algorithms and economic analysis for the use of optimal …

ALGORITHMS AND ECONOMIC ANALYSIS FOR THE USE

OF OPTIMAL POWER FLOW AND UNIT COMMITMENT IN

WHOLESALE POWER MARKETS

by

Brent C Eldridge

A dissertation submitted to The Johns Hopkins University in conformity with the

requirements for the degree of Doctor of Philosophy

Baltimore Maryland

October 2020

copy 2020 Brent C Eldridge

All rights reserved

Abstract

Optimal power ow (OPF) and unit commitment (UC) are two of the most important op-

timization problems underlying both daily and minute-to-minute wholesale power market

operations However both problems are complex and require modeling simplications in

order to be used for market clearing purposes This dissertation provides three main con-

tributions to improve the delity of modeling simplications to the more dicult problems

that market operators would prefer to solve Formulating OPF with the physically correct

Kirchhos laws results in the alternating current (AC) OPF a non-convex and NP-hard

problem Market operators instead solve a simpler linear model called the direct current

(DC) OPF The UC problem includes binary variables and is also NP-hard Due to the UC

problems complexity market operators cannot solve UC to full optimality but only within

a tolerance of optimality

The rst contribution in this thesis is an iterative algorithm that improves the physical

accuracy of the DC OPF model The main advantage of the proposed algorithm is that it uses

the same DC OPF formulation that is used in current practices and does not substantially

increase the number of computations that must be performed by the market operator

ii

ABSTRACT

The second contribution is a set of three novel OPF formulations that are linear like

the DC OPF but are derived directly from the AC OPF Unlike DC OPF formulations the

proposed models include linear constraints for reactive power and voltage that are ignored

in DC OPFs and results show that the proposed formulations provide substantially better

power ow accuracy than the OPF models used in current practice

Finally new properties are proven for UC pricing methods based on convex hull pric-

ing Traditional UC pricing models are known to be unstable which causes the market

settlements of the near-optimal UC schedules used in practice to be signicantly dierent

from the settlements of an optimal UC schedule that would be ideally used I prove that

the aggregate change in settlements can be bounded by implementing convex hull pricing

methods including a wide range of computationally simple approximations

Readers

Benjamin Hobbs (primary advisor)Theodore M and Kay W Schad Professor in Environmental ManagementDepartment of Environmental Health amp EngineeringDepartment of Applied Mathematics and Statistics (joint)Johns Hopkins University

Dennice GaymeAssociate Professor and Carol Croft Linde Faculty ScholarDepartment of Mechanical EngineeringDepartment of Environmental Health amp Engineering (joint)Department of Electrical amp Computer Engineering (joint)Johns Hopkins University

Enrique MalladaAssistant ProfessorDepartment of Electrical amp Computer EngineeringJohns Hopkins University

iii

Acknowledgments

This PhD has been a long journey that I could not have completed without the help of many

people First and foremost I am so grateful for my advisor Benjamin Hobbs for his guidance

and for sharing his expertise I also thank Dennice Gayme and Enrique Mallada for their

help throughout my PhD and for lending their critical thought and support in my entrance

exam and dissertation committees My studies also would not have been possible without the

WindInspire program funded by the National Science Foundation (NSF) the Advanced Grid

Modeling (AGM) program funded by the US Department of Energys Oce of Electricity

and Sandia National Laboratory and the Young Scientists Summer Program (YSSP) with

funding from the US National Academy of Sciences and hosted by the International Institute

for Applied Systems Analysis (IIASA)

Many others also helped shaped my trajectory and deepened my understanding of sys-

tems modeling I would especially like to thank Natarajan Gautam and Sergiy Butenko for

introducing me to operations research Justin Yates and Jose Vazquez for their wealth of

practical advice Sauleh Siddiqui for his enthusiasm for teaching equilibrium models Shmuel

Oren for all that I learned by attending his seminar Steven Gabriel for encouraging me to

iv

ACKNOWLEDGMENTS

continue to pursue PhD study and of course Richard ONeill for familiarizing me with the

important connections between optimization and electricity markets I also thank my main

collaborators Anya Castillo Ben Knueven and Robin Hytowitz whose help has made this

dissertation possible

Thank you to all of all of the friends Ive met along the way particularly my classmates

at UC Berkeley FERC colleagues the IIASA sta the 2019 YSSP cohort (especially my

ASA lab mates and fellow members of the IIASA music club band) fellow members of Dr

Hobbs research group and other Hopkins classmates and of course my old friends in Houston

and my new friends in Baltimore Thank you all for sharing study sessions game nights

lighthearted discussions helpful advice and generally keeping me occupied with a healthy

amount of non-academic activities over the past ve or so years

For my family I absolutely could not have made it this far without you always encouraging

me to do my best I also owe a huge thanks to the love of my life Yana Gurova for

your unconditional support through tough times and your constant supply of enthusiasm

optimism and perspective to keep going Lastly I have been deeply saddened in the last

year and half by the loss of my father Barney Eldridge and my aunt and Godmother Pat

Ann Dawson Both were incredibly proud to see me become the rst member of our family

to enter a PhD program

v

Dedicated to my dad

vi

Contents

Abstract ii

Acknowledgments iv

Dedication vi

List of Tables ix

List of Figures x

1 Introduction 111 Brief Background 512 Research Questions 1013 Contributions and Scope 11

2 Mathematical Preliminaries and Literature Review 1421 Power Flow 1522 Optimal Power Flow 2523 Unit Commitment 3424 State-of-the-Art and Current Gaps 46

3 An Improved Method for Solving the DC OPF with Losses 7631 Introduction 7632 Power Flow Derivations 8633 Model 9334 Proposed SLP Algorithm 9935 Conclusion 111

4 Formulation and Computational Evaluation of Linear Approximations ofthe AC OPF 11341 Introduction 11342 Model Derivations 12143 Simplication Techniques 13644 Computational Results 14245 Conclusion 164

vii

CONTENTS

5 Near-Optimal Scheduling in Day-Ahead Markets Pricing Models andPayment Redistribution Bounds 16651 Introduction 16652 Unit Commitment and Pricing 17253 Theoretical Results 18254 Example 19155 Test Cases 19456 Conclusion 204

6 Conclusion 20661 Discussion 20862 Looking Forward 211

A Sensitivity Factor Calculations 215A1 Parametric Descriptions 215A2 Implicit Sensitivity Solutions 218

B Self-Commitment Equilibrium 221B1 Nash Equilibrium in a Small Market 224B2 Simulating Equilibria Heuristically with a Greedy Algorithm 229

Vita 253

viii

List of Tables

31 ISO line loss approximation methodologies 8033 IEEE 300-bus test case solution statistics 9734 Two node example 10035 Solutions for initial and nal bids 10136 Solution comparison of SLP and AC OPF 10837 Computational comparison of SLP QCP and AC OPF 109

41 OPF case study sources 14442 Normalized objective function values default model implementations 14643 Model speedup compared to AC OPF by implementation settings 163

51 Pricing model denitions 17852 Generator attributes simple example 19253 Test case summary 19554 Mean payment redistribution quantities 199

B1 Optimal schedules given self-commit oers 225B2 Expected prots given self-commit oers 227B3 Pricing model eect on expected production cost and price 228

ix

List of Figures

11 Simplications should be reasonably consistent with underlying complexity 4

21 Convex relaxations of a non-convex region 4822 Linear approximations of a non-convex region 56

31 Accuracy comparison of DC OPF formulations 9832 SLP algorithm convergence 10733 Error sensitivity analysis in the IEEE 24-bus test case 110

41 LMP comparison in the 118-bus IEEE test case with nominal demand 14842 LMP errors in Polish test cases with nominal demand 14943 Real power ow errors in Polish test cases with nominal demand 15244 Real power ow error statistics in Polish test cases with nominal demand 15345 Solution times in IEEE and Polish test cases with and without lazy algorithm 15446 Solution times in IEEE and Polish test cases with factor truncation tolerances 15547 Real power ow error in Polish test cases with factor truncation tolerances 15648 Detailed error sensitivity analysis of the IEEE 118-bus test case 15849 Summary error sensitivity analysis of the IEEE test cases 160410 Solution times in all test cases and model implementations 161

51 Hourly price mean and coecient of variance in the PJM test case 19752 Make-whole payments and lost opportunity costs 19853 Redistribution quantity cdf normalized by Corollary 53 bounds 20154 Generator prot coecient of variance cdf 203

B1 Self-commitment strategies under dierent pricing models and market sizes 232B2 Comparison of competitive and simulated self-commitment strategies 234

x

Chapter 1

Introduction

Wholesale power markets rely on solving various complex optimization problems that deter-

mine not only how much power to produce and at what price but perhaps more importantly

when and where to produce it This is a very dicult problem that requires simplications

I have written this dissertation to discuss the modeling of nonlinear and non-convex con-

straints for alternating current (AC) power ow and binary operating status constraints of

thermal generators in wholesale power markets In particular I address the question of

how such constraints can be simplied in order to calculate implementable and economically

ecient electricity generation schedules

High voltage AC transmission lines allow the transfer of electric power from generation

sources to end use In an interconnected power grid the power ow on any specic trans-

mission line depends on the ows on all other components of the system through a set of

AC power ow equations known as Kirchhos laws Because AC power ow is nonlinear

1

CHAPTER 1 INTRODUCTION

and non-convex there is currently no computationally ecient method for power market

operators to satisfy the AC power ow equations in the optimization software that performs

market clearing Market clearing software for power markets is instead based on a sim-

plication of AC power ow called direct current (DC) power ow DC power ow does

not literally model direct current power but it is a linear approximation of AC power ow

that can be computed very quickly This simplication allows market clearing software to

consider many more aspects of power systems operation such as contingency scenarios How-

ever the DC power ow simplication can also result in inecient use of the power grids

physical infrastructure since the market results may either be suboptimal (if cheap resources

are under-utilized because a network constraint is not actually binding) or infeasible (when

remedial actions must be taken because a network constraint is actually violated) In either

case approximation errors in the DC power ow cause ineciencies in the power market

because the DC power ow model is not a perfect representation of AC power ow

In addition to network constraints nearly all thermal power generation technologies

include various situations that create non-convex cost structures or operating regions Mixed

integer programming (MIP) software has progressed over the past few decades so that these

lumpy (0 or 1) decisions can be determined very nearly to optimality However there is recent

interest in how the cost of these lumpy decisions should be reected in market prices Many

market clearing price formulations have been applied or proposed for non-convex electricity

markets but all rely on modifying the standard competitive equilibrium conditions to include

some form of side-payments Unfortunately it is currently unknown if there exists any market

2


clearing price formulation that can be guaranteed to support a competitive equilibrium in

these markets

In a general sense ecient power production scheduling is a large-scale stochastic nonlin-

ear mixed integer problem Ignoring epistemic problems to this approacheg what are the

correct probability distributions and what is known with absolute certaintysuch a detailed

scheduling problem cannot currently be solved within the tight time constraints enforced by

the rolling basis of continuous power market operations Modeling simplications free up the

market operators computational budget which allows computational time to be spent on

modeling the most salient aspects of power scheduling as realistically as possible In addition

to balancing supply and demand current power market software also considers factors such

as operating reserves contingency scenarios conguration transformations of combined cycle

gas turbines (CCGTs) and many other factors Recent growth in renewable wind and solar

generation technologies has increased the amount of variability and uncertainty in power

production making it more important than ever to increase the level of detail included in

power production scheduling software The future of ecient power market operation will

almost certainly require modeling simplications with ever higher levels of delity to the

complex systems that they approximate

The main topic of this dissertation is the ecient operation of wholesale power mar-

kets In broader terms the theoretical framework of this dissertation might be generally

summarized by Figure 11 Current practices often use simplied models that have minor

inconsistencies with the complex problem that they are attempting to solve Of course the

3


Figure 11 Simplications should be reasonably consistent with underlying complexity

hope is that small inconsistencies in a modeling simplication will only result in small de-

viations from the solution to the complex problem but is this actually what happens The

world is full of complex problems and simple solutions can often lead us far astray On the

other hand ensuring that all aspects of a model are consistent with the original problem

would prevent the use of any simplications The goal of this dissertation is to nd simple

and computationally practical methods that can guide us within a step or two from the right

solutions to complicated problems especially those problems that are essential for the safe

ecient and reliable production of electricity

The remainder of this introductory chapter provides brief electricity market design back-

4


ground in Section 11 Section 12 describes the research questions and Section 13 describes

the main contributions and scope of the dissertation

11 Brief Background

This thesis addresses the formulation and use of optimization tools for the market-based

dispatching and scheduling of electricity production as currently performed in the US by

organizations called Regional Transmission Organizations (RTOs) and Independent System

Operators (ISOs) RTOs and ISOs are synonymous and for brevity we will often only refer

to ISOs This background discussion will briey describe the features of electricity market

design that are salient to the proceeding chapters but more in-depth reviews can be found

elsewhere (see Cramton 2017) What follows below is a brief overview of the general ISO

market design and description of the optimal power ow (OPF) and unit commitment (UC)

problems Additional mathematical details of OPF and UC are provided in Chapter 2

ISOs operate a real-time market (RTM) that clears every 5 minutes The clearing engine

for the RTM is called the security constrained economic dispatch (SCED) model and it

is formulated to minimize the cost of energy supply subject to technological limitations

(for example the maximum output of a generator or the maximum power ow across a

transmission line) In addition the SCED model includes security constraints that ensure

that system reliability can be maintained during generator or transmission outage scenarios

When these scenarios are not considered the SCED model is called the OPF problem Even

5


though the OPF is a simplied version of the SCED model it retains most of the basic

properties of the SCED model most of all network constraints that ensure that power ows

do not exceed the physical limitations of the power grids physical infrastructure SCED and

OPF can both be solved using computationally ecient and reliable linear programming

(LP) software

One day in advance of the RTM the ISO clears a day-ahead market (DAM) that prepo-

sitions generation resources to eciently participate in the RTM Whereas the ISO solves a

SCED model in the RTM the ISO solves a security constrained unit commitment (SCUC)

model to operate the DAM The SCUC model without security constraints is simply called

the UC problem and like the OPFs relation to SCED the UC problem contains the most

salient feature of SCUC the binary-valued (or lumpy) decisions of whether to change a

generators operating status Thermal generators often require a xed start up cost to begin

producing energy cannot stably produce energy below a certain threshold quantity unless

they are shut o andor cannot shut down or start up too soon after the previous start up

or shut down The all-or-nothing nature of these decisions adds signicant complexity to

the UC problem as often it can require cheaper resources to be resources to be dispatched

down because another generator has been committed and must meet its lumpy constraints

These conditions are a common feature to most power generating technologies and impose

a signicant complexity on ISO market clearing activities especially the determination of

market-clearing prices

Market prices in the DAM and RTM are set by the marginal cost to deliver power to

6


each network location called locational marginal prices (LMPs) This idea came from control

theorists at MIT in the early 1980s (Caramanis et al 1982 Bohn et al 1984 Schweppe

et al 1988) The main idea behind LMP is that in power markets power is routed not like

delivery vehicles over roadways but according a complex set of physical laws called Kirchos

laws that govern AC power ow (Glover et al 2008 Ch 6) This makes LMPs an important

aspect of electricity market design because over- or under-supply at certain locations in the

network can result in overloading transmission lines causing those lines to overheat and

eventually fail (or in the case of security constraints possibly causing line failure if there is a

transmission or generator outage) LMPs promote market eciency because each generator

only has the incentive to produce energy if it is part of the most ecient dispatch that avoids

exceeding the networks physical limits

Hence OPF is properly formulated as the AC OPF (Cain et al 2012) by including

AC power ow constraints based on Kirchhos laws However these equations are highly

nonlinear and non-convex so current approaches to solving the AC OPF tend to have slow

convergence and poor computational performance for the large-scale OPF problems that are

relevant to ISOs Instead ISOs use software that solves the DC OPF a linear approximation

of AC power ow that is reasonably accurate and can be solved quickly and reliably (Overbye

et al 2004 Stott and Alsaccedil 2012) ISOs presently iterate their DC OPF solutions with

various network security analysis tools (see Table 31 in Chapter 3) that help generate con-

straints and ensure solution feasibility However this process can cause market ineciency

since the DC OPF solution may under-utilize the network or require remedial actions that

7


are not determined through optimization Chapter 2 introduces the mathematical details of

AC and DC power ow in Section 21 and the AC and DC OPF problems in Section 22

In favor of simplicity the following chapters will specically discuss the OPF problem

rather than SCED It will still be understood however that improvements to the OPF model

are also applicable to SCED as the only dierence is the inclusion of security constraints

Therefore by showing that the OPF solutions are faithful to the the original nonlinear and

non-convex AC power ow equations we can also conclude that the SCED model that is

actually implemented by an ISO would also be physically accurate

ISOs also use a two-settlement market design that couples the outcomes of the DAM

and RTM Within this system the SCUC model does not determine physical quantities in

the DAM but nancially-binding forward positions that are subsequently closed out in the

RTM This set-up helps incentivize ecient participation from resources that cannot respond

quickly enough to price signals in the RTM as the RTM prices can often become volatile

due to changes in weather demand or other system conditions

An important aspect of the ISOs markets is that SCUC and SCED are solved to minimize

the cost of supply oers and in some cases minus the value of demand bids Assuming

that the two-settlement market design incents all participants to oer or bid truthfully ie

there is no market power then the ISOs markets will minimize production costs1 while

simultaneously maximizing the prots of market participants These two conditions satisfy

1Or more generally maximizes market surplus dened as the value of demand bids minus productioncost oers

8


a competitive equilibrium which is later dened more precisely (see Denition 22)

Various pricing methods have been proposed for non-convex electricity markets that mod-

ify the standard competitive equilibrium conditions to include some form of side-payments

(ONeill et al 2005 Gribik et al 2007 among others) In addition ISOs are only able to

solve the SCUC problem to near-optimality rather than the full optimality that would be

required for a competitive equilibrium The outcomes from the market settlement process

can therefore dier signicantly even from the outcomes that satisfy the modied competi-

tive equilibrium conditions (Johnson et al 1997 Sioshansi et al 2008b) This inconsistency

creates concerns that the market outcomes may be arbitrary or could be inecient due to

gaming opportunities

The above issues can be modeled with just the UC problem rather than SCUC and hence

the following chapters will specically discuss the UC problem rather than SCUC Like for

the OPF problem it will also be understood that the analysis of the UC problems economic

properties are also applicable to market settlements based on SCUC Chapter 2 introduces

the UC problem and its competitive equilibrium properties in Section 23

Given the above background in ISO processes this dissertation addresses the research

questions described in the following section

9


12 Research Questions

The rst question addresses the use of sensitivity factors to approximate network line losses

in the DC OPF Line loss sensitivities are calculated from an AC power ow that is used

as the base-point for the DC OPF Current practices do not consider that the line loss

sensitivities that are input to the DC OPF are dependent on power ows and therefore may

be inconsistent with the change in power ows after the DC OPF is solved The ISOs

dispatch instructions may therefore be suboptimal or infeasible In addition it may not be

possible to update the base-point with a new AC power ow since it may be too costly to

re-run the AC power ow software

Q1) How well can iterative methods improve line loss approximations in DC OPF-based

models given an initial AC base-point and no subsequent AC power ow solutions

The second question addresses the the fact that reactive power and voltage are completely

ignored by the standard DC power ow assumptions that are applied to the OPF models

presently used by ISOs Various tight convex relaxations of the AC OPF have recently been

proposed to determine high quality OPF solutions without directly solving the non-convex

AC OPF problem However these formulations use a sparse network constraint structure

and nonlinear solution methods that are signicantly dierent than the linear OPF models

and solution methods presently used by ISOs ISOs presently use a compact and linear OPF

formulation that can be solved very quickly in SCUC and SCED software so formulating a

10


compact and linear approximation of the AC OPF may help to improve the physical accuracy

of the ISOs OPF solutions without being too costly in terms of solution times

Q2) How could reactive power and voltage constraints be formulated to create a compact and

linear OPF model with similar structure to the OPF models presently used by ISOs

What is the eect on solution speed and power ow approximation error

The last question addresses the potential eects of dierent UC pricing methods As

previously discussed ISOs do not solve the UC model to full optimality but only determine

a near-optimal solution that is within a small tolerance of the optimal cost Current pricing

methodologies are premised on modied competitive equilibrium conditions that require an

optimal UC solution so the near-optimal solutions that are found in practice may result in

market outcomes that are inconsistent with the premised competitive equilibrium

Q3) Which pricing methods can provide guarantees that the market outcomes of near-optimal

UC solutions do not signicantly dier from the market outcomes of optimal UC solu-

tions How do such guarantees aect generator oer incentives

13 Contributions and Scope

Chapter 2 presents mathematical preliminaries and relevant literature and is included for

completeness of the dissertation For readers already familiar with OPF and UC a quick

skim will suce to review the basic ideas used in Chapters 3-5

11


Chapter 3 proposes a new iterative algorithm for improving the line loss approximation

used in the ISOs generator dispatch software The proposed approach oers several advan-

tages over existing methods It uses the OPF formulation that is currently used by ISOs a

formulation that is more compact and solves faster than other OPF models The proposed

algorithm also uses very light data requirements as it only takes an AC power ow solution

at the beginning of the algorithm and unlike other approaches the line loss approximation is

updated with simple rules that do not require additional AC power ow solutions Through

extensive computational experiments we show that the proposed approach converges within

very few iterations typically two or three and results in dispatch solutions that are very

close to the ideal AC OPF dispatch solution Most importantly ISOs can implement the

proposed algorithm with only minimal changes to their current software

Chapter 4 proposes three novel linear OPF models that directly linearize the AC OPF and

are therefore able to approximate line losses as well as reactive power and voltage constraints

