]. According to the first principle of Wardrop, the journey time on all actually used routes is equal and less than the expected journey time on the unused routes. Each driver decides independently which route to choose to reduce his/her costs for a journey. An appropriate traffic assignment is the user equilibrium. The desired equilibrium is reached when not one driver could reduce his/her costs for the journey. Consider a road network presented by directed graph G = (V, E), where V is the set of nodes and E is the set of edges. Let W be the set of OD-pairs. Introduce the following notations: R w is the set of routes between w; xe is traffic flow on the edge e ∈ E; te (xe ) is travel time through the edge e ∈ E. Traffic flow through the route r ∈ R w is frw , while trip demand between w ∈ W is F w . We use the following Boolean function:  1, if edge e belongs to route r ∈ R w , w δe,r = 0, otherwise. The mathematical formalization of the first Wardrop principle is a mathematical program with the following constraints:

8.4 Time-Dependent Vehicle Routing in a Congested Urban Area



199

frw = F w , ∀w ∈ W,

(8.72)

r ∈R w

frw  0, ∀r ∈ R w , w ∈ W,

(8.73)

with definitional constraints  

xe =

w∈W

While goal function is min x

w frw δe,r , ∀e ∈ E.

(8.74)

r ∈R w

 e∈E

xe

te (u)du,

0

where te (xe ) is the delay of traffic xe on the edge e ∈ E. We use the BPR-function here [8]:  βe

xe 0 te (xe ) = te 1 + αe , ce where te0 is the free travel time on the edge e, ce is the capacity of the edge e. The minimization problem (8.72)–(8.74) takes a form T ( f ) = min x

 e∈E

xe 0

te0

1 + αe



u ce

βe

du.

The solution is the link-flow assignment x in the graph G. Then we can find the travel time on all congested edges (e ∈ E) of the network, since delay-functions are determined for all these arcs. Apply the Dijkstra algorithm, which finds the shortest paths from a vertex of a graph to all others. Applying the Dijkstra algorithm, we form matrix D ∗ = {di∗j }, where di∗j is the travel time from node i to node j in a congested road network, i, j ∈ E. D 0 is the same matrix but for a non-congested road network. Split the routing period from 9 till 17 into subperiods Z k = [z k−1 ; z k ], where k = 2, . . . , K . The travelling salesman problem was firstly formulated as an optimization problem in 1930 [51]. In a classical form of the problem, the following assumptions hold: 1. The salesman must visit all cities from the test sample at least once; 2. The salesman begins his journey in a fixed city (depot) and at the end has to return to it; 3. The length of the route should be minimal. The travelling salesman problem could be considered in the graph G = (V, E). The set of vertices V = {v1 , v2 , . . . , vn } is the set of cities, the set of edges

200

8 Transportation Processes Modelling in Congested Road Networks

E = {(vi , v j ) : vi , v j ∈ V, i = j} is the roads between cities. Each edge may be associated with benefit criterion. Depending on the problems, the weight of (vi , v j ) of G may be time, cost or distance between vi and v j . Let I = {1, . . . , n} be the set of vertices’ indexes from test problem. We define the goal function as the total distance of the route, covering all the vertices of the considered problem. The parameters of the problem are the elements of the distance matrix D 0 = {di0j }, i, j ∈ I . The length of the route between two vertices vi and v j is defined as the shortest path in the non-congested graph. When solving the vehicle routing problem in real networks we use D ∗ instead of the distance matrix D 0 . The variables are the binary elements of a transition matrix between the vertices X = {xi j }, i, j ∈ I . xi j is 1, if the found route includes the edge (vi , v j ), 0 — otherwise. Then the traveling salesman problem can be formulated and solved in the framework of linear integer programming [52–54]:  

min X

di0j xi j ,

i∈I j=i, j∈J

subject to 0 ≤ xi j ≤ 1, ∀i, j ∈ I, u i ∈ Z, i ∈ I, 

xi j = 1, ∀i ∈ I,

j∈I, j=i



xi j = 1, ∀ j ∈ I,

i∈I,i= j

u i − u j + nxi j ≤ n − 1, 1 ≤ i = j ≤ n. We use a dynamically improved genetic algorithm for solving the travelling salesman problem. Consider graph G = (V0 , E), where V0 = {0, 1, 2, ..., n} is the set of nodes, E = {(i, j) | i, j ∈ V0 } is the set of edges. Node 0 corresponds to the depot, while nodes V = {1, ..., n} are clients. There is a set of vehicles V in the depot with the same tonnage. Assume that one vehicle can make only one trip with total tonnage D. Each client i ∈ V has demand di to be satisfied. We assume that the demand of a client could be satisfied by one visiting. The quantity of the goods at the depot is assumed to be sufficient to meet the demand of all customers. We are going to consider the vehicle routing problem with the time-dependent transition matrix. This means that the transition time from one customer to another is supposed to be changing during the routing.

8.4 Time-Dependent Vehicle Routing in a Congested Urban Area

201

Fig. 8.6 Function of transition time through the edge (i, j)

We associate each edge (i, j) ∈ E with the function ti j (bi ) which is the travel time on this edge, depending on departure time bi from the node i. Matrices T (b) were obtained above. The variables in the time-dependent problem are bi , i ∈ V , under b(v) 0 , v ∈ V we mean the start of motion of the vehicle v. Peace-wise linear function ti j (bi ) has a form:  ti j (bi ) =

dikj , z k−1 + k−1 ≤ bi ≤ z k + k dikj +

k (dik+1 j −di j )(bi −z k + k ) , zk 2 k

− k < bi < z k + k ,

when k = 2, . . . , K : 0 = K = 0, otherwise k = 5. Illustration of the function is available in Fig. 8.6. The goal of optimization is to find such routes for vehicles which satisfy the constraints. The overall journey time of all vehicles should be minimal. The described problem (TDVRP) may be formulated in terms of integer programming. Let {xi j }i, j∈V0 be Boolean variables that equal to 1 if client i is visited directly after client j, 0 otherwise. Introduce Boolean variables {yiv }i∈V,v∈V that equal to 1 if client i belongs to the route v, 0 otherwise. Then we have the following problem: min X

subject to

 i∈V0 j∈V0

ti j (bi )xi j ,

202

8 Transportation Processes Modelling in Congested Road Networks



xi j =

i∈V0



x ji ,

∀ j ∈ V0 ,

i∈V0



xi j = 1,

∀ j ∈ V,

i∈V0



di yki ≤ D,

k = 1, K ,

i∈V

bj ≥

 i∈V

xi j (bi + ti j (bi )) + x0 j



y jv (bv0 + ti j (bv0 )),

∀ j ∈ V.

v∈V

References 1. Allsop RE, Charlesworth JA (1977) Traffic in signal-controlled road network: an example of different signal timings inducing different routings. Traffic Eng Control 18:118–132 2. Smith MJ, Vuren T (1993) Traffic equilibrium with responsive traffic control. Transp Sci 27(2):118–132 3. Wong SC (1995) Derivatives of the performance index for the traffic model from TRANSYT. Transp Res Part B 29(5):303–327 4. Clegg J, Smith MJ, Xiang Y, Yarrow R (2001) Bilevel programming applied to optimizing urban transportation. Transp Res Part B 35(1):41–70 5. Yang H, Yagar S (1995) Traffic assignment and signal control in saturated road networks. Transp Res Part A 29(2):125–139 6. Dempe S (2002) Foundations of bilevel programming. Kluwer Academic Publishers, Dordrecht 7. Bard JF (2002) Practical bi-level optimization: algorithms and applications. Kluwer Academic Publishers, Dordrecht 8. U.S. Bureau of Public Roads, editor. Traffic Assignment Manual. U.S. Department of Commerce, Washington, D.C. (1964) 9. Krylatov AY, Zakharov VV, Malygin IG (2015) Signal control in a congested traffic area. In: 2015 International Conference on “Stability and Control Processes” in Memory of V.I. Zubov (SCP), pp 475–478 10. Hazelton M (2001) Inference for origin-destination matrices: estimation, prediction and reconstruction. Transp Res Part B 35:667–676 11. Yang H, Sasaki T, Iida Y, Asakura Y (1992) Estimation of origin-destination matrices from link traffic counts on congested networks. Transp Res Part B 26(6):417–434 12. Bianco L, Cerrone C, Cerulli R, Gentili M (2014) Locating sensors to observe network arc flows: exact and heuristic approaches. Comput Oper Res 46:12–22 13. Bierlaire M (2002) The total demand scale: a new measure of quality for static and dynamic origin-destination trip tables. Transp Res Part B 36:282–298 14. Castillo E, Menedez JM, Jimenez P (2008) Trip matrix and path flow reconstruction and estimation based on plate scanning and link observations. Transp Res Part B 42:455–481 15. Medina A, Taft N, Salamatian K, Bhattacharyya S, Diot C (2002) Traffic matrix estimation: existing techniques and new directions. In: Proceedings of the 2002 SIGCOMM conference on computer communication review, vol 32, pp 161–174 16. Minguez R, Sanchez-Cambronero S, Castillo E, Jimenez P (2010) Optimal traffic plate scanning location for OD trip matrix and route estimation in road networks. Transp Res Part B 44:282– 298

