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Graduate Studies The Vault: Electronic Theses and Dissertations

2016

Long Term Planning and Modeling of Ring-Radial

Urban Rail Transit Networks

Saidi, Saeid

Saidi, S. (2016). Long Term Planning and Modeling of Ring-Radial Urban Rail Transit Networks

(Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26787

http://hdl.handle.net/11023/3036

doctoral thesis

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UNIVERSITY OF CALGARY

Long Term Planning and Modeling of Ring-Radial Urban Rail Transit Networks

by

Saeid Saidi

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN CIVIL ENGINEERING

CALGARY, ALBERTA

MAY, 2016

© Saeid Saidi 2016

ii

Abstract

Extensive work exists on regular rail network planning; however, few studies exist on the

planning and design of ring-radial rail transit systems. With more ring transit lines being

planned and built in Asia, Europe and the America’s, a detailed study on ring transit lines

is timely. This thesis is based on idealizing transit network in perfect ring-radial transit

lines. An analytical model using the continuum approximation approach is first introduced

to find the optimal number of radial lines considering a city with a radio-centric street

network. An approximate analytical model for ring-radial rail network planning is then

introduced allowing analysis of the feasibility and optimal alignment of a ring transit line

in a city. The city of Calgary‘s light rail transit network and Shanghai metro network are

used to illustrate the applicability and transferability of the model. The model is then

extended to allow simultaneous consideration of radial and ring lines and analyzing a

transit network with partial ring and radial lines. This extension allows a more realistic

idealization and analysis of rail transit networks. A benchmark analysis of cities with ring

transit lines is used to identify prominent types of lines in idealized ring-radial transit

networks. The cities are then assessed based on their unique network patterns using

identical model inputs such as length of rail transit network and trip distribution patterns.

This thesis provides a decision support tool for transit planners to compare the performance

of different rail transit network extension alternatives for long-term rail transit planning. It

can also be used for cost- benefit analysis to compare total generalized passenger cost

savings versus the cost of network extension. Unlike simulations and agent-based models,

this model is shown to be easily transferable to many ring-radial transit networks.

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Therefore, with a daily OD trip matrix and transit network supply characteristics and

parameters as input, the model can be implemented for many radio-centric cities. The

benchmark analysis using the combined universal ring-radial rail transit network model is

a mathematically sound platform to compare different rail transit networks and propose the

best examples of rail network topologies.

iv

Acknowledgements

I would like to express my appreciation to my advisor, Dr. Chan Wirasinghe, for his

continuously insightful leadership and dedication in supervising and mentoring his

students. His insightful comments and guidance in my research, and his well thought out

way of training me on how to perform my research were the key for completion of this

Ph.D. thesis. One of the lessons I learned from him was to “always remember what is the

question” before jumping into finding solutions or solving problems. I am proud to work

under his supervision.

I would also like to express my deepest gratitude to my co-supervisor, Dr. Lina

Kattan for her constant support, patience, dedication, and encouragement throughout my

entire graduate studies at the University of Calgary for both Masters and Ph.D. degrees. I

am very fortunate and honored to have Dr. Kattan as my advisor.

I am very grateful to the members of my Ph.D. defense committee, Drs. Nigel

Waters, Paul Schonfeld, Alex de Barros and Janaka Ruwanpura for their valuable advice

and helpful suggestions. Many thanks go to Drs. John D. Hunt, and Andrew Hunter for

serving in my candidacy exam committee. I would also like to thank my advisors at Tongji

University Drs. Yuchuan Du and Yuxiong Ji for their support during my three months

study abroad in China.

I had the opportunity to work and collaborate with many of my friends and

colleagues at University of Calgary and Tongji University. I would like to thank Shahab

Nejad for his support and help with my Ph.D. dissertation. I would also like to thank Dr.

Patrick Miller, Matuir Rahman, Mohammad Ansari, Willem Klumpenhouwer, Mostafa

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Salari, Cheng Cheng, Shengchuan Jiang, and Jinping Guan for our research collaborations

and numerous interesting discussions. I would also like to thank Spatial and Numeric Data

Services at the University of Calgary and especially Mr. Peter Peller for his kind and patient

guidance with my never ending questions with ArcGIS software.

My research would not have been practical without valuable data provided by the

City of Calgary and Calgary Transit to test and validate my model and use the City of

Calgary as the case study of this dissertation. I would like to acknowledge Mr. Chris Jordan

for his support and very practical comments on this research. I would also like to thank Mr.

Doug Morgan, Director of Calgary Transit, for providing support and funding for graduate

students through the PUTRUM Program to conduct cutting-edge research on Public

Transport and making Calgary a more sustainable and transit friendly city.

I would also like to acknowledge the support from the management and colleagues

at HDR Consulting, especially Mr. John Hubble for his encouragement and providing the

flexibility on my work schedule to continue my research while working at HDR.

Thanks to all current and former staff of the Department of Civil Engineering

especially Kate McGillis, Julie Nagy Kovacs, and Catherine Barrett who were always

ready to help with a smile on their faces.

I am very grateful for the support I got from the Natural Science and Engineering

Research Council of Canada (NSERC) through the Alexander Graham Bell Canada

Scholarship (NSERC CGS) and Michael Smith Foreign Study Supplement. It was an honor

to also receive support from the prestigious Izaak Walton Killam Memorial Scholarship

from the Killam family. I would also like to thank the family of Robert B. Paugh for the

University of Calgary Ruby Doctoral Scholarship. Many thanks to Alberta Motor

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Association for providing an excellent research location, the Active Traffic and Demand

Management Laboratory. Without all this generous support, this work would not have been

possible.

I owe who I am and anything I have achieved to my parents, Dr. Mohammad Reza

Saidi and Sedigheh Moini. They are my mentors, my role models, and my inspiration; and

I will be in debt to them forever. I would also like to thank my brother and sister in law,

Majid and Sara. I will never forget their support, kindness, and hospitality from the time I

came to Calgary for my graduate studies. I will not forget all the help and advice I received

from my two brothers Hamid and Majid since childhood until this very moment!

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Dedications

Dedicated to my parents,

Mohammad Reza Saidi and Sedigheh Moini

viii

Table of Contents

Abstract ............................................................................................................................... ii Acknowledgements ............................................................................................................ iv Dedications ....................................................................................................................... vii Table of Contents ............................................................................................................. viii

List of Tables .......................................................................................................................x List of Figures and Illustrations ......................................................................................... xi

CHAPTER 1: INTRODUCTION ....................................................................................1 1.1 Background ..............................................................................................................1

1.2 Objectives of Study ..................................................................................................6 1.3 Proposed Methodology and Research Contributions ...............................................7

1.4 Organization of Thesis .............................................................................................9

CHAPTER 2: RAIL TRANSIT – AN EXPLORATION ...............................................12

2.1 Rail Transit Network Planning Literature..............................................................12 2.2 Transit Network Assessment .................................................................................16 2.3 Rail Transit Networks Review ...............................................................................19

2.4 Ring Transit Line Review ......................................................................................27 2.5 Regression Analysis for Ring Transit Lines ..........................................................31

2.6 Discussion ..............................................................................................................38

CHAPTER 3: RADIAL LINE RAIL TRANSIT NETWORK MODEL .......................41

3.1 Optimal number of radial lines in a city ................................................................41 3.2 Application .............................................................................................................49

3.3 Discussion ..............................................................................................................52

CHAPTER 4: RING-RADIAL RAIL TRANSIT NETWORK MODEL ......................53 4.1 Passenger route choice ...........................................................................................53

4.1.1 All travels to the CBD......................................................................................54 4.1.2 Route choice for any zone pair for a network with a single ring line ..............64

4.2 Single Ring Line ....................................................................................................71 4.2.1 Feasibility and Optimal Radius ........................................................................71

4.2.2 Results ..............................................................................................................73 4.2.3 Sensitivity Analysis .........................................................................................80

4.3 Double Ring Line ...................................................................................................83 4.3.1 Feasibility and Optimal Radius ........................................................................83 4.3.2 Results ..............................................................................................................86 4.3.3 Sensitivity Analysis .........................................................................................94

4.4 Discussion ..............................................................................................................97

CHAPTER 5: COMBINED UNIVERSAL RING-RADIAL RAIL TRANSIT

MODEL ...............................................................................................................101 5.1 Combined Universal Ring Radial Transit Modelling and Analysis Framework .101

5.2 Model Output .......................................................................................................112

ix

5.3 Applications for Ring Radial Rail Transit Network Model .................................115

5.3.1 Future Rail Transit Network Scenario Comparison for the City of Calgary .116 5.3.2 Ring Radial Rail Transit Network Improvement Cost-Benefit Analysis ......121

5.4 Discussion ............................................................................................................123

CHAPTER 6: RING-RADIAL RAIL TRANSIT NETWORK BENCHMARK

ANALYSIS .............................................................................................................126

CHAPTER 7: SUMMARY AND CONCLUSIONS ...................................................145 7.1 Research Summary and Contributions.................................................................145

7.1.1 Macro Scale Parameter Assessment of Rail Transit Networks .....................145 7.1.2 Radial Only Transit Network .........................................................................146

7.1.3 Optimization of Ring Line in a Full Ring-Radial Line Transit Network ......146 7.1.4 Combined Universal Ring-radial Rail Transit Network Model .....................147

7.1.5 Model Transit Network Benchmark Analysis Using Combined Universal

Ring-radial Rail Transit Network Model .......................................................148

7.2 Research Contributions ........................................................................................149 7.3 Future Work .........................................................................................................151

REFERENCES ................................................................................................................154

x

List of Tables

Table 2.1 –Logistic regression analysis for the existence of a ring line ........................... 37

Table 2.2 – Model accuracy and classification table for logistic regression .................... 37

Table 3.1 –Unit cost values for the City of Calgary ......................................................... 50

Table 3.2 – Sensitivity analysis for the optimal number of radial lines for the City of

Calgary ...................................................................................................................... 51

Table 4.1 – Unit cost values for the City of Calgary ........................................................ 74

Table 4.2 – Unit cost values for the City of Calgary ........................................................ 89

Table 5.1 –Comparison of model output for Badia et al. (2014) and the Ring-radial

transit network model ............................................................................................. 114

Table 5.2 –Sensitivity analysis of total passenger cost with respect to transfer penalty 122

Table 6.1 – Summary of the 6 rail transit idealized network benchmark analysis ......... 129

Table 6.2 – Comparison of network length for actual and idealized networks .............. 130

Table 6.3 – Normalized network parameters for the idealized ring radial networks ...... 131

Table 6.4 – Average and standard deviation of the network parameters for the

benchmarked networks ........................................................................................... 132

Table 6.5 – Total transit passenger cost output for uniform and exponential demand

distribution .............................................................................................................. 135

Table 6.6 – Total transit passenger cost output for uniformly random distributed OD

pattern ..................................................................................................................... 136

Table 6.7 – State, form and Structure network parameters for the benchmarked

networks (Derrible and Kennedy 2009) .................................................................. 142

xi

List of Figures and Illustrations

Figure 2.1 – Three Simple Networks: Star, Triangle and Cartwheel Configuration

(Laporte et al. 2011) .................................................................................................. 17

Figure 2.2 – Relationship of Population Density and Ridership per Unit Length of

Network ..................................................................................................................... 20

Figure 2.3 – Population versus Annual Ridership for Europe, Asia and North America . 22

Figure 2.4 – Age of Transit System versus Annual Ridership for Asia, Europe, and

North America .......................................................................................................... 23

Figure 2.5 – Metro Network Topology of Cities with Ring Line: Seoul, Moscow, and

Beijing ....................................................................................................................... 30

Figure 2.6 –Length of Ring Line (km) versus Population Density per km2 ..................... 32

Figure 2.7 –Annual Ridership of the Ring Line versus Length of the Line ..................... 33

Figure 2.8 – Logistic regression analysis for cities with rail transit network – cities

based on the highest probability of having a ring line .............................................. 37

Figure 3.1 – Schematic view of a transit trip and its cost factors between from an

origin to a destination ................................................................................................ 42

Figure 3.2 – City of Calgary land use zones and centroid location .................................. 50

Figure 4.1 – Schematic view of passengers’ route choices for a. r > R and b. for r ≤ R .. 55

Figure 4.2 – Associated lowest passenger cost areas for different transit route choices

for a city with radial line spacing of 2π/3. ................................................................ 62

Figure 4.3 – Associated lowest passenger cost areas for different transit route choices

for a city with radial line spacing of (a) π/2 and (b) π/4. .......................................... 63

Figure 4.4 –Transit route choice for a network with destination shown in black: A –

area with radial line route choice; and B – area with ring line route choice; C –

area with direct access to destination. ....................................................................... 70

Figure 4.5 – Total cost of the ring line for a network with uniform demand and

exponential demand distribution. .............................................................................. 75

Figure 4.6 –City of Calgary long-term light rail transit network and the recommended

ring line. .................................................................................................................... 77

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Figure 4.7 – Total cost of ring line for different scenarios based on distance away

from the city center. .................................................................................................. 78

Figure 4.8 –Sensitivity analysis of ring line cost with respect to different access costs. . 81

Figure 4.9 –Sensitivity analysis of ring line cost with respect to different access costs. . 82

Figure 4.10 – Total cost (Capital and operating cost minus total passenger benefit) of

the ring line for different scenarios based on the distance from the city centre. ...... 91

Figure 4.11 –Recommended range of a second ring line in Shanghai.............................. 92

Figure 4.12 – Recommended second ring line and the population concentration of cell

phone users on a typical workday in Shanghai. Dark blue shading represents the

highest concentration and light green shading denotes the lowest concentration. ... 93

Figure 4.13 –Stations with the highest total passenger cost saving after introducing

the second ring line at a radius of 11 km from the CBD. ......................................... 94

Figure 4.14 –Sensitivity analysis of the changes in the total cost of a second ring line

for changes in Value of Time .................................................................................... 96

Figure 4.15 –Sensitivity analysis of the changes in the total cost of a second ring line

for changes in Demand ............................................................................................. 96

Figure 5.1 –Ring-radial rail transit network model algorithm ........................................ 102

Figure 5.2 – Schematic view of ring and radial lines and continuity feature for

crossing nodes. ........................................................................................................ 105

Figure 5.3 – Different possibilities for zone centroid projections for zone centroid (a)

between inner ring line and CBD, (b) between two ring lines, (c) outside of outer

ring line ................................................................................................................... 108

Figure 5.4 – Total passenger cost to/from each area of the network for a Radial

network scenario ..................................................................................................... 113

Figure 5.5 – Total passenger cost to/from each area of the network for a combined

ring and radial network scenario ............................................................................. 113

Figure 5.6 – Optimal network for different transit mode technology in Badia et al.

(2014) reproduced in ring-radial model .................................................................. 115

Figure 5.7 – Different transit network extension alternatives for City of Calgary from

base ......................................................................................................................... 117

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Figure 5.8 – City of Calgary from base Calgary Future Scenarios with projected long

term OD .................................................................................................................. 118

Figure 5.9 – Comparison of four future alternatives for City of Calgary Light Rail

transit network with full ring line ........................................................................... 119

Figure 5.10 – Comparison of Calgary’s 6 alternatives assuming randomly distributed

OD ........................................................................................................................... 120

Figure 5.11 – Comparison of Calgary’s 6 alternatives assuming exponential

distributed OD ......................................................................................................... 120

Figure 5.12 –Transit network improvement for Moscow – adding a second ring line ... 121

Figure 5.13 – Moscow’s network improvement cost benefit analysis............................ 123

Figure 6.1 – Comparison of network length for actual and idealized networks ............. 130

Figure 6.2 –Normalized benchmarked networks used in the ring-radial rail transit

network model ........................................................................................................ 133

Figure 6.3 –Total Transit Passenger cost output for uniform demand distribution ........ 135

Figure 6.4 –Total transit Passenger cost output for exponential demand distribution

exponential demand distribution ............................................................................. 136

Figure 6.5 – Total transit passenger cost output for randomly distributed OD pattern –

Black line showing the average, other colors showing each of the 10 runs ........... 137

Figure 6.6 – Sensitivity Analysis for different transfer disutility values for uniform

demand distribution ................................................................................................ 138

Figure 6.7 –Sensitivity analysis for different transfer disutility values for random

demand distribution ................................................................................................ 138

Figure 6.8 –Sensitivity analysis for different transfer disutility values for exponential

demand distribution ................................................................................................ 139

Figure 6.9 –Example of two runs with randomly distributed travel demand. Moscow

lowest total passenger cost for Run 1 and Tokyo lowest for Run 2 ....................... 140

1

CHAPTER 1: INTRODUCTION

1.1 Background

Following decades of focus on automobiles and planning for the use of personal

vehicles as the main means of transportation, public transit has experienced growing

attention. Increased travel and traffic congestion; increased cost of car ownership, operation,

and parking; and environmental concerns and sustainability issues have shifted the focus of

city planners and politicians more towards public transportation. In addition, this trend is

also occurring in society in general. Similar to when mass transit systems were first

introduced approximately 150 years ago, public transit is now the primary method of

transportation within many cities, especially in large cities in Asia and Europe; the trend is

also true in some North American cities. Thomson (1977) categorizes city topologies into

five main archetypes: full motorization, weak centre, strong centre, low cost, and traffic

limitation strategy. Each archetype has certain features including travel demand distribution;

employment settlements and densities; transit network topology; employment settlements;

and their efficiency.

Vuchic (2005) classified transit lines and discussed the advantages and disadvantages of

radial, diametrical and ring (or circumferential) lines. Radial lines, which traditionally

address the heaviest travel demand from suburbs to downtown, have termini in the city centre

and in a suburb. Diametrical lines connect two suburbs while passing through the city center.

These lines are supposed to be designed in a way that the two sections on different sides of

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city center are balanced in terms of passenger demand. They can have several geometrical

forms including a straight or diagonal line, L-shaped lines (e.g., New York) and U-shaped

lines (e.g., Washington, Toronto).

Ring lines are laid out around the central city, intersecting radial lines, creating transfers and,

thus, making an integrated network. They can be a whole circle with no terminus or only part

of a circle. Circumferential lines are less popular in North American cities than in European

and Asian cities, mainly because typical grid street patterns do not provide easy right-of-way

for surface circumferential alignments. The other reason may be due to the land-use

characteristics of North American cities, many of which are highly concentrated in a central

business district (CBD) and surrounded by more distant residential neighborhoods; whereas

European and Asian cities are mostly characterized by larger and mixed land use in their city

centres surrounded by several cross-town activity centres. (Vuchic 2005)

In a totally radial transit network, public transit trips with destinations away from the centre

have to be made along the radial lines passing through the city centre. For cross-town trips,

using radial lines can cause unnecessary passenger loads on some transit corridors, high

transfer loads in the city centre and additional passenger travel distance, travel time and

transfers. For instance, passengers have to go to the centre city, possibly make a transfer and

take another rapid transit line to go to the other side of the city. A ring line provides direct

trips and better connectivity and decreases network’s vulnerability, since there are alternative

routes to each destination. Laporte et al. (1994), Derrible and Kennedy (2009), and Yi and

3

Chao (2010) showed that a ring or circumferential transit line can greatly improve network

connectivity, directness, and operational efficiency of the network.

In addition to the possible savings in transit riders’ travel and waiting times, increased transit

network connectivity, reliability and the reduction of the transit load in the downtown core,

the presence of ring lines plays an important role in increasing accessibility to suburban areas

and, thus, the development of new satellite centres. While an auto ring road usually

encourages development to take place on the outskirts of a city beyond the ring (Miller,

2012), a transit ring line can encourage higher density, mixed-use, transit oriented

development along an inner city ring corridor. A ring transit line is also important for public

transit users, in terms of trip directness, mobility, and accessibility.

Many major cities such as London, Moscow, Tokyo, Berlin, Paris, Shanghai, and Seoul have

a rail ring transit lines. Some cities with ring line networks have built or are planning to build

additional ring lines. Beijing currently has two full ring lines surrounding the CBD, making

the connections between lines easier and more convenient, while also providing cross-town

transit corridors within the network. The second (Line 10) was completed in 2013 (China

Daily 2013). In addition, Paris is currently building a second ring line on the outskirts of the

city with an expected completion in 2025 (Freemark 2011). Moscow expects its second ring

line to be completed by 2018 (Panin, 2014). Shanghai is also considering the construction

of a second ring transit line as a long-term plan (Bai and Yan 2013). Chicago (Warade, 2007),

4

Washington, DC (Reed, 2013), Boston (Boston Region Metropolitan Planning Organization,

2011), and Bangkok (Urbanalyse, 2012) are also considering constructing a ring line.

Vuchic (2014) lists three important functions of ring lines: 1) improving connectivity among

radial lines and distributing the congestion away from radial lines in the Central Business

District (CBD); 2) making trips between radial lines shorter through bypassing the CBD; and

3) serving the busy areas in a ring around the CBD. A ring line has other characteristics in

addition to Vuchic’s list:

a) Ring lines allow for a high number of transfers relative to radial lines, as observed in

the inner Beijing ring line and the Shanghai, London, and Moscow ring lines. While

radial lines primarily connect the CBD and major attraction or employment centres

along the transit corridor to the residential areas, a ring line has several intersecting

radial lines, providing more transfer opportunities. Therefore, a ring line has a higher

ratio of transfers and a higher number of boarding and disembarking passengers with

respect to through passengers.

b) For most rail transit networks, topology and density of the network is different inside

and outside the full ring line. Typically, the network is relatively dense within the

inner ring unlike outside the ring where the density of rail lines is much lower.

c) For many networks, some radial lines exist only outside of the ring line between the

ring and periphery of the city and do not continue inside the ring toward CBD. This

is the case of networks such as London and Tokyo have suburban railway lines that

were forced to terminate service at the ring line.

5

d) For networks with relatively high radius of the ring line, it provides coverage to areas

not well serviced by a purely radial transit system.

e) For cities with more than one ring line (an inner and outer ring with the same centre),

the defining characteristic of each line can be based on different factors. For example,

the Beijing inner ring (Line 2) surrounds the high-density CBD and is characterized

by a high transfer load, whereas the outer ring (Line 10) provides coverage to

passengers who are distant from the radial lines. A similar observation will exist for

the current (inner) and future planned (outer) ring line in Moscow. Where the existing

inner ring has a low radius close to the CBD and a dense rail network inside. Whereas

the planned outer ring line is located far from the CBD and will serve areas closer to

the periphery of Moscow city boundary.

With the high level of attention being paid to ring transit lines, the main questions that arise

are the following: when and where should a ring line (if any) be developed in a metro

network? If it is feasible to build a ring line, what is the best alignment? How can planners

consider different alternatives for extending rail transit networks? Under what circumstances

would adding a ring line (or a partial ring line) be more beneficial compared with adding a

radial line and vice versa. If a ring line is to be built what are the contributing factors (e.g.

OD demand patterns, waiting costs) that affect its alignment. These questions are

fundamental motivations for this thesis which presents step by step, a process for

understanding of urban rail transit networks with emphasize on the ones with a ring line;

optimization models for radial line only, and with complete ring lines; and eventually a

6

general model as a tool for decision makers to evaluate different radio-centric future

alternatives with and without ring or radial lines, complete or partial.