Despite inherent approximation error of the LP-based approach the linear OPF model so-

lutions are nearly AC-feasible and can be solved substantially faster than the AC OPF The

chapter also presents three simplication techniques that further improve the computational

performance of the models without signicantly increasing power ow approximation errors

Computational experiments with both simple and realistically-sized systems show that the

proposed formulations provide higher quality power ow solutions than what can be obtained

in standard DC OPF-based models Appendix A provides implementation details for the

models proposed in both Chapters 3 and 4

12


Chapter 5 discusses how near-optimal UC schedules tend to result in unstable market

settlements in the sense that small ineciencies in the schedule can result in vastly dierent

market settlements This was long thought to be an unavoidable aspect of the UC prob-

lem but I prove theoretically that pricing methods based on convex relaxations of the UC

problem can bound the change in market settlements due to the schedules suboptimality

A consequence of the bound is that a pricing method called convex hull pricing essentially

removes incentives for inexible generators to self-commit (ie physically produce energy in

the RTM without being committed by the ISO) which is currently a common practice among

coal-red power plants in some ISO markets The chapter concludes with realistically-sized

test cases that demonstrate that unlike the pricing methods that are currently standard

practice tight convex hull pricing approximations result in market settlements that are very

close to the outcome of the optimal schedule Appendix B presents an equilibrium analysis

and a simulation to further demonstrate the reduction in self-commitment incentives

Lastly Chapter 6 concludes the dissertation by reviewing the signicance of the contri-

butions and presenting ideas for further research

13

Chapter 2

Mathematical Preliminaries and

Literature Review

The following material introduces in-depth mathematical statements of the power ow op-

timal power ow (OPF) and unit commitment (UC) problems For readers already familiar

with these topics this chapter is only presented for the sake of completeness and a care-

ful reading of this material is not necessary to appreciate the results and contributions in

Chapters 3-5

Power ow is introduced in Section 21 OPF in Section 22 and UC in 23 Sections

21 and 22 are based on physical properties of the transmission system These two sections

adopt a notation appropriate for modeling these physical details Section 23 switches to

a dierent notation that is more commonly used for general mathematical programming

This change in notation will be used to help introduce economic notions such as competitive

equilibrium that are more related to the general mathematical structure of problem than

14

CHAPTER 2 MATHEMATICAL PRELIMINARIES AND LITERATURE REVIEW

any physical details

Finally Section 24 provides a review of literature relevant to the research questions in

Section 12 Readers who are already familiar with OPF and UC may wish to skip directly

to this section for a review of the state-of-the-art current gaps in the literature and some

historical context to the problems

21 Power Flow

This section introduces the set of physical equations called Kirchhos laws that govern

alternating current (AC) power ow After presenting the full AC power ow simplica-

tions are used to derive the linear direct current (DC) power ow approximation and the

distribution factor formulation of DC power ow that is widely used in practice The DC

power ow can be solved much quicker than AC power ow yet DC power ow does not

accurately model the AC power ows because it ignores certain aspects of AC power ow

such as line losses reactive power and voltage magnitude

The development that follows is largely standard and similar presentations of this ma-

terial may be found in Glover et al (2008) and Wood et al (2013)

15


211 AC Power Flow

We begin with the AC real power ow equations for a single branch k In steady state

AC power uctuates in a sinusoidal signal that can be conveniently represented by complex

numbers Math and engineering literature often uses the symbols i or j to express the

imaginary numberradicminus1 but these letters will be used in later chapters for the locations

(nodes) in the transmission system I will instead adopt ȷ for the imaginary number

noting that the possible confusion between j and ȷ will be conned to the discussion of

complex numbers that only appears in this section

For a number V in the complex number system the following exponential polar and

rectangular representations are all equivalent

V = |V |eȷθ⏞ ⏟⏟ ⏞exponential

= |V |angθ⏞ ⏟⏟ ⏞polar

= |V | (cos θ + ȷ sin θ)⏞ ⏟⏟ ⏞rectangular

(21)

where |V | and θ denote the magnitude and angle of the vector V isin C respectively The star

notation is adopted for the complex conjugate dened below

V ⋆ = |V |eminusȷθ = |V |angminus θ = |V | (cos θ minus ȷ sin θ) (22)

The derivation for AC power ow begins with the linear equations relating current and

voltage by Kirchhos current law (KCL) and voltage law (KVL) (Glover et al 2008 Ch 2)

Assume that the network is operating at sinusoidal steady state and the elements of network

with N nodes and K branches are described by an N times N nodal admittance matrix Y =

16


G+ ȷB where G is the branch conductance and B the branch susceptance Current in the

system is I = YV the product of admittance and voltage V isin C The apparent power

S isin C consists of real and reactive components P and Q where S = P + ȷQ and is the

product of voltage and the complex conjugate of current S = VI⋆

Using the relations above dene the real and reactive power net injections into bus i

pi and qi with the rectangular notation for branch admittance and polar notation for bus

voltages

pi + ȷqi = viangθi

(sumj

(Gij + ȷBij)vjangθj

)⋆

(23)

where vi is the RMS voltage magnitude and θi is the voltage angle at bus i

Real and reactive components are then expressed separately with θij = θiminusθj the voltage

angle dierence between buses i and j (see Glover et al 2008 Sec 64)

pi = visumj

vj (Gij cos(θij) + Bij sin(θij)) foralli isin N (24a)

qi = visumj

vj (Gij sin(θij)minus Bij cos(θij)) foralli isin N (24b)

Cain et al (2012) formulates the following power ow equations to include transformer

tap settings phase shifters and shunt devices Let k isin K be the set of branches connecting

the nodes i j isin N in the network and let pfk and qfk be the real and reactive power

respectively owing from node i to j on branch k Similarly let ptk and qtk be the opposing

real and reactive power ows from node j to i on branch k Lastly assume the presence of

17


an ideal transformer with turns ratio τki and phase shift ϕki and shunt susceptance Bski

pfk = Gkτ2kiv

2i minus τkivivj (Gk cos (θij minus ϕki) + Bk sin (θij minus ϕki)) forallk isin K (25a)

ptk = Gkτ2kjv

2j minus τkjvivj (Gk cos (θij minus ϕki)minus Bk sin (θij minus ϕki)) forallk isin K (25b)

qfk = minus (Bk +Bski) τ

2kiv

2i minus τkivivj (Gk sin (θij minus ϕki)minus Bk cos (θij minus ϕki)) forallk isin K (25c)

qtk = minus (Bk +Bski) τ

2kjv

2j + τkjvivj (Gk sin (θij minus ϕki) + Bk cos (θij minus ϕki)) forallk isin K (25d)

Next redene the left hand side of equations (24a) and (24b) into terms of power supply

and demand Let pgm qgm isin Gi be the real and reactive power respectively produced by

generators at node i and let P di and Qd

i be the real and reactive power demand respectively

at node i The power balance constraints are formulated as follows

summisinMi

pgm minus P di minusGs

iv2i minus

sumkisinKfr

i

pfk minussumkisinKto

i

ptk = 0 (26a)

summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i

qfk minussumkisinKto

i

qtk = 0 (26b)

Combining (25) and (26) results again in AC power ow equations in the same form

as (24) the only dierence being the greater level of detail in modeling transformer tap

settings phase shifters and shunt devices in the transmission system This formulation is

commonly called the polar AC power ow formulation of the bus-injection model

Although others have also proposed and implemented other AC power ow formulations

that also satisfy Kirchhos laws the proceeding material in this chapter and Chapters 3

and 4 are derived only from the polar formulation above The bus-injection model can

18


be isomorphically reformulated in rectangular and current-voltage forms (see Molzahn and

Hiskens 2019 Sec 21) If the network has a radial or tree structure common in distribution

networks for example then a piar of seminal papers by Baran and Wu (1989ab) propose

that the AC power ow equations can be further simplied to the branch-ow (also called

DistFlow) model also reviewed by Molzahn and Hiskens (2019)

212 Solving AC Power Flow

Equations (25a) (25b) (25c) (25d) (26a) and (26b) constitute 4K + 2N nonlinear

equations There are 4K unknown power ow variables 2N unknown voltage angle and

magnitude variables and 2M unknown dispatch variables Throughout this dissertation I

will assume that each bus i contains at most one generator m and therefore M le N The

standard engineering approach categorizes three types of buses (Glover et al 2008 Sec 64)

bull Slack or reference bus Fixed voltage magnitude vref and angle θref (ie two

additional constraints)

bull Load (PQ) bus Fixed real and reactive power demands P di and Qd

i (ie no additional

constraints)

bull Generator (PV) bus Fixed real power dispatch pgm and voltage magnitude vm (ie

2(M minus 1) additional constraints)

The slack and PV bus designations above constitute an additional 2M equations that

19


would be required by the simple analysis equating the number of equations with the number

of variables resulting in a system of AC power ow equations with 4K+2N +2M equations

and 4K + 2N + 2M variables If there exists a solution to the power ow equations then

it can typically be found by Gauss-Seidel or Newton-Raphson methods for solving nonlinear

equations (see Glover et al 2008 Sec 64-6)

213 DC Power Flow

The idea behind DC power ow is to approximate the AC power ow using linear equations

The close approximation is achieved by exploiting the near-linearity of (25a) and (25b) with

respect to the voltage angle dierence θij Along with a few other simplifying assumptions

the AC power ow constraints (25a) (25b) (25c) and (25d) can be reduced to what is

called the DC power ow approximation For simplicity assume that there are no transformer

taps phase shifters or shunt susceptances Linear approximations for these components can

be analogously derived

First assume there are no line losses in the network or equivalently that Gk ≪ Bk

pfk = minusvivjBk sin (θij) (27)

qfk = minusBk + v2i + vivjBk cos (θij) (28)

Next assume that the voltage magnitudes are close to their nominal values ie vi = 1

20


when expressed using the per unit system (see Glover et al 2008 Sec 33)

pfk = minusBk sin (θij) (29)

qfk = minusBk +Bk cos (θij) (210)

Finally assume that the voltage angle dierence is small so the small angle approxima-

tions sin(θij) asymp θij and cos(θij) asymp 1minus θ2ij2can be applied Because θij is close to zero θ2ij asymp 0

is accurate to a second degree error

pfk = minusptk = minusBk (θij) (211)

qfk = qtk = 0 (212)

Hence the DC power ow approximation is a linear relationship between pfk and θij The

system power balance assumes voltages are normalized to 1 per unit (pu)

summisinMi


i minussumkisinKfr

i

pfk +sumkisinKto

i

pfk = 0 (213)

Equations 211 and 213 are both linear and can be equivalently stated in matrix notation

Let A isin RKtimesN be the network incidence matrix dened as Aki = 1 if node i is on the from

side of branch k -1 if node i is on the to side of branch k and 0 otherwise and let

B isin RKtimesK be the branch susceptance matrix with diagonal entries Bk In addition the

notation for generators and loads can be simplied by assuming a single generator at each

bus and dening a vector of net withdrawals pnw = Pd + Gs minus pg The DC power ow

21


equations can now be written in matrix form

pnw +A⊤pf = 0 (214a)

pf = minusBAθ (214b)

θref = 0 (214c)

The main insight behind the DC power ow is that it exploits the near-linearity of power

ow with respect to voltage angles (Stott et al 2009) Many linear and nonlinear power

ow approximations have been proposed that are elaborations of this standard DC power

ow formulation A key drawback of the standard DC power ow is that the approximation

results in a lossless transmission network Schweppe et al (1988) uses the small angle

approximation cos(θij) asymp 1 minus θ2ij2

to derive the standard quadratic line loss approximation

that is often applied to DC power ow studies Section 32 in Chapter 3 includes a similar

derivation

214 Solving DC Power Flow

The approximated ows have no line losses and reactive power ows are completely ignored

The AC power ows set of 4K + 2N nonlinear equations can be replaced by K +N linear

equations in the DC power ow Repeating the simple analysis of equations and variables

from the previous section there are now K power ow variables pfk N voltage angle variables

θi and M real power dispatch variables pgm As before one reference or slack bus can be

22


dened that xes the voltage angle and leaves the real power dispatch unconstrained and

M minus 1 generator or PV buses can be dened that xes the real power dispatch but leaves

the voltage angles unconstrained

215 Power Transfer Distribution Factors

In many instances it is advantageous to compute power ows as a function of real power

injections and withdrawals pnw instead of voltage angles θi Voltage angle variables can be

substituted out of the DC power ow equations by solving the following system Combine

equations 211 and 213 to rewrite the linear DC power ow equations

minuspnw +A⊤BAθ = 0 (215)

Solving for θ

θ =(A⊤BA

)minus1pnw (216)

From (211) the power ow is pf = minusBAθ It then follows that power transfer distri-

bution factors (PTDFs) can be dened as F = minusBA(A⊤BA

)minus1 which provides a direct

computation of pf = Fpnw Instead of solving(A⊤BA

)minus1explicitly to calculate F the

following linear algebra simplies the calculation

(A⊤BA

)F

⊤= minus (BA)⊤ (217)

Strictly speaking the linear system above cannot be solved because the nodal susceptance

23


matrix A⊤BA is a singular matrix Full rank is restored by adding equation (214c) into the

system Let W isin RN be a vector of weights to dene the reference bus (eg Wref = 1 or

more generallysum

i Wi = 1 and Wi ge 0 foralli) Then dene the reference column of the PTDF

matrix F to be zero (or more generally FW = 0)⎡⎢⎢⎣A⊤BA

W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (218)

In shorthand the above equation is equivalent to eliminating the reference bus row of the

nodal susceptance matrix A⊤BA Let this revised nodal susceptance matrix be B To prove

that F provides the correct calculation for DC power ow we observe the following

pf = Fpnw = minusBABminus1pnw

= minusBAθ

where θref = 0 is implied by (218)

The PTDF formulation eliminates the need forN voltage angle variables so the DC power

ow equations can also be reduced by N One clearly unnecessary equation is θref = 0 The

remaining N minus 1 equations can be eliminated by summing together the nodal power balance

(213) Since 1⊤A = 0 the summed power balance constraints become a simple system

power balance equation

1⊤pnw = 0 (219)

24


Overbye et al (2004) show that the DC PTDF approximation is a substantially faster

computation than AC power ow methods and only results in a small level of approximation

error that is acceptable in most practical power ow applications Baldick et al (2005)

conrms the same result in much larger test cases yet adds that the approximation performs

less well in systems with long instability-prone transmission lines such as in the Western

Interconnection In such situations Independent System Operators (ISOs) use incremental

PTDFs by linearizing from the current operating point instead of from the initial point θij = 0

that is used in the DC PTDFs described above Similar incremental PTDF calculations are

also described in Section 42 of Chapter 4

22 Optimal Power Flow

As described in the Chapter 1 OPF is an optimization problem that minimizes the cost of

energy production subject to the capabilities of power generators the physical limits of the

electric grid and power ow according to Kirchhos laws There are two classic forms of

OPF AC OPF and DC OPF Both OPF problems are formulated below rst as the AC

OPF a non-convex nonlinear program (NLP) using the AC power ow equations and then

it is simplied to the DC OPF a linear programming (LP) approximation of the AC OPF

The cost curve of each generator will be assumed to be a convex and quadratic expression

25


of the following nonlinear quadratic function

Cm(pgm) = C0

m + C1mp

gm + C2

m(pgm)

2 (220)

where pgm is the real power output of generator m and C0m C

1m and C2

m are the coecients

of the generators cost function Assuming that Cm(middot) is convex is equivalent to C2m ge 0

In practice any convex function Cm(pgm) could be approximated by a set of piece-wise

linear constraints without considerable computational diculty (Bertsimas and Tsitsiklis

1997 Sec 13) Let pgml l isin Lpgm be a set of sample points of feasible production quantities

of generator m If the cost of generator m is convex then it can be estimated by cm and the

following constraints

cm ge C0m + C1

mpgml + C2

m(pgml)

2 +(C1

m + 2C2mp

gml

) (pgm minus pgml

) foralll isin Lpgm (221)

Minimizing costs in the OPF objective ensures that (221) will be a binding constraint for

at least one l isin Lpgm as long as the cost function being linearized is convex The linear

approximation can be made arbitrarily close to (220) by adding additional sample points

The optimization models in this document will be formulated using the constraints of

(221) to approximate the actual cost function Cm(pgm) but it will often only be written as

Cm(pgm) to reduce notational clutter

26


221 AC Optimal Power Flow

The OPF problem constrained by AC power ow is called the AC OPF It has long garnered

academic interest as a large-scale nonlinear non-convex problem that is exceptionally dif-

cult to solve while also being economically and practically important Carpentier (1962)

rst presented the AC OPF as an optimization problem yet ecient algorithms to optimally

solve the AC OPF still have not been developed (Cain et al 2012)

The standard polar form AC OPF is formulated as below

max zAC =summisinM

Cm(pgm) (222a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i

ptk = 0 foralli isin N (222b)

summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i

qtk = 0 foralli isin N (222c)

pfk = Gkτ2kiv

2i minus τkivivj

(Gk cos (θij minus ϕki)

+Bk sin (θij minus ϕki)) forallk isin K (222d)

ptk = Gkτ2kjv

2j minus τkjvivj


minus Bk sin (θij minus ϕki)) forallk isin K (222e)


2kiv

2i minus τkivivj

(Gk sin (θij minus ϕki)

minus Bk cos (θij minus ϕki)) forallk isin K (222f)

27



2kjv

2j + τkjvivj


+Bk cos (θij minus ϕki)) forallk isin K (222g)

(pfk

)2+(qfk

)2le T 2

k forallk isin K (222h)(ptk

)2+(qtk

)2le T 2

k forallk isin K (222i)

minusθij le θij le θij forall(i j) isin K (222j)

V i le vi le V i foralli isin N (222k)

Pm le pgm le Pm forallm isinM (222l)

Qmle qgm le Qm forallm isinM (222m)

θref = 0 (222n)

In additon to the previously discussed power balance and power ow constraints the

OPF formulation also includes constraints (222h) and (222i) to prevent power ows from

exceeding certain engineering limits given by a steady state limit on apparent power Tk

for each branch k isin K Constraint (222j) helps to ensure stability of the network by

keeping voltage angle dierences within limits Constraint (222k) similarly keeps bus voltage

magnitudes within rated values Generator limits (222l) and (222m) ensure that dispatch

does not exceed feasible generation levels For simplicity only box constraints are considered

but more general D-curve constraints (Kundur et al 1994) could also be constructed if

desired Lastly constraint (222n) denes the voltage angle at the reference bus to be zero

Given lack of eent solution algorithms and the time constraints of the ISOs daily

28


hourly and sub-hourly operating periods the AC OPF is still impractical for market-based

implementation (Stott and Alsaccedil 2012) The following section presents a common approx-

imation that allows ecient computation of OPF solutions that are typically within an

acceptable range of AC feasibility We then explore some of the economic properties of the

approximated OPF problem

222 DC Optimal Power Flow

Despite the name the DC OPF does not really model direct current power ow but is

a linearization of the AC OPF (Stott et al 2009) RTOs and ISOs rely on DC OPF

formulations in their security constrained economic dispatch (SCED) models (Stott and

Alsaccedil 2012) because it can be solved orders of magnitude faster than the AC OPF (Alsac

et al 1990 Stott et al 2009) and usually provides solutions with an acceptable degree of

approximation error (Overbye et al 2004) However a key drawback of the DC OPF is

that the formulation inherently ignores line losses reactive power and voltage magnitudes

and therefore may provide system operators with an infeasible dispatch solution (Stott and

Alsaccedil 2012) Chapters 3 and 4 provide methods for reducing the approximation error of

DC OPF models while maintaining signicantly faster computational performance than AC

OPF solution methods

In addition to replacing AC power ow constraints (222d) (222e) (222f) and (222g)

with the DC power ow equation (211) the variables pt = minuspf are also substituted in

29


the real power balance constraint and the conductive shunt is approximated as Gsiv

2i = Gs

i

Reactive power and voltage constraints are dropped from the formulation

To simplify notation it will be assumed that each bus contains a single generator ie

M = N and generator costs will be assumed to have a single marginal cost term iesumi Ci(p

gi ) = C⊤pg The DC OPF is then written below with each constraints dual variables

shown in brackets to the right

max zDC = C⊤pg (223a)

st pg minusPd minusGs minusApf = 0 [λ isin RN ] (223b)

pf = minusBAθ [micro isin RK ] (223c)

minusT le pf le T [micro micro isin RK ] (223d)

minusθ le Aθ le θ [microθ microθ isin RK ] (223e)

P le pg le P [β β isin RM ] (223f)