References

203

17. Zakharov V, Krylatov A (2014) OD-matrix estimation based on plate scanning. In: Veremey EI (ed) 2014 International Conference on Computer Technologies in Physical and Engineering Applications (ICCTPEA), pp 209–210 18. Krylatov AYu, Shirokolobova AP, Zakharov VV (2016) OD-matrix estimation based on a dual formulation of traffic assignment problem. Informatica (Slovenia) 40(4):393–398 19. Proposal for a regulation of the European parliament and of the Council - Setting Emission Performance Standards for New Passenger Cars as Part of the Community’s Integrated Approach to Reduce CO2 Emissions from Light-Duty Vehicles. Commission of the European Communities, Dossier COD/2007/0297 (2007) 20. Krautzberger L, Wetzel H (2012) Transport and CO2 : productivity growth and carbon dioxide emissions in the european commercial transport industry. Environ Resour Econ 53:435–454 21. Most carmakers must further improve carbon efficiency by 2015 (Retrieved September 28, 2012) European Environment Agency (2011) — URL: http://www.eea.europa.eu/highlights/ most-carmakers-must-further-improve/ (date: 01.05.2015) 22. U.S. Transportation Sector Greenhouse Gas Emissions: 1990–2011. EPA, U. – Office of Transportation and Air Quality. – EPA-420-F-13-033a (2013) 23. CO2 emissions from fuel combustions. International Energy Agency (2012) 24. Ahn K, Rakha HA (2013) Network-wide impacts of eco-routing strategies: a large-scale case study. Transp Res Part D Transp Environ 25:119–130 25. Ahn K, Rakha HA (2008) The effects of route choice decisions on vehicle energy consumption and emissions. Transp Res Part D Transp Environ 13(3):151–167 26. Aziz HMA, Ukkusuri SV (2014) Exploring the trade-off between greenhouse gas emissions and travel time in daily travel decisions: route and departure time choices. Transp Res Part D 32:334–353 27. Boriboonsomsin K, Barth MJ, Weihua Z, Vu A (2012) Eco-routing navigation system based on multisource historical and real-time traffic information. IEEE Trans Intell Transp Syst 13(4):1694–1704 28. Guo L, Huang S, Sadek AW (2012) An evaluation of environmental benefits of time-dependent green routing in the greater Buffalo Niagara region. J Intell Transp Syst 17(1):18–30 29. Hensher DA (2008) Climate change, enhanced greenhouse gas emissions and passenger transport what can we do to make a difference? Transp Res Part D Transport Environ 13(2):95–111 30. Stanley JK, Hensher DA, Loader C (2011) Road transport and climate change: stepping off the greenhouse gas. Transp Res Part A Policy Pract 45(10):1020–1030 31. Ben-Akiva M, De Palma A, Isam K (1991) Dynamic network models and driver information systems. Transp Res Part A General 25(5):251–266 32. Gaker D, Vautin D, Vij A, Walker JL (2011) The power and value of green in promoting sustainable transport behavior. Environ Res Lett 6(3):1–10 33. Mahmassani HS (1990) Dynamic models of commuter behavior: experimental investigation and application to the analysis of planned traffic disruptions. Transp Res Part A General 24(6):465–484 34. Lin J, Ge YE (2006) Impacts of traffic heterogeneity on roadside air pollution concentration. Transport Res Part D Transp Environ 11(2):166–170 35. Nagurney A (2000) Congested urban transportation networks and emission paradoxes. Transp Res Part D Transp Environ 5(2):145–151 36. Zhang Y, Lv J, Ying Q (2010) Traffic assignment considering air quality. Transp Res Part D Transp Environ 15(8):497–502 37. Aziz HMA, Ukkusuri SV (2012) Integration of environmental objectives in a system optimal dynamic traffic assignment model. Comput-Aided Civil Inf Eng 27(7):494–511 38. Yin Y, Lawphongpanich S (2006) Internalizing emission externality on road networks. Transp Res Part D Transp Environ 11(4):292–301 39. Boroujeni BY, Frey HC (2014) Road grade quantification based on global positioning system data obtained from real-world vehicle fuel use and emissions measurements. Atmos Environ 85:179–186

204

8 Transportation Processes Modelling in Congested Road Networks

40. Wyatt DW, Li H, Tate JE (2014) The impact of road grade on carbon dioxide (CO2 ) emission of a passenger vehicle in real-world driving. Transp Res Part D 32:160–170 41. Zhang KS, Frey HC (2006) Road grade estimation for on-road vehicle emissions modeling using light detection and ranging data. J AirWaste Manag Assoc 56(6):777–788 42. Jovanovic AD, Pamucar DS, Pejcic-Tarle S (2014) Green vehicle routing in urban zones—a neuro-fuzzy approach. Expert Syst Appl 41:3189–3203 43. The alternative fuels and advanced vehicles data center/Serbian Department of Energy (SDE), 2013. http://www.afdc.energy.gov.rs/afdc/locator/stations/state (date: 24.02.13) 44. Krylatov, A.Yu., Zakharov, V.V.: Competitive traffic assignment in a green transit network. Int Game Theory Rev 18(2) (2016) 45. Dantzig GB, Ramser RH (1959) The truck dispatching problem. Manag Sci 6:80–91 46. Lenstra J, Rinnooy Kan A (1981) Complexity of vehicle routing and scheduling problems. Networks 11:221–228 47. Cordeau J-F, Gendreau M, Laporte G, Potvin J-Y, Semet F (2002) A guide to vehicle routing heuristics. J Oper Res Soc 53:512–522 48. Clarke G, Wright J (1964) Scheduling of vehicles from a central depot to a number of delivery points. Oper Res 12(4):568–581 49. Cordeau JF, Laporte G, Mercier A (2001) A Unified tabu search heuristic for vehicle routing problems with time windows. J Oper Res Soc 52:928–936 50. Wardrop JG (1952) Some theoretical aspects of road traffic research. Proc Inst Civil Eng 2:325–378 51. Schrijver A (2005) On the history of combinatorial optimization (till 1960). Handbook on modelling for discrete optimization. Elsevier, Amsterdam 52. Dantzig GB (1963) Linear programming and extensions. Princeton University Press, Princeton 53. Miller CE, Tucker AW, Zemlin RA (1960) Integer programming formulation of traveling salesman problems. J ACM 7(4):326–329 54. Papadimitriou CH, Steiglitz K (1982) Combinatorial optimization: algorithms and complexity. Prentice-Hall, Inc, Upper Saddle River

Chapter 9

Load Flow Estimation in a Transmission Network

Abstract This chapter includes an investigation on a power smart grid with multiple suppliers and consumers. The first section is devoted to the corresponding model description. The power load flow estimation is presented in a form equivalent to a link-flow traffic assignment problem. The second section concentrates on the competition of consumers, while the third section concentrates on the cooperation of suppliers. From mathematical perspectives, both defined models of economic interaction imply bi-level optimization problems. Thus efficient computational algorithms are required. The last section is devoted to pricing mechanisms for transmission networks with multiple suppliers and consumers. Possible techniques for the arising complex problem of transmission cost sharing are proposed.