1.2 Objectives of Study

This thesis has two main objectives relative to transit network planning research. The

first objective is to develop an analytical model to optimize the number of radial lines and

location of ring line(s) if a ring line is found to be beneficial if added to a network. The

second objective is to combine the models developed for the ring and radial networks in the

first objective and develop a decision support tool for transit planners and authorities to

measure total passenger cost for a given rail transit network. Using this tool, planners will be

able to compare different transit network improvement alternatives and measure total

passenger cost savings for each of the alternatives. Comparing benefits arising with

passenger cost savings with cost of each alternative, a decision maker can make a more

informed decision to choose the alternative with the highest social benefit with respect to the

cost.

Using the developed tool also provides an opportunity to compare and evaluate performance

of different rail transit networks in terms of total passenger cost and thus provides a tool to

benchmark the most efficient transit networks keeping all factors other than network

topology the same.

7

1.3 Proposed Methodology and Research Contributions

Although many cities have radio-centric networks where the CBD is in the centre and

the city developed around that central area forming a (imperfect) circle type urban pattern,

very few transit network planning studies with some exceptions (Saidi et. al 2016, Chen et.

al 2015, Badia et. al 2014, Vaughan 1984, and Wirasinghe and Ho 1982) use the assumption

of ring-radial models. Using the features of polar coordinates in the calculation and

mathematical modeling and ease of coding of transit networks idealized with ring and radial

are among advantages of assuming fully radio-centric systems for transit network planning.

In addition, although extensive work exists on regular rail network planning, few studies

exist on the planning and design of ring-radial rail transit systems. With more ring transit

lines being planned and built in Asia, Europe and the America’s, a detailed study on ring

transit lines is timely.

This research takes important steps in developing a new long range transit planning

transferable tool that can be applied to many radio-centric transit networks. It develops a

transit network model idealized into perfect ring and radial lines. This thesis contributes to

an efficient public transportation system which is more sustainable and environmentally

friendly. The outcome of this work will be useful for urban and regional transportation

planners.

In order to determine the feasibility of a ring line for a city, a detailed economic study is

required based on population and job distribution in the city, transit mode share, value of ride

8

time, wait time for the passengers, capital, and operating costs of the rail transit line. Such

variables, especially future trip patterns in a city and the current network topography of the

radial rail transit system, could greatly impact the feasibility and optimal layout of a ring

line.

This thesis makes the following contributions to ring-radial transit network modelling

literature:

1. It introduces first a radial line only model; and then analyses the optimal ring

line to be embedded in an existing radial network. It also considers an urban

transit network with radial lines and a ring rail line for a long-range planning

horizon. Since many cities have a Central Business District (CBD) at the

centre and radial lines that connect the suburban areas to the centre, a

mathematical model for the optimal number of radial lines is developed.

2. It develops passenger route choice for different rail networks for a many-to-

many origin-destination (OD) demand distribution, based on a total travel

time cost per passenger basis. The routes considered are: (1) radial lines only;

(2) ring line only or radial lines and ring line combined; or (3) direct access

to a destination without using the rail system. A cost-benefit optimization

model to identify the feasibility of a ring line is proposed.

3. A universal ring-radial model is developed to address the concern of

combined planning of partial and full ring and radial lines in the broader

context. This extension increases the applicability of the model to cities that

do not necessarily need a complete ring or radial line. A benchmark analysis

9

of cities with ring transit lines is used to identify prominent types of lines in

transit networks of cities categorized as: full radial lines (connecting CBD to

periphery); full ring line (a complete circle at radius R from CBD); partial

radial lines (either starting at CBD and ending at a ring line; constrained

between two ring lines; or connecting an outer ring line to the periphery); and

partial ring line (constrained between two radial lines; or only connected to

one radial line). Each city has a unique network topology in terms of length

of the network for each line category defined above. The cities are then

assessed based on their unique network patterns using identical model inputs

such as length of rail transit network and trip patterns.

1.4 Organization of Thesis

This thesis consists of seven chapters as follows:

Chapter Two describes the literature review on rail transit planning and covers transit

network assessment of around 100 rail transit networks using regression analyses. The

relationship between rail transit network parameters, such as length rail lines, the age of the

system, and network topology, with city parameters, such as population, area, and population

density are investigated. Cities that have ring rapid transit services are assessed. Parameters

that justify the implementation of a rail transit network in a city, particularly a ring line,

through the investigation of cities that have implemented such services, are analyzed.

10

Chapter Three describes an analytical model to find the optimal number of radial lines in a

city for any many to many demand distribution. An Analytical model using Continuum

Approximation (CA) approach is considered to find the optimal number of radial lines

considering a city with a radio-centric street grid with the city centre at the origin.

Chapter Four introduces a passenger route choice model that is used for the analysis of total

passenger cost. It then uses the method for optimizing a ring transit line in an existing

network for a many-to-many Origin-Destination (OD) demand distribution, based on a total

passenger cost and operating and capital cost of building the ring line. City of Calgary’s

Light Rail transit network and Shanghai’s metro network are used as case studies to illustrate

the applicability of the model.

Chapter Five is an extension of the model developed in Chapter four into a combined

universal ring-radial rail transit network allowing partial ring and radial lines. The chapter

covers the developed ring-radial transit network model algorithm. It then illustrates some

general model output. The chapter also demonstrates the applicability of the model for

performance measurement of future City of Calgary Light Rail Transit network alternatives

and cost-benefit analysis for network extension of Moscow’s metro network.

Chapter Six demonstrates a benchmark analysis of 6 mature cities that have a transit network

older than 50 years of age and contain a ring transit line. Passenger-based transit network

performance of London, Tokyo, Paris, Moscow, Berlin, and Madrid are compared using the

11

universal combined ring-radial model. The networks are normalized in terms of total network

length so that they all have the same total transit network supply with the only different factor

being network topology.

Chapter Seven provides a summary of the research findings and concluding thoughts on this

research. The contributions of this research are described along with limitations, potential

applications, and future research.

12

CHAPTER 2: RAIL TRANSIT – AN EXPLORATION1

2.1 Rail Transit Network Planning Literature

Approximately 90 years of transit network research started with attempts at a solution

to a line planning problem by Patz (1925). Transit network design has been used for various

modes, such as bus (Newell, 1979; Wirasinghe and Ho, 1982; Wirasinghe and Vandebona,

2011; Cipriani et al., 2012; Szeto et al., 2012), light rail (Liu et al., 1996; Shin et al., 2004;

Samanta et al., 2011), heavy rail (Wirasinghe and Vandebona, 1999; Laporte, 2011), and

mode combinations (Wirasinghe, 1980; Chien and Schonfeld, 1998; Uchida et al., 2005;

Wan and Lo, 2009; Mohaymany and Gholami, 2010) to produce an optimal transit network

configuration.

Depending on the ownership of the urban transit system (i.e., private or public), the

objective function might be formulated differently. Some consider user travel time

minimization as the objective and the transit agency’s cost and revenue as the constraints.

For a private transit agency, profit maximization is the main objective and user travel time

might be considered as the constraint. For social benefit maximization, a bi-objective

optimization framework can be adopted, in which optimizing users’ and agency’s cost is

considered simultaneously (Lee and Vuchic, 2005).

1 The essential contents of this chapter have been published in:

- Saidi, S., Wirasinghe, S.C., Kattan, L., 2014. Rail Transit: Exploration with Emphasis on Networks with

Ring Lines. Transportation Research Record: Journal of the Transportation Research Board 2419. 23–32.

- Saidi, S., Wirasinghe, S.C., Kattan, L., 2016. Long Term Planning for Ring-Radial Urban Rail Transit

Networks. Journal of Transportation Research Part B: Methodological 86. 128-146.

13

Few studies exist on the specific planning and design of ring transit lines. There is a

similar issue in the design of ring roads. Zitron (1974) proposed a continuous model of the

optimal cost of roads in a circular city. An Euler-Lagrange equation was derived for the

general radial symmetric case for the position-dependent cost. The analysis determined the

minimal cost routes between two diametrically opposite points at the city limits. Tan (1966)

estimated the average trip length for different road networks (e.g., direct, ring, rectangular,

radial, and arc-radial) in a circular city. He found the average area of road space required and

average travel time for each configuration. Additionally, Li et al. (2014) presented an

analytical model to optimize circular cordon toll locations in a monocentric city. It was

shown that population distribution has a significant effect on the optimal location of cordon

radius and social efficiency of the urban system.

Continuum Approximation methods have been used by several researchers (Newell 1973,

Wirasinghe and Ghoneim 1981, Wiraisnghe and Ho 1982, Vaughan 1984, Daganzo 1984,

Chang and Schonfeld 1991, Chen et. al. 2015) using “slowly varying” continuous variables

instead of discrete variables for mathematical modeling and optimization. Parameters such

as station spacing, line spacing, headway, and demand are treated as continuous functions

which may vary over time and/or space. Cipriani et al. (2012) developed a heuristic algorithm

to solve a bus network design problem based on road network topology and current rapid rail

transit in a city. The model is adopted to avoid generating loops or too-long routes with

respect to the shortest paths between terminals. Tirachini et al. (2010) developed a radial

14

transit network system to determine the optimal number of lines and mode of transit (i.e.,

light rail, heavy rail, and bus rapid transit) for a uniform demand distribution destined to a

CBD. They considered two different uniform demand densities for unit distance inbound to

the CBD and outbound from the CBD. Badia et al. (2014) considered cities characterized by

a radial street pattern. They examined radial/circular transit network schemes in the central

area, and hub and spoke in the periphery. They considered demand distribution to be either

uniform or centripetal, with maximum demand at the centre, and the minimum demand is at

the city edge.

In order to simplify continuum mathematical models to conduct analysis of cities,

some common assumptions have been made by many authors. They include:

1) uniform demand distribution (Mun et al., 2003; Chu and Tsai, 2008; Badia et al.,

2014);

2) specific zones of inner core and periphery, with a uniform population and job pattern

in each zone (Tirachini et al., 2010; Tsekeris and Geroliminis, 2013); and

3) a population that is decreasing with increasing distance from the CBD with a

linear or a known non-linear function such as exponential (Berry et al., 1963; Casetti, 1967;

Li et al., 2012; Badia et al., 2014).

Discrete OD-based transit network design has been proposed by Baaj and Mahmassani

(1995), Lee and Vuchic (2005) and Marín and García-Ródenas (2009), who used a

mathematical programming approach by first generating a large set of feasible routes

connecting all nodes in the network and applying a multi-objective analysis to find the most

15

suitable subset. Metaheuristic techniques, such as the Genetic Algorithm or simulated

annealing, have been used to analyze transit networks by Baaj and Mahmassani (1995),

Fusco et al. (2002), and Beltran et al. (2009). A common practice in these methods is to

generate a set of feasible routes and then use a Genetic Algorithm to select the optimal or

near-optimal route networks. Excellent reviews of the transit network design problem are

given by Laporte et al. (2000), Guihaire and Hao (2008), and Kepaptsoglou and Karlaftis

(2009).

None of these approaches considers a ring transit line, and most of the developed

models are not applicable to a ring transit line. With many cities that have existing rail transit

systems provisioning expansion of their system and considering extensions to include a ring

line, a particular focus on the feasibility of a ring transit line and its optimal alignment

constitutes a gap in the literature on rail transit planning.

In this study, transit network of a city with a network of ring and radial lines are being

assumed. Previous works on radio-centric networks include Wirasinghe et al. (1977) and

Wirasinghe and Ho (1982) who analyzed radial bus systems for CBD commuters. Tirachini

et al. (2010) and Badia et al. (2014) used similar polar coordinate systems for optimizing

radial transit corridors. Vaughan (1984), and Chen et al. (2015) have considered a ring-radial

transit system by finding the optimal spacing of radial and ring lines respectively. Both

Vaughan (1984) and Chen et al. (2015) have made important contributions by finding the

optimal spacing of ring lines. However, attempts to find the optimal alignment of a single

rail ring line, which is the most common case, is still not well researched. As discussed in

16

the introduction, cities usually have one or at most two full ring rail lines in their network.

The optimal alignment of a single ring rail line is not their focus; they are more suitable for

bus networks with multiple ring corridors. In addition, both assumed a route choice model,

which is based only on distance, that does not include total passenger cost.

2.2 Transit Network Assessment

Various studies have defined different performance measures or efficiency factors

for the assessment of rail transit networks, which can be used for better designs or

comparisons for new or extended transit networks. Laporte et al. (1994) assessed three simple

networks with star, triangle and cartwheel configurations (Figure 2.1) with similar total rail

lengths and compared different factors for subway networks. They defined directness and

connectivity as indicators in assessing the efficiency of a transit network, along with some

other parameters. Directness was defined as the number of origin/destination paths that can

be traveled without transfer; and, connectivity was described as the ratio of the number of

edges (or links) to the maximum number of edges possible in the network also known as

gamma index (Waters 2006). Table 1 presents modified results based on Laporte et al. (1994)

where the number of stations and the total length of the three networks are similar.

As indicated in the study’s summary, a cartwheel network, which has a ring line, had the

highest directness and connectivity among the network patterns, while all three networks had

similar numbers of stations, except for triangle network, and similar rail lengths. It should

also be noted that these performance measures were based on uniform demands at all nodes;

17

however, the demand pattern on different sections and nodes on each network may have been

different. A different number of stations was used for the triangle network. It should also be

noted that for our purposes, a triangle network similar to Figure 2.1 can also be a ring line

when the three central links are double tracked.

Figure 2.1 – Three Simple Networks: Star, Triangle and Cartwheel Configuration (Laporte et al. 2011)

In a more recent study, Derrible and Kennedy (2009) used other definitions for

connectivity and directness. Directness was defined as the maximum number of transfers

between any vertices divided by the total length of the network. Connectivity was described

as the degree of mobility or the density of transfer possibilities, which was an attempt to get

an overall view of the transfer possibilities to travel from one station to another in the

network. They used these parameters, along with coverage, which was based on the total

number of stations and land area. They computed the indicators by analyzing 19 subway

systems around the world and showed the important role of network topology in attracting

more transit ridership. The relationship between the ridership and network configuration was

studied using graph theory concepts. Ridership was considered as the annual number of

boardings per capita. Strong, statistically significant relationships between the three

18

indicators (directness, connectivity, and coverage) and ridership were observed using

multiple regression analysis:

Bpc 44.96log(7.57992.32102.98 (2.1)

where Bpc is the number of boardings per capita, is the log coverage is the

directness, and is the connectivity. They did not, however, explain the reason for using a

logarithmic term for coverage.

There is zero rail ridership when no rail network exists; therefore, we can assume a

zero ridership when coverage, directness, and connectivity are zero. Thus, RTO (regression

through origin) is performed which results in a higher coefficient of determination, with the

coverage and connectivity parameters statistically significant at a 95% level of confidence

and the directness parameter significant at 85% level of confidence. It should be noted that

coverage parameter is not the log in equation (2.2) unlike equation ((2.1). Thus, the equation

can be re-estimated as follows:

Bpc 358.649.1448.07 (2.2)

In another study, Derrible (2012) introduced other transit network parameters such

as degree, closeness, and betweenness centrality to assess transfer stations in public transport

system.

In addition to the above network parameters that were found to be statistically

significant in generating ridership, city-specific characteristics, such as transit use culture

and demand parameters, such as population, city area, land-use distribution, and population

density, can also impact transit ridership. This chapter strives to include city, transit supply,

demand and topological parameters in estimating transit ridership. Ko et al. (2011) and Yi

19

and Chao (2010) made important contributions in addressing topological parameters.

However, the studies were limited to a specific region (South Korea and China), and the

conclusions may not be transferable to other geographic regions.

The next section discusses the relevant parameters in driving transit ridership in different

regions in the world and summarizes the results of the regression models to uncover these

relationships.

2.3 Rail Transit Networks Review

Cities with rail transit networks were reviewed to find the relationship between

parameters that shape the network and transit ridership of a system. In order to perform the

analysis on transit systems, an extensive database was required, which is available online

Metrobits (2013). Information from Global Metro Monitor (Istrate and Nadeau 2012) were

also used for GDP per capita and ‘City Mayors’ (2013) for city area and population density.

The categorization of ridership density based on regions resulting in clustering for different

continents. Figure 2.2 shows the clusters for North American, European and Asian cities.

North American cities have the lowest range of density and transit ridership, and European

cities are between North America and Asia clusters in terms of the density but having a

similar range with Asia in terms of ridership per unit length. The wide range of observations

for Asia can be explained by the rapidly expanding economies and infrastructure spending

of different cities in Asia, compared with North American and European cities.

20

Figure 2.2 – Relationship of Population Density and Ridership per Unit Length of Network

Tokyo

Hong KongGuangzhou

Seoul

Manila

OsakaTaipei

Beijing SingaporeShanghai

NagoyaSapporo

FukuokaNanjing

Bangkok

Chengdu BusanDaeguXian

Kuala Lumpur

Wuhan KaohsiungGwangjuDalian

ShenzhenTianjin

BudapestPrague

Lyon

Rome KievMoscow

ViennaParis Saint PetersburgAthens

Warsaw

BrusselsLisbonMilanMunichToulouse

Marseille BerlinNuremberg BarcelonaHelsinki Stockholm London

Amsterdam Copenhagen TurinBucharestMadridLille Bilbao

RotterdamHamburg

GlasgowNaplesSofia

Newcastle

New York

Toronto

Montreal

Boston

Philadelphia

ChicagoWashington

Los AngelesBaltimore San FranciscoMiamiAtlanta

0

5

10

15

20

25

30

0 2000 4000 6000 8000 10000 12000 14000 16000

Rid

ers

hip

pe

r K

M L

en

gth

of

Ne

two

rkTh

ou

san

ds

Popualtion Density

Asia

Europe

North America

AsiaEurope

North America

21

European cities have highly variable ridership values per unit length, within a limited

population density range of 2000 to 6000: this behavior is not exhibited in Asia and North

America. Asia and Europe have a more or less equal average ridership per unit length.

However, Asia has a much higher average population density than Europe.

Figure 2.3 shows the best-fit line for the population-ridership relationship based on

continents. Unlike North America, Asian and European cities show similar behavior.

Although all lines show a positive correlation between population and ridership, the fitted

line for North America is less steep than those of European and Asian cities. This can be

possibly interpreted as the sensitivity of ridership with respect to changes in population in

North America is lower compared to that in Europe and Asia; thus, an increase in population

does not necessarily increase the ridership the same way as in Asia and Europe. The fitted

lines for Asia and North America have a lower R2 value than that of Europe. The reason for

this pattern may be that Asian cities have greater differences in terms of socio-economic and

city pattern characteristics and that there is a lack of sufficient observations for North

American cities.

22

Figure 2.3 – Population versus Annual Ridership for Europe, Asia and North America

Age of the transit system can be used as a possible surrogate measure of transit use

culture affecting ridership; since macro scale measurements of such parameters are not easy

to find. It should be noted that there may be different interpretations for transit use culture.

Parameters such as land use, auto ownership, auto usage, parking management and

transportation policies can all impact utility or attractiveness of using personal vehicles

compared to public transport use. In addition, when supported by proper land-use and

transport policies, transit can shape the distribution and intensity of land uses in a city,

thereby affecting transit use. A long-duration mass transit system in a city can influence

people’s choices of transit mode and auto ownership and also impact city development, urban

y = 175.46x - 327.92R² = 0.5536

y = 177.97x - 113.68R² = 0.8333

y = 50.428xR² = 0.5333

0

500

1,000

1,500

2,000

2,500

3,000

3,500

0 2 4 6 8 10 12 14 16 18 20

An

nu

al R

ide

rsh

ip

City Population

Asia

Europe

North America

Linear (Asia)

Linear (Europe)

Linear (North America)

Mill

ion

s

Millions

23

design, and land-use. Thus, how a city has evolved over the time, especially near transit

corridors, can be part of the age of transit system parameter, as representative of transit use

culture. For a comprehensive study of the role of history on evolving transit ridership and

auto ownership, especially in the North America, the reader is referred to Jones (2010).

Figure 2.4 – Age of Transit System versus Annual Ridership for Asia, Europe, and North America

Figure 2.4 illustrates the relationship between the age of the transit system and

ridership. As apparent from the figure, the networks of Asian cities are relatively newer, due

to the rapid expansion of cities in China and Korea and more recent introductions of rail

transit. European cities have a much older average transit age than Asian cities, with

London’s very first underground urban rail transit line dating back to 1863.

y = 15.914x1.0195

R² = 0.481

y = 4.6275x0.9805

R² = 0.3531

y = 0.4782x1.3821

R² = 0.451

0

500

1,000

1,500

2,000

2,500

3,000

3,500

0 20 40 60 80 100 120 140 160

An

nu

al R

ide

rsh

ip

Age of Transit System (year)

Asia

Europe

North America

Power (Asia)

Power (Europe)

Power (North America)

Mill

ion

s

24

Another interesting observation is the two clusters of transit age for North America:

one around 100 years ago (e.g., New York City, Boston, and Chicago), and the other around

40 years ago (e.g., Los Angles, Philadelphia, and Washington DC).

Increases in ridership can be observed with increasing age of Asian transit systems. This

may be because Asian transit networks are at the beginning of their maturity and are

expanding at a higher rate and attracting more ridership. Another plausible explanation can

be attributed to the lower average GDP per capita or auto ownership levels in Asian cities.

Interestingly, Europe and North America show a similar behavior to ridership when

compared with age of transit system. From Figure 2.3 and

Figure 2.4, there is similar behavior in terms of density-ridership in Europe and Asia;

however, North America and Europe have similar behavior in terms of age-ridership. We

can conclude that it is the urban form and population of the cities that creates the

distinction between transit ridership in Europe and North America. However, over a long

period, Europe and North America may perform similarly, in terms of attracting ridership

by providing rail transit service.

In order to find all the parameters of a city that affect ridership, a multivariable linear

regression analysis can be performed. Various parameters that are thought to be important to

estimate transit ridership are considered: city population, population density, city area, the

length of transit network, age of transit system, and GDP per capita. Obviously, there are

25

correlations between some of these parameters. Thus, the parameters with high correlation

over 0.6 were not used in the same model.

Two dependent variables were tested: ridership per unit length of the transit network; and,

the length of the network. By conducting this analysis, the expected ridership or length of

the transit network can be estimated with specific city parameters.

Using ridership per unit length as the dependent variable, the population density and age of

transit system were found to be statistically significant with an R2 value of 0.7. This

observation makes sense since the age of the network affects the transit ridership culture and

city land-use characteristics; thus, it is expected that older systems have a higher ridership

per unit length. Population density also showed a better relationship with ridership per unit

length than population, as it had a higher R2 value.

Ridership per Unit Length = 1.10Population Density (population per square

kilometer) + 106.95Age (in years) (2.3)

Due to the fact that there would be no ridership with a zero population density or age

of transit system, we performed an RTO analysis, forcing the model through the origin.