θref = 0 [δ isin R] (223g)

Note that constraint (223e) could be rewritten

minusBθ le minusBAθ le Bθ (224)

Since the middle expression is simply pf the constraint (223e) can be dropped from the

formulation by replacing the transmission limitT by taking the element-by-element minimum

T = minTBAθ Although angle dierence constraints will not be made explicit in our

30


model formulations it will be assumed that a suitable transformation of the transmission

limits has been made so that the angle dierence constraints are still satised

The DC OPF can also be formulated with PTDFs


st 1⊤pg = 1⊤(Pd +Gs) [λ isin R] (225b)

pf + Fpg = F(Pd +Gs) [micro isin RK ] (225c)


P le pg le P [β β isin RN ] (225e)

The DC OPF formulations (223) and (225) are isomorphically equivalent The pro-

ceeding chapters will focus on OPF formulations based on (225) for ease of analysis and

similarity to the models used in ISO market clearing software

The DC OPF can be solved very quickly and reliably within the ISOs market clearing

windows not only because it has fewer variables and constraints than the AC OPF but also

because the formulation is typically linear1 Whereas infeasibility is dicult to prove for an

NLP and often causes NLP solvers to time out LP solvers can implement pre-processing

steps such as the big-M method (Bertsimas and Tsitsiklis 1997 Sec 35) that quickly

and reliably provide a feasible initial solution or determine that the LP is infeasible By

quickly determining that a DC OPF problem is infeasible the ISO can quickly shift to

determining contingency plans rather than feeding new starting points into an NLP solver

1DC OPF variants can be nonlinear such as if a quadratic line loss approximation is included

31


and waiting When a solution does exist LP solvers can rely on ecient simplex and interior

point methods that provide optimal solutions within a reasonable time period

223 Locational Marginal Pricing

Locational marginal prices (LMPs) can be derived from the dual of (225)

max yDC = (Pd +Gs)⊤1λ+ (Pd +Gs)⊤F⊤micro

minusT⊤(micro+ micro) +P⊤β minusP⊤β

(226a)

st λ1+ F⊤micro+ β minus β = C [pg isin RN ] (226b)

micro+ microminus micro = 0 [pf isin RK ] (226c)

micro micro β β ge 0 (226d)

where λ isin R is the dual variable on the power balance constraint micro micro micro isin RK are dual

variables on the power ow denition lower limit and upper limit constraints and β β isin RM

are the dual variables to the power generation lower and upper limit constraints

Accordingly LMPs are dened from (226b) as the vector λ isin RN

λ = λ1+ F⊤micro

where λ is commonly called the LMP energy component and F⊤micro is the congestion compo-

nent A third component for line losses is discussed in Chapter 3

For now it will suce to show that λ provides the correct economic signal for each gener-

32


ator to produce the the quantities describe by the solution to (225)2 From complementary

slackness conditions for (225e)

βi(P i minus pgi ) = 0 (227a)

βi

(pgi minus P i

)= 0 (227b)

Suppose that λi lt ci Then from (226b) and nonnegativity of β and β we must have

λi = λ + F⊤i micro = c minus β Since this implies β gt 0 then (227a) implies that the optimal

solution must have pgi = P i

Next suppose that λi gt ci Then we must have λi = λ+ F⊤i micro = c+β Since this implies

β gt 0 then (227b) implies that the optimal solution must have pgi = P i

The last possibility is if λi = ci In this case generator i is the marginal generator

and is indierent to supplying any dispatch between its minimum and maximum limits its

generation level is a basic variable in the LP Thus determining dispatch quantities and

prices from the primal and dual OPF problems ensures that each generator has the correct

incentive to follow the optimal dispatch solution pglowast that was determined by the ISOs OPF

software

If the market possesses the standard conditions of perfect competition (eg perfect in-

formation no externalities zero transaction costs large number of buyers and sellers price-

taking behavior no increasing returns to scale etc (Kreps 2013)) then the LMPs derived

2An additional issue is that there may be multiple solutions to the OPF dual if there is degeneracy inthe primal OPF solution Hence the LMP denition may not be unique This issue will not be focused onin this thesis but Hogan (2012) provides a fuller discussion

33


above satisfy the conditions of a competitive equilibrium Section 232 denes competi-

tive equilibrium more precisely but in short it means that supply equals demand and all

participants produce the quantities that maximize their prot

The next section discusses how this analysis breaks down when trying to dene prices for

the solution to the UC problem

23 Unit Commitment

In contrast to the OPF problem the UC problem considers binary or lumpy ono deci-

sions that are a common feature to most power generating technologies For example most

thermal generators require a xed start up cost to begin producing energy cannot stably

produce energy below a certain threshold quantity unless they are shut o andor cannot

shut down or start up too soon after the previous start up or shut down These constraint

can often require cheaper generators to be dispatched down after another generator has been

committed (switched on) In other words these constraints prevent ISOs from committing

resources in a strict merit order from lowest to highest cost and this imposes a signicant

complexity on the determination of market clearing prices

This section emphasizes the UC problems economic properties rather than the search

for computational eciency that was emphasized for OPF With this in mind Section 231

formulates UC in a very general form that is more conducive to economic analysis Section

232 provides economic denitions and proofs and implications for ISO market design are

34


discussed in Section 233

231 Formulation

To introduce the change in nomenclature that will be used for UC and to develop an under-

standing of the UC problem in very general terms we will rst note that the OPF problems

(223) and (225) are both LPs and can therefore be presented in general LP form (Bertsimas

and Tsitsiklis 1997 Sec 11)

min z = c⊤x

st Ax ge b

x ge 0

where c isin RN is a vector of cost coecients from the piece-wise linear cost function (221)

x isin RN is a vector of dispatch quantities A isin RPtimesQ is a matrix of all constraint coe-

cients and b isin RP is a vector of system demands transmission limits and generator limits

Although formulated with solely greater-than inequality constraints it should be apparent

that both equality and less-than inequality constraints are easily included (as Akx le bk and

minusAkx le minusbk create the equality constraint)

The UC problem can be formulated using similar notation

min z = c⊤x+ d⊤y (228a)

35


st A0x ge b0 (228b)

(xi yi) isin χi foralli isin G (228c)

where the decision variables are the dispatch quantities x isin RN commitment decisions

y isin RN total cost z isin R and xi isin RL and yi isin RL are the components of x and y associated

with generator i the parameters are marginal costs c isin RN xed costs d isin RN system

constraint coecients A0 isin RKtimesN constraint limits b0 isin RK All system-level constraints

are now represented by the linear constraints A0x ge b0 and all generator-level constraints

have been dropped from the system Ax ge b and placed in the constraint set χ =prod

i χi where

χi is the set of generator is constraints on production quantities xi and binary variables yi

The set χi includes non-convex integer constraints

χi = (xi yi) Aixi +Biyi ge bi yi isin 0 1

where Ai isin RLtimesN and Bi isin RLtimesN are the generator constraint coecients and bi isin RL

is the constraint limit With a minor abuse of notation note that xi and yi need not be

scalar values but typically will be vectors describing generator is production quantities with

elements that might reect a stepped supply curve production in dierent time periods or

dierent binary operating status indicator variables

The core diculty of the UC problem is to determine the ys ie the on or o status

of each generator Because these decisions are binary UC is a non-convex and NP-hard

problem However the UC problem has a benecial structure in that all non-convexities are

36


conned to the separable generator constraint sets χi This allows a Lagrangian relaxation

(LR) formulation that is convenient for dening and proving economic properties of the UC

problem Section 243 discusses the use of LR in ISO scheduling software

232 Competitive Equilibrium

Although the UC problems Lagrangian formulation is no longer used for scheduling deci-

sions it is convenient for performing economic analysis of the UC problem The fundamental

issue the UC problem is that it includes binary variables so the previous duality analysis

from Section 223 cannot be used to calculate LMPs and analyze their economic properties

Instead this section will use Lagrangian relaxation to derive a Lagrangian dual problem

(Bertsimas and Tsitsiklis 1997 Sec 114) that can be used to analyze the UC problems

economic properties I will prove a short theorem regarding the competitive equilibrium for

the UC problem based on standard convex programming results (Boyd and Vandenberghe

2004 Sec 54) The competitive equilibrium often does not exist so the end of the section

discusses the modied equilibrium conditions that are satised in practice

First the Lagrangian is dened as

L(x y λ) = c⊤x+ d⊤y + λ⊤ (b0 minus A0x)

The Lagrangian function L(λ) is dened as the following minimization

L(λ) = inf(xy)isinχ

c⊤x+ d⊤y + λ⊤ (b0 minus A0x)

(229)

37


The Lagrangian function above is parameterized by a price vector λ ge 0 that prices

out the system-level constraints A0x ge b0 Given any feasible solution to (228) (x y) the

Lagrangian function will result in L(λ) le c⊤x + d⊤y regardless of the value of λ ge 0 The

Lagrange function L(λ) is therefore a relaxation of (228)

Denition 21 (Lagrangian dual) The Lagrangian dual problem is dened as Llowast = supλge0 L(λ)

In relation to the LMPs dened in Section 223 note that the previous LMP denition

1λ + micro⊤F can be translated to the term A⊤0 λ in this sections notation Given this new

representation of the LMP let the generator prots be dened by πi(λ xi yi)

πi(λ xi yi) =(A⊤

0iλminus ci)⊤

xi minus d⊤i yi

We now introduce denitions for competitive equilibrium and supporting prices

Denition 22 (Competitive Equilibrium) A competitive equilibrium is a set of prices λlowast

and production quantities xlowast such that

a) Each generators schedule xlowasti is prot maximizing

xlowasti isin arg max

(xiyi)isinχi

πi(λlowast xi yi) foralli isin G

b) The market clears A0xlowast ge b0

Denition 23 (Supporting Prices) If a set of prices λlowast and production quantities xlowast satisfy

the conditions of a competitive equilibrium then the prices λlowast are called supporting prices

and we can say that λlowast supports xlowast

38


The ideas of competitive equilibrium and supporting prices are tightly connected to the

Lagrangian dual Llowast Suppose there is an optimal solution to (228) (xlowast ylowast) such that

Llowast = c⊤xlowast + d⊤ylowast In this case the solution to the Lagrangian dual λlowast has the following

important economic property

Theorem 21 (Ecient Competitive Equilibrium) If (xlowast ylowast) is an optimal UC solution and

L(λlowast) = c⊤xlowast + d⊤ylowast then λlowast is a vector of supporting prices in a competitive equilibrium

Before Theorem 21 can be proven we will need the following two lemmas

Lemma 21 (Feasible UC Solutions) For any feasible UC solution (xprime yprime) the objective

function value is c⊤xprime + d⊤yprime = supλge0 L(xprime yprime λ)

Proof Since (xprime yprime) is feasible then b0 minus A0x le 0 This implies that

c⊤xprime + d⊤yprime ge L(xprime yprime λ) forallλ ge 0

The maximum of L(xprime yprime λ) can therefore be obtained by constructing a λprime such that λprimek = 0

for any constraint k such that A0kxprime gt b0k This results in λprime⊤(b0 minus A0x

prime) = 0 so that

supλge0 L(xprime yprime λ) = L(xprime yprime λprime) = c⊤xprime + d⊤yprime

Lemma 22 (Max-Min Inequality) Given the solution to the Lagrangian dual Llowast and the

solution to the UC problem zlowast the following inequality holds

Llowast = supλge0

L(λ) le inf(xy)isinχ

supλge0

L(x y λ) le zlowast

Proof For all (x y) isin χ λ ge 0 we have L(λ) le L(x y λ)

39


Then forall(x y) isin χ we have supλge0 L(λ) le supλge0 L(x y λ)

This implies that supλge0 L(λ) le inf(xy)isinχ supλge0 L(x y λ)

The nal inequality inf(xy)isinχ supλge0 L(x y λ) le zlowast follows from Lemma 21

Proof of Theorem 21 L(λlowast) is a solution to the Lagrangian dual problem which can be

rearranged like so

Llowast = L(λlowast)

= inf(xy)isinχ

c⊤x+ d⊤y + λlowast⊤ (b0 minus A0x)

= sup

(xy)isinχ

(A⊤

0 λlowast minus c

)⊤xminus d⊤y minus λlowast⊤b0

= sup

(xy)isinχ

sumi

πi(λlowast xi yi)minus λlowast⊤b0

= sup(xy)isinχ

sumi

πi(λlowast xi yi)

minus λlowast⊤b0

=sumi

sup

(xiyi)isinχi

πi(λlowast xi yi)

minus λlowast⊤b0

where the exchange of summation and supremum follow from the separability of χ The

production quantities found in the solution to the Lagrangian dual therefore must satisfy

the rst condition of a competitive equilibrium

Next we must show that the optimal UC solution (xlowast ylowast) also satises the solution to the

Lagrangian dual Since (xlowast ylowast) is a feasible solution then Lemma 21 implies the following

c⊤xlowast + d⊤ylowast = supλge0

L(xlowast ylowast λ)

40


And Lemma 22 implies that

Llowast = supλge0

inf(xy)isinχ

L(x y λ) le inf(xy)isinχ

supλge0

L(x y λ) le c⊤xlowast + d⊤ylowast

Further the inequalities above must hold at equality due to the premise of Theorem 21

Combining the results of Lemmas 21 and 22 we have the following

Llowast = inf(xy)isinχ

supλge0

L(x y λ) = supλge0

L(xlowast ylowast λ) = L(xlowast ylowast λlowast)

Then the optimal UC solution (xlowast ylowast) is a feasible UC solution that also satises the

solution to the Lagrangian dual and λlowast supports xlowast in a competitive equilibrium

233 Market Design Implications

There are two practical realities that limit the application of Theorem 21 First there is

said to be a duality gap if Llowast lt zlowast and in this case a supporting price might not exist

Although some network problems (Bertsimas and Tsitsiklis 1997 Theorem 75) are known

to satisfy the zero-duality-gap condition extensive industry experience shows that realistic

UC problems almost always have a positive duality gap Gribik et al (2007)

A second practical concern is that the UC problem is almost never solved to optimality

(Streiert et al 2005 Sioshansi et al 2008a) Instead the mixed integer programming

(MIP) software terminates after it identies a solution that satises a predetermined opti-

mality tolerance3 or after a time limit has been reached Therefore the primal UC solution

3Although solution algorithms are not detailed here note that the Lagrangian function (229) and Lemma

41


is almost always suboptimal so even if there exists a price λlowast that supports the optimal so-

lution the ISO will likely direct its resources to produce at some other quantities xprime that are

not supported by λlowast In almost every case the ISO will also calculate some other prices λprime

that satisfy a modied version of the equilibrium conditions and there is signicant debate

about precisely which modied equilibrium conditions should be adopted

The possibility that the optimal UC solution has no supporting prices is not a new issue

but has been discussed for decades (Scarf 1990 1994) Broadly there are two schools of

thought as to how to modify the equilibrium conditions to determine market prices when

there is a non-zero duality gap to x the UC problem at its optimal solution or to apply a

convex relaxation

ONeill et al (2005) presents the rst method called integer pricing In this method

an ISO rst solves the UC problem and then xes the values of all binary variables to their

optimal value Then the UC problem becomes an LP with an optimal solution that is equal

to the original non-convex UC problem and the dual of this LP can be used to determine

LMPs In addition although the restriction is placed on yi = ylowasti in practice the restriction

will be placed on an integer solution that may not be optimal Assuming generators have

no incentive to change the solutions integer values then the resulting LMPs will satisfy the

competitive equilibrium conditions in the same manner as presented in Section 223

In relation to Theorem 21 ONeill et al (2005) replaces the denition of χi with the

21 can be used to determine lower and upper bounds to the cost of the optimal UC solution Both LRand MIP solution algorithms use variations on this idea to determine which solutions satisfy the optimalitytolerance

42


following restriction

χRi = (xi yi) Aixi +Biyi ge bi yi = ylowasti

In the analysis of ONeill et al (2005) the restriction yi = ylowasti is also included in the

system constraints A0x ge b0 so that the price vector λlowast contains commitment prices for

ylowast in addition to the LMPs that are calculated for xlowast ONeill et al (2005) notes that the

commitment prices will often be negative in eect charging generators to come on line and

suggests that ISOs do not need to collect payments due to negative commitment prices The

remaining positive commitment prices are the formal analogs to make-whole payments that

are paid by the ISO to generators in order to cover any positive dierence between the as-bid

cost of the generator and its revenue from the LMP

Another interpretation of integer pricing is that the analogous development of Theo-

rem 21 does not include yi = ylowasti in the system constraints and therefore does not require

commitment prices Then the economic interpretation the restricted constraint set χRi is

that generators can deviate from the ISOs dispatch quantities xlowast but have no ability to

deviate from the commitment schedule ylowast Later Chapter 5 will show that this is in fact

a heroic assumption and in practice this assumption is belied by common ISO tari pro-

visions that allow generators to self-commit and self-schedule (see reports of uneconomic

self-commitments in MISO 2020b Morehouse 2020) Chapter 5 discusses self-commitment

in more detail and provides examples to show how this market design can lead to market

ineciency by encouraging generators to self-commit

43


The main benet of the ONeill et al (2005) pricing method is that the LMPs provide

exactly the correct signal for all generators to produce at the ecient quantities given

that the ISO also provides them with the ecient commitment schedule and necessary

make-whole payments That is each generators scheduled quantity xlowasti will also be their

prot-maximizing quantity when their feasible outputs are restricted by χRi However this

approach may require the ISO to collect and pay out a large sum of make-whole payments

This greater reliance on side-payments dilutes the LMPs ability to eciently signal ecient

participation and investment in the market

Gribik et al (2007) presents the other major UC pricing methodology and views the

pricing problem through the perspective of reducing the markets reliance on side payments

This approach called convex hull pricing denes a broader category of side-payments called

uplift that compensate generators for the dierence in prot between their prot maximizing

schedule and their prot from following the ISO-determined schedule4 Convex hull pricing

minimizes this set of side-payments by attempting to solve the optimal λlowast in the Lagrangian

dual problem Although this minimizes uplift the uplift payments could theoretically be

larger than the make-whole payments in integer pricing Other aspects of convex hull pricing

may also be problematic The prices are not connected to the physical dispatch solution so

generators may have incentives to deviate from the ISOs schedule (ie the analysis from

Section 223 no longer holds) Uplift payments may also be awarded to generators that are

4For example consider a generator that is scheduled to start up when its LMP revenues will be less thanits production costs This generator could alternatively decide to stay oine and incur no costs so its upliftpayment would be equal to its make-whole payment

44


not scheduled to produce any energy which consumers may object to

It is currently unknown if any UC pricing method necessarily maximizes the total mar-

ket surplus under standard idealized absence of market power conditions5 Instead the

approaches by ONeill et al (2005) and Gribik et al (2007) rely on two dierent axiomatic

modications to the competitive equilibrium analysis Making a grand ceteris paribus as-

sumption ONeill et al (2005) provides the best possible incentive for generators to produce

the correct quantities xlowast but perhaps there are incentives to deviate from the optimal ylowast

commitment schedule Gribik et al (2007) likely does not provide perfect incentives for

either xlowast or ylowast and instead attempts to solve a best compromise where any remaining devia-

tion incentives are removed though uplift payments Each pricing method likely also aects

the incentives for truthfully revealing actual costs in the oers submitted to the ISO yet

very little is presently known about how these incentives might dier

Chapter 5 proves novel bounds on the incentives for deviating from the optimal UC sched-

ule under the various pricing methods However these bounds have to do with diculties

that arise from the practical reality of near-optimal UC scheduling in ISO markets This

issue requires more discussion and will be picked back up in Section 243

5Specically the absence of market power might be variously dened as a market with an asymptoticallyinnite number of participants or one where no individual participant can inuence the market clearing price

45


24 State-of-the-Art and Current Gaps

The following sections provide a more comprehensive background discussion than the brief

literature reviews included in Chapters 3 4 and 5 in order provide a full overview of this

dissertations contributions Most importantly this section will clarify what is the current

state-of-the-art in the models referred to in each research question in Section 12 and what

holes or gaps are present in the literature that prevent the practical implementation of

higher delity models The literature reviews in the later chapters of this thesis are brief

restatements of relevant portions of this chapters comprehensive review and are provided

to remind the reader of the contributions of individual chapters

First Section 241 discusses the state-of-the-art in modeling OPF based on convex relax-

ation of the AC OPF In particular recent semi-denite second order cone and quadratic

convex relaxations have shown promising results in terms of providing a tight approxima-

tion of AC power ows However as will be discussed further the main convex relaxations

rely on variable lifting techniques that do not scale very well in large-scale problems These

relaxations also rely on NLP and semi-denite programming (SDP) software that currently

cannot solve large scale problems as quickly and reliably LP solvers Consequently the aim

of Chapters 3 and 4 is to formulate OPF models with similar physical accuracy of the convex

relation models yet using linear model formulations that can be solved with commercially

available LP software

46


Section 242 reviews the state-of-the-art in linear OPF approximations There are a

plethora of dierent approximations that can made within this class of OPF formulations of

which this section aspires to provide a small glimpse The main drawback of many of these

approaches is the limited use of information about the initial state of the system (ie an AC

power ow) Better use of this information is one of the main advantages of the proposed