9.1 Multi-supplier and Multi-consumer Power Grid System This section describes the multi-supplier, multi-consumer congestion model of a power grid. Each supplier (energy producer) has a production cost function that depends on the amount of energy to be generated. These costs are covered by consumers that make bilateral contracts with the producers. All consumers also face energy transmission costs that depend on the congestion in the corresponding links. In order to determine the loads in the links, we need to compute the current flow distribution in the network according to the Kirchhoff’s laws. The goal of consumers is to minimize their total costs while meeting their energy demand. We define a power network as an undirected connected graph G = (V, A), where V is a set of nodes and A is a set of arcs connecting some pairs of nodes. Although the power network graph is naturally undirected, in this work we represent it as a directed graph by splitting each link into two arcs with the same pair of nodes but different directions. The only condition we need to check is that a current flow is not positive in both directions of the link. Set V consists of three subsets: VQ is a set of m consumers, V P is a set of n producers, and Vint represents the rest of the nodes that we denote as intermediate. Let us enumerate the nodes {i} ∈ V in the following way: VQ = {1, 2, . . . , m}, V P = {m + 1, m + 2, . . . , m + n} Vint = {m + n + 1, . . . , |V |}. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 A. Krylatov et al., Optimization Models and Methods for Equilibrium Traffic Assignment, Springer Tracts on Transportation and Traffic 15, https://doi.org/10.1007/978-3-030-34102-2_9

205

206

9 Load Flow Estimation in a Transmission Network

We assume that each consumer i ∈ VQ has a fixed energy demand di < 0 that they need to meet by making bilateral contracts with producers. A contract between consumer i ∈ VQ and producer j ∈ V P is defined by the amount of energy ei j to be delivered. By contract profile e we denote a set of all contracts {ei j , 1 ≤ i ≤ m, m + 1 ≤ j ≤ m + n}, where (9.1) ei j ≥ 0 for any consumer-producer pair (i, j). We also define ei = {ei j , m + 1 ≤ j ≤ m + n} as a contract profile of consumer i. The following constraints must be fulfilled for each consumer: m+n  ei j , 1 ≤ i ≤ m. (9.2) di = − j=m+1

Similarly, we define an amount of energy d j that each producer j ∈ V P needs to generate: m  dj = ei j , m + 1 ≤ j ≤ m + n. i=1

For the intermediate nodes we assign zero value to dk : dk = 0, m + n + 1 ≤ k ≤ |V |. Let us call the value dk an energy input in node k ∈ V . It corresponds with the fact that in consumer nodes the values of {dk } are negative, since consumers withdraw energy from the network. Each producer j ∈ V P has a unit production cost function α j (·) ≥ 0 that reflects the unit production cost depending on the amount of energy to be generated d j . Each consumer i ∈ VQ pays their part of the cost proportionally to the contract with producer j:    ci j (e) = ei j · α j (d j ) = ei j · α j ei j . 1≤i≤m

The total production cost each consumer i ∈ VQ must pay is a function of a contract profile e: n  ci (e) = ci j (e). j=m+1

Consumer costs also include transmission part. Therefore, we need to define the flow distribution in the network. One crucial difference between electrical and transportation networks is the impossibility of flow routing in the former case. Once we set input and output currents in supplier and consumer nodes respectively, and define the resistances of arcs, current flow distribution is defined uniquely by the Kirchhoff’s laws.

9.1 Multi-supplier and Multi-consumer Power Grid System

207

By f i j we denote a current flow, or simply a flow, in a directed arc (i, j) ∈ A. According to our definition of set A, there are two versions of each arc with different directions, and we can define f i j to be non-negative. Indeed, if energy flows from j to i, we assign positive value to f ji and zero value to f i j . Let us define a flow profile f = { f i j , (i, j) ∈ A} as a vector of all flows in the arcs. By Wi = { j | (i, j) ∈ A} we denote the set of all nodes adjacent to node i. Therefore, the first Kirchhoff’s law takes the following form: 

fi j −

j∈Wi



f ji = di , ∀i ∈ V,

(9.3)

j∈Wi

where di is an energy input in node i defined above. We assume that each arc (i, j) ∈ A has a voltage change function Θi j ( f i j ) depending on the flow f i j in this arc and its resistance (for instance, if the electrical device in arc (i, j) is a linear resistor, then Θi j ( f i j ) = Ri j f i j , that corresponds to Ohm’s law). According to the second Kirchhoff’s law, the following condition must be fulfilled for any arc (i, j) ∈ A: (9.4) πi − π j = Θi j ( f i j ), where πi is a potential in node i ∈ V . As mentioned above, due to [1], the mathematical programming formulations of the problem of currents and voltage search in an electrical network appeared. Now, we are able to state that Kirchhoff’s laws in this setting correspond to the conditions of user equilibrium in the non-atomic transportation problem [2]:  

min f

fi j

Θi j (s)ds,

(9.5)

f ji = di , ∀i ∈ V,

(9.6)

f i j ≥ 0, ∀(i, j) ∈ A.

(9.7)

(i, j)∈A

0

subject to  j∈Wi

fi j −

 j∈Wi

Since this paper is devoted to the routing methods in power flow problems, we give an explicit proof of an above statement to establish a clear relationship between Kirchhoff’s laws (9.35), (9.36) and the nonlinear optimization problem (9.5)–(9.7). Theorem 9.1 ([3]) Kirchhoff’s circuit laws (9.35), (9.36) may be obtained through the solution of nonlinear network flow problem (9.5)–(9.7). Proof Consider the constrained nonlinear optimization problem (9.5)–(9.7). The Lagrangian of this problem is

208

L( f, π, η) =

9 Load Flow Estimation in a Transmission Network   fi j (i, j)∈A 0

Θi j (s)ds +

 i∈V

⎛ πi ⎝di −



fi j +

j∈Wi



⎞ f ji ⎠ +

j∈Wi



(− f i j )ηi j ,

(i, j)∈A

where {πi , i ∈ V } and {ηi j ≥ 0, (i, j) ∈ A} are Lagrange multipliers. Differentiation of the Lagrangian with respect to the f i j yields: ∂L = Θi j ( f i j ) − πi + π j − ηi j = 0 ∂ fi j and, consequently, Θi j ( f i j ) = πi − π j + ηi j .

(9.8)

Lagrange multipliers {ηi j , (i, j) ∈ A} are non-negative and the complementary slackness condition is fulfilled: (− f i j )ηi j = 0. Therefore if f i j > 0 then ηi j = 0 but if f i j = 0 then ηi j ≥ 0, and, hence, from (9.8) we obtain

Θi j ( f i j ) =

= πi − π j , for f i j > 0, ≥ πi − π j , for f i j = 0,

(9.9)

We can conclude that the solution of the optimization program (9.5)–(9.7), first of all, satisfies the first Kirchhoff’s law, since (9.35) coincides with the set of constraints (9.6) and, secondly, it satisfies the second Kirchhoff’s law according to (9.9). Remark 9.1 According to the form of problem (9.5)–(9.7), Θi j for any (i, j) ∈ A is mathematically equal to the delay function in the transportation network. Thereby, the nonlinear optimization program (9.5)–(9.7) can be interpreted as a model of equilibrium voltage and currents flow assignment in an electric network, when the energy loss of the devices is minimized. Nevertheless, there is no energy interpretation of the objective function (9.5), which is obtained as a sum of integrals of the voltage-current relations {Θi j , (i, j) ∈ A}. We now define a path r between two nodes i, j ∈ V as an ordered sequence of arcs {(i, k1 ), (k1 , k2 ), . . . , (kw , j)} that contains no cycles. Let R denote the set of all possible paths {r } in the network. Remark 9.2 Consider an arbitrary path r ∈ R between two nodes p, q ∈ V with potentials π p and πq respectively, consisting of sequentially connected electrical devices (resistors). Summing the Eq. (9.9) over all links defining the path r , we obtain:

  = (i, j)∈r πi − π j = π p − πq , if ∀ (i, j) ∈ r f i j > 0,  Θi j ( f i j ) = ≥ (i, j)∈r πi − π j = π p − πq , if ∃ (i, j) ∈ r : f i j = 0. (i, j)∈r

(9.10) If objective function (9.5) is convex then Kuhn–Tucker conditions are necessary and sufficient. In such a case, (9.10) has a clear interpretation: between any two

