Nuclear Waste Management and Road Networks Case Studies

175 Pages·2017·4.15 MB·English

Rule Derivation for Agent-Based Models of
Complex Systems: Nuclear Waste Management
and Road Networks Case Studies

By

Jorge Andrés García Hernández

A thesis
presented to the University of Waterloo
in fulfillment of the
thesis requirement for the degree of
Doctor of Philosophy
in
Systems Design Engineering

Waterloo, Ontario, Canada, 2018
ã Jorge Andrés García Hernández 2018
Examining Committee Membership
The following served on the Examining Committee for this thesis. The decision of the
Examining Committee is by majority vote.

External Examiner Ph.D. Ziad Kobti
Professor and Director School of Computer
Science, University of Windsor.

Supervisor(s) Ph.D. Kumaraswamy Ponnambalam
Professor, Systems Design Engineering,
University of Waterloo.

Internal Member Ph.D. Michele Bristow
Adjunct Assistant Professor, Systems
Design Engineering, University of Waterloo.

Internal Member Ph.D. Shi Cao
Assistant Professor, Systems Design
Engineering, University of Waterloo.

Internal-external Member Ph.D. Mark Crowley
Assistant Professor, Electrical and Computer
Engineering, University of Waterloo.

ii
Author’s declaration
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,
including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.

iii
Abstract
This thesis explores the relation between equation-based models (EBMs) and agent-based
models (ABMs), in particular, the derivation of agent rules from equations such that
agent collective behavior produces results that match or are close to those from EBMs.
This allows studying phenomena using both approaches and obtaining an understanding
of the aggregate behavior as well as the individual mechanisms that produce them. The
use of ABMs allows the inclusion of more realistic features that would not be possible (or
would be difficult to include) using EBMs.
The first part of the thesis studies the derivation of molecule displacement probabilities
from the diffusion equation using cellular automata. The derivation is extended to include
reaction and advection terms. This procedure is later applied to estimate lifetimes of
nuclear waste containers for various scenarios of interest and the inclusion of uncertainty.
The second part is concerned with the derivation of a Bayesian state algorithm that
consolidates collective real-time information about the state of a given system and
outputs a probability density function of state domain, from which the most probable
state can be computed at any given time. This estimation is provided to agents so that
they can choose the best option for them. The algorithm includes a diffusion or diffusion-
like term to account for the deterioration of information as time goes on. This algorithm
is applied to a couple of road networks where drivers, prior to selecting a route, have
access to current information about the traffic and are able to decide which path to
follow.
Both problems are complex due to heterogeneous components, nonlinearities, and
stochastic behavior; which make them difficult to describe using classical equation
models such as the diffusion equation or optimization models. The use of ABMs allowed
for the inclusion of such complex features in the study of their respective systems.

iv
Acknowledgements
I would like to express my sincere gratitude to my supervisor, Prof. Ponnu, for giving me
the opportunity to pursue a PhD is Systems Design Engineering, the knowledge shared,
the continuous support, and for setting some interesting problems from which I learned a
lot.
I would also like to express my gratitude to my Committee Members for all the time,
commitment, and insightful commentaries: Prof. Ziad Kobti, the External Examiner, from
the University of Windsor; Prof. Mark Crowley, the Internal-External Examiner, from the
Department of Electrical and Computer Engineering; and Prof. Michele Bristow and Prof.
Shi Cao, Internal Examiners, from the Department of Systems Design Engineering.
During my time at the University of Waterloo, I’ve met some amazing people. Christel &
Guido Weber have welcomed me countless times in their house for wine-tasting classes
and delicious dinners. Thank you for all the experiences shared and the warmth of your
company. I’d like to acknowledge my peers at the Design Optimization Under
Uncertainty Group for sharing thoughts, ideas, an occasionally a pint of beer: Shankai,
Mythreyi, Vimala, Ahmed and many others. Thanks to everyone who has received me
with the warmth of a smile.
The support of my family has been essential in the termination of this thesis; my wife
Den, who has been a source of continuous motivation; my father and my siblings George
and Mónica, who have always encouraged me to pursue my goals. Big motivation has
come from three schnauzers that were and have been a source of joy: Oliver, Jozen, and
Kyla.
This thesis has been possible thanks to the financial support of the following institutions:
Consejo Nacional de Ciencia y Tecnología (CONACYT) through the scholarship
CONACYT-Ciudad de Mexico; Beca SEP Complemento; my family; and the NSERC
CRD and the Nuclear Waste Management Organization.

v
Dedication

To my family.

