ADVANCING REGIONAL WATER SUPPLY
PORTFOLIO MANAGEMENT AND
COOPERATIVE INFRASTRUCTURE
INVESTMENT PATHWAYS UNDER DEEP
UNCERTAINTY
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Bernardo Carvalho Trindade
December 2019
©c 2019 Bernardo Carvalho Trindade
ALL RIGHTS RESERVED
ADVANCING REGIONAL WATER SUPPLY PORTFOLIO MANAGEMENT
AND COOPERATIVE INFRASTRUCTURE INVESTMENT PATHWAYS
UNDER DEEP UNCERTAINTY
Bernardo Carvalho Trindade, Ph.D.
Cornell University 2019
Water utilities globally are facing a great pressure to proactively address grow-
ing demands, higher levels of resource contention, and increasingly uncertain
water availability. A tension exists between improving the efficiency and co-
ordination of regional water supplies to delay or eliminate major infrastruc-
ture investments and the eventual effects of demand growth rates that neces-
sitate water supply capacity expansions. The hard- and soft-path views for
urban water portfolio planning, namely infrastructure construction and effi-
cient regionally-coordinated management (e.g., restrictions, water transfers, fi-
nancial instruments, etc.), should be integrated while minimizing costly and
contentious infrastructure expansion. However, both the soft and hard ap-
proaches may be detrimental to utilitiesThis ’ finances. Integrating long-term
water infrastructure investment and short-term management is a difficult task
because of the potentially large number of decisions that must be considered
over decadal timescales as well as the significant uncertainties inherent to the
problem. This dissertation contributes a new uncertainty-sampling formulation
for the stochastic simulation and optimization of water-infrastructure manage-
ment policies (soft path alone) for regionally-coordinated water utilities. This
sampling formulation accounts not only for well-characterized uncertainties,
usually hydrological or climate extremes, but also for key urban-systems fac-
tors such as demand growth, effectiveness of water-use restrictions, construc-
tion costs, and financing uncertainties. This sampling scheme is later integrated
into WaterPaths, a generalizable, cloud-compatible, open-source exploratory
modeling framework designed to simulate and inform long-term regional in-
vestments in water infrastructure while simultaneously aiding regions to im-
prove their short-term weekly management decisions. Uniquely, WaterPaths
has the capability to identify the challenges and demonstrate the benefits of
regionally-coordinated planning and management based on risk-of-failure de-
cision metrics for groups of water utilities sharing water resources. The Water-
Paths platform is introduced here through the hypothetical and realistic Sedento
Valley test case, a test case designed to serve as a universal test case for decision-
making frameworks and water-resources systems simulation software. Lastly,
WaterPaths is integrated into the Deep Uncertainty (DU) Pathways frame-
work. The DU Pathways framework is a decision-making framework devel-
oped for bridging long-term water supply infrastructure investments and im-
proved short-term water portfolio management to yield robust regional water
supply. The DU Pathways framework allows for elaborate uncertainty analysis
to clarify how uncertainty may determine the appropriateness of infrastructure
construction and the effectiveness of supply and financial drought-mitigation
instruments used by regionally-coordinated water utilities. The ultimate goal
of the DU Pathways is to aid stakeholders in discovering pathway policies that
attain high performance levels across challenging and deeply uncertain futures
and to guide robustness compromises that may be necessary between regional
actors. As demonstrated in the Research Triangle region in North Carolina,
DU Pathways clarifies how to identify robust infrastructure investment and
management policies across the municipalities of Raleigh, Durham, Cary, and
Chapel Hill.
BIOGRAPHICAL SKETCH
Bernardo Trindade grew up in Brası́lia, the capital of Brazil. He attended the
Federal University of Santa Catarina in the south of Brazil for one semester but
then returned to Braslia to study at University of Brası́lia, where he received a
B.S. in Civil and Environmental Engineering in 2009. After an six months as
an intern on a construction site, Bernardo spend another six months designing
sewage collection systems in Brazil before moving to the United States in 2010.
Bernardo earned a M.S. in Civil and Environmental Engineering in 2012 from
Auburn University (AL) with a concentration in computational pipe hydraulics,
which led him to work at Bechtel Oil, Gas and Chemicals in Houston, TX, with
natural-gas-liquefaction plants from 2012 to 2014. In 2014, he started started
his Ph.D. at Cornell Environmental & Water Resources Systems program, after
which he expects to apply his research in an industry position.
iii
To my family, whose example illuminates the road leading me to ever greater
success. O que faz o homem o trabalho.
iv
ACKNOWLEDGEMENTS
I would not have succeeded in my Ph.D. without the intellectual and personal
support of countless great people. I would like to first thank my advisor, Prof.
Patrick Reed, for stirring my ideas in the right direction through his own while
constantly providing me with the feedback and generous resources I needed
to succeed. I am particularly grateful for him having brought me to this great
university and encouraged me to persue my interests in topics ranging from
machine learning to existentialism, Marxism, and wines, all of which changed
my worldview and the professional life ahead of me. I also would like to thank
John Foote, whose professional advice, friendship, introductions, inspiring ex-
ample, and selfless commitment to my success have stirred my future career in
a direction I could never have anticipated. I would also like to thank my current
committee members: Professors Rich Geddes and Daniel Selva for the early and
industry-focused advice, Prof. Rich Geddes and Prof. Oliver Gao for reviewing
my five years worth of work, Prof. Geddes for introducing me for countless
high-level professionals in the infrastructure industry, and Prof. Gao for gra-
ciously accepting the last-minute invitation to replace Prof. Daniel Selva as my
committee member after the latter left Cornell. I would like to thank Prof. Pete
Loucks for the constant mentorship about engineering and leadership, for the
squash games, and for all the shared meals, Prof. Jery Stedinger for all conver-
sations that made me see statistics as the fascinating discipline it is despite my
initial prejudices, and Prof. David Bindel for helping me during the develop-
ment of WaterPaths when he had no reason to do so. I would also like to thank
my collaborators starting from Prof. Jon Herman for teaching me the rudiments
of my future research, Dr. H. B. Zeff and Prof. Gregory Characklis for providing
a pillar for my work at Cornell, David Gorelick for all the e-mail about bugs
v
and improvements on WaterPaths, and David Gold for all the help close to the
finish line. I would like to thank Dr. Jared Smith and Professors John Lamon-
tagne and Julie Quinn for all the intellectual insight that so much improved the
quality of my work and education. Lastly, I would like to thank all students I
had the honor to help as a Graduate Teaching Assistant, who through intellec-
tual rigor and while maintaining and friendly and fun environment pushed me
to master the disciplines I was in charge of, and to Nadine Porter, Jeannette Lit-
tle, Charissa King-O’Brien, Carl Cornell, Megan Keene, Tania Sharpsteen, Beth
Korson, Joe Rowe, and Alita Howard for being such great department staff.
Besides great mentors and collaborators, Cornell has provided me with
friends and colleagues of the highest quality and a fantastic and affectionate
social life. I have been fortunate to spend time in and outside Hollister with
and receive great friendship and support from Dr. Jared Smith, Prof. John La-
montagne, Prof. Julie Quinn, Prof. Jon Herman, David Gold, Jazmin Zatarain,
Kwang-Bae Choi, Rohini Gupta, Peter Storm, Sarah Kayali El-Alem, Tori Ward,
Dr. Min Pang, Dr. Tom Wild, Calvin Whealton, Candice Wu, Xiaotong Guo,
Bernardo Casares, Justice Sefas, and Fábio Guilhon. I have also greatly appreci-
ated the time spent with great music and swing dance with Renée Staffeld, my
eternal swing-dance partner, Amanda Bartley, Furkat Mukhtarov, Leah Hoff-
mann, Aleksa Basara, Addie Lederman, Allison Stilin, Aimee Owens, Katie
Spoth, Tres Reid, Sierra Cook, Chris Westersund, Renée Sifri, Yifei Zheng,
YangYang Sun, Colin Bundschu, Mari Tader, Andrea Staffeld, Bill Staffeld,
Anna-Katharina von Krauland, Irine Sarri, Cesare Paolucci, Céline Hollfelder,
Tamilla Kassenova, Destiny Meador, Erica Watson, Gustavo Ribeiro, Erica Shaw,
Howard Chong, Kevin Lin, Rachel Vanicek, Tommi Schieder, Schuyler Duffy,
Vishnu Govindaraj, Yudi Pardo, Yupeng Lin. My adversaries inside the tennis
vi
court but friends and colleagues outside have also much supported me dur-
ing my time at Cornell. They are Ian Bailey, Bruno Santarelli, Kevin O’Keefe,
Andrew Fields, Evan Gadreau, Luca Moreschini, Chloé Albietz, Ricardo Paes,
Chris Levine, Milo Johnson, Anda Perianu, Silviu Tanasoiu, Lev Kazakov, Cate-
rina Kurz, Rachel Allen, Anita Jain, Allison Vizgaitis, Dina Rigas, and Rodrigo
Viteri. I would like to also thank my great friends Ilse and Jon Jalvin for all
the great wine and conversations, Sara Vandenbroek for her support during the
long development of WaterPaths, and my great Katyusha Potemkina for all her
support and patience as I try to close my work at Cornell.
This research was funded by the National Institute of Food and Agriculture,
U.S. Department of Agriculture (WSC Agreement No. 2014-67003-22076). Ad-
ditional support was provided by the U.S. National Science Foundation’s Water,
Sustainability, and Climate Program (Award No. 1360442). I also received fund-
ing from the Cornell School of Civil & Environmental Engineering as a Gradu-
ate Teaching Assistant. High-performance computing resources were provided
by the Texas Advanced Computing Center (TACC), Blue Waters, and by the
always-prompt support from the folks at the Cornell Center for Advanced Com-
puting.
Finally, I would like to thank my parents and sister for providing me with
wings as my duties tied my feet. They have always been a constant source of
uplifting inspiration and love, making their presence felt at every hour from
the other side of the equator. And none of my great achievements over the
last ten years would have been possible were it not for the great country that
is the United States of America, which has provided me with more love and
opportunities than I will ever deserve.
vii
TABLE OF CONTENTS
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 Introduction 1
1.1 Current Challenges for Water Infrastructure Planning and Man-
agement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Water Portfolio Planning and Management Problem . 3
1.1.2 Considering Uncertainty . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Existing Water-Infrastructure Planning and Management
Software and Decision-Making Framworks and Their
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Dissertation Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Scope and Organization . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Chapter 2 — Background on Methodological Components 13
1.3.2 Chapter 3— Reducing Regional Drought Vulnerabili-
ties and Multi-city Robustness Conflicts Using Many-
objective Optimization Under Deep Uncertainty . . . . . . 13
1.3.3 Chapter 4 — WaterPaths: A Stochastic Simulation Frame-
work for Water Supply Infrastructure Investment Man-
agement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.4 Chapter 5 — Deeply Uncertain Pathways: Integrated
Multi-City Regional Water Supply Infrastructure Invest-
ment and Portfolio Management . . . . . . . . . . . . . . . 16
1.3.5 Chapter 6 — Contributions and Future Work . . . . . . . . 16
1.3.6 Author Contributions for Published Work . . . . . . . . . 17
2 Methodological Components 18
2.1 Borg Multi-Objective Evolutionary Optimization Algorithm . . . 18
2.1.1 Adaptive Operator Selection . . . . . . . . . . . . . . . . . 20
2.1.2 Solution population . . . . . . . . . . . . . . . . . . . . . . 22
2.1.3 Solution Archive and -Dominance . . . . . . . . . . . . . . 23
2.1.4 Parallelization Strategies for the Borg MOEA . . . . . . . . 25
2.2 Scenario Generation for Streamflow and Evaporation Rates Time
Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Scenario Generation for Synthetic Demand Time-series . . . . . . 30
viii
3 Reducing regional drought vulnerabilities and multi-city robustness
conflicts using many-objective optimization under deep uncertainty 32
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Methodological Framework . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 States of the World . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.3 Alternatives Generation . . . . . . . . . . . . . . . . . . . . 40
3.3.4 Robustness Measures . . . . . . . . . . . . . . . . . . . . . 43
3.3.5 Robustness Controls . . . . . . . . . . . . . . . . . . . . . . 44
3.4 The Research Triangle Test Case . . . . . . . . . . . . . . . . . . . . 46
3.4.1 Triangle Region Management Model . . . . . . . . . . . . . 49
3.5 The Mathematical Formulation of the ROF Metric . . . . . . . . . 53
3.6 Computational Experiment . . . . . . . . . . . . . . . . . . . . . . 60
3.6.1 Multiobjective Optimization Algorithm . . . . . . . . . . . 60
3.6.2 Hydrologic Scenarios and DU Optimization Runs . . . . . 60
3.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7.1 Optimization under Deep Uncertainties . . . . . . . . . . . 62
3.7.2 Robustness Comparison . . . . . . . . . . . . . . . . . . . . 64
3.7.3 The Value of Regionally Coordinated Demand Management 70
3.7.4 Scenario Discovery . . . . . . . . . . . . . . . . . . . . . . . 72
3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 WaterPaths: An open source stochastic simulation framework for wa-
ter supply portfolio management and infrastructure investment path-
ways 81
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Software Availability . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 WaterPaths Framework . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4.1 WaterPaths Overview . . . . . . . . . . . . . . . . . . . . . 91
4.4.2 ROF-based decision rules . . . . . . . . . . . . . . . . . . . 96
4.4.3 Mass-Balance Model . . . . . . . . . . . . . . . . . . . . . . 99
4.4.4 Capturing Regional Decision Making . . . . . . . . . . . . 100
4.4.5 Drought Mitigation Instruments . . . . . . . . . . . . . . . 101
4.4.6 Financial instruments . . . . . . . . . . . . . . . . . . . . . 102
4.4.7 Infrastructure Investment . . . . . . . . . . . . . . . . . . . 103
4.4.8 Flexible Representation and Evaluation of Uncertainties . 105
4.4.9 Currently Implemented Objective Functions . . . . . . . . 107
4.4.10 Reducing Runtime With Local Parallel Computing . . . . 109
4.4.11 Facilitating Decision Analytics . . . . . . . . . . . . . . . . 112
4.4.12 WaterPaths Parallelization Strategy for Computational
Policy Search . . . . . . . . . . . . . . . . . . . . . . . . . . 114
ix
4.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.5.1 The Sedento Valley: An Illustrative Test Case . . . . . . . . 116
4.5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . 120
4.5.3 Discovering Robust Pathways for the Sedento Valley . . . 126
4.5.4 Identifying Tradeoffs . . . . . . . . . . . . . . . . . . . . . . 126
4.5.5 Evaluating Robustness . . . . . . . . . . . . . . . . . . . . . 131
4.5.6 Pathway Analytics . . . . . . . . . . . . . . . . . . . . . . . 133
4.5.7 Scaling Performance on Clusters and in the Cloud . . . . . 135
4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
4.6.1 Performance and Robustness Tradeoffs . . . . . . . . . . . 140
4.6.2 Policy Rules in the Robustness Compromises . . . . . . . . 145
4.6.3 Scenario Discovery for Compromise Policies . . . . . . . . 149
4.6.4 Infrastructure Ensemble Pathway Analysis . . . . . . . . . 153
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5 Deeply Uncertain Pathways: Integrated Multi-City Regional Water
Supply Infrastructure Investment and Portfolio Management 157
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.3 The Long-Term Version of the Research Triangle Test Case . . . . 163
5.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.4.1 Identifying Infrastructure Investment and Drought Man-
agement Policy Tradeoffs . . . . . . . . . . . . . . . . . . . 168
5.4.2 Defining and Evaluating Robustness . . . . . . . . . . . . . 181
5.4.3 Discovering Uncertain Scenarios that Control Robustness 184
5.4.4 Visual Infrastructure Pathway Analysis . . . . . . . . . . . 187
5.5 Computational Experiment . . . . . . . . . . . . . . . . . . . . . . 188
5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.6.1 Comparing Tradeoffs After Re-evaluation . . . . . . . . . . 194
5.6.2 Scenario Discovery for Compromise Policies . . . . . . . . 199
5.6.3 Policy Rules in the Robustness Compromises . . . . . . . . 204
5.6.4 Infrastructure Pathway Ensemble . . . . . . . . . . . . . . 207
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6 Contributions and Future Work 217
6.1 Contributions and Conclusions . . . . . . . . . . . . . . . . . . . . 217
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.2.1 Strategies for Deep- and Well-Characterized-Uncertainty
Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
6.2.2 Metrics for Long- and Short-Term Decision Making for
Water Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.2.3 Design of New Financing Mechanisms and Insurance In-
struments for Water Infrastructure Operators . . . . . . . . 224
x
6.2.4 Conflict Resolution Among Resource-Sharing Water Util-
ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
A Notation 227
B Time-series Output for Compromise Solutions under selected scenar-
ios 231
C Deeply Uncertain Factor Sensitivities 235
D Computational Experiment Hypervolume Plots 236
E The Boosted Trees Algorithm 237
E.1 Classification and Regression Trees . . . . . . . . . . . . . . . . . . 237
E.2 General Idea of Booosting . . . . . . . . . . . . . . . . . . . . . . . 238
E.3 Geometric Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
E.4 Minimizing a Loss Function . . . . . . . . . . . . . . . . . . . . . . 241
E.5 AdaBoost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
E.6 Sensitivities of Uncertainty Factors . . . . . . . . . . . . . . . . . . 244
E.7 Exponencial Convergence . . . . . . . . . . . . . . . . . . . . . . . 245
E.8 Reduced Overfitting . . . . . . . . . . . . . . . . . . . . . . . . . . 245
xi
LIST OF TABLES
3.1 Uncertainty Factors and Sampling Ranges for Many-Objective
Robust Decision Making (vector Ψ in Equations 3.5 and 3.10). . . 42
3.2 Summary of reservoir storage capacities, demands, and alloca-
tions for the Research Triangle Region . . . . . . . . . . . . . . . . 48
3.3 Values used for -dominance. . . . . . . . . . . . . . . . . . . . . . 62
4.1 Select modeling software for water resources planning and man-
agement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.2 Select modeling software for water resources planning and man-
agement (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Uncertainties included in extended version of the regional wa-
ter utility planning and management problem presented in Zeff
et al. (2016). Regional uncertainties are those for which the same
value was used for all utilities, versus a value different value for
each utility, as in the case uncertainties described as individual. . 106
4.4 Current infrastructure capacity of Sedento Valley Water Utilities. 118
4.5 Demand to Capacity Ratios for the Sedento Valley Water Utilities 118
4.6 Potential new infrastructure options in the Sedento Valley. . . . . 120
4.7 Decision variables pertaining to the short-term drought mitiga-
tion instruments: water-use restrictions, transfers, contingency
fund and drought insurance. . . . . . . . . . . . . . . . . . . . . . 124
4.8 Values of the long-term ROF trigger and daily demands that
function as thresholds to trigger infrastructure construction by
the utilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.9 The ordinal variables below determine the infrastructure con-
struction order and adjusts them for the total number of options
available to each utility. The volumetric variable represents the
volume of available storage within Falls Lake to be re-allocated
from the water quality to Raleigh’s municipal supply pool. . . . 125
4.10 Values used for -dominance. . . . . . . . . . . . . . . . . . . . . . 125
4.11 Deep uncertainties considered for the Sedento Valley test prob-
lem. Unless specified otherwise the same minimum and max-
imum values for each uncertainty were applied for all utilities
and infrastructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.12 Architectures and configurations used for optimization scalabil-
ity and cross-platform performance tests, followed by total run
times and efficiencies for each optimization seed, ran on Stam-
pede 2 and on the cluster The Cube with different parallel con-
figurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.13 Timings and efficiency of running WaterPaths with 1,000 real-
izations split among different numbers of cores on a 4-core Xeon
Desktop and on a 48-core SkyLake node of Stampede 2. . . . . . 139
xii
5.1 Summary of proposed water supply infrastructure for the Re-
search Triangle (Zeff et al., 2016; Triangle J, 2014). . . . . . . . . . 165
5.2 Uncertainties included in extended version of the regional wa-
ter utility planning and management problem presented in Zeff
et al. (2016). Regional uncertainteis are those for which the same
value was used for all utilities, versus a value different value for
each utility, as in the case uncertainties described as individual. . 167
5.3 Decision variables pertaining to the short-term drought mitiga-
tion instruments: water-use restrictions, transfers, contingency
fund and drought insurance. . . . . . . . . . . . . . . . . . . . . . 189
5.4 Values of the long-term rof trigger and daily demands that func-
tion as thresholds to trigger infrastructure construction by the
utilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.5 The ordinal variables below determine the infrastructure con-
struction order and adjusts them for the total number of options
available to each utility. The volumetric variable represents the
volume of available storage within Falls Lake to be re-allocated
from the water quality to Raleigh’s municipal supply pool. . . . 192
5.6 Values used for -dominance. . . . . . . . . . . . . . . . . . . . . . 193
A.1 List of symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
C.1 Percent of the combined impurity the the entire boosted trees
ensemble due to each uncertain factor for Durham, Cary and
Raleigh under both buyer preferred policies D/B1 and R/B2. . . 235
xiii
LIST OF FIGURES
2.1 Flowchart of the Borg MOEA main loop adapted from (Hadka
and Reed, 2013). First, one of the recombination operators is se-
lected based on its performance. After, for an operator requiring
k parents to generate an offspring, Borg selects k − 1 from the
population and one from the archive. The resulting offspring
is evaluated and considered for inclusion in the population and
archive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Example distributions of offspring solutions (small dots) pro-
duced from parents solutions (large dots) by the six operators
included in Borg MOEA. From Hadka and Reed (2014). . . . . . 21
2.3 Illustration of two-objective optimization problem on an  grid,
adapted from Kasprzyk et al. (2013). Only one solution within
each tolerance band 1 and 2 is kept in the Pareto set. Follow-
ing the concept of -dominance, B, C, and E are the only non-
dominated points. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Panel (2) shows a diagram of the master-worker implementation
of the Borg MOEA. The master node maintains the -dominance
archive and runs the main loop of the serial Borg MOEA shown
in Figure 2.1. The decision variables generated through the six
operators inplemented in the master node are transmitted to the
worker nodes, and the evaluated objective function values and
constraints are returned to the master node. Panel (b) shows a
diagram of the multi-master implementation of the Borg MOEA.
The multi-master Borg MOEA consists of two or more mastere-
worker instances (the diagram depicts three) periodically com-
municating with one controller node, which maintains a global
-archive. (1) Each master node periodically transmits its local
-dominance archive to the controller, which updates the global
archive. (2) When a master node is not advancing its pareto-
approximate front, it sends a help message to the controller.
(3) The controller responds with guidance, which includes the
global -dominance archive and global operator probabilities.
From Hadka and Reed (2014). . . . . . . . . . . . . . . . . . . . . 26
2.5 Panel (a) shows examples of flow duration curves for histori-
cal inflows, synthetically inflows with stationary log-mean, and
synthetically inflows with non-stationary log-mean. Panel (b)
shows all sinusoid-multiplier series of the log-mean of inflows
and evaporation rates used to generate the panel (a). . . . . . . . 29
xiv
3.1 Schematics of the deep uncertainty re-evaluation of candidate
water portfolio solutions (termed DU re-evaluation). A set of ob-
jectives is calculated for each 1,000 SOW’s that share the same Ψ.
These objectives are in turn used to calculate overall robustness
and overall objectives for alternative θ. . . . . . . . . . . . . . . . 41
3.2 WCU (a) and DU (b) optimization factor sampling schemes. The
former creates states of the world by coupling synthetically gen-
erated set of streamflows with a base projection for 13 deeply
uncertain factors, while the latter couples each set of syntheti-
cally generated streamflows with a different projection for the
13 deeply uncertain factors, shuffling the coupling at every func-
tion evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Map of the North Carolina Research Triangle region. Adapted
from Zeff et al. (2014). . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Average performance objective values for WCU and DU op-
timization alternatives over the 10 million SOWs from DU
re-evaluation. The vertical axes denote average performance
in each objective and each polyline represents an alternative.
Brown tones represent alternatives from WCU optimization and
blue tones represent alternatives from DU optimization. Trans-
parency is applied to alternatives that do not meet a relaxed
version of the performance criteria established by the studied
utilities: Reliability > 98.5% (relaxed from 99%), Restriction Fre-
quency < 20%, and Worst-Case Cost < 5%). The color gradient
is based on the restriction frequency objective (light for low fre-
quency, dark otherwise, not applicable to highlighted solutions).
None of the brown (WCU) alternatives meet the criteria, while 7
out of 2,954 DU solutions do. . . . . . . . . . . . . . . . . . . . . . 63
3.5 Alternatives (x-axis) were rank-ordered according to robustness
(y-axis) delivered to each utility. An alternative’s robustness was
defined as the percentage of the DU re-evaluation’s 10,000 Ψ
samples where an alternative meets the original robustness cri-
teria elicited from water utilities. The brown bars represent the
robustness of the original 84 WCU optimization’s alternatives,
while the blue bars represent the robustness of all 2,954 DU op-
timization’s alternatives. Each group of bars is rank ordered by
robustness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
xv
3.6 Robustness tradeoffs among utilities. Each polyline represents
an alternative, while the location where a polyline intersects an
axes denotes the robustness value of the corresponding alterna-
tive for the corresponding utility. The blue lines indicate four
illustrative compromise DU alternatives with fairly high robust-
ness for all utilities, the brown line indicates the best compromise
WCU alternative from prior work by Herman et al. (2014), while
the gray lines represent the remaining DU alternatives. . . . . . . 67
3.7 Decision variables of the 4 compromise solutions for each utility.
Different colors represent the different alternatives, and corre-
spond to the ones in Figure 3.6. The ”y” axis represent a scale
from low use to high use. . . . . . . . . . . . . . . . . . . . . . . . 68
3.8 Ranges of deeply uncertain factors which should be avoided in
order to improve the robustness of each compromise solution.
Demand growth (left box) and Falls Lake municipal supply al-
location (right box) are the two most important sources of un-
certainty for the four chosen compromise solutions. Utilities are
encouraged to incorporate measures in their short-term manage-
ment plan seeking to mitigate demand and be granted a greater
allocation from Falls Lake. . . . . . . . . . . . . . . . . . . . . . . 71
3.9 Falls Lake Supply allocation vs. Demand growth pass/fail. A so-
lutions performance is more impacted by variations in demand
growth than in Falls Lake allocation. . . . . . . . . . . . . . . . . . 72
3.10 Robustness improvement due to demand growth control. De-
mand growth management may lead to a robustness gain of
up 30% for the four utilities studied here. Cary, Durham and
OWASA have the potential to reach 100% calculated robustness,
while Raleigh may reach over 90%. . . . . . . . . . . . . . . . . . 74
4.1 WaterPaths main simulation loop. The main loop includes the (1)
calculation of long-term ROF metric if at first week of the year,
(2) calculation of the short-term ROF, (3) application of drought
mitigation and financial policies, (4) system mass-balance mod-
eling, and (5) system-state data collection. . . . . . . . . . . . . . 95
xvi
4.2 For the calculation of the short-term ROF metric for the current
week, 50 year-long simulations are ran for the coming year with
all reservoir storage levels starting at their current levels, while
the streamflows and evaporation rates for each simulation are
those of one of the previous 50 years of recorded data and the
demand are those of the previous year. For the calculation of
the long-term ROF metric, the 50 year-long simulations are ex-
tended to one and a half years, with the last 6 months of the last
simulation (the last year) are filled by synthetic data. The addi-
tion of 6 months to the 50 ROF simulations allows for capturing
consecutive drought years in which the reservoirs are not filled
during the wet season. . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Schematics of (a) shared-memory parallelization and (b) hybrid
shared- and distributed-memory parallelization for distributed
heuristic optmization. Hyperthreading is not used in either
panel. The collored and semi-transparent tiles represent threads,
the dark solid tiles represent physical cores mounted on desk-
tops or computing nodes (medium grey large and thin tiles serv-
ing as base for the cores), and the light grey sheet underneath
the two nodes on panel (b) represents a cluster or cloud environ-
ment hosting the two depicted nodes. Blue represent the master
threads, green the worker threads, and read the threads (one of
which still being the master) of a master process. Threads con-
nected by dashed black lines belong to the same process, with
its own exclusive memory block, parallelized internally with
OpenMP for shared-memory parallelization. Straight, full black
two way arrows in panel (b) represent distributed-memory com-
munications between four-core, four-thread master and worker
processes, respectively represented by the red threads and the
green threads in the yellow contour as an example. Panel
(b) shows one master and seven worker processes performing
a heuristic optimization run in parallel under the distributed-
memory parallelization paradigm. . . . . . . . . . . . . . . . . . . 111
4.4 The Sedento Valley Regional Water Supply System. . . . . . . . . 117
4.5 Current Demand to Capacity ratios for the three utilities. A
higher demand to capacity ratio indicates more stress on the sys-
tem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.6 Policy design and analysis performed with WaterPaths within
the DU Pathways framework. . . . . . . . . . . . . . . . . . . . . 127
4.7 Optimization loop and single-simulation input/output informa-
tion flow. WaterPaths can be used to simulate a single policy,
allowing its use for policy re-evaluation and robustness calcula-
tion, as demonstrated in Section 4.5.5, and for policy optimiza-
tion adn tradeoff identification. . . . . . . . . . . . . . . . . . . . . 128
xvii
4.8 Evolution of the hypervolume for each of the nine sets. The over-
lap across the lines shows that the number of seeds was sufficient
to assure consistency, while the gradual plateauing of the hyper-
volumes of all seeds shows that the number of function evalua-
tions were enough to achieve maximum attainable convergence. 129
4.9 Performance and robustness tradeoffs in the DU Re-Evaluation
space. Panel (a) illustrates the tradeoffs across the performance
objectives quantified either using the mean or the worse first per-
centile across the tested SOWs, in which an ideal solution would
be represented by a horizontal line at the bottom of the plot.
Panel (b) illustrates the robustness tradeoffs across the three utili-
ties. Here, robustness is calculated as the percent of RDM SOWs
in which a policy met the performance criteria defined by the
utilities and an ideal solution would be represented by a hori-
zontal line at the top of panel (b). The highlighted policies rep-
resent the policies with best robustness for each utility (BRO-X),
with the best robustness compromise across utilities (RoC), and
best performance compromise (PC). . . . . . . . . . . . . . . . . . 141
4.10 Robustness of pre-selected 229 policies for each utility. Each bar
represents the attained robustness for one policy for each utility.
The policies are sorted by robustness (bar height) for each utility.
The policies with the highest value of the satisficing robustness
for each utility, the policy with the best robustness compromise,
and the policy with the best compromise of objectives values are
highlighted in each subplot. . . . . . . . . . . . . . . . . . . . . . . 143
4.11 Parallel axes plots for the policy variables for the policies repre-
senting the best robustness for each utility, the best robustness
compromise, and the best performance compromise. The table
indicates the infrastructure construction order for each utility for
each policy in the order shown in the legend. . . . . . . . . . . . . 145
4.12 Factor maps for the robustness compromise policy (RoC) show-
ing the key uncertainty factors controlling the robustness the
policy for all utilities. Success (grey dots and zone) required reli-
ability ≥ 98%, restriction frequency ≤ 10% and annual financial
worst 1st percentile cost≤ 10%, while a failure to meet either cri-
teria results in the scenario being considered a failure (red dots
and region). White zones denote regions with a mixture of gray
and red dots for which it is uncertain whether all criteria would
be met. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
xviii
4.13 Capacity expansion pathways and ROF dynamics for the most
favorable scenario. Demand growth rate scenarios considered
include the Most Favorable SOW, the SOW projected by the util-
ities, and the Least Favorable SOW. All infrastructure invest-
ments are illustrated as stacked color lines designating specific
infrastructure investments across 1,000 hydrologic scenarios. . . 152
5.1 Map of the Research Triangle Region test case. Potential Infras-
tructure refers to infrastructure options presented in Triangle J
(2014), that are being considered by the utilities. . . . . . . . . . . 164
5.2 The Research Triangle instance of the WaterPaths simulation
model takes as inputs a policy to be evaluated, the system com-
ponents and its connectivity, hydrological time series, infrastruc-
ture construction order, information about infrastructure to be
built, different types of demands, and uncertainty. It outputs
time series of system states, an objectives vector and pathways. . 166
5.3 Key steps of the DU Pathways approach and their key points . . 169
5.4 Illustrative portfolio of decisions related to short-term drought
mitigation instruments and long-term priorities for infrastruc-
ture investments pathways that define a system’s water supply
policy. The optimization problem consists in finding the best po-
sition for each knob corresponding to each decision variable for
each utility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.5 (a) In the WCU sampling scheme, sets of time-series samples of
synthetically generated streamflows, evaporation rates and de-
mands are generated for each realization and appended to the
historical data. The best projected values (or other assumption)
of the deeply uncertain sources of uncertainty are then coupled
with the resulting sets of time series form all SOWs. (b) In DU
sampling, the same sets of time-series are instead combined with
samples of deep uncertainties to form the ensemble of SOWs. . . 175
5.6 Re-evaluation sampling scheme. The same sets of Nr hydrolog-
ical and demand time series are combined with one vector sam-
ple of deeply uncertain factors one at a time, resulting in a num-
ber of SOWs equal to Nr times the number of samples of deeply
uncertain factors. The resulting set of objectives vectors are used
to back-calculate the objectives and robustness of the whole en-
semble, while the pathways are used to analyze the dependency
of infrastructure investments on deeply uncertainty factors. . . . 182
xix
5.7 Comparative performance analyses based on the DU re-
evaluation. Panel (a) illustrates the tradeoffs across the perfor-
mance objectives quantified either using the mean or the worse
first percentile across the tested SOWs. Panel (b) illustrates the
robustness tradeoffs across the four utilities (satisficing metric).
In both panels policies from the WCU formulation are brown
and those from the DU optimization are blue. . . . . . . . . . . . 195
5.8 Robustness of the 100 top most robust policies attained from the
WCU and DU optimization schemes for each utility. Each bar
represents the attained robustness for one policy for each util-
ity. The policies are sorted by robustness (bar height) for each
utility. The policies with the highest value for the satisficing ro-
bustness for each utility is highlighted each subplot. One of the
most robust policy for Cary called the “supplier-preferred” pol-
icy while the most robust policies for Durham and Raleigh were
called “buyer-preferred” 1 and 2. . . . . . . . . . . . . . . . . . . . 196
5.9 Factor maps for two policies — buyer preferred 1 (Durham) and
2 (Raleigh) — showing the key factors controlling the robustness
the solutions for all utilities but OWASA. Success required reli-
ability ≥ 99%, restriction frequency ≤ 20% and annual financial
worst 1st percentile cost ≤ 10%. A list with the combined impu-
rity decreases for all factors can be found in Table C. . . . . . . . 200
5.10 Parallel axis plots for the decision variables for the D/B1 and
R/B2 buyer preferred policies. The red lines designate Durham’s
preferred buyer solution (D/B1). The green lines designate
Raleigh’s preferred buyer solution (R/B2). . . . . . . . . . . . . . 205
5.11 Capacity expansion pathways and ROF dynamics for the
Durham preferred policy DB/1. Demand growth rate scenar-
ios considered include the Most Favorable SOW, the SOW with
demand growth rates reduced to 80% of their original nominal
values, and the Least Favorable SOW. All infrastructure invest-
ments are illustrated as stacked color lines designating specific
infrastructure investments across 1,000 hydrologic scenarios. . . 208
5.12 Capacity expansion pathways and ROF dynamics for the
Durham preferred policy DB/1. Demand growth rate scenar-
ios considered include the Most Favorable SOW, the SOW with
demand growth rates reduced to 80% of their original nominal
values, and the Least Favorable SOW. All infrastructure invest-
ments are illustrated as stacked color lines designating specific
infrastructure investments across 1,000 hydrologic scenarios. . . 211
xx
B.1 System dynamics for Durham and Raleigh under policy D/B1
for multiple hydrologic realizations under Durham’s most unfa-
vorable scenario. Transparent lines indicate that few realizations
had similar dynamics and the horizontal colored lines represent
ROF trigger values for the corresponding policy. . . . . . . . . . . 231
B.2 System dynamics for Durham and Raleigh under policy D/B1
for multiple hydrologic realizations under Raleigh’s most unfa-
vorable scenario. Transparent lines indicate that few realizations
had similar dynamics and the horizontal colored lines represent
ROF trigger values for the corresponding policy. . . . . . . . . . . 232
B.3 System dynamics for Durham and Raleigh under policy R/B2
for multiple hydrologic realizations under Durham’s most unfa-
vorable scenario. Transparent lines indicate that few realizations
had similar dynamics and the horizontal colored lines represent
ROF trigger values for the corresponding policy. . . . . . . . . . . 233
B.4 System dynamics for Durham and Raleigh under policy R/B2
for multiple hydrologic realizations under Raleigh’s most unfa-
vorable scenario. Transparent lines indicate that few realizations
had similar dynamics and the horizontal colored lines represent
ROF trigger values for the corresponding policy. . . . . . . . . . . 234
D.1 Evolution of hypervolume for each master of each of two seeds
ran for the WCU and DU searches. Since each seed had 4 is-
land and each island ran 125,000 function evaluations, the total
number of function evaluation for each of the 2 WCU and 2 DU
seeds was 500,000. WCU seeds advanced faster at early NFE and
converged slighly faster than the DU seeds, although the long
plateaus at later NFE show that all seeds were given more than
enough NFE to find the best Pareto approximate set of policies. . 236
E.1 Classification tree fit on illustrative data set. The first split to
be foudn was the horizontal split, which applies to the entire
data set. The three vertical splits were foudn branching off of the
horizontal split and then off of each other. . . . . . . . . . . . . . 237
E.2 Sequence of addition of single-split classification trees until all
points are correctly classified. Panel (a) shows just the data, with-
out trees, panel (b) shows data with one tree, panel (c) with two,
and panel (d) with three. Variables x1 and x2 are two uncertain
factors, the blue crosses represent a success scenario and a red
circle represents a failure, and the blue and red regions corre-
spond to regions classified as success or failure regions, with in-
termediate colors indicating uncertainty in the classification. . . 240
xxi
CHAPTER 1
INTRODUCTION
1.1 Current Challenges for Water Infrastructure Planning and
Management
Water utilities globally are facing increasing pressure to proactively address
growing demands, higher levels of resource contention, and increasingly uncer-
tain water availability (Bonzanigo et al., 2018; Hall et al., 2014). A tension exists
between improving the efficiency and coordination of regional water supplies to
delay or eliminate major infrastructure investments and the eventual effects of
demand growth rates that necessitate water supply capacity expansions. Soft
water management strategies (conservation, inter-utility water transfers, de-
mand management, etc.) are a key option to cope with growing domestic water
demand (Gleick, 2002a, 2003). In the United States (US), a key driver increasing
the importance of soft water management strategies is that most large projects
of federal and/or state interest have already been built (Lund, 2013). However,
growing urban water demands and increasingly uncertain climate conditions
are motivating efforts on the hard side, meaning redesigning system capacity
by upgrading existing and adding new infrastructure to maintain reliable water
supply systems (Shafer and Fox, 2017).
The hard and soft path views for urban water portfolio planning should
be treated as complementary, as even though infrastructure expansion may be
needed, it is not desirable and may be mitigated by soft water management
strategies. However, both the soft and hard approaches may be detrimental to
1
utilities’ finances. Together with outstanding debt, decreasing cash flows are
significant factors that negatively impact utilities credit ratings, which can in-
crease the cost debt when seeking to build new infrastructure (Moody’s, 2017;
Zeff et al., 2014; Zeff and Characklis, 2013). The financial impacts of drought
mitigation strategies, the current negative financial outlook for the water utility
sector (Moody’s, 2019), and the lack of appropriate financing for water utilities
(Gleick et al., 2014) all stress the need for new water infrastructure planning and
management frameworks. More specifically, utilities and planners would ben-
efit from a framework to facilitate improved water portfolio planning that ef-
fectively combines short-term drought mitigation, instruments for managing fi-
nancial risks, and long-term infrastructure investment pathways while account-
ing for diverse sources of uncertainty.
Such framework needs to address the following key questions: (1) how
should regional decision makers devise infrastructure investment and drought
management strategies that adequately capture their interdependencies to im-
prove water supply reliability and financial stability?, (2) how can stakehold-
ers develop management and investment policies that are sufficiently adaptive
and robust over decades given the impossibility of predicting climate, popula-
tion shifts, political landscapes, and other sources of uncertainty?, (3) how do
decision-makers know when to make substantial and irreversible investments
in new infrastructure while avoiding stranded assets?, and (4) how should wa-
ter utilities sharing water resources (e.g. watersheds, reservoirs, pipes) coor-
dinate individual actions to avoid exacerbating vulnerabilities and conflicts?.
These questions capture the defining challenges of the water utility planning
and management problem.
2
1.1.1 The Water Portfolio Planning and Management Problem
Several studies have focused on integrating efficient short-term management
strategies (soft path) into existing infrastructure to improve supply reliability
at lower expected costs Helping to avoid unecessary infrastructure expansion
(Lund and Israel, 1995; Wilchfort and Lund, 1997; Palmer and Characklis, 2009;
Padula et al., 2013; Mozenter et al., 2018). Recent studies have explored de-
mand management (Hall et al., 2019; Zeff et al., 2014; Borgomeo et al., 2018;
Huskova et al., 2016; Haasnoot et al., 2013), treated water transfers (Caldwell
and Characklis, 2014; Zeff et al., 2014; Watkins and McKinney, 1997; Hall, 2019),
upstream reservoir releases (Borgomeo et al., 2018; Gorelick et al., 2019), and
operation rules (Wu et al., 2017; Tian et al., 2018). Efforts over the last several
years focused on regionalized management seek to identify coordinated man-
agement operations where the impacts from the measures taken by each utility
has a predictable impact on the operations of the other utilities (Sheer, 2010;
Zeff et al., 2014; Herman et al., 2014; Zeff et al., 2016; Borgomeo et al., 2018;
Gorelick et al., 2019; Hall et al., 2019). For example, if multiple utilities rely
on treated or raw water transfers as a drought mitigation measure for improv-
ing reliability, each transfer request has the potential to be directly impacted by
constrained conveyance and water treatment capacities. The short-term water
portfolio problem refers to how to identify the mixtures of actions that can be
taken by the individual cooperating utilities that improve water-supply reliabil-
ity and financial stability for the whole region.
Despite aiding utilities to attain higher levels of supply reliability, the use
of drought management instruments such as water transfers or restrictions are
likely be detrimental to utilities’ finances due to potentially significant increases
3
in the variability of their costs (Watkins and McKinney, 1997; Zeff et al., 2014).
Together with outstanding debt, decreasing cash flows are significant factors
that negatively impact utilities credit ratings, which can increase the cost of bor-
rowing to build new infrastructure (Moody’s, 2017; Zeff et al., 2014; Zeff and
Characklis, 2013). This emphasizes the need to study existing and develop of
new financial instruments, such as index insurance against droughts, contin-
gency funds, and pricing structures to hedge water utilities’ finances against
droughts (Zeff and Characklis, 2013; Lopez-Nicolas et al., 2018). The financial
impacts of drought mitigation strategies, the current negative financial outlook
for the water utility sector (Moody’s, 2019), the lack of appropriate financing
for water utilities (Gleick et al., 2014), and the strong impact of operating deci-
sions on the appropriateness of capacity expansion (Tian et al., 2018) stress the
need for new frameworks that effectively combines short-term drought mitiga-
tion instruments backed by financial instruments and long-term infrastructure
investment planning.
Under extreme demand increase or droughts, even well designed soft-path
solutions and financial instruments may not eliminate the need to invest in the
costly hard-path approach (Gleick, 2003; Hall et al., 2019). Furthermore, aging
infrastructure may further exarcebate the need for costly investments in infras-
tructure (Hall et al., 2019), making even more pressing the need for a method-
ology focusing on minimizing investments in infrastructure without compro-
mising water-supply services. Acknowledging this need, researchers have been
developing methodologies to balance reliance on the soft- and hard-path ap-
proaches for addressing water supply planning and management. One exam-
ple is the real option analysis (ROA) (Cox et al., 1979) literature of the last 15
years (Wang and de Neufville, 2005; Erfani et al., 2018; Fletcher et al., 2017; Hui
4
et al., 2018; Fletcher et al., 2019), which is based on flexible decision-tree logic
based on local conditions and found through single-objective optimization (nor-
mally cost) that emphasizes near-term information to inform decisions about in-
frastructure planning and policy implementation. Other researchers have pre-
sented frameworks that consider multiple objectives while seeking to recom-
mend a sequence of infrastructure investments coupled with soft-path strate-
gies (Zeff et al., 2016; Borgomeo et al., 2018; Huskova et al., 2016; Beh et al.,
2015a; Mortazavi-Naeini et al., 2014). Alternatively, the Dynamic Adaptive
Policy Pathways approach (DAPP) takes pre-specified adaptive pathways that
combine management and infrastructure expansion actions based on monitored
action triggers over time (Haasnoot et al., 2013; Kwakkel et al., 2014; Kingsbor-
ough et al., 2016, 2017; Zandvoort et al., 2017). Across all of these frameworks,
one important and still unresolved issue is expanding their capacity to address
a broad range of uncertainties.
1.1.2 Considering Uncertainty
Hydro-climatic uncertainties such as near-term streamflows and evaporation
rates affect both short-term management and long-term planning problems, and
have been extensively studied and characterized with well-established statis-
tical techniques (Stedinger, 1993; Martins and Stedinger, 2000; Herman et al.,
2016; Lall, 1995; Kirsch et al., 2013). However, water infrastructure invest-
ment and management problems include a multitude of uncertainties referred
to hereafter as “deep uncertainties” (Kwakkel et al., 2016b; Knight, 1921). Deep
uncertainties cannot, as described by Knight (1921, p. xiv), be treated as “a gam-
ble on a known mathematical chance”. In other words, they lack consensus on
5
their underlying probability distributions as well as on individual or collective
outcomes. Such uncertainties have also been termed Knightian, or scenario un-
certainties, by multiple authors, including Walker et al. (2013); Lempert et al.
(2006) and Kwakkel et al. (2016b). Examples include regional demand growth,
political and economic uncertainties, and climate change (Herman et al., 2014;
Milly et al., 2008).
Uncertainty is often included in water utility planning and management ex-
ercises by sampling or designing representative states-of-the-world (SOWs) that
shape water supply risks (Watson and Kasprzyk, 2016; Kwakkel et al., 2014), al-
though how to perform such design or sampling is an important decision in any
drought mitigation application, defining perceptions and preferences related to
key performance objectives, vulnerabilities, and/or the robustness of a system.
Fundamental to this challenge is the choice of what uncertain factors should be
sampled or included in generating alternative scenarios. Next is a review of ex-
isting frameworks, how they solve the water utility planning and management
problem, and how they handle the underlying uncertainties.
1.1.3 Existing Water-Infrastructure Planning and Management
Software and Decision-Making Framworks and Their
Limitations
There is an extensive body of literature dealing with water infrastructure plan-
ning and management under uncertainty. Broadly, there are three dominant
classes of methodologies that are at present used in water supply planning con-
6
texts: (1) pre-specified deterministic scenarios/narratives, (2) scenario trees, and
(3) globally sampled stochastic scenario analysis. As an example of determin-
istic scenario-based planning, Liu et al. (2008) analyzed four semi-arid river
basins in the south-western US seeking to find water management strategies
based on integrated modeling that would minimize lasting or irreversible envi-
ronmental impacts (mostly on local vegetation) while allocating enough water
for competing uses. The authors analyzed a pre-specified set of scenarios based
on their a priori specified problem definition and input from stakeholders as
well as water management experts. Scenario trees is used in ROA approaches
such as Erfani et al. (2018), in which scenarios are created simulating a stochastic
process so that as the simulation of future decisions progresses, all future possi-
bilities share a common past. As an alternative example of stochastic scenarios,
Brown et al. (2012), Moody and Brown (2013), and Zeff et al. (2016) use ensem-
bles of reservoir inflows estimated using regression techniques to estimate and
assess climate vulnerability. Climate vulnerability analysis focuses on reduc-
ing risks, or losses, calculated as the product of a loss function times subjective
probabilities for the climate scenarios. The probabilities are elicited by contrast-
ing a mixture of global climate model projections, paleodata (if available), and
synthetically generated weather. These approaches are termed Decision Scaling,
and have been recently expanded by Ghile et al. (2014) and Lownsbery (2014)
to include hydroeconomic sensitivities.
The Decision Scaling example relates to a broader body of literature focused
on planning under deep uncertainty. Exploratory modeling (Bankes et al., 2001),
which involves running planning models across broad global samples of hy-
pothesis, has emerged as a primary strategy for discovering which combina-
tions of deeply uncertain factors are of high consequence in decision problems,
7
as opposed to considering one most likely future projection (Walker et al., 2003;
Lempert et al., 2006; Walker et al., 2013; Zeff et al., 2014; Herman et al., 2014). As
noted by Dessai et al. (2009), the intent of exploratory modeling is to shift focus
from predicting future conditions to understanding which conditions lead to de-
cision relevant consequences. Given a better understanding of the consequences
related to these discovered conditions, decision makers can then evaluate their
relative importance or plausibility.
As reviewed by Herman et al. (2015), the selection of decision alternatives
to analyze in exploratory modeling frameworks can either be done a priori or
with computational search. In the case of decision alternatives that are spec-
ified a priori, the decision analysis then takes the form of a classical discrete
choice among those alternatives based on their interpreted robustness. This ap-
proach has been commonly employed across the full range of decision support
frameworks in this area (Brown et al., 2012; Moody and Brown, 2013; Hipel and
Ben-Haim, 1999; Lempert et al., 2006; Bryant and Lempert, 2010). Conversely,
a growing number of studies are using search strategies to discover high per-
forming design alternatives in their initial stage of analysis and then evaluat-
ing their robustness to inform design selection (Korteling et al., 2013; Kasprzyk
et al., 2013; Hamarat et al., 2014; Kwakkel and Haasnoot, 2015; Giuliani and
Castelletti, 2016; Singh et al., 2015; Zeff et al., 2014; Herman et al., 2016). None
of the questions presented thus far are new, although utilities and researchers
still have difficulties in answering them through existing water-infrastructure
planning and management frameworks (Bonzanigo et al., 2018; Paulson et al.,
2018).
Considering the frameworks coupling soft- and hard-path approaches, the
8
ROA is limited in its accounting of stakeholders with diverse interests as a
single-objective approach and the scope limiting challenges in addressing a
broader suite of of uncertainties pose by its curse of dimensionality (Dittrich
et al., 2016). Borgomeo et al. (2018) address some of these concerns by propos-
ing a multi-objective evaluation of candidate water supply investments. They
maximize robustness and minimize risk and cost of a 30-year long fixed con-
struction schedule encompassing multiple infrastructure options. However, the
derived sequence of investments is not adaptive and does not account for evolv-
ing information feedback. This limitations also exists in the recent studies by
Beh et al. (2015b) and Huskova et al. (2016), although the former seeks to im-
prove adaptivity by incorporating a similar emphasis on near term informa-
tion as done in ROA. Alternatively, the DAPP approach (Haasnoot et al., 2013;
Kwakkel et al., 2014; Kingsborough et al., 2017) adds flexibility to the planning
process by making use of potentially diverse metrics (signposts) to inform adap-
tive action pathways. DAPP, however, still maintains a strong reliance on the
use of limited numbers of predefined action sequences that require high levels
of institutional stability and coordinated consensus over the long term (30 to
100 years in case studies).
The adaptive planning frameworks reviewed above all must confront issues
related to institutional uncertainty. They all make the “omniscient and omnipo-
tent lawgiver” assumption (Walker et al., 2001), ignoring all inherent “policy
myopia” (Nair and Howlett, 2017). This is similar in effect to the “planning
fallacy” described by Scruton (2013), in which a regimented plan prevents the
necessary use of live information to effectively realize the needs that motivated
its implementation in the first place. Additionally, as pointed by Scruton (2013)
and Bosomworth et al. (2017), decades-long plans inherently assume that cur-
9
rent goals and values are uncontested and will remain so for the duration of
the plan, together with the required institutional context. These limitations, ac-
knowledged by (Borgomeo et al., 2018; Beh et al., 2015b) though not addressed,
make the case for a framework that maximizes the use of current data and
minimizes the over-specification of decisions to be made in the future. Lastly,
when commenting on the DAPP approach Bosomworth et al. (2017) makes the
broadly valid point that the lack of flexibility in an approach may clash with
case-specific institutional issues that may render the approach unfeasible. More
specifically, planning approacha lacking adaptivity require higher degrees of
abstraction of the particularities of individual and concrete institutional con-
texts, some of which in direct conflict with the structural assumptions that are
core to planning frameworks. In light of these limitations, an approach that
takes in current information at short time scales throughout the life of the plan
and fits within the annual budget approval of most municipalities is desirable.
Zeff et al. (2016) introduced an integrated infrastructure planning and man-
agement approach, in which infrastructure investment and water portfolio
management policies specify short- and long-term decision-making rules based
on the risk-of-failure metric (ROF), as well as candidate orderings forsequencing