9.1 Multi-supplier and Multi-consumer Power Grid System

209

nodes p, q ∈ V of the electric network, the current utilizes paths with total potential differences equal to the potential difference of nodes p and q, and the current does not utilize paths of resistors with total potential differences greater than the potential difference of nodes p and q. In a large network it can be useful for the evaluation of the loop flows. A given contract profile e determines a set of energy inputs d = {dk , 1 ≤ k ≤ |V |} and, therefore, a flow profile f according to optimization problem (9.5)–(9.7). We denote this flow profile by f (e), while f kl (e) denotes the corresponding flow on arc (k, l). For each arc (k, l) ∈ A we define a transmission cost function βkl ( f kl ) that depends on the energy flow in this arc f kl . We assume that this function is the same for both directions of a link, i.e. βkl (x) = βlk (x). Though it appears to be natural that this cost must be covered by the consumers which use the arc, the calculation of transmission cost shares is not simple. Indeed, flows are distributed according to Kirchhoff’s laws and, as was shown in Remark 9.2, may use many parallel lines in circuitous routes. Generally, it is non-trivial to define the accordance between bilateral energy contracts {ei j } and their contributions to the actual current flows in the network links. Therefore, transmission pricing is an extremely obscure issue for a multi-supplier power grid network of general topology. The problem of transmission pricing is widely discussed in the literature. The transmission cost defined as the difference between nodal prices at the points of withdrawal and injection seems to be acceptable amidst the lack of reasonable alternatives [4]. In [5] it is shown that the concept of contract path, well implemented in rail lines or highways, does not fit with power grids, since the nature of electricity leads to the emergence of loop flows. Therefore, he offered to consider a concept of a contract network instead of a contract path, where the power grid should be viewed as the network of one owner to whom all network users have to pay a total transmission cost. This idea clarifies the side of actual network owners: each owner obtains the cost according to their capacity rights. However, the question of how to divide the total cost between consumers despite the well investigated capacity market and transmission rights auctions is still uncertain [6]. In order to assign appropriate shares of transmission costs for each arc (k, l) ∈ A and each consumer i ∈ VQ , we define a set of functions {δkli (e), (k, l) ∈ A, 1 ≤ i ≤ m} that fulfills following conditions: m 

δkli (e) = 1 ∀(k, l) ∈ A,

i=1

δkli (e)

(9.11)

≥ 0 ∀(k, l) ∈ A, 1 ≤ i ≤ m.

Hence, for a given contract profile e, each consumer i pays the cost tkli (e) for the use of each arc (k, l) ∈ A as follows: tkli (e) = δkli (e) · βkl ( f kl (e)).

210

9 Load Flow Estimation in a Transmission Network

Fig. 9.1 e14 = 0, f 56 = 5

Fig. 9.2 e23 = 0, f 65 = 4

Fig. 9.3 e14 = 4, e23 = 5, f 56 = 1

A given energy input profile d = {dk , 1 ≤ k ≤ |V |} determines a flow profile f according to conditions (9.35)–(9.36). For each arc (k, l) we define a transmission cost function βkl ( f kl ) that depends on the energy flow in this arc f kl . It appears to be fair if this cost is shared by those consumers who use this arc, but it is not simple to determine which consumers actually utilize it and what are the appropriate transmission cost shares. For illustrative purposes, let us consider an example of a power network with six nodes and five arcs. There are two consumers VQ = {1, 2} and two generators V P = {3, 4} in this network. Generator 3 intends to transfer 5 MW of power to consumer 2, whereas generator 4 must transfer 4 MW to consumer 1, according to bilateral contracts e23 and e14 . We want to determine the transmission cost shares for arc (5, 6). If this arc is used by a single consumer-generator contract pair, this pair covers the total cost β56 ( f 56 ) (or β65 ( f 65 )). Truly, if e14 = 0, contract pair (2, 3) is the only user of the network and it pays the transmission costs for using arcs (3, 5), (5, 6) and (6, 2) (see Fig. 9.1). On the other hand, if e23 = 0, contract pair (1, 4) pays the transmission costs for using arcs (4, 6), (6, 5) and (5, 1) (see Fig. 9.2). However, if both pairs use the network at the same time (see Fig. 9.3), flow f 56 = 1, although virtually both pairs transmit more power through this arc (e14 = 4, e23 = 5). It is unclear how to share this cost β56 (1) fairly between them. The previous example considers a setting where two contract pairs use the same arc in different directions. An even more sophisticated case of cost-sharing arises if multiple contract pairs use multiple parallel arcs. Indeed, flows are distributed according to Kirchhoff’s laws when routed directly, and they may use several parallel

9.1 Multi-supplier and Multi-consumer Power Grid System

211

paths. The relation between bilateral energy contracts and actual flow parts that contribute to the congestion in a specific link is not trivial for networks with multiple generators and consumers. We now formulate a technique that helps determine the contribution of each user to the transmission cost in particular network arcs. If we know the values of electric potentials in the network, we are able to reconstruct the actual current paths between a generator and a consumer. Due to Corollary 9.2, we check all arcs of a particular path, and if πi − π j < 0 for any arc, the current does not flow along this arc. Therefore, we need to compute potentials, either by solving an optimization problem (9.5)–(9.7) and finding flows in arcs, or we can reformulate the primal optimization problem (9.5)–(9.7) in a dual form and find potential differences directly. Indeed, the

Lagrange multiplier πi is the electric potential at the node i ∈ V , while πi − π j is the potential difference between nodes (i, j) ∈ A. The dual problem for (9.5)–(9.7) takes the following form: 

max π

qi j (πi − π j ),

(9.12)

(i, j)∈A

where qi j (·) is defined by

 qi j (πi − π j ) = min

f i j ≥0

fi j

 Θi j (s)ds − (πi − π j ) f i j .

(9.13)

0

In particular cases, finding potentials by solving the dual problem may be easier than solving the primal problem [7]. Therefore, in order to estimate the shares of transmission costs for a particular arc (i, j) ∈ A, we can apply the following method: 1. Solve the optimization problem (9.5)–(9.7) or, alternatively, dual problem (9.12)– (9.13), and find the potential differences of all adjacent pairs of nodes. 2. Find actual current paths using (9.10) for each contract pair (k, l) ∈ VQ × V P . 3. Determine all paths that pass through arc (i, j), and the contract pairs these paths belong to. All other contract pairs do not utilize this arc, and should not share the costs for its use. 4. Charge each of the contract pairs determined in the previous step proportionally to the value of contracts and number of paths traversing arc (i, j). The developed technique is based on the similarities between electric current flow distribution and transportation flow assignment. Our work does not differentiate between the main flow and loop flows in electric networks, but rather considers all possible routes as alternative. Thus, the problem of routes reconstruction between producer (origin) and consumer (destination) allows the use of corresponding ideas and techniques from transportation theory.

212

9 Load Flow Estimation in a Transmission Network

9.2 Competition of Consumers in Smart Grid Systems Let us describe a game of consumers. Each consumer i tries to minimize their total cost that consists of production and transmission parts while meeting the energy demand di . Therefore, the total cost of each consumer can be determined as a function of contract profile e. By Ci (e) we denote a total cost of consumer i ∈ VQ for a given contract profile e. It can be defined in the following form: Ci (e) =

m+n  j=m+1

ci j (e) +



tai (e),

(9.14)

a∈A

i.e., the total costs paid to producers and the total transportation costs assigned by corresponding arcs of the network. Note that the second summand in (9.14) depends on flow profile f (e) obtained as a solution of optimization problem (9.5)–(9.7). The competitive game of consumers can now be formulated as a constrained set of minimization problems: min Ci (e), ei

1 ≤ i ≤ m, (9.15)

subject to (9.1)-(9.2), where vector ei can be viewed as a strategy of consumer i, and contract profile e is a set of the consumers’ strategies. The most used and well-studied solution concept for competitive games is Nash equilibrium, i.e. such strategy profile that the players cannot reduce the costs by unilaterally changing their strategy. In game (9.15), contract profile e∗ is a Nash equilibrium, if Ci (e∗ ) ≤ Ci (ei , e∗−i ), ∀ei , 1 ≤ i ≤ m, where by (ei , e∗−i ) we denote a contract profile that differs from e∗ only in components of ei . There are three main questions we need to answer when we study the Nash equilibrium of any game: whether it exists, is it unique, and how to find it. Though the first two questions do not assume a positive answer in a general game setting, there are many results in this field for some specific settings. Existence of equilibria in a competitive game strongly depends on the properties of cost functions. In some particular cases, positive results have been established. For example, if cost functions are convex, the existence of Nash equilibria was proven in [8]. Moreover, the uniqueness of an equilibrium for settings with some stronger conditions on cost functions is also proved. Consider game (9.15) for a network with general topology. As it was discussed in Sect. 3.2, contract profile e determines flow profile f via optimization problem (9.5)– (9.7). Hence, cost functions depend on a contract profile implicitly, and determining