vi
Table of contents
EXAMINING COMMITTEE MEMBERSHIP ..................................................................................................... II
AUTHOR’S DECLARATION ............................................................................................................................. III
ABSTRACT .......................................................................................................................................................... IV
ACKNOWLEDGEMENTS .................................................................................................................................... V
DEDICATION ...................................................................................................................................................... VI
LIST OF FIGURES ............................................................................................................................................... XI
LIST OF TABLES .............................................................................................................................................. XIV
LIST OF ACRONYMS ....................................................................................................................................... XVI
LIST OF SYMBOLS ......................................................................................................................................... XVII
CHAPTER 1. INTRODUCTION .......................................................................................................................... 1
MOTIVATION......................................................................................................................................................................... 1
NUCLEAR WASTE MANAGEMENT ....................................................................................................................................... 2
ROAD NETWORKS ................................................................................................................................................................ 3
RESEARCH OBJECTIVES ........................................................................................................................................................ 3
ORGANIZATION ..................................................................................................................................................................... 3
PART I ................................................................................................................................................................... 5
CHAPTER 2 CELLULAR AUTOMATA AND NUCLEAR WASTE MANAGEMENT ..................................... 6
2.0 LITERATURE REVIEW .................................................................................................................................................... 7
2.1 NUCLEAR WASTE MANAGEMENT ................................................................................................................................ 8
2.1.1 Problem Description ......................................................................................................................................................... 8
2.1.2 Canadian DGR Design Concept ................................................................................................................................... 8
2.1.3 Normal Evolution Scenario........................................................................................................................................... 9
2.1.4 Definition of failure ........................................................................................................................................................ 10
2.1.5 Copper corrosion ............................................................................................................................................................. 10
2.1.6 Groundwater composition.......................................................................................................................................... 11
2.1.7 Bentonite Density ............................................................................................................................................................ 11
2.2 CELLULAR AUTOMATA ............................................................................................................................................... 12
2.2.1 Definition and development ...................................................................................................................................... 12
2.2.2 Neighborhood ................................................................................................................................................................... 12
2.2.3 Rules of behavior ............................................................................................................................................................. 13
2.2.4 Collective behavior ......................................................................................................................................................... 15
2.2.5 Conway’s Game of “Life” .............................................................................................................................................. 15
2.2.6 Relation between CA and PDEs ................................................................................................................................ 16
2.2.7 Chemical Reaction .......................................................................................................................................................... 17
2.2.8 Diffusion Equation .......................................................................................................................................................... 17
2.2.9 Wave Equation ................................................................................................................................................................. 19
2.2.10 Reaction Diffusion Equation ................................................................................................................................... 20
CHAPTER 3 DIFFUSION USING CELLULAR AUTOMATA ......................................................................... 21
3.1 DIFFUSION .................................................................................................................................................................. 22
3.2 DISCRETIZED DIFFUSION ........................................................................................................................................... 22
vii
3.3 PROBABILITY OF DISPLACEMENT .............................................................................................................................. 23
3.4 INITIAL AND BOUNDARY CONDITIONS ...................................................................................................................... 26
3.5 RELATION TO BROWNIAN MOTION .......................................................................................................................... 27
3.6 RELATION TO MARKOV CHAINS ................................................................................................................................ 28
3.7 PARTICLE DISPLACEMENT ......................................................................................................................................... 31
3.8 DIFFUSION IN MIXED MEDIUMS ................................................................................................................................ 33
3.9 REACTION PROBABILITY ............................................................................................................................................ 34
3.10 REACTION-DIFFUSION PROBABILITY ..................................................................................................................... 37
3.11 ADVECTION TERM ................................................................................................................................................... 38
3.12 CONSISTENCY, ORDER, STABILITY, AND CONVERGENCE ...................................................................................... 38
CHAPTER 4 ESTIMATING LIFETIMES OF NUCLEAR WASTE CONTAINERS ....................................... 42
4.0 ASSUMPTIONS AND LIMITATIONS .............................................................................................................................. 43
4.1 SULPHATE-REDUCING BACTERIA.............................................................................................................................. 43
4.1.1 Reaction Rate .................................................................................................................................................................... 44
4.1.2 Optimal Reduction Rate Coefficient ...................................................................................................................... 45
4.1.3 Sulphide Production ...................................................................................................................................................... 47
4.1.4 Density-Dependent Reaction Rate.......................................................................................................................... 49
4.2 REACTION – DIFFUSION MODEL ............................................................................................................................... 51
4.2.1 Cellular Automata Model ............................................................................................................................................ 52
4.2.2 Diffusion ............................................................................................................................................................................... 53
4.2.3 Reaction ............................................................................................................................................................................... 54
4.2.4 Reaction-Diffusion .......................................................................................................................................................... 54
4.2.5 Initial Conditions ............................................................................................................................................................. 54
4.2.6 Boundary Conditions ..................................................................................................................................................... 56
4.3 LIFETIME CALCULATION ............................................................................................................................................ 57
4.3.1 Sulphide Flux ..................................................................................................................................................................... 57
4.3.2 Corrosion depth ............................................................................................................................................................... 58
4.3.3 Determination of the lifetime of canisters ......................................................................................................... 58