of infrastructure investments. By combining infrastructure planning and man-
agement rules in a single policy, the infrastructure pathway approach aids the
design of drought mitigation instruments that are carefully coupled with new
infrastructure investments to minimize to help better balance system reliability
and reduce debt burden. Despite the high adaptivity on both the short and long
term, the infrastructure pathways framework introduced Zeff et al. (2016) is
limited in its treatment of uncertainties. The limitation comes from its reliance
on creating SOWs based only on well-characterized stationary hydro-climatic
10
uncertainties while using expert-elicited assumed values for deep uncertainties
(e.g., demand growth rates, pricing changes, behavioral responses to water re-
strictions, etc.), limiting the scope of the analysis and potentially limiting the
robustness of the resulting policies. Furthermore, although the infrastructure
pathways framework introduced by Zeff et al. (2016) proposes a fixed suite of
major candidate investments over decades to come making it still subject to
the same challenges that emerge changes in societal or institutional context, the
plans proposed by this approach includes mechanisms for incorporating envi-
ronment and economical information throughout its life, adding an opportunity
for endogenous learning. Therefore, the joint-infrastructure pathways and wa-
ter portfolio management introduced by Zeff et al. (2016) serves as the basis
for the Deeply Uncertain (DU) Pathways methodology and for the WaterPaths
simulation software, both developed in this thesis research.
1.2 Dissertation Objectives
This dissertation contributes the WaterPaths open-source stochastic simulation
software and the DU Pathways methodology to reduce the computational and
conceptual barriers to integrated infrastructure investment pathways and wa-
ter portfolio management. In combination, WaterPaths and the DU Pathways
framework introduced in this dissertation provide the first formal integration of
dynamic and adaptive water infrastructure pathways, financial and supply- and
reliability-focused drought-mitigation instruments, and the recent extensions
of the Many-Objective Robust Decision-Making (MORDM) framework that in-
corporate deep uncertainties in search-based identification candidate actions
(Kasprzyk et al., 2013; Kwakkel et al., 2014; Watson and Kasprzyk, 2016). For
11
this, WaterPaths incorporates decision making at two time scales: weekly, for
operational management decisions, and annual, for infrastructure investment
decisions. Expanding on the approach recommended by Zeff et al. (2016), the
integrated infrastructure investment and management policies designed with
DU Pathways specify short- and long-term decision-making rules based on the
risk-of-failure metric (ROF), as well as candidate orderings for infrastructure in-
vestments. By combining infrastructure planning and management rules in a
single policy, the DU Pathways approach aids, here through its use of Water-
Paths, the design of drought mitigation instruments tailored to new infrastruc-
ture to minimize the need of further investments.
To assure a broad uncertainty analysis, the DU Pathways methodology ex-
ploits WaterPaths to account for multiple well-characterized and deep uncer-
tainties in each function evaluation during search, improving the search-based
identification of candidate infrastructure investment and management policies.
Additionally, the framework formalizes a detailed evaluation of the multi-city
policies’ robustness tradeoffs. DU Pathways aids stakeholders in navigating
these tradeoffs through a combination of visual decision analytics (Woodruff
et al., 2013) and enhanced scenario analysis in which boosted trees (Schapire,
1999) aid in the identification of the uncertainties most strongly shape the in-
frastructure investment and management policies’ vulnerabilities. The appli-
cation of the DU Pathways framework is demonstrated on the Research Trian-
gle test case (Zeff et al., 2016; Herman et al., 2014; Zeff et al., 2014), where this
dissertation will explore how the region’s vulnerabilities evolve under alterna-
tive policies, identify significant interdependencies between the region’s four
utilities, and show the importance of carefully balancing supply and financial
instruments.
12
1.3 Scope and Organization
1.3.1 Chapter 2 — Background on Methodological Components
This chapter presents an overview of computational methods that used in all
of the subsequent chapters. These are the Borg multiobjective evolutionary al-
gorithm (Borg MOEA) and the synthetic streamflow, evaporation rate, and de-
mand time-series scenario generation technique. The Borg MOEA is used in
all of the subsequent chapters to discover Pareto-efficient infrastructure invest-
ment and water portfolio management policy alternatives, while the time-series
generation methods were used to generate SOW to evaluate performance.
1.3.2 Chapter 3— Reducing Regional Drought Vulnerabili-
ties and Multi-city Robustness Conflicts Using Many-
objective Optimization Under Deep Uncertainty
Chapter 3 introduces a multi-stakeholder extension of the MORDM framework
that informed the formal model base development for WaterPaths (Chapter 4)
and the broader DU Pathways framework (Chapter 5). This chapter focused
on better accounting for deeply uncertain factors when identifying cooperative
drought management strategies based on soft-path approaches for cooperative
regional drought management. The analysis is applied in the Research Trian-
gle area of North Carolina, with a focus on the water utilities within Raleigh,
Durham, Cary and Chapel Hill. Prior analysis of this region through the year
2025 has identified significant regional vulnerabilities to volumetric shortfalls
13
and financial losses. Moreover, efforts to maximize the individual robustness
of any of the mentioned utilities also have the potential to strongly degrade
the robustness of the others. The results in Chapter 3 show that appropriately
designing adaptive and high-efficacy drought mitigation instruments required
stressing them with a comprehensive sample of deeply uncertain factors in the
computational search phase of MORDM. Search under the new ensemble of
SOWs is shown to fundamentally change perceived performance tradeoffs and
substantially improve the robustness of individual utilities as well as the over-
all region to water scarcity. The results in Chapter 3 show that search under
deep uncertainty enhanced the discovery of how cooperative water transfers,
financial risk mitigation tools. The results highlight that coordinated regional
demand management employed jointly strongly improves regional robustness
and decreases robustness conflicts between the utilities. Insights from this work
have general merit for regions where adjacent municipalities can benefit from
cooperative regional water portfolio planning.
1.3.3 Chapter 4 — WaterPaths: A Stochastic Simulation Frame-
work for Water Supply Infrastructure Investment Man-
agement
Chapter 4 contributes the WaterPaths model: an open-source simulation soft-
ware that implements, extends, and generalizes the short-term water portfolio
management problem and extended uncertainty sampling presented in Chap-
ter 3 and combines it with the integrated infrastructure pathways presented
in Zeff et al. (2016). WaterPaths is designed to simulate stochastic risk and
14
demand-based sequencing of long-term regional water infrastructure construc-
tion in parallel with short-term operational decisions (e.g., restrictions, trans-
fers, financial instruments, etc), bridging the gap between weekly management
and yearly planning for water utilities. The solution framework is designed to
flexibly represent new conservation and financial mitigation strategies, such as
schemes for allocating raw and treated transfer water among different utilities
or different ways of structuring drought insurance policies. Moreover, Chap-
ter 4 introduces a hypothetical Sedento Valley test case to demonstrate Water-
Paths capabilities. The Sedento Valley test case is proposed as a new bench-
mark problem for testing optimization algorithms to be used in water system
engineering and aid the development of improved planning methodolgies. The
code, written in C++, makes use of parallelization strategies to perform scalable
ensemble-based exploratory analyses ranging from small tests on desktops to
larger runs on clusters or cloud computing platforms. It should be noted that
the software development leading to WaterPaths occurred simultaneously with
the formalization of the DU Pathways framework illustrated for the Research
Triangle region in NC in Chapter 5, resulting in the formulations of both be-
ing partly indissociable. Therefore, a small amount of forward-referencing in
Chapter 4 was innevitable.
15
1.3.4 Chapter 5 — Deeply Uncertain Pathways: Integrated
Multi-City Regional Water Supply Infrastructure Invest-
ment and Portfolio Management
Chapter 5 contributes the Deep Uncertainty (DU) Pathways framework, in
which WaterPaths is used to combine ROF decision rules presented in Chapter
3, dynamic adaptive policy pathways concepts, and a careful consideration of
time-evolving information feedbacks to yield management-conditioned infras-
tructure pathways for regions. The DU Pathways’ framework has been devel-
oped to carefully consider multi-actor regional contexts with the goal of aiding
stakeholders in discovering pathway policies that attain high performance lev-
els across challenging, deeply uncertain futures and to guide robustness com-
promises that may be necessary between regional actors. As demonstrated
again in the Research Triangle region of North Carolina, DU Pathways clari-
fies how to identify robust infrastructure investment and management policies
across the municipalities of Raleigh, Durham, Cary, and Chapel Hill. Our re-
sults provide insights about the most cost-effective infrastructure options to be
pursued in the near-term, and clarify which sources of uncertainty drive the
performance and robustness of the regional system and of the individual ac-
tors.
1.3.5 Chapter 6 — Contributions and Future Work
Finally, Chapter 6 summarizes the contributions of this dissertation to the body
of existing knowledge in the field of water resources systems and planning. It
16
also provides recommendations for future work, such as features to be added to
WaterPaths and for different treatments of deep uncertainty for water resources
planning and management.
1.3.6 Author Contributions for Published Work
Chapter 3: borrowing from previous studies by H.B. Zeff and J.D. Herman under
the supervision of Professor Patrick Reed and Gregory Characklis, Bernardo
Trindade conceived the studies, modified the inherited model, performed the
optimization runs and data analysis, and wrote the paper under the supervision
of Professors Patrick Reed and Gregory Characklis.
Chapters 4: borrowing from previous studies by H.B. Zeff and J.D. Her-
man under the supervision of Professor Patrick Reed and Gregory Characklis,
Bernardo Trindade conceived the study, wrote a new model, performed the op-
timization runs and data analysis, and wrote the paper under the supervision
of Professors Patrick Reed and Gregory Characklis.
Chapters 5: borrowing from previous studies by H.B. Zeff and J.D. Her-
man under the supervision of Professor Patrick Reed and Gregory Characklis,
Bernardo Trindade conceived the study, implemented the test case conceived in
partnership with David Gold, performed the optimization runs and data anal-
ysis, and wrote the paper together with David Gold under the supervision of
Professors Patrick Reed and Gregory Characklis.
17
CHAPTER 2
METHODOLOGICAL COMPONENTS
2.1 Borg Multi-Objective Evolutionary Optimization Algo-
rithm
In the recent literature, multiobjective evolutionary algorithms are emerging
as a primary tool used for solving water supply infrastructure management
and/or planning problems (Zeff et al., 2016; Quinn et al., 2018; Kwakkel et al.,
2014; Borgomeo et al., 2016, 2018; Beh et al., 2015a). Zeff et al. (2016) employed
multiobjective evolutionary algorithms in the context of the Research Triangle to
identify water infrastructure management and planning policies effective across
an ensemble of synthetically generated streamflows, evaporation rates, and de-
mands (all well-characterized uncertainties). Such policies combine water trans-
fers, financial risk instruments, demand management and infrastructure con-
struction strategies. Seeking to find more robust candidate solutions, Kwakkel
et al. (2014) and Watson and Kasprzyk (2016) explored simpler problem for-
mulations while also sampling deep uncertainties in their search for optimal
tradeoff policies. Combining the problem formulation presented in Zeff et al.
(2016) and sampling strategies similar to Kwakkel et al. (2014) and Watson and
Kasprzyk (2016) as desired in this dissertation creates a potential for more ef-
fective policies but in turn increases the complexity of the problem. Mathemat-
ically speaking, the policy structure, based mostly on discrete decisions and on
permutations of infrastructure options, makes the resulting objectives (perfor-
mance metrics) non-convex and discontinuous functions of the policy variables,
while the uncertainty sampling adds a high degree of non-linearity to them.
18
In order to handle these difficulties and the high number of objective func-
tions (up to six) of the the water portfolio planning problem, we used in the
following chapters the master-worker and multi-master variants of the par-
allel Borg multi-objective evolutionary algorithm (MS- and MM-BorgMOEA)
(Hadka and Reed, 2013; Hadka et al., 2013; Hadka and Reed, 2014). All versions
of the Borg MOEA have demonstrated high levels of performance when applied
to multiobjective problems in a variety of applications, including water supply
portfolio planning, pollution control given ecological thresholds, groundwater
monitoring design, and reservoir control (Bode et al., 2019; Reed et al., 2013;
Hadka et al., 2012; Ward et al., 2015; Zatarain Salazar et al., 2016). Figure 2.1
shows a schematic flow chart of the main loop of the serial version (original) of
the Borg MOEA.
Initially, a number of solutions (here infrastructure planning and manage-
ment policies) are randomly generated and stored in the population (top-left
box) after being checked for Pareto-dominance, described in Section 2.1.2. Next,
an operator (dashed ellipses) is chosen to generate an offspring solution based
on k = 2, 3 from the population and potentially of the archive, as explained in
Section 2.1.1. This offspring is then evaluated with a model (here WaterPaths)
to be potentially circled back to the population and/or archive, as described
in Sections 2.1.2 and 2.1.3, closing the loop which is performed a pre-specified
number of times. The following subsections will introduce some of the key
concepts behind Borg MOEA highlighted in Figure 2.1 and both parallelization
strategies.
19
Figure 2.1: Flowchart of the Borg MOEA main loop adapted from (Hadka
and Reed, 2013). First, one of the recombination operators is
selected based on its performance. After, for an operator re-
quiring k parents to generate an offspring, Borg selects k − 1
from the population and one from the archive. The resulting
offspring is evaluated and considered for inclusion in the pop-
ulation and archive.
2.1.1 Adaptive Operator Selection
In order to be a suitable algorithm across a wide range of problems, Borg MOEA
relies on adaptive operator selection (Vrugt and Robinson, 2007) based on prob-
abilities calculated from solutions in its archive (Hadka et al., 2013) . An evo-
lutionary optimization operator is a function that combines aspects of parent
solutions to create an offspring solution in a semi-randomized fashion. The six
operators implemented in Borg MOEA, shown in Figure 2.2, are: simulated bi-
nary crossover (Deb and Agrawal, 1994), differential evolution (Storn and Price,
1997), parent-centric crossover (Deb et al., 2002a), unimodal normal distribution
crossover (KITA et al., 2000), simplex crossover (Tsutsui et al., 1999), polynomial
20
mutation (PM), and uniform mutation (UM). Figure 2.2 shows the distributions
of offspring solutions (small dots) produced from parent solutions (big dots) by
all six operators in plemented in the Borg MOEA.
Figure 2.2: Example distributions of offspring solutions (small dots) pro-
duced from parents solutions (large dots) by the six operators
included in Borg MOEA. From Hadka and Reed (2014).
At the beginning of an optimization run, all of Borg MOEA’s K operators
(here K = 6) are assigned equal probability Qi = 1/K of being selected to gen-
erate a new offspring solution. However, as the search progresses, meaning
after a few iterations over the loop in Figure 2.1, the selection probabilities Qi
of each operator are adjusted according to how many Pareto-efficient solutions
each operator has generated until then Hadka and Reed (2013); Hadka et al.
(2013); Hadka and Reed (2014). More specifically, the probabilities Qi are up-
dated over time with a higher probability assigned to the operators with more
solutions admitted into the archive at that stage of the search, as shown in Equa-
tion 2.1.
21
∑ Ci + ςQi = (2.1)K
j=1(Cj + ς)
where Ci is the number of solutions generated by operator i that was admitted
into the archive and ς is a constant to ensure even operators that are perform-
ing poorly at a given time have a small probability of being chosen—Hadka
and Reed (2013) suggests ς = 1. This allows the Borg MOEA to not only se-
lect operators that provide better offspring on each region of the solution space
based on mathematical properties of both, but also to focus the later stages of the
search on operators better at exploitation rather than exploration for the prob-
lem at hand, given sufficient search time. The offspring solutions generated by
the operators may be discarded or stored in the population and/or archive, as
described next.
2.1.2 Solution population
The Borg MOEA has two groups of solutions stored in memory at a given time:
the population and the archive. The population is the most current pool of
strongly non-dominated solutions, found by the Borg MOEA. A Pareto-efficient
solution is one that perfoms better than all solutions in a set in a pairwise com-
parison in at least one objective. A set of Pareto-efficient solutions is called a
Pareto set or Pareto front. Below is a definition of strong non-dominance:
F (a)  F (b) ⇐⇒ Fi(a) ≤ Fi(b) ∀i ∈ {1, ..., D} ∧ ∃i s.t. Fi(a) < Fi(b) (2.2)
where F is a vector of D objectives in an all-minimization problem, and a and b
are different vectors of decision variables. In other words, a strongly dominates
b if all objective values in F (a) are equal or smaller than in F (b) and at least
22
for one objective i, Fi(a) is strictly smaller than Fi(b) (Pareto, 1896; Coello et al.,
2006).
A new offspring solution is admitted into the archive if it strongly dominates
one or more population members, in which case replaces a randomly selected
solution it dominates. If this offspring is dominated by at least one solution in
the population (i.e. if it performs worse in every objective than at least one solu-
tion in the population), it is discarded. If it does not dominate nor is dominated
by any solution in the population, it is either added to the population or re-
places a member of the population at random so that the size of the population
is maitained smaller than 125% of the size of the archive—and never smaller
than 100% either.
2.1.3 Solution Archive and -Dominance
As explained in Section 2.1.2, Pareto-efficient solutions may be randomly
deleted from the population, potentially resulting in the loss of a good solu-
tion. The goal of archive is to keep Pareto-efficient solutions on record in case
of one is deletied from the population. The archive is a set of Pareto-efficient
solutions found by the Borg MOEA from the start of the optimization process
and kept on record until found to be dominated by a new solution. For an off-
spring solution to be included in the archive, it must not be dominated by any
solution already in the archive by margins i for all objectives i ∈ {1, ..., D}. This
is because following the definition in Equation 2.2, it is possible for set of non-
dominated solutions to contain dozens of thousands (up to an infinite number)
of solutions, which is not desirable for decision-making purposes. Such stricter
23
type of dominance, called -dominance (Laumanns et al., 2002), is used to keep
the size of the archive resonably small (ideally a few hundreds) by preventing
solutions non-dominated by a negligible fraction of the value of an objective
from being included in the archive. In -dominance, solutions are placed on a
grid as the one shown in Figure 2.3 and checked for dominance for each objec-
tive i according to i tolerances. Laumanns et al. (2002) defined -dominance
as:
F (a)F (b) ⇐⇒ (1−i)·Fi(a) ≤ Fi(b) ∀i ∈ {1, ..., D} ∧ ∃i s.t. (1−i)·Fi(a) < Fi(b)
(2.3)
Figure 2.3: Illustration of two-objective optimization problem on an  grid,
adapted from Kasprzyk et al. (2013). Only one solution within
each tolerance band 1 and 2 is kept in the Pareto set. Fol-
lowing the concept of -dominance, B, C, and E are the only
non-dominated points.
The goal to be attained by running the Borg MOEA is therefore to find a
best approximation of the true Pareto front for the problem, called the Pareto-
approximate front. Beyond constraining the size of an Pareto-approximate so-
lution set, the use of -dominance also guarantees two desirable properties of
a multiobjective optimization algorithm given sufficient search time: conver-
24
gence to the true Pareto-efficient solution set and diversity across the set. All i
values are specified by the user, observing the constraint i > 0 ∀i ∈ {1, ..., D}
(Laumanns et al., 2002).
2.1.4 Parallelization Strategies for the Borg MOEA
The Borg MOEA was originally developed as a serial search tool where func-
tion evaluation is done in the sequence for each member of its search popula-
tion (Hadka and Reed, 2013). To make use of modern parallel computing clus-
ters, two parallelization strategies have been developed for it in recent years:
the Master-Worker Borg MOEA (MS-Borg MOEA) and the Multimaster Borg
MOEA (MM-Borg MOEA), both of which are used in the subsequent chapters
of this dissertations. In the MS-Borg MOEA algorithm, the master process dis-
tributes function evaluations across worker processes, each running on one or
on a group of cores (processors), allowing for the search to be run in parallel.
This distribution of work in MS-Borg MOEA is shown in Figure 2.1.4a.
On the other hand, the multi-master variant of the Borg MOEA generalizes
the original Borg MOEA’s adaptive operators across coordinating instances of
the master process each with its own suite of workers, as shown in Figure 2.1.4b.
This hierarchical parallelization scheme (Cantu-Paz, 2000; Hadka and Reed,
2014) takes advantage of both the multiple population and master-worker paral-
lelization strategies to increase scalability and the difficulty of problems that can
be addressed (Tang et al., 2007; Reed and Hadka, 2014). The MM-Borg MOEA’s
coordinating instances automatically detect search stall and share archived so-
lutions and operator probabilities to enhance the efficiency and effectiveness of
25
Figure 2.4: Panel (2) shows a diagram of the master-worker implemen-
tation of the Borg MOEA. The master node maintains the -
dominance archive and runs the main loop of the serial Borg
MOEA shown in Figure 2.1. The decision variables gener-
ated through the six operators inplemented in the master node
are transmitted to the worker nodes, and the evaluated objec-
tive function values and constraints are returned to the master
node. Panel (b) shows a diagram of the multi-master imple-
mentation of the Borg MOEA. The multi-master Borg MOEA
consists of two or more mastere-worker instances (the diagram
depicts three) periodically communicating with one controller
node, which maintains a global -archive. (1) Each master node
periodically transmits its local -dominance archive to the con-
troller, which updates the global archive. (2) When a master
node is not advancing its pareto-approximate front, it sends a
help message to the controller. (3) The controller responds with
guidance, which includes the global -dominance archive and
global operator probabilities. From Hadka and Reed (2014).
26
search for difficult problems (Giuliani et al., 2018; Quinn et al., 2017, 2018). The
next section explains the synthetic streamflow generation engine used to sample
streamflows and evaporation rates for simulating the water portfolio planning
problem.
2.2 Scenario Generation for Streamflow and Evaporation Rates
Time Series
All synthetic inflows and evaporation rates for this work were generated using
the methodology proposed by Kirsch et al. (2013). More specifically, this process
was used initially to generate cross-site-correlated matrices with time series of
log-normally distributed synthetic weekly inflows NI ∈ RNys·52×Nrs for eachk
reservoir and gauge k. Secondly, it was used to generate the corresponding
matrices E ∈ RNys·52×Nrs of evaporation rates for reservoirs only. Given thek
procedure for generating streamflow and evaporation series is nearly identical,
matricesNI and E will be referred to asQ hereafter.
The first step is to create matrices of years of historical (subscript h) stream-
flows and evaporation ratesQ ∈ RNhr·52×Nrh , one of each for each(wate)r sourcek
k. Next, these matrices go through the transformation Y y,w = ln Qy,wh h to cre-k k
ate matrices Yhs for all sites. These are next converted into standard-normally-
distributed matrices Z ∈ RNhr×52h , in which:k
Y y,w
Zy,w = h
− µ̂
h (2.4)σ̂
Matrix Zy,wh is next used to create matrix C ∈ RNys·52×Nrs . Each column ofk k
27
Cs is filled by sampling with replacement from the corresponding week of thek
year (column) across all historical years in Zh . The column-wide sampling isk
performed by creating 52 Nys-long vectors of bootstrap samples ranging from
1 to Nhr, to preserve cross-site correlation across reservoirs and gauges — the
samples correspond to the index of the historical year from which the specific
streamflow/evaporation sample is drawn. The samples MatrixCs is, therefore,k
a matrix of uncorrelated standard-normally-distributed historical streamflows
or evaporation rates. Temporal autocorrelation is then imposed on site k by
using Choleskly decomposition to create an upper-triangular matrix Uk such
that corr(Zh ) = UTk Uk, and multiplying it by Cs , or Ss = Cs Uk, resulting ink k k k
matrices Ss streamflows and evaporation rates. As a way of improving inter-k
annual correlation, the described process is performed on matrix ′Qs , a versionk
of matrixQs shifted by 27 weeks Kirsch et al. (2013), resulting in matrices
′
S
k s .k
Matrix ′Ss has auto and cross-site correlated values of standard, log-normalk
streamflows with the same sample means and variances are the same as the his-
torical data. The last step is therefore to transform matrices ′Ss from log backk
to real space using the log-means and log-variances of the historical data. Ad-
ditionally, the log-mean of the historical data was scaled in the transformation
to simulate climate change. The scaling was done by multiplying the histori-
cal log-mean and log-variance by weekly sinusoid-time-series, the parameters
of which deeply uncertain themselves, before performing the reverse log trans-
formation Quinn et al. (2018). The mathematical formulation of the described
transformation is presented below:
′
Qy,w
y,w
= eSs ,r σ̂+µ̂·[Ar·sin(φk r+frw)]s ,r (2.5)k
28
where Ar is the amplitude assigned to that exploratory SOW, φr the phase, and
fr the frequency of the sinusoids. Examples of sinusoids used to create the
streamflows and evaporation ensembles and the corresponding flow duration
curves are presented in Figure 2.5.
Figure 2.5: Panel (a) shows examples of flow duration curves for historical
inflows, synthetically inflows with stationary log-mean, and
synthetically inflows with non-stationary log-mean. Panel (b)
shows all sinusoid-multiplier series of the log-mean of inflows
and evaporation rates used to generate the panel (a).
The generation of evaporation rates time series happens concomitantly. Cor-
relation between inflows and evaporation rates for a single reservoir is enforced
by using the same years samples to build both inflow and evaporation matri-
ces Cs . The only difference in the synthetic time series generation processesk
for inflows and evaporation is that the historical evaporation rates used in this
problem are normally distributed, which eliminates the need to work with log-
transform the data.
29
The process for creating the re-evaluation series was nearly identical for op-
timization and re-evaluation series. The only differences are that that Ar, φr,
and fr are the same for all scenarios r for each of the 5,000 re-evaluation series
sets. Therefore, all the variability across scenarios in each function evaluation
comes from sampling process alone and not from the sinusoidal trends. Fig-
ure 2.5 shows the flow duration curves and sinusoid mean-multipliers for all
exploratory scenarios.
2.3 Scenario Generation for Synthetic Demand Time-series
For generating stochastic demand time series correlated with the corresponding
synthetic streamflows, the historical weekly deviations of demands and stream-
flows from the corresponding annual means are used to to build empirical joint
probability distributions from which synthetic demands can be sampled based
on future synthetic inflows. The first step is to whiten the historical inflows
and demands. For this, the historical data is divided by the corresponding an-
nual historic mean, as shown below for demand (the same logic applies to the
log-streamflows):
y,w
Dy,w
D
= hrmf (2.6)
D̄yh
where Dy,wmf is the recorded demand as a fraction of the corresponding annual
mean D̄y for week w of historical year y, and Dr is the value of the correspond-
ing recorded weekly demand. This process should eliminate changes in annual
demands and inflows (mostly resulting from changes in population and local
climate) so that all years of historical data are used to build empirical demand
30
distributions for each week of the year, whose means and standard deviations
are calculated as below (again, the same logic applied to log-streamflows):
∑Nhr
Dy,wrmf
w y=0µ√D = (2.7)√h N√√√∑
hr
Nhr
√ ( )Dy,w w 2rmf − µDh
w y=0σD = (2.8)h Nhr − 1
The historical demands and streamflows display a negative correlation dur-
ing the months of April through September because of lawn irrigation and no
correlation from October to March. The historical weekly streamflows are then
split into sixteen bins for each week of the year based on their values. To gen-
erate a demand sample for a given future week during irrigation season, the
generator selects the bin corresponding to the value of the whitened synthetic
streamflow for that same week of the year, samples a whitened demand value
from the corresponding historical whitened demands, and converts the value
back to volume per time in real-space based on the corresponding projected
annual mean. For non-irrigation season, sampling occurs across any of the his-
torical whitened demands independently of the streamflows.
31
CHAPTER 3
REDUCING REGIONAL DROUGHT VULNERABILITIES AND
MULTI-CITY ROBUSTNESS CONFLICTS USING MANY-OBJECTIVE
OPTIMIZATION UNDER DEEP UNCERTAINTY
This chapter is drawn from the following peer-reviewed journal article: B.
Trindade, P. Reed, J. Herman, H. Zeff, G. Characklis, Reducing regional drought
vulnerabilities and multi-city robustness conflicts using many-objective optimiza-
tion under deep uncertainty, Advances in Water Resources 104 (2017) 195-209.
doi:10.1016/j.advwatres.2017.03.023. Funding for this work was provided by the Na-
tional Institute of Food and Agriculture, U.S. Department of Agriculture (WSC Agree-
ment No. 2014-67003-22076). Additional support was provided by the U.S. Na-
tional Science Foundation’s Water, Sustainability, and Climate Program (Award No.
1360442). The views expressed in this work represent those of the authors and do not
necessarily reflect the views or policies of the NSF or the USDA.
3.1 Abstract
Emerging water scarcity concerns in many urban regions are associated with
several deeply uncertain factors, including rapid population growth, limited co-
ordination across adjacent municipalities and the increasing risks for sustained
regional droughts. Managing these uncertainties will require that regional wa-
ter utilities identify coordinated, scarcity-mitigating strategies that trigger the
appropriate actions needed to avoid water shortages and financial instabilities.
This chapter focuses on the Research Triangle area of North Carolina, seeking
to engage the water utilities within Raleigh, Durham, Cary and Chapel Hill
in cooperative and robust regional water portfolio planning. Prior analysis of
32
this region through the year 2025 has identified significant regional vulnerabili-
ties to volumetric shortfalls and financial losses. Moreover, efforts to maximize
the individual robustness of any of the mentioned utilities also have the poten-
tial to strongly degrade the robustness of the others. This research advances a
multi-stakeholder Many-Objective Robust Decision Making (MORDM) frame-
work to better account for deeply uncertain factors when identifying coopera-
tive drought management strategies. Our results show that appropriately de-
signing adaptive risk-of-failure action triggers required stressing them with a
comprehensive sample of deeply uncertain factors in the computational search
phase of MORDM. Search under the new ensemble of states-of-the-world is
shown to fundamentally change perceived performance tradeoffs and substan-
tially improve the robustness of individual utilities as well as the overall region
to water scarcity. Search under deep uncertainty enhanced the discovery of how
cooperative water transfers, financial risk mitigation tools, and coordinated re-
gional demand management must be employed jointly to improve regional ro-
bustness and decrease robustness conflicts between the utilities. Insights from
this work have general merit for regions where adjacent municipalities can ben-
efit from cooperative regional water portfolio planning.
3.2 Introduction
Understanding and exploring the combined states-of-the-world (SOWs) that
shape water supply risks is a fundamental challenge in any drought mitigation
application, defining perceptions and preferences related to key performance
objectives, vulnerabilities, and/or the robustness of a system. Fundamental
to this challenge is the choice of what uncertain factors should be sampled
33
or included in generating alternative scenarios. Broadly, there are two domi-
nant methodologies that are at present used in water supply planning contexts:
(1) pre-specified deterministic scenarios/narratives and (2) globally sampled
stochastic scenario analysis. As an example of deterministic scenario-based
planning, Liu et al. (2008) analyzed four semi-arid river basins in the south-
western United States seeking to find water management strategies based on
integrated modeling that would minimize lasting or irreversible environmental
impacts (mostly on local vegetation) while allocating enough water for compet-
ing uses. The authors analyzed a pre-specified set of scenarios based on their a
priori specified problem definition and input from stakeholders as well as water
management experts. As an alternative example of stochastic scenarios, Brown
et al. (2012) and Moody and Brown (2013) use ensembles of reservoir inflows
estimated using regression techniques to estimate climate vulnerability. Cli-
mate vulnerability analysis focuses on reducing risks, or losses, calculated as the
product of a loss function times subjective probabilities for the climate scenarios.
The probabilities are elicited by contrasting a mixture of global climate model
projections, paleodata (if available), and synthetically generated weather. These
approaches are termed Decision Scaling, and have been recently expanded by
Ghile et al. (2014) and Lownsbery (2014) to include hydroeconomic sensitivities.
The Decision Scaling example relates to a broader body of literature focused
on planning under deep uncertainty. Deep uncertainties cannot, as described
by Knight (1921, p. xiv), be treated as “a gamble on a known mathematical
chance”. In other words, they lack consensus on their underlying probability
distributions as well as on individual or collective outcomes. Such uncertain-
ties have also been termed Knightian, or scenario uncertainties, by multiple au-
thors, including Walker et al. (2013); Lempert et al. (2006) and Kwakkel et al.
34
(2016b). Exploratory modeling (Bankes et al., 2001) — which involves running
planning models across broad global samples of hypothesis — has emerged
as a primary strategy for discovering which combinations of deeply uncertain
factors are of high consequence in decision problems, as opposed to consider-
ing one most likely future projection (Walker et al., 2003; Lempert et al., 2006;
Walker et al., 2013; Zeff et al., 2014; Herman et al., 2014). As noted by Dessai
et al. (2009), the intent of exploratory modeling is to shift focus from predict-
ing future conditions to understanding which conditions lead to decision rele-
vant consequences. Given a better understanding of the consequences related
to these discovered conditions, decision makers can then evaluate their relative
importance or plausibility.
As reviewed by Herman et al. (2015), the selection of decision alternatives
to analyze in exploratory modeling frameworks can either be done a priori or
with computational search. In the case of decision alternatives that are spec-
ified a priori, the decision analysis then takes the form of a classical discrete
choice among those alternatives based on their interpreted robustness. This ap-
proach has been commonly employed across the full range of decision support
frameworks in this area (Brown et al., 2012; Moody and Brown, 2013; Hipel and
Ben-Haim, 1999; Lempert et al., 2006; Bryant and Lempert, 2010). Conversely, a
growing number of studies are using search strategies to discover high perform-
ing design alternatives in their initial stage of analysis and then evaluating their
robustness to inform design selection (Korteling et al., 2013; Kasprzyk et al.,
2013; Hamarat et al., 2014; Kwakkel and Haasnoot, 2015; Giuliani and Castel-
letti, 2016; Singh et al., 2015; Hadka et al., 2015).
This chapter advances the Many-Objective Robust Decision Making
35
(MORDM) framework initially introduced by Kasprzyk et al. (2013) by
demonstrating the value of progressively transitioning from sampling well-
characterized uncertainties only to also sampling deeply uncertain factors dur-
ing the MORDM search phase. The inclusion of deeply uncertain factors in the
search phase of the MORDM framework seeks to design portfolios of drought
mitigation actions that perform well under a wide range of SOWs — where a
SOW is defined here as a fully specified world, comprised of one fully specified
sampled vector of deeply uncertain factors (Ψ) and one streamflow time series
for each reservoir. With this change, deep uncertainty sampling is performed
for both generating candidate management alternatives and evaluating their
robustness. This chapter builds on the hypothetical western water market test
case explored by Watson and Kasprzyk (2016), which shows the value for in-
creasing robustness by including deeply uncertain factors in the computational
search and alternative generation phases of MORDM. Their study demonstrates
that there were potential robustness benefits for a hypothetical water supply
portfolio analysis for a single city in the Lower Rio Grande. In contrast, this
chapter focuses on the Eastern U.S. using the Research Triangle region of North
Carolina, engaging the water utilities serving Raleigh, Durham, Cary, Chapel
Hill and Carborro in cooperative and robust regional water portfolio planning
under deep uncertainty.
The Research Triangle region poses a multi-jurisdictional decision making
context, where each municipality faces its own performance supply reliability
and financial tradeoffs. Moreover, each of the utilities’ water portfolio plan-
ning actions can have strong negative impacts on the neighboring systems. Ef-
forts to maximize the individual robustness of any of the mentioned utilities
have been shown to potentially significantly degrade the robustness of the oth-
36
ers. Therefore, this chapter focuses on three core contributions: (1) demonstrate
that adequately combining dynamic risk-triggered conservation and transfers
with financial hedging requires the inclusion of deeply uncertain factors in the
search phase of MORDM; (2) demonstrate that searching adaptive risk-based
action triggers under a comprehensive sample of deeply uncertain factors may
promote the discovery of a larger and more diverse suite of candidate solu-
tions that meet stakeholder requirements in a broader variety of hypothetical
future SOWs; and (3) illustrate how the number and diversity of robust water
portfolios can aid in reducing planning conflicts among utilities (i.e., robustness
tradeoffs across the municipalities).
3.3 Methodological Framework
3.3.1 Overview
As mentioned in Section 3.2, this chapter advances the MORDM framework ini-
tially introduced by Kasprzyk et al. (2013) by demonstrating the value of pro-
gressively transitioning from sampling well-characterized uncertainties only to
also sampling deeply uncertain factors during search. The inclusion of deeply
uncertain factors in the search phase of the MORDM framework seeks to iden-
tify alternatives that perform well under a wide range of SOWs. With this
change, deep uncertainty sampling is performed for both generating and evalu-
ating the robustness of candidate management alternatives. Following the tax-
onomy presented in Herman et al. (2015), the proposed MORDM methodology
has the following four core component steps:
37
1. Sampling States of the World: Selection and definition of well-characterized
and deeply uncertain factors that will be used to generate ensembles of
SOWs.
2. Generation of alternatives: For one or more candidate problem formula-
tions (i.e., objectives, decisions, constraints, and SOWs), exploiting many-
objective evolutionary search under uncertainty to discover the Pareto ap-
proximate solutions.
3. Choosing Measures of Robustness: Evaluation of tradeoff alternatives’ ro-
bustness across a broader ensemble of Ψ vectors using stakeholder elicited
measures of acceptability (e.g., regret or satisficing-based robustness mea-
sures, as in Lempert et al. 2006 and Herman et al. 2014, respectively). Mul-
tiple candidate robustness measures may be elicited over iterations with
stakeholders as part of a constructive learning feedback process (Tsoukias,
2008).
4. Discovering the Factors that Control Robustness: Use global sensitivity
analysis and/or factor mapping to discover which uncertain factors con-
trol the robustness of Pareto approximate alternatives and warrant fur-
ther potential monitoring or actions [e.g., scenario-discovery presented by
Bryant and Lempert (2010)]. Sensitivity analysis methods can be used
to rank order the most important factors (i.e., factor prioritization, as in
Saltelli et al. 2008, 2006), or for factor mapping to find specific factors’
value ranges that cause conditions of concern (i.e., categorical classifica-
tion as introduced by Friedman and Fisher 1999; Breiman et al. 1984, re-
spectively).
More detailed descriptions for each of these four component steps for the
38
MORDM framework are provided in Sections 3.3.2 to 3.3.5, below.
3.3.2 States of the World
Related to the present study, Zeff et al. (2014) accounted for a single source
of well-characterized uncertainty (WCU) in their analysis of the Research Tri-
angle’s water supply tradeoffs through 2025, namely streamflows. Uncertain
streamflows were included in the model using synthetically generated stochas-
tic streamflow time series following the methodology presented by Kirsch et al.
(2013). In Zeff et al. (2014), multiobjective search was used to identify candidate
water portfolio designs. During the search phase of their study, all other fac-
tors such as regional demand trajectories, consumer response to water restric-
tions, prices for water transfers, etc., however, were treated deterministically.
Transitioning the Research Triangle from planning solely based on determinis-
tic historical streamflow records to synthetically generated streamflows showed
that the region’s planners were underestimating the severity and impacts of
droughts. The initial use of the streamflow WCU served two benefits: (1) it
highlighted that the region needs to more carefully consider transfers, financial
risk mitigation, and conservation actions and (2) it illustrated consequential re-
sults that motivated the consideration of a broader range of uncertainties (e.g.,
demand growths, transfer pricing, imperfect responses to restrictions, demand
hardening, etc.).
Building on the WCU optimization by Zeff et al. (2014), Herman et al. (2014)
evaluated the robustness of the candidate water portfolios using a Latin Hy-
percube Sampling (LHS) of the 13 deeply uncertain parameters shown in Table
39
3.1, as a means of ensuring a good representation of possible futures. Figure 3.1
illustrates what we term the deep uncertainty re-evaluation (DU re-evaluation)
used in the robustness calculations. For the Research Triangle test case, the DU
re-evaluations pair the original synthetically generated stream flows, as used
in Equation 3.17, with each LHS sampled set of deeply uncertain factors. For
this chapter, the 1,000 original synthetic inflows used in Zeff et al. (2014) and
Herman et al. (2014) are used in conjunction with 10,000 LHS sampled sets of
deeply uncertain factors (Ψ) to create 10 million SOWs. The 13 deeply uncertain
factors in Table 3.1 can be grouped into four categories representing different
aspects of the problem. The categories account for future hydrologic uncer-
tainties, uncertainties in demand projections regarding timing and magnitude,
legal and physical factors related to future storage capacities, and fluctuations
in the water transfers’ pricing. Ranges for deeply uncertain factors were cho-
sen collaboratively with the Research Triangle utilities. As noted by Bryant and
Lempert (2010), global sampling of deeply uncertain factors, with subsequent
participatory assessments that facilitate the discovery of consequential combi-
nations, allows for an improved a posteriori assessment by stakeholders of where
to focus their attention and actions.
3.3.3 Alternatives Generation
In the context of the Research Triangle, Zeff et al. (2014) employed multiobjec-
tive evolutionary algorithms to identify water supply portfolios combining wa-
ter transfers, financial risk instruments, and demand management. Each port-
folio evaluation was based on multiple realizations of synthetically generated
streamflows coupled with a set of deterministic base values for each of the 13
40
Figure 3.1: Schematics of the deep uncertainty re-evaluation of candidate
water portfolio solutions (termed DU re-evaluation). A set of
objectives is calculated for each 1,000 SOW’s that share the
same Ψ. These objectives are in turn used to calculate overall
robustness and overall objectives for alternative θ.
deeply uncertain factors presented in Table 3.1, representing what we term in
this dissertation the WCU optimization was used initially to identify design
alternatives (see Figure 3.2a). Transitioning the Research Triangle from plan-
ning solely based on the deterministic historical streamflow record to synthetic
streamflows (Kirsch et al., 2013) as a well established uncertainty characteriza-
tion tool showed that the region’s planners were underestimating the severity
and impacts of droughts.
The study transitions from the WCU optimization used in the prior pub-
lished analysis of the Research Triangle to include deep uncertainties in the
search and identification of candidate water portfolios. This transition poses
a challenging multi-objective search problem that motivated our use of the MS-
Borg algorithm, described in Section 2.1. The coupling of sampling of deeply
uncertain factors with the computational search provided by the Borg MOEA
41
Table 3.1: Uncertainty Factors and Sampling Ranges for Many-Objective
Robust Decision Making (vector Ψ in Equations 3.5 and 3.10).
Category Name Current Lower Upper
Value Bound Bound
Inflows multiplier 1.0 0.8 1.2
Climate
Evaporation multiplier 1.0 0.8 1.2
Consumer reductions multiplier 1.0 0.8 1.2
Consumer reductions lag (weeks) 0 0 4
Demand Mean peaking factor 1.0 0.5 2.0
Demand growth multiplier 1.0 0.5 2.0
Standard deviation of demand varia- 1.0 0.5 2.0
tions
Falls Lake municipal supply allocation 1.0 0.8 1.2
Jordan Lake municipal supply allocation 1.0 0.8 1.2
Capacity
Cary treatment capacity multiplier 1.0 1.0 2.0
Transfer connection capacity multiplier 1.0 1.0 2.0
Transfer cost ($/MG) 3000 2500 5000
Costs
Insurance premium multiplier 1.2 1.1 1.5
provides direct feedbacks between the alternative generation and selection of
SOWs steps of the MORDM framework. More specifically, it results in an alter-
native generation scheme that evaluates each candidate solutions against differ-
ent SOWs, each with its own streamflow time series and vectors samples of 13
DU factors, for each function evaluation performed during the computational
search. This alternative generation scheme, illustrated in Figure 2b, is termed
here Deep Uncertainty (DU) optimization. In Figure 3.2b, the green boxes rep-
resent one alternative water portfolio, each red box represent a different set of
42
Figure 3.2: WCU (a) and DU (b) optimization factor sampling schemes.
The former creates states of the world by coupling syntheti-
cally generated set of streamflows with a base projection for
13 deeply uncertain factors, while the latter couples each set of
synthetically generated streamflows with a different projection
for the 13 deeply uncertain factors, shuffling the coupling at
every function evaluation.
synthetic reservoir inflows that are randomly paired with one of 1,000 samples
of 13 DU factors. The new alternative generation scheme seeks to yield more ro-
bust alternatives, as Borg MOEA will search for alternatives whose performance
should be acceptable under a broad range of SOWs. The DU optimization’s
sampling scheme shown in Figure 3.2b approximates the much broader and
computationally demanding sampling used in the DU re-evaluation illustrated
in Figure 3.1. It would be intractable to include all 10 million DU re-evaluation
SOWs to evaluate each candidate water portfolio during search.
3.3.4 Robustness Measures
As reviewed by Herman et al. (2015), the three most common types of robust-
ness measures are: expected value, regret and satisficing; all three can be em-
ployed in either univariate or multivariate contexts. The three measures are de-
43
scribed in detail in Lempert and Collins (2007), the latter two being further de-
veloped and tested in the context of water supply portfolio analysis in Herman
et al. (2015). Herman et al. (2014) have used the domain criterion robustness
measure (i.e., percent of states of the world in which given performance criteria
are met) on the Research Triangle problem, which was used in this work again
because it best aligned with utilities preferences (Herman et al., 2015). As was
discussed above in Section 3.3.2, the implementation of the domain criterion
robustness measure for the Research Triangle problem is illustrated in the DU
re-evaluation approach in Figure 3.1. This choice of robustness calculation was
the motivation behind the computational search sampling scheme presented in
this work (DU optimization, Figure 3.2b), which was intended to approximate
the computationally intensive DU re-evaluation scheme (Figure 3.1) during the
Borg MOEA search phase described in Section 3.3.3 for the sake of computa-
tional tractability.
3.3.5 Robustness Controls
Scenario discovery is a key step in many RDM frameworks and represents a
factor-mapping sensitivity analysis used to determine under which ranges of
the deep uncertain factors each alternative may be vulnerable to failures (Bryant
and Lempert, 2010; Herman et al., 2014, 2015; Kasprzyk et al., 2013; Lempert
et al., 2006; Halim et al., 2015; Kwakkel and Jaxa-Rozen, 2016; Groves and Lem-
pert, 2007). Based on these ranges and on their experience with their system,
decision makers can better understand where candidate actions perform accept-
ably and inform monitoring efforts to detect conditions of concern. PRIM and
CART trees are suggested by Lempert et al. (2008), instead of more sophisticated
44
statistical learning algorithms, for assessing scenario discovery due to their high
interpretability.
PRIM works by gradually shrinking (peeling) prespecified ranges of factors
according to a set confidence interval in order to find multi-dimensional boxes
that encompass as high of a proportion of true positives as possible (Friedman
and Fisher, 1999). In this application, positives are Ψ vectors for which an al-
ternative fails to meet a given performance criteria. The algorithm returns a
series of candidate boxes, each one devised after one peel and described by the
following standard box metrics:
1. Density: the ratio of the number cases of interest (fails) inside that box
to the total number of cases inside that box. A value of 1 is ideal to this
metric.
2. Coverage: the ratio of the number of cases of interest inside that box to the
total number of cases of interest in the entire domain. A value of 1 is ideal
to this metric.
3. Equivalent Type II error: the ratio of the number of cases of interest out-
side the box to the total number of cases outside the box (ratio of false
negatives). A value of 0 is ideal to this metric.
The analyst should choose a box according to stakeholders’ interests. For
example, relative to the water utilities in the Research Triangle, low type II error
and high coverage are more important than high density, given the high risk
aversion to any failures of the system. The type II error represents the relative
perceived risk water utilities are willing to take of being wrong in their expected
system performance. There is not a known universal density versus coverage
45
ratio that can be used to all problems, as they are problem and stakeholder spe-
cific (Lempert et al., 2008). The described process of scenario discovery (robust-
ness controls) is described in more detail in Bryant and Lempert (2010).
3.4 The Research Triangle Test Case
The Research Triangle region, as illustrated in Figure 3.3, is comprised of several
municipalities, whose water supplies are provided by public utilities in Raleigh,
Durham, Cary, and Chapel Hill/Carrboro (i.e., the Orange Water and Sewer
Authority, or OWASA). Cary and Raleigh receive water from allocations of Jor-
dan and Falls Lakes, respectively, two large flood control reservoirs operated
by the United States Army Corps of Engineers. Volumetric allocations from
each lake are designated by the state for municipal water supply. Raleigh has
been granted 100% of the municipal water supply allocation to Falls Lake, and
Cary holds 39% of the municipal water supply allocation to Jordan Lake. Al-
though Cary is the only utility to operate an intake on Jordan Lake, other utilities
also have allocations that can be accessed by purchasing water from Cary’s wa-
ter treatment plant during conditions of scarcity, subject to Cary’s availability.
Rapidly growing demands in the region have accelerated competition for the re-
maining unallocated portion of Jordan Lake’s water supply pool. A number of
municipalities, including Raleigh, the largest in the region, have requested new
allocations, and it is likely that the pool will be completely allocated after the
most recent round of allocation requests is approved by the state, even without
Raleigh’s request (Triangle J, 2014).
Table 3.4 summarizes the region’s reservoir storage dedicated to municipal
46
Figure 3.3: Map of the North Carolina Research Triangle region. Adapted
from Zeff et al. (2014).
water supply as well as allocations for all four utilities. The Research Triangle
is projected to experience a total demand growth of 33% through the planning
horizon year of 2025 considered in this chapter (average of annual growth of
2.4%) (NCDENR, 2002). Given that the regional municipalities cannot bring
new sources online before 2025, Jordan Lake represents the only major supply
source that still has unallocated water for municipal use. Consequently, it is a
current source of regional resource conflicts.
Although the municipalities are in close geographic proximity and share the
47
Table 3.2: Summary of reservoir storage capacities, demands, and alloca-
tions for the Research Triangle Region
48
Utility Reservoir Reservoir Percentage of 2013 De- 2025 De- Demand/
capacity reservoir capac- mand mand Capac-
reserved for ity reserved for (Million (Esti- ity Ratio
municipal municipal sup- gallons mated) (2025)
supply (Bil- ply allocated /day) (Million (10−3year−1)
lion gallons) to each utility gallons
currently in use /day)
Cane Creek 100%
Stone Quarry 100%
OWASA 3.0 8.0 9.0 8.2
University Lake 100%
Jordan Lake 5%
Little River 100%
Durham Lake Michie 6.4 100% 27.5 34.9 14.9
Jordan Lake 10%
Raleigh Falls Lake 14.7 100% 57.4 76.2 14.2
Cary Jordan Lake 14.9 39% 23.1 34.0 6.2
same hydroclimatic conditions, their different demand-to-capacity ratios yield
highly variable drought impacts across the region’s utilities. In severe droughts,
transferring water from Jordan Lake via the existing Cary water treatment plant
offers a means of more efficiently using the region’s existing supply capacity to
overcome water scarcity challenges (Caldwell and Characklis, 2014; Zeff et al.,
2014). Transfers from existing sources can help to delay or avoid the costs of
building new supply infrastructure, but they do create financial risks for utili-
ties. Utilities typically set rates with the aim of returning revenues that cover
their costs, which are dominantly fixed annual debt payments. The additional
costs of water transfers during periods of drought can destabilize this budgetary
balance. If they are large enough, these swings could harm the ability of a util-
ity to repay their debt on schedule and increase future borrowing costs (Hughes
and Leurig, 2013). The costs and financial risks associated with these trans-
fer agreements for several types of financial instruments were studied in de-
tail in Palmer and Characklis (2009), Zeff and Characklis (2013), Caldwell and
Characklis (2014) and Zeff et al. (2014).
3.4.1 Triangle Region Management Model
Zeff et al. (2014) is the first study to present a fully detailed water supply simula-
tion for the Research Triangle region’s water utilities using drought mitigation
instruments based on short term conservation, water transfers, and financial
hedging. This work represents the culmination of several prior efforts (risk-
of-failure triggers in Palmer and Characklis 2009, index insurance in Zeff and
Characklis 2013, and water transfers in Caldwell and Characklis 2014). Zeff
et al. (2014) used the model to find tradeoffs between reliability focused choices
49
and their inherent financial risks when formulating drought management poli-
cies for the utilities in the Research Triangle. The WCU optimization formula-
tion in that study considered hydrologic variability as the sole source of uncer-
tainty. The policies consisted of different combinations of financial instruments
and risk-of-failure (ROF, described in equations 3.8 below) triggered drought
mitigation instruments. Herman et al. (2014) then used the same model to as-
sess the robustness of the solutions found by Zeff et al. (2014) by testing them
against a broader ensemble of uncertainties, some of them deep, finding that
multiple utilities had significantly low robustness values. This chapter builds
on these prior studies, seeking to discover if including deep uncertainties in the
search phase of MORDM will result in improving the individual utilities as well
as the whole Triangle region’s robustness to drought. Given the growing com-
plexity of the Research Triangle problem formulations as well as the evolving
treatments of uncertainty, this section provides a careful mathematical descrip-
tion of the model linking the prior studies to the proposed DU optimization
explored here.
Minimization Problem
Here, we define the optimization problem as finding the optimal policy θ∗ that
minimizes the objective function vector F, i.e.
θ∗ = argmin F (3.1)
θ
50
where,
 − fREL(xs, θrt, θtt, θjla) 

