Table Of ContentEssays on the ACOPF Problem: Formulations,
Approximations, and Applications in the Electricity Markets
by
Anya Castillo
A dissertation submitted to The Johns Hopkins University in conformity with the
requirements for the degree of Doctor of Philosophy.
Baltimore, Maryland
February, 2016
⃝c Anya Castillo 2016
All rights reserved
Abstract
The alternating current optimal power flow (ACOPF) problem, also referred to as
the optimal power flow (OPF) problem, is at the core of competitive wholesale elec-
tricity markets and vertically integrated utility operations. ACOPF simultaneously
co-optimizes real and reactive power. First formulated over half a century ago in 1962
by Carpentier, the ACOPF is the most representative mathematical programming-
based formulation of steady-state operations in AC networks. However the ACOPF is
not solved in practice due to the nonconvex structure of the problem, which is known
to be NP-hard. Instead, least-cost unit commitment and generation dispatch in the
day-ahead, intra-day, and real-time markets is determined with numerous simplifica-
tions of the ACOPF constraint set.
This work presents a series of essays on the ACOPF problem, which include for-
mulations, approximations, and applications in the electricity markets. The main
themes center around ACOPF modeling fundamentals, followed by local and global
solution methods for a variety of applications in the electricity markets. Original
contributions of these essays include an alternative formulation of the ACOPF, a suc-
cessive linear programming algorithm to solving the ACOPF for the real-time energy
ii
market, an outer approximation method to solving integrated ACOPF-unit commit-
ment as a mixed-integer linear program for the day-ahead market, and applications of
convex relaxations to the ACOPF and its approximations for the purpose of globally
optimal storage integration. These contributions are concluded with a discussion of
potential future directions for work.
Academic Co-advisor and Reader:
Dennice F. Gayme, Assistant Professor
Department of Mechanical Engineering
The Johns Hopkins University
Academic Co-advisor and Reader:
Benjamin F. Hobbs, Schad Professor of Environmental Management
Department of Geography and Environmental Engineering
The Johns Hopkins University
Reader:
Sauleh Siddiqui, Assistant Professor
Department of Civil Engineering
The Johns Hopkins University
iii
Dedication
For my husband Mashama, our son Kamali, and our baby girl: A dedication isn’t
enough, but it’s a start.
iv
Acknowledgements
The struggle has been real and my success is not mine alone.
I thank my family for their motivation and belief in me, because behind every
worthwhile success are lifelong lessons and failures, many that have challenged me to
work harder and stay persistent.
I have had the rare, enlightening, and fortunate opportunity to work with a pow-
erhouse trifecta: my co-advisor Dennice Gayme, my other co-advisor Ben Hobbs and
my supervisor Dick O’Neill. I thank you for giving profusely in your mentorship role,
which has been principle to my success and has inspired me to pay it forward.
I thank Jean-Paul Watson for facilitating research development and strong syner-
gies with other practitioners in the field.
IthankmycoauthorsDenniceGayme, XinyiJiang, CarlLaird, PaulaLipka, C´esar
Silva-Monroy, Dick O’Neill, Shmuel Oren and Jean-Paul Watson for their excellence
in the publications related to the enclosed works.
I thank my third dissertation committee member Sauleh Siddiqui who has pro-
videdencouragingandinsightfulfeedbackonmyresearchpresentationsandproposals
throughout the years.
v
I thank Brent Eldridge, Roderick Go, Robin Hytowitz, Saamrat Kasina, Francisco
Mun˜oz, Venkat Prava and Andrea Staid for their companionship and collegiality in
this grad school journey.
I am eternally grateful to my husband Mashama McFarlane, who says to me when
life gets hard: “to those whom much is given, much is expected.” And so, true to
our undergrad alma mater, ‘my heart is in the work’ (Andrew Carnegie). I love my
profession, I thank those individuals mentioned here along with those along the way
who also gave me a chance, and I hope to have the opportunity to make meaningful
contributions in the lives of others.