Correlations between the parameters were also investigated. The highest correlation was

found between the network length and the population, with a value of 0.58. The t statistics

for each coefficient are given below the values of the coefficient in the equations.

(7.79)

(6.38)

26

The length of the network was also tested as a dependent parameter. With similar statistical

analyses, population and GDP per capita were found to be statistically significant

explanatory parameters, showing a positive coefficient and an R2 value of 0.87.

Length of Network (in kilometers) = 1.17GDP per Capita (in thousands of

dollars) + 11.40*Population (in millions) (2.4)

Two multivariable regression analyses were performed here. The first multivariable

regression model, with ridership per unit length as the dependent variable, showed the age

of the transit system and population density. These parameters were found to be significant

in prompting enough ridership to have an operationally efficient transit system. Higher urban

population density is a city parameter that is relevant and has a positive impact on the level

of ridership per unit length. The age of the transit system is both a network parameter and a

behavioral parameter, which showed that the longer the existence of a transit system, the

greater the change in attitude toward using transit. The age of the transit system also affected

changes in city development, land-use and planning policies, encouraging transit or

discouraging personal auto use.

In the second regression analysis, network length was used as the dependent variable and

found that GDP per capita and population were relevant parameters. The population

indicated the demand required to warrant an increase in the length of the transit system, and

(4.15)

(7.53)

27

GDP per capita was representative of the funding availability for the expansion of the rail

transit system.

2.4 Ring Transit Line Review

Ring transit lines, also called circular or circumferential lines, may not necessarily be

whole circles (regular or irregular) and may only connect a portion of a circle around the

central core of the city. In addition to the possible savings in transit riders’ travel and waiting

times, increased transit network reliability and the reduction of the transit load in the

downtown core, the presence of circumferential lines also increases accessibility and, thus,

development of new satellite centres. While an auto ring road usually encourages

development to take place on the outskirts of a city beyond the ring, a transit ring line can

encourage higher density, mixed-use development to take place along an inner city corridor.

A ring transit line is also important for public transit users, in terms of their mobility and

accessibility needs.

Findings from the literature (Laporte et al., 1994; Derrible and Kennedy, 2009; and

Yi and Chao, 2010), as described earlier, showed that a ring or circumferential transit line

can greatly improve network connectivity, directness, and operation efficiency of the

network. However, a combination of transit network demand, supply, and transit cultural

effect parameters should also be investigated.

The average circumference of existing ring lines is 28.18 kilometers with a standard

deviation of 11.59, ranging from as little as 10 kilometers (Glasgow) to 57 kilometers

28

(Beijing). Within big cities, most ring lines orbit a city centre, connecting peripheral railway

stations.

Seoul’s Subway Line 2 is one of the most heavily used and longest ring subway lines

in the world. This line was built between 1978 and 1984. It has 53 stations and connects

urban centres, such as office clusters, colleges, and recreational centres. The daily ridership

of the Seoul Line 2 subway is over 2 million, which is the highest ridership among all subway

lines within the 16 billion annual passenger load of the Seoul subway system.

Another successful circular rail transit line is the London Underground’s Circle line at 27

kilometers in length, with 36 stations and about 218,000 daily passengers (Transport London

2009). The success of this inner circle line in London led to the operation of three other semi-

circumferential routes using the existing underground lines in the network. However, these

routes failed to attract the desired passenger ridership, and the operation of all three is now

interrupted (London Overground & Orbirail 2013, Tubeprune 2013).

Madrid has two ring metro lines: one circles the central core of the city; and, the other

connects five suburban towns and one small village south of Madrid. Planners of the Paris

Metro envisioned a ring line similar to that of the London Underground; however, difficulties

in operations forced the formation of two separate semicircular lines on the north and south

sides of Paris (Lines 2 and 6). Line 6 is 13.6 kilometers in length, 6.1 kilometers of which

are above ground, with more than 100 million passengers per year. Line 2 is 12.4 kilometers

in length with about 92 million passengers per year.

29

Transit planners are not the only people who have dealt with the design of subway

systems. An urban legend has Joseph Stalin himself suggesting the Koltsevaya Line in

Moscow (Figure 2.5) by placing a coffee cup on the original development plan, lifting it and

leaving a circular stain around the centre of the city. It is thought that this is the reason for

the use of the color brown to designate this line on maps. The passenger flow patterns

underwent massive changes with the introduction of the Koltsevaya Line. Initially a total of

7 lines began at the rings, four of them later extended to the centre to make two diameter

lines (Moscow Metro 2007).

The Beijing metro (Figure 2.5) also has two complete rectangular ring lines: Numbers

2 and 10. The Beijing subway network was originally planned to have only one ring line

(Xin, 2013); however, the rapid economic and population expansion in Beijing caused a huge

passenger load on the first ring line, exceeding its capacity. Two L-shaped lines (Numbers

10 and 11) were proposed to be connected to create the second loop around Beijing and

relieve passenger demand on Number 2. This line is now one of the worlds’ largest subway

routes with a length of 57.1 km. After only 5 years of operation, it is now the busiest subway

line with a daily ridership of 1.69 million; and, it is expected to grow to 2 million, easing the

passenger load on the city’s first ring line. All other subway lines in Beijing that pass through

the city centre intersect with Number 10, making for 24 transfer stations along the route and

a total of 45 stations. The first ring line in Beijing (Number 2) is the second oldest and the

only line that connects to the Beijing railway station (Xin, 2013).

30

The Yamanote Line in Tokyo (Figure 2.5) is another example of ring line. It is 34

kilometers long, has 29 stations and was completed in 1914. It connects major stations and

urban centres. Out of 29 stations, only two do not connect to other railway or subway lines.

An estimated 3.68 million passengers ride this line daily. Its annual ridership of 3.68 million

can be compared to that of the New York City subway network, which carries 5.08 million

passengers on its 26 lines and 468 stations, and the London Underground with 2.7 million

daily passengers on 12 lines and 275 stations. (“Yamanote Line,” 2016)

During the prewar era in Japan, no permits were issued to private suburban railway

companies for new lines crossing the Yamanote Line to the central districts of Tokyo, forcing

the lines to terminate service at Yamanote Line stations. New urban centres around major

transfer points on the Yamanote Line are the result of this policy. Examples of this policy

include the Shinjuku and Ikebukuro stations, which are now two of the busiest railway

stations in the world. (“Yamanote Line,” 2016)

Figure 2.5 – Metro Network Topology of Cities with Ring Line: Seoul, Moscow, and Beijing

31

The list of cities with ring rail transit lines is longer, but there are no insightful

published studies to be found on their actual performance measures. However, some studies

have focused on the feasibility and cost-benefit analyses of projected ring transit services,

which are presented in the next section.

2.5 Regression Analysis for Ring Transit Lines

An analysis similar to that of the Rail Transit Review section was conducted for ring

transit lines to investigate the macro-level city parameters that warrant a ring line. Thus, the

length of the circular line was considered to be the dependent variable. There are cities with

more than one ring line; however, it is argued that the second line could be less important,

i.e., have a lower priority, compared to the first line. For all the second ring lines investigated,

they were built years after the first line was operational. Thus, for the length of the circle as

the dependent variable, the following equation is used:

Length of Ring Line = Length of the First Line + kLength of Second Line (2.5)

where k is an unknown constant. Different k values were tested, ranging from 0 to 2; and,

the value of 0.2 showed the best fit with the highest R2. A value more than one may mean

the second line is more important and has a higher priority than the first line.

The most significant relationship was found between the population density and the length

of the line, as shown in Figure 2.6, where an increase in the density of cities also increased

the expected length of a ring line. This supports the arguments made earlier that ring lines

32

are better for connectivity between major high density activity centres that complement the

role of the downtown core. The greater the overall density of the city, the more likely there

are higher density activity centres at some distance from the city centre. The regression

analysis shows an R2 value of 0.46 with the following equation:

Length of Ring Line (in kilometers)=0.0018*Population Density (population per

square kilometer)+17.39 (2.6)

Figure 2.6 –Length of Ring Line (km) versus Population Density per km2

Data on the most recent ring line ridership of eight cities have been obtained through

various reports and websites (Moscow Metro 2007; Seoul Metro 2011; Simens 2013). A

positive relationship between the length of the line and the annual ridership per line unit

Beijing

Madrid

Tokyo

Seoul

BerlinShanghai

Nagoya

ParisBucharestLondon

OsakaMoscow

Hamburg

Glasgow

y = 0.0018x + 17.396R² = 0.4608

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Len

gth

of

Rin

g Li

ne

(km

)

Pouplation Density

33

length with a high R2 square can be observed in Figure 2.7. The fitted line shows an R2 value

of 0.73 with the following equation:

𝐴𝑛𝑛𝑢𝑎𝑙 𝑅𝑖𝑑𝑒𝑟𝑠ℎ𝑖𝑝 (𝑖𝑛 𝑚𝑖𝑙𝑙𝑖𝑜𝑛𝑠) = 10.17 ∗ 𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒 𝐿𝑒𝑛𝑔𝑡ℎ (𝑖𝑛 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠) (2.7)

Figure 2.7 –Annual Ridership of the Ring Line versus Length of the Line

In order to find relevant parameters that can be used as justification for introducing a

ring line to a city’s transit network, multivariate regression analyses were performed. Similar

to Equations (2.3) to (2.5), all potentially relevant parameters were identified and their

correlations were tested to make sure that parameters in the same model are not highly

correlated. The regression model that had all parameters statistically significant with a 95%

Seoul - Line 2

London - Cricle Line

Paris - Line 2

Paris - Line 6

Moscow -Koltsevaya Line

Singapure - cricle Line

Bejing - Line 10

Glasgow

y = 10,174,264.75xR² = 0.73

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50 60

An

nu

al R

ide

rsh

ipM

illio

ns

Length of Circle Line (km)

34

level of confidence and significant F test value was adopted. Only cities with full ring lines

were investigated in the model. It appears that two parameters – city density and length of

the network – were the statistically significant parameters in determining the length of a ring

line with an R2 value of 0.86 in Equation (2.8). The correlation between parameters was not

found to be high; and, this may mean that a ring line is warranted when a city’s transit

network passes a certain maturity level, in terms of the length of the network and population

density.

𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒 (𝑖𝑛 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠) = 2.22 ∗ 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 (𝑖𝑛 𝑡ℎ𝑜𝑢𝑠𝑎𝑛𝑑𝑠) +

0.053 ∗ 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑁𝑒𝑡𝑤𝑜𝑟𝑘 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒 (𝑖𝑛 𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟𝑠) (2.8)

We should note that the network length parameter used in this equation is different

from that of Equation (2.3), as this one does not include the ring line length in the network.

Although the two relevant parameters were found to be important for the existence and

possible length of a ring line, there are also other parameters that were not considered in this

analysis. As indicated in the introduction section, one parameter may be the presence of

major activity centres around a city, not just in the downtown core. Population density in this

analysis was defined based on the urban population of the city divided by the urban city area

based on the city’s boundaries. It should also be emphasized that the range of city areas

observed in this study was between 228 km2 (Bucharest) to 2875 km2 (Nagoya) with an

average city area of 1473.71 km2 and a standard deviation of 956.67. Thus, the model cannot

estimate the length of the ring line for a city area that does not fall within this range.

(2.59)

(1.91)

35

Since the analysis is investigating macro-level parameters, only one value was used for the

population density of a city. But population density could differ from one part of the city to

another. However, in macro scale analysis, the average population density can be a good

representative of how the city is spread out, whether it sprawls out to the suburbs or there is

a high population density all around the city. It can be argued that the model will show a high

value of ring length for a hypothetical city with a very high population concentrated in a

small area. However, in such city the length of the transit network should be relatively low;

therefore, the model will not estimate a high value for the ring length.

It was also investigated whether cities with a ring line have a higher valued performance

measure than cities without a ring line. The dataset was provided by (Derrible and Kennedy

2009), which includes 19 cities with network data. The number of boardings per capita was

used. With a confidence level of 95%, the results showed that cities with ring lines had a

higher directness and connectivity index compared to cities without a circular line. This

conclusion sounds logical since a ring line provides more direct service between certain

stations and more alternatives for route choices than radial only networks. Derrible and

Kennedy (2010) indicated that in the investigated subway networks with high connectivity

and directness, all have radial or diametrical lines connected by complete, semi-

circumferential, or tangential lines. It is also worth noting that connectivity is also directly

related to network reliability since higher connectivity provides more choices between

stations in case of an interruption in a section of the network.

36

Derrible and Kennedy (2009) also found a positive relation between these performance

measures and the annual number of boardings. Thus, it can be argued that a ring line can

increase network connectivity, directness and ridership, due to the provision of better, faster

and more reliable services. We also found that networks with ring transit lines have a

significantly higher number of lines compared to other networks.

Logistic regression was performed to identify parameters that justify a ring line in a city by

considering macro-scale parameters found through the World Metro Database (Metro

Orbits, 2013), such as population, age of transit system, length of transit line, and additional

information, such as city area, population density, and GDP per capita. We identified cities

with a ring line with a dependent value of 1 and cities with no ring line with a dependent

value of 0, and tested all parameters from 97 cities around the world.

The length of the rail transit network, age of transit system, population, and city area

(negative relation) are found to be statistically significant parameters with a Nagelkerke R

Square of 0.66. Table 2.1 shows the parameters that are statistically significant, and the

associated model coefficient and significance value. As shown in Table 2.2, the model

demonstrates 92.6% accuracy.

The parameters “length of transit network” and “age of transit system” show that a

transit network should be mature enough before introducing a ring line. A transit network in

a city would always start with radial or diametrical lines and, later on, the need for a ring line

would be determined. The positive sign for population and the negative sign for city area

37

represent the population density of the city. If a city is sprawled with a low or medium

population, it is less likely that a ring line may be justified.

Table 2.1 –Logistic regression analysis for the existence of a ring line

Parameters Coefficient (B) Significance Exp (B)

Length of Network (km) 0.015 0.008 1.015

Age of Transit System (years) 0.058 Less than 0.001 1.059

Population (millions) 0.221 0.025 1.247

City Area (squared km) -0.001 0.004 0.999

Constant -6.792 Less than 0.001 0.001

Table 2.2 – Model accuracy and classification table for logistic regression

Observed Predicted

Ring or not? Percentage

Correct 0 (No Ring) 1 (Ring)

Ring or not? 0 77 3 96.3

1 4 10 71.4

Overall Percentage 92.6

Figure 2.8 – Logistic regression analysis for cities with rail transit network – cities based on the highest

probability of having a ring line

38

Figure 2.8 shows the cities that were ranked above a 10% likelihood of having ring lines.

Only 14% of the cities in the database showed a 50% or greater likelihood of having a ring

line. The cities colored in red already have one or more ring lines, and cities shown in green

are the cities that are considering building a ring line.

In order to determine the feasibility of a ring line for a city, a detailed economic study is

required based on population and job distribution in the city, transit mode share, value of ride

time, wait time for the passengers, and unit values of operations and fixed cost of the rail

transit line. Such variables, in particular, current and future trip distribution in a city and the

current configuration of the radial rail transit system, would greatly impact the feasibility and

optimal layout of a ring line.

2.6 Discussion

This chapter used regression analysis to study the relationship between the transit

network with different city parameters, such as population, city area, and population density.

In the analysis of ridership per unit length, the age of the transit system was found to have a

positive impact. A long duration of mass transit system in a city can affect people’s choices

of transit mode and auto ownership and can also affect city development, urban design, and

land use in cities. Since many of these parameters are not quantitative, the authors used the

age of the transit system as representative of such factors.

39

Comparisons of the relationships of transit ridership for different continents were

made. It was found that Europe and Asia perform similarly regarding population and

ridership. The relationship between ridership and the age of the transit system for Europe

and North America showed very similar behavior, which was quite unlike that of cities in

Asia.

As indicated, general city parameters, such as population density, city area and

network size, are not the only parameters that indicate the necessity of a ring line. A feasible

circumferential line should connect higher density areas with activity centres that are not in

the city centre. Auto ownership is another important factor that could impact transit ridership

Jones (2010). GDP per capita, which was used in this study, and auto ownership were thought

to have a high correlation; thus, the use of one of these parameters was assumed to show the

impact of the availability of a personal mode option. In fact, the higher transit ridership

observed in European cities than North American cities with similar auto ownership levels

showed that not only car availability, but also other travel demand management parameters,

such as parking cost and parking availability, fuel taxes and tolls, have a major impact on

transit ridership.

The chapter also investigated the macro-level city parameters that could warrant a

ring line. Population, city area, the length of the network, and age of the system were found

to be statistically significant parameters. The result shows that a city should have a certain

level of population density reasonably spread around the city, not only in a single area. In

addition, transit networks should first have some radial lines and a certain level of network

maturity before a ring transit line is introduced.

40

The analysis will be of interest to cities considering the extension of their transit

networks and provision of a ring transit line service. Although the question of the location,

radius, length and mode of the service depends on more detailed information, such as

origin/destination patterns and topology of the current network, this chapter gives a general

idea of when cities should consider adding a circumferential line to the network and the

parameters that warrant such a line. It also provides evidence that such lines can improve the

efficiency of the system and, thus, help to improve ridership. Regression analyses are

performed to find the statistically significant parameters and their coefficients, in order to

determine the level of magnitude for each parameter.

The analysis in this chapter allows for a better understanding of how a ring rail transit line

can revitalize the transit system for an entire city. The results suggest a starting point for

more comprehensive analysis to design a circumferential light rail transit line that meets the

sustainable transportation goals for North American cities.

One important question is whether a ring line is a better investment than additional radial

lines. Although there are studies comparing these different network topologies (Laporte et

al. 1994, Laporte et al. 2011), more in-depth and detailed study needs to be conducted, as

priorities for each city or even regions within a city may be different. These questions are the

motivation for the study described in the next chapters in this thesis.

41

CHAPTER 3: RADIAL LINE RAIL TRANSIT NETWORK MODEL2

This chapter describes an analytical model to find the optimal number of radial lines

in a city for any transit OD demand distribution. An analytical model using Continuum

Approximation (CA) approach is considered to find the optimal number of radial lines

considering a city with a radio-centric street grid with the city center at the origin. The

chapter starts with the description of the mathematical model. Then a case study for the City

of Calgary is considered.

3.1 Optimal number of radial lines in a city

As mentioned earlier, a continuum approximation method is used to find the optimal

number of radial lines considering a city with a radio-centric street grid with the city center

at the origin. Consider a city with radial or diametrical rail lines that merge in a CBD similar

to Wirasinghe and Ho (1982) who analyzed radial bus systems for CBD commuters. It is

assumed that all passengers access the nearest radial line, and then use those radial lines;

thus, no passenger would directly access a destination without using the rail transit system.

All passengers access the radial rail lines circumferentially and not radially. Any radial

access to a station is considered independent of the number and spacing of radial lines. In

this model, station spacing is independent of the optimal number of radial lines, as we are

2 The essential contents of this chapter have been published in:

- Saidi, S., Wirasinghe, S.C., Kattan, L., 2016. Long Term Planning for Ring-Radial Urban Rail Transit

Networks. Journal of Transportation Research Part B: Methodological 86. 128-146.

42

optimizing the trade-off between access cost and number of radial lines. Figure 3.1 shows

schematic view of a transit trip and its cost factors from an origin to a destination.

Figure 3.1 – Schematic view of a transit trip and its cost factors between from an origin to a destination

We can allow any transit OD combination by defining two density functions: trip

production and trip attraction, for each unit area in the network. Since all transit riders will

use a radial rail transit line for their travel, there is only one rational route choice for each

passenger and, thus, passenger ride cost (the cost associated with the riding time by train) is

independent of the number of radial lines. Each passenger has to take the nearest radial line,

43

and either continues on the same radial line to reach his or her destination, or transfer to

another line at the CBD. Obviously, similar to circumferential access movement to radial

lines, passengers would make circumferential egress movement to reach their destination. A

portion ξ of originating passengers at (r,θ) will have to make a transfer and, therefore, incur

another wait time, assuming that trains on different lines are not coordinated. Continuous

function u(θ) is defined as the density of radial lines per radian at θ. The summation of u(θ)

over the network will give the total number of radial lines in the network. The other

parameters are listed below:

𝑝𝑚(𝑟, 𝜃), demand production per unit area per day for access mode m;

P (r,θ), total demand production in passengers per unit area per day;

Q (r,θ), total demand attraction in passengers per unit area per day;

g(θ), train dispatch rate per unit time at θ;

T, operating hours of the system per day;

𝛾𝑎 𝑚, access cost with mode m to reach destination or rail line per kilometer per passenger;

𝛾𝑎, generalized access cost to reach destination or rail line per kilometer per passenger as

defined in equation (3.1);

𝛾𝑤, wait cost per hour per passenger;

𝜆𝑜, operating rand capital cost of trains per unit distance per day;

𝜆𝐿, line cost per unit distance per day;

L(θ), length of radial line at angle θ;

C(θ), city boundary at angle θ; and

α, a sector in which the analysis is being conducted with α = 2π, if we examine an entire city.

44

The length of access is the multiplication of the angle between the origin with the

radial line, by the radius of the origin (or destination for egress). The average angular spacing

between the radial lines near θ is 1/u(θ). Therefore, the average circumferential distance from

the origin (or destination for egress) with radius r to the nearest radial line will be r/4u(θ).

Passengers will use different modes of transportation such as walking or biking, feeder buses,

park and ride or kiss and ride to access the nearest rail radial line. Since there are different

passenger costs for each of the access modes (e.g., wait time and feeder bus ride time, car

operating cost and parking, walking time), a generalized access unit cost value is used, such

that:

γa(r, θ) =∑pm(r, θ)γa m p(r, θ)

m

(3.1)

Assuming that the above generalized cost is invariant with location, generalized access cost

is set

γa(r, θ) = γa (3.2)

Consider a section of the network with sector angle α. Access cost from each origin to the

nearest radial line, and egress cost from each destination to its nearest radial line are shown

in Equations (3.3) and (3.4), respectively:

∫ ∫1

4u(θ)rγap(r, θ)rdrdθ

C(θ)

0

α

0

(3.3)

45

∫ ∫1

4u(θ)rγaq(r, θ)rdrdθ

C(θ)

0

α

0

(3.4)

Different assumptions for a passenger’s average wait time can be used for a train,

which depends on factors such as whether the schedule of the system is known or there is a

fixed headway strategy. Moreover, if the train is scheduled, it is important whether

passengers have prior knowledge of the schedule. In addition, for commuting trips,

passengers may experience schedule-delays at their destination since they may need to start

their work at a specific time. Thus, various ranges of average wait time can be used here. For

example, if the schedule is not known and we consider schedule-delay at the destination,

passengers’ wait time would be equal to one headway. However, it will be half-a-headway

either if we assume that they know the schedule, or they do not know the schedule but their

trip is for recreational purposes. Since the analysis can be conducted for different types of

trips (i.e., commuting, recreational), we can use a parameter k, which is a value between 0.5

and 1, for estimating the fraction of headway that is equal to the mean wait time.