fRF (xrof , θrt, θtt, θjla) 

F = fJLA(θjla)  (3.2)fAC(x rof , θrt, θtt, θjla, θacfc, θirt) 
fWCC(xrof , θrt, θtt, θjla, θacfc, θirt)
 
xrofX =  (3.3)
xs
θ = [θrt, θtt, θjla, θacfc, θirt] (3.4)
where F is the vector-based objective function, fREL is the reliability objective,
fRF is the restriction frequency objective, fJLA is the Jordan Lake allocation ob-
jective, fAC is the average cost objective, fWCC is the worse case cost objective, X
is the state vector across the utilities and time, xrof is a vector of approximately
calculated risk of failures (Equation 3.8), xs is a vector of combined utility stor-
age, θrt is a vector of restriction triggers, θtt is a vector of transfer triggers, θjla is
a vector of Jordan Lake Allocations, θacfc is a vector of annual contingency fund
contributions, and θirt is a vector of insurance restriction triggers. The storage
state xs can be defined as:
(
x w w−
)
1 w w w w w
s = f xs ,C,D ,TF ,NI ,E ,W |Ψ (3.5)
Dw = f(x wrof , θrt) (3.6)
51
TFw = f(x wrof , θtt) (3.7)
where C is a vector of reservoir capacities (7 reservoirs, some utilities have
more than 1), D are the demands for each utility, TF are the transfer volumes
for each utility, NI are the natural inflows in each reservoir, E the evapotranspi-
ration in each reservoir, W the spillage of each reservoir, and Ψ is a vector of
deeply uncertain factors such as population growth, future allocations yet to be
negotiated, etc.
The approximate risk-of-failure (ROF) state xrof is used in conjunction with
decision variables θrt and θtt to trigger drought mitigation instruments. The pri-
mary drought mitigation instrument currently used by utilities is water use re-
strictions, which is triggered independently by each utility according to rules
based on either days of supply remaining, minimum storage levels, or sim-
plified risk calculations (Department of Water Management, 2009; The City of
Raleigh Public Utilities Department, 2011; Goodwin, 2009; Orange Water and
Sewer Authority, 2010). Despite being important system state variables, days
of supply remaining and storage levels do not comprehensively represent the
water supply systems’ dynamics. Therefore, action triggers should be based
on metrics that incorporate more representative dynamics in available capacity
and evolving demands (Palmer and Characklis, 2009). The proposed approach
based on ROF calculations allows instead for the inclusion of information from
a wider range of state variables while making stakeholder decisions focus on
risk-based action triggers. The scenarios for the ROF calculations are described
by past recorded inflows and by assumptions about the deeply uncertain fac-
tors in Ψ presented in Table 3.1. The inflows are assumed to be statistically
52
stationary, given this is a short-term planning study through 2025.
3.5 The Mathematical Formulation of the ROF Metric
The drought mitigation measures are triggered in a given week if the calculated
value of the ROF state xrof reaches the trigger values θrt and θtt. Equations 3.8
to 3.10 mathematically define how the ROF metric is computed:
N∑rof
xw
1
= fw
′ ′
srof,j y′,j(NI
y ,Ey ) (3.8)
Nrof
y′=0
where,
 y,w′0 ∀w′ ∈ {(y′, w), ..., (y′ x ′, w + T s ,jrof )} : ≥ sc
fwy′,j =  C j (3.9)1 otherwise
and, ( )
y′x ,w
′ ′ ′ ′ ′ ′ ′
s′,j = f Cj,UD
w
j ,NI
y ,w ,Ey ,w ,W y ,wj j j |Ψs (3.10)
In Equations 3.8 to 3.10, w′ and y′ mean a week and a year simulated with past
data for the calculation of the ROF. Variable xwrof,j is the ROF for utility j in
current week w, and fy′,j is a binary variable for which 0 denotes a failure hap-
pened during the corresponding ROF simulation with data from past year y′. In
the ROF metric, NIy′ , Ey′ and W y′ are the recorded natural reservoir inflows,
evaporation rates and reservoir spillage, respectively, in year y′ prior to current
week w used in one of theNrof simulations — note that in a simulation in which
the ROF metric is calculated for a week 20 years from now, 20 of the Nrof years
of hydrological data will belong to synthetically generated data. Variable Trof
53
equals 52 weeks for this chapter, so that single-year droughts are captured. A
′ ′
failure is defined as the combined storage xy ,ws′,j for any realization y
′ divided
by combined storage capacity Cj for utility j falling below critical storage sc.
′ ′
Variable xy ,ws′,j is the vector of storage states calculated in one of the year-long
ROF simulations using recorded hydrologic data from past year y′. In the cal-
′
culations of xy,w ws′,j , UD is the unrestricted demand (no restrictions enacted or
transfers purchased) in week w, and Ψs is a matrix of sampled deeply uncertain
factors obtained through WCU or DU optimization. Given a year has 52.178
weeks, every 6 years (when the rounded number of weeks in a year would reach
53) the first week of the following historical year will be used.
54
Financial Model
Financial losses are accumulated when a utility purchases transfers or enacts
conservation, and are equal to the total transfer costs plus the revenue losses
resulting from reduced water sales. Utilities can attempt to offset these losses
through payouts from third party contracts, and/or annual contributions made
to contingency funds (self-insurance). Both were jointly explored in this disser-
tation.
By purchasing third-party financial insurances, a utility pays an upfront fee
— a premium — in exchange for a payout in the event of a drought. For con-
tractual purposes, the drought conditions that trigger insurance payouts are de-
fined as inflow values that would trigger a pre-determined demand restrictions
schedule. To define these inflows, system simulations are run with projected
demand values for the following year. The insurance triggers are set as the
maximum cumulative values, through each week from the beginning of the
simulation, where storage drops sufficiently (and, consequently, for the ROF
to increase enough) for water use restrictions corresponding to θirt to be trig-
gered in the simulations. Inflows rather than the enactment of restrictions are
used as insurance criterion in order to avoid moral hazards, given that utilities
would be able to artificially enact restrictions on times of financial need in order
to get a payout. More details about the insurance model can be found in Zeff
and Characklis (2013). The insurance payout (IPO) is set for each utility in the
beginning of the simulation as the expected value of revenue losses if restriction
stage θirt is enacted given the historical inflows:
IPO = E [RLw] = E [UDw ·UWP−Dw ·WPw] (3.11)
55
WPw = f(x
w
rof , θrt) (3.12)
where RL is the vector of revenue losses, UWP is the vector of unrestricted
water prices (independent of time), andWP is the vector of water prices subject
to restrictions.
The insurance premium (IP ) is calculated based on the expected total pay-
outs made during each historical year (Y IPO) plus a loading (IPR) based on
the variance of such total payouts to account for volatility.
IP = E [YIPO] + IPR (3.13)
( )
IPR = f σ2YIPO (3.14)
The total payouts Y IPO that inform the insurance price can be expressed as:
YIPOy = f(θirt, IPO,NI
y) (3.15)
where y denotes a year.
The annual contingency fund contribution decision variable is used to set
aside a percentage of the annual revenue into a contingency fund, which rolls
over to future years indefinitely. The amount of money in a contingency fund is
updated yearly according to the following expression:
CFy+1 = CFy · (1 + r) + θ yacfc ·ATRy (3.16)
where CF is the total amount of money in the contingency fund, ATR is the an-
nual total revenue, and r is the fund’s interest rate. Contingency funds work as
56
a utility’s savings account for protection against financial instabilities that can
arise from multiple causes. Utilities are expected to contribute annually to their
own contingency funds with a percentage of the annual revenue. Although a
contingency fund can be set large enough to attempt to provide financial pro-
tection against any drought or sequence of droughts, large funds would require
high annual contributions resulting in opportunity costs relative to more effi-
cient financial management strategies. Moreover, large contingency funds could
be appropriated by municipal governments for other purposes. The annual con-
tingency fund contribution is the percentage of annual revenue a utility should
deposit in its contingency fund.
Objectives
1. Reliability: The reliability objective is an approximation of the system re-
liability calculated as the fraction of considered states of the world which
may cause the storage level of any reservoir used by a given utility to drop
below 20% of its maximum capacity in[any g(iven∑week ()fa]ilure condition):N1 r
maximize fREL = min min g
y
i,j (3.17)
j y Nr i=1
where,
 xw,y0 ∀w :
s,i,j ≥ S
y  C c jgi,j = 1 otherwise
where gi,j,y = 0 if there was a week in a given year of a particular re-
alization where reservoir storage falls below Sc of capacity (20% in this
chapter), and 1 otherwise, Nr is the number of realizations in one func-
tion evaluation, y is the simulation year, Nys is the number of years in the
57
project horizon, i is the simulation realization index, and j is the utility
index. Although the term reliability generally denotes a probability, this
is not the case for fREL, as the inclusion of the deep uncertainty factors
in the calculation makes the SOWs not equally likely and not guaranteed
to exhaust all possible future scenarios. However, fREL can be used as a
proxy for reliability for decision making purposes as a way of assessing
how a utility can rely on each alternative relative to each other, given the
uncertainty space defined by the utilities. Therefore, fREL will be further
referred to in this work as reliability for the sake of readability.
2. Restriction Frequency: Restriction frequency represents the fraction of years
over the course of the planning horizon for which at least one week of
restricted water use is expected: [
1 ∑∑ ]Nr Nys
minimize fRF = max h
y
i,j (3.18)
j Nys ·Nr i=1 y=1
where,
0 ∀w : x
y,w
 rof,i,j
≤ θrt,j
hyi,j = 1 otherwise
where hi,j,y = 0 if there was a week in a given year of a particular realiza-
tion where water use restrictions were enacted, and 1 otherwise.
3. Drought Management Cost: The average cost objective represents the ex-
pected yearly cost of all non-infrastructure water portfolio assets used to
manage droughts over the planning horizon. These costs are revenue
losses from restrictions, transfer costs, contingency fund contributions,
58
and third party insurance contract c[osts:
1 ∑ ]Nr ∑Nys
minimize fAC = max SY C
y (3.19)
j Nys ·N i,jr i=1 y=1
where,
θ · ATRy + IP y +max(RLy + TCy y yy acfc,j i,j i,j i,j i,j − Y IPOi,j − CFi,j, 0)SY Ci,j = ATRyi,j
where CFC is the one time contribution to a contingency fund in a given
year y, IP is the insurance contract cost, RL is the revenue losses from
water use restrictions, TC is the transfer costs, Y IPO is the total insurance
payout over year y, CF is the available contingency funds, andATR is the
total annual volumetric revenue. All these variables are dollar values.
4. Exposure to Financial Risk (worse case cost): The worse case cost objec-
tive represents the 1% highest single-year drought management costs ob-
served across all analyzed SOWs ove{r the planning horizon}:
minimize fWCC = max quantile(SY Ci,j, 0.99) (3.20)
j i∈Nr
5. Jordan Lake allocation: The Jordan Lake allocation objective represents the
allocation of the Jordan Lake water municipal supply pool to be granted
to a utility, such that:
∑Nu
minimize fJLA = JLAj (3.21)
j=1
where JLA is the percent of the pool allocated to utility j, and Nu is the
total number of utilities in the system.
The regional water supply portfolio problem formulated in equations 3.1 -
3.21 represents a challenging multi-jurisdictional planning problem, given its
59
number of stakeholders and objectives, and its stochastic and non-linear nature.
The transition to searching for appropriate decision triggers that effectively use
a balanced portfolio of conservation, transfers, and financial risk measures has
broad value beyond the Research Triangle region. Section 3.3.1 describes our
extensions of the multi-stakeholder MORDM framework in more detail.
3.6 Computational Experiment
3.6.1 Multiobjective Optimization Algorithm
As mentioned in Section 2.1, the Borg MOEA (Hadka et al., 2013) has been
shown to provide competitive or superior performance relative to other top
performing MOEAs across a broad suite of challenging problems (Reed et al.,
2013; Hadka and Reed, 2012a). The Borg MOEA combines -dominance based
archiving (Laumanns et al., 2002) with an adaptive operator selection (Vrugt and
Robinson, 2007) based on probabilities calculated from solutions in its archive
(Hadka et al., 2013), making it suitable across a wide range of problems. The
standard values of parameters of the Borg MOEA v1.4 Master-Slave were used
in this work (see Hadka et al. 2013 for the specific values).
3.6.2 Hydrologic Scenarios and DU Optimization Runs
The number of Monte Carlo realizations used to estimate the objective func-
tions’ evaluations was 1,000 — empirical assessments with ensemble sizes vary-
ing from 100 to 5,000 showed that 1,000 evaluations per run is sufficient for
60
approximating the mean and variances of the Monte Carlo distributions used
for computing candidate solutions’ objectives. This approach for determining
sampling rates originates from early studies of metaheuristic search dynamics
given a single noisy objective function (e.g., Miller and Goldberg 1996; Smalley
et al. 2000), where it was shown that relatively small Monte Carlo samples per
function evaluation can provide good approximations when verified with much
larger verification samples post search. These early results were translated to
multi-objective evolutionary algorithms in subsequent studies (see Goh and Tan
2007; Singh and Minsker 2008; Kasprzyk et al. 2012; and Reed et al. 2013), which
empirically test a step-wise increase in Monte Carlo sampling rates per func-
tion evaluation with subsequent verification tests with much larger samples.
The deeply uncertain factors were sampled using the Latin Hypercube Sam-
pling technique, based on the ranges presented in Table 3.1, while the synthetic
streamflows, evaporation rates, and demands were generated using the meth-
ods described in Section (2.2) to create stationary time series by not modifying
the log-means with the sinusoid described in Equation 2.5.
Running 1,000 realizations per function evaluation, however, significantly
increases computational burden, which in turn set a requirement for higher
computing capacity. The optimization was performed on the Texas Advanced
Computing Center’s Stampede, where 25 random seed Borg MOEA trial runs
each performed a total of 1,000,000 function evaluations while exploiting 2,000
cores (AMD Opteron Quad-Core processors @ 2.3 GHz). The Pareto approxi-
mate fronts of all seeds were then combined to form a best known approxima-
tion of the global Pareto front (i.e., the reference Pareto set). Table 3.3 shows the
values of  (i.e., significant precision) used for the optimization and for reference
set generation, which are the same as in Zeff et al. (2014).
61
Table 3.3: Values used for -dominance.
Objective Reliability Restriction Jordan Lake Average Worse
Frequency Allocation Cost Case Cost
 0.005 0.02 0.002 0.05 0.01
3.7 Results and Discussion
3.7.1 Optimization under Deep Uncertainties
A key motivation of this dissertation is to use the search phase of the MORDM
framework to identify more robust water portfolios for the Research Triangle re-
gion. The DU optimization scheme illustrated in Figure 3.2b was formulated to
approximately sample the more complex DU re-evaluation space shown in Fig-
ure 3.1. As discussed in Section 3.3.3, it would not be computationally tractable
to use the DU re-evaluation scheme within the optimization itself. Figure 3.4
shows a comparative verification of the tradeoffs attained by the WCU and DU
optimization alternatives by showing their average performance across the 10
million SOWs that define the DU re-evaluation space.
In Figure 3.4, the points where each line (water portfolio alternative) inter-
sects the vertical axes represents the average performance in each objective in
the DU re-evaluation space. The plot is oriented such that the ideal alternative
would be a horizontal line at the bottom of all of the axes. When comparing
alternatives, a downward shift along the axes represents the direction of higher
preference and diagonal lines capture pairwise objective tradeoffs. The ranges
of the averaged objective values as well as their relative tradeoffs highlight the
62
Figure 3.4: Average performance objective values for WCU and DU op-
timization alternatives over the 10 million SOWs from DU
re-evaluation. The vertical axes denote average performance
in each objective and each polyline represents an alternative.
Brown tones represent alternatives from WCU optimization
and blue tones represent alternatives from DU optimization.
Transparency is applied to alternatives that do not meet a re-
laxed version of the performance criteria established by the
studied utilities: Reliability > 98.5% (relaxed from 99%), Re-
striction Frequency < 20%, and Worst-Case Cost < 5%). The
color gradient is based on the restriction frequency objective
(light for low frequency, dark otherwise, not applicable to high-
lighted solutions). None of the brown (WCU) alternatives meet
the criteria, while 7 out of 2,954 DU solutions do.
adaptivity of ROF-based transfers and restrictions in combination with the fi-
nancial risk instruments in still attaining high levels of average performance
despite the increased difficulty of the DU optimization’s formulation. The most
striking feature of Figure 3.4 is the increased use of Jordan Lake allocations,
which indicates that the Research Triangle region’s water utilities could face
strong contention for this resource.
Figure 3.4 also illustrates which of the high performing DU optimization al-
ternatives have high levels of performance on average when evaluated against
the 10 million SOWs within the DU re-evaluation. In Figure 3.4, all alternatives
63
whose values failed on average to meet the performance criteria established by
the utilities — to attain at least 98.5% average reliability, average restriction fre-
quencies no greater than 20% and average worst case costs no greater than 5%
— were filtered out (transparent polylines). The reliability criteria set by the
Research Triangle utilities was relaxed by 0.5% from the original 99% because
none of the solutions from either the DU or the WCU optimization runs met this
criterion without violating the restriction frequency and worse case cost criteria
when evaluated against the broader ensemble of 10 million SOWs. Confirming
the robustness assessment results in Herman et al. (2014), the alternatives from
WCU optimization significantly deteriorated in their worse case costs, reliabil-
ity and restriction frequencies; therefore all of the re-evaluated brown WCU
solutions where filtered in Figure 3.4 (i.e., shown with a high level of trans-
parency) — the re-evaluated averages for the reliability and worse case cost
criteria would have to be relaxed to 95% and 25% respectively for any solutions
from WCU optimization to avoid being brushed from Figure 3.4. The simplified
but more comprehensive sampling of uncertainties used in the DU optimization
proved to be helpful for generalizing to the DU re-evaluation space.
3.7.2 Robustness Comparison
Transitioning from the performance tradeoffs to satisficing frequencies, Figure
3.5 illustrates the significant differences in the robustness of the supply portfo-
lios obtained with the WCU (brown) and DU (blue) optimization runs based on
their calculated objectives values for each of the 10,000 samples of Ψ (deeply
uncertain factors in Table 3.1). In this figure, the alternatives from both opti-
mization schemes are rank-ordered (horizontal-axis) based on their robustness
64
(vertical-axis). Figure 3.5 defined robustness as the percentage of the 10,000 Ψ
samples where an alternative was able to meet all of the utilities’ requirements
as calculated with equations 3.17-3.21 for all 1,000 synthetically generated sets of
natural inflows. The results show that DU optimization provides far more can-
didate solutions that are also significantly more robust. Both traits make it more
likely that robust satisfactory compromise solutions can be found. However, it
cannot be inferred from these results that DU optimization will always lead to
more solutions, as the presented results may reflect particular characteristics of
this problem’s specific decision space.
The use of DU optimization improved the maximum attainable robustness,
as calculated by the model, of all utilities but OWASA, who was already able
to attain nearly 100% maximum attainable robustness in a majority of the WCU
optimization solutions. Figure 3.5 shows that Cary, who had maximum attain-
able robustness of 86% for the WCU solutions reached 100% with the DU opti-
mization’s results. Likewise, Durham’s maximum robustness went from 39% to
85%, and Raleigh’s went from 18% to 59%. Although Durham’s and Raleigh’s
robustness may not be as high as desired, their improvement from the DU op-
timization is substantial. It is important to notice that Figure 3.4 shows that
such improvements in robustness comes at the expense of average expected
performance, notably in average costs and Jordan Lake allocation — the trans-
parent blue lines show consistently higher values for these two objectives than
the transparent brown lines.
As noted by Herman et al. (2014), beyond their individual levels of attained
robustness, it also important to understand and mitigate potential robustness
conflicts where one of the utilities strongly benefits by degrading one or more
65
Figure 3.5: Alternatives (x-axis) were rank-ordered according to robust-
ness (y-axis) delivered to each utility. An alternative’s robust-
ness was defined as the percentage of the DU re-evaluation’s
10,000 Ψ samples where an alternative meets the original ro-
bustness criteria elicited from water utilities. The brown bars
represent the robustness of the original 84 WCU optimization’s
alternatives, while the blue bars represent the robustness of
all 2,954 DU optimization’s alternatives. Each group of bars
is rank ordered by robustness.
of the others. Figure 3.6 shows the robustness tradeoffs among all the utilities
for the DU optimization alternatives. The ideal solution would be a horizontal
line at 100% robustness for all of the utilities. The gray lines are all the solutions
from the DU optimization run, while the blue and brown represent illustrative
compromise solutions from the DU and WCU optimization runs, respectively,
attained by filtering out solutions of high robustness sacrifice for any of the
66
utilities. The gray diagonal crossing lines and color gradient for the DU opti-
mization results suggest that increases in Raleigh’s and Durham’s robustness
are often at the expense of Cary’s, due to the amount of transfers that are being
demanded. The brown line representing the candidate compromise alternative
from the WCU optimization solutions shows low attainable system robustness,
which confirms the value of DU optimization.
Figure 3.6: Robustness tradeoffs among utilities. Each polyline represents
an alternative, while the location where a polyline intersects an
axes denotes the robustness value of the corresponding alterna-
tive for the corresponding utility. The blue lines indicate four il-
lustrative compromise DU alternatives with fairly high robust-
ness for all utilities, the brown line indicates the best compro-
mise WCU alternative from prior work by Herman et al. (2014),
while the gray lines represent the remaining DU alternatives.
Figure 3.6 highlights four compromise alternatives from the DU optimiza-
tion results that were able to maintain the utilities’ robustness levels close to
their maximum attained values. The DU optimization effectively exploited the
flexibility of the ROF-based transfers and financial instruments under high un-
certainty to identify compromise portfolios that strongly reduce regional robust-
67
ness conflicts. This shows that a more comprehensive approach to uncertainty
in the search phase has the potential to not only yield more robust alternatives,
but also to decrease regional inter-utility conflicts during negotiations.
Figure 3.7 shows the decision variables for each utility for the four com-
promise alternatives. According to the figure, under a compromise scenario,
Durham and Raleigh (panels a and c) have low restriction and transfer trigger
values, which means they would often make use of these two instruments. In
order to compensate for the financial losses due to restrictions and transfers,
both utilities would have to rely on substantial contingency funds and possibly
on third party insurances even for mild droughts.
Figure 3.7: Decision variables of the 4 compromise solutions for each util-
ity. Different colors represent the different alternatives, and
correspond to the ones in Figure 3.6. The ”y” axis represent
a scale from low use to high use.
Being a small utility in comparison with the others, OWASA (Figure 3.7d)
is in a more comfortable position, not needing to rely often on restrictions or
transfers. As a whole, OWASA seems to be largely insensitive to any of the
instruments, as it performs well in most of the tested states of the world. How-
ever, due to its small budget and size, a more severe drought poses significant
financial risk, which makes it have to rely on relatively low insurances trigger
68
values.
Cary (Figure 3.7b) would require a significant Jordan Lake allocation to meet
its demand because it relies on Jordan Lake as its primary water source. How-
ever, being the source of all transfers increases Cary’s revenue — Durham and
OWASA have allocations from Jordan Lake but have to pay a fee to Cary to
use them, given Cary owns the Jordan Lake intake and treatment plant. This
would put Cary in a relatively safe position against mild droughts, not strongly
needing to set up a large contingency fund — which is mainly an instrument to
mitigate financial impacts of mild droughts — or insurances that would be trig-
gered often. However, Cary may not have enough revenue from water transfers
in exceptionally dry years, given its water sales can be no greater than the al-
locations possessed by Durham and OWASA, making it have to rely on contin-
gency funds as a complement to the insurance payouts. Under such scenario,
the tradeoffs observed in Figure 3.6 between Raleigh and Durham would be
more noticeable, given their low transfers trigger values and resulting in possi-
ble competition for transfers from Cary, which are limited by conveyance capac-
ities. From these results, it is clear that an increased Falls Lake allocation would
be paramount for maintaining Raleigh’s reliability and regional robustness dur-
ing low frequency extreme events, since Raleigh has the highest demands and
Falls Lake is its only direct supply source (see Table 1).
The addition of the deeply uncertain factors in the search phase significantly
impacted the underlying decisions that would result from the robust compro-
mise policies. Both Raleigh and Durham are shown to require a substantially
greater Jordan Lake allocation relative to the results of the WCU optimization
shown in Herman et al. (2014). Additionally, the low values of the transfers ROF
69
trigger for both utilities imply that the allocations are going to be frequently
used (i.e., more risk averse ROFs). However, the increased transfers do not
prevent both utilities from requiring relatively frequent restrictions when com-
pared to previous results. Increased use of transfers and restrictions also serves
to increase the required annual contingency fund contributions for Raleigh and
Durham. Another important difference between WCU and DU optimization re-
sults is that OWASA transitioned from a very limited to a more significant use
of restrictions, transfers, and the contingency funds.
3.7.3 The Value of Regionally Coordinated Demand Manage-
ment
Beyond assessing the robustness of the utilities, it also important to understand
the key deeply uncertain factors that control the success or failure of alternative
regional water portfolios. Following prior MORDM studies (Kasprzyk et al.,
2013; Herman et al., 2014), factor mapping sensitivity analysis, as described in
section 3.3.5, was performed to identify the ranges deeply uncertain factors that
were most related to system failures, and the results are shown in Figure 3.8.
This is analogous to the scenario discovery approach as described by Bryant
and Lempert (2010). A conservative approach was taken when selecting the
PRIM box. Out of the over 80 boxes for each alternative, boxes with high cover-
age and low type II error were chosen at the expense of the density metric. The
implication of this choice is that the bars on Figure 3.8 emphasize the ranges of
parameters where the compromise alternatives are likely to perform satisfacto-
rily.
70
Figure 3.8: Ranges of deeply uncertain factors which should be avoided
in order to improve the robustness of each compromise solu-
tion. Demand growth (left box) and Falls Lake municipal sup-
ply allocation (right box) are the two most important sources
of uncertainty for the four chosen compromise solutions. Utili-
ties are encouraged to incorporate measures in their short-term
management plan seeking to mitigate demand and be granted
a greater allocation from Falls Lake.
The two deeply uncertain factors that most strongly influence the robustness
of the four compromise alternatives performances highlighted in Figure 3.8 are
regional demand growth rates and the Falls Lake municipal supply allocation.
The relative importance of Falls Lake municipal supply allocation in compar-
ison to the deeply uncertain factors other than demand growth reemphasizes
the importance of new water allocations/sources for the region, which is also
expressed in Figure 3.7 by the high Jordan Lake allocation needed for Cary to be
able to provide for frequent transfer requests from Raleigh and Durham. Gener-
ally speaking, these results show that utilities should carefully consider regional
demand growth rates and possibly monitor the adequacy of the Falls Lake sup-
ply as well, depending on the chosen compromise solution. It is important to
notice that demand growth multiplier and Falls Lake allocation had a much
71
stronger effect on the system’s performance than any climate or inflow factor
through the 2025 planning horizon.
3.7.4 Scenario Discovery
Figure 3.9 provides a more illustrative view of a representative PRIM box with
a low angle well-defined boundary for success versus failure in attaining the
utilities performance requirements along the demand growth vertical axis. This
clear performance boundary indicates that even though demand growth is not
the only influential parameter determining the acceptability of an alternative’s
performance under a given Ψ, it tends to be the most important one. Any fur-
ther improvement in robustness for any of the compromise alternatives would
then have to be accomplished by first trying to mitigate demand growth.
Figure 3.9: Falls Lake Supply allocation vs. Demand growth pass/fail. A
solutions performance is more impacted by variations in de-
mand growth than in Falls Lake allocation.
72
Following this result, a suite of robustness evaluations was performed in or-
der to further understand the effects of changing the demand growth. Figure
3.10 shows the robustness results of robustness evaluations for new LHSs with
the demand growth multiplier factor varying from 0.5 to 0.8, 1.0, 1.2, 1.4, 1.6,
and 2.0 (results for 1.8 were similar to 2.0). The 0.8 multiplier represents a 20%
reduction of the Research Triangle’s demand growth rates relative to their as-
sumed nominal value (i.e., the 1.0 multiplier). From a policy perspective, this
would require that the utilities coordinate demand growth reductions from ap-
proximately 3% through 2025 to approximately 2.4% (0.8 · 3%) instead. As seen
in Figure 3.10, demand growth management would be an effective means of
mitigating droughts under various tested SOWs. Under scenarios of highly con-
trolled demand growth, Cary, OWASA, and Durham could attain 100% robust-
ness across several water portfolio options. In several of the candidate portfolios
Raleigh’s robustness exceeds 80%, which represents a drastic gain compared to
allowing the demand to grow twice as much as the projected increase.
This work, as well as the prior motivating study (Herman et al., 2014) high-
light the value of the water utilities monitoring and innovating their demand
management schemes. A key contribution of this chapter relative to prior work
is that the proposed DU optimization facilitated the discovery far more water
portfolios with improved robustness. Our results highlight that a high level of
adaptivity exists even when the system is confronted with the most challeng-
ing SOWs. A more detailed understanding of candidate demand management
schemes holds significant value for future research. Cooperative water trans-
fers, financial risk mitigation tools, and coordinated regional demand manage-
ment must be explored jointly to decrease robustness conflicts between the util-
ities. The insights from this work have general merit for regions where adjacent
73
municipalities can benefit from cooperative regional water portfolio planning.
Figure 3.10: Robustness improvement due to demand growth control. De-
mand growth management may lead to a robustness gain of
up 30% for the four utilities studied here. Cary, Durham and
OWASA have the potential to reach 100% calculated robust-
ness, while Raleigh may reach over 90%.
74
3.8 Discussion
While prior studies (e.g., Kwakkel et al. 2014, 2016a; Watson and Kasprzyk 2016)
have also advocated for including deep uncertainties into the search for robust
management actions, the core of the computational contributions in this work
lie in the dynamic and adaptive nature of our ROF-triggered water supply port-
folio instruments. Our DU optimization is seeking better risk-based manage-
ment rules for the Research Triangle’s interdependent regional actors. Each of
the utilities is balancing supply reliability focused actions (i.e., transfers and
restrictions) with their intrinsic financial risks (i.e., extreme tail costs). The com-
bined use of ROF decision triggers and financial hedging is critical for DU op-
timization to realize its full value. In contrast, commonly employed average
cost water supply formulations that focus on searching a space of non-adaptive
decisions (e.g., see recent examples in Beh et al. 2015a; Borgomeo et al. 2016;
Huskova et al. 2016) are strongly limited in their ability to respond to challeng-
ing combinations of deeply uncertain factors. More specifically, as formulated
in equations 3.8 and 3.10, ROF is actually a dynamic representation of each util-
ity’s evolving capacity to meet their demand. Each utility’s dynamic ROF state
is subject to its ROF decision thresholds, the conditions of the specific SOW sam-
pled, and the choices of the neighboring utilities. A key distinguishing feature
of this portfolio formulation is that every sampled SOW yields a different suite
of tailored actions taken in each management period subject to the ROF trig-
gers that compose a given candidate solution. Akin to the concept of a closed
loop feedback, each observed ROF state has a state-dependent action triggered.
Put simply, this represents acting according to the world you are presently ob-
serving. In contrast, the more typical abstraction of drought decisions in the
75
water supply literature (see literature in Beh et al. 2015a; Borgomeo et al. 2016;
Huskova et al. 2016) is for each candidate solution to implement the same fixed
set of actions in every sampled SOW and evaluate their expected performance
measures (i.e., taking actions based on the average performance measures over
possible future worlds). This difference is critical to the success of the DU opti-
mization extension demonstrated in this chapter. For each candidate DU opti-
mization solution, its ROF triggers yield water portfolio actions that are specific
to the conditions observed in each sampled SOW. Consequently, a much higher
degree of adaptivity and enhanced exploitation of information feedbacks are
employed when the DU optimization confronts challenging SOWs. Moreover,
robustness here requires the financial risk instruments to successfully mitigate
extreme tail costs, making higher levels of restrictions and water transfers finan-
cially tenable in the most challenging SOWs encountered in the DU optimiza-
tion (see Figures 3.4-3.6).
Successfully balancing robustness and regret is another potentially more
subtle computational contribution in our transition from the WCU optimiza-
tion to the DU optimization. In both cases, the same risk-based action rules
where searched but the resulting action triggers are very different. Challeng-
ing SOWs in the DU optimization stressed the ROF-based rules to more fully
exploit their potential adaptivity and yielded an strong increase in the number,
diversity, and robustness of candidate solutions (see Figure 3.5). However, our
results also highlight that increasing the robustness of individual Research Tri-
angle utilities frequently increased the potential for regional resource conflicts
(see Figure 3.6). If not properly understood and managed, these increased re-
source conflicts represent a significant unintended regret that emerges from the
DU optimization. As highlighted by Giuliani and Castelletti (2016), robustness
76
focused formulations abstract very high levels of risk aversion by designing for
severe SOWs. Consequently, they can be mathematically prone to high levels of
regret if solutions are not sufficiently adaptive to take advantage of less severe
SOWs. Our results aid in better balancing these types of robustness and regret
concerns. Although our most robust solutions are capable to confronting severe
SOWs, their underlying ROF-based decision triggers would take actions rela-
tive to the world observed. Moreover, our scenario discovery shows that if the
Research Triangle’s utilities are willing to coordinate in modestly reducing their
demand growth rates by 20%, their collective robustness increases dramatically
while reducing large, potentially contentious Jordan Lake allocation requests.
Beyond the Research Triangle, our results have implications for rapidly
growing urban regions that are now confronting increasing water scarcity con-
cerns. Planning by nearly all water utilities at present, still relies solely on his-
torical streamflow records, although it is common knowledge that even long
streamflow records are poor representations of extreme drought quantiles (e.g.,
Fiering et al. 1971; Lettenmaier and Burges 1978; Potter 1976; Stedinger and Tay-
lor 1982). Shifting from planning based solely on the historical streamflow to
synthetic hydrology while also actively accounting for the diversity of factors
included in our DU optimization holds significant promise as a means of im-
proving the robustness of regional water supplies. That being said, this rec-
ommendation also highlights a rapidly growing gulf between emerging capa-
bilities and the traditional planning frameworks that are currently commonly
employed. This chapter also highlights that high density urban regions with
multiple water supply utilities can and likely must carefully combine of highly
adaptive reliability-driven actions that are supplemented with careful financial
hedging. DU optimization was critical for discovering a soft path (Gleick, 2002a)
77
mixture of conservation, transfers, and financial hedging in this chapter that
could successfully overcome very severe combinations of stressors. This result
has broad value for urban systems around the globe confronting climate change,
rapid population growth, economic expansion, and limits on new source devel-
opment.
3.9 Conclusions
This chapter contributes computational advances for a multi-stakeholder vari-
ant of the MORDM framework demonstrating how to include deep uncertain-
ties in the generation of candidate policies to improve robustness and reduce re-
gional robustness conflicts. In the water resources context, this work contributes
a clear demonstration of the value of integrated regional water supply portfolio
planning for mitigating drought risks, particularly for densely populated mu-
nicipalities that are in close proximity to one another. The Research Triangle
test case demonstrates that significant potential drought risks and inter-utility
supply conflicts exist between utilities in the planning period of analysis from
2015 to 2025. This region currently encompasses more than 2 million users and
trends of very rapid population growth.
This chapter highlights the need for flexible and scalable modeling frame-
works to simulate the broad array of candidate management actions and un-
certain stressors in coupled human-natural systems. The contributions of this
work extend beyond computational algorithms. The Research Triangle man-
agement model that is core to this work requires detailed human system data
(reservoir rules, finances, demands, rules for restriction, conditions for trans-
78
fer, pricing, etc.) as well as natural system data (multiple ensembles of stream-
flows, evaporation, etc.). Having sufficient abstractions of the human systems
strongly constrains the scope of candidate management actions (i.e., transfers,
restrictions, and financial instruments) that can be simulated and explored un-
der deep uncertainty. In terms of a broader guidance for future studies, densely
populated urban regions with multiple utilities must be carefully abstracted as
coupled humannatural systems. A multitude of factors fundamentally shape
the degree to which droughts impact water supplies (e.g., magnitude and du-
ration of rainfall deficits, coordination of crisis management, water treatment
and conveyance capacities, population growth, financial stability, infrastructure
maintenance, etc.). These interdependent factors yield complex dynamically
evolving risks. Moreover, the overall systems’ behavioral responses to these
evolving risks are strongly shaped by asymmetries in regional impacts, highly
diverse regional priorities, and conflicting stakeholder preferences. The DU op-
timization extensions of the MORDM framework contributed in this chapter act
in combination with ROF-based decision triggers to provide a broad ability to
explore the coupled humannatural systems’ dynamics of the Research Triangle.
More generally, this work demonstrates that understanding urban water sys-
tems’ management tradeoffs, vulnerabilities, and dependencies requires mod-
eling and analysis frameworks that are capable of capturing dynamic stocks and
flows of risk itself.
The inclusion of deep uncertainties in the search phase of the MORDM
framework made it possible to both discover a much more diverse range of
drought mitigation actions that the municipalities could take while also improv-
ing their overall robustness. Robustness in this work is defined in collaboration
with regional stakeholders and encompasses requirements for supply reliability,
79
the frequency of demand restrictions, and the stability of financial risks. Across
all of these concerns, our results show that regional demand growth rates dom-
inate all other factors in controlling the robustness of alternative portfolios of
drought mitigation actions. In combination, this chapter’s improved search un-
der deep uncertainty and coordinating regional demand management helps to
eliminate many key tradeoffs and potential inter-utility conflicts. Future work
should consider innovative strategies for coordinating individual and regional
demand growth rates. In terms of the broader global water resources field, in-
sights from this work have general merit for regions where adjacent municipal-
ities can can introduce cooperative regional water portfolios.
80
CHAPTER 4
WATERPATHS: AN OPEN SOURCE STOCHASTIC SIMULATION
FRAMEWORK FOR WATER SUPPLY PORTFOLIO MANAGEMENT AND
INFRASTRUCTURE INVESTMENT PATHWAYS
Funding for this work was provided by the National Institute of Food and Agri-
culture, U.S. Department of Agriculture (WSC Agreement No. 2014-67003-22076).
Additional support was provided by the U.S. National Science Foundation’s Water,
Sustainability, and Climate Program (Award No. 1360442). The views expressed in
this work represent those of the authors and do not necessarily reflect the views or poli-
cies of the NSF or the USDA.
4.1 Abstract
Financial risk, access to capital, and regional competition for limited water
sources represent dominant concerns in the US and global water supply sec-
tor. This chapter introduces the WaterPaths framework: a generalizable, cloud-
compatible, open-source exploratory risk-modeling framework designed to in-
form long-term regional investments in water infrastructure while simultane-
ously aiding regions to improve their short-term weekly operational decisions.
Uniquely, WaterPaths has the capability to identify the challenges and demon-
strate the benefits of regionally coordinated planning and management for
groups of water utilities sharing water resources. As a platform for decision
making under deep uncertainty, WaterPaths accounts for uncertainties not only
related to hydrological or climate extremes, but also to key urban systems fac-
tors such as demand growth, effectiveness of water-use restrictions, construc-
tion costs, and financing uncertainties. The WaterPaths platform is introduced
81
here through the fictional and realistic Sedento Valley test case, in which three
resources-sharing water utilities improve their individual and joint infrastruc-
ture investments and weekly operations subject to various sources of deep un-
certainty to attain higher supply and financial performance. The Sedento Val-
ley test case was designed to serve as a universal test case for decision-making
frameworks and water-resources systems simulation software.
4.2 Software Availability
• Name of Software: WaterPaths
• Description: WaterPaths is an open-source C++ model for the stochastic
simulation of decision-making policies by water utilities concerning the
use and upgrade of single- and jointly-owned infrastructure. It was de-
signed to be easily customizable and to be used on high-performance-
computing clusters and cloud for single and batch simulations and for
policy optimization when coupled with a black-box multiobjective opti-
mization algorithm. WaterPaths can export detailed time-series output of
various system states and the values of objective functions (performance
metrics) defined by the user. Included is also an example test case named
Sedento Valley.
• Developer: B. Trindade (bct52@cornell.edu) with contributions by P. Reed.
D. Gold contributed to the development of the Sedento Valley test case.
• Funding Source: Funding for this work was provided by the National
Institute of Food and Agriculture, U.S. Department of Agriculture (WSC
Agreement No. 2014-67003-22076). Additional support was provided by
82
the U.S. National Science Foundation’s Water, Sustainability, and Climate
Program (Award No. 1360442).
• Source Language: C++
• Supported Systems: Unix, Linux, Windows, Mac
• License: Apache 2.0
• Availability: https://github.com/bernardoct/WaterPaths
The code used for the WaterPaths optimization runs can be found in the git
commit 1d07e944654569236dc496e5d6fa9933c75de9cd.
The code used for the WaterPaths re-evaluation runs can be found in the git
commit 5b612cc8eae39acf4411d9674d897c03943f1f27.
4.3 Introduction
Recent projections estimate that the United States (US) will require over one
trillion dollars of investment in water supply infrastructure in the next 20 years
(ASCE, 2017). This investment represents a difficult challenge as water fees,
which often fund water infrastructure projects, are already rising above the Con-
sumer Price Index (USWA, 2019b,a; Hall et al., 2019) and initiatives such as the