4.4 VERIFICATION AND VALIDATION .............................................................................................................................. 59
4.4.1 Validation Scenario I ..................................................................................................................................................... 59
4.4.2 Validation Scenario II ................................................................................................................................................... 60
4.5 SCENARIOS .................................................................................................................................................................. 61
4.6 RESULTS ...................................................................................................................................................................... 62
4.6.1 Scenario I. SRB active in clays and rock interface ......................................................................................... 62
4.6.2 Results Scenario II. SRB active in rock interface............................................................................................. 65
4.6.3 Results Scenario III. SRB active, homogeneous densities ........................................................................... 68
4.7. SENSITIVITY ANALYSIS.............................................................................................................................................. 70
4.7.1 Scenario I. SRB active in clays and rock interface ......................................................................................... 71
4.7.2 Scenario II. SRB active only at rock interface .................................................................................................. 72
4.7.3 Scenario III. SRB active, homogeneous densities ............................................................................................ 73
4.8 PART I CONCLUSIONS ................................................................................................................................................. 73
PART II ................................................................................................................................................................ 76
CHAPTER 5 AGENT-BASED MODELING ...................................................................................................... 77
5.0 LITERATURE REVIEW ................................................................................................................................................. 78
5.1 AGENT-BASED MODELING ........................................................................................................................................ 81
5.1.1 Properties of agents ....................................................................................................................................................... 81
viii
5.1.2 ABM implementation .................................................................................................................................................... 82
5.2 AGENT-BASED MODELS VS. EQUATION-BASED MODELS ....................................................................................... 82
5.2.1 ABM vs. ODEs: A Microeconomics Example ....................................................................................................... 83
5.3 ROAD NETWORKS ...................................................................................................................................................... 88
5.3.1 Traveling Time ................................................................................................................................................................. 88
5.3.2 Road Segment Speed ..................................................................................................................................................... 89
5.3.3 Road density ...................................................................................................................................................................... 90
CHAPTER 6 BAYESIAN STATE ESTIMATION USING COLLECTIVE INFORMATION .......................... 91
6.1 DEFINITIONS AND ASSUMPTIONS .............................................................................................................................. 92
6.2 AGENT PERCEPTION................................................................................................................................................... 93
6.3 ADDITION OF NEW INFORMATION ............................................................................................................................ 94
6.4 PROCESS OF FORGETTING INFORMATION ................................................................................................................. 99
6.4.1 Diffusion-like process .................................................................................................................................................... 99
6.4.2 Diffusion process ........................................................................................................................................................... 101
6.5 MOST PROBABLE STATE .......................................................................................................................................... 103
6.6 NUMERICAL IMPLEMENTATION .............................................................................................................................. 106
6.6.1 Diffusion-like Process Implementation .............................................................................................................. 106
6.6.2 Diffusion Process Implementation ....................................................................................................................... 108
6.6.3 Normalizing a Function ............................................................................................................................................. 110
6.7 AGENT LEARNING..................................................................................................................................................... 110
CHAPTER 7 TRAVELING TIME ESTIMATION IN ROAD NETWORKS ................................................ 113
7.0 ASSUMPTIONS AND LIMITATIONS ............................................................................................................................ 114
7.1 ROAD NETWORKS .................................................................................................................................................... 114
7.1.1 Road network I (RN-I) ................................................................................................................................................ 114
7.1.2 Road network II (RN-II)............................................................................................................................................. 115
7.1.3 Road network III (RN-III) ......................................................................................................................................... 116
7.2 ROAD NETWORK OPTIMIZATION MODEL............................................................................................................... 116
7.3 AGENT-BASED MODEL ............................................................................................................................................ 117
7.3.1 Pseudo code...................................................................................................................................................................... 118
7.4 DERIVATION OF AGENT RULES ............................................................................................................................... 119
7.4.1 Example I. Solving RN-I ............................................................................................................................................. 120
7.4.2 Example II Solving RN-II ........................................................................................................................................... 122
7.5 AGENT DECISION-MAKING ...................................................................................................................................... 124
7.6 BAYESIAN STATE ESTIMATION IMPLEMENTATION ............................................................................................... 125
7.6.1 Input Values ..................................................................................................................................................................... 126
7.6.2 Comparison Bayesian vs. True values for RN-I .............................................................................................. 127
7.6.3 Comparison between different types of agents.............................................................................................. 128
7.6.4 Implementing Learning ............................................................................................................................................. 129
7.6.5 Implementing RN-III ................................................................................................................................................... 130
7.7 PART II CONCLUSIONS ............................................................................................................................................. 132
CHAPTER 8. SUMMARY AND CONCLUSIONS .......................................................................................... 134
8.1 SUMMARY .................................................................................................................................................................. 134
8.2 CONTRIBUTIONS ....................................................................................................................................................... 135
8.3 CONCLUSIONS ........................................................................................................................................................... 135
8.4 FUTURE WORK .......................................................................................................................................................... 137
REFERENCES .................................................................................................................................................. 138
ix
APPENDIX A ................................................................................................................................................... 144
APPENDIX B ................................................................................................................................................... 149
APPENDIX C .................................................................................................................................................... 151
APPENDIX D ................................................................................................................................................... 154
APPENDIX E .................................................................................................................................................... 156

x