Some passengers might have to transfer from one line to another. They will spend a wait

time equal to twice that of others. A proportion of ξ passengers transfer from one line to

another at the CBD. The ratio can be estimated, for example, to be 1-1/(n-1), where n is the

number of radial lines in the network, and each line is equally likely to be transferred to.

However, the value of ξ would depend on the passenger trip distribution and ranges from 0

to 1. We will compare different ranges of ξ in a sensitivity analysis. Hence, the wait cost

function will be:

46

∫ ∫kγwg(θ)

(1 + ξ)p(r, θ)rdrdθ

C(θ)

0

α

0

(3.5)

The operating cost involves the dispatching rate and length of travel in two directions. It is

assumed that each train has sufficient capacity to handle the demand. During peak times,

additional cars can be added to each train. g(θ) is taken as the average dispatching rate per

day, since it could be a different value for peak and off-peak hours. Thus, operating costs

would depend on the length of each line and the number of radial lines:

∫2Tu(θ)L(θ)g(θ)λo

α

0

dθ (3.6)

Line cost is the capital cost per unit distance discounted per day for the radial lines for the

entire network (the length of the radial line times the density of the radial lines):

∫u(θ)L(θ)λL

α

0

dθ (3.7)

Summing up all of the above costs, total cost function Z can be obtained, which includes

passenger access cost, passenger wait cost, operating cost, and line cost:

47

Z = ∫ ∫1

4u(θ)rγa{p(r, θ) + q(r, θ)}rdrdθ

C(θ)

0

α

0

+∫ ∫kγwg(θ)

(1 + ξ)p(r, θ)rdrdθ

C(θ)

0

α

0

+∫2Tu(θ)L(θ)g(θ)λo

α

0

dθ + ∫u(θ)L(θ)λL

α

0

dθ

(3.8)

It is denoted that:

S1(θ) = ∫ rp(r, θ)dr

C(θ)

0

(3.9)

Sp(θ) = ∫ r2p(r, θ)dr

C(θ)

0

(3.10)

Sq(θ) = ∫ r2q(r, θ)dr

C(θ)

0

(3.11)

S1(θ) is the total demand, and Sp(θ) and Sq(θ) can be interpreted as the second moment of

demand. Thus, the wait time cost is a function of the total demand, and the access cost is a

function of the first moment of demand.

Therefore, the total cost function is defined as the summation of passenger costs

(access and wait cost), and operating and line cost. As mentioned previously, it is assumed

that all passengers take radial lines for their travel and, thus, passenger ride cost is

independent of this optimization. We assume the lengths of radial lines are equal to the city

limit, i.e. L(θ)= C(θ), for each segment of the city.

48

Z = ∫1

4u(θ)γa(Sp(θ) + Sq(θ))dθ

α

0

+∫kγwg(θ)

(1 + ξ)S1(θ)dθ

α

0

+∫2Tu(θ)L(θ)g(θ)λo

α

0

dθ + ∫u(θ)L(θ)λL

α

0

dθ

(3.12)

Let first optimize Z with respect to dispatching rate g(θ):

−kγwg2(θ)

S1(θ)(1 + ξ) + 2Tu(θ)L(θ)λo = 0 (3.13)

Thus, the optimal dispatch rate is:

g(θ) = [kγw(1 + ξ)S1(θ)

2TL(θ)λou(θ)]1/2 (3.14)

Substituting g(θ) in Z:

Z = ∫[γa(Sp(θ) + Sq(θ))

4u(θ)+ 2[2kTγw(1 + ξ)L(θ)λoS1(θ)u(θ)]

1/2

α

0

+ u(θ)L(θ)λL]dθ

(3.15)

Now, optimizing with respect to u(θ), we can obtain:

−γa(Sp(θ) + Sq(θ))

4u2(θ)+ [2kTγwL(θ)λo(1 + ξ)S1(θ)

u(θ)]1/2 + L(θ)λL = 0 (3.16)

Equation (3.16) can be rearranged as:

u1/2(θ) + [2kTγwλo(1 + ξ)S1(θ)

𝜆L2L(θ)

]1/2 =γa(Sp(θ) + Sq(θ))

4L(θ)λL∗ u−

32(θ) (3.17)

To find the optimal value of u(θ), Equation (3.17) should be solved numerically.

49

If we assume a constant given headway for all radial lines, as commonly practiced in many

systems, we obtain the closed form solution given in Equation (3.18). The optimal density

of radial lines is directly proportional to the square root of unit access cost and total second

moment of demand, and inversely proportional to the length. We observe a dampening effect

on the relationship between line density and cost factors due to the square root formulation:

u(θ) =1

2[

γaλL + 2Tgλo

∗Sp(θ) + Sq(θ)

L(θ)]1/2 (3.18)

3.2 Application

We can use Equation (3.17) to calculate the optimal number of radial lines for the City

of Calgary for the current demand and for the long range horizon (the year 2076). Figure 3.2

shows land use zones for the City of Calgary. The travel demand to and from each zone is

represented based on its centroid. The coordinates of each zone are with respect to the city

center as the origin. The first and second moment of demand introduced in Equations (3.9)

to (3.11) can be calculated with the above information. Thus, the proposed methodology can

be used with any demand distribution, i.e., assumptions need not be made about any

particular functional form for the demand as a function of location.

50

Figure 3.2 – City of Calgary land use zones and centroid location

Table 3.1 –Unit cost values for the City of Calgary

Based on the City of Calgary’s actual geometry, and the different distances to the boundary

from the center, we proceed with the analysis by dividing the city into 4 different sectors.

The results reported in Table 2 are the summation of the optimal number of radial lines for

all sectors. With the current population and job distribution data, assuming that all peak hour

Parameters Unit Cost

𝛾𝑤 $5/hr/passenger

𝜆𝐿 $6000/km/day

𝜆𝑜 $2000/km/per direction/day

𝛾𝑎 $2/km/passenger

𝜉 0.7

𝑘 0.5

51

transit trips are using rail radial lines, there should be 5 radial (rounded down from 5.2) lines

for the City of Calgary, which is close to the current number of 4 radial lines. Using projected

2076 transit trip OD, we can obtain a value of 7 (rounded down from 7.2) as the optimal

number of radial lines, compared with the currently planned 6 radial lines. It should be noted

that we only considered peak transit trips in this case study. A sensitivity analysis is s shown

in Table 3.2. The rounded number of radial lines is given in parentheses.

Table 3.2 – Sensitivity analysis for the optimal number of radial lines for the City of Calgary

As observed from Table 3.2, access cost and travel demand are highly sensitive

parameters for finding the optimal number of radial lines. If access cost is decreased, the

optimal number of radial lines is decreased significantly, meaning that alternative travel costs

become cheaper and, thus, it is less efficient to provide a high number of radial lines to

balance access to the radial lines. If demand for travel with rail transit is decreased, a lower

number of passengers would take the radial lines and, obviously, such lines would be less

efficient due to the cost of operations and construction. Line cost per unit length is also an

important factor but shows lower sensitivity with respect to the number of radial lines

compared to the other parameters discussed.

𝛾𝑎 u (2π) 𝛾𝐿 u (2π) Demand Factor

u (2π) 𝜉 u (2π)

0.5 2.2(2) 1000 6.9(7) 0.1 2.4(2) 0 6

1 3.2(3) 3000 5.7(6) 0.5 4.4(4) 0.3 5.6(6)

2 5.2(5) 6000 5.2(5) 1 5.2(5) 0.5 5.5(6)

5 9.3(9) 10000 4.9(5) 1.5 6.4(6) 0.8 5.2(5)

10 15.2(15) 15000 4.3(4) 2 7.3(7) 1 5.1(5)

52

3.3 Discussion

This chapter presented an analytical model to find the optimal number of radial lines in

a city for any demand distribution. An analytical model using a continuum approximation

method is considered to find the optimal number of radial lines considering a city with a

radio-centric street grid with the city center at the origin. Since all trips on radial direction

are assumed to use the radial line, ride cost is independent of optimal radial line spacing.

Thus, optimization is basically a trade-off between access cost to radial line versus the cost

of operation and construction of radial lines to find the optimal number of radial lines or

optimal radial line spacing. City of Calgary’s long-term projected transit trip demand was

considered to optimize the number of radial lines for the long term. This chapter only looked

into radial lines and no ring line was considered in the mathematical model. The next chapter

presents optimizing of a ring line in a fully radial network. As showed for City of Calgary’s

case study, the optimal number of radial lines may be a real number and not necessarily an

integer. However, in reality, an integer number should be applied in the network. Thus, the

optimum value should be rounded up or down based on the lower total cost.

53

CHAPTER 4: RING-RADIAL RAIL TRANSIT NETWORK MODEL3

This chapter presents an optimization analysis of adding a ring transit line in a fully radial

network or a network with an existing ring line. A passenger route choice model that is used

for the analysis of total passenger cost is introduced in this chapter. It then uses the method

for optimizing ring transit line in an existing network for a many-to-many Origin-Destination

(OD) demand distribution, based on a total passenger cost and operating and capital cost of

building the ring line. City of Calgary’s Light Rail transit network and Shanghai’s metro

network are used as case studies to illustrate the applicability of the model. Also the model

is extended to allow a second ring line following the same assumptions. Shanghai metro is

considered to compare the current alignment of its urban rail transit network with the

optimized model output, and recommend a potential second ring transit line for the future

Shanghai network.

4.1 Passenger route choice

As described earlier a comparison between a network with a ring line and a radial-only

network is considered in this section to identify the scenarios (the number of radial lines) in

3 The essential contents of this chapter have been published in:

- Saidi, S., Wirasinghe, S.C., Kattan, L., 2016. Long Term Planning for Ring-Radial Urban Rail Transit

Networks. Journal of Transportation Research Part B: Methodological 86. 128-146.

- Saidi, S., Ji, Y., Cheng, C., Guan, J., Jiang, S., Kattan, L., Du, Y. Wirasinghe, S.C., 2016. Planning an

Urban Ring Rail Transit Line: A Case Study of Shanghai, China, Transportation Research Record: Journal of

the Transportation Research Board 2540. DOI 10.3141/2540-07 (in press)

54

which a ring line will be used. Passenger cost parameters are assumed to be reflected in their

route choice. A full ring line is assumed with two directions of train movement so that

movement along a ring line will follow the shortest path. The fare is assumed to be flat, and

thus independent of the chosen path.

4.1.1 All travels to the CBD

Assuming that all passengers are traveling to the CBD, there are only three options

for the passengers in the network. They are:

1: direct access from the origin to the CBD;

2: access the radial line and take it to the CBD; and

3: access the ring line, take it, transfer to the radial line, and take it to the CBD.

In the case of all trips being to the CBD, the ring line is only used as a faster access path to

the radial lines. The following costs to calculate total passenger cost for each alternative are

considered; the one with lower passenger cost will be the alternative chosen. The intention

is to find the route catchment area for the different alternatives for passengers based on their

origin location (r,θ), assuming a mean value of unit costs: access cost, ride cost (𝛾𝑅), headway

for radial and ring lines (𝐻𝑅𝐿 𝑎𝑛𝑑 𝐻𝑅𝑁 ), wait time cost, and transfer penalty(𝛾𝑇).

55

(a)

(b)

Figure 4.1 – Schematic view of passengers’ route choices for a. r > R and b. for r ≤ R

56

The additional required parameters are defined below:

R radius of the ring line;

𝛾𝑅 ride cost per unit distance per passenger;

𝐻𝑅𝐿 headway on the radial line;

𝐻𝑅𝑁 headway on the ring line; and

𝛾𝑇 transfer penalty that passengers will suffer in case they need to transfer from one rail

line to another in dollars per transfer per passenger.

Two cases are analyzed where (i) r ≥ R and (ii) r < R. Figure 4.1 shows the passenger route

choices available to travel to the CBD for the case of an origin outside the ring line

(Figure 4.1.a) and inside the ring line (Figure 4.1.b).

1. Case (i) r ≥ R

Since 𝛾𝑎 > 𝛾𝑅, one would expect that no passenger would directly access the CBD;

they would rather take a circumferential access route to the nearest radial line, or take a radial

access route to the ring line and then take the ring line to the same radial line. In addition,

since all passengers should eventually take a radial line to access the CBD, only the passenger

cost to where the ring and radial line cross is found, as indicated in Figure 4.1.a. The

passenger costs after taking the radial line at point A are independent of the presence of a

ring line.

57

The alternative 1 (radial line) cost per passenger will be the cost of access to the

nearest radial line plus the ride cost on the radial line along with the average wait time

experienced to take the radial line. Similar to equation (3.5), parameter k is used, which is a

value between 0.5 and 1, for the fraction of headway that passengers need to experience as

waiting time to take the train:

𝛾𝑎𝑟𝜃 + 𝑘𝐻𝑅𝐿𝛾𝑤 + (𝑟 − 𝑅)𝛾𝑅 (4.1)

The alternative 3 (ring line) passenger cost will be related to access to the ring line plus ride

cost on the ring line to transfer to the radial line along with two wait costs, i.e., one for taking

the ring line and one for taking the radial line. There will also be an additional transfer

penalty, 𝛾𝑇:

(𝑟 − 𝑅)𝛾𝑎 + 𝑘𝐻𝑅𝑁𝛾𝑤 + 𝑅𝜃𝛾𝑅 + 𝑘𝐻𝑅𝐿𝛾𝑤 + 𝛾𝑇 (4.2)

If (𝑟 − 𝑅)𝛾𝑎 + 𝑘𝐻𝑅𝑁𝛾𝑤 + 𝑅𝜃𝛾𝑅 + 𝑘𝐻𝑅𝐿𝛾𝑤 + 𝛾𝑇 < 𝛾𝑎𝑟𝜃 + 𝑘𝐻𝑅𝐿𝛾𝑤 + (𝑟 − 𝑅)𝛾𝑅,

then alternative 3 will be used, and the passenger will take the ring line.

Simplifying, we have:

𝜃 >(1 −

𝑅𝑟)(1 −

𝛾𝑅𝛾𝑎)

1 − (𝑅𝑟)(

𝛾𝑅𝛾𝑎)

+𝑘𝐻𝑅𝑁𝛾𝑤𝛾𝑎𝑟 − 𝑅𝛾𝑅

+𝛾𝑇

𝛾𝑎𝑟 − 𝑅𝛾𝑅 (4.3)

The above inequality shows the changes in the angle of the boundary with respect to

changes in r. In order to draw the boundary line, we find its slope:

58

𝜕𝜃

𝜕𝑟=𝑅(𝛾𝑎 − 𝛾𝑅)

2 + 𝛾𝑎2 − 𝑅𝛾𝑎𝛾𝑟

𝛾𝑎𝑟 − 𝑅𝛾𝑅 (4.4)

If 𝜕𝜃

𝜕𝑟> 0, with increase in r θ will also increase. On the other hand, with

𝜕𝜃

𝜕𝑟< 0,

increase in r would cause θ to be decreased. This is a less likely scenario, and we expect the

angle boundary line to increase with a higher r. Thus, the following condition should be met

to have a positive derivative:

𝑅

𝐻>

𝑘𝛾𝑤𝛾𝑎

+ 𝛾𝑇/(𝑘𝐻𝛾𝑎)

(1 − 𝛾𝑅/𝛾𝑎)2 (4.5)

Also from equation (4.4), increase in r will lead to a lower slope. Thus, the boundary

line starts with a higher slope and ends with a lower slope for high values of r.

For r = R:

∝=𝑘𝐻𝑅𝑁𝛾𝑤 + 𝛾𝑇𝛾𝑎𝑟 − 𝑅𝛾𝑅

(4.6)

For a very large r, such as at the city boundary, ∝will be:

∝= 1 −𝛾𝑅𝛾𝑎

(4.7)

𝜕𝜃

𝜕𝑟< 0 is very unlikely since headway should be very high and 𝛾𝑤or 𝛾𝑇 should be

unusually high compared with 𝛾𝑎 and 𝛾𝑟.

The observation about the high range of r is worth some discussion. The radial line

angular spacing has a high impact on the shed area of ring line use. In case the angular

59

spacing of radial lines is lower than 2(1 −𝛾𝑅

𝛾𝑎), it will result in the boundary lines of the two

radial lines inefficiently overlapping, thus, giving a lower area for ring line use.

2. Case (ii) r < R

For r < R, three options exist:

Alternative 1: direct access to the CBD, the cost of which per passenger will be unit

access cost times the radius of the zone considered:

𝑟𝛾𝑎 (4.8)

Alternative 2: access via the radial line; passengers take the radial line to the CBD,

which will involve the costs of circumferential access to the radial line plus ride on the radial

line and the wait experienced to take the radial line:

𝛾𝑎𝑟𝜃 + 𝑘𝐻𝑅𝐿𝛾𝑤 + 𝛾𝑟𝑟 (4.9)

Alternative 3: access via the ring line; passengers take the ring line and transfer to the

radial line; then take the radial line to the CBD, which will amount to the costs of the travel

distance to access the ring line; ride on the ring line and radial line; wait to take the train on

both ring and radial lines; and the transfer penalty to switch from one line to another:

(𝑅 − 𝑟)𝛾𝑎 + 𝑅𝜃𝛾𝑅 + 𝑘𝐻𝑅𝑁𝛾𝑤 + 𝑅𝛾𝑅 + 𝑘𝐻𝑅𝐿𝛾𝑤 + 𝛾𝑇 (4.10)

Since there are three alternatives, we will have three boundaries, i.e., one between

each pair of alternatives. For direct access and radial line alternatives (alternative 1 and 2), a

radial line will be used if: 𝜃 < 1 −𝛾𝑟

𝛾𝑎−𝑘𝐻𝑅𝐿𝛾𝑤

𝑟𝛾𝑎

60

The derivative of θ with respect to r is a positive value as given in equation above,

and it is always a positive value:

𝜕𝜃

𝜕𝑟=𝑘𝛾𝑤𝐻𝑅𝐿𝛾𝑎𝑟2

> 0 (4.11)

The derivative of θ with respect to r shows that at very low r, i.e., close to 0, we will

have an infinite slope; with an increase in r, the slope will be reduced substantially at a very

high rate.

For alternative 2 and 3, two situations exists:

𝑅𝑎𝑑𝑖𝑎𝑙 𝑙𝑖𝑛𝑒 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑢𝑠𝑒𝑑 𝑤ℎ𝑒𝑛:

{

𝜃 >

(1 − 𝑟/𝑅)(1 + 𝛾𝑎/𝛾𝑟)𝛾𝑎𝛾𝑅⁄𝑟𝑅⁄ − 1

+𝑘𝐻𝑅𝑁𝛾𝑤 + 𝛾𝑇𝛾𝑎𝑟 − 𝑅𝛾𝑟

𝑖𝑓 𝑟/𝑅 < 𝛾𝑟/𝛾𝑎

𝜃 <(1 − 𝑟/𝑅)(1 + 𝛾𝑎/𝛾𝑟)

𝛾𝑎𝛾𝑅⁄𝑟𝑅⁄ − 1

+𝑘𝐻𝑅𝑁𝛾𝑤 + 𝛾𝑇𝛾𝑎𝑟 − 𝑅𝛾𝑟

𝑖𝑓 𝑟

𝑅> 𝛾𝑟/𝛾𝑎

(4.12)

The first case only occurs with very low values of r. However, in such a case, a direct

access alternative will be more appealing. Thus, we would only consider it when 𝑟

𝑅>

𝛾𝑟

𝛾𝑎 in

the analysis. The derivative for this inequality is a negative value:

𝜕𝜃

𝜕𝑟=𝑅(𝛾𝑅

2 − 𝛾𝑎2) − 𝑘𝛾𝑎𝛾𝑤𝐻𝑅𝑖𝑛𝑔 − 𝛾𝑇𝛾𝑎

(𝛾𝑎𝑟 − 𝛾𝑅𝑅)2< 0 (4.13)

The slope of the line is decreasing with a decrease in r. At r = R, similar to the r > R

scenario, then:

∝=𝑘𝐻𝑅𝑁𝛾𝑤 + 𝛾𝑇𝛾𝑎𝑟 − 𝑅𝛾𝑅

(4.14)

61

As mentioned previously, with a lower value of r, alternative 1 will have a lower cost.

Therefore, although the boundary line of alternative 2 and 3 will have a higher angle with

decrease in r, at some point it will be cut off by the alternative 1 boundary with higher r.

For alternative 1 and 3, direct access or ring line, the ring line will be used if:

𝜃 <2𝑟𝛾𝑎𝑅𝛾𝑅

−𝑘𝐻𝑅𝑁𝛾𝑤𝑅𝛾𝑅

−𝑘𝐻𝑅𝑁𝛾𝑤𝑅𝛾𝑅

−𝛾𝑇𝑅𝛾𝑅

−𝛾𝑎𝛾𝑅− 1 (4.15)

The slope will be a positive value, and it is constant:

𝜕𝜃

𝜕𝑟=2𝛾𝑎𝑅𝛾𝑅

> 0 (4.16)

Thus, the boundary line for 1 and 3 will not be a curve. The boundary line will start

from a point between the CBD and R with zero angle when θ = 0, and will gradually increase.

The point where the angle of the boundary line of 2 and 3 is crossing the radial line

(θ = 0) is given in equation (4.17), although alternative 2 will have the lower cost in this area:

𝛼 =𝑅

2+𝑘𝐻𝑅𝑁𝛾𝑤 + 𝑘𝐻𝑅𝐿𝛾𝑤 + 𝛾𝑇 + 𝑅𝛾𝑅

2𝛾𝑎 (4.17)

Figure 4.2 below shows the three boundaries of route alternatives (Ring, Radial,

Direct access) for r < R and r > R for a ring line between the CBD and city boundary, and

with two radial lines with spacing 2π/3. Each area with a certain color represents a particular

transit route alternative with the lowest passenger cost.

62

Figure 4.2 – Associated lowest passenger cost areas for different transit route choices for a city with

radial line spacing of 2π/3.

Area A shows the region from which passengers will use a radial line to travel to the CBD,

area B shows the region from which passengers will directly access the CBD, and area C

represents the region from which passengers first use the ring line to transfer to the radial

lines and then travel to the CBD. Figure 4.3 shows the three boundaries for r < R and r > R

for a ring line in the middle between the CBD and city edge, and with radial lines spacing of

π/2 and π/4. As shown in Figure 4.3, at a lower spacing of radial lines, we will have a smaller

area B and, thus, fewer passengers will take the ring line.

63

(a)

(b)

Figure 4.3 – Associated lowest passenger cost areas for different transit route choices for a city with

radial line spacing of (a) π/2 and (b) π/4.

64

Moreover, a similar behavior is observed for area C where fewer passengers will

directly travel to the CBD without any radial or ring line use. Thus, if trips are only made to

the CBD from the rest of the network, an increase in the number of radial lines is a better

solution compared with building a ring line. In addition, it is observed that the main area of

ring line use will be at the outer section of the ring line. Furthermore, trips originated from

the area inside the ring line would mostly either use the radial line or directly access the

CBD.

As seen in Figure 4.3, when the spacing between the radial lines decreases to less

than 2 (1 −𝛾𝑅

𝛾𝑎), the boundary lines between the radial and ring lines for r > R start crossing

each other, consequently, resulting in a smaller area of ring line use compared to the previous

cases.