Water Infrastructure Finance and Innovation Act (commonly known as WIFIA)
(Copeland, 2016) have only been able to address a fraction of the required in-
vestment (EPA, 2019). This challenge is mirrored around the world as water
managers face the task of maintaining supply reliability under climatic, social,
and financial uncertainties (Bonzanigo et al., 2018; AWWA, 2019). In many re-
gions, rapid shifts in seasonal climate, drought patterns, and the frequency of
83
extreme events threaten to overwhelm existing water supply capacity (Shafer
and Fox, 2017). These threats are felt more acutely in large metropolitan areas,
which have observed strong and sustained population growth in recent decades
(McNabb, 2019; Census, 2019). To face these challenges, water managers must
move beyond historic paradigms of water resources planning and management
and transition to new methods of sustainable freshwater management (Gleick,
2018).
Historically, water managers have relied on infrastructure expansion to con-
front long-term supply risks and utilized water use restrictions to manage short-
term drought crises (Gleick, 2002b). This approach often yields high financial
burdens as utilities are forced to accumulate large amounts of debt. Revenue
disruptions resulting from water use restrictions may cause utilities to miss pay-
ments on this debt, threatening their financial stability, and reducing their abil-
ity to make future investments (Hughes and Leurig, 2013; Moody’s, 2017, 2019).
Additionally, the land or water resources most suitable for new water supply
infrastructure have largely been developed (Lund, 2013) and new water infras-
tructure projects (such as large dams) are often environmentally destructive and
unpopular. These challenges have led to the proposal of soft path strategies to
water resources management that seek to compliment centralized infrastructure
with non-structural measures to improve efficiency and manage water demand
(Gleick, 2002b).
Exploring synergies between long-term infrastructure investment pathways
and non-structural drought mitigation instruments is one promising method for
incorporating soft path strategies into water supply planning and management.
Potential short-term management instruments include demand management
84
(Hall, 2019; Zeff et al., 2014; Borgomeo et al., 2018; Huskova et al., 2016; Haas-
noot et al., 2013), treated water transfers through regional shared infrastructure
(Caldwell and Characklis, 2014; Zeff et al., 2014; Watkins and McKinney, 1997;
Hall, 2019), and raw water transfers through upstream reservoir releases (Bor-
gomeo et al., 2018; Gorelick et al., 2019). Integrating these short-term manage-
ment instruments with long-term infrastructure sequencing allows utilities to
create adaptive management strategies that maintain reliability under drought
conditions while minimizing the need for infrastructure expansion (Zeff et al.,
2016).
The coupling of long-term infrastructure investment strategies with short-
term drought mitigation necessitates innovations in decision support systems
for water resources management problems. Broadly, decision support sys-
tems are composed of the suite of analytical, mathematical, and/or simulation-
focused tools that aid planners in the design and operation of water resources
systems (Loucks and Da Costa, 2013). There is large historical body of litera-
ture water resources planning and management where simulation models are
a central focus for analysts developing insights on how a system behaves un-
der varying system states or decision maker actions (see review in Loucks and
van Beek 2017). Since the inception of the field in the early 1960s, these wa-
ter systems models have typically consisted of networks of storage and junc-
tion nodes linked by conveyance structures such as pipelines, canals and river
reaches (Maass et al., 1962; Harou et al., 2009). There a large range of mod-
ern example simulation software that have held substantial benefits for decision
support applications including MODSIM (Labadie, 2011), Water Evaluation and
Planning (WEAP) (Sieber, 2006), Interactive River-Aquifer Simulation (IRAS-
2010) (Matrosov et al., 2011), WATHNET (Kuczera 1992), RIBASIM (Hydraulics,
85
2004), MIKE BASIN (Jha and Gupta, 2003), Source (Welsh et al., 2013), CALVIN
(Draper et al., 2003) and OASIS (HydroLogics, 2009). For a summarized feature
comparison between WaterPaths and other mentioned software/framework,
see Tables 4.3 and 4.3.
More recently, there has been a growing interest in multiobjective simu-
lationoptimization applications where models such as MODSIM, IRAS and
WATHNET are coupled with multiobjective evolutionary algorithms (MOEAs)
to aid in the discovery of key water supply planning and management tradeoffs
(Matrosov et al., 2015; Borgomeo et al., 2018; Basdekas, 2014). MOEAs are global
population-based search algorithms that evolve sets of Pareto approximate so-
lutions to multiobjective problems through processes of mating, mutation and
selection (for reviews see Nicklow et al. 2010, Maier et al. 2014 and Coello et al.
2006). However, analytical decision support tools such as MOEAs require the
ability to develop an effective and efficient software coupling with simulation
software. Moreover, typical applications substantially increase the computa-
tional demands of analyses because MOEA search typically requires thousands
to tens of thousands of simulation-based evaluations of performance objectives
to guide their search processes.
Two more nuanced concerns with regard to modern water systems simula-
tion software is that (1) they often do not provide users with flexibility in rep-
resenting complex state-aware actions as either an artifact of their approach to
water balance modeling (e.g., optimization or rule-based allocations that can-
not capture information feedbacks) and (2) they contain limitations that arise
from a lack of access to their underlying source code bases (e.g., commercial
software packages). Likewise, despite the growing recognition of that a broad
86
87
Table 4.1: Select modeling software for water resources planning and management
Framework Purpose Incorporated Routing Financial Modeling Ca- MOEA Represented
Uncertainty Method pability Ready Time-scales
IRAS-2010 Computationally efficient simulation model for Demand and in- Rule Based Capital costs, operating Yes Daily to decadal
flows and storages, multi-reservoir release, water flow costs, energy costs and
consumption, hydropower and pumping energy use hydropower revenue
OASIS River Basin Management, Hydropower, Water Sup- Inflow and de- Optimization None No Minute to
ply, Conflict Resolution mand Driven decadal
WEAP Simulation of water supply systems through Inflow, de- Optimization None No Daily to decadal
watershed-scale hydrologic processes, water de- mand, land Driven
mands and envrionmental requirements use
MODSIM River Basin Management for analysis of long term Inflow and de- Optimization None Yes Minute to
planning, midterm management and short term op- mand Driven decadal
erations
WATHNET Simulation model for streamflow storage and trans- Inflow and de- Optimization None Yes Daily to decadal
fer and water demand mand Driven
RIBASIM Multi sector planning to allocate water resources at Inflow and de- Rule Based Crop yeild and produc- No Daily to decadal
the river basin level mand tion costs
WaterPaths Flexible simualtion and optimization to aid in Inflow, de- None Water revenues, de- Yes Weekly to
long term infrastructure sequencing and short term mand, cus- mand mangement and decadal
drought mitigation tomizable suite drought mitigation
of uncertainty cost. Long term capital
costs and infrastructure
financing
88
Table 4.2: Select modeling software for water resources planning and management (continued)
Framework Batch pro- Endogenous Represented deci- GIS inte- Custom GUI Extendable Access Illustrative citations
cessing Infrastructure sions gration
Expansion
IRAS-2010 Yes No Static infrastructure Yes, exter- Link to hy- Yes, Fortran Free (Matrosov et al., 2011,
and demand manage- nal droplatform 2015)
ment
OASIS No No Custom River Basin No Yes No Commercial (HydroLogics, 2009)
Operating Rules
WEAP No No Custom River Basin Yes Yes No Free (Sieber, 2006; Yates
Operating Rules et al., 2005)
MODSIM No No Custom River Basin Yes Yes Yes, C# .NET Free (Labadie, 2011; Bas-
Operating Rules or Visual Ba- dekas, 2014)
sic .NET
WATHNET Yes No Static infrastructure Yes Yes Yes, custom Free (Kuczera, 1992; Bor-
and demand manag- scripting gomeo et al., 2018)
ment
RIBASIM No No River Basin Operating Yes Yes No Free (Hydraulics, 2004)
Rules
WaterPaths Yes Yes Dynamic state-based No No Yes, C++ Free This Manuscript
risk triggers, Custom
metrics
array of uncertainties strongly shape water resources systems (e.g., financial
risks, behavioral responses, changing hydro-climatic conditions, shown Chap-
ter 3), currently available water resources systems simulation software are again
highly constrained in their representation and scalable computational support
for including these concerns in water resources infrastructure investment and
management applications.
Beyond limitations in current water systems simulation software, broader
conceptual challenges must be considered in our decision support framework’s
treatment of uncertainties, in particular the presence of deep uncertainties (for
recent reviews see Moallemi et al. 2019; Dittrich et al. 2016; Kwakkel and Haas-
noot 2019; Herman et al. 2015). Deep uncertainty refers to conditions when deci-
sion makers do not know or cannot agree upon probability distributions of key
system parameters and/or the system boundaries (Kwakkel et al., 2016a; Lem-
pert, 2002). Planning and management under deep uncertainty shifts the task
of discovering optimal management strategies for water resources systems, to
crafting robust and adaptive strategies that maintain performance across a wide
array of potential future conditions (Walker et al., 2013; Dittrich et al., 2016). Re-
cent work in water resources systems planning and management has focused on
bottom up decision. Frameworks such as Decision Scaling (Brown et al., 2012),
Robust Decision Making (RDM) (Lempert et al., 2006), Many Objective Robust
Decision Making (MORDM) (Kasprzyk et al., 2013) and Info-gap (Ben-Haim,
2006) provide methods to aid decision makers in the discovery of key uncer-
tainties control system vulnerability. These methods are often computationally
demanding exploratory Monte Carlo simulation software, that strongly benefit
from highly scalable simulation software that can be adapted across a range of
state-of-the-art computing architectures.
89
The challenges and needs summarized above motivated this chapter’s con-
tribution of the open-source WaterPaths is a stochastic simulation software that
has been specifically develop to support applications of the DU Pathways deci-
sion support framework. WaterPaths allows for a flexible and efficient represen-
tation of multi-actor water resources systems while providing advanced com-
putational support for multiobjective optimization algorithms and exploratory
analyses of a broad range of uncertainties. WaterPaths has been specifically
developed to focus on water supply infrastructure and water portfolio manage-
ment applications for systems confronting water scarcity or increasingly severe
droughts. WaterPaths provides decision makers with ability to flexibly abstract
water infrastructure portfolio problems that contain measures for addressing
short-term supply and financial impacts of droughts along with long-term ca-
pacity expansions. WaterPaths is developed to run on desktops, cloud com-
puting systems, and on high performance computing resources, utilizing both
shared memory and distributed memory parallelization to enable decision mak-
ers to efficiently design, optimize and evaluate candidate water supply portfo-
lios over large ensembles of potential future states of the world.
We demonstrate WaterPaths by introducing the Sedento Valley Test Case, a
new highly detailed and realistic hypothetical multi-actor regional water sup-
ply planning test case. WaterPaths is used to represent three hypothetical water
utilities in the south eastern US that are facing the prospect of water shortage
due to growing demand and changing climate. The utilities seek to craft both
short-term drought mitigation responses and long-term infrastructure sequenc-
ing pathways that maintain reliable supply and financial stability. The utilities
have the potential to cooperate using treated transfers through existing infras-
tructure and through shared infrastructure development. The DU Pathways
90
framework is used to assess the utilities tradeoffs, robustness, and infrastructure
investment pathways as a means of showcasing the key features of WaterPaths.
The framework is demonstrated on the Sedento Valley Test Case, a new
generic multi-actor test bed for regional water supply planning. The model
represents three hypothetical water utilities in the South Eastern United States
facing the prospect of water shortage due to growing demand and changing cli-
mate. The utilities seek to craft both short-term drought mitigation responses
and long-term infrastructure sequencing pathways that maintain reliable sup-
ply and financial stability. The utilities have the potential to cooperate using
treated transfers through existing infrastructure and through shared infrastruc-
ture development. The DU pathways explored through this test case showcase
the features of WaterPaths.
This chapter is organized as follows: Section 4.4 details the WaterPaths
framework, Section 4.5 introduces the Sedento Valley test case, presents its de-
velopment in WaterPaths and overviews the computational experiment, Section
4.6 presents results and discussion and details and Section 4.7 provides conclud-
ing thoughts.
4.4 WaterPaths Framework
4.4.1 WaterPaths Overview
Planned adaptation has been studies as an effective way to cope with deep un-
certainty (Walker et al., 2003). One approach to adaptive infrastructure plan-
91
ning is Dynamic Adaptive Policy Pathways (DAPP). DAPP uses adaptation tip-
ping points, system conditions that cause an alternative to fail to meet speci-
fied objectives (Kwadijk et al., 2010) to design signposts over 30 to 100 year-
long schedules that trigger pre-specified adaptive actions(Haasnoot et al., 2013;
Kwakkel et al., 2014; Kingsborough et al., 2016, 2017; Zandvoort et al., 2017).
DAPP, however, still maintains a strong reliance on the use of limited num-
bers of predefined action sequences that require high levels of institutional sta-
bility and strong consensus over the long-term. Real Option Analysis (ROA)
(Cox et al., 1979) is a probabilistic decision process for flexibility and adaptabil-
ity in the context of irreversible decisions based on decision trees, lattices and
Monte Carlo analysis (Erfani et al., 2018). Although recently applied in analy-
sis of reservoir development and expansion (Erfani et al., 2018; Fletcher et al.,
2019), drought planning (Fletcher et al., 2017) and flood protection (Hui et al.,
2018), ROA is limited in its accounting of stakeholders with diverse interests for
being a single-objective approach and its underlying mathematical tree logic
quickly becomes computationally intractable when a large number of uncer-
tainties are considered (Dittrich et al., 2016). Borgomeo et al. (2018) propose a
multi-objective evaluation of candidate water supply investments as a way to
consider diverse interests in which the authors maximize robustness and min-
imize risk and cost of a 30-year long fixed construction schedule for multiple
infrastructure options. However, the resulting static sequence of investments
is not adaptable to regional developments and to the local political process, a
limitation only partly addressed by Beh et al. (2015b) and Huskova et al. (2016).
The recently proposed Deeply Uncertain Pathways (DU Pathways) frame-
work, presenteded in detail in Chapter 5, utilizes state-adaptive portfolio based
strategies that employ risk based rule systems to define infrastructure sequenc-
92
ing, drought mitigation response and financial instruments and tailor actions
to observed system states. The framework builds upon the Many MORDM
framework to include the design of rule systems that react to observed system
states. DU Pathways provides a means of implementing closed loop control
policies (Bertsekas et al., 1995) for infrastructure sequencing and drought miti-
gation. The simultaneous optimization of risk-based infrastructure triggers and
drought-mitigation instruments effectively bridges the gap between long-term
infrastructure investment and short-term water portfolio management. Chapter
5 demonstrates that the DU Pathways approach can produce robust and adap-
tive infrastructure pathways that perform well across broad sets of uncertainty
and balance conflicting interests within a multi-actor systems. However, DU
Pathways framework involves upgrades of the infrastructure system over time,
is computationally intensive due to its required stochastic simulations, and re-
quires large numbers of function evaluations during MOEA search, making it of
difficult implemention with existing decision-support systems. To aid in the de-
velopment and application of DU Pathways a new decision support framework
is needed for their design and evaluation. This need motivated the development
of WaterPaths.
WaterPaths is a open-source and modular model for the simulation and op-
timization of water infrastructure planning and management policies. Water-
Paths designed to facilitate the application of the DU Pathways methodology
and for easy addition of new features and other customization by an user with
basic knowledge of C++. WaterPaths includes in one simulation stochastic risk
calculations, stochastic uncertainty-based objective calculations, evolution of in-
frastructure system, and regionally coordinated planning operations of a group
of water utilities. On a high level, the simulation of a system of cooperating
93
water utilities and water infrastructure on WaterPaths is organized in terms of
mass-balance performed for each piece of water infrastructure, utilities drawing
water from and building new water infrastructure, and drought mitigation and
financial policies acting on utilities to ensure safe operations from a supply and
financial points of view. Each time it is run for a system, WaterPaths simulates a
number ranging from hundreds to thousands of realizations (independent sce-
narios) and calculate the final performance metrics (objectives) based on aver-
ages and min-max rules of various system-state variables across all realizations.
Each time step (day, week, or month) follows the following sequence, shown in
Figure 4.4.1: (1) calculation of long-term ROF metric if at first week of the year,
(2) calculation of the short-term ROF, (3) application of drought mitigation and
financial policies, (4) system mass-balance modeling, and (5) system-state data
collection.
The addition of new features such as new types of water infrastructure, cus-
tomized reservoir control rules, and new drought mitigation policies is done
by extending already implemented abstract classes. Similarly, any system met-
ric for decision-making purposes can be implemented with the addition of one
function to the existing code. The mass-balance model, ROF calculation model,
and other aspects of the system-wide simulation are designed to work with any
child class of the existing abstract classes, making it possible for a user to add
various features to WaterPaths without venturing into the existing WaterPaths
code.
Lastly, WaterPaths was designed for centralized system-state information
recording to allow for easy implementation of custom objective functions. It
currently has five objectives implemented, but more can be easily added to the
94
Figure 4.1: WaterPaths main simulation loop. The main loop includes
the (1) calculation of long-term ROF metric if at first week of
the year, (2) calculation of the short-term ROF, (3) application
of drought mitigation and financial policies, (4) system mass-
balance modeling, and (5) system-state data collection.
MasterDataCollector class. The currently implemented objectives are reliability,
restriction frequency, net present value of infrastructure built during the life of
the policy, maximum annual financial cost of drought mitigation and infrastruc-
ture construction, and worse-first-percentile annual financial cost of drought
mitigation and infrastructure construction. Each of WaterPaths features are dis-
cussed in the subsequent sections.
95
4.4.2 ROF-based decision rules
WaterPaths formulates investment and management policies as state-aware
rules that serve as action triggers based on short- and long-term ROF metrics
(Palmer and Characklis, 2009; Zeff et al., 2016). More specifically, the default
representation of decision policies in WaterPaths trigger each component action
in a utility’s investment and management portfolio in a given week if the calcu-
lated value of the ROF state xrof reaches trigger values. These decision triggers
are the core decision variables and allow for consistent and adaptive consider-
ation of short-term temporary management actions or longer-term permanent
water supply capacity expansions. As discussed in prior studies (Zeff et al.,
2016) and in Chapter 3, the ROF-basis allows for low dimensional closed-loop
rules that take actions tailored to the state of the world (SOW) being experienced
(i.e., different action sequences for wet versus dry scenarios). Additionally, the
ROF metric can be parametrized to capture only single-year droughts, the fo-
cus of short-term drought management instruments (here water use restrictions,
transfers and drought insurance), and multiple-year drought, the focus of long-
term actions (infrastructure construction). Equations 4.1 to 4.3, partly repeated
here from Section 3.5 for completeness but now expandaded, mathematically
define how the short- and long-term ROF metrics are computed. Figure 4.2
presents an equivalent graphical representation while making the distinction
between the short- and long-term ROF metrics.
Nrof
w 1
∑
w ′ ′xsrof,j = fy′,j(NI
y ,Ey ) (4.1)
Nrof
y′=0
where,
96
Figure 4.2: For the calculation of the short-term ROF metric for the cur-
rent week, 50 year-long simulations are ran for the coming year
with all reservoir storage levels starting at their current levels,
while the streamflows and evaporation rates for each simula-
tion are those of one of the previous 50 years of recorded data
and the demand are those of the previous year. For the calcula-
tion of the long-term ROF metric, the 50 year-long simulations
are extended to one and a half years, with the last 6 months of
the last simulation (the last year) are filled by synthetic data.
The addition of 6 months to the 50 ROF simulations allows for
capturing consecutive drought years in which the reservoirs
are not filled during the wet season.
97
 y,w′∀ ′ ∈ { ′ ′ x} s′0 w (y , w), ..., (y , w + Trof ) :
,j ≥ sc
w  C jfy′,j = (4.2)1 otherwise
and, ( )
xy
′,w′ ′ ′ ′ ′ ′ ′
s′,j = f Cj,UD
w,NIy ,w ,Ey ,w y ,wj j j ,Wj |Ψs (4.3)
In equations 4.1 to 4.3, w′ and y′ mean a week and a year simulated with past
data for the calculation of the ROF. Variable xwrof,j is the ROF for utility j in
current week w, and fy′,j is a binary variable for which 0 denotes a failure hap-
pened during the corresponding ROF simulation with data from past year y′. In
the ROF metric, ′ ′ ′NIy , Ey and W y are the recorded natural reservoir inflows,
evaporation rates and reservoir spillage, respectively, in year y′ prior to current
week w used in one of theNrof simulations — note that in a simulation in which
the ROF metric is calculated for a week 20 years from now, 20 of the Nrof years
of hydrological data will belong to synthetically generated data. Variable Trof
equals 52 weeks for the short-term ROF, so that single-year droughts are cap-
tured, and 78 for the long-term ROF, so that each ROF year-long simulation po-
tentially captures two dry seasons. A failure is defined as the combined storage
′
xy ,w
′ ′
s′,j for any realization y divided by combined storage capacity Cj for utility
′ ′
j falling below critical storage sc. Variable x
y ,w
s′,j is the vector of storage states
calculated in one of the year-long ROF simulations using recorded hydrologic
′
data from past year y′. In the calculations of xy,w ws′,j , UD is the unrestricted de-
mand (no restrictions enacted or transfers purchased) in week w, and Ψs is a
matrix of sampled deeply uncertain factors obtained through WCU or DU op-
timizatio. Given a year has 52.178 weeks (365.25 / 7), every 6 years (when the
rounded number of weeks in a year would reach 53) the first week of the fol-
lowing historical year will be used. Although the ROF metric been proved to
98
be an effective metric for water-systems decision making, other metrics such as
the more conventional days-of-supply-remaining or new metrics can be imple-
mented and tested with WaterPaths for the Sedento Valley test problem. The
calculation of the ROF metric as well as the mass-balance simulation for the
system at hand are dependent on WaterPath’s mass-balance model, described
next.
4.4.3 Mass-Balance Model
The core of WaterPaths is its mass-balance model. The mass-balance
model solves mass-balance equations for all water infrastructure following its
upstream-to-downstream order, found through topological sorting based on in-
frastructure connectivity information provided by the user as a directed graph.
The mass-balance equation for reservoirs is shown in equation 4.4:
xw+1 ws = xs +NI
w + SEw + UROw − ERw ·RA(xws )− EOw − Sw −RDw (4.4)
where xw+1s is the volume of water stored in the reservoir at the week after the
current week w, NI is the natural inflow into the reservoir from all its tribu-
taries, SE is a treated sewage effluent discharged either on a tributary or directly
on the reservoir, URO is the upstream reservoir total outflow (if such reservoir
exists, mandated outflow plus spillage), ER is a non-dimensional evaporation
rate, RA is the reservoir area as a function of stored volume, EO is the environ-
mental outflow, RDw is the total municipal demand drawn from that reservoir
by one or more independently modeled utilities, and S is the reservoir spillage,
which is set to zero unless the reservoir is completely full. If a reservoir is al-
99
located for multiple uses (municipal supply to multiple utilities, water quality,
etc.), each with a designated percentage of the total storage capacity, the inflow
is split across all uses proportionally to their allocated volumes. Water in excess
of any allocated capacity is redistributed among the others.
To allow for more realistic simulation of reservoirs, WaterPaths also provides
an abstract class for reservoir control rules named MinEnvFlowControl and one
for controls of other discharges such as treats waste water discharges (the lat-
ter flowing into a reservoir or stream), named ControlRules. WaterPaths also
contemplates non-storage water infrastructure such as water intakes and wa-
ter re-use stations, both of which already implemented, although others such
as desalinization plants can be easily implemented by extending the abstract
class WaterSources based on a custom mass-balance function. WaterPaths also
has implemented upgrades of existing infrastructure, namely reservoir expan-
sions and treatment capacity expansions, both meant to be potentially triggered
throughout the simulation by one or multiple utilities based on the long-term
ROF or a custom metric.
4.4.4 Capturing Regional Decision Making
Being designed for regional decision making, WaterPaths can simulate multiple
water utilities, potentially with interconnected networks shared infrastructure,
as separate elements within the system. Each utility in WaterPaths uses infor-
mation about the allocated available volume of water from its infrastructure
available (e.g. stored volumes, weekly volume allowed to be taken with an in-
take on a river, re-use station’s treatment capacity in a given week) to split its
100
demand among them, drawing as much from non-storage infrastructure before
drawing from reservoirs. The volume of water in storage owned by a given
utility is also used to calculate its short- and long-term ROF metrics described
in Section 4.4.2, based on which each utility triggers the construction of new
infrastructure and drought mitigation policies. The demand each water utility
is required to fulfill may be mitigated or increased drought mitigation policies,
described in Section 4.4.5, before it is drawn its infrastructure.
Additionally, water utilities in WaterPaths track their own finances. The fi-
nances as modeled as cost fluctuations due to drought mitigation and financial
instruments and debt repayment, described respectively in Sections 4.4.5, 4.4.6,
and 4.4.7, around operations and maintenance costs. The total annual revenue
is also calculated based demand, consumer tiers, and corresponding tariffs, and
is used to support objectives calculations and financial instruments.
4.4.5 Drought Mitigation Instruments
Building from prior works (Zeff and Characklis, 2013; Zeff et al., 2014, 2016;
Palmer and Characklis, 2009; Caldwell and Characklis, 2014) and from Chapter
3, WaterPaths has implemented water-use restrictions and inter-utility, treated-
water transfers, although other instruments can be easily added by creating a
new child class of the DroughtMitigationInstrument class. Water-use restric-
tions as currently implemented in WaterPaths can be implemented on multi-
ple tiers, each with different percentages of demand reduction based on stricter
measures (reductions in lawn irrigation, car and sidewalk washing, etc.) and
triggered by a different value of the ROF metric. As currently implemented,
101
revenue losses due to restricted water sales can be mitigated by adjusted tariffs
as provided by the user.
Treated-water transfers in WaterPaths are performed from a source utility to
requesting utilities. As water use restrictions, requests for treated transfers are
contingent on the current value of the short-term ROF metric for each request-
ing utility. However, treated-water transfers often suffer from two constraints:
treatment capacity at the source utility and limited inter-utility conveyance ca-
pacity. To calculate the volume granted to each utility, WaterPaths solves a con-
strained allocation problem using a quadratic-programming algorithm (Gold-
farb and Idnani, 1983; Gaspero, 2007). The problem is set up to minimize the
mean-square error between the volumes requested by each utility adjusted for
their ROF values (a utility with a higher ROF receives proportionally more wa-
ter) and the volumes that can be transferred through the inter-utility network
subject to conveyance constraints. Funds are then transferred from all request-
ing to the source utility. Drought mitigation instruments, however, can be costly,
and utilities may benefit from setting in place financial instruments to absorb
and stabilize such costs. More details about water-use restrictions and treated-
water transfers are presented in Equations 3.6 and 3.7.
4.4.6 Financial instruments
WaterPaths allows utilities to hedge against the negative financial effects of the
drought mitigation instruments by using financial instruments, such as the cur-
rently implemented drought insurance and contingency funds (Zeff et al., 2014).
The drought insurance currently implemented in WaterPaths triggers a payout
102
of a percentage of the previous year’s annual revenue whenever the value of the
ROF metric reaches a set value. The policy is updated, priced, and bought by
the utilities every year based on the most current values of annual revenue and
on a premium of 20% of the expected cost of the policy for the following year.
Other types of insurance policy based on state variables other than the ROF can
be easily added to WaterPaths by creating a child class of the DroughtMitiga-
tionPolicy class.
On the other hand, contingency funds are not dependent on the ROF metric,
but on a fixed percentage of the annual revenue. On the last week of every year,
each utility that has a contingency fund adds a fixed percentage of that year’s
total revenue to its contingency fund. These funds are then used during the
following year to pay for drought mitigation policies, other financial policies
and for debt issued to build new infrastructure. The drought mitigation and
financial instruments, however, may not be enough to protect a utility against
the unexpected negative financial effects of droughts, in which case construction
of new infrastructure may be needed. More details and drought insurance and
contingency funds are presented in Section 3.5.
4.4.7 Infrastructure Investment
As with the drought mitigation instruments and insurance, infrastructure con-
struction is also triggered by an ROF value, although that of the long-term ROF
metric. If a utility or region has several candidate infrastructure investment op-
tions, users or MOEA-based search must provide a prespecified construction
sequence (i.e., a infrastructure pathway across time). To allow for infrastruc-
103
ture construction over the course of the simulated planning period, WaterPaths
relies on information passed by the user or optimization algorithm about all in-
frastructure options (storage capacity, treatment capacity, cost, etc.) and about
the sequence in which a utility is to build such options if the value of the long-
term ROF metric reaches its trigger value. This is a unique feature of Water-
Paths, which allows the user to prioritize options in the near term and define
requirements.
Infrastructure options to be considered for construction can be located any-
where in and outside of the reservoir connectivity network. Infrastructure op-
tions in WaterPaths may take form of altogether new projects, such as new reser-
voirs, water intakes, treatment plants, and others, and of expansion of current
infrastructure such as storage treatment capacity expansions and reservoir re-
allocations. This allows WaterPaths to account for flexible infrastructure devel-
opment, justly emphasized in the Real Options Analysis literature as important
elements in the long-term planning for water utilities (Fletcher et al., 2017, 2019;
Erfani et al., 2018; Wang and de Neufville, 2005; Cox et al., 1979).
Infrastructure development is often financed over decades rather than paid
upfront. WaterPaths allows for multiple types of bonds to be issued for financ-
ing new infrastructure, resulting in different possible debt repayment streams
with different net present values for the same infrastructure option. Cur-
rently implemented in WaterPaths are level-debt-service, balloon-payment, and
variable-interest bonds. However, WaterPaths allows for the design of creative
finance mechanisms by allowing the user to create new types of bonds by cre-
ating children classes of the Bond class and including those in optimization and
simulation exercises. Infrastructure development and policy design are highly
104
dependent on uncertainty analysis, deemed as a central concern during the de-
sign of WaterPaths. The uncertainty handling within WaterPaths is described
next.
4.4.8 Flexible Representation and Evaluation of Uncertainties
One of the key features that differentiate WaterPaths from existing simulation
systems is its stochastic treatment of a broad array of uncertainties. As discussed
in the Section 4.3, multiple commercial frameworks can only run one realization
(a scenario fully specified by one time series of stream flows, evaporation rates
and demands, when pertinent, and one value for other uncertainty factors) at
a time, which implies in no explicit consideration of uncertainties (see Tables
4.3 and 4.3). Others, most notably the ROA frameworks, include uncertainty
in policy optimization in a Bayesian fashion using decision trees and stochastic
dynamic programming over expected costs (Fletcher et al., 2017, 2019; Hui et al.,
2018). Lastly others consider uncertainty in a stochastic fashion by simulating
a policy over hundreds or thousands of realizations every time the model is
called, with policy optimization being performed by attaching the model to a
black-box optimization algorithm (Zeff et al., 2014; Kwakkel et al., 2014; Watson
and Kasprzyk, 2016; Zeff et al., 2016; Borgomeo et al., 2018). WaterPaths was
designed based on the latter approach.
WaterPaths allows for both WCU and DU uncertainty sampling schemes
(see Figure 3.2 and the broader Section 3.3.3). It currently requires as minimum
uncertainty-related input one time series of inflows, evaporation rates, and de-
mands for each piece of infrastructure and utility, when applicable, for each
105
realization to be run. In addition, the user has the option of providing a series
of multipliers representing deeply uncertainty factors, which if not provided
will assume a default value of the unit. All time-series and multipliers are to
be sampled externally from any desired distribution and passed to WaterPaths
as input data. The user can add as many deeply uncertain factors as desired
when setting up a problem or in new classes of water infrastructure, controls,
and drought mitigation policies. The deep uncertainties currently included in
WaterPaths are presented in Table 4.3. Simulating and optimizing a policy using
a stochastic approach to modeling uncertainty is a mathematically complex and
can be computationally costly, so care was taken to allow WaterPaths to both
be computationally efficient and to make use of any scale of computational re-
sources available to the analyst.
Type Uncertainty Source Regional or Individ-
ual for Utility/ In-
frastructure
Supply/Demand Growth of Mean An- Regionalnual Demand
Growth of Mean An- Regional
nual Evaporation
Financial/ Interest Rate Regional
economics Bond Term RegionalDiscount Rate Regional
Policy Effective- Water Use Stage Restric- Individual
ness tion Efficacy
Infrastructure Permitting Time Individual
Construction Construction Cost Individual
Table 4.3: Uncertainties included in extended version of the regional wa-
ter utility planning and management problem presented in Zeff
et al. (2016). Regional uncertainties are those for which the same
value was used for all utilities, versus a value different value for
each utility, as in the case uncertainties described as individual.
106
4.4.9 Currently Implemented Objective Functions
WaterPaths has implemented by default five objective functions, which can be
deleted, modified, and added for specific problem. The first two are the Reli-
ability (fREL), Restriction Frequency (fRF ), objectives presented in Section 3.5.
However, a new Infrastructure Net Present Cost was added and the Drought
Management Cost and Exposure to Financial Risk objectives presented in Sec-
tion 3.5 were replaced by the Financial Cost and Worse First Percentile Cost
below.
• Infrastructure Net Present Cost (fNPV ): The average net present cost of all
new infrastructure build across all realizations:
∑N1 r ∑BM PMT
minimize fNPV = (4.5)
Nr (1 + d)yi=1 y=1
where BM is the bond term, d is the discount rate (5%), y is the year of the
debt service payment PMT since the bond was issued, with PMT being
calculated as (assuming a level debt service bond):
[ ]
P BR(1 +BR)BM
PMT = (4.6)
[(1 +BR)BM − 1]
where P is the principal (construction cost), BR is the interest rate to be
paid to the lender BT is the bond term. The stream of payments is then
discounted to present values.
• Financial Cost (fAC): The financial cost objective represents the expected
yearly cost of all water portfolio assets used to manage droughts over the
planning horizon. These costs are revenue losses from restrictions, trans-
fer costs, contingency fund contributions, third-party insurance contract
107
costs, and debt repayment: [ ∑ ]Nr ∑Nys1
minimize fAC = max SY C
y
i,j (4.7)
j Nys ·Nr i=1 y=1
where, ∑
PMT y y
y c∈Cj i,j,c + θacfc,j · ATRi,j + IPi,jSY Ci,j = ATRyi,j
where IP is the insurance contract cost in a given year y, PMTi,j,c is the
debt payment for infrastructure option c if it belongs to the set Cj of in-
frastructure options to be built by utility j and is built in realization i, and
ATR is the total annual volumetric revenue. All these variables are dollar
values.
• Worse First Percentile Cost (fWFPC): The worse case cost objective rep-
resents the 1% highest single-year drought management costs observed
across all analyzed SOWs over the planning horizon:
max(RLyy i,j + TC
y y y
i,j − θacfc,j · ATRi,j − Y IPOi,j, 0)
SY Ci,j = y (4.8)ATRi,j
where IP is the insurance contract cost in a given year y,RL is the revenue
losses from water use restrictions, TC is the transfer costs, Y IPO is the
total insurance payout over year y, CF is the available contingency funds,
and ATR is the total annual volumetric revenue. All these variables are
dollar values. The worse case cost objective is then:
{ }
minimize fWCC = max quantile(SY Ci,j, 0.99) (4.9)
j i∈Nr
All six objectives are calculated based on the hundreds to thousands of re-
alizations described in Section 4.4.8. Given the high computational demands
108
of calculating such high numbers of realizations, WaterPaths implements con-
cepts of parallel computing to decrease the simulation and policy optimization
times. The focus of the remainded of this section is on the parallel structures of
simulation and policy optimization with WaterPaths.
4.4.10 Reducing Runtime With Local Parallel Computing
WaterPaths’ stochastic design allows for detailed uncertainty analysis but is
computationally expensive due to the required high number of realizations and
the stochasticity of the ROF calculations. However, given each realization can be
simulated independently from all others, they can be easily distributed across
multiple computational cores (CPUs) within a processor of a modern desktop,
laptop, or cloud computing instance. This parallelization was achieved by using
the shared-memory parallelization scheme (several cores working on the same
process, meaning sharing the same block memory) implemented by means of
OpenMP 5.0 code directives (Klemm et al., 2019). The OpenMP directives allow
WaterPaths to create a given number of computational threads and distribute
them across available cores. A thread is a set of computing instructions created
to execute calculations independently of each other while accessing the same
block of memory containing the problem’s data. One of these threads is the
master thread, which coordinates all worker threads among which it distributes
all realizations, retaining some for itself to run, and runs the serial parts of the
code (e.g. objectives calculation). WaterPaths creates by default a number of
threads equal to the number of cores in the computer, although the user is given
the option of manually setting the number of threads (a value of one means that
all realizations will be run in serial). Figure 4.4.10a shows a WaterPaths run
109
parallelized with shared-memory parallelization.
Manually setting a number of threads different than the number of cores may
have four advantages. First, it allows the user to prevent WaterPaths from using
cores already in use by other programs (or other WaterPaths instances) running
simultaneously on the same computer, which may potentially significantly slow
down all applications and maybe even freeze the computer. The second advan-
tage is that although Hyperthreading1 often results in performance gains for
WaterPaths, it sometimes has the opposite effect, which and can be avoided by
manually setting the number of used cores to the number of physical cores (e.g.
four in case of a quad-core desktop). Thirdly, running one realization per core in
some processors with higher cores counts (e.g. Intel’s Knight’s Landing series
with its 68 cores) may reduce memory-access efficiency and negatively affect
performance, in which case the smallest runtimes can be achieved by setting
the number of used cores to a number slightly smaller than that of available
physical cores. In fact, if running WaterPaths for batch simulations, it is often
more efficient to run two to four WaterPaths instances simultaneously, each with
fewer threads than the total number or cores and with all instances amounting
to two threads per core when Hyperthreading is available. The last reason for
manually setting the number of used cores is related to coupling WaterPaths
to distributed optimization algorithms, in which multiple function evaluations
(one fully stochastic WaterPaths run) are performed simultaneously. This use
case is discussed next, following the exposition of a typical policy optimization
exercise performed with WaterPaths.
1Hyperthreading allows for two threads to run simultaneously in one core by taking advan-
tage of the high number of circuits within a core. It does so by feeding independent instructions
from each thread to circuits not being utilized at a time by the other thread, potentially resulting
in efficiency gains to the order of 30% although sometimes decreasing performance. The opera-
tional system of a computer with Hyperthreading enabled will count each physical core as two
virtual cores.
110
Figure 4.3: Schematics of (a) shared-memory parallelization and (b) hy-
brid shared- and distributed-memory parallelization for dis-
tributed heuristic optmization. Hyperthreading is not used in
either panel. The collored and semi-transparent tiles represent
threads, the dark solid tiles represent physical cores mounted
on desktops or computing nodes (medium grey large and thin
tiles serving as base for the cores), and the light grey sheet
underneath the two nodes on panel (b) represents a cluster
or cloud environment hosting the two depicted nodes. Blue
represent the master threads, green the worker threads, and
read the threads (one of which still being the master) of a mas-
ter process. Threads connected by dashed black lines belong
to the same process, with its own exclusive memory block,
parallelized internally with OpenMP for shared-memory par-
allelization. Straight, full black two way arrows in panel (b)
represent distributed-memory communications between four-
core, four-thread master and worker processes, respectively
represented by the red threads and the green threads in the
yellow contour as an example. Panel (b) shows one master
and seven worker processes performing a heuristic optimiza-
tion run in parallel under the distributed-memory paralleliza-
tion paradigm.
111
4.4.11 Facilitating Decision Analytics
Given the diversity and number of policy actions that can be evaluated with Wa-
terPaths, the resulting water infrastructure investment and management path-
ways are typically composed of mixtures of discrete variables, continuous val-
ues, and permutations of infrastructure options. As a consequence, objective
measuring system performance across different metrics are non-convex and dis-
continuous functions of the policy variables. Moreover, the uncertainty sam-
pling enabled by WaterPaths has the potential to make the performance metric
functions noisy and highly heterogeneous in their behavior across water supply
or financial concerns. All these traits make the problem of manually or auto-
matically designing policies for water infrastructure planning and management
particularly difficult and attractive to researchers (Zeff et al., 2016; Fletcher et al.,
2019; Borgomeo et al., 2018; Huskova et al., 2016; Kwakkel et al., 2014; Beh et al.,
2017).
WaterPaths policy input and objectives output was designed for easy inte-
gration with any black-box multiobjective optimization algorithm and currently
has out-of-the-box integration with the Master-Worker Borg Multiobjective Op-
timization Evolutionary Algorithm (MS Borg MOEA) (Hadka and Reed, 2013,
2014), although support other algorithms, written in C++ or other languages,
can be easily added. The typical optimization problem formulation solved by
WaterPaths coupled with a multiobjective optimization algorithm is shown be-
low in Equations 4.10 through 4.15:
θ∗ = argminθ F (4.10)
112
s.t.
|ME| ≤ 1 ∀ME ⊆ BI (4.11)
Where:
 