9.2 Competition of Consumers in Smart Grid Systems

213

their properties is a difficult task. Indeed, the transmission part of consumer i’s total cost has a form:   tai (e) = δai (e) · βa ( f a (e)), a∈A

a∈A

and even if we guarantee the convexity of functions {δai (·)} and {βa (·)}, flow functions { f a (e)} are generally not convex. Therefore, for a general network topology, the existence of equilibria cannot be easily determined, and we need to restrict the class of network topologies in order to establish the existence of equilibria. As an example, we study the case of a network with a tree structure, i.e. when a network graph does not contain any cycles. Let us consider a network G = (V, A) that has a tree structure. It is clear that for each consumer-producer pair (i, j), a unique path r ji exists from j to i. The solution to optimization problem (9.5)–(9.7) is trivial in this case, since the second Kirchhoff’s law conditions are fulfilled automatically in a network without loops, and we only need to solve the constrained linear system (9.6)–(9.7) to determine a flow profile f (e). Therefore, all components of this flow profile can be found as linear combinations of the contracts {ei j }. ij We define a set of coefficients {qkl } as follows:  ij qkl

=

1, (k, l) ∈ r ji 0, (k, l) ∈ / r ji ,

where each coefficient determines whether a path r ji contains an arc (k, l) or not. Then a flow in an arbitrary arc (k, l) ∈ A can be found as a linear combination of {ei j }:  ij qkl ei j . (9.16) fˆkl = r ji ∈R

However, if both fˆkl and fˆlk are positive, it contradicts our definition of { f kl }. Therefore, we refine (9.16) in the following way: f˜kl = max{0, ( fˆkl − fˆlk )}.

(9.17)

Lemma 9.1 ([3]) In a network with a tree structure, flow profile { f˜kl } is determined according to (9.16)–(9.17) fulfills the set of conditions (9.6)–(9.7). Proof It is clear that (9.7) is fulfilled due to (9.17). Hence, we need to show that for each k ∈ V the condition   (9.18) f˜kl − f˜lk = dk , ∀i ∈ V, l∈Wk

l∈Wk

214

9 Load Flow Estimation in a Transmission Network

is also fulfilled. Note that (9.17) implies f˜kl − f˜lk = fˆkl − fˆlk , and we can rewrite the left part of (9.18): 

f˜kl −

l∈Wk



⎛ ⎝

 l∈Wk



 l∈Wk



ij

qkl ei j −

r ji ∈R

l∈Wk

f˜lk =

   f˜kl − f˜lk = fˆkl − fˆlk =

⎞



qlk ei j ⎠ = ij

r ji ∈R

r ji ∈R

⎛

l∈Wk



⎝ei j ·

⎞ (qkl − qlk )⎠ ij

ij

l∈Wk

Let us consider three cases, one for each of the sets VQ , V P , Vint . If k ∈ Vint , any route r ji that passes node k must contain two arcs (l1 , k) and (k, l2 ), ij ij since no route starts or ends in an intermediate node. Therefore, qkl1 = ql2 k = 1 and ij ij all other coefficients {qkl }, {qlk } equal zero. The right part of (9.11) takes the form: 

⎛ ⎝ei j ·

r ji ∈R



⎞ (qkl − qlk )⎠ = ij

ij



(ei j · (1 − 1)) = 0 = dk .

r ji k

l∈Wk

When k ∈ VQ , there are exactly n routes {r jk , m + 1 ≥ j ≥ m + n} ending at k. kj Each of these routes contains arc (l1 , k) with ql1 k = 1, but no arcs (k, l). There might also be some routes passing node k. Again, the right part of (9.11) can be rewritten: 

⎛ ⎝ei j ·

r ji ∈R



⎞ (qkl − qlk )⎠ = ij

ij

r jk

l∈Wk

+

 (ek j · (0 − 1)) +

m+n  

ek j = dk . ei j · (1 − 1) = − r ji k

j=m+1

Finally, if k ∈ V P , there are m routes {rki , 1 ≥ i ≥ m} starting from k. Each of these routes contains arc (k, l2 ) with qklik2 = 1 and no arcs (l, k). Some routes can also pass node k. Rewriting (9.11), we obtain  r ji ∈R

⎛ ⎝ei j ·



⎞ (qkl − qlk )⎠ = ij

ij

rki

l∈Wk

+

 (eik · (1 − 0)) +



m

 ei j · (1 − 1) = eik = dk ,

r ji k

i=1

and (9.18) is fulfilled for all nodes in V . ij

If the topology of the network does not change, we need to find the set {qkl } only once, and for any given contract profile e we can calculate the components of the

9.2 Competition of Consumers in Smart Grid Systems

215

resulting flow profile f as a linear combination of contracts using (9.16)–(9.17) for each arc of the network. Let us assume that transmission costs are shared evenly by the consumers that may use this arc. In other words, consumer i shares the transmission cost in arc (k, l) if at least one route r ji exists that crosses this arc. Therefore, functions {δa (e)} do not depend on contract profile e, but only on the structure of the network. In other words, they are constant for a fixed network: δkli (e) = δkli ,

∀(k, l) ∈ A, i ∈ VQ ,

(9.19)

and it is clear that these functions fulfill conditions (9.11). Though this cost sharing rule may seem unfair, its simplicity allows us to efficiently operate with the transmission costs. Theorem 9.2 ([3]) Assuming that a network contains no cycles, functions {α j (·)} are convex and increasing, functions {βa (·)} are convex, and transmission costs are shared according to rule (9.19). Then game (9.15) has a Nash equilibrium contract profile e∗ . Proof According to [8], an equilibrium exists for any n-person game with concave payoff functions. Since we consider cost functions rather than payoff functions, the same statement is true for games with convex cost functions. Therefore, we need to check whether a cost function Ci (e) = Ci (e1 , . . . , em ) is convex in ei for each consumer i ∈ VQ . Function Ci (e) consists of several summands: Ci (e) =

m+n 



ci j (e) +

tkli (e).

(k,l)∈A

j=m+1

If we show that each summand in this sum is convex, the convexity of the whole sum will be established as well. First, we study function ci j (e):  ci j (e) = ei j · α j (d j ) = ei j · α j



 ei j .

(9.20)

1≤i≤m

When we fix the contract profiles of all consumers except i, function α j (ei j + λ) remains convex and increasing, and function in (9.20) isconvex as a product of two non-negative increasing convex functions. Here λi = k =i ekl is some constant value. Second, we rewrite the function tkli (e) with fixed contract profiles of all consumers except i applying Lemma 9.1: ⎛ tkli (e) = δkli · βkl ( f a (e)) = δkli · βkl ⎝μi +

m+n  j=m+1

⎞ (qkl − qlk )ei j ⎠ , ij

ij

(9.21)

216

9 Load Flow Estimation in a Transmission Network

Fig. 9.4 5-node network with no cycles

The argument of βkl (·) in (9.21) is a linear combination of {ei j , j ∈ V P }, components of consumer i’s contract profile, and constant μi determined by fixed profiles of other consumers. Therefore, βkli (e) remains convex in ei , as well as tkli (e). Example 9.2.1 Let us consider a 5-node network with a structure depicted in Fig. 9.4. There are two consumers VQ = {1, 2}, two producers V P = {3, 4} and one intermediate node Vint = {5}. The interaction between consumers and producers can be formulated as a game (9.15). We assume that all nodes are located in the same area except for node 3 that depicts a larger energy generator, e.g., a power plant. Therefore, arc (3, 5) is longer than all other arcs in the network, and transmission costs are assumed to be higher for this arc. Due to Lemma 9.1, a flow on each arc is a linear combination of the contracts {ei j }: fˆ35 = e13 + e23 ,

fˆ42 = e23 + e24 ,

fˆ51

= e13 + e14 ,

fˆ54 = e23 − e14 . (9.22) The direction of the flow in arc (5, 4) may differ depending on the values of the contracts. If fˆ54 < 0, we assign f 54 = 0 and f 45 = − fˆ54 . According to rule (9.19), we 1 2 2 2 1 1 = δ35 = 1/2, δ42 = δ54 = δ51 = δ45 = 1, whereas all other δ-coefficients obtain δ35 are equal to zero. We assume that functions {α j (·)} and {βkl (·)} have the form α j (x) = λ j · x 1+ + μ j , βkl (x) = λkl · x 1+ζ ,

j ∈ VP , ∀(k, l) ∈ A,

(9.23)

where all coefficients are non-negative. We now find an equilibrium point in game (9.15) with specific values of demands d1 , d2 and coefficients {λ}, {μ}. In this example we assign = ζ = 0.2. We compare the costs of consumer 1 and consumer 2 in the case of single energy producer 3 (C1 and C2 ) and in Nash equilibrium (C1∗ and C2∗ ). The simulation results for different values of the parameters (the Table 9.1) are given in the Table 9.2. It is clear that the distribution of contracts between different suppliers reduces the costs of consumers. Our tests show reduction of costs from 6 up to 17%. The reduction rate strongly depends on the demand values of consumers and the parameters of the cost functions.