OPF formulations in Chapters 3 and 4

Finally Section 243 examines how a long-running economic debate about marginal

pricing in markets with non-convexities and its relation to unresolved ISO market design

issues The beginning of this section reviews the history of this debate showing that the

possible absence of supporting prices in markets with non-convexities has been discussed

over much of the past century without a denitive conclusion Subsequently I discuss how

this very old issuethe lack of market clearing pricesis particularly important in centrally

dispatched electricity markets like ISOs and some of the current approaches that attempt

to solve this dilemma Section 243s broad overview provides background for the analysis

presented in Chapter 5

241 Convex Relaxations of Optimal Power Flow

Convex relaxation allows the application of many powerful convex optimization tools (Luen-

berger and Ye 2008) As shown in Figure 21 convex relaxations modify the grey non-convex

feasible region into the green convex region that includes the entire original non-convex re-

47


(a) Convex Relaxation (b) Convex Hull Relaxation

Figure 21 Convex relaxations of a non-convex region

gion The smallest possible convex region is called the convex hull and is shown in blue in

Fig 21b Non-convex problems can be eciently solved to global optimality if there is a

closed form description of the problems convex hull but such a closed form description is

almost never available in practice (Bertsimas and Tsitsiklis 1997 Sec 103) Instead cur-

rent research aims to provide stronger (also called tighter) convex relaxations by proposing

novel formulations that ideally are subsets of previously known convex relaxations

There are a number of key advantages to using convex relaxations to solve the AC OPF

First by relaxing the AC OPF to a convex problem all solutions are guaranteed to be

globally optimal However because it is a relaxation that solution might not be feasible in

the original problem Therefore if the solution to the convex relaxation happens to be a

feasible AC OPF solution then the convex relaxation results in a globally rather than locally

optimal solution to a non-convex problem In addition the convex relaxation is at least as

feasible as the AC OPF meaning that the relaxed problem is guaranteed to be feasible if

48


the AC OPF is feasible and an infeasible relaxed problem guarantees that the AC OPF is

infeasible

This section provide a brief overview of the main convex relaxation techniques and re-

sults especially semi-denite relaxation (SDR) second order conic relaxation (SOCR) and

quadratic convex relaxation (QCR) The review of these methods is breif and focuses only on

the main convex relaxation of AC OPF Zohrizadeh et al (2020) and Molzahn and Hiskens

(2019) provide more extensive surveys

Convex Relaxation Variants

SDP is a type of nonlinear convex optimization problem and that generalizes the theory

of LPs to include variables in the space of symmetric positive semi-denite (psd) matrices

(Luenberger and Ye 2008 Sec 159) It turns out that it can be applied to a strong relaxation

of the AC OPF To dene the SDP problem let SN be the space of NtimesN symmetric matrices

and w isin SN be the space of decision variables Vandenberghe and Boyd (1996) write the

general SDP problem in the following form

min z = tr(Cw)

st tr(Aiw) le bi i = 1 M

w ⪰ 0

where C isin Sn is a symmetric cost coecient matrix Ai Am isin SN a set of M symmetric

constraint coecient matrices b isin RM are the constraint limits tr(middot) is the trace function

49


(where tr(Cw) =sum

ij Cijwij) and the symbol ⪰ denotes that w must be psd The

constraints tr(Aiw) le bi are linear and the psd constraint is convex The SDP is therefore

a convex optimization problem and it can be solved using algorithms with polynomially-

bounded worst case complexity and practical performance that is typically much better than

worst case (Vandenberghe and Boyd 1996)

Early work on SDP dates back to Bellman and Fan (1963) However practical SDP

solution algorithms were not available until Karmarkar (1984) proposed the interior point

method for linear programming which was then generalized for use in SDP (Vandenberghe

and Boyd 1996 Luenberger and Ye 2008)

The SDR approach to solving AC OPF was rst proposed by Bai et al (2008) The

main transformation used in SDR and SOCR is a change of variables that lifts the decision

variables v isin RN into a higher dimensional space w isin SN Additional constraints for the

cycle condition of Kirchhos voltage law and a matrix rank constraint rank(w) = 1 would

make the problem equivalent to the AC OPF but these constraints are non-convex and

therefore relaxed in the SDR formulation (Low 2014) However the benet is that the

auxiliary variables W allow quadratic constraints to be expressed as linear constraints

The SOCR approach was rst proposed by Jabr (2006) and is similarly formulated as

will be described below Corin et al (2015) formulates the SOCR by taking the non-convex

constraint wij = vivj and relaxing it to a rotated second-order cone constraint |wij|2 le wiiwjj

As shown below this can be cast into an SDP constraint by using the property that a matrix

50


is psd if and only if its leading principle minors are all nonnegative

wii ge 0 wiiwjj minus |wij|2 ge 0 hArr

⎡⎢⎢⎣wii wij

wij wjj

⎤⎥⎥⎦ ⪰ 0

Low (2014) shows that the SOCR formulation is in fact a further relaxation of the

SDR formulation The dierence between the formulations is that SOCR only enforces

the psd constraint on the (i j) submatrices of w where nodes i and j are connected in the

electric network whereas SDR enforces the psd constraint on the full matrix w The SDR

is therefore a tighter relaxation than SOCR but SOCR can be formulated with signicantly

fewer variables (Low 2014)

Proposed by Hijazi et al (2017) QCR uses a dierent relaxation approach that is tighter

than SOCR but not necessarily tighter than SDR (Corin et al 2015) Rather than lift-

ing quadratic terms the QCR applies convex relaxations to the trigonometric functions of

the polar AC OPF formulation (Hijazi et al 2017) The remaining quadratic terms are

then reformulated with their convex envelope bilinear terms with McCormick envelopes

(McCormick 1976) and multilinear terms are handled using a sequential bilinear approach

The resulting relaxation is reasonably tight due to the near linearity of the sine function

around zero and the near linearity of bilinear voltage terms that are near nominal values

(ie vi asymp vj asymp 1) The tighter relaxation between QCR and SDR is case dependent so Cof-

frin et al (2015) note that the QCR and SDR seem to exploit dierent convexity structures

51


Considerations for Practical Implementation

Results from the SDR have spurred signicant interest in convex AC OPF relaxations

Most notably Lavaei and Low (2011) show that the SDR can solve many standard test

case problems to global optimality if each transformer is modeled with a small resistance

In other words the SDR method often performs better by discarding the common ideal

transformer assumption used in many power ow applications (Glover et al 2008 Section

31) Sojoudi and Lavaei (2012) continues this exploration of how the systems physical

characteristics aect the accuracy of the SDR showing that a small number of phase-shifters

and an allowance for load over-satisfaction can also guarantee that the SDR provides a

globally optimal AC OPF solution

However as NLPs OPF models based on these relaxations can have unreliable conver-

gence properties that are not suitable for practical application given currently available NLP

software Stott and Alsaccedil (2012) Regarding scalability in large-scale problems the SDR

SOCR and QCR approaches lift the power ow variables into a higher dimensional space

The number of model variables therefore grows quadratically with problem size and is a

signicant impediment to obtaining faster computation times in larger test cases Lavaei and

Low (2011) Hijazi et al (2017) Low (2014) and Corin et al (2015) show that the SOCR

and QCR methods reduce this problem by exploiting the sparsity of the network structure to

reduce the number of auxiliary variables but this can come at the cost of weaker relaxations

Another diculty is how to obtain a feasible AC power ow when the lifted solution

52


matrix w does not have rank equal to one (Corin et al 2015 Kocuk et al 2015)6 Lavaei

and Low (2011) and Sojoudi and Lavaei (2012) show that this is uncommon given certain

physical characteristics of the transmission grid but Lesieutre et al (2011) provide an ex-

ample that shows how SDR and other relaxed OPF formulations have diculty maintaining

the rank condition in the presence of negative LMPs The SOCR and QCR approaches may

result in larger duality gaps than the SDR and consequently could have even more diculty

in obtaining physically meaningful solutions (Low 2014 Corin et al 2015)

Advocates of linear OPF models point out that there are no general methods to prove

that a nonlinear model is infeasible so nonlinear solvers can sometimes terminate at an

infeasible solution even if a feasible solution exists (Stott et al 2009) It may also take

the solver a long time to converge to a solution for example if the problem is numerically

unstable or uses poorly designed heuristics (Stott and Alsaccedil 2012) SDR SOCR and QCR

also face implementation diculties for many OPF applications A few papers have recently

proposed tight relaxations for integer and AC power ow constraints in applications such as

UC (Bai and Wei 2009) transmission switching (Fattahi et al 2017 Kocuk et al 2017)

and transmission network expansion (Ghaddar and Jabr 2019) Solvers used for the SDR

SOCR and QCR formulations do not natively support the use of integer variables so these

works typically require heuristic methods to recover feasible integer solutions and so far have

only been implemented in the small IEEE test cases (U of Washington 1999)

6This rank condition allows the solution matrix to be decomposed into w = vv⊤

53


Discussion

Convex relaxations of the AC OPF have attracted signicant academic interest in recent

years The relaxations tend to be very tight oering a closer approximation of AC power ow

than standard approaches based on DC power ow Additionally their convex formulations

allow the use of powerful convex optimization algorithms with guaranteed convergence to

globalrather than localoptima This may one day lead to practical OPF implementations

that can provide ISO markets with huge production cost savings (Cain et al 2012)

However there are signicant impediments before such benets can be realized ISOs

currently implement the PTDF formulation of the DC OPF The PTDF formulation is more

compact than the B-theta formulation and is a considerable computational advantage when

implementing security-constrainted problems like SCED The previous convex relaxations

require signicantly more variables than the B-theta formulation due to the inclusion of

voltages and the use of variable lifting techniques PTDFs are also linear and easy to im-

plement with integer-constrained problems such as UC or transmission switching To date

there are still no reliable and computationally ecient mixed integer nonlinear program-

ming (MINLP) solvers that would be necessary for solving industry-relevant problems with

nonlinear convex relaxations (Kronqvist et al 2019)

The next section discusses linear approximations for OPF that have long been used to

solve large-scale industrial applications and bypass many of the practical hurdles faced by

nonlinear formulations

54


242 Linear Optimal Power Flow

Approximation methods do not possess the same feasibility properties as the convex relax-

ation methods and so at rst glance may seem to be either less reliable less accurate or

otherwise less advantageous compared to convex relaxation In fact this is far from being

the case Linear approximations have been the dominant method of formulating OPF in

practical applications for many decades and under most system conditions can obtain good

solutions quickly (Stott and Alsaccedil 2012)

Figure 22 compares the linear approximation and convex relaxation approaches A non-

convex region is shaded grey in each subgure and is shown with a linear approximation in

Fig 22b and a convex relaxation in Fig 22a The dotted line and normal vector denote

the location of the optimal solution Whereas the optimal point in the linear approximation

accurately identies the optimal solution in the original non-convex problem the convex

relaxation causes the optimizer to nd a solution that is infeasible Of course this is not

guaranteed to be the case if for example the convex relaxation were tighter or the linear

approximation was poorly constructed Instead we use the gure to illustrate the idea that

well-constructed linear approximations can be extremely accurate The challenge is to how

to nd such an approximation

Assuming that the physical approximations are reasonably accurate computational per-

formance has always been the main advantage of using linear OPF models (Caramanis et al

1982) Happ (1977) notes that the DC power ow approximation was used as far back as the

55


(a) Linear Convex Relaxation (b) Linear Approximation

Figure 22 Linear approximations of a non-convex region

1920s and was later implemented in an LP by Wells (1968) LP has since been recognized

as one of the most practical methods for solving OPF problems (Stott and Marinho 1979

Alsac et al 1990 Stott and Alsaccedil 2012)

Linear Formulation Variants

There are many variations of the DC OPF7 Stott et al (2009) describes the wide breadth

of cold-start hot-start and incremental model variants of the DC OPF The simplest clas-

sical DC OPF is called the B-theta model (previously introduced in Section 213) and

approximates power ow as a linear function of the susceptance of each branch times the

phase angle dierence between the nodes on either side of the branch (Stott et al 2009)

A result of the linearization is that the DC OPF does not model reactive power or voltage

magnitudes ISOs use the PTDF variant of the DC OPF also variously called a distribution

7In addition to the papers discussed here other surveys give a more comprehensive review of the variousmodels based on the DC OPF (see Chowdhury and Rahman 1990 El-Hawary 1993 Huneault and Galiana1991 Momoh et al 1999ab Qiu et al 2009)

56


factor or shift factor model (Litvinov et al 2004 Eldridge et al 2017) which can be de-

rived isomorphically from the B-theta model (as described in Section 215) In contrast to

the B-theta model the distribution factor model uses dense sensitivity matrices to calculate

ows across each element of the system

Houmlrsch et al (2018) present a third isomorphic DC OPF variant called the cycle-ow

formulation in addition to six other isomorphic equivalents of the DC OPF each based on the

B-theta distribution factor and cycle-ow formulations Computational testing by Houmlrsch

et al (2018) shows that the computational speed of the B-theta and cycle-ow variants are

typically much faster than for the distribution factor models Their results are consistent

with standard results from numerical analysis that show advantages to sparsity in matrix

calculations (Kincaid et al 2009) yet the results are inconsistent with the preference among

ISOs for the distribution factor DC OPF formulation

As explained by Eldridge et al (2017) the distribution factor DC OPF formulation

allows many constraints to be suppressed in large-scale OPF problems resulting in signi-

cant computational advantages that are not captured in many computational studies For

instance consider a system with 8000 buses (N) 10000 lines (K) and only 10 binding

transmission constraints The Btheta model would need to model all N balance constraints

and K line ow constraints for a total of 18 000 equations in order to resolve the 10 binding

transmission limits The cycle-ow formulation uses a graph-theoretic interpretation8 of the

8The cycle-ow formulation is based on dening a cycle basis the electric network that consists ofKminusN+1loops in the network

57


Btheta model to reduce the number of required constraints to K minus N + 1 = 2 001 If the

10 binding constraints are known in advance then the PTDF model can be implemented

with just those 10 transmission constraints and ignoring the power ows on the other 9990

lines The main challenge in the PTDF formulation is to identify which lines are binding

but this is usually not dicult in practical situations since highly-loaded transmission lines

are closely monitored in real time

Today all ISOs implement some form of the PTDF formulation of the DC OPF The

formulation is computationally ecient but introduces power ow error due to the inherent

errors in linear power ow approximations Most power ow error in the DC approximation

can be attributed to line losses As discussed below there are a few dierent ways of

modifying the PTDF model to incorporate line losses in the the OPF problem

Incorporating Line Losses

Although the DC OPF can be quickly solved by standard LP software the assumption

of no line losses can result in inecient generator dispatch prices that diverge from the

marginal cost to deliver power and the need for out-of-model adjustments to achieve power

balance Line losses occur because power ow across the transmission system causes the

conductive material to heat up and dissipate energy These line losses are nonlinear so they

must be somehow approximated to be included in an LP model

A common naive approach to include losses in the DC OPF is to simply increase demand

in proportion to the expected amount of line losses but this method results in suboptimal

58


dispatch since generators are not penalized if their production causes a marginal increase

in line losses or conversely rewarded if their production causes a marginal reduction in line

losses

To correct for each generators marginal contribution to line losses a second common

approach is to calculate marginal loss sensitivities for each bus in the network that will

appropriately penalize or discount the cost of power at each location Because line losses

cause total generation to exceed total demand the marginal sensitivities are also used to

approximate total line losses which are then included in the system power balance constraint

When implemented naively this approach leads to distorted power ows and a KCL violation

at the reference bus (Eldridge et al 2017 Section 31) However is still a commonly used

simplication (see discussions in Litvinov et al 2004 Li and Bo 2007 Li 2011 Santos and

Diniz 2011)

More sophisticated DC OPF models are able to improve the accuracy of line loss ap-

proximations without causing power ow inconsistencies like the previously mentioned KCL

violations One method is to perform iterations to resolve KCL violations at the reference

bus that is to solve a successive linear program (SLP) Li and Bo (2007) propose a DC OPF

model with ctitious nodal demand (FND) FND is a xed power withdrawal that allocates

the expected line losses of each branch evenly to its two connected buses The algorithm

proposed in (Li and Bo 2007) iteratively updates the FND values and once the algorithm

converges results in no KCL violation at the reference bus location Although the iterative

FND model satises the reference bus KCL constraints Bharatwaj et al (2012) points out

59


that the reference bus selection still aects which solution the algorithm will converge to

Bharatwaj et al (2012) then proposes an algorithm to improve the solution by dynamically

changing the reference bus denition in each iteration More recently Garcia et al (2019)

derives the FND formulation of the DC OPF without using the standard DC power ow

assumptions (see Section 213) but only the assumption that all voltages are held xed In

this case Garcia et al (2019) derives linear OPF formulations that closely approximate the

AC OPF

Litvinov et al (2004) proposes a DC OPF model with losses that does not require an

iterative proces and produces LMPs that are independent of the reference bus Instead of

FND this model uses loss distribution factors (LDFs) that distribute the models system loss

estimation into nodal withdrawals Although the solution is independent of the reference bus

Hu et al (2010) points out the solutions dependence on LDFs and the loss function These

parameterizations are typically derived from historical data so Hu et al (2010) proposes an

iterative algorithm analogous to the approach by Li and Bo (2007) to update LDFs and

the loss function based on an AC power ow solution between each iteration

Garcia and Baldick (2020) derive both formulations from Li and Bo (2007) and Litvinov

et al (2004) as part of a series of linearizations of the AC OPF Unlike DC power ow

models the derivation by Garcia and Baldick (2020) linearizes the AC power ow equations

directly rather than by taking the common DC assumptions This results in a more accurate

approximation of the AC OPF and allows Garcia and Baldick (2020) to prove conditions for

when solutions to the linear OPF models will satisfy optimality conditions of the AC OPF

60


However although Garcia and Baldick (2020) uses linear power ow constraints that are

direct linearizations of the AC power ow equations the formulation assumes voltages are

xed and therefore does not model reactive power or voltage

Others have proposed to model line losses with piece-wise linear inequality constraints

especially in long term transmission planning applications One of the rst models was

from Alguacil et al (2003) which sets piece-wise linear constraints a priori in the model

formulation As pointed out by Hobbs et al (2008) one downside to this approach is

that it quickly adds a large number of variables and associated bounds to approximate the

pieces of each quadratic function Santos and Diniz (2011) later proposed to add line loss

inequality constraints iteratively in an algorithm called dynamic piece-wise linearization

Some piece-wise linear formulations allow load over-satisfaction through non-physical line

losses and similar to the SDP limitations discussed by Lesieutre et al (2011) the over-

satisfaction of demand also causes such models to perform poorly in situations where LMPs

should be negative Hobbs et al (2008) and Oumlzdemir et al (2015) avoid this problem by

implementing SLP algorithms to account for system losses and Fitiwi et al (2016) avoids

load over-satisfaction by formulating line losses with integer SOS type-2 constraints The

SLP approach is very eective as Hobbs et al (2008) found that only a handful of iterations

were required for SLP convergence in a test case for the Western North American power

grid

61


Incorporating Reactive Power and Voltage

The downside to linear models is that it becomes more dicult to accurately model

reactive power and voltage Convex relaxation approaches (Bai et al 2008 Jabr 2006

Hijazi et al 2017) are more successful in this regard since the nonlinear convex power ow

approximation will typically be more accurate than the linear approximation Some linear

approximations include reactive power and voltage variables but are very crude approxi-

mations of AC power ow For example Taylor and Hover (2011) applies a similar lifting

technique as the SDR and SOCR approaches to derive a linear relaxation of AC power ows

Corin et al (2016) shows that this relaxation is signicantly less accurate than linear net-

work ow9 and copperplate10 models that can be derived by further relaxing the SOCR

Other linear power ow approximations are also derived from the SDR SOCR and

QCR models Most notably Corin and Van Hentenryck (2014) applies many of the same

relaxations from the QCR formulation (Corin et al 2015) to formulate a piece-wise linear

approximation of AC power ow The linear approximations can be extremely accurate and

can be implemented without pre-specifying a base-point AC power ow solution Bienstock

and Munoz (2014) applies the same variable lifting technique used in SDR and SOCR to

formulate a linear relaxation of the AC OPF Although these linear relaxations result in a

considerably larger objective function gap than the nonlinear convex relaxation models their

9A network ow model models power balance at each bus and transmission constraints but does notinclude any constraint or approximation to satisfy Kirchhos voltage law

10Further copperplate models assume unlimited transmission capacity and therefore all injections andwithdrawals can be assumed to take place at the same bus