The difference in route choices for different numbers of radial lines is also analyzed.

Mathematica and MATLAB are used to further understand the mathematical model. It is

clear that increasing the number of radial lines would result in a smaller number of

passengers taking the ring line for the case of all trips to the CBD. This is logical since ring

lines only serve as access to the radial lines in this case and, thus, they are not needed when

the access distance to the nearest radial line is short.

4.1.2 Route choice for any zone pair for a network with a single ring line

Let’s consider a route choice model for all OD pairs to find the optimal location of a

single ring line. This would help to implement the model for any city with radial lines and a

CBD in the center. Therefore, it is essential to find the passenger route choices for any zone

65

pair in the network and not only for trips destined to the CBD. Depending on the angles

between radial lines, radius of the ring line, and unit cost values, we can demonstrate the

route choice problem for any coordinate (r,θ) in a network or particular study area. The

preferred alternative of a given passenger would be the route with the minimum cost.

By identifying the coordinates of the radial line and the destination, the route choice is

identified by the minimum total passenger cost alternative for each origin. We later use this

model to estimate the total benefit to the passengers by introducing a ring line in a city.

While Vaughan (1984) and Chen et al. (2015) used a similar radial-ring transit network for

many to many travel demand for a bus network, it is important to highlight that a simplified

route choice model which is only distance based was used. Parameters such as wait cost and

access cost were not considered for route choice although these parameters were used for

optimization of line spacing. In addition, route options such as taking a circumferential rail

line only (route alternative 3.1 in the list of route alternatives below) are not allowed. It is

important to also consider route alternatives where passengers would directly take to reach

their destination (route Alternative 1) using access modes other than rail (walk, feeder bus,

etc). Other alternatives also include direct access to CBD or egress from CBD combined with

rail transit (route Alternatives 2.3, 3.3, and 3.5). Such alternative are not included in Vaughan

(1984) as the model is to optimize bus networks. For the same reason, capital cost was also

not considered.

66

All alternatives are based on an individual choosing an alternative with the minimum cost.

An all-or-nothing assignment for each route choice with minimum passenger cost are

considered. A deterministic route choice model is more suitable for this study. Nielsen (2000)

classified the error components of stochastic transit route choice for different modes of

transit. He found that for the modes light rail, rail, metro, and regional train, the error

component of waiting and in-vehicle time are categorized as either low or very low. Such a

lower level of error would result in a stochastic model being closer to deterministic models.

In addition, Lee and Vuchic 2005, Marín and García-Ródenas (2009), and others used

random utility models for route choices only (either between transit and personal vehicles or

among transit modes). A headway-based (Spiess and Florian, 1989) or scheduled-based

transit assignment (Tong and Wong, 1998) proposes an optimal cost strategy method to

determine the probability of the choice between different lines. However, the fundamentals

of these models are mainly based on different headways and the amount of wait time of the

alternatives and are, thus, more suitable for bus networks rather than high frequency, more

reliable rail transit. Overall, a random utility model for rail passenger route choice will only

add to the complexity of our analysis, but will not greatly improve its accuracy. In addition,

using the deterministic and, thus, a continuum mathematical model can provide useful

insights for long-term planning of major ring transit routes.

All of the following alternatives can be the minimum cost option for some OD pairs:

1. Direct access from a given origin to a given destination

2. Use only radial line

67

2.1 Access the nearest radial line, take the radial line, access destination (1

wait cost);

2.2 Access the nearest radial line, take the radial line, access destination, if

needed transfer to another radial line in the CBD, or direct access to the

destination (either 1 wait cost, or 2 wait costs and 1 transfer); and

2.3 Direct access to the CBD and then take the radial line, access the

destination (1 wait cost).

3. Use ring line (radial line may also be used)

3.1 Access the ring line, take the ring line, access destination (1 wait cost);

3.2 Access the ring line, take the ring line, transfer to the radial line, access

destination (2 wait costs and 1 transfer);

3.3 Access the ring line, transfer to the radial line, access destination, if

needed transfer to another radial line in the CBD, or direct access to

destination from CBD (either 2 wait costs and 1 transfer, or 3 wait costs

and 2 transfers);

3.4 Access the nearest radial line, take the radial line, transfer to the ring line,

access destination (2 wait costs and 1 transfer);

3.5 Access the nearest radial line, take the radial line, transfer to the ring line,

transfer to the radial line, access destination (3 wait costs and 2 transfers);

and

68

3.6 Direct access to the CBD, then take radial line, then transfer to the ring

line, take the ring line and access the destination (2 wait costs and, 1

transfer).

The general passenger cost equation for the above alternatives are:

Direct Access Alternative:

𝛾𝑎(𝐷) (4.18)

Radial Line Alternative:

𝛾𝑎𝐷𝑜 + 𝛾𝑎𝐷𝑑 + 𝜁𝑅𝐿𝑘𝐻𝑅𝐿𝛾𝑤 + (𝜁𝑅𝐿 − 1)𝛾𝑇 + 𝛾𝑅𝐷𝑅 (4.19)

Ring Line Alternative:

𝛾𝑎𝐷𝑜 + 𝛾𝑎𝐷𝑑 + 𝜁𝑅𝑁𝑘𝐻𝑅𝑁𝛾𝑤 + 𝜁𝑅𝐿𝑘𝐻𝑅𝐿𝛾𝑤 + (𝜁𝑅𝐿+𝜁𝑅𝐿 − 1)𝛾𝑇 + 𝛾𝑅𝐷𝑅 (4.20)

Where:

D is access distance (radial, circumferential, or both) travelled between origin to

destination;

𝐷𝑜 is distance (radial, circumferential, or both) between transit line and origin;

𝐷𝑑 is radial (radial, circumferential, or both) distance between transit line and destination;

𝐷𝑅 is ride (radial, circumferential, or both) distance travelled;

R is radius of the ring line;

𝛾𝑅 is ride cost per unit distance per passenger;

𝐻𝑅𝐿 is headway on the radial line;

𝐻𝑅𝑁 is headway on the ring line; and

69

𝛾𝑇 is transfer penalty that passengers will suffer in case they need to transfer from one rail

line to another in dollars per transfer per passenger.

𝛾𝑤 is wait cost per hour per passenger;

𝜁𝑅𝐿= 0, 1, or 2 in case transfer is needed on radial lines; and

𝜁𝑅𝑁= 1 or 0 in case transfer is needed on ring lines.

We can implement this model for the entire OD trip matrix of a city by using the centroidal

coordinates of all zones. The smaller the zones, the higher the accuracy of the estimates. It

should be noted that once the destination is not necessarily the CBD, the route choices of

passengers will exhibit a different behavior. For illustration, Figure 4.4 is shown with 4 radial

lines, all π/2 away from each other; thus, the study area for radial line number 1 would be

from -π/4 to π/4. If the destination is at (r = 5, θ = π/8), the route choice of passengers is

shown at different locations on the plane. Each letter is associated with different transit route

choices.

70

Figure 4.4 –Transit route choice for a network with destination shown in black: A – area with radial line

route choice; and B – area with ring line route choice; C – area with direct access to destination.

71

This model will be used in the next section to identify the route choices and to obtain

passenger costs, in order to complete a cost-benefit analysis and optimization of ring line

radius. It is important to identify the anticipated passenger route choices once a ring line is

introduced to determine the feasibility of implementation of such a ring line. Thus, it is

essential to examine the route choices of passengers with and without a ring line. In addition,

the absolute difference between passengers’ cost for ring and no ring scenario also impacts

the magnitude of total passenger benefit arises through the ring line. In the next section, we

will discuss the feasibility of introducing a ring line for a network considering the passenger

route choice model.

4.2 Single Ring Line

This section presents feasibility analysis of whether passenger cost saving justifies

construction and operation of a ring line. Also the optimal radius of the ring line that

minimizes total passenger and operator’s cost can be measured.

4.2.1 Feasibility and Optimal Radius

In the case of all trips being to the CBD, the ring line is only used as a faster access

path to the radial lines. Following costs are considered to calculate total passenger cost for

each alternative; the one with lower passenger cost will be the one chosen. It is intended to

find the boundary area for the different alternatives for passengers based on their origin

location (r,θ), assuming values of unit costs: access cost, ride cost (𝛾𝑅), headway

72

The passenger route choice model introduced in section 4.2 is used to calculate passenger

costs for different route options for each OD pair. Thus, it will calculate the minimum cost

option for different available routes, and then use the passenger cost of the path with the

minimum cost for each OD for alternatives with a ring line, or no ring line, in place.

For the input data of this model, we need a transit OD matrix which shows the demand for

travel between each zone pair. We also require coordinates of each zones’ centroid. With this

input data, we can obtain a matrix which shows the minimum passenger cost option for each

OD pair. Multiplying these two matrices will give us the total passenger cost for the entire

network. The two scenarios of existing conditions and network with ring line can be

measured. Thus:

𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐶𝑜𝑠𝑡 𝑤𝑖𝑡ℎ 𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒(𝑅) =∑[𝑃𝐶(𝑅)𝑚𝑖𝑛𝑖,𝑗∗ 𝑂𝐷𝑖,𝑗]

𝑁

𝑖,𝑗

(4.21)

𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐶𝑜𝑠𝑡 𝑓𝑜𝑟 𝑅𝑎𝑑𝑖𝑎𝑙 𝑂𝑛𝑙𝑦 𝑁𝑒𝑡𝑤𝑜𝑟𝑘 =∑ [𝑃𝐶𝑚𝑖𝑛𝑖,𝑗 ∗ 𝑂𝐷𝑖,𝑗]

𝑁

𝑖,𝑗

(4.22)

where:

PC(R)mini,j

is the minimum passenger cost for zone i to zone j for a network with ring line

at radius R;

PCmini,j is the minimum passenger cost from zone i to zone j for the radial-only network;

and

ODi,j is transit demand from zone i to zone j.

73

𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐵𝑒𝑛𝑒𝑓𝑖𝑡(𝑅)

= 𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐶𝑜𝑠𝑡 𝑓𝑜𝑟 𝑅𝑎𝑑𝑖𝑎𝑙 𝑜𝑛𝑙𝑦 𝑁𝑒𝑡𝑤𝑜𝑟𝑘

− 𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐶𝑜𝑠𝑡 𝑤𝑖𝑡ℎ 𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒 (𝑅)

(4.23)

Then, by considering total operators’ cost, which contains capital and operating cost,

objective function for cost-benefit analysis can be built:

𝑂𝑝𝑒𝑟𝑎𝑡𝑜𝑟′𝑠 𝐶𝑜𝑠𝑡 (𝑅) = 2𝜋𝑅𝛾𝐿 + 4𝜋𝑅𝛾𝑂 (4.24)

The optimization problem is to minimize:

𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡 (𝑅) = 𝑂𝑝𝑒𝑟𝑎𝑡𝑜𝑟′𝑠 𝐶𝑜𝑠𝑡 (𝑅) − 𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐵𝑒𝑛𝑒𝑓𝑖𝑡(𝑅) (4.25)

A negative total cost would mean that building a ring line is beneficial. The radius with

minimum total cost would show the optimum location of the ring line.

4.2.2 Results

In order to illustrate the model before applying it to a real OD table, two hypothetical

demand patterns are first introduced, in which all trips are destined to the CBD, but with two

different demand distribution patterns: uniform demand and exponential demand.

The following parameter values were utilized with the same total demand. The

following function from Casetti (1967) and Berry et al. (1963) was used, in which demand

would decrease exponentially with increase in distance from the CBD:

𝑇𝑝(𝑟) = 𝑃0 ∗ 𝑒−𝑏∗𝑟 (4.26)

b = 0.3 was assumed in this analysis. The value of P0 is chosen to have an equal total demand

for uniform and exponential demand.

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Table 4.1 – Unit cost values for the City of Calgary

The results, as shown in Figure 4.5, demonstrate that a decreasing travel demand over r will

give a lower total cost value of a ring line compared with a uniform demand. This observation

is anticipated since we have lower population further away from the center and, thus, a ring

line is not expected to be feasible at such locations. It is also worthwhile to note that even

though the population density much closer to the CBD is higher, we can still see that the

optimum ring line is not expected to be feasible very close to the CBD; rather, it is further

away (8 km, in this example). This is because a ring line in the middle of a network could

potentially have better coverage for both passengers inside and outside of the ring line.

Parameters Unit Cost

Value of wait cost 𝛾𝑤 $5/hr/passenger

Capital cost 𝜆𝐿 $6000/km/per direction/day

Operating cost 𝛾𝑜 $2000/km/per direction/day

Passenger access cost 𝛾𝑎 $2km/passenger

Passenger ride cost 𝛾𝑅 $0.5/km/passenger

Transfer disutility 𝛾𝑡 $2/passenger/transfer

Wait time factor 𝑘 0.5

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Figure 4.5 – Total cost of the ring line for a network with uniform demand and exponential demand

distribution.

In the hypothetical example above, we only had a simplified OD, in which all passengers

travel to or from the CBD. In order to apply the model to a realistic network, land use OD

matrix of the City of Calgary for the current and the projected horizon of 2076 was obtained.

The information contains total trips and transit trips. In order to incorporate the zonal

information, ArcGIS software is used to calculate geometries, measurements, and selections.

The following steps are implemented:

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1) Obtain the center of each zone area featured to a single node;

2) Obtain coordinates of each zone, assuming that the CBD is the origin;

3) Transform all cartesian coordinates into a polar system;

4) Generate the cost-benefit equation for different values of R from i = 1 to C (edge

distance from the CBD):

o Build equations (4.18) to (4.20) to calculate route choice and passengers’ cost

for ring and radial-only networks;

o Find total cost value for R = i according to equation (4.25).

5) Find R = i with minimum total cost value.

The analysis for current and 2076 OD data was performed for the City of Calgary’s land use

zones with their current and projected radial light rail transit systems. Figure 4.6 shows the

City of Calgary’s approved light rail transit network and the range of the optimum ring line

radius. The figure also reflects the range of combined projected population and employment

densities with the darker zones being the higher density values.

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Figure 4.6 –City of Calgary long-term light rail transit network and the recommended ring line.

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Figure 4.7 shows total cost value of a ring line with respect to different radii of a ring

line centered on the CBD. The lower the value, the more efficient it is to build the ring line.

Values below zero would show that a ring line with such a radius could be potentially

feasible. Two different years have been considered: the current transit OD and the targeted

horizon of transit OD by 2076. With a larger population (2076 scenario) and with expected

changes in population and job distributions, a ring line will be more feasible with a lower

total cost value.

Figure 4.7 – Total cost of ring line for different scenarios based on distance away from the city center.

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For the current population, a radius of 6 km away from the CBD has the minimum

cost-benefit value; however, for 2076, the optimal location of the ring line will be shifted to

a radius 9 km away from the CBD with the same number of radial lines in the city (N = 4).

By adding future North Central and South East LRT lines by 2076, the benefit of a ring line

will be a lower value compared with the 2076 scenario with four radial lines. This is what

one would expect and there is agreement with the findings of the route choice model, which

indicate that increases in the number of radial lines would decrease the ridership and route

choices for taking the ring line. With an increase in population and changes in OD

distribution in the future, we observe a shift from the location of the optimal ring line from

6 km to 9 km away from the CBD. A ring line is not usually an exact circle and based on

street networks, major attraction centers, city constraints and other planning issues, the radius

for each part can be different. Based on the observation for the City of Calgary, a ring line

with a radius varying between 6 km and 9 km away from the city center is suggested.

Typically a long-term planning model for an urban rail network does not require a capacity

constraint as the fleet size is mainly a medium term matter. The capacity can be addressed

by increasing the fleet size, car size, passenger spaces per car and train size. The flow on

each line can be estimated using our model and checked against the capacity. For long term

projected peak demand, for a ring line headway range of 5 to 10 minutes and ring line radius

of 6 to 9 km we found total peak hour boardings to be between 42,000 to 49,000 for both

directions. Light rail transit systems have a capacity ranged between 18,000 to 24,000 pass/hr

and heavy rail transit systems carry up to 55,000 to 75,000 pass/hr (Vuchic 2007 and

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Thomson 1977). Taking a mean passenger travel length on the ring line (of ¼ of the ring),

the maximum average flow on the ring line for each direction will be between 10,500 to

12,250.

In addition, even in the case of transit flow reaching the capacity of the line, it is

unlikely that passengers will change their route due to the high volume or longer wait times

to board a train. Changing route choice due to a capacity constraint is more unlikely for the

case of rail transit systems with one ring line as there is usually no better available alternative.

4.2.3 Sensitivity Analysis

A sensitivity analysis was performed to find the changes in cost-benefit values and

optimal location of the ring line for the current data. As expected, an increase in the unit

value of access cost, assuming all other unit costs to be equal, would make a ring line more

feasible, as shown in Figure 4.8. This is because passengers would have much lower ride

cost compared with their access and, thus, a ring line would be more beneficial to substitute

access movements to radial lines or directly to the destination. We should also expect the

optimum location of the ring line to be shifted further away from the CBD so that more

passengers can be covered by the ring line against the high cost of access. Under this

scenario, passengers in the inner ring, as well as in the periphery of the ring line, can also

access the ring line to avoid the high cost of direct access to the destination. We can observe

the changes in the optimal location of the ring line for a higher access cost factor, which will

be increased from 6 km in the base scenario to 9 km with a higher access cost. It is observed

that with lower access cost values, the optimum ring line shows a flat minimum typically for

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a certain range of 6 km to 9 km. However, with high access cost, the optimum ring line

shows a sharp trend with the global minimum location at 9 km away from the CBD.

Figure 4.8 –Sensitivity analysis of ring line cost with respect to different access costs.

Changes with respect to ride cost would also show shifts in the location of the ring line.

Assuming that unit access cost is equal to $2/km, the closer the value of the ride cost is to

this number, the lower will be the feasibility and efficiency of a ring line as shown in

Figure 4.9.

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Figure 4.9 –Sensitivity analysis of ring line cost with respect to different access costs.

The value of ride time is an important factor in the optimal location of a ring line. If it is less

costly to travel by transit, passengers would tend to be only sensitive to their access to the

nearest transit line. Such a trend and decrease in the value of travel cost would also be

expected more with the continued proliferation of smart phones, tablets, and laptops; so that

a passenger can still use his or her time effectively while commuting to or from work. Ettema

and Verschuren (2007) have shown multitasking to have a significant impact on the value of

time and, as applied to this model, the ride cost. By decreasing the value of ride cost, the

minimum ring line alternative will also shift further away from the centre since such a line

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can better serve the city, in terms of coverage and decrease in access cost. However, if the

ride cost is increased to $2 per kilometer, a ring line will not be feasible as the ride cost is

equal to access cost and thus passengers will directly access their destination.

4.3 Double Ring Line

The model presented in this section is an extension to allow a second ring line

following the same assumptions. Shanghai has the largest metro system in the world with a

network length of more than 550 kilometers and is among cities with a complete ring transit

line. Shanghai metro is considered to compare the current alignment of its urban rail transit

network with the optimized model output, and recommend a potential second ring transit line

for the future. Also, current alignment of the Shanghai ring line is compared with the

optimized model output.

4.3.1 Feasibility and Optimal Radius

Depending on the type of planning and optimization, we can assume a network with

no ring line and add the operator’s cost of having one ring line (optimization for the first ring

line in Shanghai). The equations for total passenger benefit, total passenger cost, total

operator’s cost is similar to equations (4.21) to (4.25) in section 4.2.1 with only one

difference that total passenger benefit is the difference between two scenarios (one ring line

versus two ring lines or one ring line versus no ring line) as follows:

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𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐵𝑒𝑛𝑒𝑓𝑖𝑡(𝑅)

= 𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐶𝑜𝑠𝑡 𝑓𝑜𝑟 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑇𝑟𝑎𝑛𝑠𝑖𝑡 𝑁𝑒𝑡𝑤𝑜𝑟𝑘

− 𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑠𝑠𝑒𝑛𝑔𝑒𝑟 𝐶𝑜𝑠𝑡 𝑤𝑖𝑡ℎ 𝐸𝑥𝑡𝑒𝑛𝑑𝑒𝑑 𝑇𝑟𝑎𝑛𝑠𝑖𝑡 𝑁𝑒𝑡𝑤𝑜𝑟𝑘(𝑅)

(4.27)

Passenger route choice model will also have more alternatives compared with the case of one

ring line introduced in section 4.1.

Having a combination of many radial lines and two ring lines creates different route

choices. Passengers will choose an alternative route with a minimum total travel cost.

Different route alternatives in the ring-radial model are listed below:

1. Direct access from a given origin to a given destination without using the metro

network.

2. Use only a radial line:

2.1 Access the nearest radial line, take the radial line, access destination (1 wait cost);

2.2 Access the nearest radial line, take the radial line, access destination; if needed

transfer to another radial line in the CBD, or direct access to the destination (either 1 wait

cost, or 2 wait costs and 1 transfer); and

2.3 Direct access to the CBD and then take the radial line, access the destination (1 wait

cost).

3. Use one ring line (radial line may also be used).

3.1 Access the ring line, take the ring line, access destination (1 wait cost);

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3.2 Access the ring line, take the ring line, transfer to a radial line, access destination; if

needed transfer to another radial line in the CBD, or direct access to destination from CBD

(either 2 wait costs and 1 transfer, or 3 wait costs and 2 transfers);

3.3 Access the nearest radial line, take the radial line, transfer to a ring line, access

destination (2 wait costs and 1 transfer);

3.4 Access the nearest radial line, take the radial line, transfer to a ring line, transfer to

another radial line, access destination (3 wait costs and 2 transfers); and

3.5 Direct access to the CBD, take radial line, transfer to a ring line, take the ring line,

access destination (2 wait costs and, 1 transfer).

4. Use both ring lines (a radial line may also be used).

4.1 Access a ring line, take the ring line, transfer to a radial line, if needed transfer to

another radial line in the CBD, transfer to the other ring line, access destination (3 wait cost,

2 transfers or 4 wait costs, 3 transfers);

4.2 Access a ring line, take the ring line, transfer to a radial line, transfer to another ring

line, transfer to another radial line, access destination (4 wait costs, 3 transfers);

4.3 Access a radial line, take the radial line, transfer to a ring line, take the ring line,

transfer to another radial line, take the radial line, access destination (4 wait costs and 3

transfers);

Alternatives 4.1, 4.2, and 4.3 are unlikely to be the options with the minimum

passenger cost. The main changes to the total passenger cost for different route alternatives

occur for alternative 3, where for each OD pair, one of the ring lines can be in the route with

the minimum passenger cost.

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Similar to Section 4.1 deterministic passenger route choice is considered to calculate

passenger costs for different route options for each OD pair. The passenger route choice

model calculates the minimum cost option for each of the different available routes. The

model then uses the passenger cost of each route, with the minimum cost for each OD pair,

for alternatives with a ring line or no ring line in place. The general passenger cost equations

for the above alternatives are the same as equations (4.18) to (4.20) in section 4.2.1.