f1 (xsrof, Ψs, xlrof, θ1, ..., θm, OPV)  f2 (xsrof, Ψs, xlrof, θ1, ..., θ , OPV)mF =  ... 
 (4.12)

fn (xsrof, Ψs, xlrof, θ1, ..., θm, OPV)
θ = [θ1, ..., θm, OPV] (4.13)
 
xsrof 

X = xlrof (4.14)
xs
       

ψinflows, 1   ψevaporation rates, 1 

ψ ψ 
ψdemands, 1   ψDU,1 
 inflows, 2   evaporation rates, 2  
ψdemands, 2   ψDU,2 Ψs = 
, , , 

. .  .   .  (4.15)
 ..   ..   ..   .. 
ψinflows, N ψR evaporation rates, N ψ ψ R demands, NR DU,NR 
(Optional)
Where F is a vector based objective function containing regional objectives
f1 through fn. The management and investment policies are represented in θ, a
vector containing all of the decision variables for all utilities. Decision variables
113
are usually comprised of but not limited to decision ROF triggers θ1 through θm
plus other policy variables OPV such percentages of annual revenue to be paid
by insurance in case of a drought. Matrix X has values of decision-relevant state
variables for all utilities and is comprised of xsrof , xlrof and xs, vectors of short-
and long-term values of the ROF metric (or any other custom decision metric),
described in Section 4.4.2, and system states, respectively. Matrix Ψs contains
vector samples of well-characterized and deeply uncertain time series and pa-
rameters. The matrix with samples deeply uncertain factors is an optional pa-
rameter. Matrix X has values of decision-relevant state variables for all utilities
and is comprised of xsrof , xlrof and xs, vectors of short- and long-term values
of the ROF metric, described in Section 4.4.2, and system states, respectively.
Matrix Ψs contains vector samples of well-characterized and deeply uncertain
time series and parameters. Well characterized uncertainties are contained in
the vector ΨWCU and deep uncertainties are contained in ΨDU.
4.4.12 WaterPaths Parallelization Strategy for Computational
Policy Search
The optimization problem described in Equations 4.10 through 4.15 can be
solved with various heuristics techniques that can in theory rely on a master
process coordinating multiple worker processes running simultaneously and in
charge of performing model evaluations. A process is an instance of the exe-
cutable comprised of WaterPaths plus the optimization algorithm running on
its own separate calculations and enclosed block of memory. After a worker
process finishes performing its function evaluation of a policy generated by
114
the master process, it returns the corresponding objective values to the mas-
ter process. The master process then compares the objective values it just re-
ceived to those of previously run policies, generates a new policy to be eval-
uated, and sends it to that worker process for evaluation, repeating the cycle.
To allow for massive parallelization across hundreds to thousands of nodes,
each with dozens of cores, this parallelization scheme must be developed with
a distributed-memory parallelization paradigm, in which the master and each
worker process have their own independent blocks of memory. An optimization
algorithm written in C/C++ or Fortran for distributed-memory parallelization
will likely be implemented with a library based on the MPI standard (Forum,
1994), such as OpenMPI (Gabriel et al., 2004) or Intel MPI (Intel, 2019), although
other languages may have their own libraries.
Shared-memory parallelization, described in Section 4.4.10, plays an impor-
tant role in enabling efficient use of parallel heuristic-optimization algorithms.
The naı̈ve parallelization strategy to make maximum use of all available cores
would be to have one worker process per computing core, resulting in all cores
being used simultaneously, each running one model evaluation. However, us-
ing this naı̈ve strategy across tens of cores within a node or a desktop would
likely result either on a freeze due to lack of RAM, as each WaterPaths func-
tion evaluation may require more RAM than available per core, or in inefficient
memory access. To solve this problem, each computing node can have a small
number of processes (distributed memory, each process has its own block of
memory), each running one model evaluation at a time and distributing real-
izations across a smaller number of cores. Figure 4.4.10 illustrates this hybrid
shared-/distributed-parallelization in action with WaterPaths.
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4.5 Methodology
4.5.1 The Sedento Valley: An Illustrative Test Case
To demonstrate the functionality of WaterPaths, we have created a hypothetical
test case that reflects many of the challenges facing regions with several inde-
pendently operated urban water utilities. Furthermore, this test case was de-
veloped with other researchers in mind as a benchmark problem against which
upcoming methodologies for water infrastructure planning and management
can be tested. The test case consists of three utilities in close geographic prox-
imity, facing increasing vulnerability to drought conditions due to climatic and
population-based stressors. The close proximity of the three urban utilities in
the test case provides an opportunity for regional cooperation and shared in-
frastructure development, but also creates the potential for inter-utility conflicts
over scarce resources. This study demonstrates WaterPaths through a demon-
strative implementation of the DU Pathways methodology (Chapter 5) to ex-
amine infrastructure investment and drought management strategies for the
Sedento Valley. The DU Pathways analysis combines the key WaterPaths capa-
bilities: (1) Monte Carlo evaluation of financial, institutional, and hydro-climatic
uncertainties, (2) ROF-based adaptive infrastructure pathways (Zeff et al., 2016),
and (3) a mix of short-term water supply portfolio instruments for drought mit-
igation (Characklis et al., 2006; Zeff et al., 2014).
The Sedento Valley, shown in Figure 4.4, is home to two medium sized cities,
Dryville and Fallsland and a smaller municipality, Watertown. The population
of the valley is near 1.5 million residents. The three municipalities are each
supplied by independent water utility. The cities of Dryville and Fallsland cur-
116
rently receive water from the Autumn Lake reservoir, a large flood control reser-
voir owned and operated by the US Army Corps of Engineers (USACE). Water-
town owns and operates College Rock Reservoir and receives water from Lake
Michael, another large USACE operated reservoir. Dryville and Fallsland also
have emergency allocations to Lake Michael to which they have access by pur-
chasing water from Watertown’s treatment plant and transferring via shared
pipelines. Current supply capacities of each water utility can be found in Table
4.4.
Figure 4.4: The Sedento Valley Regional Water Supply System.
A growing population has reduced each of the utilities’ capacity-to-demand
ratios, increasing their vulnerability to drought conditions. This problem is
117
Table 4.4: Current infrastructure capacity of Sedento Valley Water Utilities.
Utility Infrastructure Percent Allocation Supply capacity
(MG)
Watertown College Rock Reservoir 100 % 1049
Lake Michael 60 % 13,940
Dryville Autumn Lake Reservoir 38 % 23,839
Fallsland Autumn Lake Reservoir 62 % 23,839
Table 4.5: Demand to Capacity Ratios for the Sedento Valley Water Utili-
ties
Utility Baseline Capac- 2025 De- Demand-Cap Ratio
ity (MG) mand
(MG/Year)
Watertown 6629 12410 1.87
Dryville 9058.82 12738.5 1.41
Fallsland 14780.18 27813 1.88
compounded by the increased difficulty of large infrastructure investments (i.e.,
new reservoirs) due to scarcity in feasible sites, higher costs and reduced toler-
ance of environmental impacts. Increased hydro-climatic variability and uncer-
tainties stemming from land use and land cover changes that effect reservoir
inflows pose additional stresses to the region. Currently, the water utilities’
drought mitigation strategies rely entirely on the water use restrictions, a mea-
sure that is expensive and deeply unpopular with local residents. While the
three utilities are each facing increased water stress, their capacity-to-demand
ratios and access to new supply options differ as shown in Figure 4.5 and Table
4.5.
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Figure 4.5: Current Demand to Capacity ratios for the three utilities. A
higher demand to capacity ratio indicates more stress on the
system.
The asymmetry across the utilities capacity-to-demand ratios, their close ge-
ographic proximity, and interconnected infrastructure represents an opportu-
nity for cooperative regional water management strategies. There is an inter-
est in cooperative infrastructure investment pathways and coordinated drought
mitigation using a regionalized portfolio approach (Zeff et al., 2014) to improve
regional and individual performance of water supply systems. Each of the three
utilities has identified potential new infrastructure investments to improve their
water supply reliability. Watertown has the option to expand college rock reser-
voir, this can either be a large expansion or a small expansion. Dryville has
two small potential water supply options, the development of Granite Quarry
into a reservoir and the construction of Sugar Creek Reservoir. Fallsland and
Watertown have also been investigating a joint infrastructure investment in the
construction of the New River Reservoir. A listing of potential infrastructure
improvements can be found in Table 4.6. The three utilities are seeking to find a
cooperative strategy that links infrastructure pathways planning with drought
119
Table 4.6: Potential new infrastructure options in the Sedento Valley.
Infrastructure Utility (allocation %) Cost Storage/ Permitting
($MM) produc- Period
tion (years)
College Rock Reservoir Watertown 50 500 5
expansion (Low)
College Rock Reservoir Watertown 100 1000 5
expansion (high)
Watertown Reuse Watertown 50 35 MGD 5
Granite Quarry Dryville 22.6 200 MG 17
Sugar Creek Reservoir Dryville 150 2909 MG 17
Dryville Reuse Dryville 30 35 MGD 5
New River Reservoir Fallsland (50%), Wa- 263 3700 MG 17
tertown (50%)
Fallsland Reuse Fallsland 50 35 MGD 5
mitigation policy. As part of this policy, the utilities are interested in incorpo-
rating newly available financial tools to hedge against unexpected costs from
drought mitigation.
4.5.2 Problem Formulation
The three utilities seek to solve the many objective optimization problem shown
in Equations 4.16 through 4.21:
120
θ∗ = argminθ F (4.16)
s.t.
|ME| ≤ 1 ∀ME ⊆ BI (4.17)
Where:
 
 −fREL (xs, θrt, θtt, θlma, θit, ICO, Ψs)   fRF (xsrof, θrt, θtt, θlma, θit, ICO, Ψ  s
) 
  fNPV (xlrof, ICO, Ψs)F =  

(4.18)
 fFC (xsrof, θrt, θlma, θarfc, θirt, θit, ICO, Ψs, xlrof)  fWFPC (xsrof, θrt, θtt, θlma, θarfc, θirt, θit, ICO, Ψ s, xlrof)
flma (θlma)
θ = [θrt, θtt, θarfc, θirt, θit, ICO, θlma] (4.19)
 xsrof

X = x lrof (4.20)
xs
121
   
ψinflows, 1  
    
ψevaporation rates, 1   ψdemands, 1   ψDU,1  

ψ   ψ  inflows, 2   evaporation rates, 2Ψ =  .  , . 