9.3 Integrated Smart Energy System Table 9.1 Parameters of cost-functions d1 d2 λ35 −10 −10 −20 −20

−12 −12 −22 −22

0.005 0.01 0.01 0.02

Table 9.2 Simulation results C1∗ C1 0.935 1.133 2.824 4.526

1.134 1.236 3.054 5.271

217

λ42 = λ51 = λ54

λ3 , μ3

λ4 , μ4

0.001 0.001 0.001 0.001

(0.002, 0.02) (0.002, 0.02) (0.001, 0.04) (0.002, 0.04)

(0, 0.12) (0.005, 0.12) (0.005, 0.12) (0.01, 0.12)

C2∗

C2

1.206 1.374 3.116 4.942

1.361 1.463 3.356 5.751

9.3 Integrated Smart Energy System The question of collaboration in the energy market has long been irrelevant and, in contrast to smart grid consumer collaboration, the competitive market forces of power consumption have been deeply investigated. Nowadays, the newly established circumstances force us to investigate the collaborative mechanisms in power consumption. Indeed, due to modern technologies, generators and consumers are now able to be integrated spatially in a smart local (nodal) power system. Nodal integration could be especially promising in case of generators of renewable energy, which deeply depends on weather. First of all, a smart system could decide how to assign power loads among consumers. Secondly, a smart grid power system could guarantee the conditions of optimal nodal demand-side consumption by virtue of the agreement between consumers to collaborate in energy consumption. Let us formulate the power grid model with multiple generators and consumers. Each generator has a production cost function, and it depends on the total amount of energy this generator is assigned to produce. Consumers are price-taking agents with an energy demand that they need to meet. Both production and transmission costs are covered by the consumers. Consider a directed connected graph G = (V, A), where V is a set of nodes and A is a set of arcs. We enumerate the nodes in V in a specific way, so that set V consists of three subsets: a set of m consumer nodes VQ = {1, . . . , m}, a set of n producer nodes V P = {m + 1, . . . , m + n}, and a set of all other nodes VO = {m + n + 1, . . . , |V |}. Consumers make bilateral energy contracts with multiple producers in order to meet their energy demands. Demand constraints take the following form:

218

9 Load Flow Estimation in a Transmission Network m+n 

di +

ei j = 0, i ∈ VQ .

(9.24)

j=m+1

For all other nodes k ∈ V P ∪ VO : dk =

m 

eik ,

k ∈ VP ,

i=1

dk = 0,

k ∈ VO .

For each producer j ∈ V P we define a unit production cost α j (d j ) ≥ 0. Consumers share production costs proportionally to their contracts: ci j (e) = ei j · α j (d j ) = ei j · α j

 m 

 ei j .

(9.25)

i=1

Therefore, we can determine the overall production cost of each consumer i ∈ VQ for a given contract profile e: ci (e) =



ci j (e) =

j∈V P



 ei j · α j

j∈V P

 m 

 ei j

.

(9.26)

i=1

As shown above, we assume that once the contract profile is determined, the flow profile can be found as a solution of a non-linear optimization problem of flow assignment. Usually, the participants of the energy market are assumed to be selfish and independent, which leads us to a system formulated in terms of competitive games. However, if energy resources are scarce and/or uncontrollable (such as renewable energy generation), it may be more profitable for consumers to cooperate. In order to reduce the total costs for energy production and transmission, we need to solve a total cost minimization problem. Let us consider the total cost of all consumers:  (ci (e) + ti (e)) = C(e) = i∈VQ

 i∈VQ j∈V P



j∈V P

ei j · α j

 m  i=1

 ei j

+

 

δikl (e) · βkl ( f kl (e)) =

i∈VQ (k,l)∈A



d j (e) · α j d j (e) + βkl ( f kl (e)) (k,l)∈A

The minimization problem can be formulated in the following form:

(9.27)

9.3 Integrated Smart Energy System

minimize e

subject to



219



d j (e) · α j d j (e) + βkl ( f kl (e))

(9.28)

(k,l)∈A

j∈V P

(9.24) − (9.25),

(9.29)

where flow profile f (e) is found as a solution of optimization problem (9.5)–(9.7). Note that the solution of the constrained optimization problem (9.28)–(9.29) does exist. However, it is not a trivial task to define the uniqueness of the solution in a general case. Nevertheless, the properties of a solution could be established via differentiating the Lagrangian with respect to ei j and using Kuhn-Tucker conditions. The only thing to be stated is that Kuhn-Tucker conditions are necessary and sufficient when product d j (e) · α j d j (e) and βkl ( f kl (e)) are convex functions. There are different ways of fair cost sharing among consumers. Ideas of the cooperative game theory may be applied, such as the concept of Shapley value for a coalition of consumers (see, e.g., [9]). Another simplified policy to encourage the cooperation between consumers is to share the total cost proportionally to the demand of each consumer: (9.30) Ci (e) = di · C(e), ∀i ∈ VQ . In this case each consumer i minimizes the same function (up to a constant coefficient {di }). Example 9.3.1 ([10]). Consider a network with 7 nodes that is depicted in Fig. 9.5. There are 3 consumers (red nodes), 3 producers (green nodes) and one intermediate node. Therefore, VQ = {1, 2, 3}, V P = {4, 5, 6}, and VO = {7}. Since there are no cycles in the network, we only need to check the first Kirchhoff’s law (9.35). A flow on each arc is a linear combination of {ei j }: fˆ25 = e34 + e36 − e15 − e25 ,

fˆ53 = e34 + e35 + e36 ,

fˆ42 = e14 + e24 + e34 ,

fˆ27 = e14 + e15 − e26 − e36 ,

fˆ71 = e14 + e15 + e16 ,

fˆ67 = e16 + e26 + e36 .

The direction of the flow in arcs (2, 5) and (2, 7) may differ depending on the values {ei j }. If fˆ25 < 0, we assign f 25 = 0 and f 52 = − fˆ25 . The same is true for fˆ27 . Let us assume that functions {α j (·)} and {βkl (·)} have the following form:

Fig. 9.5 7-node network with no cycles

220

9 Load Flow Estimation in a Transmission Network

α j (x) = λ j · x 1+ + μ j , ∀ j ∈ V P ,

(9.31)

βkl (x) = λkl · x 1+ζ , ∀(k, l) ∈ A.

where all coefficients are non-negative. We now solve the problem (9.28)–(9.29) with specific values of demands {di , i ∈ VQ } and coefficients in (9.31), and evaluate the total cost reduction. Let us assume that consumers have similar demand values, and that the transmission cost in arc (4, 2) is higher due to longer distance, while the other five arcs have identical coefficients of transmission function (i.e., λ42 > λa , ∀a = (4, 2)). We also assign = ζ = 0.2. We compute and compare the minimal total costs of consumers in two different cases: Csin if each consumer buys energy from a single supplier, and Copt for a case when consumers distribute their contracts between several producers. Some of the simulation results can be found in the tables: d1 , −20

d2 , −22

d3 , −23

λ42 0.005

λa 0.001

−10

−12

−9

0.005

0.001

−10

−12

−9

0.005

0.001

Copt Csin 15.81 17.21 5.43 5.5 3.98 4.33

λ4 , μ4 (0.002, 0.2) (0.002, 0.2) (0.002, 0.2)

λ5 , μ5 (0.001, 0.12) (0.001, 0.12) (0, 0.12)

λ6 , μ6 (0.005, 0.1) (0.005, 0.1) (0.01, 0)

Gain (%) 8.1 1.2 7.9

It is clear that the cooperative contract distribution reduces the total cost significantly (up to 8%). However, this reduction strongly depends on the demand values of the consumers and the cost functions of the producers.