4.3.2 Results

To conduct a cost-benefit analysis and optimize the alignment of the ring lines in

Shanghai, one should use transportation or land use zonal OD data for an accurate and

realistic analysis. However, due to the limited ability to access such data for Shanghai, peak

period metro station OD data for a typical workday in 2013, and the 2013 network was used

for the analysis. Total OD demand is 3.6 million per day for 263 zones; some zones produce

as high as 60,000 and some as low as 1,000 trips. The average demand produced is 13,637

trips with a standard deviation of 11,686. Metro lines other than the ring lines are assumed

to be straight and are modeled as radial lines merging at the CBD, or are connected to another

radial line via a diametrical line. The actual location of the stations and the current metro

network is obtained in polar coordinates. Using this data and coding the radial and ring line

of the Shanghai network, an idealized network is assumed so that the network is in perfect

ring and radial lines. Due to the limitation of the perfect and complete ring radial network,

information for two lines are removed (one whole line and half of another line), as they could

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not be coded as radial nor as a full ring line on the developed perfect and complete ring-

radial model. The following questions are:

- Assuming that the current Shanghai metro had no ring line, what would be the best

alignment of a ring line in Shanghai? The result found from the model can be compared with

the current ring line in Shanghai.

- Assuming Shanghai’s current metro network with the ring line, what is the optimal

alignment of a second ring line? Will such a line be beneficial when compared to the cost of

construction and operation?

The costs of constructing a metro line in different cities in China ranges from 500 million

CNY (75 million USD) per kilometer in Shanghai (Smith 2015) up to 800 million CNY (120

million USD) and, in some cases, 1 billion CNY (150 million USD) per kilometer for one of

the newest metro lines in Xi’an (Dingding 2012). Since such an investment will be used for

many years, we should calculate the investment for the annual or daily benefits of having a

metro line. Shang and Zhang analyzed the operating cost of the Shanghai and Hangzhou

metro systems by considering different factors, such as wages, maintenance fees, business

cost, taxes, and surcharges (Shang and Zhang 2013). According to this study, the cost of the

Shanghai metro operation is approximately 5.7 billion Yuan per year, or 12 million CNY

(1.8 million USD) per kilometer per year. To estimate the average passenger access cost to

the metro stations, the composition of different access modes for the Shanghai metro was

determined. Pan et al. (2010) investigated the average mode share of passenger access to, or

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egress from, a metro station in Shanghai. Different modes were considered, and the average

percentages for access and egress modes, respectively, were estimated as follows: walk

(51%, 81%), bus (29%, 13%), bike (11%, 1%), motorcycle (6%, 1%), and car (3%, 4%).

Using the average speed of the different modes in Shanghai based on the fifth travel survey

conducted by the Shanghai City Comprehensive Transportation Planning Institute (Lu and

Gu 2011), we are able to calculate the average speed of access/egress and metro ride for

passengers. Value of time is also important for this analysis to estimate passenger ride and

wait costs. The average value of time in Chinese cities, as reported in the literature, varies

from 9 to 34 Yuan/hr (1.35 to 5.1 $/hr.) (Guan and Yang 2013). In a study conducted by

Guan (2015), a median value of time of 24.6 Yuan/hr. (3.7 $/hr.) was found for large-scale

residential areas in Shanghai. In another study, a value of time of 9.67 Yuan/hr. (1.5 $/hr.)

was determined for metro transit users in Tianjin (Jiang et al. 2009). Therefore, we can

employ a value of time of 20 Yuan/hr. (or 3 $/hr.) for this study, and also test different values

as part of the sensitivity analysis. A transfer disutility of 2.17 Yuan/passenger (0.33

$/passenger) is used for the Shanghai metro system based on the information provided in

(Guan 2015). Table 4.2 displays the unit costs used in this case study.

For the input data of this model, we required a transit OD matrix that showed the demand for

travel between each zone pair. We also required coordinates of each zone’s centroid. With

this input data, minimum passenger cost matrices for the current and extended network are

obtained. Table 1 shows the unit cost factors used for the model.

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Table 4.2 – Unit cost values for the City of Calgary

Figure 4.10 shows the cost-benefit analysis and the optimal alignment of ring lines in

Shanghai for three different scenarios. The horizontal axis is the distance from the CBD, and

the vertical axis represents the total cost value of a ring line (operating and capital cost minus

total passenger benefit, as per Equation 1). The section of the graph with a negative vertical

axis shows the radius of a ring line for the alternative where the benefit exceeds the cost of

construction. For Scenario 1, all lines on the existing network without the ring line were

considered and the ring-radial model was applied to compare the optimum location of a

single ring line with the alignment of the current ring line. This scenario only allows one ring

line in the network. We can observe that R=7 km, centered at the CBD, shows the minimum

cost-benefit value. The optimum radius of the ring line was found to be 1.7 km larger than

the current Shanghai ring line (Line 4), which has an average radius of 5.3 km. Although the

specific process of planning and selecting the current ring line in Shanghai has not been

published to date, it is known that the line is partially located on a previously constructed

Parameters Unit Cost

Value of wait cost 𝛾𝑤 20 CNY/hr./passenger*

Capital cost 𝜆𝐿 59.3 thousand CNY/km/day*

Operating cost 𝜆𝑜 32.8 thousand CNY/km/day*

Passenger access cost 𝛾𝑎 2.8 CNY/km/passenger*

Passenger ride cost 𝛾𝑅 0.56 CNY/km/passenger*

Transfer disutility 𝛾𝑡 2.17 CNY/transfer/passenger*

Wait time factor 𝑘 0.5

* 1 CNY is equivalent to ~0.15 USD

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track and the existing road network. However, the difference between the optimal ring radius

and the current radius can still be examined. The discrepancy could be attributed to the

existence of latent transit demand; Shanghai has experienced a recent unprecedented

economic and population boom and, thus, the network and transit demand in 2004, when the

first ring line was constructed, was very different from the current population and job

distribution. Also, envisioning a possible second ring line can potentially impact the optimal

alignment of the inner ring line, and shift it closer to the CBD, as in Scenario 3.

Scenario 2 is a cost-benefit analysis for the second ring line considering the existing ring

line. The result provides a range of 10 to 11 km from the CBD as the global minimum cost-

benefit value for the alignment. The construction of the proposed second ring line will make

the current ring line (Line 4) the inner ring and the proposed second ring line an outer ring.

We can also observe another local optimal value of 16 km from the CBD. The results of this

model were communicated to Shanghai City Comprehensive Transportation Planning, which

is the governmental body responsible for planning the future Shanghai metro network. They

expect the average radius of the second ring line in Shanghai to have a similar range of 10 to

12 km, although the plans are not yet finalized (Shanghai City Comprehensive

Transportation Planning Institute 2015).

Scenario 3 tests the impact of the presence of a first ring line of radius 11 km (similar to the

optimal alignment of the second line in Scenario 2) on the optimal alignment of a second

ring line. The ring-radial model should be able to suggest either an inner or another outer

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optimal ring as the case may be. The results revealed a radius of 6 km from the CBD as the

optimum location of the second ring line, a much closer location to the radius of the current

ring line (Line 4) in Shanghai.

Figure 4.10 – Total cost (Capital and operating cost minus total passenger benefit) of the ring line for

different scenarios based on the distance from the city centre.

Comparing Scenario 1 and 3, it is observed that the optimal inner ring shifts closer to the

CBD when an outer ring exists in the network. This finding suggests that plans for a second

ring line can impact the optimal location of an inner ring line. The alignment of the original

first ring line cannot be changed, but it is important to understand whether the future location

of a second ring line will negatively impact the ongoing operation of the first ring. Thus, this

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scenario helps to test whether the first ring line is still located at, or close to the best possible

alignment. Figure 4.11 shows the recommended second ring line range of 10 to 11 km on the

Shanghai metro network.

Figure 4.11 –Recommended range of a second ring line in Shanghai

Using cellular location data has recently become a practice for transportation and evacuation

planning (Oxendine and Waters 2014, and Oxendine et. al 2012). Since zonal population

data was not available for the analysis, locations of cell phone users on a typical workday in

Shanghai is obtained as an indicator of population and employment concentration. As shown

in Figure 4.12, the suggested second ring line range mainly covers high-density (darker

shadow) areas, except for the southeast section. The optimum range of the second ring line

is shown to pass through high-density areas, supporting the third characteristic of ring lines

introduced by Vuchic (2014).

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Figure 4.12 – Recommended second ring line and the population concentration of cell phone users on a

typical workday in Shanghai. Dark blue shading represents the highest concentration and light green

shading denotes the lowest concentration.

We can find zones that would obtain the greatest total benefit from introducing the second

ring line. Considering the OD flow in the analysis, we represent the nodes with the greatest

benefits by black dots in Figure 4.13. The benefits are calculated based on the demand for

each station and total passenger cost saving by introducing the second ring line. The results

demonstrate that the stations beyond the outer ring stand to gain the greatest benefit in terms

of total passenger cost savings. This observation is highly dependent on the OD patterns, and

anticipated changes to the route choices of passengers will be altered following the

introduction of the second ring line in Shanghai. This observation is consistent with the

expectation of an outer ring line, which will primarily benefit trip-end passengers located

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close to the ring line or passengers on an outer section of a radial line who transfer to another

radial line in the outer section. By introducing an outer ring line, these passengers bypass the

additional ride time toward the CBD to make their transfer.

Figure 4.13 –Stations with the highest total passenger cost saving after introducing the second ring line

at a radius of 11 km from the CBD.

4.3.3 Sensitivity Analysis

Sensitivity analysis can be performed to identify the changes in total cost values and

optimal alignment of the ring line for Scenario 2, as shown in Figure 4.14 and Figure 4.15.

Figure 4.14 presents the changes in the value of time and the result on the total cost plot. The

increase in the value of time will result in changing the global optimum to a previously local

optimum radius of 16 km. By decreasing the value of time, the ring line will not be very

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desirable, and the cost of operation and construction will exceed the total benefit of having

the ring line.

All reported data and analysis are based on existing metro OD observations. As such,

induced demand from other transportation modes or zones not well served by the current rail

transit network cannot be considered. A new rail line, especially in areas not served well by

high speed transit can induce new demand. However, the current model shows that, with the

same OD distribution, there are still situations where a new ring line would be beneficial.

The induced demand can make the ring line even more desirable.

Consequently, we can test different OD demand factors for the sensitivity analysis while

keeping trip distribution constant. Although this sensitivity analysis will not explicitly

address induced demand, analysing the sensitivity of the optimal ring line with respect to a

total demand factor is still worthwhile. 150%, 125%, and 110% increases in total demand

are tested with the same OD distribution and the decrease in total demand to 80% of the

baseline. Similar to previous cases, an increase in total demand makes the second ring line

more attractive, as shown in Figure 4.15. In addition, a 150% increase in total demand also

changes the global minimum radius of the second ring line to a 16 km radius, similar to the

case of a decrease in ride cost. Assuming a similar growth rate of the total metro OD demand

to the population growth rate, 1.67% per year, of Shanghai in 2014 (Cox 2015), we expect

the total metro demand to reach 125% by 2027, and 150% by 2038. Therefore, conducting

this sensitivity analysis will prove essential in determining the potential benefit of the new

ring line considering future demand and its potential effect on the optimum transit network.

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Figure 4.14 –Sensitivity analysis of the changes in the total cost of a second ring line for changes in Value

of Time

Figure 4.15 –Sensitivity analysis of the changes in the total cost of a second ring line for changes in

Demand

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4.4 Discussion

In this chapter, an approximate analytical model for ring radial rail network planning

was developed. Feasibility and optimal alignment of a ring line in a city was analyzed in this

chapter based on a route choice model that identifies circumstances under which passengers

would choose to take the ring line.

Factors, such as capital and operating cost, ride cost, OD patterns, and existing transit

network configuration are found to play an important role in the feasibility and the

circumference of a ring line. However, the most important factors are OD patterns and the

existing radial network configuration. As expected, the feasibility and optimal alignment of

a ring line has a direct relationship to line cost and operating cost. However, this study also

shows the potential net benefit of introducing a ring line by assessing anticipated reductions

in total passenger costs. Thus, a joint passenger route choice/cost model is developed to find

the route choices that passengers would make. This model is used in conjunction with a

transit OD trip table to estimate the value of total passenger costs for two different scenarios:

no ring, or ring line with a specific radius.

A decrease in unit ride cost, as reflected by the reduction of the value of travel time, made

the ring line more attractive and shifted it further away from the CBD. This is an important

factor regarding the possible decrease in value of travel time cost resulting from the riders'

ability to use their travel time more productively during commuting. Reduction in the value

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of disutility associated with in-vehicle travel time will have important implications for

shaping a transit network.

The OD travel patterns and the number of radial lines also impact the attractiveness of a ring

line. The more the number of radial lines above a certain threshold, the less attractive it will

be to have a ring line. This factor is also highly dependent on OD travel patterns. If most

passengers are travelling to or from the CBD, the ring line would only act as a feeder service

to the radial lines. However, for an OD trip table with more cross-town movements (i.e.,

corridors between satellite centers), a ring will likely be more attractive.

Depending on the demand and unit cost factors considered, the optimal radius of the ring line

could be a range or a single value. However, with the exception of underground lines,

building a rail line at the exact optimum radius may not be feasible due to right of way

constraints and trip generators at the local level. Thus, the recommended alignment could be

a range close to the optimal radius. As a result, the final estimate of the total generalized

passenger cost (or the total social cost) function might be slightly higher than the idealized

value obtained by the model.

The model was tested for the City of Calgary’s current and projected OD trip table, and

recommendations were made for the optimal range of radii for a single ring line, in terms of

distance from the CBD. Also, Shanghai metro network was considered to compare the

optimal location of first ring line in Shanghai using the model with the actual alignment of

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current Shanghai ring line. Also, the model was extended to optimize a possible second ring

line for the purpose of long-term planning in Shanghai.

Taken together, the results for all scenarios demonstrate that the location of one ring line will

impact the optimal location of the second ring line known as dynamic location problem.

Therefore, if an outer ring line is planned for construction, the optimal alignment of the inner

ring line may change. The alignment of the original inner ring line cannot be changed after

construction and, thus, it is important to ensure that while a second ring line may be planned

later, the original ring line is located at the best possible alignment and will not be negatively

impacted by the second ring line. This problem can be handled in two ways. First, it is

important to consider the possibility of a second ring line in the original long-term transit

network plan. Or, if such long-term planning has not occurred, the objective should be to

optimize the alignment of the new line and to maintain the effective operation of the inner

ring line.

Sensitivity analysis conducted showed that an increase in value of time and an

increase in demand will create a more desirable ring line, while shifting the optimal radius

of the ring line further away from the CBD. Additionally, by analyzing all of the attraction

and production nodes, it was found that by introducing an outer ring line, nodes located

outside of this ring will obtain the greatest benefit in terms of total passenger cost savings.

The developed model is capable of testing a perfect and complete ring radial rail network

given any OD demand pattern, coordinates of transportation zone centroids, and coordinates

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of radial lines. This model can be used by transit planners to test the potential benefits of

building a ring line in any monocentric city.

However, one of the main limitations of Shanghai case study is the use of the metro station

OD data as the input for the model. One of the primary purposes of a ring line (or any metro

line in general) is to effectively service new catchment areas not served well by the current

radial transit system. The use of transportation or land-use zonal OD data can address this

issue, model the passenger route choices, and provide detailed trip-end movements and thus

result in a more accurate cost-benefit analysis for different locations of the ring line. Since

only station OD data (not transportation zone OD data) for the Shanghai case study was

available, trip attraction and production nodes remain the same with or without a ring line.

Thus, the new ring line is only optimized to create faster routes for the same passengers with

the same station origin or destination. This limitation could have been resolved by using

transportation zone OD as the input.

The major limitation of the developed model is that only complete ring and radial lines could

be considered to allow an approximate analytical model. This limitation will greatly impact

practicality of the model since many networks include partial ring or radial lines. Chapter

five is an extension to a comprehensive Ring-radial rail transit network model addressing

this issue.

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CHAPTER 5: COMBINED UNIVERSAL RING-RADIAL RAIL TRANSIT

MODEL

One of the major limitations of the ring-radial model described in the previous

chapter is the lack of flexibility in simultaneously considering partial ring and radial lines.

In reality, transit networks have many partial ring and radial lines. Also, a single rail transit

line can consist of one or more partial ring and radial lines forming a complete line.

Incorporating partial lines would help to improve the applicability of the model to cities that

do not necessarily need or have complete ring or radial lines.

This chapter which considers more general ring-radial networks includes four

sections. Section 5.1 describes a universal framework capable of modelling and analysing

any general network that consists of a combination of (full or partial) ring and radial lines.

Section 5.2 illustrates specific model outputs and compares the combined universal ring-

radial rail transit network model with outcomes from similar studies in the literature.

Section 5.3 demonstrates the applicability of the model for the future LRT network of the

City of Calgary and compares selected possible future scenarios. This section also highlights

the applicability of the model to perform cost-benefit analysis to compare passenger travel

cost savings versus the cost of operation and construction of new rail transit lines. Finally,

Section 5.4 summarizes the proposed framework and its contribution.

5.1 Combined Universal Ring Radial Transit Modelling and Analysis Framework

The combined universal ring-radial transit model consist of six main steps to estimate

the cost of transit trips from each origin to each destination of the network, and total

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generalized transit passenger cost as explained below and summarized in Figure 5.1. Similar

to the previous chapter, polar coordinates are used to model (full or partial) ring and radial

transit lines. Similarly, all passenger movements are assumed to be made along a radius or

circumference line.

Figure 5.1 –Ring-radial rail transit network model algorithm

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Step 1 – Processing data inputs: The algorithm has three main input components:

1) Input Passenger unit cost factors (including unit access cost 𝛾𝑎, unit ride cost 𝛾𝑟, transfer

disutility 𝛾𝑡, wait time cost 𝛾𝑤) and network parameters (headway wait time factor k,

average headway for ring line 𝐻𝑅𝑁, average headway for radial line 𝐻𝑅𝐿).

2) Input transit trip origin-destination matrix and transportation zone centroid coordinates.

Coordinates should be defined in the polar coordinate system.

3) Input the rail transit network. The network is defined based on ring or radial line

categories:

- Radial Line: A complete radial line is defined with start terminus at r=0 and end

terminus is at r=C (the edge of the city). Any other value for start and ending terminus is a

partial radial line. Each radial line component has 4 features: radial line angle, terminus radii

(one radius for the start point and one radius for the end point), and a feature value which

represents continuous travel on the line. This parameter is defined for modeling the

connectivity of lines and we call it ‘continuity feature’ or ‘connection level’. For example,

as shown in Figure 5.2, ‘radial line a’ might be connected to ‘radial line b’ at the CBD

forming a diametrical (or traversal) line. By assuming same connection level value, the

model considers a direct seamless movement between the two lines at the CBD without any

cost of transferring and changing the line, while passengers transferring from ‘radial line a’

to ‘radial line c’ have different connection level (e.g. one line have connection level of 1 and

another one of 2) to reflect the disutility of transfer and wait time to board the train at ‘radial

line c’ at CBD. Also, if a line is a combination of a few partial radial and ring lines (‘line d’

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shown in Figure 5.2), the nodes at which the direction of the line changes between radial or

ring does not have any transfer cost when the continuity feature for these partial ring and

radial lines are the same. Radial and ring lines with their angle, radius and their continuity

feature can also be explained as a three-dimensional network where each line is featured with

a particular value of ‘connection level’. If at a particular radius and angle intersect at two

points that have the same connection level value, the two points are connected seamlessly

and no transfer cost is associated at this point. If they have different values, the two points

are connected with a virtual link that represents the total transfer cost that includes: transfer

disutility and wait cost to board the transfer train. This approach will ease the possibility of

including diametrical or traversal lines (two radial lines connected at CBD) or lines that are

a set of partial radial and ring line similar to ‘line d’ shown in Figure 5.2. Each rail transit

line will have a unique integer value greater than zero for its connection level. It is also

assumed that nodes connected by access/egress mode have connection level equal to zero.

- Ring Line: similarly, each ring line has four components: radius of the ring line,

terminus angles of the ring line which shows the circumference within which the ring line

exists; and the feature value representing the elevation of the (partial) ring line similar to

radial line. If the ring line is complete, there is no terminus point on the ring line (which

means ring line exist from angle zero to 2π).

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Figure 5.2 – Schematic view of ring and radial lines and continuity feature for crossing nodes.

Step 2 - Connect Projection Nodes to Origins and Destination Zone Centroids

(Access/Egress Cost Calculation):

Each transportation (or land use) zone is represented as a node in the center of the

zone with a radius and angle in polar coordinates. Each node in the transportation zones set

should be connected to other nodes to create a path. Similar to the previous section, we

assume passengers will choose the route with the minimum passenger cost. Thus, each node

will be connected to the rest of the nodes in either one of two possible ways: direct connection

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without any use of rail transit network; or using the rail transit network. The first option is

calculated as follows:

𝐷𝑖𝑟𝑒𝑐𝑡 𝐴𝑐𝑐𝑒𝑠𝑠 𝐶𝑜𝑠𝑡 = 𝛾𝑎min [𝑟𝑖 + 𝑟𝑗 , ∆𝜃 ∗ min(𝑟𝑖, 𝑟𝑗) + |𝑟𝑖 − 𝑟𝑗|] (5.1)

Where

∆𝜃 = min (|𝜃𝑖 − 𝜃𝑗|, 2𝜋 − |𝜃𝑖 − 𝜃𝑗|) (5.2)

For calculation of the latter scenario, each node should be connected to the nearby

ring and radial rail lines. Each node is bounded between a two consecutive radial lines (two

angles) and two ring lines (two radii). Thus, each node should be projected on the two

consecutive ring and radial lines to form access (or egress) route between the origin or

destination node and the rail network (as illustrated in Figure 5.3(a)). There are two

exceptions for projection of points on ring line: if the node is between CBD and the first ring

line, one of the projection point will be in the CBD (and thus not on a line) (Figure 5.3 (b)).

Also, for nodes located outside the outer ring line, there will be only 1 projection point on

the ring line (Figure 5.3 (c)). All projected points are also connected to each other considering

unit access cost (Similar to direct link between origin-destination nodes).

The cost of access between the projected point and each node is calculated as follows:

𝐴𝑐𝑐𝑒𝑠𝑠 𝐶𝑜𝑠𝑡 𝑡𝑜 𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒 = 𝛾𝑎(𝑟𝑖 − 𝑟𝑗) (5.3)

𝐴𝑐𝑐𝑒𝑠𝑠 𝐶𝑜𝑠𝑡 𝑡𝑜 𝑅𝑎𝑑𝑖𝑎𝑙 𝐿𝑖𝑛𝑒 = 𝛾𝑎∆𝜃𝑟𝑖 (5.4)

107

where

∆𝜃 = min (|𝜃𝑖 − 𝜃𝑗|, 2𝜋 − |𝜃𝑖 − 𝜃𝑗|) (5.5)

Figure 5.3 illustrates different possibilities for node projections or a particular origin

or destination node.