, ψdemands, 2   ψDU,2  

. ,s


..   ..   ..   ...  (4.21) ψinflows, N ψR evaporation rates, N ψR demands, N ψR DU,NR 
(Optional)
Where F is a vector based objective function containing regional objectives
fRel, reliability, fRF , restriction frequency, fNPV , net present value of infrastruc-
ture investment, fFC , financial cost of drought mitigation and debt payment,
fWFPC , worst-first-percentile cost of the fFC and fLMA, Lake Michael alloca-
tion (detailed descriptions of the objectives are provided in Section 4.4.9 of the
supplement). The management and investment policies are represented in θ, a
vector containing all of the decision variables for the three utilities. Decision
variables controlling supply stability are θrt, a vector of restriction triggers, and
θtt, a vector of transfer triggers. Decision variable regulating financial stability
are θarfc, a vector of annual reserve fund contributions, and θirt, a vector of insur-
ance restriction triggers. Lastly, decision variables θit, a vector of long-term-ROF
infrastructure construction triggers, ICO a matrix containing infrastructure con-
struction ordering for each utility and θlma, a vector of Lake Michael allocations
for the three utilities control infrastructure construction and allocations.
The bounds of the decision variables of the Sedento Valley problem are
shown in Tables 4.7 through 4.9. The reader may find strange that values of
1.0 were accepted as the the ROF trigger values for restrictions and transfers for
all utilities. The logic behind this choice is that more than providing utilities
with precise values of ROF triggers to be implemented by the utilities, the goal
of a study such as this one is to provide overarching strategies, in which a value
122
close to 1.0 for restriction trigger would provide a clear signal that other alter-
natives should be explored besides restrictions so that utilities do not over-rely
on it. Lastly, the -dominance precisions for the objectives of the Sedento Valley
problem are shown in Table 4.10. The next subsection will describe how Wa-
terPaths was designed not only for standalone simulations but also to facilitate
decision making for regionally-coordinated water utilities.
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Table 4.7: Decision variables pertaining to the short-term drought mitigation instru-
ments: water-use restrictions, transfers, contingency fund and drought insur-
ance.
Decision Variable Lower Bound Upper Bound
Dryville restriction ROF trigger 0% 100%
Fallsland restriction ROF trigger 0% 100%
Watertown restriction ROF trigger 0% 100%
Dryville transfer ROF trigger 0% 100%
Fallsland transfer ROF trigger 0% 100%
Fallsland Lake Michael allocation 5% 33.4%
Dryville Lake Michael allocation 5% 33.4%
Watertown Lake Michael allocation 33.4% 90%
Dryville annual contingency fund contribution as 0% 10%
percentage of annual revenue
Fallsland annual contingency fund contribution as 0% 10%
percentage of annual revenue
Watertown annual contingency fund contribution 0% 10%
as percentage of annual revenue
Dryville insurance ROF trigger 0% 100%
Fallsland insurance ROF trigger 0% 100%
Watertown insurance ROF trigger 0% 100%
Dryville insurance payment as percentage of rev- 0% 2%
enue
Fallsland insurance payment as percentage of rev- 0% 2%
enue
Watertown insurance payment as percentage of rev- 0% 2%
enue
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Table 4.8: Values of the long-term ROF trigger and daily demands that function as thresh-
olds to trigger infrastructure construction by the utilities.
Decision Variable Lower Bound Upper Bound
Dryville infrastructure construction long-term ROF 0% 100%
trigger
Fallsland infrastructure construction long-term 0% 100%
ROF trigger
Watertown infrastructure construction long-term 0% 100%
ROF trigger
Table 4.9: The ordinal variables below determine the infrastructure construction order
and adjusts them for the total number of options available to each utility. The
volumetric variable represents the volume of available storage within Falls
Lake to be re-allocated from the water quality to Raleigh’s municipal supply
pool.
Decision Variable Lower Upper
Bound Bound
New River Reservoir ranking 1st 8th
Sugar Creek Reservoir ranking 1st 8th
College Rock Expansion Low ranking 1st 8th
College Rock Expansion High ranking 1st 8th
Watertown Reuse I ranking 1st 8th
Watertown Reuse II ranking 1st 8th
Dryville Reuse 1st 8th
Fallsland Reuse 1st 8th
Table 4.10: Values used for -dominance.
Objective Reliability Restriction Infrastructure Financial Worse First
Frequency Net Present Cost Percentile
Cost Cost
Value 0.1% 2% $10MM 2.5% of 1% of AR*
AR*
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4.5.3 Discovering Robust Pathways for the Sedento Valley
We will illustrate the core functionality of WaterPaths through an exercise for
the design and analysis of a water-infrastructure planning and management
policy. Such exercise consists in an application of the DU Pathways framework
for the Sedento Valley example. The DU Pathways framework, as illustrated in
Figure 4.5.3, has three core steps besides the problem formulation, all of which
forming a loop repeated until a satisfactory solution has been found: (1) identify
tradeoffs, (2) evaluating robustness, and (3) infrastructure pathway analytics
(more details will be given on Chapter 5). Our application of these steps for the
Sedento Valley test is described next.
4.5.4 Identifying Tradeoffs
The Sedento Valley water portfolio management and infrastructure investment
pathway demonstration is a challenging high-dimensional (in terms of deci-
sions and objectives) stochastic multiobjective problem. Our demonstration of
WaterPaths capabilities and use for discoverying the Sedento Valley test case’s
tradeoffs exploits the MS Borg MOEA (Hadka et al., 2013). The MS Borg MOEA
combines adaptive operator selection (Vrugt and Robinson, 2007) based on
probabilities calculated from solutions in its -dominance archive (Laumanns
et al., 2002) described in Equation 2.3 and Figure 2.3. This combination makes
it suitable to solve problems with a wide range of mathematical characteristics.
The Borg MOEA has demonstrated superior performance over a diverse set of
multiobjective problems, such as benchmarking test problems, water supply
portfolio planning, pollution control given ecological thresholds, groundwater
126
Figure 4.6: Policy design and analysis performed with WaterPaths within
the DU Pathways framework.
monitoring design, and reservoir control (Hadka and Reed, 2012b; Reed et al.,
2013; Zatarain Salazar et al., 2016; Ward et al., 2015). Furthermore, the MS Borg
MOEA has demonstrated that it is capable of solving infrastructure pathway
problems Zeff et al. (2016) without the need of parameter tuning, which makes
it the ideal choice for engineers not familiar with the technical aspects of evolu-
tionary optimization. The standard values of parameters of the Borg MOEA
v1.8 Master-Worker were used in this work (see (Hadka et al., 2013) for the
127
specific values). Although WaterPaths can be readily used with any available
modern MOEA or other search tools, the MS-Borg MOEA parallel design facil-
itates our demonstration of the framework’s ability to support state-of-the-art
massively parallel analytical capabilities involving input and output of poten-
tially high amounts of data. Figure 4.7 shows the optimization loop and the
single-simulation input/output flow.
Figure 4.7: Optimization loop and single-simulation input/output infor-
mation flow. WaterPaths can be used to simulate a single pol-
icy, allowing its use for policy re-evaluation and robustness cal-
culation, as demonstrated in Section 4.5.5, and for policy opti-
mization adn tradeoff identification.
For this work, we performed all optimization runs on Texas Advanced Com-
puting Center’s (TACC) Stampede 2. The runs were performed on the SKX com-
puting nodes, comprised of two Intel Xeon Platinum 8160 (“Skylake”), with 48
128
cores @ 2.1 GHz and 192GHz of RAM. We found that nine different optimiza-
tion seeds ran with MS-Borg MOEA, each with 125,000 function evaluations and
a unique number of nodes, was sufficient to converge to the best attainable ap-
proximate Pareto set based on the evolution of the hypervolume metric, seen in
Figure 4.5.4.
Figure 4.8: Evolution of the hypervolume for each of the nine sets. The
overlap across the lines shows that the number of seeds was
sufficient to assure consistency, while the gradual plateauing of
the hypervolumes of all seeds shows that the number of func-
tion evaluations were enough to achieve maximum attainable
convergence.
Each function evaluation was performed using the Monte Carlo functional-
ity of WaterPaths by sampling 1,000 realizations. Each realizations fully defines
one full draw of candidate values for all of the deeply uncertain factors of con-
cern presented in Table 4.11. Each single draw of these DU values are then cou-
pled to one realization of a 45-year synthetically generated record for stream-
flows, evaporation rates, and demands. Overall we augment the limits of the
129
Table 4.11: Deep uncertainties considered for the Sedento Valley test prob-
lem. Unless specified otherwise the same minimum and max-
imum values for each uncertainty were applied for all utilities
and infrastructure.
Category Factor name Min [-] Max [-]
Sinusoid amplitude 0.8 1.2
Synthetic series trend Sinusoid frequency 0.2 0.5
Sinusoid phase −π/2 π/2
Demand growth multiplier 0.5 2.0
Bond interest rate multiplier 1.0 1.2
Utilities
Bond term multiplier 0.6 1.0
Discount rate multiplier 0.6 1.4
Drought mitigation Watertown 0.9 1.1
instruments (restriction Dryville 0.9 1.1
effectiveness multiplier) Fallsland 0.9 1.1
Permitting time multiplier 0.75 1.5
New infrastructure
Construction time multiplier 1.0 1.2
80-year historical record for the Sedento Valley system by creating 1,000 cross-
correlated synthetic streamflows and evaporation-rate time series (realizations)
for each reservoir and stream gauge. These natural times series are also used
to inform the generation of 1,000 synthetic demand time-series for each utility,
which account for correlations with changing levels of water scarcity for each
utility. The simulation of all 1,000 fully specified realizations and subsequent
calculation of the specified objectives constitutes one function evaluation. Wa-
terPaths’ software design permits significant flexibility for stochastic simulation
as well as stochastic simulation-optimization.
130
The resulting output Pareto approximation sets from each of the MS Borg
MOEA random seed trials on TACC Stampede 2 were combined and sorted
using -dominance sorting to build a best-known reference set for the Sedento
Valley demonstration problem. The use of -dominance provides a convenient
means of preserving high-quality representations of key tradeoffs sets while
reducing solution sets to a user preferred size to improve interpretability of
tradeoff assessments and/or limit the computational demands of the robust-
ness assessments described below (readers interested in more details can refer-
ence Reed et al. 2013; Kollat and Reed 2007). For the Sedento Valley example
presented here, we filtered the reference set of solutions using two steps: (1)
-dominance sorting of all solutions based with our specified  values and (2)
filtering solutions based onperformance goals for specific objectives.
4.5.5 Evaluating Robustness
The next step was to further stress test solutions in the final reference set, gen-
erated by -dominance sorting all solutions based on specified  values, by re-
evaluating them against a larger independent sampling of 5,000 vectors of the
DU factors presented in Table 4.11. For each of the 5,000 re-evaluation runs, we
created one set of 1,000 streamflows, evaporation rate, and demand time-series
based on the first three deeply uncertain factor in Table 4.11 time series for each
reservoir and gauge, resulting in 5,000,000 SOWs. The processes for the gener-
ation of inflows, evaporation rate, and demand time series is presented in the
supplemental Section B.
Before proceeding with such computationally expensive re-evaluation and
131
robustness evaluation, the initial reference set resulting in 830 non-dominated
solutions was reduced in size to 229 solutions before being re-evaluated over
5,000 re-evaluation scenarios for computational tractability, as mentioned in
Section 4.5.4. This was achieved by keeping in the reference set only solutions
whose objective values as calculated during optimization displayed reliability
greater than 98%, restriction frequency on less than 30% of the years, and worse
first percentile cost smaller than %10 of the annual revenue. These criteria were
chosen for (1) being similar to minimum performance standards deemed ac-
ceptable based on the authors’ prior experience with real utilities, and (2) for
reducing the number of solutions limit computational demands.
The resulting 229 · 5, 000 = 1, 145, 000 simulations required for re-evaluation
were split into 25 independent jobs on Stampede 2. Each job consisted in one
Python script using MPI4Py (distributed memory parallelization) to use dis-
tribute blocks of (229 solutions) · (200 RDM scenarios) = (22,900 function eval-
uations) across 50 nodes, resulting in (229 solutions) · (4 RDM scenarios) = 916
function evaluations per node per job for each of the 25 jobs. The reason for
the split was to improve the fault tolerance of the re-evaluation exercise by pre-
venting a crashing function evaluation from crashing the entire re-evaluation
exercise, as a crash in one function evaluation would crash the Python script
and in turn all its function evaluations. Each of the 50 nodes ran one function
evaluation at a time distributing its 1,000 realizations across all 48 cores with 2
realizations running simultaneously on each core.
The objective values for each of the 5,000 re-evaluation function evaluations
for each of the policies in the abbreviated combined reference set were used to
re-calculate the policies’ objectives, this time based on 1, 000 · 5, 000 = 5, 000, 000
132
realizations rather than the initial 1,000 using during the MOEA search phase.
Additionally, the 5,000 sets of objective values obtained for each policy were
used to calculate the value of the satisficing robustness metric (Starr, 1962; Her-
man et al., 2014) for each utility under each policy, defined as the percentage of
the 5,000 sets that met the performance criteria defined by the utilities.In short,
the satisficing measure of robustness measures the percent of sample worlds
where decision makers deem performance acceptable based specified perfor-
mance requirements.
4.5.6 Pathway Analytics
Based on the robustness values, a solution was selected for a detail scenario dis-
covery analysis (Bryant and Lempert 2010 and Section 3.3.5) performed with
the Boosted Trees algorithm (Schapire, 1999), as further discussed in Section
5.4.3 and Appendix E. Scenario discovery is an effort to determine the uncer-
tainties (factor prioritization) and/or combinations of their values (factor map-
ping) that most closely relate to performance and robustness. This informa-
tion is presented in terms of a map of the space of uncertainty showing which
combinations of values for uncertain factors would likely cause a policy to fail.
Included in the axes of the maps is the uncertain factors that are most influen-
tial in that policy’s performance, with their relative performances displayed in
percentage metric analog to explained performance variance (see Section 5.4.3
and Appendix E for details). The Boosted Trees was used for scenario discov-
ery for the Sedento Valley test case because of two advantages it holds over the
other more easily interpretable but limited scenario discovery methods such as
PRIM (see Section 3.3.5, Figure 3.8, and Bryant and Lempert 2010; Friedman
133
and Fisher 1999), CART (Bryant and Lempert, 2010; Breiman et al., 1984), and
logistic regression (Quinn et al., 2018; Gold et al., 2019): (1) Boosted Trees cap-
tures non-differentiable boundaries typical from threshold-based rules such as
WaterPaths’ ROF-based action triggers, (2) it captures non-linear boundaries
without explicitly modeling variable interactions while being resistant to over-
fitting, helping assure scenario discovery maps that are as simple as possible to
interpret.
To obtain the maps and factor priorities, a Python script provided with Wa-
terPaths’ source code reads all samples 5,000 of deeply uncertain factors and
corresponding objectives, and for each solution fits a Boosted Trees classifier to
the uncertain factors and objective values for each utility for each policy. The
surface is then presented in maps indicating the regions of the space of uncer-
tainties in which that policy is likely to fail for each utility, as well as the sources
of uncertainty that most impact the performance of each utility under that pol-
icy. The script for fitting the Boosted Tree classifier and the plotting the maps is
based on the Scikit-Learn machine-learning library (Pedregosa et al., 2011) and
provided in the repository listed in Section 4.2.
Lastly, all infrastructure-pathway data output for each of the 5,000 function
evaluations performed for the policy being analyzed is used to create plots rep-
resenting infrastructure construction as a function of time. The data is com-
prised of the realization ID, the utility that triggered a project, the infrastructure
option that was built, and the week in which the option became operational.
Three pathway plots are created for each utility under the policy being ana-
lyzed: one presenting infrastructure construction over time for all realizations
under the least favorable sample of deeply uncertain factors, one for the pro-
134
jected values for each deeply uncertain factor, and one for the most favorable.
These plots can be used to understand which projects are likely to be suitable
for construction in the near-term and how dependent on uncertainty the need
for construction is. Next we provide details about the parallel architectures and
configurations we exploited to provide an assessment of WaterPaths parallel
performance.
4.5.7 Scaling Performance on Clusters and in the Cloud
In this comparative assessment, replicate optimization random seed trial runs
were run on small and large traditional high-performance computing clusters as
well as a virtual cloud cluster. In the scaling analysis, all WaterPaths model runs
(or function evaluations) were distributed across computing nodes on Stam-
pede 2 with Intel MPI for Linux v2019u5 (distributed memory parallelization
library) (Intel, 2019) and on the Cornell Red Cloud and on the research clus-
ter using OpenMPI v3.1.4 (Gabriel et al., 2004), while the sampled realizations
that composed the function evaluations were distributed across cores within a
node using OpenMP v5.0 (shared memory parallelization library) (Stallman and
Community, 2017; Klemm et al., 2019). This hybrid parallelization scheme as
discussed in Section 4.4.11 and in Figure 4.4.10b has the advantage of allowing
the user to scale optimization runs up to as many computing nodes as are avail-
able while avoiding core idle time due to memory access and size limitations
within a node. The within-node shared memory parallelization of realizations
within a function evaluation also has the advantage of allowing the user to run
more realizations simultaneously. This feature is particularly useful as the num-
ber of cores per processor is increasing and as cloud computing with nodes of
135
various sizes are becoming more readily available.
In addition to optimization runs performed on Stampede 2 described above
in Section 4.5.4, performance was assessed on a research cluster termed The
Cube, whose nodes are comprised of two Intel Xeon E5-2680 (Sandy Bridge), @
2.7 GHz and 128 GB of RAM (see Table 4.12). The cloud results were attained
on Cornell’s Red Cloud with nodes with 28 cores and variable architecture. The
scaling analysis consists of two types of tests to evaluate search performance on
each platform: search scalability and cross-platform wall clock search time com-
parisons. Our goal for the multi-architecture comparison tests is to benchmark
how performance varies for typical small clusters, elite class high performance
computing systems, and emerging cloud platforms. Our scalability benchmark-
ing focuses on efficiency as a function of the number of nodes/cores as defined
in equation 4.22 derived from Amdahl’s law assuming negligible, necessarily-
serial work. Our core demonstration results are based on nine random seed
search trials on Stampede 2, where the seed trial count allowed our experiment
to exploit the maximum allowed by the system administrators. The Cube and
Red Cloud analyses are based on three random seed trial runs for the sole pur-
pose of informing this computational scalability analysis. Each seed had one
MS Borg MOEA master process and a number of worker processes based on the
number of nodes and of processes per node.
N0 · T arch
E ≈ N0 (4.22)
N · T archN−nodes
In equation 4.22 E is parallel efficiency, T archN is the total execution time on0
the base case (here, eight nodes) for a given architecture and N is the number
of nodes to be compared against the base case. Equation 4.22 assumes the vast
136
majority of the execution time is spent on tasks run in parallel. Table 4.12 shows
which computer configurations were used for each test. The configurations of
each seed can be found in Table 4.12 together with details about the architecture
of the compute nodes used. Table 4.12 also shows that the exercise of optimizing
a model built with WaterPaths using MS-Borg MOEA is nearly perfectly scalable
when running function evaluations in parallel for optimization on either low or
high numbers of nodes in high performance computing applications. These
results highlight that when using WaterPaths that investments in high perfor-
mance computing power will yield returns linearly proportional to the number
of nodes, which corresponds to ideal scalability.
137
Table 4.12: Architectures and configurations used for optimization scalability and cross-
platform performance tests, followed by total run times and efficiencies for
each optimization seed, ran on Stampede 2 and on the cluster The Cube with
different parallel configurations.
Platform Node Type # of # of Total Processes Times Efficiency
Nodes Cores/ # of per [min] [-]
Node Cores Node
8 384 1720 —
16 768 882 98%
32 1536 425 100%
2 x Intel Xeon Platinum 40 1920 337 100%
Stampede 2 8160 (SkyLake), 2.1 48 48 2304 4 289 99%
GHz, 192 GB of RAM 56 2688 245 100%
64 3072 219 98%
96 4608 143 100%
128 6144 109 99%
Aristotle Cloud Haswell (variable), 236 8 28 224 2 3665 —
GB of RAM
The Cube 2 x Intel Xeon E5-2680 8 128 8274 —
(Cornell Research (Sandy Bridge), 2.7 16 16 256 3 4177 100%
Custer) GHz, 128 GB of RAM 32 512 2020 98%
In addition to the results above, we also analyzed performance losses from
distributing realizations of a single function evaluation across multiple cores, as
opposed to the previously described scalability experiment which distributed
multiple function evaluations across nodes for optimization. These tests con-
sidered 1, 2, and 4 cores within a single Intel Xeon CPU E5-2680 0 @ 2.70GHz
standard desktop workstation and across 2, 4, 8, 16, 32, and 48 cores of a single
Stampede 2 Skylake node. This scalability study is aimed at users running sin-
gle Monte Carlo simulations or ensembles of Monte Carlo simulations in batch
mode, as opposed to MOEA search runs. Table 4.13 shows the results of timing
138
WaterPaths running 1,000 realizations on different number of cores on a four-
core Xeon desktop and on Stampede 2.
Table 4.13: Timings and efficiency of running WaterPaths with 1,000 re-
alizations split among different numbers of cores on a 4-core
Xeon Desktop and on a 48-core SkyLake node of Stampede 2.
— Xeon Desktop Stampede 2 Skylake
Number of Cores Time [s] Efficiency Time [s] Efficiency
1 176 — — —
2 88 100% 128 —
4 44 100% 103 62%
8 — — 63 51%
16 — — 37 43%
32 — — 20 40%
48 — — 15 36%
Table 4.13 shows that WaterPaths also scales its realizations across cores on
the Xeon desktop, which has a processor with few a few strong cores, at 100%
efficiency. The same, however, does not happen on Stampede 2 Skylake nodes,
which is a processor with several weaker cores, that become progressively more
impacted with memory access issues with core count (i.e., less active mem-
ory per core for computations). Despite the losses in efficiency, a user can still
achieve a 10-fold runtime reduction using all available cores, which is a much
desirable feature to have at production scale.
139
4.6 Results
4.6.1 Performance and Robustness Tradeoffs
Figure 4.9 shows the objectives (panel “a”) and robustness (panel “b”) tradeoffs
among Sedento Valley utilities following re-evaluation as output by WaterPaths.
In Figure 4.9a, each axis represents an objective and each line represents a pol-
icy. The location where each line (policy) intersects a vertical axes represents it
performance value in the corresponding objective. The highlighted policies rep-
resent the policies with best robustness for each utility (BRo-X), with the best ro-
bustness compromise across utilities (RoC), and best performance compromise
(PC). The axes in Figure 4.9a are oriented such that the ideal policy would be a
horizontal line at the bottom of all axes.
Figure 4.9a highlights moderately strong tensions between reliability and re-
striction frequency, which is expected given restrictions are a supply-reliability
instrument. Likewise, significant tradeoffs exist between Infrastructure NPV
and Financial Cost for higher reliabilities, indicating that the construction of
infrastructure successfully offsets the need (and therefore the cost) of costly
drought mitigation and financial instruments. Figure 4.9a also shows that the
low-reliability policies tend to generally have between 5% and 15% of restriction
frequency and that policies with low restriction use rank high in infrastructure
NPV, further suggesting that building the suggested infrastructure successfully
offsets the use of drought mitigation instruments. However, despite the general
trends, a small group of policies with the highest reliability still make use of
moderate restrictions and investments in new infrastructure. Such conclusions
would be difficult to reach with existing frameworks and were made possible
140
Figure 4.9: Performance and robustness tradeoffs in the DU Re-Evaluation
space. Panel (a) illustrates the tradeoffs across the performance
objectives quantified either using the mean or the worse first
percentile across the tested SOWs, in which an ideal solution
would be represented by a horizontal line at the bottom of
the plot. Panel (b) illustrates the robustness tradeoffs across
the three utilities. Here, robustness is calculated as the per-
cent of RDM SOWs in which a policy met the performance cri-
teria defined by the utilities and an ideal solution would be
represented by a horizontal line at the top of panel (b). The
highlighted policies represent the policies with best robustness
for each utility (BRO-X), with the best robustness compromise
across utilities (RoC), and best performance compromise (PC).
141
by WaterPaths’ unique combination of supply and financial models, policies im-
plemented such that their ideal operation is discovered instead of prespacified,
and possibility of infrastructure construction during the course of a simulation.
Transitioning to robustness tradeoffs, Figure 4.9b uses a similar parallel axes
plot to display the percentages of SOWs where each utility meets their goal
performance requirements: reliability > 98%, restriction frequency < 10% and
annual worst-first-percentile cost < 10%. In Figure 4.9b, the direction of prefer-
ence is upward, so that the ideal policy would be represented by a horizontal
line at the top of the axes (i.e., 100% for all utilities). The most robust policies for
each utility and the best robustness and objectives-performance compromises
across all utilities are again highlighted in different colors in Figure 4.9b. The
highlighted best-robustness policies in Figure 4.9b suggest inter-utility robust-
ness tradeoffs for the highest levels of attained robustness, which indicate com-
plex interdependencies between the utilities caused by utilities attempting to
simultaneously use constrained regional resources. Furthermore, performance
compromise solution shows that focusing on performance levels may degrade
robustness for all utilities, which would likely go against the utilities’ risk aver-
sion. The capability of simulating resources shared by utilities to allow for re-
gional conflict analysis is increasingly important given how the distance be-
tween areas serviced by water utilities is decreasing, and is a distinct capability
of WaterPaths.
Figure 4.10 supplements Figure 4.9b by comparing utilities’ robustness pro-
files across all of the policies found through optimization. One important result
highlighted in Figures 4.10 and 4.9b is the strong source of potential regional
tension depending on if the utilities seek to focus on overall performance trade-
142
Figure 4.10: Robustness of pre-selected 229 policies for each utility. Each
bar represents the attained robustness for one policy for each
utility. The policies are sorted by robustness (bar height) for
each utility. The policies with the highest value of the satisfic-
ing robustness for each utility, the policy with the best robust-
ness compromise, and the policy with the best compromise of
objectives values are highlighted in each subplot.
offs versus individual robustness tradeoffs. For Dryville and Fallsland, the il-
lustrated distances between the blue (robustness compromise) and the fuchsia
(performance compromise) solutions in both figures implies that seeking a re-
gional management and investment policy compromise for performance trade-
offs could make Dryville and Fallsland more vulnerable individually. Figure
4.10 highlights that there are very few solutions that are simultaneously close
to the highest attainable robustness for each individual utility, as illustrated by
the steep downward slope in bar heights on the leftmost region of all three bar
charts, which is concern that should be considered explicitly in any regional
143
water portfolio management and infrastructure investment policy negotiation.
WaterPaths provides detailed analyses that can be helpful for these types of re-
gional decision-making contexts. The simulation framework allows stakehold-
ers to explore how the component policy decision variables themselves are driv-
ing these tradeoffs.
144
4.6.2 Policy Rules in the Robustness Compromises
Figure 4.11: Parallel axes plots for the policy variables for the policies rep-
resenting the best robustness for each utility, the best robust-
ness compromise, and the best performance compromise. The
table indicates the infrastructure construction order for each
utility for each policy in the order shown in the legend.
145
Figure 4.11 detailed illustration of the rules that compose the policies high-
lighted in Figures 4.9 and 4.10. Recall that each policy is comprised of: (1) ROF
triggers for short-term mitigation instruments [Figures 4.11a, 4.11b, and 4.11c]
and for long-term infrastructure investments [Figure 4.11e], (2) allocations to the
water supply pool in Lake Michael allocation [Figure 4.11f], (3) annual contin-
gency fund contributions and insurance payouts defined as percentages of an-
nual revenue [Figures 4.11d and 4.11g], and (4) infrastructure construction order
[table within Figure 4.11]. The first striking feature of Figure 4.11 is the similar-
ities between all highlighted solutions regarding the use of supply-reliability
tools, namely restrictions, transfers, infrastructure construction, Lake Michael
Allocation, and infrastructure construction order, the latter for Dryville and Fall-
sland. In all policies, all utilities make extensive use of restrictions due to its low
trigger value, build considerable infrastructure, and almost the entirety of the
Lake Michael is allocated to Watertown. In addition, Dryville also has high re-
liance on transfers from Watertown. In comparing Figure 4.11b to Figure 4.10, it
is possible to notice a possible relation between transfer obligations for Water-
town and its robustness, as the most robust policy for Watertown implies less
treated water transfers and robustness for the other two utilities. The variabil-
ity in all utilities’ financial strategies across solutions, however, requires better
scrutiny. All policies except for Watertown’s most robust have values between
2.5% and 5% for the contingency fund variable, suggesting this is an instrument
to be explored by all utilities. In addition, Dryville insurance strategy seems to
gravitate around infrequent use of insurance with low payments, which effec-
tively means insurance is not of use for Dryville. Watertown’s and Fallsland’s
financial strategies varied across policies with no discernible pattern.
Figure 4.11 provides a detailed illustration of the rules that compose the wa-
146
ter portfolio management and infrastructure investment policies highlighted in
Figures 4.9 and 4.10. Recall that each policy is comprised of: (1) ROF triggers for
short-term mitigation instruments [Figures 4.11a, 4.11b, and 4.11c] and for long-
term infrastructure investments [Figure 4.11e], (2) Lake Michael allocations [Fig-
ure 4.11f], (3) annual contingency fund contributions and insurance payouts de-
fined as percentages of annual revenue [Figures 4.11d and 4.11g], and (4) infras-
tructure construction order [table within Figure 4.11]. The first striking feature
of Figure 4.11 are the similarities between all of the highlighted solutions regard-
ing the use of supply focused instruments that are strongly influential for reli-
ability, namely restrictions, transfers, infrastructure construction, Lake Michael
Allocation, and infrastructure construction order, the latter for Dryville and Fall-
sland. In all policies, all utilities make extensive use of restrictions due to its low
trigger value, build considerable infrastructure, and almost the entirety of the
Lake Michael is allocated to Watertown. In addition, Dryville also has high re-
liance on transfers from Watertown. Comparison of Figure 4.11b to Figure 4.10
highlights a possible relation between transfer obligations for Watertown and its
robustness, as the most robust policy for Watertown implies less treated water
transfers [higher triggers in Figure 4.11] and robustness for the other two utili-
ties [smaller robustness for Dryville and Fallsland in Figure 4.10]. Transitioning
to the utilities use of financial instrument, there are significant differences in
their financial decisions across the highlighted solutions. Figure 4.10 shows that
all of the highlighted policies except for Watertown’s most robust have values
between 2.5% and 5% for the contingency fund variable, suggesting this is an
instrument to be explored by all utilities. In addition, Figure 4.11 shows that
Dryville’s insurance strategy tends to gravitate towards either unusually infre-
quent use of insurance and/or low payments, both effectively meaning insur-
147
ance is less useful for Dryville. Watertown’s and Fallsland’s financial strategies,
on the other hand, varied across policies depending their focus on performance,
robustness, and alternative compromises.
A key feature of WaterPaths is providing a self-consistent means of simu-
lating and evaluating how short-term management actions shape sequences of
infrastructure investments. In Figure 4.11 the ROF-based management and in-
vestment action triggers interact with the ordering or prioritization of infras-
tructure sequences for each of the utilities. As Figure 4.11 shows, the infrastruc-
ture construction order is stable for Dryville and Fallsland across solutions but
varies for Watertown. In all policies, Watertown builds early on the New River
reservoir, which is shared with Fallsland and for whom it is also the first option
to be built, indicating the importance of this joint project. Also, all utilities in all
solutions prioritize the construction of added storage as opposed to water reuse
stations except for Watertown in its most robust policy, which indicates high
storage’s higher efficacy in improving supply and financial performance. The
inversion in priority between expanding the College Rock Reservoir and build-
ing reuse stations for Watertown in Watertown’s most robust policy may also
been connected to its high robustness in this policy and to the decrease of Falls-
land robustness, given the College Rock Reservoir helps stabilize the flows into
the New River Reservoir. The similarities in the use of reliability-focused in-
struments across utilities also suggests that their differences in robustness may
be substantially dependent on the utilities’ differences in their use of their fi-
nancial instruments. This leaves room for fine-tuning of policies using Water-
Paths, given that financial instruments are part of individual (as opposed to
regional) planning and that the during optimization the financial instruments
were optimized only for the worse performing utility for each objective. With
148
an understanding of the decisions being made by the utilities under each policy,
the next step is to understand which uncertainty factors and their values drive
robustness.
4.6.3 Scenario Discovery for Compromise Policies
WaterPaths as a flexible Monte Carlo simulation framework enables careful ex-
ploratory modeling to better understand what deeply uncertain factors most
strongly influence robustness. Each map in Figure 4.12 shows the performance
attained across scenarios for the two most important uncertainties for each of
the three utilities when the Sedento Valley implements the policy with the best
robustness compromise, highlighted in Figures 4.9 and 4.10. In each panel of
Figure 4.12, the percentages shown in the axes denote the importance of each
source of uncertainty in determining whether the mentioned policy would or
would not meet the utilities performance criteria in the samples scenarios, as
identified by the boosted trees algorithm as mentioned in Section 4.5.3. More
specifically, the percentages represent the decrease in impurity of the tree en-
semble from splits on that factor, with a higher percentage indicating a higher
importance of that factor. The red dots denote scenarios in which the utility
fails to meet its performance targets and gray dots represent scenarios a utility
performs as or better than required. Red shading represents regions of the un-
certainty space where a utility under the mentioned policy will likely not meet
the performance criteria with P(meeting criteria|scenario) ≤ 0.5, white regions
denote inconclusive regions in which 0.98 ≥ P(meeting criteria|scenario) ≥ 0.5
(inconclusive regions), and gray regions represent scenario regions in which the
utility will likely meet the performance criteria with P(meeting criteria|scenario)
149
≥ 0.98. All probabilities were calculated using Boosted Trees with 500 trees of
depth 4. In each panel of Figure 4.12, the stars represent the most favorable
scenario (Gray star), the least favorable (red), and the future according to pro-
jections (blue star).
Figure 4.12: Factor maps for the robustness compromise policy (RoC)
showing the key uncertainty factors controlling the robust-
ness the policy for all utilities. Success (grey dots and zone)
required reliability ≥ 98%, restriction frequency ≤ 10% and
annual financial worst 1st percentile cost ≤ 10%, while a fail-
ure to meet either criteria results in the scenario being con-
sidered a failure (red dots and region). White zones denote
regions with a mixture of gray and red dots for which it is
uncertain whether all criteria would be met.
150
Figure 4.12 shows that all of the utilities’ robustness are mostly contingent
on demand growth, but that contrary to expectations both low and high val-
ues of demand growth may cause the three utilities to fail. However, low- and
high-demand failures happen for very different reasons. Failures due to low de-
mand result for all three utilities when low long-term ROF values do not trigger
infrastructure construction leading to over-dependence on financially challeng-
ing restrictions (i.e., high restriction frequencies and volatile cost swings). The
low-demand failures result when restrictions and short-term transfers are not
sufficient for preventing supply failures, causing low reliabilities and high re-
striction frequencies. In low-demand scenarios, all of the utilities struggle with
financial stability and fail to meet the robustness criterion for the worse first
percentile cost objective, which is worse first percentile cost smaller ≤ 10%.
Overall, Figure 4.12 provides some important regional insights for the
Sedento Valley. The dependence on failures in all three objectives on demand
growth rates indicate an intimate relationship between supply reliability and
financial stability. Overall, keeping demand growth rates at 20% above the util-
ities’ projected value would be an important step for all of the utilities to en-
sure satisfactory regional performance while avoiding stranded assets, which
requires high demands because of their high costs. The next section will ex-
plore how the scenarios highlighted above by the stars drive infrastructure con-
struction, as a way of understanding the relationship between infrastructure
construction and robustness, performance, uncertainty factors.
151
Figure 4.13: Capacity expansion pathways and ROF dynamics for the
most favorable scenario. Demand growth rate scenarios con-
sidered include the Most Favorable SOW, the SOW projected
by the utilities, and the Least Favorable SOW. All infrastruc-
ture investments are illustrated as stacked color lines desig-
nating specific infrastructure investments across 1,000 hydro-
logic scenarios.
152
4.6.4 Infrastructure Ensemble Pathway Analysis
Figure 4.13 illustrates the detailed analyses that WaterPaths enables for each
of the utilities, which includes the joint outputs for capacity expansions, short-
term ROF dynamics, and pathways for each of the utilities under their indi-
vidual most favorable, projected, and least favorable scenarios highlighted by
starts in Figure 4.12. Figures 4.13a - 4.13c plot Watertown’s storage capacity
and short-term ROF dynamics for each of the three scenarios. Likewise, Figures
4.13d - 4.13f show Watertown’s resulting ensemble of infrastructure pathways
across the three highlighted demand growth scenarios. The same panel layout
is repeated for the Dryville and Fallsland pathways. The horizontal axes in the
pathway panels represent time in weeks from 2015 to 2060, the planning hori-
zon. Each infrastructure pathway plot displays a stack of results attained for
1,000 SOWs and the horizontal multicolored lines, each represent the construc-
tion sequence that would be triggered under a specific hydrological scenario
on the vertical axis. For each horizontal line in the stack, the color of a seg-
ment represents the infrastructure option built last in time. That being the case,
a transition in color indicates the time when an infrastructure option becomes
operational. For example, the orange horizontal segment starting at the bottom-
center (significant and early infrastructure, year 26) of the panel “p” indicates
that in that realization, one with more build infrastructure options, Fallsland
finished the construction of the New River Reservoir 9 years after finishing the
construction of its reuse station, brought online in year 17.
Figure 4.13, panels “d” through “f” indicate that the amount of infrastruc-
ture built by Watertown varies significantly with demand growth. While the
scenario with the highest demand growth triggers infrastructure investments
153
for all hydrological realizations and still fails to minimize short-term ROF val-
ues, the projected future triggers infrastructure investments in less than 10% of
the hydrological realizations. However, the low investments in infrastructure
observed in the projected scenario, shown in panel “e,” makes Watertown more
susceptible to swings in revenue in more challenging realizations due to the
need for frequent restrictions. Such financial swings cause Watertown to fail the
worse-first-percentile-cost target, resulting in a red zone around the blue star in
Figure 4.12a. On the other hand, all three scenarios for Dryville and Fallsland
see the same infrastructure-investment patterns with small timing variations,
which combined with the short-term drought mitigation and financial instru-
ments are generally enough to make utilities meet their robustness performance
criteria in the regions highlighted in gray in Figure 4.12a and (b).
An important aspect of the infrastructure investment problem is the effect of
permitting times on the scheduled capacity expansions, which may cause the
construction of an infrastructure option with high permitting time to be post-
poned in favor of another option with already granted permits. In the path-
ways illustrated in Figures 4.13 panels “d” through “f” and “p” through “r” for
Watertown and Fallsland, the occurrence of purple (high-capacity College Rock
expansion) and green lines (Fallsland reuse) segments, respectively, before the
orange regions (New River reservoir) shows that this issue often happen to the
jointly-built New River reservoir, resulting in an observed infrastructure con-
struction order that does not match the order specified in the corresponding
policy (robustness compromise policy) in Figure 4.11. More specifically, in sce-
narios of higher stress, when either Watertown or Fallsland trigger the News
River reservoir its permit may be strongly delayed by the extended permitting
period, so Watertown may trigger instead the College Rock reservoir expansion
154
and Fallsland may trigger the construction of its water reuse station, both with
substantially reduced permitting times.
4.7 Conclusion
Recent projections estimate that the United States (US) will require over one tril-
lion dollars of investment in water supply infrastructure in the next 20 years,
which can potentially be minimized by exploring synergies between infras-
tructure investment and non-structural drought mitigation instruments, both
shared among regional actors. The coupling of long-term infrastructure invest-
ment strategies with short-term drought mitigation necessitates a model with
capabilities not all found in a single currently available software: joint simu-
lation of multiple utilities involved in joint planning and operations of shared
water resources and infrastructure, joint financial-supply simulation, broad ar-
ray of uncertainties within a single simulation, easy customizability, and ready
integration with multiobjective optimization algorithms for use with high per-
formance computing. WaterPaths was designed and shown to fill these gaps.
We present the design of WaterPaths, an open-source model written in the
C++ programming language and designed to fit the DU Pathways. In ad-
dition, we demonstrate the application of WaterPaths on a hypothetical wa-
ter regionally-coordinated infrastructure planning and management problem,
called the the Sedento Valley. Using WaterPaths, we modeled the decision mak-
ing and system upgrades of the regionally-coordinating water utilities in the
Sedento Valley under well-characterized and deep uncertainty to explore policy
options, discovered inter-dependencies between utilities, and discovered vul-
155
nerabilities revealed under deep uncertainty. Additionally, WaterPaths allowed
us to discover policy alternatives for all utilities that land in different location
of the objectives-tradeoff curve by integrating the MS Borg MOEA and running
the optimization exercises in different high-performance-computing platforms.
The Sedento Valley test case was designed to be itself a contribution to the
field of water systems engineering. It was designed to be a minimalist test
case for the direct comparison of competing regional-decision-making frame-
works, for the design and comparison of drought mitigation strategies, and for
the evaluation of decision-making metrics. While minimalist for computational
tractability and interpretability, the Sedento Valley test case maintains a high
degree of complexity due to the number of utilities involved, to the unbalanced
distribution of resources and liabilities across utilities, and to the utilities lo-
cations in relation to each other in the local shared basins. The code reposi-
tory provided in WaterPaths’ code repository contains all the data needed to
reproduce the test case in WaterPaths or other frameworks and to compare new
methodologies and policy instruments against the ones presented this paper.
156
CHAPTER 5
DEEPLY UNCERTAIN PATHWAYS: INTEGRATED MULTI-CITY
REGIONAL WATER SUPPLY INFRASTRUCTURE INVESTMENT AND
PORTFOLIO MANAGEMENT
This chapter is drawn from the following peer-reviewed journal article: Trindade,
B., Reed, P. M., and Characklis, G., “Deeply Uncertain Pathways: Integrated Multi-
City Regional Water Supply Infrastructure Investment and Portfolio Management.”,
Advances in Water Resources, v134, DOI:10.1016/j.advwatres.2019.103442, (2019).
Funding for this work was provided by the National Institute of Food and Agriculture,
U.S. Department of Agriculture (WSC Agreement No. 2014-67003-22076). Additional
support was provided by the U.S. National Science Foundation’s Water, Sustainability,
and Climate Program (Award No. 1360442). The views expressed in this work represent
those of the authors and do not necessarily reflect the views or policies of the NSF or the
USDA.
5.1 Abstract
This chapter contributes the Deep Uncertainty (DU) Pathways framework for
bridging long-term water supply infrastructure investments and improved
short-term water portfolio management (e.g., restrictions, water transfers, fi-
nancial instruments, etc.) to yield robust regional water supply. The DU Path-
ways framework combines flexibility-providing risk-of-failure (ROF) decision
rules presented in Chapter 3.5, dynamic adaptive policy pathways concepts,
and a careful consideration of time-evolving information feedbacks to yield
management-conditioned infrastructure pathways for regions. The DU Path-
ways’ framework has been developed to carefully consider multi-actor regional
157
contexts with the goal of aiding stakeholders in discovering pathway policies
that attain high performance levels across challenging, deeply uncertain futures
and to guide robustness compromises that may be necessary between regional
actors. As demonstrated in the Research Triangle region of North Carolina,
DU Pathways clarifies how to identify robust infrastructure investment and
management policies across the municipalities of Raleigh, Durham, Cary, and
Chapel Hill. Our results provide insights about the most cost-effective infras-
tructure options to be pursued in the near-term, and clarify which sources of
uncertainty drive the performance and robustness of the regional system and of
the individual actors.
5.2 Introduction
Water utilities globally are facing a growing pressure to proactively address
growing demands, higher levels of resource contention, and increasingly uncer-
tain water availability (Bonzanigo et al., 2018; Hall et al., 2014). A tension exists
between improving the efficiency and coordination of regional water supplies
to delay or eliminate major infrastructure investments and the eventual effects
of demand growth rates that necessitate water supply capacity expansions. The
soft water management strategies presented in Chapter 3 (conservation, inter-
utility water transfers, demand management, etc.) are a key option to cope
with such growing domestic water demand (Gleick, 2002a, 2003). In the United
States (US), a key driver increasing the importance of soft water management
strategies is that most large projects of federal and/or state interest have already
been built (Lund, 2013). However, growing urban water demands and increas-
ingly uncertain climate conditions are motivating efforts for redesigning system
158
capacity by upgrading existing and adding new infrastructure to maintain reli-
able water supply systems (Shafer and Fox, 2017).
The hard and soft path views for urban water portfolio planning should
be treated as complementary, as even though infrastructure expansion may be
needed, it is not desirable and may be mitigated by soft water management
strategies. However, both the soft and hard approaches may be detrimental to
utilities’ finances. Together with outstanding debt, decreasing cash flows are
significant factors that negatively impact utilities credit ratings, which can in-
crease the cost of borrowing to build new infrastructure (Moody’s, 2017; Zeff
et al., 2014; Zeff and Characklis, 2013; Trindade et al., 2017). The financial im-
pacts of drought mitigation strategies studied in Chapter 3, the current nega-
tive financial outlook for the water utility sector (Moody’s, 2019), the lack of
appropriate financing for water utilities (Gleick et al., 2014) all stress the need
for new frameworks that facilitate improved water portfolio planning that effec-
tively combines short-term drought mitigation instruments for managing finan-
cial risks and long-term infrastructure investment pathways while accounting
diverse sources of uncertainty. These needs motivated the creation of the Deep
Uncertainty Pathways (DU Pathways) framework proposed and demonstrated
in this study.
Integrating long-term water infrastructure investment and short-term man-
agement is a difficult task because of the potentially large number of decisions
that must be considered over decadal timescales as well as the significant uncer-
tainties inherent to the problem. Hydro-climatic uncertainties such as near-term
streamflows and evaporation rates have been extensively studied and character-
ized with well-established statistical techniques (Stedinger, 1993; Martins and
159
Stedinger, 2000; Herman et al., 2016; Lall, 1995; Kirsch et al., 2013). However,
water infrastructure investment and management problems include a multi-
tude of deep uncertainties — uncertainties for which decision makers cannot
agree on their boundaries, importance, and probability distributions (Kwakkel
et al., 2016b; Knight, 1921), see Section 1.1.2. Examples include regional de-
mand growth, political and economic uncertainties, and climate change (Her-
man et al., 2014; Trindade et al., 2017; Milly et al., 2008).
Although these are not new issues, utilities are still struggling with them
(Bonzanigo et al., 2018; Paulson et al., 2018). Although WaterPaths was de-
velopped with these issues in mind, existing commercial frameworks for wa-
ter infrastructure management modeling (e.g., HydroLogics, 2009; Sieber and
Purkey, 2015; Labadie, 2011) focus on existing infrastructure, have strong lim-
itations on their inclusion of uncertainties, and do not connect to key concepts
from the infrastructure investment literature
Our DU Pathways framework broadens the water portfolio problem to also
consider long-term planning and the construction of new infrastructure into the
short-term management framework presented in Chapter 3. The DU Pathways
framework uses some features of ROA (Wang and de Neufville, 2005; Erfani
et al., 2018; Fletcher et al., 2017; Hui et al., 2018) such as a focus on flexible
decision rules and a careful consideration of short-term versus long-term infor-
mation. However, ROA as a single-objective approach is limited in its account-
ing of stakeholders with diverse interests and its underlying mathematical tree
logic becomes computationally intractable when large number of uncertainties
are considered [curse of dimensionality, (Dittrich et al., 2016)]. Borgomeo et al.
(2018) address some of these concerns by proposing a multi-objective evaluation
160
of candidate water supply investments. They maximize robustness and mini-
mize risk and cost of a 30-year long fixed construction schedule encompassing
multiple infrastructure options. It should be noted that the derived sequence
of investments is not adaptive and does not account for evolving information
feedbacks. These limitations also exist in the recent studies by Beh et al. (2015b)
and Huskova et al. (2016), although the former seek to improve adaptivity by
incorporating a similar emphasis on near term information as done in ROA. Al-
ternatively, the recently introduced dynamic adaptive pathways policy (DAPP)
approach (Haasnoot et al., 2013; Kwakkel et al., 2014) adds flexibility to the
planning process by making use of potentially diverse sets metrics (signposts).
DAPP, however, still maintains a strong reliance on the use of limited numbers
of predefined action sequences that require high levels of institutional stability
and coordinated consensus over the long term.
The DU Pathways framework introduced in this work study is first formal
integration of dynamic and adaptive water infrastructure pathways (Zeff et al.,
2016; Haasnoot et al., 2013) (annual, long-term planning), financial and supply-
reliability drought mitigation instruments [weekly, short-term decisions dis-
cussed in Chapter 3 and in Zeff et al. 2014 and Trindade et al. 2017], and the re-
cent extensions of the MORDM framework that incorporate deep uncertainties
in search-based identification candidate actions (Kasprzyk et al., 2013; Trindade
et al., 2017; Kwakkel et al., 2014; Watson and Kasprzyk, 2016). DU Pathways
addresses decision making at two time scales: weekly, for operational manage-
ment decisions, and annual, for infrastructure investment decisions. Following
the approach recommended by Zeff et al. (2016), the integrated infrastructure
investment and management policies specify short- and long-term decision-
making rules based on the risk-of-failure metric (ROF), as well as candidate
161
orderings for infrastructure investments. By combining infrastructure planning
and management rules in a single policy, the water portfolio planning approach
aids the design of drought mitigation instruments tailored to new infrastructure
to minimize the need of further investments. The main limitation of the infras-
tructure pathways work by Zeff et al. (2016) is its limited treatment of uncer-
tainties. The DU Pathways framework contributed here significantly expands
the suite of deep uncertainties considered in the search-based identification of
candidate infrastructure investment and management policies. Additionally,
the framework formalizes a detailed evaluation of the multi-city policies’ ro-
bustness tradeoffs. DU Pathways aids stakeholders in navigating these trade-
offs through a combination of visual decision analytics Woodruff et al. (2013)
and enhanced scenario discovery in which boosted trees Schapire (1999) aid in
the identification of the uncertainties most strongly shape the infrastructure in-
vestment and management policies’ vulnerabilities. The application of the DU
Pathways framework is demonstrated the Zeff et al. (2016) long-term version of
Research Triangle test case described in Section 3.4 and prior works (Trindade
et al., 2017; Herman et al., 2014; Zeff et al., 2014), where we have discovered
how the region’s vulnerabilities evolve under alternative policies, clarified sig-
nificant interdependencies between the region’s four utilities, and show the im-
portance of carefully balancing supply and financial instruments. Furthermore,
this is chapter also serves to test the WaterPaths framework in a challenging
real-world regional decision context.
162
5.3 The Long-Term Version of the Research Triangle Test Case
The DU Pathways framework introduced in this study is demonstrated on the
long-term version of Research Triangle region in North Carolina presented in
Section 3.4. As mentioned in Section 3.4, the present test case is comprised of
the four inter water utilities (Figure 5.1) interconnected through water mains
and currently responsible for meeting the majority of the region’s growing de-
mand in the Research Triangle portion of the Neuse and Cape Fear river basins:
Raleigh, Durham, Cary and Orange Water and Sewer Authority (OWASA, rep-
resenting Carrboro and Chapel Hill). The Triangle Water Supply Partnership
(TWSP) of which the four utilities participate has focused on developing collab-
orative short-term drought management and infrastructure planning solutions
for its members with specific focus on meeting projected 2060 water supply de-
mands (Triangle J, 2014). The two key challenges for the Research Triangle Re-
gion are to develop (1) collaborative plans for short-term regional management
of the existing water infrastructure, as mentioned in Section 3 and (2) plan long-
term infrastructure investments to meet future demands.
The the long-term version of the Research Triangle test case includes all cur-
rent infrastructure systems providing water to the four utilities presented in
Section 3.4 (Table 3.4) as well as recently proposed candidate investments (Ta-
ble 5.1) including additional storage, conveyance and treatment projects across
the four utilities laid out by the JLP in Triangle J (2014). OWASA and Durham
own allocations in the Jordan Lake that can currently only be accessed through
Cary’s water treatment plant on eastern shore of Jordan Lake, subject to treat-
ment and inter-utility conveyance availabilities (Figure 5.1). Currently, Jordan
Lake’s water supply storage is still not fully allocated and several utilities in the
163
Legend 0 2.5 5 10 15 20 25 miles
Potential Infrastructure
Existing Sources
Suppy Water Bodies Lake Michie
Existing Treated Water (Expansion)
Pipe Connections
Little River Reservoir
Eno Teer Quarry
Falls Lake
Cane Creek Reservoir Reclaimed (Reallocation)
(Expansion) Water
Durham
Richland Creek
OWASA QuarryStone Quarry
(Expansion)
University Lake Neuse River Intake
(Expansion)
Raleigh Little River
Reservoir
Western Jordan Cary
Lake WTP
Lake Wheeler
Jordan Lake
Lake Benson
Figure 5.1: Map of the Research Triangle Region test case. Potential Infras-
tructure refers to infrastructure options presented in Triangle J
(2014), that are being considered by the utilities.
Research Triangle are planning on submitting new allocation requests, which
gives rise to potential resource contention that the the four studied water utili-
ties want to avoid.
Key short-term (i.e. drought) regional management instruments include wa-
ter use restrictions and treated water transfers that are employed using weekly
operational rules with the goal of making better use of the Research Triangle re-
gion’s shared resources. Chapter 3 focused on defining management operations
collaboratively so that the impacts from the measures taken by each utility has
a predictable impact on the operations of the other utilities, while this chapter
will solve the same problem but considering added infrastructure and over a
longer period of time (through 2060).
164
Utility Infrastructure Option Storage/ Cost
Production ($MM)
OWASA University Lake expansion 2550 MG 107
OWASA Cane Creek Reservoir expansion 3000 MG 127
OWASA Stone Quarry shallow expansion 1500 MG 1.4
OWASA Stone Quarry deep expansion 2200 MG 64.6
Durham Teer Quarry expansion 1315 MG 22.6
Durham Reclaimed water (low) 2.2 MGD 27.5
Durham Reclaimed water (high) 11.3 MGD 104.4
Durham Lake Michie expansion (low) 2500 MG 158.3
Durham Lake Michie expansion (high) 7700 MG 203.3
Raleigh Little River Reservoir 3700 MG 263
Raleigh Falls Lake WQ Pool Reallocation 4100 MG 68.2
Raleigh Neuse River Intake 16 MGD 225.5
Regional Western Jordan WTP (low capacity) 33 MGD 243.3
Regional Western Jordan WTP (high capacity) 54 MGD 73.5
Table 5.1: Summary of proposed water supply infrastructure for the Re-
search Triangle (Zeff et al., 2016; Triangle J, 2014).
As a first step in our development of the DU Pathways framework con-
tributed in this study, we borrowed from the model presented in Section 3.4.1
and developed a flexible regional model using WaterPaths that can encompass
a breadth of system uncertainties and support our evaluation of the diverse
portfolios of short-term management actions as well as long-term infrastructure
investment pathways. At its most basic core, our simulation of the Research Tri-
angle is a water mass-balance model with the inputs and outputs illustrated in
Figure 5.2 below.
The model is used in a set of optimization runs whose results were further
assessed to identify robust infrastructure management and investment policies
for each of the four Research Triangle utilities. Beyond management actions,
made at a weekly timestep, the utilities must also define infrastructure invest-
ment pathways that specifically prioritize and sequence major projects (among
165
Figure 5.2: The Research Triangle instance of the WaterPaths simulation
model takes as inputs a policy to be evaluated, the system
components and its connectivity, hydrological time series, in-
frastructure construction order, information about infrastruc-
ture to be built, different types of demands, and uncertainty. It
outputs time series of system states, an objectives vector and
pathways.
options specified in Triangle J, 2014). As with the short-term drought mitiga-
tion instruments (restrictions and transfers), such infrastructure sequences and
corresponding decision-making rules are sought that will positively impact the
Research Triangle region’s supply reliability and financial stability. The man-
agement and investment actions are critically interdependent, given that effec-
tive short-term drought mitigation instruments can serve to dramatically delay
and/or reduce capital investments (Zeff et al., 2016).
166
Type Uncertainty Source Regional or Individ-
ual for Utility/ In-
frastructure
Supply/Demand Growth of Mean An- Regionalnual Demand
Growth of Mean An- Regional
nual Evaporation
Financial/ Interest Rate Regional
economics Bond Term RegionalDiscount Rate Regional
Policy Effective- Water Use Stage Restric- Individual
ness tion Efficacy
Infrastructure Permitting Time Individual
Construction Construction Cost Individual
Table 5.2: Uncertainties included in extended version of the regional wa-
ter utility planning and management problem presented in Zeff
et al. (2016). Regional uncertainteis are those for which the same
value was used for all utilities, versus a value different value for
each utility, as in the case uncertainties described as individual.
5.4 Methodology
Expanding the methodology presented in Section 3.3, the major steps of the DU
Pathways methodology as illustrated in Figure 5.3 are as follows:
1. Identifying Infrastructure Investment and Drought Management Policy
Tradeoffs: Candidate Pareto-approximate infrastructure investment and
water portfolio management policies are identified using robust multiob-
jective optimization (Reed et al., 2013; Hadka and Reed, 2014; Deb et al.,
2002b; Shah and Ghahramani, 2016; Hernández-Lobato et al., 2016).
2. Defining and Evaluating Robustness: Exploiting a more expansive sam-
pling of the potential deep uncertainties that could pose challenging
states-of-the-world (SOWs) in which robustness metrics can be used to
167
evaluate and rank-order solutions for their robustness. For reviews and
comparisons of alternative definitions of robustness, see (McPhail et al.,
2018; Herman et al., 2015; Giuliani and Castelletti, 2016).
3. Discovering Uncertain Scenarios that Control Robustness: The Pareto ap-
proximate of policies of interest once evaluated against a broader sam-
ple of SOWs undergo further diagnostic assessment by means of sen-
sitivity analysis (Sobol, 2001; Saltelli, 2002; Saltelli et al., 2008; Morris,
1991; Campolongo et al., 2007; Herman and Usher, 2017; Jaxa-Rozen and
Kwakkel, 2018) and/or scenario discovery (Bryant and Lempert, 2010;
Quinn et al., 2018; Kwakkel, 2017) to determine which specific uncertain-
ties dominantly impact system’s performance (factor prioritization) for
specific combinations of values (factor mapping).
4. Visual Infrastructure Pathway Analysis: Use of visual analytics to deter-
mine how infrastructure investment decisions are impacted by deviations
from base projections for the most important sources of uncertainty.
The four DU Pathways framework steps listed above and summarized
graphically in Figure 5.3 are described in more detail in the remainder of this
section.
5.4.1 Identifying Infrastructure Investment and Drought Man-
agement Policy Tradeoffs
As mentioned in Section 5.2, DU Pathways framework bridges three active areas
of research in recent literature: (1) dynamic adaptive pathway planning (Haas-
168
Figure 5.3: Key steps of the DU Pathways approach and their key points
noot et al., 2013; Kwakkel et al., 2014; Zandvoort et al., 2017), (2) ROF-based
urban water portfolio planning (Palmer and Characklis, 2009; Zeff et al., 2014,
2016; Herman et al., 2014), and (3) multi-objective optimization under deep un-
certainty (Kwakkel et al., 2014; Watson and Kasprzyk, 2016). In DU Pathways,
action triggers based on the ROF dynamics are the basis of infrastructure invest-
ment and management policies. These policies are in turn what determine the
169
infrastructure investment pathways. The ROF-based action triggers (or decision
variables) are based on the premise of monitoring storage-to-demand dynam-
ics to yield a time-continuous assessment of when the risks of unacceptably low
water supply capacity requires action. As demonstrated by Zeff et al. (2016), the
ROF-based action triggers have the advantage of providing system adaptabil-
ity across planning time-scale decisions (i.e., annual infrastructure construction)
and short-term management time-scale decision (i.e., the weekly use of drought
mitigation instruments). In either case, actions are triggered when the given
ROF metric (Palmer and Characklis, 2009) crosses ROF threshold values stated
in a fully specified infrastructure investment and management policy (Zeff et al.,
2016). The set of all regional ROF-triggers, other financial policy variables and
infrastructure construction order for all utilities in the problem defines the op-
erations of all drought mitigation instruments and the strategy for the construc-
tion of infrastructure. This set is what is in this work called an infrastructure
investment and management policy, shown in Figure 5.4. With that, the mini-
mization problem to be solved then becomes a matter of finding positions for
the decision levers in Figure 5.4, where the levers represent the decision/policy
variables, that lead to Pareto-approximate policies.
Problem Formulation
The multi-objective minimization problem formulation presented below in
Equations 5.1 - 5.6 is a general mathematical abstraction for search-based iden-
tification of regional water infrastructure investment and management policies,
denoted by θ, for all members of a cooperative group of water utilities. The op-
timization problem focuses on finding Pareto-optimal policies θ∗ that minimize
170
Figure 5.4: Illustrative portfolio of decisions related to short-term drought
mitigation instruments and long-term priorities for infrastruc-
ture investments pathways that define a system’s water supply
policy. The optimization problem consists in finding the best
position for each knob corresponding to each decision variable
for each utility.
the objective function vector F :
θ∗ = argmin F (5.1)
θ
where,
 −fREL(xs,θrt,θtt,θjla,θit, ICO,Ψs)


fRF (xsrof ,θrt,θtt,θjla,θit, ICO,Ψs) 
 

fNPV (xlrof , ICO,Ψ

 s
) 
F =   (5.2)fFC(xsrof ,θrt,θtt,θjla,θacfc,θirt,θit, ICO,Ψs,xlrof ) fWFPC(xsrof ,θrt,θtt,θjla,θacfc,θirt,θit, ICO,Ψs,xlrof )
fJLA(θjla)
s.t.
|ME| ≤ 1 ∀ME ⊆ BI (5.3)
171
where
 xsrof 

X = xlrof  (5.4)
xs
θ = [θrt,θtt,θacfc,θirt,θit, ICO,θjla] (5.5)
   — ψ WCU
—
 



— ψ —ΨWCU =  WCU  . .. 
   
 
— ψWCU —
Ψs =    (5.6)— ψ —

 DU,1  
— ψDU,2 —

ΨDU =  . .. 