62


computational speed is a considerable advantage

Various linear approximations can also be obtained by dening the linear voltage mag-

nitude v squared magnitude v2 or other substitutions as the independent variables in the

linearization The linear voltage magnitude variable is utilized by Zhang et al (2013) which

formulates the power ow equations in terms of deviations from nominal voltage This for-

mulation results in linear approximations for real and reactive power ow after assuming the

squared deviation terms are approximately zero Real and reactive losses are assumed to be

quadratic and are approximated by piece-wise linearization Yang et al (2017) proposes a

linear formulation with independent variables for v2 by performing a substitution on bilin-

ear terms vivj Other substitution approaches have also shown promising results Fatemi

et al (2014) proposes a formulation using a heuristically-derived approximation formula and

substituting independent variables for v2θ

Based on an extensive error analysis Yang et al (2018) nds that the substitution for

squared voltages v2 is empirically the most accurate of the above approaches However Li

et al (2017) derives a linear OPF formulation using a logarithmic transform of the power ow

equations in exponential form (see Equation (21)) that appears to provide more accurate

power ows than the other voltage substitutions

Like for line losses linear OPF models with reactive power and voltage have also been

solved iteratively using SLP Castillo et al (2015) applies an SLP algorithm to solve the

current-voltage (IV) formulation of the AC OPF based on earlier work by Pirnia et al

(2013) Yang et al (2016) points out that the SLP by Castillo et al (2015) often requires

63


too many iterations and that the IV-AC OPF formulation cannot take advantage of the

near-linear relationship between real power ow and the voltage angle Therefore Yang et al

(2016) proposes an SLP based on the standard polar AC OPF formulation (222) by applying

the same v2 variable substitution also applied in Yang et al (2017) Because of this change

of variables the formulation by Yang et al (2016) typically introduces more approximation

error than the approach by Castillo et al (2015) Nonetheless Lipka et al (2016) notes

that the SLP approach has many practical advantages including the ease of implementation

into existing market clearing processes and ability to leverage computationally ecient and

reliable LP solvers that are widely available

Discussion

Iterative procedures have been shown to be eective for accurately modeling nonlinearities

in AC power ow while maintaining a linear model formulation that can be solved with

commercially available LP solvers Similarly Chapter 3 proposes a novel SLP that can

be readily implemented with the same DC OPF-based dispatch model used in most ISOs

Unlike other models from literature the model proposed in Chapter 3 is the rst model

that takes accurate AC power ow data as input and does not require any other AC power

solutions in subsequent iterations

The SLP approach tends to show slower convergence when more aspects of AC power

ow are considered such as reactive power and voltage In general SLPs are known to

exhibit poor performance in modeling nonlinear equality constraints (Bazaraa et al 2013

64


Sec 103) and they have no known globally optimal convergence guarantees for non-convex

problems SLP may therefore be better suited for modeling line losses than for reactive

power and voltage

A single-shot linear OPF may be a better approach for linear OPF models that include

reactive power and voltage and this is the approach taken by the three novel OPF formu-

lations that are proposed in Chapter 4 One of the proposed models is much more compact

(requiring fewer variables) than the models proposed by others (Corin et al 2015 Bien-

stock and Munoz 2014 Zhang et al 2013 Yang et al 2017) The compact formulation

uses dense power ow constraints that are similar to current ISO dispatch models that ap-

ply the PTDF model of DC power ow described in Section 215 This formulation is also

linear and requires signicantly fewer constraints and variables than other formulations that

include reactive power and voltage so it may be a good candidate for future implementation

in security-constrained OPF and UC models

243 Pricing in Markets with Unit Commitment

This section continues the earlier discussion of UC and competitive equilibria from Section

23 The discussion is split into two parts First I discuss the roots of the problem through

classic economic literature dating from around the infancy of mathematical economics and

operations research from around 1930-1960 I then review contemporary discussions on

electricity market design from about 1990 to present and review how certain unresolved

65


issues from the early period still aect important market design topics today

Classic economic literature (1930-1960)

Diculties regarding the UC pricing problem reviewed in Section 23 have in fact been

discussed for a very long time and do not have any obvious solution Nearly any introductory

economics textbook will assert that marginal cost pricing is a necessary component of market

eciency (eg Kreps 2013 Ch 15) The diculty is that market clearing prices become ill-

dened in the presence of xed costs or indivisibilities (eg the non-convexities that appear

in the UC problems integer constraints) In such cases average production costs may exceed

the marginal cost of the last good produced leading to economies-of-scale

The solution from Lerner (1937) and Hotelling (1938) was to maintain a strict marginal

cost pricing policy for all goods in an economy arguing that marginal cost pricing would

lead to the least dstortion in ecient economic activity in particular by resulting in under-

consumption due to ineciently high marginal prices Industries with high xed costs but

low marginal costs (such as railways telecommunications or electric power) would be unable

to make any prot in such a case so Lerner (1937) and Hotelling (1938) therefore propose

that a government agency should determine which activities are economically ecient and

then provide subsidies to recompensate the xed costs of production Coase (1946) suggests

that this particular marginal cost pricing approach may not be so ecient because it would

impede the discovery of economically ecient activity by buyers and sellers and the prices

66


themselves may be dicult or impossible to determine administratively (applying an argu-

ment from von Hayek (1935 pg 226-231) that marginal prices may reect an intricate and

subjective discounting of future opportunity costs) Coase (1946) therefore proposes the use

of a multi-part pricing system in which producers could charge a marginal cost price plus

a xed charge that is invariant to the level of consumption This arrangement would align

with many economic intuitions such as that the xed costs of production should be directly

paid by the consumers of the product or service

An alternative approach was also proposed by Ramsey (1927) suggesting that the optimal

xed cost recovery is obtained by increasing prices in proportion to the inverse elasticity of

the demand from dierent consumer sectors Boiteux (1956) independently derived the

same conclusion However this approach causes price discrimination among consumers

and Eacutelectriciteacute de France (EDF) instead later adopted a peak-load pricing policy (Boiteux

1960) that pays for xed costs by including a capacity adder to marginal costs during peak

periods11

Interest in the marginal pricing issue continued as the elds of operations research and

economics became more advanced Giants in the two elds Gomory and Baumol (1960)

investigated the use of cutting plane algorithms for computing prices of integer-valued prob-

lems Unfortunately the prices do not seem wholly satisfactory For one the cutting plane

algorithm does not determine a unique set of cuts so the resulting prices may change depend-

ing on which arbitrary cuts are added Additionally constraints on integer-valued quantities

11Marcel Boiteux later became President of EDF from 1979-1987

67


may be determined to have a zero price even if increasing the constraints limit by one would

result in a more ecient solution so the prices may be poor guides for determining e-

cient investment in new resources Such peculiar economic properties limited the practical

applicability of the pricing methodology

The elds of operations research and economics have grown more and more distant in the

subsequent decades resulting in very little further progress in the pricing of integer-valued

problems (Scarf 1990)

Electricity market liberalization (1990-present)

Interest in energy market liberalization in the early 1990s then led to further attempts to

connect the UC scheduling problem with fundamental economic concepts but challenges

remained Scarf (1990 1994) notes that there seem to be no good methods for pricing

integer-constrained problems like UC Unlike convex problems integer problems often derail

the use of marginal analysis see Kreps (2013 Sec 98) for the standard approach with convex

production technologies Using a very simple UC example and the standard marginal cost

pricing method Scarf (1994) illustrates that small changes to demand can result in large

positive or negative changes to prices Hence although the marginal cost price provides

the correct price signal to the marginal generator it does not provide a good signal for

investment or the overall level of production In contrast Scarf (1994) also shows that

average cost pricing provides a more accurate signal for investment and overall production

68


levels but does not incentivize the correct production amount from the marginal generator

Thus neither approach is wholly satisfactory

An apparently unrelated diculty is that large scale UC problems cannnot be solved

to optimality Up until the early 2000s were solved heuristically by LR methods Despite

spirited defense of LR by many in the power industry (see Guan et al 2003) the MIP for-

mulation of the UC problem (228) is now ubiquitous Modern MIP solvers have transformed

industry practice thanks to rapidly improving solution times and the ability to handle gen-

eral side constraints such as network power ow (Hobbs et al 2006) and regularly solve

the UC problem within a reasonable amount of time (Streiert et al 2005) However MIP

software is also based on optimality tolerances and will typically terminate before the true

optimal solution is found

Johnson et al (1997) explain that the LR method determines many UC solutions with

essentially the same total cost so the proposed ISO format would involve an arbitrary

selection of which UC solution to send to market participants Using a marginal cost pricing

methodology Johnson et al (1997) shows that the selection of one UC solution over another

can have a large inuence on the protability of dierent resources This was seen as a

disadvantage of the ISO market design as the market operator could have considerable

ability to arbitrarily choose winners and losers After the implementation of MIP-based

UC software Sioshansi et al (2008a) shows that the improved solution quality does not

avoid the price volatility shown by Johnson et al (1997) Further Sioshansi and Tignor

(2012) show that prot volatility over the long run tends to be highest for the most exible

69


generation technologies possibly distorting long-term investment incentives

Integer and Convex Hull Pricing

The main pricing methodologies by ONeill et al (2005) and Gribik et al (2007) have

already been discussed in Section 233 but will now be placed in a wider context

Integer pricing as proposed by ONeill et al (2005) is considered the standard formulation

for LMPs in ISO markets However Hogan and Ring (2003) note that ISOs make certain

deviations from the formal model presented by ONeill et al (2005) For example commit-

ment prices are almost never charged as described formally Negative commitment prices

are ignored because they would have the eect of conscating generator prots When these

commitment prices are positive they are similar to the standard ISO provisions to provide

make-whole payments to generators that do not receive enough market revenue to cover their

as-bid costs Make-whole payments are part of a broader category of uplift payments which

are dened as generally any out-of-market payment used that is required to support ecient

behavior by market participants

The analyses by Johnson et al (1997) and Sioshansi et al (2008a) mostly follow the

integer pricng methodology described by ONeill et al (2005) with the main distinction

that Sioshansi et al (2008a) nds that the make-whole payments suggested by ONeill et al

(2005) lead to a small decrease in the redistribution of economic surpluses due to near-optimal

solutions whereas Johnson et al (1997) does not consider the eects of side-payments

Eldridge et al (2018b) presents the rst analysis of this surplus redistribution that compares

70


outcomes from integer and convex hull pricing and is the basis for Chapter 5

The convex hull pricing methodology proposed by Gribik et al (2007) grew out of earlier

work in Brendan Rings PhD thesis (Ring 1995) As discussed by Hogan and Ring (2003)

an issue with the ONeill et al (2005) prices is that the market settlements may rely heavily

on make-whole payments that distort the prices in the short term electricity spot market

and consequently do not support ecient long term investment incentives Hogan and Ring

(2003) therefore formulate the make-whole payments as part of a set of uplift payments and

they then propose setting prices by a solving minimum-uplift problem This proposal was

then rened by Gribik et al (2007) dening an uplift minimization instead based on the

Lagrangian dual

Schiro et al (2016) describe a number of challenges to implementing convex hull pricing

in an ISO market These challenges include (a) cost allocation for a new uplift payment

category called Product Revenue Shortfall (PRS) (b) no explicit convex hull formulation

or specic amortization of xed costs (c) prices set by o-line resources or physically non-

binding constraints and (d) technical diculties with a rolling horizon dispatch Cadwalader

et al (2010) discusses more detail into the issues with collecting PRS to fund FTR payouts

Finding an ecient algorithm to compute convex hull prices is indeed a dicult math-

ematical problem Wang et al (2013a) and Wang et al (2013b) propose gradient descent

algorithms to solving the Lagrangian dual directly but such solution methods tend to show

poor convergence Gribik et al (2007) explains that solving the Lagrangian dual is very dif-

cult to do in general so instead proposes an approximation called the dispatchable model

71


in which all integer variables are relaxed to be continuous Hua and Baldick (2017) demon-

strates a more rened version of this approach by formulating a tight convex relaxation of

the UC problem This approach is motivated by the fact that the dual of the Lagrangian

dual is the convex hull of the UC problem (see Bertsimas and Tsitsiklis 1997 Ch 11) so

consequently convex hull prices can also be dened as the optimal Lagrange multipliers of

the UC problems convex hull relaxation The conjecture is that tighter convex relaxations

of the UC problem result in closer approximations of the true convex hull price

Although an exact representation of the UC convex hull is exceedingly dicult numerous

mathematical studies have provided relatively simple constraint and tight and compact UC

constraint formulations Here tightness refers how close the UC relaxation is to dening the

actual convex hull and compactness refers to requiring fewer constraints Various formu-

lations include minimum up- and down-time by (Takriti et al 2000 Rajan et al 2005)

thermal unit operation (Carrioacuten and Arroyo 2006 Ostrowski et al 2011 Morales-Espantildea

et al 2012 2013 2015) two-period ramping constraints (Damc-Kurt et al 2016 Ostrowski

et al 2011) generator variable upper bound constraints (Gentile et al 2017) and convex

envelope of generator cost functions (Hua and Baldick 2017) Knueven et al (2017) shows

that a general convex hull formulation of generator ramping constraints is possible but is

impractical because the number of constraints is O(T 3) for a T -period problem

The focus in this section has been on the pricing proposals by ONeill et al (2005) and

Gribik et al (2007) but it should be emphasized that these are not the only approaches that

have been proposed to address the pricing of the UC problems integer constraints Some of

72


these alternatives are discussed below

Other Pricing Proposals

Most similar to the ONeill et al (2005) pricing method is Bjoslashrndal and Joumlrnsten (2008)

which proposes a modication to the ONeill et al (2005) prices based on a decentralized

UC formulation For each demand quantity dprime the Bjoslashrndal and Joumlrnsten (2008) method

sets prices equal to the minimum prices of all solutions where d ge dprime Although this results

in a set of stable and non-decreasing prices it can still result in high uplift payments

Ruiz et al (2012) and Huppmann and Siddiqui (2018) propose EPEC-based approaches

that search for integer solutions that are close to satisfying equilibrium conditions How-

ever these equilibrium-based approached have an inherent disadvantage in that the solution

algorithms may explicitly reject optimal UC schedules

Araoz and Joumlrnsten (2011) and ONeill et al (2016) propose methods that support the

optimal UC schedule by determining prices optimization-based pricing models that augment

the economic conditions constrained in the dual problem An additional approach is pro-

posed by Motto and Galiana (2002) to eliminate uplift payments and replace them with a

potentially complex set of internal zero-sum side payments between market participants

Although these methods support the optimal UC schedule a major obstacle to their imple-

mentation comes from the potential complexity and non-transparency of the methods

73


Discussion

Today all ISOs in the US use a multi-part pricing format similar to the proposal by Coase

(1946) and the oer-based auction format also generally allows oers to reect subjective

assessments of opportunity costs as suggested by von Hayek (1935 pg 226-231) For

example the oer-based auction system in the US can be contrasted with the ineciencies

found in cost-based auction formats commonly used in Latin America (Munoz et al 2018)

However a fully decentralized ISO market design in which all supply is self-committed would

also create ineciencies (Sioshansi et al 2008b) so the dicult task is to design a market

with centralized UC scheduling that still provides good incentives for individual participation

Towards that end Chapter 5 makes a novel analysis of the incentives for deviating from

the optimal UC schedule under the various pricing methods I show that there exists a bound

on the increase or decrease in prots that can result from ineciencies in the UC schedule

and that this bound applies to many computationally simple approximations of convex hull

pricing Appendix B demonstrates that this theoretical result has an important application

for reducing incentives for generators to self-commit (come online) or self-schedule (produce

a specic quantity) without rst being selected in the the ISOs commitment schedule Self-

commitments and self-scheduling account for over 85 of coal generation dispatch in MISO

(MISO 2020b) and are argued to result in costly ineciencies (Daniel et al 2020) so the

theory from Chapter 5 may have substantial real-world benets

I conclude this section by remarking that there are many quite complex issues with imple-

74


menting UC-based market clearing software including the correct representation of generator

capabilities and various algorithmic heuristics that are not discussed here Descriptions of

the above UC formulations and pricing methodologies have also been by necessity quite

brief12 However we have seen that the issue of pricing in UC-based markets goes back to

early in the previous century at least and yet the tools for analyzing this problem are only

in their infancy

12Knueven et al (2018) provides a more detailed review of UC formulations Liberopoulos and Andrianesis(2016) review electricity pricing methods in more detail and Van Vyve (2011) discusses those pricing systemsin relation to the methods used in Europe

75

Chapter 3

An Improved Method for Solving the

DC OPF with Losses

31 Introduction

Almost all Independent System Operators (ISOs) include the marginal cost of line losses

to optimize system dispatch and all include marginal losses in the calculation of locational

marginal prices (LMPs) used for settlements (see tari references CAISO (2020) ISO-NE

(2019) MISO (2020a) NYISO (2020c) PJM (2010) SPP (2020)) Assuming a competitive

This chapter was previously published with co-authors Richard ONeill and Anya Castillo Although co-authors include members of FERC sta the views expressed in the chapter do not necessarily represent theviews of FERC or the US Government The previous publication has been edited for clarity and consistencywith the rest of the dissertation and can be cited as B Eldridge R ONeill and A Castillo An ImprovedMethod for the DCOPF With Losses IEEE Transactions on Power Systems 33(4)37793788 2018a

76

CHAPTER 3 IMPROVED METHOD FOR SOLVING THE DC OPF WITH LOSSES

market the LMP market design is economically ecient1 because the price signal to each

node reects the increase in system cost required to serve the next unit of demand

As discussed in Section 22 ISOs determine generator dispatch by solving a security

constrained economic dispatch (SCED) model and the core problem solved by SCED is

called optimal power ow (OPF) The ideal OPF problem is the alternating current (AC)

OPF (Sec 221) which optimizes over the actual ow of AC power in the transmission

system but this formulation is a nonlinear non-convex optimization problem that cannot

be eciently solved by existing methods ISOs instead solve linear direct current (DC) OPF

models (Sec 222) since they can be solved within the time limitations of the day-ahead

and real-time market (DAM and RTM) clearing windows (Cain et al 2012) Making the

linear approximation as close as possible to the actual physics can help prices to accurately

reect the marginal cost of electricity so most DC OPF models by implemented by ISOs

use power ow sensitivities that are determined from a base-point solution (Stott et al

2009) which may be from a state estimator AC power ow analysis or dispatch solution

To improve upon those current practices this chapter rst derives an accurate linear loss

function approximation from the AC power ow equations then proposes a novel method

for updating the loss approximation without solving additional AC power ow problems

1See Kreps (2013 Sec 86) for a discussion on the desirability of market eciency as the main marketdesign criteria Other criteria such as environmental externalities transparency simplicity fairness or otherstakeholder concerns may also be relevant to good market design

77


311 Current Practices

ISOs typically implement the DC OPF with power ow sensitivities called power transfer

distribution factors (PTDFs) Section 215) and line loss sensitivities called loss factors

(Litvinov et al 2004) Calculating PTDFs requires selecting a reference bus that is assumed

to be the marginal source (or sink) of any changes in power consumed (or produced) A

common alternative to the distribution model approach is called the B-theta model (Stott

et al 2009) and also results in a linear model However the B-theta model takes a few orders

of magnitude longer to solve the security-constrained OPF problem used in ISO software

and therefore is not used to clear ISO markets

This chapter will therefore focus on distribution factor implementations of the DC OPF

In particular the DC OPF model can be used to dene the LMP vector λ isin RN which is

commonly decomposed into three components

λ = λE + λL + λC (31)

where λE is the marginal cost of energy at the reference bus and λL and λC are respectively

the marginal cost of losses and the marginal cost of transmission congestion both with

respect to the reference bus

Line losses can contribute signicantly to marginal costs and this has a number of im-

plications for ISO applications Because physical line losses are a convex and approximately

quadratic function the marginal cost of line losses is about twice the average cost of losses

78


This results in an over-collection for the total cost of line losses that is then generally re-

funded to load on a pro rata basis In 2019 PJMs total cost of marginal line losses was $642

million and included a $204 million marginal loss surplus while total congestion costs were

about the same magnitude at $583 million (Bowring 2020) A study by ERCOT (2018)

which does not currently use marginal losses in dispatch showed that total generator rev-

enues would change by $216 million and -$332 million in its Houston and North geographic

zones respectively if marginal losses were included in ERCOTs dispatch model Financial

participants have also been able to exploit inconsistencies in the modeling of losses in the

DAM and RTM (Patton 2012 FERC 2015) Although losses are typically 1-2 of ISO

billings the dollar amounts can be signicant and similar in magnitude to congestion costs

Table 31 summarizes the processes used by each ISO Standard ISO power ow applica-

tions are shown in bold The rst standard application the state estimator uses a statistical

model and AC power ow equations to t real-time power ow measurements to the system

topology and determine the current state of the system The second application the network

model solves a DC power ow based on the current system topology and a set of real power

injections and withdrawals such as from a security constrained unit commitment (SCUC)

or SCED solution Despite approximately $114 million in projected annual production cost

savings ERCOT is the only ISO in the US that does not include marginal losses in its sys-

tem dispatch models (ERCOT 2018) As shown in Table 31 each ISO implements slightly

dierent loss factor methodologies

79

CHAPTER3

IMPROVEDMETHODFORSO

LVING

THEDCOPFWITHLOSSE

S

Table 31 ISO line loss approximation methodologies

ISO(Source)

Used inSCED

Base-point (DAM) Base-point (RTM) Update Frequency

CAISO(2020)

Yes Network model withSCUC solution

Network model withinput from state estima-tor

Every hour in DAM andevery 5 minutes in RTM

ERCOT(2016)

No Linear interpolation orextrapolation of forecastdemand with ono peakloss factors