9.4 Pricing Mechanisms in Multi-generator and Multi-consumer Power Grid Classical power grids have a hierarchical structure, with few large generators such as power plants on one end, transmission and distribution networks in between, and multiple end-consumers on the other end. This matter is changing now due to the growing number of local small-scale generators. Therefore, new power grid architecture is required in order to efficiently incorporate these new actors into the network

9.4 Pricing Mechanisms in Multi-generator and Multi-consumer Power Grid

221

system without causing disruptions in its operation. As mentioned above, one of the challenges is to maintain a balance between generation and consumption, which becomes more complicated in the presence of weather-dependent renewable energy sources. Since generators cannot respond effectively to the changing demand of consumers, or even keep the rate of energy generation constant, it becomes necessary to promote production-oriented consumption. In other words, consumers respond to the changes in generation and adjust their consumption accordingly. However, this consumption shift may cause discomfort and, therefore, needs to be incentivized. Therefore, advanced pricing mechanisms can be used to provide the necessary motivation. This section studies a multi-generator power grid model, with consumers as rational decision-makers. Each consumer-generator pair signs a bilateral contract that determines a day-ahead profile of energy amounts to be generated and delivered from generator to consumer in this particular pair during next 24 h. We consider local area network and, therefore, assume transmission costs to be negligible (or constant and not dependent on the consumers’ decisions). However, we still need to ensure that the transmission links are not overloaded. We describe consumers’ costs as functions of their strategies with transmission constraints and formulate a competitive game of consumers. A power network is represented by a directed graph G = (V, A), with a set of nodes V and a set of arcs A that connect some pairs of nodes. Node set V consists of three subsets: set VC of m consumers, set VG of n generators, and set VI of intermediate nodes that do not consume or generate energy. We enumerate all nodes in V as follows: VC = {1, 2, . . . , m}, VG = {m + 1, m + 2, . . . , m + n} and VI = {m + n + 1, . . . , |V |}. Pair (i, j) ∈ VC × VG signs an energy contract E i j = (ei1j , . . . , eiHj )T , a vector that determines the amounts of energy to be generated and delivered from generator j to consumer i during each of the H time slots. Generally, the choice of H is arbitrary, but in this work we consider a day-ahead time span with one hour intervals, hence H = 24. Let us denote the set of time intervals {1, . . . , H } by H. We also assume that eihj ≥ 0 for any time slot h and pair {i, j}. By Ei we denote a H × n matrix of all consumer i’s contracts, where Ei = (E i(m+1) , . . . , E i(m+n) ). Furthermore, a total energy profile E denotes a set of all consumers contacts: E = {E1 , . . . , Em }. According to energy contracts, we define an order vector Oi = (oi1 , . . . , oiH )T that determines the amount of energy delivered to consumer i during each of the time slots 1 ≤ h ≤ H . It has the following form: Oi =

m+n 

E i j , ∀i ∈ VC .

(9.32)

j=m+1

Every consumer i ∈ VC has a total day-ahead energy demand Di ≥ 0 as well as a minimal demand d ih for each time slot h, which needs to be covered by the set of orders Oi . Therefore, the following conditions must be fulfilled for each i ∈ VC :

222

9 Load Flow Estimation in a Transmission Network

d ih ≤ oih , 1 ≤ h ≤ H, Di ≤

H 

(9.33)

oih .

h=1 T Similarly to an order vector, we define a supply vector S j = (s 1j , . . . , s H j ) that represents the amount of energy to be delivered by generator j during each of the time slots 1 ≤ h ≤ H . This vector is determined by the following expression:

Sj =

m 

E i j , ∀ j ∈ VG .

(9.34)

i=1

The current flow distribution is determined by Kirchhoff’s laws, once the order and supply vectors of the nodes are assigned and the resistances of the links are defined. Let f klh ≥ 0 denote a flow in a directed arc (k, l) ∈ A. We define a flow profile f h as a set of flows in all arcs of the grid during time interval h: f h = { f klh , (k, l) ∈ A}. By Wk we denote a subset of nodes that are adjacent to node k, i.e. Wk = {l | (k, l) ∈ A}. Then the first Kirchhoff’s law takes the following form:  ( f klh − flkh ) + okh = 0, if k ∈ VC , l∈Wk



l∈Wk

( f klh − flkh ) − skh = 0, 

( f klh − flkh ) = 0,

if k ∈ VG ,

(9.35)

k ∈ VI .

l∈Wk

For each arc (k, l) ∈ A, the voltage function is an increasing positive function Θkl (·) that has flow f klh as its argument and implicitly depends on the resistance of this arc. The second Kirchhoff’s law can now be formulated for every arc (k, l) ∈ A as follows: πk − πl = Θkl ( f kl ), if f kl > 0, if flk > 0, πl − πk = Θlk ( flk ), (9.36) πk = πl , if f kl = flk = 0, where πk is an electric potential in node k ∈ V . The problem of finding current flows in a power grid could be formulated as a mathematical program: minimize fh

subject to

  (k,l)∈A



l∈Wkout

f klh

0

f klh −

Θkl (s)ds, 

flkh = bkh , ∀k ∈ V,

(9.37) (9.38)

l∈Wkin

f klh ≥ 0, ∀(k, l) ∈ A,

(9.39)

9.4 Pricing Mechanisms in Multi-generator and Multi-consumer Power Grid

where bkh is

bkh = −okh , bkh = skh , bkh = 0,

if k ∈ VC , if k ∈ VG , if k ∈ VI .

223

(9.40)

Theorem 9.1 proves that the solution of (9.37)–(9.39) satisfies Kirchhoff’s laws for any h. We assume that each consumer tries to minimize their total cost over a 24-hour time span while fulfilling their demand requirements. Let cihj denote consumer i’s payment to generator j during time interval h. In a simple case of constant energy price p hj it has a form cihj = p hj · eihj . Each consumer pays for their share of energy to each generator node, and the total payment for a time interval h is determined as follows: m+n  Cih (E) = cihj . (9.41) j=m+1

In turn, the total cost for consumer i is Ci (E) =

H 

Cih (E).

(9.42)

h=1

Thus, the game of consumers without arc capacity constraints has the following form: minimize

Ci (E),

1 ≤ i ≤ m,

(9.43)

subject to

eihj ≥ 0,

∀i ∈ VQ , ∀ j ∈ V P , ∀h ∈ H,

(9.44)

∀i ∈ VQ , ∀h ∈ H,

(9.45)

∀i ∈ VQ .

(9.46)

Ei

oih

≥

d ih ,

Oi ≥ Di ,

In this game, matrix Ei is a strategy profile of consumer i. We denote by i a set of strategies fulfilling conditions (9.44)–(9.46). Let us define the capacity restrictions. For each link (k, l) ∈ A we define the value of maximal flow allowed to be transmitted during any time interval, independent of h. We denote this arc capacity by fˆkl , and the fourth set of constraints can be written as follows: (9.47) f klh ≤ fˆkl , ∀(k, l) ∈ A, ∀h ∈ H. Note that the second form of a game takes into account the arc capacities: minimize

Ci (E),

subject to

(9.44) − (9.47).

Ei

1 ≤ i ≤ m,

(9.48) (9.49)

224

9 Load Flow Estimation in a Transmission Network

We assume that each generator can estimate their generation costs and capacities, and chooses price functions accordingly in order to balance their costs. For simplicity reasons, we consider generators that do not try to maximize their profits, since otherwise it would lead to another competitive game among the generators. More formally, let a generator j have a generation capacity of sˆ hj at each time interval h, and after crossing this limit the energy generation becomes significantly more expensive or even impossible. This might reflect the weather forecasts, sunlight hours etc. Therefore, we can define a generation cost function α hj (s hj ) that depends on the capacity sˆ hj . One possible form of this function is piecewise linear: α hj (s hj )

 a hj · s hj , if s hj ≤ sˆ hj , = h h h h h a j · sˆ j + a j · (s j − sˆ j ), if s hj > sˆ hj ,

(9.50)

where a hj and a hj are two parameters representing the lower and the higher generation cost per unit. There is no central price coordinator in this setting, and each generator determines their own price policy. We utilize and study two classes of pricing mechanisms. The first one is two-tariff pricing inspired by [11], and the second one uses linear price functions [12]. The goal of each generator is to cover their costs without excessively increasing prices. In other words, generator j solves the following optimization problem:  H  m    h   h h Δj =  ci j (E) − α j s j  ,  

minimize pj

h=1

i=1

where p j is a set of price functions determined by this generator, and cihj (E) = p hj (E) · eihj is a payment to the generator from consumer i at time interval h. In the case of two-tariff structure, a price function for an arbitrary generator j at time interval h has the following form:  p hj

=

p hj , if s hj,th , p hj , if s hj > s hj,th ,

(9.51)

where p hj and p hj are lower and higher tariffs, and s hj,th is a load threshold that switches

the tariffs. More precisely, in the case of s hj > s hj,th the payment cihj (E) has a nonlinear form:   eh ij (9.52) cihj (E) = p hj · s hj,th + p hj · (s hj − s hj,th ) · h . sj The disadvantage of this scheme is the nonlinear structure of consumers prices. As an alternative we may suggest a quota-based pricing mechanism that is clearer for consumer agents. Each consumer i has an equal right for buying a low-price energy

9.4 Pricing Mechanisms in Multi-generator and Multi-consumer Power Grid

share q hj =

s hj,th m

225

from a generator j, and the cost cihj (E) takes the following form:  cihj (E) =

p hj · eihj ,

eihj ≤ q hj ,

p hj · q hj + p hj · (eihj − q hj ), eihj > q hj .