 

— ψDU,N —R
In the equations above, X is the time-varying state matrix across all of the util-
ities, where the components xsrof and xlrof are vectors of short- and long-term
values for the dynamic tracking of ROF. This distinction refers to the short-term
ROF metric calculated over 52 weeks and used for the short-term instruments,
as presented in Section 3, and the long-term version of the same metric using
TROF = 78 weeks in Equation 3.8 used for infrastructure triggering. The deci-
sion variable θacfc is a vector of annual contingency fund contributions, which
are percentages off annual revenue saved in a utility’s drought mitigation fund.
As mentioned above, restrictions, transfers, insurance, and infrastructure in-
vestment decisions are all formulated consistently through time using ROF trig-
172
gers. The overall regional policies are composed of the following decisions for
all utilities: the θrt vector of restriction triggers, the θtt vector of transfer trig-
gers, the θjla vector of Jordan Lake Allocations, the θirt vector of insurance re-
striction triggers, the θit vector of long-term-ROF infrastructure construction
triggers, and the ICO vectors with the infrastructure construction ordering for
each utility. Overall, all decision variables are encompassed by the triggers θ
plus the infrastructure construction order ICO. A Pareto optimal management
and investment policy is denoted by θ∗ in Equation 5.1. Equation 5.3 enforces
the requirement that two infrastructure options, such as high and low capac-
ity expansions of a reservoir, cannot be built in the same realization (Zeff et al.,
2016; Erfani et al., 2018). In Equation 5.3, ME represents a generic subset of
mutually exclusive infrastructure options within the set of built or prospective
infrastructure BI . Matrix Ψs is comprised of the vector samples of the deeply
uncertain variables of concern where each element corresponds to a specific un-
certainty. F is the vector-values objective function where fREL is the supply re-
liability objective, fRF is the restriction frequency, fNPV is the net present cost of
infrastructure, fFC is the total cost of financial drought mitigation instruments
(contingency fund and insurance loading) plus debt repayment, fWFPC is the
worse-first-percentile of financial variability caused by the supply drought miti-
gation instruments (restrictions and transfers), and fJLA is the combined Jordan
Lake allocation objective, representing the allocation of the municipal supply
pool of the Jordan Lake, in the Research Triangle, NC, to be granted to a utility,
such that: ∑Nu
minimize fJLA = JLAj (5.7)
j=1
where JLA is the percent of the pool allocated to utility j, and Nu is the total
number of utilities in the system. Utilities want to minimized requested allo-
173
cation due to local political contention surrounding the Jordan Lake allocation
requests. Such objective is specific to the test case presented in this study, al-
though groups of utilities in different regions may go through similar conflicts.
A detailed formulation of the other five objectives is presented in Sections 3.5
and 4.4.9. Both sampling processes for matrix Ψs, well-characterized uncer-
tainty (WCU) sampling (ΨWCU ) and deep uncertainty (DU) sampling (ΨDU ),
are described in detail in the next section.
Sampling States-of-the-World for Alternatives Generation
All of the objectives presented above in Equation 5.2 except for the Jordan Lake
allocation are stochastically evaluated within a Monte Carlo simulation frame-
work across an ensemble of candidate SOWs comprised of time series of in-
flows, eraporation rates, and demands, and potentially several deeply uncer-
tain factors. The DU Pathways framework distinguishes between two formu-
lations of the water portfolio management and infrastructure investment prob-
lem. These formulations differ in their forward Monte Carlo-based evaluations
based on sampling the matrices of natural inflows (NI), evaporation series (E),
unrestricted demands (UD), and samples of deeply uncertain factors (Ψs). The
Well-Characterized Uncertainty (WCU) and Deep Uncertainty (DU) sampling
schemes employed in this study are illustrated in Figure 5.5 and were imported
from Section 5.4.1.
Figure 5.5a shows that the WCU sampling scheme specifies SOWs (rows of
rectangles) by sampling uncertainties for which historical-data-driven distribu-
tions are available from their historical observations, represented by colored
rectangles (e.g., natural stream inflows, evaporation rates, and demand fluctu-
174
Figure 5.5: (a) In the WCU sampling scheme, sets of time-series samples
of synthetically generated streamflows, evaporation rates and
demands are generated for each realization and appended to
the historical data. The best projected values (or other assump-
tion) of the deeply uncertain sources of uncertainty are then
coupled with the resulting sets of time series form all SOWs.
(b) In DU sampling, the same sets of time-series are instead
combined with samples of deep uncertainties to form the en-
semble of SOWs.
ations around projected mean demands). For all other factors, represented by
grey rectangles in Figure 5.5a, deterministic best projections are used. Alter-
natively, the DU sampling strategy shown in Figure 5.5b increases the number
of uncertain factors that are varied across SOWs by sampling uncertainties not
only from data-derived distributions but also from expert elicitation. More for-
mally, the WCU and DU sampling schemes can be abstracted mathematically
as a forward Monte Carlo approximation of the system performance (see Equa-
tion 5.8). Table 5.2 shows the deep uncertainties considered in this study, and
as represented in Equation 5.8 are assumed to be uncorrelated with the well-
characterized uncertainties.
175
NIs,Es,UDs,Ψs ∼ f(NIh,Eh,UDh,S,R) · D(·) (5.8)
Within Equation 5.8 the subscript s denotes synthetic and h denotes historical
time series. Synthetically generated series compose matrices for natural inflows
(NI), evaporation (E), and unrestricted demands (UD) where their dimen-
sions are the number of synthetic series samples (Nr) by the total number of
weeks in the full planning horizon (Nw). The historical data used to derive the
synthetic series (subscript h) are represented in respective matrices in equation
5.8 of dimension of the number years in the historical record (Nhr) and the num-
ber of weeks in a year (Nw). Cross-site correlation and temporal auto-correlation
in the historical natural inflows and evaporation time series are captured with
the S andRmatrices, respectively. The joint-pdf represented by f characterizes
the inflows, evaporation rates, and demand fluctuations around annual means,
all well-characterized uncertainties, whileD(·) is an assumed pdf for the deeply
uncertain factors that can take the following forms, depending on the sampling
scheme: ∏Nf
DWCU(X ) = δ(Xi) (5.9)
∏i=0Nf
DDU(l,u) = U(li, ui) (5.10)
i=0
Within Equation 5.9, δ(·) is a Dirac function, X is a vector of best projections
for the deeply uncertain factors considered in the problem, i is the index of a
deeply uncertain factor, and Nf is the total number of deeply uncertain factors.
Within Equation 5.10, l and u are sets of lower and upper bounds for the deeply
uncertain factors, and U denotes a uniform distribution. The initial probabilis-
tic management and infrastructure investment pathways concepts introduced
by Zeff et al. (2016) used only the WCU sampling strategy. The DU Pathways
176
framework significantly expands the sampling scope to consider deep uncer-
tainties and increase their role in the discovery of alternatives. Non-stationary
hydro-climatology trends are accounted for in this work by applying a weekly
series of multipliers to inflow and evaporation log-mean and log-variance mul-
tipliers, increasing and/or decreasing them over time (Quinn et al., 2018). In
this work, the series of multipliers was created from sinusoid functions, to as-
sure the inclusion of scenarios in which trends go up and/or downwards (more
details in Section (2.2)). The ROF metric, used as a basis of the infrastructure
investment and management decision triggers, is presented next.
Tracking Risk-of-Failure Dynamics
The ROF dynamics for each utility are tracked for each week by following the
procedure described in Section 4.4.2. The ROF metric presented in Section 4.4.2
simultaneously encompasses information about current demands and storages,
and of historical inflows and evaporation rates, improving therefore on infor-
mation use relative to traditional take-or-pay agreements and days-of-supply-
remaining metric (Palmer and Characklis, 2009) and providing the system with
more adaptability. We recommend an Nrof (one-year long simulations used in
the ROF calculation for one week) of at least 50 ROF simulations so that the ROF
metric has a precision of at least 1/50 = 2%.
Drought Mitigation Instruments and Financial Model
The drought mitigation instruments explored in this study build from Chapter 3
and are focused on water use restrictions and treated inter-utility water transfers
177
(Zeff et al., 2014, 2016; Caldwell and Characklis, 2014; Palmer and Characklis,
2009; Zeff et al., 2014). Obviously, the overall DU Pathways framework could
consider other short-term management actions as would be appropriate in other
regions of application. The use of restrictions and transfers are contingent on the
value of the short-term ROF metric, as shown in Equations 3.6 and 3.7. Restric-
tions are tiered based on the progressive severity of a drought, so each tier may
have its own short-term ROF trigger value. The transferred volumes requested
by utilities are capped for two reasons: (1) limited conveyance capacity and (2)
competition between two or more utilities for available volume for transfer, in
which case each utility receives a volume proportional to its short-term ROF.
Inter-utility water transfers in combination with water use restrictions have
negative financial impacts on the water utilities by reducing revenues and in-
creasing costs. Drought-focused index insurance instruments provide a means
to hedge utilities against the financial risks associated with drought mitigation
measures. Index insurance, in which payouts depend on an index value, have
been studied in literature to hedge utilities against droughts (Zeff and Charack-
lis, 2013; Zeff et al., 2014), hydropower generators against climate risks (Foster
et al., 2015), climate-related government regulation (Denaro et al., 2018), and
groundwater irrigated agriculture against government groundwater pumping
regulations during droughts (Blanco and Gómez, 2013). The ROF-triggered
drought index insurance structure used to hedge utilities against financial losses
in this study exploits a binary payout structure (Foster et al., 2015; Zeff et al.,
2016). Details about the insurance and contingency models used in this work
can be found in Section 3.5.
178
Prioritized Sequencing of Supply Infrastructure Investment Pathways
As with drought mitigation and financial instruments, infrastructure construc-
tion or permitting in the DU Pathways framework is triggered rather than pre-
scheduled in time (Walker et al., 2001). Triggering occurs in years when the
long-term ROF metric, defined in Equations 3.8 to 3.10 as an annual calculation
using TROF = 78, crosses a set threshold (Zeff et al., 2016). The DU Pathways
framework specifies that When infrastructure triggering occurs for a given util-
ity, construction begins for the next infrastructure option in the construction
sequence specified in the policy.
When infrastructure is triggered, utilities issue debt to finance the new in-
frastructure. The infrastructure and debt service payments stream for all utili-
ties are then updated every year according to Equation 5.11.
OIy,DSy = f(x y−1 y−1lrof ,θirof , ICO,OI ,DS ) (5.11)
In Equation 5.11, OIy is the vector with the IDs of all online supply infrastruc-
ture, DS is a matrix with the streams of debt service payments for all utilities,
xlrof is the vector of long-term ROFs for all utilities, θirof is the vector of long-
term ROF triggers for all utilities, and ICO is a matrix with the infrastructure
construction order for each utility.
Another distinguishing feature of the DU Pathways formulation approach
is the direct accounting for the possibility that infrastructure may be built and
operated jointly by multiple utilities (Zeff et al., 2016). If triggered by one util-
ity, such projects are built by all utilities who have pledged to be part of the
project with the cost proportionally split, meaning that associated risks would
179
be shared among utilities. The supply infrastructure construction logic, together
with the short-term drought mitigation instruments, are integrated with a mass-
balance model and the WCU and DU uncertainty sampling into a combined
framework to allow for the evaluation of drought mitigation policies, as de-
scribed next.
Evolutionary Multi-Objective Search
The complexity of the DU pathways decisions, the large number of objectives,
and the breadth of modeled uncertainties motivated our use of the multi-master
parallel Borg multi-objective evolutionary algorithm (MM-BorgMOEA) Hadka
and Reed (2013, 2014) described in Section 2.1. Given that the Borg MOEA is
a stochastic global optimizer, in both the WCU and DU optimization formula-
tions, multiple random seed search trials of the MM-Borg MOEA are used to
attain Pareto-approximate sets of policies (see equation 5.1 in Section 5.4.1). Af-
ter running all the seeds, we then identify the best known Pareto-approximate
sets (also termed the reference sets) by epsilon non-dominated sorting (Hadka
and Reed, 2013; Laumanns et al., 2002) across all of the attained approximate
sets. More details on the specific MM-Borg MOEA parametrization and parallel
configuration used in this work are provided subsequently in Section 5.5. All
of our subsequent results, including our more detailed robustness assessments,
focus on the best known reference sets obtained for the WCU and DU sampling
schemes optimization formulations.
180
5.4.2 Defining and Evaluating Robustness
Deep Uncertainty Re-Evaluation
The DU Re-Evaluation illustrated in Figure 5.6 defines a broader and more chal-
lenging set of SOWs that are used to assess the robustness of candidate infras-
tructure investment and management policies. As noted in prior studies Groves
and Lempert (2007); Bankes (1993), the intent of this exploratory modeling and
analysis is to shift away from predicting to instead refocus on discovering con-
sequential scenarios. The DU Re-Evaluation sampling scheme illustrated in
Figure 5.6 also provides a challenging but consistent space for comparing the
performance objectives and robustness of policies found with WCU and DU
optimization formulations. Scenario generation for the DU Re-Evaluation sam-
pling scheme consists of two steps. The first step is to generate a number Nr of
new synthetic inflows and evaporation-rates annual time series (NIs,Es,UDs)
following the approach presented in Kirsch et al. (2013) and correlated synthetic
demands — both procedures are described in Section (2.2). The second step is
to sample a matrix Ψs of vectors of deeply uncertain factors Ψ from distribution
DDU(·). Lastly, all of time series are coupled with one vector of deeply uncertain
factors Ψs at a time for all vectors in matrix Ψs, as illustrated in Figure 5.6. Each
Monte Carlo evaluation (Nr sets of inflow, evaporation rate and demand time
series paired with the same sample of deeply uncertain factors, or one row Ψ of
matrix Ψs) returns one value of each objective and Nr pathways. The objective
values of all Monte Carlo evaluations are later used to back-calculate the objec-
tive values of the entire re-evaluation exercise for a given solution, as described
in the subsection below.
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Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond TermGrowth
Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond TermGrowth
Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond TermGrowth
Objectives
Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond TermGrowth
Demand 
Policy NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond TermGrowth
Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond TermGrowth
Pathways
Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond Term
Growth
Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond Term
Growth
Demand 
NI, E, UD, 𝝋𝒊𝒆 Discount Rate Bond Term Robustness
Growth
Figure 5.6: Re-evaluation sampling scheme. The same sets of Nr hydro-
logical and demand time series are combined with one vector
sample of deeply uncertain factors one at a time, resulting in
a number of SOWs equal to Nr times the number of samples
of deeply uncertain factors. The resulting set of objectives vec-
tors are used to back-calculate the objectives and robustness
of the whole ensemble, while the pathways are used to ana-
lyze the dependency of infrastructure investments on deeply
uncertainty factors.
Objectives Estimation Based on the DU Re-Evaluation
The re-evaluation samples provide means of verifying performance by recom-
puting the WCU or DU optimization solutions’ objectives attained from the re-
duced form Monte Carlo sampling schemes during search (see Figure 5.5). It
should also be noted the WCU and DU optimization formulations are inconsis-
tent with one another in terms of the number and severity of uncertainties that
are included in search. We overcome this inconsistency by using the DU Re-
Evaluation to verify and compare performance tradeoffs when evaluated under
a consistent suite uncertainties. The DU Re-Evaluation yields an ensemble for
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the objectives, which are aggregated as follows:
• Reliability, Restriction Frequency, Net Present Cost of infrastructure and
total Annual Financial Cost of drought mitigation costs are calculated as
their means across DU re-evaluation samples.
• the Worse First Percentile Cost is calculated as the worse first percentile
across the samples, mimicking the original objective as a measure of
supply-enabled financial risk.
• the Jordan Lake Allocation objective does not change across samples.
The next step is to use the same ensemble of objectives used to back-calculate
the overall objective values for a single policy evaluation and use it to evaluate
its robustness.
Evaluating Robustness
A myriad of robustness metrics have been developed for different applications
while being general enough to be applied in water systems engineering. In this
chapter, we again exploit the domain criterion satisficing measure (introduced
in Starr (1962), formalized in Schneller and Sphicas (1983) and used in several
prior studies, such as Herman et al. (2014), Herman et al. (2015) and Lempert
and Collins (2007)) used in Chapter 3. Satisficing metrics are preferred by stake-
holders whose primary concern is to not violate performance constraints, de-
spite losses of attainable performance (regret) (McPhail et al., 2018; Simon, 1959).
The reader is encouraged to refer to recent works comparing approaches to ro-
bustness (Lempert and Collins, 2007; McPhail et al., 2018; Dittrich et al., 2016;
Herman et al., 2015; Giuliani and Castelletti, 2016) if interested in more details.
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5.4.3 Discovering Uncertain Scenarios that Control Robustness
The SOWs sampled in the DU Re-Evaluation sampling scheme illustrated in
Figure 5.6 are next used for scenario discovery (Groves and Lempert, 2007;
Bryant and Lempert, 2010). In scenario discovery, the analyst uses the objec-
tives of a given policy calculated for the policy re-evaluation under different
scenarios (Figure 5.6) to map regions of the space of uncertainties (or scenarios)
to be sought or avoided should that policy be implemented. A classification
of regions of concern would ideally include most scenarios in which the policy
in question meets the conditions of performance measures of focus (high cover-
age) and avoid mixtures with other scenarios (high density). Scenario discovery
is the step of the DU Pathways framework that allows analysts to switch from
asking “what should decision-makers do?” to “what scenarios should decision-
makers be concerned by if they implement candidate actions?”.
Seminal studies introducing scenario discovery (Lempert et al., 2008) sug-
gested the use of the Patient Rule Induction Method, or PRIM (Friedman and
Fisher, 1999), used in Chapter 3, and Classification and Regression Trees, or
CART (Breiman et al., 1984). Pass/fail boundaries found by both methodolo-
gies are hypercubes orthogonal to all the sources of uncertainties (Dalal et al.,
2013). The advantage of both methods is that when mathematically appropri-
ate they yield interpretable and relevant scenarios, as in Section 3.7.4. The main
disadvantage of both methods is that they do not capture interactions between
sources of uncertainties, which is often significant, preventing boundaries ex-
pressed as combinations of values for multiple sources of uncertainty. The issue
of capturing interactions between uncertainties was addressed in Quinn et al.
(2018) by using logistic regression to find the pass/fail boundaries. Logistic re-
184
gression is a linear classifier, and therefore will return pass/fail boundaries that
are straight but not necessarily orthogonal to the uncertainty axes, accounting
therefore for interactions between sources of uncertainty. Logistic regression can
be used to find non-linear boundaries by adding features to the original data.
For example, if a data set of scenarios has two sources of uncertainty, x1 and x2,
adding an independent term x1 · x2 and training a logistic regression model on
all three terms will return a non-linear model. However, there are three disad-
vantages for this approach: (1) output rules are not as easy to interpret as those
from PRIM and CART, (2) the approximate mathematical representation of the
interactions between sources of uncertainty must be pre-specified, and (3) the
boundaries are necessarily smooth.
In the DU pathways framework presented in this study, the mixture of ROF-
based rules for short-term management actions and long-term large discrete in-
vestments in infrastructure yields failure regions in the uncertainty space delim-
ited by mixtures of sampled SOWs that are non-linear and multimodal. These
complexities cannot be handled by logistic regression without the manual addi-
tion of several interaction terms — kernel logistic regression (Zhu and Hastie,
2005) can be used to fit such complex boundaries, but given kernel methods
are non-parametric, no information about factors’ sensitivities would be avail-
able. Alternatively, we propose the use of boosted trees (Drucker and Cortes,
1996; Freund et al., 1999; Schapire, 1990). Boosting is a distribution-free strat-
egy to turn a weak learning model — models whose accuracy is only slightly
better than random guessing — into a strong learning model — models that
can achieve arbitrarily high accuracy (Schapire, 1990). It works by creating and
ensemble of weak classifiers and forcing part of the ensemble to focus on the
hard-to-learn parts of the problem, while other parts focus on the easy-to-learn
185
parts. A boosted ensemble model therefore has the following shape:
∑T
HT (x) = αtht(x) (5.12)
t=1
whereH is the ensemble model, T is the total number of models in the ensemble,
αt is the boosting multiplier corresponding to model ht where t is the model
index, and x is the vector of explanatory variables. Model complexity can be
adjusted by setting the number of models (T ) in the ensemble. The boosting
algorithm is described in more detail in Appendix E.
Boosting is often applied to Classification and Regression Trees (CART) trees
(Breiman et al., 1984) with small maximum depth (e.g., 3 or 4), that are high bias
(weak) classifiers, yielding an ensemble of trees. CART trees are comprised of
nested orthogonal partitions within the explanatory variables space. The parti-
tions are found one at a time by finding the variable and its value where a split
would decrease some loss function (or leaf impurity) the most. Since in sce-
nario discovery the analyst wants to find regions of the deep uncertainty space
in which a policy will fail to reach pre-established performance criteria by any
amount, classification trees are used here. Boosted trees have three advantages
over the previously mentioned scenario discovery methods: (1) the approach
captures non-differentiable boundaries that typical arise from threshold-based
rules (e.g., ROF-based action triggers), (2) it captures non-linearity without ex-
plicitly modeling variable interactions, and (3) it is resistant to overfitting, help-
ing assure scenario discovery maps that are simple to interpret. The use of
boosted trees as opposed to random forests was motivated by the preference
for simple boundaries between success/fail regions to be achieved with trees
as shallow as possible. An important concept for understanding the impact of a
186
particular uncertain factor on the performance of a policy is that of leaf impurity.
A impure leaf has a mix of success and fail scenarios, as opposed to all scenarios
within it corresponding to either one or the other. As mentioned, every time a
partition is added to a tree by splitting a leaf into two based on an uncertainty
factor at a time, it is done so decrease the impurity of the resulting leafs as much
as possible. The percentage total impurity decrease across all trees due to splits
on each uncertainty factor can be used as a measure of the impact of that deep
uncertainty factor in the performance of a policy.
5.4.4 Visual Infrastructure Pathway Analysis
Lastly in the DU Pathways framework, after key uncertainties have been iden-
tified, we suggest detailed visual diagnostic assessments of key pathway solu-
tions of interest to decision makers. In this study, we demonstrate the value
of considering how the infrastructure investment and management pathways
evolve for three scenarios: the most favorable values for the most influential
uncertainties, an intermediate case of interest, and one with the least favorable
values for the most influential uncertainties. The challenging part of this is step
is the design of a plot that displays the pathways of all hydrologic realizations
(e.g., 1,000 realizations) for a policy, while clearly showing the times at which
each infrastructure option is built.
There are in the literature examples of visualizations of infrastructure path-
ways. The first and most well-known is the “metro maps” (Haasnoot et al., 2013;
Kwakkel et al., 2016a; Zandvoort et al., 2017). The metro maps have the advan-
tages of being easy to interpret and to present to audiences. However, they
187
allow for the representation of only a handful of pathways before the pathways
become difficult to distinguish. Another option is the probabilistic pathways
Zeff et al. (2016), which was derived from the concept of metro maps (Haas-
noot et al., 2013). The probabilistic pathways visualization makes use of trans-
parency to indicate the timing and frequency that each candidate infrastructure
option is implemented over an ensemble hydro-climatic scenarios. However,
distinguishing the pathways differences in each of the ensemble realizations
is challenging. The pathways visualization proposed in this work displays the
pathways explicit temporal evolution for major investments for all hydrological
realizations as a stacked ensemble plot. The resulting plots provide a visual in-
dication of what new water supply infrastructure would be needed as function
of key hydrological or other exogenous scenario gradients. This can help plan-
ners provide resources for near-future construction and for permitting, detailed
design, public consultations, etc.
5.5 Computational Experiment
The computational experiment was divided into two phases: optimization and
re-evaluation. Given the runtime of the Research Triangle model, we used the
Texas Advanced Computing Center’s Stampede 2 for both optimization and re-
evaluation phases. The optimization phase was comprised of two MM-Borg
MOEA (see Chapter 2.1.4) seeds trials for the WCU formulation and two ran-
dom seed trials for the DU optimization. Each optimization trial was run on
256 MPI processes (1 controller, 4 master, and 251 workers) with 24 threads each
on 128 2 x Intel Xeon Platinum 8160 (Skylake) nodes (48 cores per node) with
1,000 realizations per function evaluation — see schematic or realization sam-
188
pling for each type of search in Figure 5.5. A total of 500,000 function evalu-
ations were used for each trial, 125,000 per master. Our runs were evaluated
for their search solution quality by confirming that further search had minimal
hypervolume benefits (see Figure D.1). The resulting solutions were each re-
evaluated on nodes of the same type over 1,000 sets of hydrological/demand
series against 2,000 SOWs, resulting in 2,000 re-evaluation function evaluations
for each re-evaluated solution—see Figure 5.6, but instead of 3 SOWs, as in the
figure, 2,000 were used. All time series were generated using the methods de-
scribed in Section 2.2.
The ranges of values considered for the decision variables are presented in
Tables 5.3-5.5 and the  values (i.e., significant precision) for all objectives are
presented in Table 5.6. The MM-Borg MOEA was parameterized following prior
published recommendations Hadka and Reed (2014).
Table 5.3: Decision variables pertaining to the short-term drought mitigation
instruments: water-use restrictions, transfers, contingency fund and drought
insurance.
Decision Variable Lower Bound Upper Bound
Durham restriction ROF trigger 0% 100%
OWASA restriction ROF trigger 0% 100%
Raleigh restriction ROF trigger 0% 100%
Cary restriction ROF trigger 0% 100%
Durham transfer ROF trigger 0% 100%
OWASA transfer ROF trigger 0% 100%
Raleigh transfer ROF trigger 0% 100%
Continued on next page
189
Table 5.3 – continued from previous page
Decision Variable Lower Bound Upper Bound
OWASA Jordan Lake allocation 5% 47%
Raleigh Jordan Lake allocation 0% 37%
Durham Jordan Lake allocation 5% 42%
Cary Jordan Lake allocation 35% 72%
Durham annual contingency fund contri- 0% 10%
bution as percentage of annual revenue
OWASA annual contingency fund contri- 0% 10%
bution as percentage of annual revenue
Raleigh annual contingency fund contribu- 0% 10%
tion as percentage of annual revenue
Cary annual contingency fund contribu- 0% 10%
tion as percentage of annual revenue
Durham insurance ROF trigger 0% 100%
OWASA insurance ROF trigger 0% 100%
Raleigh insurance ROF trigger 0% 100%
Cary insurance ROF trigger 0% 100%
Durham insurance payment as percentage 0% 2%
of revenue
OWASA insurance payment as percentage 0% 2%
of revenue
Raleigh insurance payment as percentage 0% 2%
of revenue
Continued on next page
190
Table 5.3 – continued from previous page
Decision Variable Lower Bound Upper Bound
Cary insurance payment as percentage of 0% 2%
revenue
Table 5.4: Values of the long-term rof trigger and daily demands that function
as thresholds to trigger infrastructure construction by the utilities.
Decision Variable Lower Bound Upper Bound
Durham infrastructure construction long- 0% 100%
term ROF trigger
OWASA infrastructure construction long- 0% 100%
term ROF trigger
Raleigh infrastructure construction long- 0% 100%
term ROF trigger
Cary infrastructure construction long-term 0% 100%
ROF trigger
Daily demand Infrastructure Trigger for 0 100 MGD
Cary WTP Expansion 1
Daily demand Infrastructure Trigger for 0 100 MGD
Cary WTP Expansion 2
191
Table 5.5: The ordinal variables below determine the infrastructure construc-
tion order and adjusts them for the total number of options available to each
utility. The volumetric variable represents the volume of available storage
within Falls Lake to be re-allocated from the water quality to Raleigh’s mu-
nicipal supply pool.
Decision Variable Lower Upper
Bound Bound
University Lake expansion ranking 1st 22nd
Cane creek expansion ranking 1st 22nd
Stone quarry reservoir expansion shallow ranking 1st 22nd
Stone quarry reservoir expansion deep ranking 1st 22nd
Teer quarry expansion ranking 1st 22nd
Reclaimed water ranking low 1st 22nd
Reclaimed water high 1st 22nd
Lake Michie expansion ranking low 1st 22nd
Lake Michie expansion ranking high 1st 22nd
Little River reservoir ranking 1st 22nd
Richland creek quarry rank 1st 22nd
Neuse river intake rank 1st 22nd
Re-allocate Falls Lake rank 1st 22nd
Western Jordan Lake treatment plant rank OWASA low 1st 22nd
Western Jordan Lake treatment plant rank OWASA 1st 22nd
high
Western Jordan Lake treatment plant rank Durham low 1st 22nd
Continued on next page
192
Table 5.5 – continued from previous page
Decision Variable Lower Upper
Bound Bound
Western Jordan Lake treatment plant rank Durham 1st 22nd
high
Western Jordan Lake treatment plant rank Raleigh low 1st 22nd
Western Jordan Lake treatment plant rank Raleigh high 1st 22nd
Western Jordan Lake treatment plant OWASA fraction 1st 22nd
Western Jordan Lake treatment plant Durham fraction 1st 22nd
Western Jordan Lake treatment plant Raleigh fraction 1st 22nd
Volume to be re-allocated within Falls Lake for Raleigh 0MG 10,000MG
Table 5.6: Values used for -dominance.
Objective Reliability Restriction Infrastructure Financial Worse First Jordan
Frequency Net Present Cost Percentile Lake Allo-
Cost Cost cation
Value 0.2% 2% $10MM 2% of AR* 1% of AR* 2.5%
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5.6 Results
5.6.1 Comparing Tradeoffs After Re-evaluation
Our results compare the impacts of changing the SOW sampling strategies (Fig-
ure 5.7a) across the WCU and DU optimization formulations. The two core ad-
vantages of carefully tracking the impacts of including more uncertainties in the
multiobjective search are (1) an ability to distinguish how the tradeoffs for the
Research Triangle infrastructure investment and management pathways change
when stressed with more diverse scenarios for the future and (2) to confirm if
switching from the more traditional WCU to the approximate SOW sampling
used in the DU optimization significantly impacts the robustness attained by the
individual utilities as expected. The DU optimization’s approximate sampling
avoids the full computational cost of the comprehensive DU re-evaluation (Fig-
ure 5.6) during search. For a fair comparison, the resulting Pareto-approximate
infrastructure investment and management policies found with the WCU and
DU optimization schemes were each fully re-evaluated to assess their expected
tradeoffs and robustness using the more comprehensive suite of SOWs captured
by the DU re-evaluation ensemble.
Figure 5.7a shows the post-re-evaluation objective tradeoffs (panel “a”) and
the utilities relative robustness tradeoffs (panel “b”) for all policies obtained
through the DU and WCU optimization formulations. In Figure 5.7a, each axis
represents an objective, while each line represents a regional infrastructure in-
vestment and management policy. The points where each line (policy) intersects
the vertical axes represents the performance value of the metric corresponding
to that axis. In Figure 5.7a, the parallel axes plot is oriented such that the ideal
194
policy would be a horizontal line at the bottom of all axes.
Figure 5.7: Comparative performance analyses based on the DU re-
evaluation. Panel (a) illustrates the tradeoffs across the perfor-
mance objectives quantified either using the mean or the worse
first percentile across the tested SOWs. Panel (b) illustrates the
robustness tradeoffs across the four utilities (satisficing metric).
In both panels policies from the WCU formulation are brown
and those from the DU optimization are blue.
195
Figure 5.8: Robustness of the 100 top most robust policies attained from
the WCU and DU optimization schemes for each utility. Each
bar represents the attained robustness for one policy for each
utility. The policies are sorted by robustness (bar height) for
each utility. The policies with the highest value for the satis-
ficing robustness for each utility is highlighted each subplot.
One of the most robust policy for Cary called the “supplier-
preferred” policy while the most robust policies for Durham
and Raleigh were called “buyer-preferred” 1 and 2.
Figures 5.7 shows that the policies from DU optimization were on average
cheaper to operate and less infrastructure intensive than their WCU optimiza-
tion counterparts despite the comparable restriction frequency. The attained
values of the ROF triggers for the policies discovered using the DU optimiza-
tion better balance the reliability benefits and financial impacts of the short-term
drought mitigation, financial instruments, and infrastructure investments while
avoiding stranded infrastructure assets. Figure 5.7a shows that search under
deep uncertainty served to shift policies towards higher levels of performance.
196
The more diverse SOWs in the DU optimization appear to sufficiently stress
the management and investment policies to force them to more effectively use
the full suite of water supply portfolio options at the utilities’ disposal. Figure
5.7a also highlights strong tensions between reliability and restriction frequency
across the WCU and DU optimization formulations. Likewise, significant trade-
offs exist between Reliability and Total Financial Cost, meaning that if a utility
wants high reliability it has to pay for it through a contingency fund and/or
drought insurance. Lastly, Figure 5.7a highlights a tension between average an-
nual cost and annual worst-first-percentile cost, which is important as the util-
ities often struggle when confronted with significant variability in their annual
costs and are often willing to pay a premium for more predictable costs.
Transitioning to robustness tradeoffs, Figure 5.7b uses a similar parallel axes
plot to display the percentages of DU re-evaluation SOWs where each utility
meets their goal performance requirements: reliability > 99%, restriction fre-
quency < 20% and annual worst-first-percentile cost < 10%. In terms of ideal
performance for robustness in Figure 5.7b, the direction of preference is upward,
where the ideal policy would be represented to a line horizontally intersecting
each of the utilities robustness axes at their top maximum values (i.e., 100%).
Figure 5.7b shows that strong robustness tradeoffs exist between Raleigh, Cary,
and Durham. In contrast, OWASA attains close to maximum robustness for
almost all policies. The specific policies that represent the most robust poli-
cies for each utility are highlighted in different colors in Figure 5.7b. Figure
5.8 supplements Figure 5.7b by more directly comparing the utilities’ satisfic-
ing robustness for each of the policies found by the WCU and DU optimization
formulations. To clarify our analysis, we designate the most robust policies for
Durham and Raleigh, respectively, as buyer-preferred policies D/B1 and R/B2,
197
given their dependence on buying water transfers from Cary. The most robust
policy for Cary is designated the supplier-preferred policy, or C/S in Figures 5.7
and 5.8. Figure 5.7a shows that the most robust policy for Durham (D/B1) was
the only policy whose DU re-evaluation objectives met the performance crite-
ria established by the utilities: reliability > 99%, restriction frequency < 20%
and annual worst-first-percentile cost < 10%. Even though WCU optimization
found 246 Pareto approximate policies versus 112 from DU optimization, only
those policies attained by including deeply uncertain factors in search attained
both high levels of objective performance and robustness against deeply un-
certain scenarios (DU Re-evaluation). Also, the most robust management and
investment policies for each individual utility in the Research Triangle were ob-
tained through DU optimization.
Overall, Figures 5.7 and 5.8 show that the policies from DU optimization
were on average more reliable, less infrastructure intensive, financially cheaper
and less risky despite slightly higher restriction frequency than their WCU op-
timization counterparts when both sets of policies are exposed to broader un-
certainty. The policies resulting from the DU optimization formulation have
combinations of ROF triggers that allow the system to respond faster and more
cost-efficiently to drought events, avoiding costly stranded infrastructure and
high capital debt levels while maintaining reliability. Even if the reliability per-
formance requirement is relaxed to 97%, only 4 policies from WCU optimization
would meet the criteria versus 34 from DU optimization.
198
5.6.2 Scenario Discovery for Compromise Policies
Building on a better understanding of the performance tradeoffs and robust-
ness conflicts as illustrated in Figures 5.7 and 5.8, it is also important to under-
stand what deeply uncertain factors most strongly influence robustness. Figure
5.9 shows scenario discovery maps for Durham, Cary and Raleigh. Each map
shows the performance attained across sampled scenarios for the two most im-
portant uncertainties for each utility when the Research Triangle implements
one of the two buyer preferred policies highlighted in Figures 5.7b and 5.8. In
each panel of Figure 5.9, the two deep uncertain factors plotted on the horizon-
tal and vertical axes are the factors identified by the boosted trees algorithm
to be the most dominant in shaping robustness performance for a given policy.
Such dominance is quantified by the percentages in parentheses next to each
factor label, which indicate the decrease in impurity of the tree ensemble from
splits on that factor. As summarized in section 5.4.3, significant changes in the
percentage of impurities from tree-based splits of a factor represents the direct
reduction of classification errors in terms of which sampled SOWs attain success
or failure. High percentage changes in impurities consequently can be inter-
preted as a measure of utilities’ performance sensitivity to a given factor. Table
C in the Supplement Section C reports the impurity decreases (or sensitivities)
for all of the sampled factors for Durham, Cary and Raleigh under both buyer
preferred policies D/B1 and R/B2. Red regions in the factor maps of Figure
5.9 represent regions in which a utility will likely not meet its performance tar-
gets, grey regions represent high performance scenarios, and white regions are
deemed inconclusive due to close proximity mixtures of failures and successes
in a given region of the scenario space. In each of the six panels of Figure 5.9, the
stars represent the most favorable scenario (grey star), the least favorable (red),
199
and a projected future where demand growth rates in the Research Triangle re-
gion are 80% of the nominal values assumed in the WCU optimization values
(blue star).
Scenario Discovery for the Two Buyer Preferred Solutions D/B1 and R/B2
a) Durham b) Cary c) Raleigh
150% 150% 110%
125% 125% 103%
100% 100% 97%
75% 75% 90%
50% 100% 150% 200% 50% 100% 150% 200% 50% 100% 150% 200%
Demand growth Demand growth Demand growth
multiplier (47%) multiplier (100%) multiplier (51%)
d) e) f)
110% 120% 110%
103% 107% 103%
97% 93% 97%
90% 80% 90%
50% 100% 150% 200% 50% 100% 150% 200% 50% 100% 150% 200%
Demand growth Demand growth Demand growth
multiplier (54%) multiplier (49%) multiplier (45%)
Success Fail 80% of demandInconclusive Most Favorable Least Favorable growth projection
Figure 5.9: Factor maps for two policies — buyer preferred 1 (Durham)
and 2 (Raleigh) — showing the key factors controlling the ro-
bustness the solutions for all utilities but OWASA. Success re-
quired reliability ≥ 99%, restriction frequency ≤ 20% and an-
nual financial worst 1st percentile cost ≤ 10%. A list with the
combined impurity decreases for all factors can be found in Ta-
ble C.
200
Buyer Preferred 2 (R/B2) Buyer Preferred 1 (D/B1)
Restriction multip. Permitting time
Durham (25%) multiplier Low WTP (18%)
Evaporation rate
multiplier (25%) All others uncertainties (0%)
Restriction multip. Restriction multip.
Raleigh (19%) Raleigh (16%)
Figures 5.9a-5.9c show the scenario discovery maps obtained for Durham,
Cary and Raleigh, respectively, under Durham’s preferred policy D/B1. Figure
5.9a shows that Durham’s failures are mostly contingent on demand growth.
The same figure also shows that the D/B1 management and investment policy
effectively hedges Durham’s robustness against demand growth, requiring a
demand growth rate more than 50% over the nominal rate assumed by Durham
before failing to meet the utility’s performance requirements. However, if de-
mand growth rates are 50% to 70% above the projected values, Durham may
still meet the performance criteria depending on how many years the utilities
take to get the permits for the low capacity Joint WTP. Figure 5.9b shows that
Cary’s failures under policy D/B1 are contingent on extreme demand growth
rates exceeding their nominal projections by 85%. Cary’s failures are due to a
hard treatment capacity boundary where demand would surpass the maximum
upgraded treatment capacity for Cary’s own water treatment plant on the Jor-
dan Lake.
Figure 5.9c, on the other hand, shows that Raleigh has a far more complex
failure dynamic. Figure 5.9c shows that the most dominant factor influencing
Raleigh robustness is demand growth, though Raleigh’s performance does not
decrease monotonically as demand growth increases. Rather, Raleigh’s displays
poor performance at demand growths between 50% and 80% higher than the
nominal projected values, but at twice the projected values Raleigh meet the
performance criteria again. The reason for the stratification of failure scenarios
in Figure 5.9c is because scenarios in the red vertical strips emerge due to poor
coordination between infrastructure construction and the effective use of short-
term drought mitigation instruments. Figure 5.9c shows that in low demand
growth scenarios, Raleigh builds storage infrastructure slowly to avoid costly
201
stranded assets while supply risks are managed effectively by short-term miti-
gation instruments. On demand growth scenarios greater than Raleigh’s nom-
inal projections, Raleigh fails to meet the performance goals (red dots) when
new storage infrastructure is not built fast enough to keep up with demand
growth rate, straining the short-term drought mitigation instruments. In Figure
5.9c, the complex non-convex fingering of failure or inconclusive scenarios re-
sult when the D/B1 investment long-term ROF rules do not trigger sufficiently
rapid capacity expansion and the short-term ROFs are not capable of mitigating
the demand growth rates (i.e., cases of slow recognition of growing risks). The
complex non-linear fingering of the failures captured in several panels of Fig-
ure 5.9 are an interesting result that provides a general insight, as this behavior
could occur for any kind of decision-making metric based on thresholds, such
as the ROF or days-of-supply-remaining. Geometrically, this allows a utility to
more explicitly recognize the scenarios their investment and management rules
do not perform as wished.
Transitioning to the Raleigh preferred buyer preferred solution (R/B2) in
Figures 5.9d - 5.9f, the demand growth rate remains a dominant factor shap-
ing the robustness performance for all of the utilities. However, Figure 5.9d
shows that the effectiveness of Durham’s water use restrictions emerges as the
second most important factor influencing Durham’s performance. Switching to
the R/B2 investment and management policy yields an appreciable increase in
the number and complexity of failure scenarios for Durham. The underlying
mechanisms that yield the complex fingering of failure scenarios in Figure 5.9d
for Durham are similar to those for Raleigh under the D/B1 policy in Figure
5.9d: poor coordination between infrastructure construction and the short-term
use of drought mitigation instruments. For the R/B2 policy, Durham has far less
202
hedging to robustness failures and is specifically more sensitive to the effective-
ness of their populations response to restrictions.
Transitioning to Cary, Figure 5.9e shows that Cary’s failure region is far
larger under policy R/B2 than under D/B1. Interestingly, evaporation emerges
as an important uncertain factor for Cary instead of for Raleigh, despite Falls
lake (Raleigh’s main supply source) being rather shallow. The emergence of
evaporation as key concern for Cary would imply that longer term warming
from climate change could pose a severe challenge to the R/B2 policy. To our
knowledge, these results are the first to show the complex emergence of vastly
different failure modes and asymmetric sensitivities that can emerge for multi-
utility regionalization of investment and management pathways.
Lastly, Figure 5.9f shows that the R/B2 policy strongly hedges Raleigh while
degrading the robustness of both Cary and Durham as shown for the D/B1
policy. Regional demand growth rate and the response of Raleigh’s customers
to restrictions dominate its robustness performance. Figure 5.9f again shows
a complex fingering of failure scenarios as has been noted above where slow
recognition of the need to expand capacity when demand growth rates exceed
nominal projections by approximately 40%. Moderate losses in the effectiveness
of restrictions serves to compound Raleigh’s robustness failures. The consis-
tent importance of Raleigh’s water use restriction effectiveness across both the
D/B1 and R/B2 policies stresses the importance of Raleigh’s readiness to effec-
tively implement water-use restrictions and make sure their population will be
responsive.
Overall, Figure 5.9 provides some important regional insights for the Re-
search Triangle. The dominant impact of demand growth rates emphasizes
203
that supply reliability is harder to achieve than financial goals for this system.
Overall, keeping demand growth rates at or below the utilities’ projected value
would be an important step for all of the utilities to ensure satisfactory regional
performance. Furthermore, in most cases the reductions by 20% of the projected
growth (suggested by the utilities themselves) would significantly improve sat-
isfactory performance while also adding a safety margin that decreases the re-
gion’s need for new infrastructure.
Figure 5.9 also contributes some general insights regarding the value of us-
ing boosted trees to guide the scenario discovery process for complex infrastruc-
ture pathway problems. Boosted trees succeeded in capturing complex and dis-
tinctive features of the factor mapping plots without inherent overfitting. Most
non-linearities and fingering within complex transition zones between success-
ful and failed scenarios were identified by the boosted trees approach. In ad-
dition, the identification of inconclusive regions between passing and failing
scenario regions are helpful for avoiding errors in complex scenario discovery
contexts.
5.6.3 Policy Rules in the Robustness Compromises
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Figure 5.10: Parallel axis plots for the decision variables for the D/B1
and R/B2 buyer preferred policies. The red lines designate
Durham’s preferred buyer solution (D/B1). The green lines
designate Raleigh’s preferred buyer solution (R/B2).
Figure 5.10 provides a more detailed illustration of the underlying decisions
or rules that compose the Durham (D/B1) and Raleigh (R/B2) preferred poli-
cies. Recall that the investment and management policies are comprised of:
(1) ROF triggers for short-term mitigation instruments (Figures 5.10a, 5.10b,
and 5.10d) and for long-term infrastructure investments (Figure 5.10e), (2) al-
locations to the water supply pool in Jordan Lake allocation (Figure 5.10c),
(3) annual contingency fund contributions and insurance payouts defined as
percentages of annual revenue (Figure 5.10f and 5.10g), and (4) infrastructure
construction order (Figure 5.10h and 5.10i). In both the D/B1 and the R/B2
policies, there is significant similarity between Durham and Raleigh in their
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high reliance on restrictions (Figure 5.10a), on their prompt triggering of new
infrastructure by utilizing low long-term ROF triggers (Figure 5.10d), and in
their moderate-to-high use of contingency funds ((Figure 5.10f). Figure 5.10a
shows that Durham makes consistently high use of transfers under Durham’s
preferred policy (D/B1) by setting the transfer trigger value low. This in turn
requires a high annual contingency fund contribution as highlighted in Figure
5.10f. Figures 5.10g and 5.10e show that Durham uses small insurance payouts
in both policies at moderate and high ROFs, which emphasizes the preferred
strategy of relying dominantly on a contingency fund.
In comparing Raleigh versus Durham’s preferred policies, Figure 5.10b high-
lights competition for transfers. This partly explains the robustness conflicts
between Durham and Raleigh, given their shared use of limited conveyance ca-
pacity for transfers from Cary. In Durham’s preferred robustness compromise
policy (D/B1), Durham consistently uses the full capacity of the intra-utility
transfer pipes from 2015 until 2027. Alternatively, in Raleigh’s preferred so-
lution (R/B2), Durham’s limited use of transfers allowed Raleigh to request
medium-to-high volumes of transfers whenever needed (see supplementary
Figures B.1 to B.4). These differences are reflected in each utility’s approach
to their finances because although transfers are an effective way of decreasing
the need for restrictions, they can be costly. Under policy R/B2 Raleigh opts for
moderate payouts at moderate ROFs to hedge against the higher use of trans-
fers because of the lower associated ROF trigger (Figure 5.10b). In fact, Raleigh’s
sparse use of transfers under policy R/B2 warrant the use of drought insurance
in contrast to policy D/B1. Most insurance payouts for Raleigh under R/B2
happen during periods of high use of transfers (see supplementary Figures B.3
and B.4). Raleigh’s use of financial insurance in the R/B2 policy serves to reduce
206
the political risks and opportunity costs associated with a large and rarely used
contingency fund.
5.6.4 Infrastructure Pathway Ensemble
The Durham preferred policy (D/B1) and the Raleigh preferred policy (R/B2)
capture alternative strategies for navigating the performance and robustness
conflicts highlighted in Figures 5.7 and 5.8 and each lead to distinctly different
future pathways for the Research Triangle. Figure 5.11 contributes a detailed
analysis of the capacity expansions and ROF dynamics for Durham and Raleigh
that emerge from the D/B1 policy. The plots in Figure 5.11 focus on the resultant
capacity expansion pathways and ROF time series corresponding to the three
highlighted scenarios from Figure 5.9: the Most Favorable SOW (left column),
the SOW with 80% of the Nominal Demand Growth Rate (middle column),
and the Least Favorable SOW (right column) for Durham and Raleigh. Fig-
ures 5.11a - 5.11c plot Durham’s storage capacity and short-term ROF dynamics
for each of the three demand growth rate scenarios. Likewise, Figures 5.11d -
5.11f plot Durham’s resulting ensemble of infrastructure pathways across the
demand growth scenarios. Figures 5.11d - 5.11f show the infrastructure path-
ways for the same scenarios. The horizontal axes of each pathway panel repre-
sents time from 2015 to 2060, encompassing planning horizon of 45 years. All
of our illustrated infrastructure pathway plots display stacks of 1,000 horizon-
tal multicolored lines, each line corresponding to the investment sequences for
one hydrological realization. For each realization line, the colors represent the
infrastructure project that was built last in time. For example, the dark green
horizontal line segment close to the center of the plot indicates that in that real-
207
Figure 5.11: Capacity expansion pathways and ROF dynamics for the
Durham preferred policy DB/1. Demand growth rate sce-
narios considered include the Most Favorable SOW, the SOW
with demand growth rates reduced to 80% of their original
nominal values, and the Least Favorable SOW. All infrastruc-
ture investments are illustrated as stacked color lines desig-
nating specific infrastructure investments across 1,000 hydro-
logic scenarios.
208
ization indicates that Durham builds the Teer Quarry typically between 2040 to
2050.
An important concern captured in this study relates to the importance of
permitting times, which vary significantly across different infrastructure invest-
ments. In the pathways illustrated in Figure 5.11, the infrastructure construc-
tion order for a policy may not match the order in which infrastructure is built
over the course of a simulation due to permitting times. This issue frequently
emerged when considering investments in the Joint WTP for Durham under all
three scenarios (Figures 5.11d - 5.11f) and for Raleigh under Raleigh’s least fa-
vorable scenario for policy D/B1 (Figure 5.11l). The infrastructure construction
order for policy D/B1 in Figure 5.10h shows that the low capacity version of
the Joint WTP was ranked first and second in the infrastructure construction
queues of Raleigh and Durham under policy D/B1. However, the colors be-
tween the salmon (status-quo) and light or medium blue (low and high capacity
Joint WTP) in Figures 5.11d through 5.11f and 5.11l show that the Joint WTP was
not the first option built by either utility under the considered demand growth
rate and hydrologic scenarios. The reason for the apparent incongruence is that
the permitting and planning period of the Joint WTP is sufficiently long as to al-
low other options — namely Teer Quarry Reservoir, Low Capacity Lake Michie
expansion, Falls Lake reallocation and in a few cases Richland Creek quarry and
reclaimed water — to be built first by Durham and Raleigh.
A more detailed analysis of the pathways resulting from Durham’s preferred
policy D/B1 shows that the Joint WTP expansion and co-investment is impor-
tant and dominantly triggered by Durham. Figures 5.11 and B.1 shows for even
the Least Favorable SOW that Durham does not need to invest in any other
209
projects in the queue after either version of the Joint WTP is built. The water
treatment plant investment successfully reduces Durham’s long-term ROF to
0 after its construction (see supplementary Figures B.1 and B.2). Transitioning
to Raleigh, despite having to still build one more project in the Least Favor-
able SOW (Figure (5)l), the short-term ROF in Figure 5.11i noticeably decreases
with the construction of the Joint WTP. This reduces the pressure on Raleigh’s
short-term drought mitigation instruments. Despite the Joint WTP being a fairly
expensive project when compared to other infrastructure alternatives, its shared
cost makes it effective reliability- and financially-wise for both Durham and
Raleigh because under both policies both utilities have a substantial allocation
at Jordan Lake (see Figure 5.10c).
Aside from the Joint WTP, most of the focus of Durham and Raleigh un-
der policy D/B1 is on storage infrastructure. This conclusion follows from the
high prevalence of storage infrastructure seen in all of the pathways (Durham
— Figures 5.11d - 5.11f and Raleigh — Figures 5.11j - 5.11l). The investments
in storage come at the expense of investments in reclaimed water and intakes,
which are comparatively cheaper but not as effective at mitigating supply short-
falls. The only appearance of reclaimed water infrastructure in the pathways
occurred for Durham under their Least Favorable SOW (Figure 5.11f) largely
as a consequence of construction times for the other water sources. Lastly, it
is also interesting to observe that even in their Least Favorable SOW (Figure
5.11l), Raleigh goes through substantial storage capacity growth without build-
ing a single reservoir or reservoir expansion. Instead, increases in their supply
capacity result from direct access to its Jordan lake water supply pool via the
Joint WTP and through the Falls Lake’s municipal pool relocation, which are
cheaper and less disruptive projects than storage infrastructure.
210
Figure 5.12: Capacity expansion pathways and ROF dynamics for the
Durham preferred policy DB/1. Demand growth rate sce-
narios considered include the Most Favorable SOW, the SOW
with demand growth rates reduced to 80% of their original
nominal values, and the Least Favorable SOW. All infrastruc-
ture investments are illustrated as stacked color lines desig-
nating specific infrastructure investments across 1,000 hydro-
logic scenarios.
211
Transitioning to Raleigh’s preferred buyer policy R/B2, Figure 5.12 shows
the ROF dynamics and infrastructure pathways for Durham and Raleigh un-
der the same three demand growth rate scenarios displayed in Figure 5.11.
Although Figures 5.11 and 5.12 show many similarities in the resulting infras-
tructure investment pathways, key differences that favor Raleigh do emerge for
the R/B2 policy beyond the utilities competition for transfer water discussed
above. Under policy D/B1, as illustrated in Figure 5.11, it is always Durham
who triggers either version of the Joint WTP. However, for the policy R/B2 un-
der Raleigh’s Least Favorable SOW, the low capacity version of the Joint WTP
is triggered 60% of the time by Raleigh. This typically happens when Raleigh
finishes implementing the Falls lake reallocation and Durham has the low ca-
pacity version of the Lake Michie expansion under construction. The Joint WTP
remains as an effective investment to reduce the ROFs of both Durham and
Raleigh under the R/B2 policy.
Differences in the D/B1 and R/B2 policies are most pronounced for high de-
mand growth scenarios. Recall from our scenario discovery results in Figure 5.9
that short-term use and competition for water transfers yields strong differences
in Durham’s and Raleigh’s robustness (and failure mechanisms). This results
despite the similarity of the investment pathways across both policies shown in
Figures 5.11 and 5.12. This further stresses that sound infrastructure investment
strategies must be complemented by a sound short-term drought mitigation in-
struments for performance levels to be maintained under extraneous scenarios.
More specifically, effective use of restrictions and transfers strategies coupled
with financial instruments to hedge utilities serve to reduce and delay for years
the need for building new infrastructure projects. Still, despite a loss in per-
formance, under policy R/B2 a combination of an earlier Falls Lake reallocation
212
and effective use of transfers before the construction of the second infrastructure
option allowed for a more gradual system expansion, resulting in less new built
infrastructure (i.e., the Neuse River intake was not built and the final storage
capacity for the system was about 3,000 MG smaller).
Despite the mentioned differences in the pathways between both solutions
shown in Figures 5.11 and 5.12, their similarities (similar timings and op-
tions built) imply general insights for the Research Triangle region. Short-term
drought mitigation instruments are critical to the evolution of the system’s in-
frastructure investment pathways. Regionally coordinated strategies for man-
aging demand growth rates are an important risk hedge for all four of the Re-
search Triangle utilities. Overall, an important take-away point from the path-
ways plots for Raleigh and Durham is that the Teer Quarry and the low-capacity
Joint WTP projects are very likely to be needed in the foreseeable future. There-
fore, Durham, Raleigh and Cary should include the Joint WTP in their bud-
geting process, perform more detailed design analysis and contemplate their
permitting requirements. The main recommendation for Raleigh would be to
initiate an application for a reallocation of the Falls Lake’s water quality pool to
its own supply pool. More broadly, DU optimization significantly reshaped sys-
tem objectives tradeoffs, robustness, scenario discovery results, portfolio mix of
decisions, and the corresponding pathways. The analyses framework as illus-
trated in our results provides a rich context for an understanding of the system.
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5.7 Conclusion
This work contributes the DU Pathways framework, which includes deep un-
certainty in the search phase of the infrastructure pathways problem. In the
context of water resources systems, this work demonstrates the value of inte-
grated regional infrastructure planning and management coupled with deep
uncertainty analysis for mitigating drought risks and their corresponding fi-
nancial impacts. Such risks are even more prevalent in areas of high population
growth, such as the Research Triangle region in North Carolina. The Research
Triangle regional system demonstrates how performance and robustness con-
flicts can emerge between geographically close utilities in their long planning.
The DU Pathways framework’s flexibility in incorporating and exploring
uncertainties as demonstrated in this study highlights some general insights
and benefits for regional water utilities seeking to coordinate their short-term
management actions (weekly) and long-term (annual) investment decisions. A
key benefit is the ability to create more flexibility and reduce required infrastruc-
ture debt burden by carefully balancing short-term decision making and long-
term infrastructure investment decisions. The consistent integration of both
management and investment decisions using ROF-based rules provides oper-
ators with ways to account for information beyond measurements of reservoir
levels and past data, allowing for decisions based on possible future conditions
as well. Another major benefit is the possibility of performing uncertainty anal-
ysis on the effectiveness of alternative infrastructure investment pathways for
individual utilities as well as broader cooperating regional coalitions. This anal-
ysis not only informs utilities about what uncertainties to monitor when plan-
ning the early studies preceding the triggering of new infrastructure, but also
214
about what investments would be more effective as the results from monitoring
emerge. Lastly, the DU Pathways framework allowed for a clear understanding
of strong interdependencies between short-term drought mitigation and finan-
cial instruments as well as their potential to determine the proper timing of in-
vestments in new infrastructure. The DU Pathways framework centralizes and
promotes all these benefits for single and regional water utility infrastructure
planning and management.
The inclusion of deep uncertainties in the search-based identification water
supply management and investment portfolios yielded pathway policies that
are more robust (according to performance criteria defined by the utilities them-
selves) and that also provide better performance and robustness compromises
between the four utilities in the Research Triangle Region. The scenario dis-
covery analysis showed that the key driver of system performance and robust-
ness is demand growth, which utilities can act on through coordinated demand
management. The infrastructure pathways analysis also showed that demand
growth may determine whether specific investments in infrastructure will be
needed by a given point in time (should the right conditions be observed), how
key uncertainties may drive the need for investments to be sooner or later, or if
some candidate projects would never be prioritized at all even under the most
adverse futures. This information is crucial for utilities to design their rate struc-
ture to accommodate for debt servicing for the maintenance of their credit rating
when issuing the required bonds. Future work could consider a wider variety
of drought mitigation instruments, new strategies for sampling and potentially
screening deep uncertainties, and short-term drought mitigation policies that
adapt their triggers with the construction of new infrastructure (a meta-policy,
so to speak). The insights of this work have broad merit for water utilities facing
215
growing pressures to better coordinate management and investment decisions
to confront resource contention, growing urban demands, and increasingly ex-
treme droughts.
216
CHAPTER 6
CONTRIBUTIONS AND FUTURE WORK
6.1 Contributions and Conclusions
As discussed in Chapters 3 and 4, water utilities are increasingly challenged
to balance the burdens from required new investments in infrastructure with
innovating their management of existing resources to confront pressures from
growing populations, changes in climate, evolving environmental regulations,
growing financial risks, and many other factors. This dissertation seeks to aid
utilities in confronting these challenges through three key contributions. First,
the Deep Uncertainty (DU) Pathways framework formalizes uncertainty-driven
infrastructure planning and management by bridging stochastic adaptive in-
frastructure pathways and multi-objective robust decision making (MORDM).
Secondly, this dissertation contributes the open source WaterPaths model for
simulating planning and management decision making by water utilities and
designed to enable the seamless application of the DU Pathways approach to
any single or regionally coordinated group of water utilities. The third major
contribution of this dissertation is the Sedento Valley hypothetical test case that
can serve as a shared training and diagnostic test bed to promote future inno-
vations in water infrastructure planning and management frameworks. These
three contributions were motivated under the unifying and specific goal of im-
proving decision-making and policy design for water-infrastructure systems
with multiple stakeholders and conflicting objectives. These contributions are
discussed in more detail below.
(1) Enabling a more sophisticated incorporation of uncertainty in infrastructure invest-
217
ment pathways that are carefully coordinated with short-term water portfolio manage-
ment instruments yields regional actions that are more robust and financially stable.
Given the long lives, high costs, and irreversibility of investments in pub-
lic infrastructure assets, decisions to pursue specific options require careful
thought to assure the population’s needs are met and that the new asset does
not become stranded (Grant et al., 2013). These risks can be mitigated by (1) per-
forming careful uncertainty analysis (Nair and Howlett, 2017), (2) by presenting
all stakeholders with multiple options, all in accordance with the specific insti-
tutional backgrounds, to foster a high-level political debate (Bosomworth et al.,
2017), by (3) using short-term management strategies tailored to the resulting
infrastructure system and vice-versa (Zeff et al., 2016), and by (4) using regional
planning including various municipalities impacted and benefitted by the as-
set. This dissertation contributes the DU Pathways framework, an extension
the MORDM framework that includes strategies for sampling deeply uncertain
factors during the computational policy search as a way of assuring the dis-
covery of regional infrastructure planning and management policies that are
robust to unforeseen conditions (Chapters 3 and 5). To improve compliance
with the timelines of utilities’ operations and local political processes, the DU
Pathways framework simultaneously employs weekly and annual rule-based
decision-making based on the ROF metric for decisions regarding short-term
drought management and infrastructure planning, respectively.
This dissertation has broadened the scope of uncertainties that can be sam-
pled during automated infrastructure planning and management policy search
as well as the robustness assessments of the discovered policies. Chapter 3 in-
troduced the concept of DU uncertainty sampling and used it to find a drought-
218
management policy for the Research Triangle test case. Such policies have
higher robustness to deeply uncertain factors for all stakeholders, fewer con-
flicts over robustness tradeoffs among stakeholders, and that nonetheless ex-
ceeds performance expectations under projected values for all analyzed deeply
uncertain factors without being explicitly optimized to do so. Chapter 5 built
on these findings to extend the framework proposed by (Zeff et al., 2016) and
achieve the goal of finding infrastructure planning and management policies for
the Research Triangle test case that are robust and adaptable to a wide variety
of uncertainties, minimizing the risk of stranded-assets and capacity shortfalls.
Furthermore, the resulting policies are designed to work with drought miti-
gation measures trusted by utilities and accepted by the public, as well as to
respect the nature and pace of local political and institutional processes. In ad-
dition, the Boosted-Trees-based scenario discovery methodology introduced in
Chapter 5 filled the last gap regarding the applicability of scenario discovery
theory, namely that in which the uncertainty-performance relationships can-
not be fully explained by models with pre-specified functional forms. By in-
corporating uncertainty sampling in all stages of the MORDM framework and
improving later uncertainty analysis, this dissertation seeks to promote water-
infrastructure supply and financial modeling under uncertainty as a way to plan
and operate water infrastructure with greater supply reliability and financial
stability.
(2) The WaterPaths model addresses limitations in existing software for modeling water
infrastructure by incorporating stochastic uncertainty analysis, infrastructure expan-
sions, and regional infrastructure planning and management.
Despite the demonstrated benefits of the DU Pathways framework for the
219
formulation of regional water infrastructure planning and management poli-
cies, performing all its steps with existing simulations models would not be
possible because the existing simulation tools lack needed features. More specif-
ically, the existing models are not designed for stochastic uncertainty anal-
ysis or for evaluating broad categories of objectives that bridge water sup-
ply reliability as well as financial concerns. Existing simulation frameworks
lack out-of-the-box integration with multiobjective optimization algorithms and
would not be capable of representing diverse infrastructure investment or wa-
ter portfolio management actions. These concerns are particularly acute for
multi-stakeholder regionally coordinated operations and planning and for joint
supply and financial modeling. Lastly, large scale computational search or
ensemble-based exploratory modeling would not be tractable with current sim-
ulation frameworks as they lack functionality on cloud and high-performance-
computing platforms and often do not provide open-source access to their un-
derlying codes. Chapter 4 demonstrates how WaterPaths fills these capability
gaps with a risk-based planning and management methodology, while Chap-
ter 5 shows its first successful real-world application in the Research Triangle,
NC. WaterPaths offers regions with interconnected utilities the possibility of
a full DU Pathways analysis with minimal model setup time. Furthermore,
as an open-source code WaterPaths is designed for ease of modification and
extension. The simulation framework provides researchers and practitioners
with a test bed for novel drought mitigation instruments, types of infrastructure
projects, decision-making metrics, and uncertainty sampling techniques. Cur-
rently, multiple researchers are employing and extending WaterPaths to study
systems in the US, Brazil, and South Korea.
(3) The Sedento Valley test case contributes a highly realistic and challenging hypothet-
220
ical system that can support educational training efforts, algorithmic benchmarking for
emerging search or scenario discovery algorithm innovations, for testing the efficacy of
new water management and planning methodologies.
In Chapter 4 the contribution of the Sedento Valley test case as a new generic
multi-actor test bed for regional water supply planning has broad value for dis-
seminating and extending the innovations presented in this dissertation. The
model represents three hypothetical water utilities in the South Eastern US fac-
ing the prospect of water shortage due to growing demand and changing cli-
mate. The utilities seek to craft both short-term drought mitigation responses
and long-term infrastructure sequencing pathways that maintain reliable sup-
ply and financial stability. The utilities have the potential to cooperate using
treated transfers and through shared infrastructure development. The Sedento
Valley test case was designed to be a realistic but computationally tractable tool
for the direct comparison of competing regional-decision-making frameworks,
for the design and comparison of drought mitigation strategies, and for the eval-
uation of decision-making metrics. While care was taken to limit computational
demands and maintain clear interpretability, the Sedento Valley test case main-
tains a high degree of complexity due to the number of utilities involved, to the
unbalanced distribution of resources and liabilities across utilities, and the util-
ities locations in relation to each other in the local shared river basins. The code
repository provided with WaterPaths in Chapter 4 contains all the data needed
to reproduce the test case in WaterPaths or other frameworks and to compare
methodologies and policy instruments against what was presented in Chapters
5 and 4.
221
6.2 Future Work
This dissertation provides a foundational basis for the study of several chal-
lenges in water systems planning. This section highlights four suggested exten-
sions: strategies for deep- and well-characterized-uncertainty sampling, metrics
for long- and short-term decision making for water utilities, design of new fi-
nancing and insurance instruments for water infrastructure operators, and con-
flict resolution among resource-sharing water utilities.
6.2.1 Strategies for Deep- and Well-Characterized-Uncertainty
Sampling
Uncertainty factors are called deep when there is no explanatory probability dis-
tribution accepted by the various parties to a decision (Kwakkel et al., 2016b).
Innovations in exploratory modeling frameworks that address deeply uncer-
tainty factors and their impacts on a water infrastructure systems should focus
on more efficient computational sampling strategies to inform, the calculation of
robustness metrics or stochastic objectives. Computational parallelization and
careful data management workflows would be highly beneficial for expand-
ing analysts ability to parse through large multi-objective solution sets of plan-
ning policies and chose a limited number of them for scenario discovery, as per-
formed in Chapters 3, 4 and 5 of this dissertation. The calculation of such robust-
ness metrics and objectives requires in turn that values for deeply-uncertainty
factors be assigned a distribution, and this choice may potentially impact the se-
lection of a policy for scenario discovery (Reis and Shortridge, 2019). However,
222
the topic of deep-uncertainty sampling has received little attention despite its
potential to significantly impact decision making.
WaterPaths provides the ideal platform for the study of new workflows
and new innovative stochastic sampling techniques when evaluating objec-
tives and system robustness. Its stochastic design for objective calculations and
batch mode provide an easy way to run sampling studies without the need
for codewrappers or other workarounds as is typical current models to run
all needed simulations. Furthermore, researchers interested in the topic would
benefit from the Sedento Valley test case, which is already implemented in Wa-
terPaths and has been extensively analyzed in this dissertation with the stan-
dard uniform-distribution approach.
6.2.2 Metrics for Long- and Short-Term Decision Making for
Water Utilities
Various studies have analyzed metrics to inform short-term decision making by
water utilities, such as the days-of-supply-remaining metric (Fisher, 1993; Fisher
and Palmer, 1997) and the ROF metric exploited in this dissertation (Palmer and
Characklis, 2009; Zeff et al., 2014) However, the field would benefit from a uni-
fied study of metrics for decision making. Researchers interested in the topic
could extend WaterPaths to include a variety of decision-making metrics and
perform MORDM analyses for the Sedento Valley test case with different met-
rics. The MORDM analyses could be followed by a detailed analysis of time
series of system states for a granular understanding of how each metric affects
decision making over time and for different conditions for water utilities. Such
223
study would have great practical impact on the field of water infrastructure
planning and management, demonstrating to water utilities the value of poten-
tially switching metrics used for their operations and planning.
6.2.3 Design of New Financing Mechanisms and Insurance In-
struments for Water Infrastructure Operators
As shown in Chapters 3 and 5 and in other works (Zeff et al., 2014, 2016; Herman
et al., 2014), analyzing the financial component of water infrastructure planning
and management is as important as the supply component to water utilities. In-
creasing pressure on water utilities to operate on ever smaller margins, the need
for infrastructure refurbishment and capacity expansions (AWWA, 2019), and of
credit rating standards for water utilities (Moody’s, 2017, 2019), the municipal
water sector would benefit from a continued effort to design innovative finan-
cial instruments and mechanisms for structuring dept. Although there are re-
cent studies that are more directly engaging with the challenges associated with
designing financial insurance programs for water utilities (Baum et al., 2018;
Zeff and Characklis, 2013) and the existing Water Infrastructure Finance and
Innovation Act (WIFIA) program for financing water infrastructure (Copeland,
2016). The area still appears to be in its technical infancy and is not at present ca-
pable of effectively addressing the global and US needs for massive investments
in water infrastructure as well as sustaining the systems with well-funded op-
erations and maintenance programs (Hallegatte et al., 2019). WaterPaths can
be an effective tool for testing financial instruments and debt structure. It can
be used by researchers and single or groups of municipalities to test general or
224
tailored financial instruments and debt structure before committing to them in
practice. Such proof-of-concepts may encourage case-specific implementation
of such financial strategies, allowing for cost savings and improved financial
stability across the municipal water industry.
6.2.4 Conflict Resolution Among Resource-Sharing Water Util-
ities
The global densification of urban systems means that water utilities that were
once isolated are becoming more dependent on water resources shared with
neighboring utilities. In such an environment, water transfers allow a group of
water utilities to share capacity in ways that are more efficient than relying on
independent sources (Vaux Jr and Howitt, 1984; Lund and Israel, 1995; Green
and Hamilton, 2000; Olmstead, 2010). The trend of cooperation among utili-
ties has also been encouraging the joint development and operation of shared
infrastructure (see Zeff et al. 2016; Triangle J 2014 and Chapter 5). Such use
and development of resources shared by multiple municipalities is explored in
Chapters 3 and 5 of this dissertation and in other works (Zeff et al., 2014; Gore-
lick et al., 2019; Caldwell and Characklis, 2014; Borgomeo et al., 2018), although
with a major simplification: the assumption that perfect contracts will be cre-
ated and followed through the lifespan of a project. Gold et al. (2019) take a
promising step by exploring how deviations in implementation of a plan can
fundamentally shift perceived tradeoffs and robustness. They touch on the po-
tential for game theory informed strategies for navigating the resulting conflicts
among utilities in the Research Triangle region. Overall the body literature on
225
the topic the stability of regional coalitions of utilities with conflicting interests
and increasingly severe deeply uncertain stressors is underdeveloped. Water-
Paths provides interested researchers with the unique ability to jointly simulate
the behavior of multiple utilities in a regional system that can be abstracted with
a broad array policy rule behaviors in highly dynamic and stochastic contexts
(e.g., stochastic dynamic mechanism design or evolutionary dynamic games).
WaterPaths can be used for the design of contracts to simulate the effects of util-
ities breaching or opting out of them, allowing for the design of policies with
more safeguards and in more agreement with local institutional and political
processes. These extensions would provide a deep and rich area of research
that could be of immediate value for aiding regions globally that are straining
to maintain the reliability and financial stability of their water supply services.
226
APPENDIX A
NOTATION
Symbol Description
ATR annual total revenue
CF total amount of money in the contingency fund
D actual (restricted) demands
Dw actual water demand (as opposed to unrestricted) on
week w
DS matrix with the streams of debt service payments for
all utilities
E evaporation series matrix
f joint-pdf of inflows, evaporation rates, and de-
mand fluctuations around annual means, all well-
characterized uncertainties
F vector-values objective function
fFC financial cost of drought mitigation instruments plus
debt repayment
fJLA combined Jordan Lake allocation objective
fNPV net present cost of infrastructure
fREL supply reliability objective
fRF restriction frequency
fWFPC worse-first-percentile of the financial-cost objective
·h historical data used to derive the synthetic series
H ensemble model
ht model with index t of the ensemble
·i index of a deeply uncertain factor
227
ICO matrix with vectors with the infrastructure construc-
tion ordering for each utility
l sets of lower bounds for the deeply uncertain factors
IPOw payout received by a utility in a given week
IPR insurance loading (1.2 for this work, representing a
loading of 20%)
Nhr number years in the historical record
NI natural inflows matrix
Nrof number of one-year long simulations used to calculate
the ROF metric
Nf total number of deeply uncertain factors
Nr number of synthetic series samples
Nw total number of weeks in the full planning horizon or
number of weeks in a year
OIy vector with the IDs of all online supply infrastructure
POy insurance payout fixed on year’s y and insurance con-
tract
R temporal auto-correlation in the historical natural in-
flows and evaporation time series
·s denotes synthetic time series
S cross-site correlation matrix
·t index of model of boosting multiplier or demand tier
(type of consumer)
T total number of models in the ensemble
TFw volume of treated water to be transferred in week w
228
u sets of upper bounds for the deeply uncertain factors
UD matrix with unrestricted demands
UWP unrestricted water prices
WP restricted water prices
x vector of explanatory variables
X time-varying state matrix across all of the utilities
xlrof long-term ROF values
xlrof vector of long-term ROFs for all utilities
xsrof short-term ROF values
·w week number
·y year number
αt boosting multiplier corresponding to model ht
δ(·) Dirac (delta) function
θacfc vector of annual contingency fund contributions,
which are percentages off annual revenue saved in a
utility’s drought mitigation fund.
θins ROF trigger for the index insurance
θipo percent of the previous year’s revenue used to calcu-
late the insurance payout
θirt vector of insurance restriction triggers
θit vector of long-term-ROF infrastructure construction
triggers
θjla vector of Jordan Lake Allocations
θrt vector of restriction triggers
θtt vector of transfer triggers
229
θirof vector of long-term ROF triggers for all utilities
θ∗ Pareto optimal management and investment policy
ΨDU matrix of DU vector samples of the deeply uncertain
variables of concern where each element corresponds
to a specific uncertainty.
Ψs matrix of vector samples of the deeply uncertain vari-
ables of concern where each element corresponds to a
specific uncertainty.
ΨWCU matrix of WCU vector samples of the deeply uncertain
variables of concern where each element corresponds
to a specific uncertainty.
D(·) assumed pdf for the deeply uncertain factors
U uniform distribution
X vector of best projections for the deeply uncertain fac-
tors considered in the problem
Table A.1: List of symbols.
230
APPENDIX B
TIME-SERIES OUTPUT FOR COMPROMISE SOLUTIONS UNDER
SELECTED SCENARIOS
Figure B.1: System dynamics for Durham and Raleigh under policy D/B1
for multiple hydrologic realizations under Durham’s most un-
favorable scenario. Transparent lines indicate that few real-
izations had similar dynamics and the horizontal colored lines
represent ROF trigger values for the corresponding policy.
231
Figure B.2: System dynamics for Durham and Raleigh under policy D/B1
for multiple hydrologic realizations under Raleigh’s most un-
favorable scenario. Transparent lines indicate that few real-
izations had similar dynamics and the horizontal colored lines
represent ROF trigger values for the corresponding policy.
232
Figure B.3: System dynamics for Durham and Raleigh under policy R/B2
for multiple hydrologic realizations under Durham’s most un-
favorable scenario. Transparent lines indicate that few real-
izations had similar dynamics and the horizontal colored lines
represent ROF trigger values for the corresponding policy.
233
Figure B.4: System dynamics for Durham and Raleigh under policy R/B2
for multiple hydrologic realizations under Raleigh’s most un-
favorable scenario. Transparent lines indicate that few real-
izations had similar dynamics and the horizontal colored lines
represent ROF trigger values for the corresponding policy.
234
APPENDIX C
DEEPLY UNCERTAIN FACTOR SENSITIVITIES
D/B1 R/B2
Uncertainty Durham Cary Raleigh Durham Cary Raleigh
Regional Demand 47 % 100 % 51 % 53 % 49 % 45 %
Growth Multiplier
Regional Bond Interest 1 % 0 % 0 % 0 % 1 % 1 %
Rate Multiplier
Regional Bond Term Mul- 0 % 0 % 0 % 0 % 0 % 0 %
tiplier
Regional Discount Rate 1 % 0 % 0 % 0 % 1 % 1 %
Multiplier
Restriction Efficacy Mul- 1 % 0 % 0 % 2 % 0 % 0 %
tiplier OWASA
Restriction Efficacy Mul- 14 % 0 % 0 % 25 % 0 % 0 %
tiplier Durham
Restriction Efficacy Mul- 0 % 0 % 1 % 0 % 0 % 0 %
tiplier Cary
Restriction Efficacy Mul- 1 % 0 % 16 % 0 % 0 % 19 %
tiplier Raleigh
Regional Evaporation 1 % 0 % 1 % 0 % 25 % 1 %
Rate Multiplier
Permitting Time Low 18 % 0 % 10 % 0 % 4 % 3 %
WJLWTP
Construction Time Multi- 2 % 0 % 1 % 1 % 1 % 0 %
plier Low WJLWTP
Permitting Time High 0 % 0 % 0 % 1 % 0 % 3 %
WJLWTP
Construction Time Multi- 1 % 0 % 0 % 0 % 0 % 1 %
plier Low WJLWTP
Table C.1: Percent of the combined impurity the the entire boosted trees ensemble due
to each uncertain factor for Durham, Cary and Raleigh under both buyer pre-
ferred policies D/B1 and R/B2.
235
APPENDIX D
COMPUTATIONAL EXPERIMENT HYPERVOLUME PLOTS
Figure D.1: Evolution of hypervolume for each master of each of two seeds
ran for the WCU and DU searches. Since each seed had 4 is-
land and each island ran 125,000 function evaluations, the to-
tal number of function evaluation for each of the 2 WCU and
2 DU seeds was 500,000. WCU seeds advanced faster at early
NFE and converged slighly faster than the DU seeds, although
the long plateaus at later NFE show that all seeds were given
more than enough NFE to find the best Pareto approximate set
of policies.
236
APPENDIX E
THE BOOSTED TREES ALGORITHM
E.1 Classification and Regression Trees
Classification and Regression Trees (CART) works by orthogonally dividing a
space into regions and attaching a label or number to all points inside each re-
gion (Breiman et al., 1984). Each tree may sequentially divide the space multiple
times, with each split being the one that minimizes the classification error on the
overal tree at the time it is found. Take as an example the tree three shown in
Figure E.2a, where the blue crosses represent a success scenario and a red circle
represents a failure. The first split to be found, the one that results in the high-
est number of correctly-classified points among all possible orthogonal splits on
both axes, would be the horizontal. The second split, a vertical one, would be
found next and branching off of the horizontal split, the third (also vertical) off
of the second (vertical as well), and so on.
Figure E.1: Classification tree fit on illustrative data set. The first split to
be foudn was the horizontal split, which applies to the entire
data set. The three vertical splits were foudn branching off of
the horizontal split and then off of each other.
237
More previsely, the ideal split at each iteration for a classification problem
(e.g., scenario discovery) can be found by checking all possible splits for the one
that minimizes one of a variety of metrics, such as the Gini impurity calculated
for a tree (Breiman et al., 1984), shown in Equation E.3:
|Sk|
pk = ← fraction of scenarios i∑n leaf Sn with success/fail label k (E.1)|S|
G(Sn) = pk(1− pk) (E.2)
k={success, fail}
with the Gini impurity after the split being:
|S1| |S2| |SN |
GT (S) = GT (S1) + G
T (S2) + ...+ G
T (SN) (E.3)|S| |S| |S|
for data set S, Sn ⊆ S, Sn,k ⊆ Sn where Sn,k = {(x, y) ∈ Sn : y = k} (k being
either success or fail), and S = Ssuccess∪Sfail. In Equations E.1-E.3, Sn is the data
in a leaf, pk is the fraction of points belonging to class k within leaf n, G(Sn) is
the Gini impurity within a leaf, GT (S) is the Gini impurity for the entire tree.
The change in the total impurity across a tree due to all splits on a particular
uncertain factor (dependent variable) is also an indicator of how sensitive to
that factor performance is.
E.2 General Idea of Booosting
The idea of boosting is to train an ensemble of shallow classification trees (or any
weak classifier), each of which an expert in a specific region of the uncertainty
space. The ensemble of trees has the form below:
∑T
H(x) = αtht(x) (E.4)
t=1
238
where H is the ensemble of trees, αt is the weight assigned to weak classifier ht
around samples x at iteration t, and T is the number of trees in the ensemble.
Boosting creates such an ensemble in a similar fashion to gradient descent.
However, instead of:
xt+1 = xt − α∇`(xt) (E.5)
as in gradient descent in real space, where ` is a loss function, Boosting is trained
via gradient descent in functional space, so that:
Ht+1(X) = Ht(X) + αt∇`(ht(X)) (E.6)
The question then becomes how to find the αt’s and the trees ht. Before
answering these questions, we should get a geometric intuition for Boosting
first.
E.3 Geometric Intuition
Figure E.2a shows a set of blue crosses and red circles denoting success and fail-
ure scenarios, respectively, we would like our ensemble of classification trees
with depth of one split (hereafter called a tree stump) to correctly classify. For
simplicity, let’s assume all trees will have the same weight αt in the final ensem-
ble.
The first tree stump, a horizontal divide in panel “b,” classified ten out of
thirteen points correctly but failed to classify the remaining three. Since it incor-
rectly classified a few points in the last attempt, we would like the next classifier
correctly classify these points. To increase the chances of that being the case,
239
Figure E.2: Sequence of addition of single-split classification trees until all
points are correctly classified. Panel (a) shows just the data,
without trees, panel (b) shows data with one tree, panel (c)
with two, and panel (d) with three. Variables x1 and x2 are
two uncertain factors, the blue crosses represent a success sce-
nario and a red circle represents a failure, and the blue and red
regions correspond to regions classified as success or failure
regions, with intermediate colors indicating uncertainty in the
classification.
boosting will increase the weight w of the points that were misclassified earlier
before training the new tree stump. The second tree stump, a vertical line on
panel “c,” correctly classifies the two blue crosses that were originally incor-
rectly classified, although it incorrectly classifies three crosses that were origi-
nally correctly classified. For the third classifier, Boosting will now increase the
weight of the three bottom misclassified crosses as well of the other misclassified
two crosses and circle because they are still not correctly classified – technically
speaking they are tied, each of the two classifiers classifies them in a different
way, but we are here considering this a wrong classification. The third iteration
will then prioritize correcting the high-weight points again, and will end up as
the vertical line on the right of panel “d.” Now, all points are correctly classified.
240
There are different ways to mathematically approach Boosting. But before
getting to Boosting, it is a good idea to go over gradient descent, which is a
basis of Boosting. Following that, the AdaBoost algorithm will be presented,
which is a boosting algorithm that assumes an exponential loss function.
E.4 Minimizing a Loss Function
Gradient Descent in Functional Space
In real-space gradient descent function minimization algorithm, the goal is to
start from an initial guess x0 and iteratively find the value of x corresponding
to the minimum value of f(x), which in machine learning is a loss function,
henceforth called `(x). Gradient descent does that by moving one step s at a
time starting from x0 in the direction of the steeped downhill slope at location
xt, the value of x at the tth iteration (Murphy, 2012; Hastie et al., 2009).
` [x t Tt+1 − αg(xt)] ≈ `(x )− αg(xt) g(xt) (E.7)
where α, called the learning rate, must be positive and can be set as a fixed
parameter. The dot product on the last term g(x )Tt g(xt) will also always be
positive, which means that the loss should always decrease.
In gradient descent for functional space, x is a function instead of a real num-
ber. This means that the loss function `(·) is a function of a function, say `(H(x))
instead of a real number x. By analogy, gradient descent in functional space ap-
plied to classification trees works by adding trees to an ever growing ensemble
of trees. Using the definition of a functional gradient, which is beyond the scope
241
of the this post, this leads us to (Murphy, 2012):
`(H + αh) ≈ `(H) + α<∇`(H), h> (E.8)
where H is an ensemble of trees, h is a single tree, and the <f, g> notation
denotes a dot product between f and g. Gradient descent in function space is
an important component of Boosting. Based on that, the next section will talk
about AdaBoost, the Boosting algorithm used in this study.
E.5 AdaBoost
Basic Definitions
AdaBoost is a Boosting algorithm, whose goal is to find an ensemble function
H of weak classifiers h, here shallow classification trees, that minimize an expo-
nential loss function below for a binary classification problem (Murphy, 2012):
∑n
`(H) = e−y(xi)H(xi) (E.9)
i=1
where x , y(x ) is the ithi i data point in the training set. The step size α can be
interpreted as the weight of each classifier in the ensemble, which optimized
for each function h added to the ensemble. The uncertain factors independent
variables x have corresponding vector of dependent variables y(xi), in which
each y(xi) ∈ {−1, 1} is a vector with the classification of each point, with -1
and 1 representing the two classes to which a point may belong (say, -1 for red
circles and 1 for blue crosses). The weak classifiers h in AdaBoost also return
h(x) ∈ {−1, 1}.
242
The weight αt of each weak classifier h is calculated as:
1 1− 
α = ln (E.10)
2 
where  is the classification error of weak classifier ht. The error  for each itera-
tion is: ∑
h(xi) = argminh wi (E.11)
i:h(xi)6=y(xi)
which is the summation of the weights of the points misclassified by h(xi)t –
e.g., in panel “b” the error would be summation the of the weights of the two
crosses on the upper left corner and of the circle at the bottom right corner. Now
let’s get to these weights.
There are multiple ways we can think of for setting the weights of each data
point at each iteration, corresponding to the sizes of the points in Figure E.2.
The way presented in Schapire (1990) is a common choice:
e−√αtht(x)y(x)wt+1 = wt (E.12)
2 (1− )
Algorithm 1 shows a pseudo-code of AdaBoost.
243
Algorithm 1: AdaBoost pseudo-code.
Input : (X , y), H (classification tree)
Output: Ensemble classifier H(x)
1 H=0
2 w~ = {1/n, ..., 1/n}
3 for t = 0 : T − 1 do
4 find ht by traini∑ng classification tree on weighed scenarios
5 calculate error: i:h(xi)6=y(x )wi i
6 if  < 1 then
2
7 calculate iteration classification tree weight: α = 1 ln1−
2 
8 add classification to ensemble: H t+1 = H t + αtht
t+1 t e−α
tht√ (~x)y(~x)9 update scenarios’ weights: w = w
2 (1−)
10 else
11 return H ;
12 end
13 end
E.6 Sensitivities of Uncertainty Factors
Each tree added to the ensemble has its own values of impurity decrease for
each uncertain factor. The average total impurity decrease for an uncertain fac-
tor x across all trees h in the ensemble H can be used as a measure of how
significant that factor is in determining the system’s performance under a given
scenario (Hastie et al., 2009), as sho∑wn in Equation E.13.∑T D1 I Td · (G (S)−GT (S))
sd = 100% · d=1 d−1 d (E.13)
T GT (S)
t=1 0
244
where sk is the sensitivity index for factor k, d in the split index (first split, sec-
ond split, etc.), D is the depth of the trees, and Id,k is a binary variable assuming
the value of 1 if split d was performed on k and 0 otherwise. Although deeper
trees will more likely report scores for more factors, the scores for factors that
only appear on deeper trees are likely to be negligible. Furthermore, Boosting is
designed for high bias classifiers, meaning shallow trees (depths between 2 and
4), so a deeper tree may hamper its performance.
E.7 Exponencial Convergence
Boosted algorithms, not only AdaBoost and independently of which weak clas-
sified is chosen, are guaranteed to converge. It is possible to derive an expres-
sion for the upper bound of the error, resulting in:
`(H) ≤ n(1− 4γ2)T/2 (E.14)
which means that the training error is bound by an exponential decay as you
add classifiers to the ensemble. This result applies to any boosted algorithm.
E.8 Reduced Overfitting
Lastly, Boosted algorithms are remarkably resistant to overfitting. According to
Murphy (2012), a possible reason is that Boosting can be seen as a form of `1
regularization, which is prone to eliminate irrelevant features and thus reduce
overfitting. Another explanation is related to the concept of margins, so that at
least certain boosting algorithms force a classification on a point only if possible
245
by a certain margin, thus also preventing overfitting.
246
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