Linear interpolation orextrapolation of actualdemand with ono peakloss factors

Seasonal

ISO-NE(2019)


Most recent state esti-mator solution


MISO(Sutton2014)

Yes Recent state estimatorsolution with similar loadand wind conditions



NYISO(2020ab)


Network model withlast dispatch solution


PJM(2010)

Yes State estimator solu-tion with estimated fu-ture operating conditions



SPP(2020)




80


312 Literature Review

As discussed in Section 242 of Chapter 2 there are a wide variety of linear OPF models

based on the DC OPF A brief overview of those methods is provided below and for more

detail the reader may refer back to Section 242

The DC OPF problem remains the standard problem for electric dispatch applications

Computational performance has always been the main advantage of using linear OPF models

and researchers have been interested in computationally ecient and physically accurate DC

OPF formulations and loss sensitivity calculations for many years (Stott and Marinho 1979

Shoults et al 1979 Alsac et al 1990 Chang et al 1994 Stott et al 2009) New DC

OPF formulations remain an active area of research One of the most important DC OPF

applications is the calculation of LMPs for electricity markets (Hu et al 2010 Litvinov

et al 2004 Orfanogianni and Gross 2007 Peng et al 2009 Li 2011 Liu et al 2009)

The DC OPF is also an important aspect in transmission expansion planning (Hobbs et al

2008 Fitiwi et al 2016) renewable energy and storage integration (Castillo et al 2014)

and other applications that are not enumerated here

Iterative approaches to the DC OPF (Hu et al 2010 Li and Bo 2007 Hobbs et al 2008

Bharatwaj et al 2012 Santos and Diniz 2011 Helseth 2012) have shown some success at

improving the physical accuracy of the model Some use additional AC power ow solutions

after each iteration (Hu et al 2010) while others (Li and Bo 2007 Hobbs et al 2008

Bharatwaj et al 2012 Santos and Diniz 2011 Helseth 2012) only use DC power ow and

81


estimate losses with the quadratic loss equation from Schweppe et al (1988 Appendix D)

Line loss constraints that model losses on individual lines (Santos and Diniz 2011) or each

node (Helseth 2012) can also improve the approximation but results in a much larger model

than using a single system-wide loss constraint (Litvinov et al 2004)

However the iterative approach has advantages compared to methodologies that either

require nonlinear solvers due to the inclusion of quadratic (Schweppe et al 1988) or conic

(Jabr 2005) contraints a large number of constraints due to piecewise linearization (Santos

and Diniz 2011 Helseth 2012) or the inclusion of binary variables (Fitiwi et al 2016)

While these approaches may have better accuracy they also increase the formulations size

and complexity Many DC OPF models use the B-theta power ow approximation which is

not suitable for ISO applications because the SCED formulation would require an additional

set of voltage angle variables for each outage scenario and therefore scales poorly in large-

scale systems (Stott and Alsaccedil 2012)

313 Contributions

This chapter proposes the rst sequential linear programming (SLP) procedure for the DC

OPF formulation with marginal line losses that uses an AC power ow as input data and

then iteratively improves the line loss approximation without the use of additional AC power

ow solutions The key advantage of the proposed methodology is that it does not introduce

any new variables or constraints to the formulation by Litvinov et al (2004) the standard

82


DC OPF formulation currently used in market software Its use of AC power ow data is

also consistent with the industry practices described in Table 31 Therefore the proposed

method improves the model from Li and Bo (2007) by incorporating an initial AC base-point

and it reduces the computational burden of the model from Hu et al (2010) because it does

not require solving a new AC power ow after each iteration Additionally the approach

shows robust and accurate performance with a wide range of starting points

The rest of the chapter is organized as follows Section 32 reviews the B-theta and PTDF

DC power ow approximations from Section 21 and then derives two sets of loss factors

rst from the AC power ow equations and then from a less accurate quadratic line loss

approximation that will be used for comparison purposes Section 33 formulates the linear

DC OPF model An example problem is presented in Section 331 to compare the LMPs

that result from three dierent DC OPF formulations a naive model the model with loss

factors based on the quadratic line loss approximation and then the model with loss factors

based on the linearized AC power ow equations Then results are presented for a one-shot

solve of each model without performing iterations Section 34 presents the proposed SLP

algorithm and demonstrates it on a selection of test cases to show that the proposed iterative

procedure is fast and accurate Section 35 concludes the chapter

83


Notation

Variables and parameters will be dierentiated by upper and lowercase letters with upper-

case letters denoting model parameters and lowercase letters denoting model variables To

dierentiate vectors and matrices with scalar values boldface will denote a vector or ma-

trix and regular typeface will denote scalar values Calligraphic text denotes a set with set

indices denoted by lowercase letters Dual variables will be denoted by letters of the Greek

alphabet as will certain variables and parameters (eg θ ϵ ) that commonly use Greek

letters in the engineering and mathematics literature

Sets

K Set of K transmission lines

k isin K

Ki Subset of K connected to

bus i Ki = Kfri cup Kto

i

Kfri Kto

i Subset of K with from or

to bus i

N Set of N nodes or buses

i j n isin N

R Set of real numbers

Ξ Power ow solutions ξ isin Ξ

Parameters

τkn isin R Transformer tap ratio at

n-side of branch k

ϕkn isin R Phase shifter at n-side of

branch k

ω isin R Damping parameter

1 Vector of ones of suitable

length

A isin RKtimesN Network incidence matrix

dened as Aki = 1 and

Akj = minus1 if k isin Kfri cap Kto

j

and 0 otherwise

B isin RKtimesK Diagonal branch susceptance

matrix with elements Bk

84


C isin RN Generator marginal costs

D isin RN Loss distribution vector

E(y)k isin R Loss approximation

adjustment for branch k

y isin 0 1 2

F isin RKtimesN Dense real power ow

sensitivity to real power

injections

Gk isin R Conductance of branch k

Gsn isin R Shunt conductance at bus n

I isin RNtimesN Identity matrix

L isin RKtimesN Branch loss factor coecient

matrix

L0 isin RK Branch loss factor oset

L isin RN System-wide loss factor

vector

L0 isin R System-wide loss factor

oset

M isin RNtimesN AC-linearized nodal

susceptance matrix

Pd isin RN Real power demand

PP isin RM Maximum and minimum

real power output

Rk isin R Resistance of branch k

Sθ isin RNtimesN Voltage angle sensitivity

matrix

T isin RK Power ow limit

U isin RN Loss penalty factor

Xk isin R Reactance of branch k

Variables

θ isin RN Voltage phase angle with

elements θi

θij isin R Voltage phase angle

dierence between buses

θi minus θj

pf isin RK Real power ow in the

from direction with

elements pfk

pt isin RK Real power ow in the to

direction with elements ptk

pg isin RN Generator real power output

with elements pgn

pℓ isin R Real power system losses

pnw isin RN Net real power withdrawals

with elements pnwi

u isin R Reference bus injection

v isin RN Voltage magnitudes with

elements vi

y isin R Dual objective function

z isin R Primal objective function

85


Dual Variables

α α isin RN Dual variables to the

generator upper and lower

limits

λ isin R Dual variable to the system

power balance constraint

λ isin RN Locational marginal price

λAC isin RN Dual variables to nodal

power balance constraints in

the AC OPF

micro micro isin RK Dual variables to the power

ow upper and lower limit

constraints

σ isin R Dual variable to the loss

function constraint

Additional Notation

Hadamard product

⊤ Matrix or vector transpose

xlowast Optimal solution

xAC AC OPF solution

xξ Fixed variable in ξ isin Ξ

32 Power Flow Derivations

This section extends the previous DC power ow derivations in Section 21 by linearizing a

feasible AC power ow solution to derive an accurate marginal line loss approximation that

can be implemented in a DC OPF model As dened in Section 21 power ows through

each branch k isin K can be dened from node i to j or from j to i

pfk = Gkτ2kiv

2i minus τkivivj (Gk cos (θij minus ϕki) + Bk sin (θij minus ϕki)) (32a)

ptk = Gkτ2kjv

2j minus τkjvivj (Gk cos (θij minus ϕki)minus Bk sin (θij minus ϕki)) (32b)

where the parameters are the branch conductance Gk branch susceptance Bk tap trans-

former tap ratio at the i side of branch k τki transformer phase shifter at i side of branch

86


k ϕki and the variables are the voltage magnitude vi and voltage angle θi Dierences in pfk

and ptk will be used to calculate line losses but otherwise it will be assumed that pfk = minusptk

The real power ow variables are stored in a vector pf isin RK and are assumed to ow in the

from direction

The amount of power generated minus the amount consumed at a node must be equal to

the amount owing out of its adjacent transmission lines Power generation (an injection)

and load (a withdrawal) are simplied using the net withdrawal pnwi at node i isin N which

by convention is positive for a net withdrawal and negative for a net injection Losses due to

shunt conductance are modeled analogously to loads For real power the network balance

equations are

pnwi +sumkisinKfr

i


i

pfk = 0 foralli isin N (33)

where Kfri is the set of branches k with from side i and Kto

i is the set of branches with

to side i The same system can also be expressed in matrix form

pnw +A⊤pf = 0 (34)

where A isin RKtimesN is a network incidence equal to 1 if branch k is assumed to ow into node

i minus1 if the branch is assumed to ow out of node i and 0 if branch k is not connected to

node i and v isin RN is a vector of nodal voltage magnitudes

87


321 DC Power Flow

Many industry applications rely on DC power ow approximations DC power ow equations

are preferable in many instances because they are linear and can be solved quickly Con-

versely AC power ow equations model the system more accurately but are nonlinear and

non-convex It can even be dicult to nd a feasible solution to AC power ow equations

in a large scale system such as one of the main US power grids As previously described in

Section 213 the standard DC power ow approximation makes three main assumptions

bull Voltage is close to one per unit (pu) at all buses

bull Voltage angle dierences are small ie sin(θi minus θj) asymp θi minus θj and cos(θi minus θj) asymp 1

bull A lossless network ie Rk ≪ Xk or equivalently Gk ≪ Bk

The B-theta power ow equation (214b) can then be derived from (32) and is repro-

duced below with the inclusion of phase shifters Φ isin RK

pf = minusB(Aθ + Φ

) (35)

where B isin RKtimesK is a diagonal matrix with values Bk asymp minus1Xk

for a lossless model A isin RKtimesN

is the network incidence matrix and θ isin RN is a vector of nodal voltage angles

To reduce solution time in practice equation (35) can be simplied using PTDFs also

called shift factors (Stott et al 2009) PTDFs describe the fraction of real power injected at

each bus that ows across each branch (Wood et al 2013) The injection (or withdrawal) is

88


assumed to be withdrawn (or injected) at the reference bus Let this reference bus be dened

by W isin RN a vector that sums to one such that the reference bus is the weighted sum of

physical bus locations The PTDF is then dened by Equation (218) which is conveniently

reproduced below ⎡⎢⎢⎣A⊤BA

W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (36)

322 Marginal Line Losses

Line losses are the sum of (32a) and (32b) and loss factors dene the linear sensitivity of

total system losses to real power injections at each bus Loss factors will be determined from

a base-point solution denoted by ξ isin Ξ where (θξvξ) denotes base-point voltage angle and

magnitude values and Ξ is the set of all possible base-points (eg from (32)) Then let

M isin RNtimesN be a more exact representation of the nodal susceptance matrix A⊤BA where

the diagonal entries are taken from the partial derivatives of pfk and the o-diagonal entries

from the partial derivatives of ptk both derivatives being with respect to voltage angles at

the base-point solution ξ

Mii =sumkisinKi

τkivξivξj

(Gk sin(θξij minus ϕki)minus Bk cos(θξij minus ϕki)

) foralli isin N

Mij = τkivξivξj

(Gk sin(θξij minus ϕki) + Bk cos(θξij minus ϕki)

) forall(i j) isin Ki

The change in voltage angles ∆θ resulting from a marginal real power injection ∆pnw is

89


given by the linear system with a bordered matrix of M and the reference bus weights W⎡⎢⎢⎣ M W

W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣∆θ

u

⎤⎥⎥⎦ =

⎡⎢⎢⎣∆pnw

0

⎤⎥⎥⎦ (37)

The bottom row W⊤∆θ = 0 constrains the voltage angle at the reference bus which is

xed at zero The variable u isin R is a reference bus injection resulting from the marginal

withdrawals ∆pnw Therefore if ∆pnwn = 1 and ∆pnwi = 0 for all i = n then the loss factor

for bus n is uminus 1

Let U isin RN be the vector of marginal reference bus injections and Sθ be the sensitivity

of voltage angles ∆θ to withdrawals ∆pnw The linear system (37) can be expanded to solve

for L and Sθ directly by replacing pnw with the identity matrix The result is essentially a

matrix inversion problem ⎡⎢⎢⎣ M W

W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣ Sθ

U⊤

⎤⎥⎥⎦ =

⎡⎢⎢⎣ I

0⊤

⎤⎥⎥⎦ (38)

Then loss factor vector is simply L = U minus 1 Loss factors for each branch can also be

computed from the matrix Sθ Let pℓ isin RK be the vector of branch line losses By summing

(32a) and (32b)

pℓk = Gk

(τ 2kiv

2i + τ 2kjv

2j minus 2τkivivj cos(θij minus ϕki)

) (39)

90


A sparse matrix L isin RKtimesN gives the partial derivatives partpℓkpartθn

Lkn = 2Gkτkvξnvξm sin(θξnm minus ϕkn) forallk isin Kfrn

Lkm =minus 2Gkτkvξnvξm sin(θξnm minus ϕkn) forallk isin Ktom

Then a dense matrix L isin RKtimesN gives individual branch loss factors where L⊤= 1⊤L

L = LSθ (310)

Lastly a constant L0 is calculated such that the line loss approximation is exact at the

base-point2 Let pℓ isin R be the total system line losses Pd isin RN be the vector of xed nodal

demands and pg isin RN be a vector of nodal generator injections such that pnw = Pd minus pg

The system loss approximation is

pℓ = L⊤(Pd minus pg) + L0 (311)

Equation (311) is linear and can be easily integrated into market optimization software

323 Alternative Line Loss Derivation

Alternatively a set of loss factors can be derived assuming a quadratic loss formula (Schweppe

et al 1988 Chang et al 1994 Hobbs et al 2008 Santos and Diniz 2011 Helseth 2012

Li 2011 Bharatwaj et al 2012 Fitiwi et al 2016) which assumes small angle dierences

and that all voltages are equal to 1 pu The derivation is originally given in the Appendix

2For now use L0 = pℓξ minus L⊤(Pd minus pg

ξ ) See Sec 42 for the formulation in terms of θξ and vξ

91


to Bohn et al (1984)

Lki =dpℓkdpnwn

=dpℓkdθijtimes dθij

dpnwn

=2Rk

R2k +X2

k

τkivivj sin θij timesdθijdpnwn

(312)

Assuming θij is small sin θij asymp θij Similarly to the standard B-theta approximation we

can also approximate pfk asymp minusBkθij asymp (R2k +X2

k)minus12θij Then make the substitution for θij

2Rk

R2k +X2

k

τkivivjθij timesdθijdpnwn

=2Rk

(R2k +X2

k)12

τkivivjpfk times

dθijdpnwn

(313)

The PTDF can be dened as Fkn = minusBkdθijdpnw

n Similar to the previous step we take the

approximation Fkn asymp (R2k + X2

k)minus12 dθij

dpnwn

and make the substitution for dθijdpnw

n Rearranging

terms and summing Li =sum

k Lki then the result from (312) is

Li = 2sumk

RkτkivivjFknpfk (314)

However due to simplied voltage and cosine assumptions the loss factor calculation

(314) loses some delity compared to the calculation in (38) The above derivation is based

on the Appendix to Bohn et al (1984) and predates the commonly cited (Schweppe et al

1988 Appendix D) Both references derive the same quadratic loss formula below

pℓ asympsumk

Rk(pfk)

2 (315)

92


33 Model

The following analysis uses the DC OPF model that was implemented by ALSTOM EAI

Corp for ISO-NE and Litvinov et al (2004) formulated below

max z = C⊤pg (316a)

st 1⊤ (pg minusPd)minus pℓ = 0 (316b)

pℓ = L⊤(Pd minus pg) + L0 (316c)

minusT le F(Pd minus pg minusDpℓ

)le T (316d)

P le pg le P (316e)

where the decision variables are power generation pg and total system losses pℓ parameters

are the generator marginal costs C (see cost function assumptions Section 22) power

demand Pd the loss function coecients L and L0 loss distribution factors D PTDFs F

transmission limits T and generator limits P and P

Each element Di of the loss distribution factor D isin RN allocates line losses into a nodal

withdrawal at node i As suggested by Litvinov et al (2004) each Di is calculated to be

proportional to the line losses in the branches connected to each bus This formulation a

violation of Kirchhos current law at the reference bus by ensuring that the injections and

withdrawals in (316d) sum to zero (Eldridge et al 2017 Sec 31) In addition Litvinov

et al (2004) shows that the resulting LMPs are independent of the reference bus selection

93


LMPs are obtained by solving the dual of (316)

max y =λ1⊤Pd + σ(L0 + L

⊤Pd

)minus micro⊤

(T+ FPd

)minus micro⊤

(Tminus FPd

)+ α⊤Pminus α⊤P

(317a)

st λ1+ σL+ micro⊤Fminus micro⊤F+ αminus α = C (317b)

minus λ+ σ +(microminus micro

)⊤FD = 0 (317c)

micro micro α α ge 0 (317d)

where λ isin R is the dual variable to the system balance constraint (316b) σ isin R is the

dual variable to the system loss constraint (316c) micro micro isin RK are the dual variables to the

transmission limits (316d) and α α isin RN are the dual variables to the generator output

limits (316e) Constraint (317b) forms the basis for LMPs with the terms commonly

decomposed into three components

λE = λ1 (318a)

λL = σL (318b)

λC =(microminus micro

)⊤F (318c)

λ = λE + λL + λC (318d)

where λE λL and λC are the marginal costs of energy losses and congestion all with respect

to the reference bus

94


331 Model Initializations

It is important to initialize the OPF model with a base-point solution that accurately ap-

proximates line losses Three initializations of (316c) are tested

bull DC OPF assumes no marginal losses L = 0 and L0 = 0 and compensates demand

Pd by a scalar factor α = 1 + pℓξ1⊤Pd proportional to total losses in the base-point

bull DC OPF-Q assumes voltages are uniformly 1 pu and uses the power ow variables

pfξ to calculate loss factors L by equation (314) and calculates L0 by solving pℓξ =

Lpnwξ + L0 from the base-point solution

bull DC OPF-L uses base-point values (θξvξ) to calculate loss factors by solving equation

(38) and calculates L0 by solving pℓξ = Lpnwξ + L0 from the base-point solution

Each initialization uses progressively more information from the base-point solution The

rst model the standard DC OPF only uses the total losses in the base-point solution to

estimate line losses and assumes zero marginal line losses The DC OPF-Q model uses the

system topology (via the PTDF F) and power ows pfξ to calculate loss factors assuming

the quadratic loss function (315) This initialization is similar to what would be computed

from a DC power ow solution in an ISOs network model (eg as described in Table 31)

Finally the DC OPF-L model linearizes the AC power ow equations directly and is similar

to what would be computed based on an ISOs state estimator (eg Table 31)

95


332 LMP Accuracy

Each model is solved using the IEEE 300-bus test case from the University of Washington

test case archive (U of Washington 1999) The analysis was implemented in GAMS based

on code available from (Tang and Ferris 2015) In this case the base-point for each model

is an AC OPF solution which is highly optimistic Later Section 344 investigates model

solutions where the base-point is less advantageous In addition all transmission line limits

in the test case have been relaxed This simplication to the test case was included so that

all dierences shown in the results will indicate how the dierent line loss approximations

aect the accuracy of the LMP calculation

The linear model solutions are compared to solutions to an AC OPF model implemented

by Tang and Ferris (2015) The AC OPF is non-convex and may not always nd the

globally optimal solution Nonetheless it is used here as a benchmark for our DC OPF

results AC OPF LMPs are the dual variable of the real power balance constraint in an AC

OPF solution (Liu et al 2009) while DC OPF LMPs are calculated from (318) The prices

from the AC OPF solution range from $3719MWh to $4676MWh Since transmission

losses are only 12 of total demand in this test case this price spread is much larger than

might be expected and underscores the importance of accurate line loss modeling

Figure 31 shows results from the solving IEEE 300-bus test case with each model initial-

ization The DC OPF-L is the most accurate model for both pricing and dispatch The most

simplistic model DC OPF is included here to demonstrate a naive approach and produces

96


Table 33 IEEE 300-bus test case solution statistics

Avg Disp LMP Rel CostModel Di (MW) MAPE () Di ()

DC OPF 259 377 -0179DC OPF-Q 93 123 -0035DC OPF-L 18 024 -0002

the same price for each node in the system The dispatch is consequently inecient because

the marginal cost of line losses is not accounted for The DC OPF-Q model does a better job

of dierentiating locations based on marginal losses but it also mis-estimates the marginal

eect by a large amount at some buses because it assumes network voltages are at their

nominal values The DC OPF-L produces prices and dispatch that are very similar to the