This work was made possible by the Heath fellowship, the NSF graduate research
supplement to the U.S. NSF EFRI Grant 0835879, the U.S. NSF Grant IIA-1243482
(WINDINSPIRE), Sandia National Laboratories, and the Federal Energy Regulatory
Commission.
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Contents
Title i
Abstract ii
Dedication iv
Acknowledgments v
Table of Contents vii
List of Tables xiii
List of Figures xv
List of Symbols xvii
List of Nomenclature xix
1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scope and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
vii
CONTENTS
I ACOPF Modeling Fundamentals 15
2 The ACOPF Problem and its Formulations 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 History of the ACOPF Problem . . . . . . . . . . . . . . . . . . . . . 18
2.3 Modeling Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Branch Admittance Matrix . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Nodal Admittance Matrix . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Power Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6.1 Alternating Current Power Flows (ACPF) . . . . . . . . . . . 28
2.6.2 Linearized Real Power Flows and Quadratic Loss Approximations 33
2.6.3 Decoupled Power Flows . . . . . . . . . . . . . . . . . . . . . 37
2.7 The Canonical Alternating Current Optimal Power Flow (ACOPF)
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7.1 Polar Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7.2 Rectangular Form . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8 Spot Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.9 The Proposed IV-ACOPF Formulation . . . . . . . . . . . . . . . . . 44
2.10 Mathematically Equivalent ACOPF Formulations . . . . . . . . . . . 48
2.11 Discussion and Extensions . . . . . . . . . . . . . . . . . . . . . . . . 53
viii
CONTENTS
II Local Solution Methods 56
3 A Successive Linear Programming Approach to Solving the IV-
ACOPF 57
3.1 Prologue and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Fundamentals of General SLP Algorithms . . . . . . . . . . . . . . . 68
3.5 Algorithm Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 Algorithm Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.1 Linearization and Reduction Methods . . . . . . . . . . . . . . 74
3.6.1.1 Piecewise Linear Interpolations . . . . . . . . . . . . 75
3.6.1.2 Taylor Series Approximations . . . . . . . . . . . . . 79
3.6.1.3 OuterApproximationswithInfeasibilityHandlingand
Penalty Factors . . . . . . . . . . . . . . . . . . . . . 82
3.6.1.4 Constraint Reduction . . . . . . . . . . . . . . . . . 85
3.6.2 LP Subproblem Formulation . . . . . . . . . . . . . . . . . . . 86
3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4 The Unit Commitment Problem with AC Optimal Power Flow Con-
straints 100
ix
CONTENTS
4.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . 101
4.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.1 ACOPF Constraint Set . . . . . . . . . . . . . . . . . . . . . . 107
4.3.2 Power Production Constraint Set . . . . . . . . . . . . . . . . 109
4.3.2.1 Binary Constraint . . . . . . . . . . . . . . . . . . . 109
4.3.2.2 Minimum Uptime and Downtime Constraints . . . . 110
4.3.2.3 Real Power Dispatch Constraints . . . . . . . . . . . 110
4.3.2.4 Reactive Power Dispatch Constraints . . . . . . . . . 111
4.3.3 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . 112
4.4 The Solution Technique . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5.1 Six-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . 121
4.5.2 IEEE RTS-79 Test System . . . . . . . . . . . . . . . . . . . . 125
4.5.3 IEEE-118 Test System . . . . . . . . . . . . . . . . . . . . . . 131
4.5.4 Computational Results . . . . . . . . . . . . . . . . . . . . . . 133
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.7 Supporting Material . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.7.1 Adaptive Step-size Region . . . . . . . . . . . . . . . . . . . . 136
4.7.2 Linearized Constraint Set . . . . . . . . . . . . . . . . . . . . 138
4.7.3 Six-Bus Data Set . . . . . . . . . . . . . . . . . . . . . . . . . 139
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