(a)

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(b)

(c)

Figure 5.3 – Different possibilities for zone centroid projections for zone centroid (a) between inner ring

line and CBD, (b) between two ring lines, (c) outside of outer ring line

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In addition to projection points, there are points on the network where radial and ring lines

cross each other. These points will have two different continuity feature values at the same

radius and angle coordinate.

Step 3 - Connecting Projection or Transfer Nodes (Transfer and Wait Cost

Calculation):

As described in Step 1, the links travelled on access/egress mode have connection

level value of zero. Thus, there is no transfer disutility considered for any access/egress

movement. Passengers who want to take any radial or ring line have to change their

‘connection level’ (defined as the continuity feature) to board the train. The change in

connection level is represented with transfer and wait cost factor. If the change of level is

from connection level equal to zero (continuity feature =0) to another level (continuity

feature value associated with a ring or radial line), passengers will incur wait cost to board

ring or radial train. If the change of connection level is from any other level to ground, no

passenger cost is considered since we assume passenger will directly egress to destination

(or next desired projection point). If change of level is between any connection level other

than 0, it means there is a transfer between two different lines. The passenger cost under this

scenario includes a transfer disutility factor associated with a transfer and the wait time cost

to board the train after the line transfer. The equation associated with each cost described is

as follows:

𝑅𝑎𝑑𝑖𝑎𝑙 𝐿𝑖𝑛𝑒 𝑊𝑎𝑖𝑡 𝐶𝑜𝑠𝑡 = 𝑘𝐻𝑅𝐿𝛾𝑤 (5.6)

110

𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒 𝑊𝑎𝑖𝑡 𝐶𝑜𝑠𝑡 = 𝑘𝐻𝑅𝑁𝛾𝑤 (5.7)

𝑇𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝐶𝑜𝑠𝑡 = 𝛾𝑡 (5.8)

The same logic of transfer possibility and passenger transfer and wait time cost

calculation is applied in connecting nodes as defined in step 2. If a ring or a radial line is

connected with the same continuity feature value, the algorithm assumes no cost for the link

connecting the two nodes. If continuity feature values are different, the two connected lines

are associated with transfer disutility and wait time cost.

Step 4: Connecting Nodes on Rail Lines (Ride Cost Calculation):

Step 2 creates a set of projected and connecting points on each ring or radial lines.

Obviously, no projection point is defined on partial ring or radial lines at coordinates where

the partial lines do not exist. Each set of projection points and transfer points on ring lines

are combined and sorted based on the angle. Similarly, each set of projection and transfer

points on radial lines are combined and sorted based on radius. Each two consecutive points

on ring and radial lines are connected with the cost equal to passenger ride time from the two

points as follows:

𝑅𝑖𝑑𝑒 𝐶𝑜𝑠𝑡 𝑜𝑛 𝑅𝑎𝑑𝑖𝑎𝑙 𝐿𝑖𝑛𝑒 = 𝛾𝑟|𝑟𝑖 − 𝑟𝑗| (5.9)

𝑅𝑖𝑑𝑒 𝐶𝑜𝑠𝑡 𝑜𝑛 𝑅𝑖𝑛𝑔 𝐿𝑖𝑛𝑒 = 𝛾𝑟𝑟𝑖∆𝜃 (5.10)

Where

∆𝜃 = min (|𝜃𝑖 − 𝜃𝑗|, 2𝜋 − |𝜃𝑖 − 𝜃𝑗|) (5.11)

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Steps 5 - Creating Node to Node Cost Matrix:

Passenger cost components created in steps 2 to 4 are combined to create a table of

all existing points that are connected and associated with a passenger’s cost to move from

one point to another. A passenger route (path cost) is a set of nodes that are associated with

a non-zero cost as defined in steps 2-4. If the travel cost value between two nodes is zero, it

means that the two nodes are not connected. The summation of the path costs in this set is

the passenger cost to move from the beginning point to the end point. A passenger route

option between an origin node to a destination node contains a set of connected nodes with

their associated passenger costs between each two OD pairs as extracted from the node to

node cost matrix.

Step 6 - Calculating Shortest Path for each OD (Transit Passenger OD Cost

Calculation):

This step is a path finder which incorporates node to node cost matrix created in step

5 and OD demand from each zone centroids given as the input. It finds all possible paths

from each origin to destination node and finds the path with minimum passenger cost using

Dijkstra’s shortest path algorithm (Dijkstra 1959). The output is a table representing

passenger cost from each origin to each destination. Multiplication of passenger cost table

and passenger OD demand table will provide the total passenger cost value based on the

examined transit network given a trip demand distribution. In addition, the algorithm

computes number of times each section of the ring and radial line is used by the passengers.

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5.2 Model Output

Figure 5.4 and Figure 5.5 are provided to better illustrate the output of the ring radial

model and its output on passenger cost performance for two different ring/radial networks.

Figure 5.4 is a radial only network with radial lines at north, north-west, west, south,

southeast, and northeast (similar to Calgary’s current approved light rail transit network).

Figure 5.5 is the same network with the addition of a full ring line. Height and color of each

area in the network represent generalized passenger cost produced or attracted to that

particular sector. It also reflects the total transit accessibility to/from each zone of the

network. As expected, the central area has the minimum cost (i.e. height) which means CBD

has the lowest passenger cost to access to any other area of the network and the lowest area

to be accessed from any other area of the network. Areas around each radial line also have

lower costs compared with other areas. Effect of the ring line can be seen in Figure 5.5.

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Figure 5.4 – Total passenger cost to/from each area of the network for a Radial network scenario

Figure 5.5 – Total passenger cost to/from each area of the network for a combined ring and radial

network scenario

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The output of this model is meant to be compared with similar works in the literature on ring-

radial networks namely Badia et al. (2014) and Chen et al. (2015). Unfortunately, none of

the papers reported total transit passenger cost or unit cost factors for their optimum transit

network to reproduce and compare their result with this model. Sensitivity analysis of

different transit modes by Badia et al. (2014) are used to compare the two model outcomes

for different modes of transit, Rapid Rail Transit (RRT), Light Rail Transit (LRT), and High

Performance Bus (HPB) using the similar model inputs (number of radial lines, number of

ring lines, location of outer ring line). Table 5.1 shows the comparison of the two model

outputs. It should be noted that the two models have differently defined user and agency

costs and thus the similar ratio of transit mode capital and operating cost and unit passenger

cost have to be considered for the ring-radial model. The optimum network for each transit

mode is reproduced as input to the Ring-radial model (shown in Figure 5.6) to calculate

‘Total user cost/total agency cost’ similar to Badia et al. (2014). Results show relatively close

ratios of user cost over agency cost although model construction and assumptions are

different. Similar comparison with Chen et. al (2015) was not possible since model output

and total passenger cost values were not reported.

Table 5.1 –Comparison of model output for Badia et al. (2014) and the Ring-radial transit network model

Results reported from Badia et al. on mode selection Ring Radial Model

Results

Optimum

number of radial lines

Optimum number of ring lines

alpha (R/C)

User Cost/Agency Cost

User Cost/Agency Cost

HPB 15 3.53 (4) 0.76 3.84 3.89

LRT 5.58 (6) 1 0.42 5.48 5.53

RRT 3.58 (4) 1 0.42 3.67 3.32

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Figure 5.6 – Optimal network for different transit mode technology in Badia et al. (2014) reproduced in

ring-radial model

5.3 Applications for Ring Radial Rail Transit Network Model

This model can be used for three different purposes: 1) to compare the performance of

different rail transit networks in terms of total passengers cost, 2) to conduct cost-benefit

analysis and compare total passenger cost savings versus the cost of extension in the network

and 3) to conduct sensitivity analysis and examine the impact of varying the OD patterns (i.e.

different land-use scenarios) and/or demand levels, cost parameters, etc. Also, it can be used

to compare different network extensions constrained with a fixed total length of extension.

In this section, two applications of the ring radial rail transit model are presented. Chapter 6

also presents another application of the model comparing six existing rail transit networks

and assessing performance of the 6 networks in terms of passenger cost using the ring-radial

rail transit network model.

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5.3.1 Future Rail Transit Network Scenario Comparison for the City of Calgary

One application of the ring radial transit model is to compare future possible

alternative scenarios of rail transit network based on analyzing potential improvement in

passenger cost savings for each scenario assuming the same total network length. Five

possible scenarios for Calgary’s future extension of its light rail transit network are

considered. All five scenarios shown in Figure 5.7, have the same total network length of

about 140 km (ranging between 137-143 km). A base scenario which is the current and

approved Light Rail Transit network for City of Calgary (Completed Red line, Blue line, and

Green line) is also included for observing the changes in total passenger cost savings with

the future proposed network extensions. The basic structure of the 5 alternative future

networks are kept constant: 6 radial lines associated with the city of Calgary’s Red, Blue and

future Green Line; another radial line between CBD and the east. Scenario 6 is a fully radial

network with no complete ring line. For scenarios 2-5 a complete ring line at radius of 6 Km;

a partial inner ring at radius of 2.5 Km; and a partial radial line connecting the partial ring to

the full ring line are considered. Future east radial line and northeast radial line have shared

track with the ring line. Scenario 2 has a partial ring line at radius of 9 km connecting north

and northeast radial lines with access to Calgary’s International Airport. Scenario 3 connects

the north and northwest radial lines with partial ring line at radius of 10 km. Similarly,

scenario 4 is connecting south and southeast radial lines with a partial ring line at radius of

10 km. Scenario 5 is similar to scenarios 2-4 in the case of full ring line but it has a partial

radial line between northwest and west radial lines. Figure 5.7 illustrates the 6 scenarios

considered.

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Figure 5.7 – Different transit network extension alternatives for City of Calgary from base

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Figure 5.8 – City of Calgary from base Calgary Future Scenarios with projected long term OD

Figure 5.8 shows total daily passenger cost performance comparing the six scenarios using

the ring radial model. City of Calgary’s 2076 land use OD data is used for the analysis. As

expected, Scenario 1 (approved light rail network) has the highest total passenger cost since

other scenarios are different extensions to this scenario. Scenario 6 (fully radial network) is

shown to have the lowest total passenger cost compare with all other scenarios. Scenarios 2-

5 have a similar range of total generalized passenger cost. Figure 5.9 shows a comparison of

these four scenarios. The difference between scenario 5 and scenario 4, the maximum and

minimum total passenger cost between the four alternatives, is only 20,000 dollars.

5.7

5.9

6.1

6.3

6.5

6.7

6.9

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6

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ion

s

Tota

l Gen

eral

ized

Pas

sen

ger

Co

st($

)

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Figure 5.9 – Comparison of four future alternatives for City of Calgary Light Rail transit network with

full ring line

Calgary’s OD pattern suggests that with the future expected OD scenario with similar trips

distribution which is mainly monocentric – to and from CBD -, more radial lines can still

outperform a ring line. However, possible changes in trip distribution with more activity

centers outside CBD can change the results. We can test uniformly random distribution and

exponential trip distribution to compare best network configuration among the 5 scenarios.

Figure 5.10 and Figure 5.11 shows uniformly random distributed trip OD and exponential

trip distribution respectively with the same total number of passenger trips as real OD data

(560,000 daily trips). Randomly distributed OD results in a different outcome with the real

OD distribution with scenario 5 (full ring line + a partial radial line) as the alternative with

the lowest passenger cost. Exponential trip distribution, however, shows a similar result to

the City of Calgary projected OD scenario. Scenario 6 (fully radial network) is shown to

6.55

6.56

6.56

6.57

6.57

6.58

6.58

6.59

Scenario 2 Scenario 3 Scenario 4 Scenario 5

Mill

ion

s

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have the lowest total passenger cost. This interesting observation shows that current

projected origin-destination trip distribution has a pattern similar to the exponential OD

demand distribution considered.

Figure 5.10 – Comparison of Calgary’s 6 alternatives assuming randomly distributed OD

Figure 5.11 – Comparison of Calgary’s 6 alternatives assuming exponential distributed OD

7.2

7.4

7.6

7.8

8.0

8.2

8.4

8.6

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6

Mill

ion

s

5.9

6.0

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6

Mill

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s

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5.3.2 Ring Radial Rail Transit Network Improvement Cost-Benefit Analysis

Another application of the ring radial transit model is to test potential benefit of

network improvement. Moscow expects its second ring line to be completed by 2018 (Panin,

2014). Thus, an idealized Moscow ring-radial network is used to test network extension and

analyze the improvement in total passenger cost. The assumed network improvement aims

to create a second full ring line by connecting the three partial outer ring lines at R=10.4 km

as shown in Figure 5.12. Uniform, exponential and uniformly random demand distributions

are considered for the analysis.

Figure 5.12 –Transit network improvement for Moscow – adding a second ring line

Different range of passenger unit cost provides a relatively equal difference in total passenger

cost between current and improved network for uniform and random demand distribution.

However, the discrepancy of current and extended networks for exponential transit demand

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is lower compared with the two other demand distribution. Since radius of the second ring

line in Moscow’s improved network is at R=10.4, the line will have less riders for the case

of exponential demand distribution and thus savings in total passenger cost are not as

significant. Sensitivity analysis illustrated in Table 5.2 for different transfer disutility shows

a similar value of total passenger cost between current and improved network for a random

demand distribution, a range between 1.71 million to 1.76 million dollars as the expected

passenger cost savings by extending the second ring line in Moscow.

Table 5.2 –Sensitivity analysis of total passenger cost with respect to transfer penalty

Demand Distribution

gamma t = 0.25 gamma t = 1 gamma t = 0.1

Random

Current network 38,845,031 40,574,689 38,255,164

Improved network 37,096,628 38,863,765 36,490,841

Difference 1,748,403 1,710,924 1,764,323

Uniform

Current network 38,987,136 40,747,208 38,390,873

Improved network 37,227,307 39,028,624 36,628,494

Difference 1,759,829 1,718,584 1,762,379

Exponential

Current network 27,455,589 28,479,960 26,963,585

Improved network 27,107,212 28,254,234 26,587,366

Difference 348,377 225,727 376,219

Similar to Chapter 4, cost-benefit analysis is conducted to find whether passenger cost

savings for the case of network extension can recover the high cost of construction and

operation of the line. Similar to the previous analysis, line capital cost and operating cost is

assumed to be a function of total network length.

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Figure 5.13 – Moscow’s network improvement cost benefit analysis

Figure 5.13 illustrates different unit value capital and operation cost of the new ring line

(blue line) and the difference between total passenger cost before and after network

extension, the value of 1.7 million dollars discussed earlier (red line). The analysis shows

that total passenger cost saving is desirable for a range of capital and operating cost of up to

57,000 $ per kilometers assuming a random or uniform demand distribution.

5.4 Discussion

This chapter presented different applications and capabilities of the ring-radial transit

network model developed. It is used to measure total transit passenger cost and it can be used

to compare different transit network alternatives in a fairly simple and quick set-up and

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Capital and Opeation Cost in Netowrk Improvement Passenger Cost Saving

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computation process where the end-user will only have to give the transit network

alternative, transit demand distribution, and passenger unit cost factors. The ring-radial

model can then compute the next steps by first finding access and egress movement of zone

centroids to nearby rail transit lines (projection nodes on the rail network) and computing

access and egress transit cost. The model computes passenger wait time to board trains on

rail network using an average headway for each line. Ride costs on the rail network are also

computed. Transfer cost is also computed by connecting lines that are crossing each other.

All transit passenger costs calculated will form a matrix of the node to node cost.

Deterministic route choice is considered in this model assuming that passengers will choose

a route option that would minimize their passenger cost. The node to node cost matrix is used

as input to the Dijkstra shortest path algorithm to compute the transit path with lowest

passenger cost to travel between each OD pair. The OD distribution input is then used to

measure total passenger cost performance of the network. The model is transferable and can

analyze different rail transit networks and measure transit passenger cost between each

origin-destination pair and thus total passenger cost of each network. This model can be used

to compare the performance of different rail transit networks. It can also be used for cost-

benefit analysis to compare total passenger cost savings versus the cost of extension in the

network. Also, it can be used compare different network extensions constrained with a fixed

total length of extension.

There are a few limitations with the ring-radial model developed. The most important

one is that the network should be an idealized ring-radial network and all rail lines should be

coded in polar coordinates. This limitation makes it difficult (and in some cases unrealistic)

125

to code a given network that does not have a concentrated merging of rail transit lines in the

CBD area or due to geographical reason (such as Hong Kong or New York networks). At the

same time, this is an advantage since assuming a ring radial model will create opportunities

for each calculation and computation of projection nodes, access and egress cost, and ride

cost on the network. The user can simply give network coordinates (angle and length of radial

lines and radius and circumference length of ring lines) and the rest of process is computed

by the algorithm. Thus, set up to test different transit network is very straightforward and

quick. The route choice model also does not consider capacity constraint. The capacity

constraint can be included in the model but it will require an iterative process of running the

model and changing appropriate headways or updating passenger cost for the links with high

volume. Since the intention of this model is for long term planning, the capacity constraint

was not included in the model and the analysis.

Overall, this model is a simple way of estimating total passenger cost for any given

radio-centric network which provides a useful tool for transit planners to measure different

network extension alternatives. Different alternatives can be measured without performing

detailed analysis using conventional transportation network planning tools with parameters

such as identifying transit stations, adding auxiliary transit link between rail transit line and

zone centroids and calibrating the model.

While this model is based on mathematical models for transit passenger cost factors,

it uses a real trip OD table and can be used in practise for many transit networks. The model

is easily transferable and is a unique tool to compare different transit network topologies

assuming similar demand and total transit network length.

126

CHAPTER 6: RING-RADIAL RAIL TRANSIT NETWORK BENCHMARK

ANALYSIS

A benchmark analysis of cities with ring transit lines is conducted to identify prominent

types of lines in transit networks categorized as: full radial lines (connecting CBD to

periphery); full ring line (a complete circle at radius R from CBD); partial radial lines (either

starting at CBD and ending at a ring line; constrained between two ring lines; or connecting

an outer ring line to the periphery); and partial ring line (constrained between two radial

lines; or only connected to one radial line). The following 6 mature rail transit networks that

are older than 50 years and have a ring transit lines are considered in the benchmark analysis:

London, Moscow, Berlin, Tokyo, Paris, and Madrid. Any actual line can be modeled into

perfect ring and radial lines by allowing partial ring and radial lines and use of the continuity

feature discussed in section 5-1. Idealization is especially relevant for networks with the

CBD at centre.

Idealizing networks into the perfect (partial or full) ring and radial lines is not an

uncommon practice and has been considered in the past by Thomson (1977) and for Berlin

back to 1931, Paris 1936, Moscow 1976, and London 2012 by Roberts 2013. Also more

recently, Dr. Max Roberts, a psychologist at the University of Essex, has focused on

designing metro maps for many networks such as London, Tokyo, Madrid, Paris, Chicago,

Berlin, Barcelona, and Moscow based on concentric circles and radials (The Guardian 2013).

The purpose of such maps, however, are mainly for better illustration of the lines in terms of

understanding the network by transit users.

127

We can find general network design patterns for all these networks in the idealized ring-

radial network. The patterns investigated are related to features of full ring, full radial, partial

ring and partial radial lines for each network. Several factors are considered to identify

network parameters:

- Number of full radial lines

- Average length of full radial lines: the parameter is estimated as the sum of the

length of all full radial lines divided by total number of full radial lines

- Number of partial radial lines: Partial radial lines are any lines that are not starting

at CBD and/or ending anywhere other than periphery of the city

- Relative radius of full ring line with respect to radial lines: the parameter is defined

as the ratio of the number of stations on the radial line connecting with the complete

ring line from the CBD over the total number of stations on the radial line. Each city

has a range (and not a single value) for this parameter since radial lines are not equal

in terms of length and number of stations (radius of full ring line are not constant

over Ɵ). The same measure can be obtained based on the radius of the ring line with

respect to the maximum length of complete radial line as well. However, since station

spacings tend to be higher at longer radii toward the edge of the city and even

sometimes to regional communities, the station number can provide a more realistic

ratio.

- Maximum circumference of partial ring line: the parameter is the maximum

observed length (in terms of radians) of partial ring line

128

- Average circumference of partial ring line: the parameter is the average observed

length of partial ring line. Two different values are found for each city: partial ring

lines located inner and outer of the full ring line.

- Number of partial ring lines: any circumference that is not complete (circumference

is less than 2π)

Table 6.1 summarizes the benchmark analysis for the six cities discussed. Total length of

networks are calculated based on the input values in this table. Total estimated idealized

network length is compared with the actual total length of the network for each city. The

comparison is shown on Figure 6.1 and Table 6.2.

129

Table 6.1 – Summary of the 6 rail transit idealized network benchmark analysis

London Madrid Paris Moscow Tokyo Berlin

Location of Full Ring Line 0.31-0.33 0.33-0.36 0.46-0.5 0.2-0.25 0.2-0.33 0.37

Number of Full Radial Lines 7 3 6 10 5 4

Average length of full radial lines (km) 15 14 12 18 12.8 22

Maximum degree of partial ring line () 150 160 180 180 120 120

inner outer inner outer inner outer inner outer inner outer inner outer

Average degree of partial ring line() 150 46 141.25 54 128.13 40 138.33 56.67 115 42.31 150 41.76

Average Radius of partial Ring Line (km) 2.42 5.08 2.42 4.58 2.88 3.12 2.03 6.98 1.71 4.69 4.07 6.93

Number of Partial Radial Lines 13 22 9 17 15 25 5 5 18 22 17 22

Average Length of Partial Radial Line (km) 2.42 5.08 2.42 4.58 2.88 3.12 2.03 6.98 1.71 4.69 4.07 6.93

Estimated Full Radial Line Length (km) 105 42 72 180 64 88

Estimated Full Ring Line Length (km) 30.43 30.36 36.24 25.44 21.44 51.19

Estimated Partial Radial Line Length (km) 31.48 111.72 21.75 77.92 43.27 77.88 10.13 34.88 30.72 103.25 69.26 152.37

Estimated Partial Ring Line Length (km) 38.04 103.56 29.79 71 32.25 18.61 14.67 76.33 17.13 41.9 42.66 76.92

Total Estimated Length (km) 420.24 272.82 280.26 341.44 278.45 480.41

130

Figure 6.1 – Comparison of network length for actual and idealized networks

Table 6.2 – Comparison of network length for actual and idealized networks

Actual Network Length

(km)

Estimated Network Length

(km) % Difference

London 402 420.24 5%

Madrid 286 272.82 -5%

Paris 308 280.26 -9%

Moscow 329 341.44 4%

Tokyo 304 278.45 -8%

Berlin 478.4 480.41 0%

Table 6.3 shows the ratio of each type of rail line defined using the information provided in

Table 6.1. The ratios are used to obtain an average pattern of each type of line category and

used to normalize each of the six rail networks in terms of total network length to compare

402.0

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272.8 280.3

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480.4

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100

200

300

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Londn Madrid Paris Moscow Tokyo Berlin

Actual Length of Rail Transit Network Total Estimated Network LengthN

etw

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Length

(km

)

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and evaluate. About 53% of Moscow’s rail line is a full radial while this ratio for other 5

cities is 21% on average. Consequently, Moscow has a much lower partial radial line ratio

compared to the other 5 networks (13% for Moscow versus 41% average for the other 5).