AC OPF and is the closest of all three linear models

Maximum relative LMP errors for each initialization are as follows DC OPF underes-

timates the LMP at bus 528 by 141 DC OPF-Q overestimates at bus 51 by 57 and

DC OPF-L overestimates at bus 250 by 38 Further comparisons of the three models are

given in Table 33 Three summary statistics are dened by

Avg Dispatch Di =1

N

sumi

|pglowasti minus pgACi | (319)

LMP MAPE =1

N

sumi

|λlowasti minus λAC

i |λACi

times 100 (320)

Rel Cost Di =zlowast minus zAC

zACtimes 100 (321)

The relative performance of the each initialization is network-specic but in most cases

97


0 50 100 150 200 250 300-15

-10

-5

0

5

10

15DC OPFDC OPF-QDC OPF-L

(a) Relative Price Dierence

0 10 20 30 40 50 60-150

-100

-50

0

50

100


(b) Dispatch Dierence

Figure 31 Accuracy comparison of DC OPF formulations

98


the DC OPF-L will perform the as well or better than the others because it can be tuned to

the current operating conditions of the network For example the DC OPF-Q model assumes

voltages are at their nominal levels and therefore may overestimate marginal losses if the

network is operating at higher than its nominal voltage (as occurs at buses 265-300 in Figure

31) The DC OPF-L computes losses with respect to the base-point voltage magnitude and

voltage angle so its loss factors reect the reduction in marginal line losses due to operating

parts of the system at higher voltages

It should be emphasized that there is no additional computational cost to the DC OPF-L

initialization ocmpared to the DC OPF-Q On the IEEE 300-bus test case the DC OPF-

L initialization decreases the cost gap with the AC OPF by more than 20x compared to

DC OPF-Q and more than 100x compared to the naive DC OPF model Average dispatch

dierence and LMP MAPE are both reduced by about 5x compared to DC OPF-Q and 15x

compared to DC OPF

The next section uses the optimal real power ow pflowast of the DC OPF-L initialization to

iteratively update the line loss approximation so that the line loss approximation error is

reduced in cases with system demands that dier from the original base case

34 Proposed SLP Algorithm

The base-point in the previous section was the AC OPF solution but such a good base-

point is not possible in practice This section presents a motivating example to show how the

99


Table 34 Two node example

GeneratorsBus Initial Oer ($) Final Oer ($) Capacity (MW)

A 1 3000 2950 10B 1 3000 2975 100C 2 3000 3000 100

Transmission LoadFrom To Resistance (Ω) Bus Demand (MW)1 2 00005 2 90

one-shot linear programming (LP) solution from the previous section can lead to inecient

dispatch That is the simplied DC OPF results in an inconsistency and therefore higher

costs than indicated by the DC OPFs optimal objective function value I then describe a

novel SLP algorithm that corrects this inconsistency Results are then presented to show

that the algorithm converges in the IEEE test cases and still obtains an accurate solution

when system conditions dier from the original base-point solution

341 Motivating Example

Consider the two node problem described in Table 34 Three generators initially have

identical costs and are connected by a resistive transmission line For simplicity it is assumed

that the voltage at both nodes is 1 so line losses are precisely equal to R12(pf12)

2

A few potential solutions are given in Table 35 When accounting for line losses Solution

3 is clearly optimal for the initial bids Suppose that in the next time period generators A

and B reduce their bids after purchasing new gas contracts on the spot market Instead of

100


Table 35 Solutions for initial and nal bids

SolutionDispatch 1 2 3Gen A 10 MW 10 MW 0 MWGen B 8446 MW 0 MW 0 MWGen C 0 MW 8005 MW 90 MWFlow 9446 MW 10 MW 0 MWLosses 446 MW 005 MW 0 MWInitial oers Total CostNo losses $270000 $270000 $270000Actual losses $283384 $270150 $270000

Final oers Total CostNo losses $267500 $269500 $270000Actual losses $280773 $269650 $270000

Presumed optimal solution

$30 the new bids are $2950 for generator A and $2975 for generator B The new costs are

shown on the `Final Oers - Actual Losses line of Table 35 and Solution 2 is optimal when

line losses are accurately modeled

However current practices miss a key point in this scenario Suppose that Solution 3 is

used as a base-point to calculate loss factors Then there are no losses in the network since

pf12 = 0 so the marginal cost of line losses is also zero The dispatch model would therefore

select the cheapest generators A and B corresponding to Solution 1 without considering

the actual cost of line losses The bottom row of Table 35 shows that the actual line losses

in Solution 1 increases dispatch cost by about 5 and the total cost is about 4 higher than

the actual optimal solution Alternatively if the ISO were to parameterize the loss function

(316c) to be consistent with Solution 2 then the ISO would have correctly identied the

optimal dispatch The key diculty is how to identify the correct base-point solution

101


342 Algorithm Description

This section proposes a novel SLP algorithm to update loss factors in such a case This

results in a more accurate representation of marginal losses which results in more accurate

prices and more ecient dispatch

To help parameterize how the loss function should be updated we note that total system

losses can be decomposed to individual branches with the losses on each branch taking a

quadratic form similar to (315)

pℓ =sumk

pℓk =sumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (322)

Any quadratic function can be given by dierent values of E(y)k y isin 0 1 2 so (322)

includes the previous quadratic approximation (315) as a special case Ignoring the sum-

mation for now rearranging (322) gives

pℓk = E(2)k (pfk)

2 +(2E

(2)k E

(1)k

)pfk +

(E

(2)k (E

(1)k )2 + E

(0)k

)(323)

The rst-order Taylors series of (323) assessed at pfk = pξfk is

pℓk asymp 2E(2)k (pξ

fk + E

(1)k )pfk + E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (324)

The core idea in the SLP methodology therefore comes from the linear approximation in

(324) This function splits into rst order linear coecients (2E(2)k (pξ

fk +E

(1)k )) that can be

summed to calculate the loss factor Ln and constant terms (E(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k )

102


that can be summed to calculate the loss oset L0 Then each time the model is solved the

line loss function can be updated with new values pξfk = pflowastk and the SLP can be terminated

when the size of the update approaches zero

Although (322) (and therefore (324)) can be parameterized by equation (315) Section

331 shows that this approximation (the DC OPF-Q model) can result in signicant pricing

errors Instead the SLP algorithm combines the quadratic approximation with the more

accurate loss factor initialization (38) from the DC OPF-L model

First let the elements Lkn L0k Ln and L0 be dened so that the linear constraint (311)

is expressed in terms of the quadratic function parameters E(y)k y isin 0 1 2

Lkn = 2E(2)k (pξijk + E

(1)k )Fkn (325a)

L0k = E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (325b)

Ln =sumk

Lkn (325c)

L0 =sumk

L0k (325d)

The initial base-point solution can only specify a point and slope of the function (322)

which is unfortunately not enough to specify all three values of E(y)k The additional degree

of freedom can be eliminated by the appearance of pkk in equation (314)

dpℓkdpnwn

= 2RkτkivivjFknpfk (326)

which suggests that E(2)k = RkτkivivjFkn in equation (322) Coecients E(1)

k and E(0)k can

103


Algorithm 1 Proposed SLP for improved line loss approximation

Input FDR L L0Pdpgξ vξp

ℓξ τki

1 pfξ larr F(pg

ξ minusPd minusDpℓξ)

2 E(2)k larr Rkτkivξivξj forallk isin K

3 E(1)k larr Lkn

(2E

(2)k Fkn

)minus pξ

fk n = argmaxm(|Fkn| m isin i j forallk isin K

4 E(0)k larr L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)

2) forallk isin K5 solve (316) h = 1

6 while |z(h)minusz(hminus1))|z(hminus1) ge tol and h le hmax do

7 pξgn larr pglowastn pξ

fk larr pflowastk foralln isin N forallk isin K

8 pξℓ larr

sumk E

(2)k (pξijk + E

(1)k )2 + E

(0)k

9 Ln larr 2sum

k

(E

(2)k (pξijk + E

(1)k )Fkn

)foralln isin N

10 L0 larr pξℓ minussum

n Ln(pξgn minus P d

n)11 solve (316) hlarr h+ 112 end while

subsequently be calculated based on the initial values of Lkn and L0k

E(2)k = Rkτkivξivξj (327a)

E(1)k =

Lkn

2E(2)k Fkn

minus pξfk (327b)

E(0)k = L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)

2) (327c)

By construction the loss function above is a rst order Taylor series approximation of

(322) and the values of E(1)k and E

(0)k can be updated when new values pξ

fk are available If

an initial AC solution is not available one can assume E(2)k = Rk and E

(1)k = E

(0)k = 0 and

the algorithm is the essentially same as the SLP described by Hobbs et al (2008)

Algorithm 1 was implemented with the following few numerical side notes First the

assignment of E(1)k requires an arbitrary selection for the index n for Lkn and Fkn This can

104


be a source of numerical errors but choosing n = argmaxm(|Fkn| m isin i j) helps to

minimize these errors Similarly a numerical issue can occur when calculating E(1)k if E(2)

k is

very small or zero due to very low resistance on the line In this case set a tolerance value

ε gt 0 and let E(1)k = 0 if E(2)

k lt ε

Lastly the update rule was implemented using a damping parameter ω isin [0 1] in Line 7

of the algorithm

pξgh+1n = ωpξ

ghn + (1minus ω)pglowastn (328a)

pξfh+1k = ωpξ

fhk + (1minus ω)pflowastk (328b)

Step size constraints may also be useful in larger or more complex networks but they

were not found to be necessary for convergence In addition their formulation requires

considerable care to avoid infeasible model solves or convergence to a suboptimal solution

Each iteration in this SLP solves an approximation of a quadratically constrained program

(QCP) This QCP is the same formulation as (316) except that the constraint (316c) is

replaced with the following relaxation of (322)

pℓ gesumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (329)

Because the is problem convex any locally optimal solution is also a global optimum

Introducing inequality loss constraints may cause articial losses when the constraint is not

binding However the loss constraint was binding in each solution of the relaxed problem

and therefore the relaxed solutions were also optimal in the unrelaxed QCP

105


343 Convergence Results

Results from Algorithm 1 are shown in Figure 32 for a selection of test cases from the

University of Washington test case archive (U of Washington 1999) as well as few other

that are available in MATPOWER (Zimmerman et al 2011) The analysis was implemented

by modifying the GAMS code from Tang and Ferris (2015) Solution times were measured

on a laptop computer with a 230 GHz processor and 8GB of RAM CPLEX 125 solved

SLP and Ipopt solved the QCP and AC OPF PTDF values less than 001 were removed

and quadratic cost functions were approximated as piecewise linear functions with ten steps

to improve solution times3

Including the damping modication (328) improved the convergence speed of all test

cases and the 118- and 300-bus cases did not converge unless the damping parameter was

used After some trial and error ω = 025 for the smaller cases (lt100 buses) and ω = 05 for

the larger cases (118- and 300-bus networks) showed good results Generally setting ω too

large can slow down convergence but setting it too small may cause solution cycling issues

in the algorithm

The results in Figure 32 were obtained by uniformly increasing demand parameters by

5 compared to the base-point solution and randomizing generator costs by multiplying by

a normal random variable N(1 002) These parameter changes led to a binding line limit in

3Both techniques have a minimal change in the dispatch solution The PTDF truncation procedure isexplained in greater detail in Sec 43 of Chapter 4

106


0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103

case6wwcase9case14case24case30case39case57case118case300

(a) Dispatch ∥pgh minus pghminus1∥2

0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103


(b) Power ow ∥pf h minus pf hminus1∥2

0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103


(c) LMPs ∥λh minus λhminus1∥2

0 5 10 15 201e-09

1e-06

0001

1


(d) Total cost ∥zh minus zQCP ∥2

Figure 32 SLP algorithm convergence

the 39-bus network but did not aect convergence The randomization step was necessary

because many of the generators have identical cost functions in the original data sets and

this can cause degeneracy issues that impede convergence

Convergence was measured with the standard L2 norm dened as the square root of the

sum of squared dierences Each iterative solution to (316) is indexed by h Values for

pghpf h and λh were compared with the previous iteration Figure 32 also shows conver-

107


Table 36 Solution comparison of SLP and AC OPF

Avg Disp LMP Rel CostNetwork Di (MW) MAPE () Di ()case6ww 0121 0725 -0135case9 0006 0375 -0007case14 0163 0270 -0379case24 0125 0406 0041case30 0035 0393 -0129case39 3551 1246 0039case57 3575 1239 -0094case118 0983 0255 -0229case300 6223 0912 -0023

gence with respect to the objective function of the QCP The objective function converges

to within 001 of the QCP solution by the sixth iteration in each test case Although there

is not a proven convergence guarantee it was fairly easy to achieve the results using a very

simple damping method

Table 36 compares the SLP and AC OPF solutions Dispatch quantities are typically

within a few MW of the AC OPF dispatch and LMPs relavtive errors are about 1 The

relative dierence in total cost was less than 04 in each case

Table 37 shows the number of iterations required for the SLP to converge and compares

solution times of the three models The convergence criterion was set at a 001 change in

the objective function between iterations All test cases met this criteria within two or three

iterations The SLP was consistently faster to solve than the QCP and AC OPF models

and would likely have signicantly better relative performance in larger test cases due to its

linear formulation and small number of constraints and variables

108


Table 37 Computational comparison of SLP QCP and AC OPF

SLP Solution time (s)Network Iterations SLP QCP AC OPFcase6ww 2 0026 0177 0171case9 3 0053 0167 0295case14 2 0042 0167 0285case24 3 0072 0241 0378case30 2 0070 0260 0264case39 3 0059 0232 0273case57 2 0068 0235 0373case118 2 0117 0458 0635case300 2 0246 0625 1157average of ten trials

344 Varying the Demand Levels

Varying the demand levels of each test case illustrates that the SLPs performance does

not depend on providing a base-point that already represents the optimal system dispatch

The demand variations are parameterized by multiplying demands by a system-wide scalar

ranging from 090 to 110 in 001 increments The base-point solution is the the AC OPF

solution when this multiplier equals one and is the same for each demand level After

the initial LP solve the marginal line loss approximation is updated according to the SLP

algorithm (Algorithm 1) using a damping parameter ω = 025 (and ω = 05 for the 118- and

300-bus networks)

The sensitivity analysis measured the eect of increased demand on accuracy of the

LMP and the loss approximation with respect to marginal prices and losses calculated by a

nonlinear AC OPF problem (Tang and Ferris 2015) LMP accuracy is again measured by

109


09 095 1 105 11Demand multiplier

001

01

1

10

100

1000LMP

MAPE


-03

-02

-01

0

01

RelativeLoss

Error

Base pointIteration 1Iteration 2Iteration 3

Figure 33 Error sensitivity analysis in the IEEE 24-bus test case

MAPE and loss accuracy was measured by relative error

Relative Loss Error =pℓlowast minus pℓAC

pℓACtimes 100 (330)

The 24-bus network tended to have poor line loss accuracy in the rst LP solve and its

results from subsequent iterations are shown in detail in Figure 33 The proposed approach

is fairly robust to non-ideal starting points as there is very little approximation error after

three iterations of the SLP The most signicant error in the 24-bus case is a 1 LMP MAPE

when the demand multiplier is 094 Relative loss errors are practically zero

In all of the test cases the average LMP MAPE at Iteration 3 was 095 (worst-case

307 in the 39-bus network) and the average relative loss error was 002 (worst-case

110


042 in the 14-bus network) These results were obtained by implementing a relatively

naive damping rule and may possibly be improved with a more sophisticated update rule

35 Conclusion

The DC OPF is at the core of many applications in todays electricity markets but compu-

tational advantages of its LP formulation come at the expense of approximating the physics

of power ow The analysis presented in this chapter therefore focuses on improving the

accuracy of the DC OPF model by implementing a high delity line loss approximation and

devising update rules to correct for changes in system dispatch Implementation of the SLP

algorithm in an ISO-scale network would be an important step in proving its computational

eectiveness which is left for future work

Additionally the proposed approach motivates a broader analysis of trade-os between

computation speed and physical accuracy of dispatch models For example speed require-

ments or the availability of a base-point solution may be dierent in real-time dispatch or

long-term planning contexts Various approximation methods such as piecewise linear ap-

proximations (Santos and Diniz 2011 Helseth 2012) or conic programming (Jabr 2005)

should be compared in each context

The loss approximation is the largest component of the DC OPF models inherent ap-

proximation error A feasible AC base-point provides valuable information about voltage

angles and voltage magnitudes that are omitted from many DC OPF formulations Adding

111


this information improves the accuracy of marginal line losses The proposed SLP algorithm

can be used to further improve the accuracy of the loss function and may be of use to re-

searchers interested in modeling electricity markets or practitioners interested in improving

the eciency of ISO market dispatch software Inaccuracy of the dispatch models marginal

loss approximation can signicantly aect generator dispatch and market pricing so the

methods explained in this chapter help to reduce this inaccuracy

This topic continues into Chapter 4 where I discuss extensions to the DC OPF formu-

lation to improve accuracy by including linear approximations for reactive power ows and

voltage levels

112

Chapter 4

Formulation and Computational

Evaluation of Linear Approximations

of the AC OPF

41 Introduction

In the previous chapter an improvement to the direct current (DC) optimal power ow

(OPF) was formulated by performing iterative updates to the models line loss approximation

until the line loss approximation is accurate and consistent with the physics of alternating

This chapter was drafted with help from my co-authors Anya Castillo Ben Knueven and ManuelGarcia Although this work was authored in part by sta from FERC Sandia National Laboratory andthe National Renewable Energy Laboratory the views expressed do not necessarily represent the views ofthose organizations or the US Government Funding provided by the US Department of Energys Oceof Electricity Advanced Grid Modeling (AGM) program

113

CHAPTER 4 LINEAR APPROXIMATIONS OF THE AC OPF

current (AC) power ow However iterative procedures come with some downsides Except

in certain cases the updated model may not be guaranteed to be feasible (the main exception

being if one iterations optimal solution can be proven to be feasible in the subsequent

iteration which can sometimes be proven for convex problems) Finally iterations also cost

time to perform An attractive alternative is to formulate the OPF so that the problem

is solved by commercial solver software in a single shot without any iterations Instead

of performing iterations this chapter formulates linear OPF models with a high degree of

physical accuracy by including very high-delity approximations for power ows line losses

and voltage magnitudes Rather than only modeling real power both real and reactive power

are included in the approximations and computational results show that including reactive

power and voltage constraints in the OPF formulation signicantly improves the models

physical accuracy

411 Problem Denition

The search for more ecient methods of solving the AC OPF problem has been an active

area of research in recent years OPF underlies many of the daily processes undertaken by

Independent System Operators (ISOs) and therefore ecient AC OPF solution methods

hold promise for large savings in annual electricity production costs (Cain et al 2012) The

AC OPF is highly nonlinear and non-convex so ISOs instead rely on a simplied linear

model called the DC OPF (Stott and Alsaccedil 2012) as discussed in Chapter 3 Whereas the

114


AC OPF co-optimizes an explicit formulation of real and reactive power ows DC OPF

models assume that voltage magnitudes are xed which results in a simplication that

completely ignores reactive power (Stott et al 2009) This chapter proposes three novel

linear OPF formulations that improve upon the physical accuracy of DC OPF models by

including voltage and reactive power constraints The models dier in the compactness and

sparsity of the formulations A fourth OPF model drops the additional voltage and reactive

power constraints for a MW-only formulation that can be solved considerably faster but is

less accurate

As discussed in Section 241 others have proposed simplifying the AC OPF by convex

relaxation methods including the semi-denite relaxation (Bai et al 2008) second order

conic relaxation (Jabr 2006) and quadratic convex relaxation (Hijazi et al 2017) Although

these methods have shown promising results (Lavaei and Low 2011 Corin et al 2015

Castillo and Gayme 2013) their implementation in large-scale ISO markets is limited by

poor scaling properties of the formulations (Lavaei and Low 2011 Hijazi et al 2017) Linear

models scale very well and benet from huge eciency improvements in linear programming

(LP) software over the past 30 years (Bixby 2015)

Although DC OPF models are often implemented using the ubiquitous B-theta DC

power ow constraints (Stott et al 2009) most ISOs have instead implemented power trans-

fer distribution factor (PTDF) or shift factor DC OPF formulations (Litvinov et al 2004

Eldridge et al 2017) of the same type implemented in the previous chapter The PTDF

formulation of the DC OPF is isomorphic to the B-theta formulation but is more compact

115


and is formulated by numerically dense transmission constraints Various studies have shown

considerable computational performance improvements of the B-theta formulation over oth-

ers (Houmlrsch et al 2018) mostly owing to standard results from numerical analysis that show

advantages to sparsity in matrix calculations (Kincaid et al 2009 Sec 46) However test-

ing these OPF models in explicit form ignores many simplications that are used in practical

implementations of the PTDF model

Linear OPF approximations are reviewed in more detail in Section 242 Recent work by