(9.53)

Let us now investigate linear price functions applied to the model under consideration. Assume that generator j’s cost function α hj (·) now has a quadratic form rather than piecewise linear as in (9.50): α hj (s hj ) = β hj · (s hj )2 + γ jh · s hj ,

(9.54)

where β hj and γ jh are time and generator dependent parameters. This cost is shared among the consumers proportionally to their orders: cihj (E) = (β hj · (s hj )2 + γ jh · s hj ) · =

(β hj

·

s hj

+

γ jh )

·

eihj s hj

=

(9.55)

eihj .

The energy unit price depends on the strategies of other consumers. Unlike the constant tariff mechanism that can present the generators with strictly limited generation, linear tariff pricing is more suitable for the generators whose marginal costs steadily increase along with the volumes of energy to be generated. As it was already mentioned, condition (9.47) cannot be easily checked on the consumer side. Therefore, we propose to consider it as an objective rather than a strategy set constraint. Generators need to modify  their price function parametersdepend h h h ˆ ing on the f kl . These are p j , p j , s j,th in the first pricing scheme and β hj , γ jh in the second one. Generally, it is not possible to determine which contract causes the flow to increase in a particular arc. However, we may estimate which generators use the link the most, and they should increase the price coefficients or decrease the low-price threshold accordingly. For example, if an arc a ∈ A is used by a subset of generators VˆG , thresholds may be proportionally lowered in the constant tariff setting as follows: s h∗ j,th = 

s hj,th j∈VˆG

s hj,th

· fˆkl , ∀ j ∈ VˆG .

(9.56)

  For the similar situation in the linear tariff setting, price coefficients β hj , γ jh should be increased for all of the generators in the subset. Therefore, load dependent pricing allows generators to balance their costs. It causes the change in the strategies of consumers, who are encouraged to shift fractions of their demand to alternative generators or time intervals.

226

9 Load Flow Estimation in a Transmission Network

Fig. 9.6 6-node network

Table 9.3 Power demand i d i1 1 2

100 100

d i2

d i3

d i4

Di

300 200

100 200

300 200

900 1200

Example 9.4.1 ([13]). The goal of this simulation is to evaluate the strategies of consumers under the discussed pricing schemes and compare it with a constant pricing scenario. We also implement simple heuristics for flow capacity condition fulfillment and estimate the respective increase in consumers costs. Let us consider a sample 6-node local network with 2 consumers and 3 generators as shown in Fig. 9.6. Two consumers (nodes 1 and 2) are located in the middle, surrounded by three generators (nodes 3, 4 and 5). Node 6 is an intermediate node that neither generates nor consumes energy. We also set H = 4 for the sake of simplicity. Time slots h = 1, 2, 3, and 4 correspond to night, morning, day, and evening quarters of the 24-hour interval. Consumer power demand conditions (in Watts) are given in the Table 9.3. It is clear that consumer 1 is less active during the day time but more active in the morning and in the evening. On the other hand, consumer 2 has a demand spread evenly throughout the day. Both consumers require less energy during the night time. We assume that each arc has the same flow capacity fˆ, and consider two cases: fˆ = 500 and fˆ = 700. Due to the tree structure of the network we have: h h h = e13 + e23 , f 31 h h h h f 21 = e14 + e15 − e23 , h h h h h f 62 = e14 + e15 + e24 + e25 , h h h f 46 = e14 + e24 , h h h f 56 = e15 + e25

at any time interval h. Moreover, we assume that generator 3 is conventional, and its generation cost does not depend on the time interval, whereas generators 4 and 5 produce wind and solar energy respectively, and so both of them have low marginal costs and limited generation capacities. However, generator 5 cannot produce energy at night, and its

9.4 Pricing Mechanisms in Multi-generator and Multi-consumer Power Grid Table 9.4 Generation (piecewise linear)  cost parameters    1 1 2 2 1 2 a j , a j , sˆ j a j , a j , sˆ j j 3 4 5

(2, 5, 900) (1, 100, 300) (1, 100, 0)

(2, 5, 900) (1, 100, 300) (1, 100, 200)

Table 9.5 Generation (quadratic)  cost parameters    1 1 βj , γj β 2j , γ j2 j 3 4 5

(0.0002, 2) (0.002, 0.2) (100, 100)

(0.0002, 2) (0.002, 0.2) (0.003, 0.3)

  a 3j , a 3j , sˆ 3j

  a 4j , a 4j , sˆ 4j

(2, 5, 900) (1, 100, 300) (1, 100, 400)

(2, 5, 900) (1, 100, 300) (1, 100, 200)





β 3j , γ j3

900 950 900 1076 1050

1 1.056 1 1.196 1.166



(0.0002, 2) (0.002, 0.2) (0.002, 0.2)

Table 9.6 Simulation results for different pricing schemes Scenario C1 C1 /C1b C2 Basic scheme Piecewise linear Quota linear Quadratic fˆ = 500

227

1200 1250 1400 1521 1350

β 4j , γ j4



(0.0002, 2) (0.002, 0.2) (0.003, 0.3)

C2 /C2b 1 1.042 1.167 1.267 1.125

capacity is lower in the morning and evening intervals. The corresponding parameters of functions {α hj } of piecewise linear form are listed in the Table 9.4. Similarly, the coefficients of {α hj } in the case of quadratic cost function are given in the Table 9.5. The following Table 9.6 shows the values of Δ j for different scenarios. Values {Ci } are total costs of consumers, whereas {Ci b} are their costs in a basic single tariff scenario. All computations are conducted using the open-source numerical computing environment Octave. We can conclude that the pricing mechanisms generally increase the costs of consumers. In the considered scenarios, the increase differs between 0 and 26%. One may note that stricter flow capacities also increase the consumers’ costs in specific scenarios. It is natural since arc flow capacities may affect the pricing schemes and decrease the effective low-price thresholds of particular generators, which in turn motivates consumers to buy energy from other generators, possibly for a higher price.

228

9 Load Flow Estimation in a Transmission Network

References 1. Duffin R (1947) Nonlinear networks. IIa. Bull Am Math Soc 53, 963–971 2. Patriksson M (2015) The traffic assignment problem: models and methods. Dover Publications Inc, Mineola 3. Popov I, Krylatov A, Zakharov V, Ivanov D (2017) Competitive energy consumption under transmission constraints in a multi-supplier power grid system. Int J Syst Sci 48(5):994–1001 4. Schweppe FC, Caramanis MC, Tabors RD, Bohn RE (1988) Spot pricing of electricity. Kluwer Academic Publishers, Norwell 5. Hogan WW (1992) Contract networks for electric power transmission. J Regul Econ 4:211–242 6. O’Neill RP, Helman U, Hobbs BF, Stewart WR, Rothkopf MH (2002) A joint energy and transmission rights auction: proposal and properties. IEEE Trans Power Syst 17(4):1058–1067 7. Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont, Massachusetts 8. Rosen JB (1965) Existence and uniqueness of equilibrium points for concave n-person games. Econometrica: J Econ Soc 520–534 9. Roth AE (1988) The Shapley value: essays in honor of Lloyd S. Cambridge University Press, Shapley 10. Popov I, Krylatov A, Zakharov V (2016) Integrated smart energy system based on productionoriented consumption. IFIP Adv Inf Commun Technol 480:265–273 11. Veit A, Xu Y, Zheng R, Chakraborty N, Sycara KP (2013) Multiagent coordination for energy consumption scheduling in consumer cooperatives. In: 27th AAAI conference on artificial intelligence, pp 1362–1368 12. Mohsenian-Rad A-H, Wong VW, Jatskevich J, Schober R, Leon-Garcia A (2010) Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid. IEEE Trans Smart Grid 1(3):320–331 13. Popov IV, Krylatov AYu, Lukina AA (2017) Pricing mechanisms for day-ahead demand management in multi-generator power grid. In: 2016 International conference on recent advances and innovations in engineering (ICRAIE)