Paris has a relatively low ratio of partial outer ring lines compared to other 5 cities (7% versus

average 16% for other 5). Thus, overall each network is shown to have a unique pattern of

ring versus radial, complete versus partial lines.

Table 6.3 – Normalized network parameters for the idealized ring radial networks

London Madrid Paris Moscow Tokyo Berlin

Ratio of Full

Radial Line Length*

25% 15% 26% 53% 23% 18%

Ratio of Full Ring

Line Length* 7% 11% 13% 7% 8% 11%

inner outer inner outer inner outer inner outer inner outer inner outer

Ratio of Partial Radial Line

Length*

7% 27% 8% 29% 15% 28% 3% 10% 11% 37% 14% 32%

Ratio of Partial

Ring Line Length*

9% 25% 11% 26% 12% 7% 4% 22% 6% 15% 9% 16%

Table 6.4 shows the average and standard deviation of the ratios for each line category. The

standard deviation for full radial line is relatively high since Moscow has a high number of

full radial lines compared with other networks. Same reason can be used to explain the

relatively high standard deviation for the ratio of partial radial line.

* Ratios provided are out of total network length

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Table 6.4 – Average and standard deviation of the network parameters for the benchmarked networks

Average Standard Deviation

Ratio of full radial line length 27% 13%

Ratio of full ring line length 10% 2%

Inner Outer Inner Outer

Ratio of partial radial line length 10% 27% 5% 9%

Ratio of partial ring line length 8% 18% 3% 7%

Normalized ratios for the network parameters are used to compare the performance of each

network in terms of passenger cost using the developed ring-radial model. An equal total

network length of 300 km and the same OD distribution is used to compare the 6 networks.

Different scenarios in terms of various OD patterns and passenger unit cost parameters are

considered.

Table 6.3 are used to reproduce the same network pattern for each city but with a

consistent total network length and city size. Figure 6.2 shows the network pattern with equal

total length of 300 kilometers for all the six cities considered.

Passenger cost performance is computed using the Ring-radial modelling/analysis

framework that was introduced in Chapter 5for each network. It is assumed as shown by

Wirasinghe and Vandebona, 1999 that the capital and operating cost of the networks are a

function of the length of the network. Using the same passenger unit cost parameters (value

of ride cost, access cost, wait cost, and transfer penalty) and the same daily trip demand

distribution, the performance of the six networks is compared with the demand loaded to the

different network patterns as described in Figure 6.2.

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Figure 6.2 –Normalized benchmarked networks used in the ring-radial rail transit network model

134

Various OD demand distribution patterns are considered with the same total demand:

Uniform demand distribution, exponential demand function with respect to the distance from

CBD, and uniformly random generated OD demand. The OD table considered was a

224*224 dimension assuming projected nodes distributed at the equal radius and angle

distances. Figure 6.3 and Figure 6.4 show uniform and exponential demand distribution and

the passenger cost performance of each city. While the two different OD pattern shows a

different total passenger cost for each city, the trend for three lowest networks in terms of

total passenger cost is the same for both scenarios.

In both scenarios, Moscow has the lowest total passenger cost with Tokyo placing after

Moscow for the lowest total passenger cost. The networks that correspond to the highest total

passenger cost change for the two OD scenarios: Paris has the highest passenger cost under

a uniform OD distribution, with Madrid for exponential demand distribution. The total daily

passenger cost for uniform demand distribution ranges between 35.5 million (Moscow) to

36.4 million (Paris) passenger dollar for a total of 4.2 million daily rail transit trips.

Exponential demand distribution with the same total daily rail transit trips total ranges

between 25.1 million (Moscow) to 27.6 million (Madrid) passenger dollars. Changing the

OD pattern from uniform distribution to exponential distribution considerably decreases

daily total passenger cost for all cities. This decrease is explained by the fact that the number

of shorter trips is higher in the case of exponential demand patterns (i.e. higher trip density

near CBD) and the rail network is denser near CBD.

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Table 6.5 – Total transit passenger cost output for uniform and exponential demand distribution

Demand Distribution

Uniform Exponential

Madrid 36,238,530 27,567,269

Paris 36,395,317 27,057,404

London 35,925,949 26,272,852

Moscow 35,485,153 25,111,503

Tokyo 35,676,997 25,444,438

Berlin 36,175,519 26,469,507

Figure 6.3 –Total Transit Passenger cost output for uniform demand distribution

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Figure 6.4 –Total transit Passenger cost output for exponential demand distribution exponential demand

distribution

A uniformly random OD demand distribution constrained with the same total number of

daily transit trips (4.2 m) is also tested. Similar demand distribution created randomly is fed

to each network to calculate total passenger cost. Each set of random OD showed a similar

trend as the uniform demand distribution scenario as shown in Figure 6.5 while average total

passenger cost for each city varies with the uniform distribution scenario. Figure 6.5 and

Table 6.6 shows set of 10 random OD demand distribution and average of the 10 runs.

Table 6.6 – Total transit passenger cost output for uniformly random distributed OD pattern

Uniformly Random Distribution

Average Standard Deviation

Madrid 36,116,089 138,080

Paris 36,269,807 134,078

London 35,809,762 121,916

Moscow 35,365,742 144,015

Tokyo 35,545,721 142,864

Berlin 36,057,133 153,255

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Figure 6.5 – Total transit passenger cost output for randomly distributed OD pattern – Black line

showing the average, other colors showing each of the 10 runs

While some runs have lower total generalized passenger cost, others have higher total

generalized passenger costs with respect to uniform demand OD. The order of magnitude of

the networks with different runs does not change in random demand distribution scenario.

Passenger unit cost parameters are shown to impact the performance in terms of total

passenger costs of the ring-radial transit networks considered. Figure 6.6, Figure 6.7, and

Figure 6.8 illustrate the sensitivity analysis of total passenger cost performance with respect

to changes to transfer disutility (for uniform, random and exponential distributions

respectively). 4 scenarios are considered (𝛾𝑡=0.1, 0.25, 1, and 2) while keeping all other

parameters constant.

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Figure 6.6 – Sensitivity Analysis for different transfer disutility values for uniform demand distribution

Figure 6.7 –Sensitivity analysis for different transfer disutility values for random demand distribution

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GammaT= 0.1 GammaT= 0.25 GammaT= 1 GammaT= 2

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Madrid Paris London Moscow Tokyo Berlin

Mill

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GammaT= 0.1 GammaT= 0.25 GammaT= 1 GammaT= 2

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Figure 6.8 –Sensitivity analysis for different transfer disutility values for exponential demand

distribution

There are interesting observations with the sensitivity analysis: For 𝛾𝑡=0.25 performance of

London and specially Moscow and Tokyo are very close; with only 32,000 passenger dollar

difference for uniform distribution. For random demand distribution, the network with

minimum total passenger cost alternates between Tokyo and Moscow for different runs.

Figure 6.9 illustrates two separate runs for random demand distribution where on one run

Moscow and on the other run Tokyo has the lowest total passenger costs among all six

networks.

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Madrid Paris London Moscow Tokyo Berlin

Mill

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s GammaT= 0.1 GammaT= 0.25 GammaT= 1 GammaT= 2

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Figure 6.9 –Example of two runs with randomly distributed travel demand. Moscow lowest total

passenger cost for Run 1 and Tokyo lowest for Run 2

Reducing transfer disutility makes London network more efficient in terms of passenger cost

performance. For Gamma T=0.1, London is shown to have the minimum passenger cost for

uniform and random demand distribution as shown in Figure 6.6 and Figure 6.7.

Changes in transfer disutility, however, does not change the pattern of passenger cost

performance of the networks for exponential demand distribution. For all scenarios, Moscow

has the lowest total passenger costs with Tokyo ranked second and Madrid ranked as the

highest total passenger cost.

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Madrid Paris Partial London Moscow Tokyo Berlin

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Comparing the 6 networks we can analyze the observations on passenger cost performance.

Moscow and Tokyo have a very dense rail network inside their full ring line compared with

other networks. This reason would justify the observation for exponential demand

distribution where Tokyo and Moscow had the lowest total passenger cost in all sensitivity

analysis scenarios. Exponential demand distribution has higher trip demand closer to the

CBD where the network is dense and fully connected for Moscow and Tokyo. Madrid has

relatively uniform rail network in its entire area and thus it would perform the worst in

exponential demand distribution.

We can also observe Moscow, Tokyo, and in most cases, London have lower total passenger

cost compared with Madrid, Paris, and Berlin. Comparing radius of full ring line shows that

the first group (Moscow, Tokyo, and London) have a relatively lower radius (average radius

of 3.8 Km) compared with the second group (Madrid, Berlin, and Paris with an average

radius of 5.6 Km). Moscow and Tokyo both have a fully connected network inside their inner

full ring line while their partial ring and radial lines outside of the inner ring are relatively

covering the outer ring area uniformly. Madrid and Paris do not have a dense inner rail line

for the full ring line and their network is distributed relatively uniform over the entire area.

Berlin and London are in between these two spectrum with a denser rail network inside their

full ring line.

We can also analyze passenger cost performance using transit network parameters.

The resulting outcome was compared with the study of characterizing metro networks by

Derrible and Kennedy (2009). The study compares network values and indicators of 33

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transit networks. Table 6.7 summarizes network indicator and characteristics of the Berlin,

London, Tokyo, Moscow, and Paris in Derrbile and Kennedy (2009). Directness is defined

as ease of travel within a network so as to avoid unnecessary transfers. Structural

connectivity is defined as how the network is connected allowing more travel paths. The

degree of connectivity describes how much a network is connected relative to how much it

could be connected. Average line length indicator describes whether a network is regionally

or locally focused. The longer average length would imply lines are typically reach further

out in the suburbs and therefore representing a regionally oriented network.

Table 6.7 – State, form and Structure network parameters for the benchmarked networks (Derrible and

Kennedy 2009)

State Form Structure

Metro

Networks Complexity

Degree Of

Connectivity

Average Line Length

(Km)

Structural

Connectivity Directness

Berlin 1.41 0.5 16.86 1.11 3.00

Paris 1.78 0.61 15.09 1.2 4.67

Madrid 1.78 0.62 17.44 1.25 4.33

London 1.87 0.64 33.75 1.00 6.5

Moscow 1.6 0.56 23.54 1.25 6.00

Tokyo 1.95 0.67 22.49 1.29 6.5

Directness is the most relevant factor to the ring-radial network pattern used in this study.

Average line length is also relevant especially in the case of radial lines. Among the 6 cities,

London, Moscow, and Tokyo has the highest directness indicator values. This parameter

shows a consistency between passenger cost performance analysis conducted in this thesis

with the directness measure found in Derrible and Kennedy. In both studies, London,

Moscow, and Tokyo outperform Madrid, Paris and Berlin.

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A few of the more interesting observations discussed in this chapter are summarized in this

section. Using the ring radial model, transit network performance of London, Tokyo, Paris,

Moscow, Berlin, and Madrid was compared using the ring radial model. The network was

normalized in terms of total network length so that they all have the same total transit network

supply and the only different factor was network topology.

Comparing the 6 networks we can analyze the observations on passenger cost performance

of the networks. In most of the scenarios of unit passenger cost and travel demand

distribution, Moscow, Tokyo, and London showed to have lower total passenger costs

compared with the other three networks. The observation was consistent with the findings of

metro network characterization by Derrible and Kennedy (2009) where directness factor for

these three networks was significantly higher than Madrid, Berlin, and Paris. Comparing

radius of full ring line show that the first group (Moscow, Tokyo, and London) have a

relatively smaller radius (average radius of 3.8 Km) compared with the second group

(Madrid, Berlin and Paris with an average radius of 5.6 Km). Moscow and Tokyo both have

a fully connected network inside their inner full ring line while their full and partial ring and

radial lines outside of the inner ring are relatively covering the outer ring area uniformly.

Madrid and Paris do not have a dense inner rail line for the full ring line and their network is

distributed relatively uniform over the entire area. Berlin and London are in between these

two spectra with a denser rail network inside their full ring line. For exponential travel

distribution, Tokyo and Moscow were shown to have the lowest total passenger cost in all

sensitivity analysis scenarios. Both networks have a very dense rail network inside their full

144

ring line compared with other networks. Exponential demand distribution has higher trip

demand closer to the CBD where the network is dense and fully connected for Moscow and

Tokyo. Madrid has relatively uniform rail network in its entire area and thus it would perform

the worst in exponential demand distribution.

Although there are variations in the ranking of the 6 networks studied in terms of passenger

cost performance, changing trips distribution does not hugely change the rankings while it

changes the total transit passenger costs for different OD patterns. However, the observation

for Calgary, where the network is not dense and neither it is mature enough showed a

contradictory result: changes in OD pattern significantly changes the network extension

alternative with the lowest cost. It was found that for real OD demand distribution (provided

by the City) and exponential demand distribution a fully radial network with no complete

ring line will be the best alternative while using a randomly distributed OD pattern will make

alternatives with complete ring lines a more desirable option.

145

CHAPTER 7: SUMMARY AND CONCLUSIONS

This chapter presents the concluding comments of this thesis and suggests directions for

future research. Overall research summary and findings of this research is presented in

Section 7.1. Section 7.2 presents the contributions of this research to transit network

modelling literature. Finally, Section 7.3 discusses potential future research directions.

7.1 Research Summary and Contributions

7.1.1 Macro Scale Parameter Assessment of Rail Transit Networks

A review of global rail transit networks was performed with special emphasis on

cities with ring transit lines. The analysis found relationships between transit ridership and

different city parameters, such as population, city area, and population density. Moreover,

cities with rail ring transit lines, such as London, Moscow, Berlin, Beijing, Tokyo, and

Shanghai were reviewed. Population density and network length, and maturity of transit

system were found to be statistically significant parameters that warrant a ring line.

Therefore, a city should have a high population density spread throughout the city rather than

a densely populated single area to justify building a ring transit line. In addition, transit

networks should first possess some radial lines and a certain level of network maturity before

a ring transit line is introduced. Also, according to Thomson (1977), the desire to develop

distributed employment centres to reduce congestion in the CBD would also justify a ring

line.

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7.1.2 Radial Only Transit Network

An analytical model to find the optimal number of radial lines in a city for any

demand distribution was introduced. An analytical model with continuum approximation

approach is used to find the optimal number of radial lines considering a city with a radio-

centric street grid with the city center at the origin. The optimization is basically a trade-off

between access cost to radial line versus the cost of operation and construction of radial line

to find the optimal number of radial lines or optimal radial line spacing. As showed in the

case study section, the optimal number of radial lines may be a real number and not

necessarily an integer. However, in reality, an integer number should be applied in the

network. Thus, the optimum value should be rounded up or down based on the lower total

cost.

7.1.3 Optimization of Ring Line in a Full Ring-Radial Line Transit Network

An approximate analytical model for ring radial rail network planning was introduced

allowing analysis of the feasibility and optimal alignment of a single ring line in a city, based

on a route choice model that identifies circumstances under which passengers would choose

to take the ring line. Factors, such as capital and operating cost, ride cost, OD patterns, and

existing transit network configuration are found to play an important role in the feasibility

and the circumference of a ring line. However, the most important factors are OD patterns

and the existing radial network configuration. As expected, the feasibility and optimal

alignment of a ring line has a direct relationship to line cost and operating cost. However,

147

this study also shows the potential net benefit of introducing a ring line by assessing

anticipated reductions in total passenger costs. Thus, a joint passenger route choice

generalized cost based model is developed to identify the potential routes that passengers

would take. This model is used in conjunction with a transit OD trip table to estimate the

value of a generalized passenger costs for two different scenarios: no ring, or ring line with

a specific radius. Depending on the demand and unit cost factors considered, the optimal

radius of the ring line could be a range or a single value. However, with the exception of

underground lines, building a rail line at the exact optimum radius may not be feasible due

to the right of way constraints and trip generators at the local level. Thus, the recommended

alignment could be a range close to the optimal radius. As a result, the final estimate of the

total generalized passenger cost (or the total social cost) function might be slightly higher

than the idealized value obtained by the model.

7.1.4 Combined Universal Ring-radial Rail Transit Network Model

The previously developed Full Ring-radial transit network model was extended to

allow simultaneous consideration of radial and ring lines and analyzing a transit network

with partial ring and radial lines. This extension allows a more realistic idealization and

analysis of rail transit networks. The model computes generalized transit passenger cost

between transportation zone centroids and thus total transit passenger cost for the entire

network can be obtained. The ability of the developed model to estimate the total transit

network passenger costs is a key step to compare different possible future network extension

scenarios and to compare the performance of these scenarios in terms of social cost. In

148

addition, the developed model can be used as a decision support tool to conduct a sensitivity

analysis that examines the factors that contribute in shaping the future transit network.

7.1.5 Model Transit Network Benchmark Analysis Using Combined Universal Ring-radial

Rail Transit Network Model

Using the ring radial model, transit network performance of London, Tokyo, Paris,

Moscow, Berlin, and Madrid was compared. The networks were normalized in terms of total

network length so that they all have the same total transit network supply capacity with the

only different factor being the network topology. The performance of the examined

normalized networks was analysed in terms of total passenger transit cost for different

scenarios corresponding to different demand level and various OD patterns and for different

values of input parameters (e.g. unit riding cost, waiting and transfer costs, etc.). In the

majority of the examined scenarios, Moscow, Tokyo, and London were shown to have the

highest performance (i.e. lowest total passenger costs) compared with the other three

networks. One of the important findings of this analysis is that while varying the OD patterns

changes the total transit passenger cost values, rankings of the six networks remained the

same for most cases. This finding indicates that the total generalized passenger transit cost

is mainly a property of the transit network and is less dependent to demand distribution

patterns. Thus, in the examined dense networks, no matter what was the examined transit

demand, the network topology in terms of line spacing and directness seems to be the main

contributing factor resulting in a lower generalized passenger transit costs. This is explained

by the fact that a transit network that is more direct (as the result offering lower generalized

149

travel costs for each OD pair) and more connected (as the result of offering more route

choices to transit riders) will result in a better performance (i.e. lower total generalized

passenger cost). Thus, network topology is the most important factor in transit passenger

performance of dense networks while transit demand pattern is marginally affecting the

performance. However, the observation for Calgary, where the network is not dense nor

mature enough showed a contradictory result: changes in OD pattern significantly changes

the desirable network extension alternative corresponding to the lowest total generalized

transits passenger cost. It was found that for the cases of both: 1) a realistic OD demand

distribution (as projected by the City of Calgary) and 2) an exponential demand distribution,

a fully radial network with no complete ring line will be the best alternative. However, using

a randomly distributed OD leads to a different desirable alternative which includes a ring

line. Due to a very mono-centric demand pattern which would require less crosstown trips

as well as fewer route choices, a fully radial network would be more desirable as found in

this case study.

7.2 Research Contributions

This analysis was developed to propose a sound generic mathematical model with

consideration of a realistic transit OD trip table. Such an approach is shown to overcome

theoretical assumptions and limitations, e.g., the assumption of uniform demand distribution,

and a fixed function of travel demand, e.g. exponential, with respect to only distance. The

model developed is capable of testing a ring radial rail network given any OD demand

pattern, coordinates of transportation zone centroids, and coordinates of radial lines. Unlike

150

simulations and agent-based models, this model is shown to be easily transferable to many

ring-radial transit networks. Therefore, with a daily OD trip matrix and transit network

supply characteristics and parameters as input, the model can be implemented for any city.

The model is transferable and can analyze different rail transit networks and estimate

generalized transit passenger cost between each origin-destination pair and thus total

passenger cost of each network. This model can be used to compare the performance of

different rail transit networks. It can also be used for cost-benefit analysis to compare total

passenger cost savings versus the cost of network extension. In addition, it can be used to

compare different network extension scenarios. The benchmark analysis using the Ring-

radial rail transit network model is a useful yet mathematically sound platform to compare

different topologies of rail transit networks and propose the best examples of rail network

topologies. It also helps to show that for relatively dense transit networks, the OD travel

pattern will not significantly change performance ranking of rail transit networks considered.

Overall, this model is a simple yet effective way of estimating total passenger cost for any

given network which provides a tool for transit planners to measure different network

extension alternatives for long-term planning. Thus, different alternatives can be analysed

and compared without detailed analysis needing too many parameters and network coding

efforts such as identifying transit station locations, adding auxiliary transit link between rail

transit line and zone centroids. In addition, there is no need to conduct cumbersome efforts

for network model calibration.

151

While this model is based on mathematical models for transit passenger cost factors, it uses

a realistic trip OD table and can be practically used for any transit network. The model is

shown to be easily transferable and is a unique tool in comparing different transit network

design alternatives.

7.3 Future Work

There are few limitations with the ring-radial model developed. The most important

one is that the network should be an idealized ring-radial network and all rail line and

movements should be coded in polar coordinates. This limitation makes it difficult (and in

some cases unrealistic) to code networks that do not have a concentrated merging rail transit

line in the CBD area or due to other geographical reasons. However, assuming a ring radial

model will create opportunities for calculation and computation of projection nodes, access

and egress cost, and ride cost on the network. The analyst can simply provide the network

coordinates (angle and length of radial lines and radius and circumference length of ring

lines) and the rest of the process is completed by the developed combined universal ring-

radial model. Thus, set up time to test different transit network is very straightforward and

computationally fast.

Another limitation of this study is that the transit route choice model does not

consider the existence of capacity constraint. Since the intention of this model is for long

term planning, capacity constraint was not included in the model and the analysis. However,

capacity constraint can be included in the model but it will require an iterative process of

152

running the model and changing appropriate headways or updating passenger cost for the

links with high volume.

This study does not consider the possible presence of induced transit demand. The

developed model is a tool for examining the best network configuration, which can be called

a supply-side model, for a given fixed OD demand. Providing a better transit service in new

areas, and the resulting improved attractiveness and convenience for transit passengers can

increase transit use and can also impact the location decisions of individuals and firms. If a

transportation-land use model (or a demand model) of a network exists, the output of the

supply model can be utilized in integration with the demand model to reflect the likely

changes in OD patterns and total demand. Consequently, this new updated OD can be fed

again into the supply model to identify the new best network configuration alternative. The

iteration can continue until an equilibrium is reached. A key question in combining both the

supply and demand models would then lie in finding the best time horizon that needs to be

considered for the described iterative process. In other words, how long would it take for a

new transportation infrastructure to impact the locations decision of residential and

employment settlements?

The developed Ring-radial model provides many opportunities and a platform for more

research. One of the possible extensions is finding a relationship between the actual and

idealized network in terms of network parameters such as network connectivity, directness,

and complexity. Including more networks in the analysis, especially younger, yet large

networks such as Beijing and Shanghai can provide more insights on the patterns or new

trends in rail transit network evolution. Comparing younger and older systems in terms of

153

their performance and their network topography can be an interesting future research topic.

Another interesting research question would be to focus on a particular network and

comparing the estimated ridership of total network and individual lines with the existing

current ridership data.

154

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