Garcia and Baldick (2020) derives linear OPF (LOPF) formulations directly from AC power

ow rather than using the common DC power ow assumptions previously described in

Section 213 This approach to formulating linear OPF models is highly applicable to ISOs

because as previously shown in Table 31 the use of an AC power ow base-point solution

is common in many ISOs This data can be obtained from from standard state estimator

software that estimates current system conditions based on real time measurements and a

model of the systems topology

412 Contribution

This chapter therefore contributes three novel LOPF formulations that include real and re-

active power constraints and voltage magnitude constraints We propose a sparse LOPF

(S-LOPF) that is derived from a Taylor series approximation of the AC power ow equa-

tions Then a dense LOPF (D-LOPF) model is derived by reformulating the S-LOPF with

116


distribution factor type constraints The D-LOPF formulation is more compact than the

S-LOPF requiring fewer constraints and few variables A third model the compact LOPF

(C-LOPF) is formulated with fewer constraints and variables by summing (ie condensing

or relaxing) the line loss constraints of the D-LOPF

After deriving the S-LOPF D-LOPF and C-LOPF models a fourth LOPF model that

we call the P-LOPF is formulated by removing all reactive power and voltage constraints

from the C-LOPF model This P-LOPF model possesses only minor dierences compared to

previous DC OPF formulations derived by linearizing the AC line loss equations (Litvinov

et al 2004 Eldridge et al 2018a Garcia and Baldick 2020) but is used to benchmark

quality of the SDC-LOPF solutions compared to a MW-only formulation

In addition the chapter proposes three simplication techniques that improve computa-

tional performance of the D-LOPF and C-LOPF models in large-scale test cases First we

describe a lazy constraint (or active set) algorithm that reduced model size by ignoring

non-binding transmission constraints Second we introduce a hybrid line loss constraint for-

mulation for the D-LOPF model that aggregates line losses from inactive branches further

reducing the size of the model Last we introduce a distribution factor truncation proce-

dure that increases the sparsity of the constraint matrices used in the D-LOPF and C-LOPF

formulations Because of the models accuracy and relatively small number of constraints

and variables after applying these simplications the C-LOPF may be a good candidate for

future unit commitment (UC) and security-constrained OPF model implementations

Later in this chapter results from a broad selection of test cases indicate that the proposed

117


formulations can be solved substantially faster than the full AC OPF and are highly accurate

with respect to providing a nearly-feasible AC power ow The proposed simplication

techniques are also shown to reduce solution times while only adding power ow errors that

are much smaller than other common approximate OPF models like the P-LOPF and both

the PTDF and B-theta formulations of the DC OPF

413 Outline

The rest of the chapter is organized as follows Section 42 provides derivations and formu-

lations for the proposed linear OPF models and is followed by a discussion of simplication

techniques in Section 43 Results for the computational performance and power ow ac-

curacy of the proposed models are presented in Section 44 and Section 45 concludes the

chapter

Notation

As in the previous chapter variables and parameters will be dierentiated by upper and

lowercase letters with uppercase letters denoting model parameters and lowercase letters

denoting model variables To dierentiate vectors and matrices with scalar values boldface

will denote a vector or matrix and regular typeface will denote scalar values Calligraphic

text denotes a set with set indices denoted by lowercase letters Dual variables will be

118


denoted by letters of the Greek alphabet as will certain variables and parameters (eg θ

ϵ ) that commonly use Greek letters in the engineering and mathematics literature

Sets


k isin K



i

Kfri Kto


to bus i

L Set of L sample points used

for linearization l isin L

M Set of M generators

m isinM

Mi SubsetM located at bus i


i j n isin N

Parameters

τki Transformer tap ratio at

i-side of branch k

ϕki Phase shifter at i-side of

branch k

1 Appropriately sized vector

of ones


dened as Aik = 1 and

Ajk = minus1 if k isin Kfri cap Kto

j

and 0 otherwise

Bsik Shunt susceptance at i-side

of branch k

Bk Susceptance of branch k

Cym Cost coecients for

generator m y isin 0 1 2

F F isin RKtimesN Real power ow

sensitivities

F0 F0 isin RK Real power ow osets

Gk Conductance of branch k

H H isin RKtimesNReactive power ow

sensitivities

H0 H0 isin RK Reactive power ow osets

I Identity matrix

K K isin RKtimesN Reactive power loss

sensitivities

K0 K0 isin RK Reactive power loss osets

K isin RN System-wide reactive power

loss sensitivity

K0 isin R System-wide reactive power

loss oset

L L isin RKtimesN Real power loss sensitivities

119


L0 L0 isin RK Real power loss osets

L isin RN System-wide real power loss

sensitivity

L0 isin R System-wide real power loss

oset

PdQd isin RN Real and reactive power

demand

PP isin RM Maxmin real power output

QQ isin RM Maxmin reactive power

output

Rk Resistance on branch k

Sθ isin RNtimesN Voltage angle sensitivity to

real power injections

Sv isin RNtimesN Voltage magnitude

sensitivity to reactive power

injections dense N timesN

matrix

S0θ S

0v isin RN Voltage angle osets

T isin RK Max power transfer limit

VV isin RN Maxmin voltage magnitude

limit

Xk Reactance on branch k

Variables

θ isin RN Voltage phase angle

pf pt isin RK Real power ow in the

from and to directions

pfαpℓ isin RK Mid-line real power ow and

branch line losses

pg isin RM Real power generation

pnw isin RN Real power net withdrawals

pℓ isin R System-wide real power loss

qf qt isin RK Reactive power ow in the


qfαqℓ isin RK Mid-line reactive power ow

and branch line losses

qg isin RM Reactive power generation

qnw isin RN Reactive power net

withdrawals

qℓ isin R System-wide reactive power

loss

v isin RN Voltage magnitude


Additional Notation


| middot | Element-by-element absolute

value

zlowast Optimal solution

[middot] Dense matrix

zξ Fixed variable in solution

ξ isin Ξ

120


42 Model Derivations

The following analysis rst reformulates the AC power ow constraints into a convenient mid-

line formulation and then performs a rst-order Taylor series expansion of each constraint

The rst linearization of of the mid-line power ow equations results in a set of sparse

constraints that are the basis for the S-LOPF Then voltage angle and magnitude variables

are substituted out of the sparse linearization to formulate the dense power ow constraints

that are the basis of the D-LOPF Next the dense constraints are made more compact to

form the C-LOPF Transmission voltage and generator limits are then introduced and the

section concludes by specifying the explicit SDCP-LOPF formulations

The LOPF model formulations are based on a simplication and linearization of the

polar formulation of the AC OPF rst formulated by Carpentier (1962) For convenience

the formulation from Section 221 is provided below

max zAC =summisinM

Cm(pgm) (41a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj

(Gk cos (θij minus ϕki) + Bk sin (θij minus ϕki)

)forallk isin K (41d)

ptk = Gkτ2kjv

2j minus τkjvivj

(Gk cos (θij minus ϕki)minus Bk sin (θij minus ϕki)

)forallk isin K (41e)

121



2kiv

2i minus τkivivj




2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T

2


)2+(qtk

)2le T

2


V i le vi le V i foralli isin N (41j)

Pm le pgm le Pm forallm isinM (41k)

Qmle qgm le Qm forallm isinM (41l)

θref = 0 (41m)

The AC OPF constraints include real and reactive power balance at each bus (41b) and

(41c) real power ows at the from (sending) and to (receiving) end of each branch (41d)

and (41e) reactive power ows at both sides of each branch (41f) and (41g) transmission

ow limits at both sides of each branch (41h) and (41i) voltage magnitude lower and

upper limits (41j) generator real and reactive power output limits (41k) and (41l) and

a reference bus voltage angle dened by (41m)

To dene the notation let a base-point solution be denoted by ξ isin Ξ so that (θξvξ)

denotes base-point voltage angle and magnitude values and Ξ is the set of all possible base-

points (eg from a state estimator or more optimistically a solution to (41))

122


421 Mid-Line Power Flow Reformulation

Reformulation of (41) begins by deriving mid-line power ows in the same manner as Garcia

et al (2019) but extending the derivation to include reactive ows Let pfk(α) = (1minusα)pfkminus

αptk be the real power ow measurement along branch k at a position α isin [0 1] where α = 0

corresponds to a measurement taken at node i and α = 1 to node j The reactive power

ow measurement is similarly dened by qfk (α) = (1minus α)qfk minus αqtk

pfk(α) = (1minus α)Gkτ2kiv

2i minus αGkv

2i

minus τkivivj((1minus 2α)Gk cos (θij minus ϕki) + Bk sin (θij minus ϕki)

) (42a)

qfk (α) = minus(1minus α)(Bk +Bski)τ

2kiv

2i + α(Bk +Bs

kj)v2j

minus τkivivj(Gk sin(θij minus ϕki)minus (1minus 2α)Bk cos(θij minus ϕki)

) (42b)

Setting α = 05 denes the real and reactive mid-line power ows variables which will

be called pfαk and qfαk in order to dierentiate from the pfk and qfk in the AC power ow

equations (41d) and (41f)

pfαk = Gk

(τ 2kiv

2i minus v2j

)2minus Bkτkivivj sin (θij minus ϕki) (43a)

qfαk =minus((Bk +Bs

ki)τ2kiv

2i minus (Bk +Bs

kj)v2j

)2minusGkτkivivj sin(θij minus ϕki) (43b)

The above expressions approximate the real and reactive power ows in terms of a voltage

dierence and a sine function Recalling the common DC power ow assumption that Bk ≫

Gk the expressions above illuminate the intuition behind tight coupling of real power with

123


voltage angles and reactive power with voltage magnitude used in the fast-decoupled load

ow (Stott and Alsac 1974) since the sine function is the dominant term in (43a) and the

dierence of squared voltages is the dominant term in (43b)

Real and reactive power loss variables pℓk and qℓk are dened as the dierence between

power ows at i and j that is pℓk = pfk(0)minus pfk(1) and qℓk = qfk (0)minus qfk (1)

pℓk = Gk

(τ 2kiv

2i + v2j

)minus 2Gkτkivivj cos (θij minus ϕki) (44a)

qℓk = minus (Bk +Bski) τ

2kiv

2i minus

(Bk +Bs

kj

)v2j + 2Bkτkivivj cos (θij minus ϕki) (44b)

Two observations can be made from the separation of mid-line power ows and line

losses First the sine functions only appear in the mid-line power ow equations and cosine

functions only appear in the line loss equations This allows these two nonlinear functions to

be conveniently handled completely separately Second the line losses expressions disappear

completely from (42a) and (42b) when α = 05 Thus half of the losses are assumed to

occur on the side of the branch closest to node i and half on the side closest to node j

4211 Power Balance

Kirchhos Current Law (KCL) implies that power entering a node must be equal to the

amount of power leaving the node and is enforced by constraints (41b) and (41c) in the

AC OPF These power balance constraints will be modied for the LOPF models to reect

the mid-line power ow denitions First real and reactive net withdrawals pnw and qnw

are dened based on power generation load and a linearization of any shunt conductance

124


or susceptance devices

pnwi = P di +Gs

i (2vξivi minus vξ2i )minus

summisinMi

pgm foralli isin N (45a)

qnwi = Qdi minus Bs


summisinMi

qgm foralli isin N (45b)

Using the mid-line power ows (43a) and (43b) and losses (44a) and (44b) we substi-

tute pfk = pfαk + 12pℓk p

tk = minusp

fαk + 1

2pℓk q

fk = qfαk + 1

2qℓk and qtk = minusq

fαk + 1

2qℓk Power balance

at each bus can then be succinctly written in linear using the network incidence matrix A

and the absolute value function | middot | applied to each element of A

pnw +A⊤pfα +1

2|A|⊤pℓ = 0 (46a)

qnw +A⊤qfα +1

2|A|⊤qℓ = 0 (46b)

4212 Sparse Linearization

The following sensitivity matrices are evaluated at a general base-point solution (vξ θξ)

using standard assumptions of decoupled power ow (Stott and Alsac 1974) that is as-

suming partpfαpartv = partqfαpartθ = partpℓpartv = partqℓpartθ = 0 Explicit denitions are located in

Appendix A1

F = partpfαpartθ H = partqfαpartv

L = partpℓpartθ K = partqℓpartv

(47)

To construct a linearization oset terms are calculated by summing the xed and con-

stant terms of the rst-order Taylors series approximation base-point values of pfαξ qfα

ξ pℓξ

125


and qℓξ are each calculated from (vξ θξ) and the equations (43a) (43b) (44a) and (44b)

F0 = pfαξ minus Fθξ H0 = qfα

ξ minusHvξ

L0 = pℓξ minus Lθξ K0 = qℓ

ξ minusKvξ

(48)

The construction of (47) and (48) constitutes rst-order Taylor series expansions of

(43a) (43b) (44a) and (44b) around the base-point solution (vξ θξ)

pfα = Fθ + F0 (49a)

qfα = Hv +H0 (49b)

pℓ = Lθ + L0 (49c)

qℓ = Kv +K0 (49d)

Power networks are generally sparsely connected so the constraints (49) constitute a

sparse linearization of the AC OPF (41) While the formal error analysis is omitted the

mid-line real power ow pfαk is typically very accurate because the second order error term

of the Taylor series is negligible the error is proportional to sin(θij) asymp 0 assuming small θij

The approximation for pℓk may not be negligible its second order errors are proportional to

cos(θij) which is not close to zero assuming small θij By similar arguments the errors to

qfαk and qℓk may be even larger than for pℓk due to a non-zero second order error term in the

Taylor series and the assumption that Bk ≫ Gk

126


4213 Dense Linearization

As previously discussed in Section 215 it is often advantageous to formulate the power ow

equations using distribution factors because it reduces the number of equations and variables

in the model making the formulation more compact Distribution factor formulations have

better scaling properties and therefore better computational performance in many OPF

applications such as UC and security-constrained OPF The downside is that these compact

formulations are numerically dense often resulting in slower computational performance in

the basic OPF model (eg without unit commitment or security constraints) This downside

will be revisited in Section 43 which presents three simplication techniques that improve

the computational eciency of dense compact OPF formulations

The compact formulation is derived by substituting the linearized power ow constraints

(49) into the real and reactive power balance constraints (46) below

pnw +A⊤(Fθ + F0

)+

1

2|A|⊤

(Lθ + L0

)= 0 (410a)

qnw +A⊤ (Hv +H0)+

1

2|A|⊤

(Kv +K0

)= 0 (410b)

Solving the resulting linear system1 denes sensitivity matrices for θ and v

Sθ = minus(A⊤F+

1

2|A|⊤L

)minus1

(411a)

Sv = minus(A⊤H+

1

2|A|⊤K

)minus1

(411b)

1Solving (411a) requires modifying the reference bus column by the method previously described in(218) which restores full rank to the matrix

127


S0θ = Sθ

(A⊤F0 +

1

2|A|⊤L0

)(411c)

S0v = Sv

(A⊤H0 +

1

2|A|⊤K0

)(411d)

The above denitions allow θ and v to be expressed by linear expressions θ = Sθpnw+ S0

θ

and v = Sθvq

nw+ S0v which will be substituted into (49) to create analogous constraints for

pfαqfαpℓ and qℓ To reduce notation we dene the following dense sensitivity matrices

F = FSθ H = HSv

L = LSθ K = KSv

(412)

As before linearization oset constants are also dened

F0 = FS0θ + F0 H0 = HS0

v +H0

L0 = LS0θ + L0 K0 = KS0

v +K0

(413)

Rather than explicitly calculating Sθ and Sv it is almost always more ecient to solve

the factors F H L and K implicitly The implicit calculation is provided in Appendix A2

By construction the resulting linear constraints implement the same rst-order Taylor

series expansion as the sparse formulation (49) but are expressed in terms of pnw and qnw

pfα = Fpnw + F0 (414a)

qfα = Hqnw + H0 (414b)

pℓ = Lpnw + L0 (414c)

qℓ = Kqnw + K0 (414d)

128


In contrast to (49) the constraints in (414) do not require explicit variables for θ and v

The power ow variables can also be calculated independently from each other That is each

power ow variable can be computed directly from pg or qg without re-solving the power

ow equations for new values of θ and v Since these variables are no longer necessary the

OPF can now be formulated with by 2N fewer variables Because of the reduction in model

size implementing the OPF in security constrained economic dispatch (SCED) or security

constrained unit commitment (SCUC) models for example would reduce the number of

variables by 2N times the number of scenarios considered

Reducing in the number of variables also allows the number of constraints to be reduced by

2N The reference bus denition θi = 0 and Nminus1 power balance constraints can be dropped

from the formulation One remaining power balance constraint is included by summing the

individual bus-level real power balance constraints to obtain a single system-level real power

balance constraint

1⊤pnw + 1⊤pℓ = 0 (415)

In physical terms all power injections and withdrawals are being balanced at the reference

bus Mathematically a system real power balance equation is required because the inverted

matrix in (410a) does not have full rank To solve the equation the row and column of

the reference bus are dropped from the matrix and therefore the calculation only eliminates

N minus 1 variables and constraints from the model (plus θref which was already eliminated

for a total of N) Then the rst N minus 1 power balance constraints can each be relaxed

129


by summing them with the N th power balance constraint resulting in a single system-wide

power balance Power ows drop out of the real power balance constraint (415) due to the

summation A1 = 0

Selecting a reference bus voltage is not required to solve the system of equations (410b)

and consequently an analogous system-level reactive power balance constraint is not math-

ematically necessary Rather the reactive power sensitivity matrices H and K are dened

such that any set of reactive power net withdrawals qnw will mathematically result in reactive

power ows and losses qfα and qℓ that implicitly satisfy the linearized reactive power balance

constraints (46b) There is no nominal reference bus voltage constraint so all N reactive

power balance constraints can therefore be dropped from the dense formulation A further

consequence of this is that the model does not produce a system-wide price for reactive

power conrming the often-repeated sentiment that reactive power is a local phenomenon

4214 Compact Linearization

The size of the dense formulation can be further reduced by condensing (ie summing) the

line loss constraints Although summation relaxes the constraints and could result in a less

accurate approximation computational results in Section 44 shows that the approximation

does not lose signicant delity compared to the S-LOPF and D-LOPF and actually is

sometimes more accurate than the D-LOPF

System-wide real and reactive losses are dened as pℓ = 1⊤pℓ and qℓ = 1⊤qℓ Real and

130


reactive loss factors are then dened by following this summation through to the sensitivity

matrices and osets from equations (412) and (413)

L = 1⊤L L0 = 1⊤L0

K = 1⊤K K0 = 1⊤K0

(416)

The resulting loss constraints are a relaxation of the original loss constraints since the

denitions in (416) are a summation of (414c) and (414d)

pℓ = Lpnw + L0 (417a)

qℓ = Kqnw + K0 (417b)

The system real power balance constraint is rewritten as follows

1⊤pnw + pℓ = 0 (418)

Implicit calculations of L and K are provided in Appendix A2

4215 Transmission Limits

Transmission limits (41h) and (41i) prevent excessive heating that may cause damage to the

transmission infrastructure In order to constrain the correct to and from quantities in

the LOPF formulations the constraints are adjusted below based on the direction of mid-line

131


power ows in the base-point solution(pfαk +

1

2pℓk

)2

+

(qfαk +

1

2qℓk

)2

le T2

k if(pfξ k

)2+(qfξ k

)2gt(ptξk

)2+(qtξk

)2(pfαk minus

1

2pℓk

)2

+

(qfαk minus

1

2qℓk

)2

le T2

k otherwise forallk isin K(419)

The above constraints are convex but nonlinear Linear constraints can be constructed

by sampling multiple operating points pfξ kl and qfξ kl for a set of l isin L sample points along

the boundary of constraint (41h) The constraint is then linearized by the rst-order Taylor

series expansion at each sample point

2pfξ kl

(pfαk +

1

2pℓk

)+ 2qfξ kl(q

fαk +

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2pℓk

)+ 2qfξ kl(q

fαk minus

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl

)2 otherwise forallk isin K l isin L

(420)

Branch-level line losses are required in (420) but are not present in the C-LOPF formu-

lation An additional step approximates branch-level losses from the system-level losses pℓ

and qℓ by calculating the following distribution factors

Ldk = pℓξk

sumκisinK

pℓξκ Kdk = qℓξk

sumκisinK

qℓξκ (421)

Substituting the approximations pℓ asymp Ldpℓ and qℓ asymp Kdqℓ constraint (420) can then

132


be reformulated for the C-LOPF

2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk +

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk minus

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl


(422)

Note that the C-LOPF is not strictly a relaxation of the D-LOPF because the approxi-

mations for pℓ and qℓ could result in a tighter constraint than (420)

In the P-LOPF reactive power is not explicitly modeled as a decision variable so reactive

power is xed at its base-point value and constraint (422) reduces to the following

2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl

)2

if(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl


(423)

Constraints (420) (422) and (423) were implemented with only one sample point l

per branch k a projection of the base-point solution onto onto the the branch limit That

is if η = Tk

(pξfkl)

2+(qξfkl)

2gt 1 then the projected sample points are pξ

fkl =

radicηpξ

fkl and

qξfkl =

radicηqξ

fkl

133


4216 Voltage Limits

Voltage constraints maintain the voltage magnitudes within rated values In the S-LOPF

v is explicit in the model and the voltage magnitude constraints (41j) does not need to be

reformulated

On the other hand the D-LOPF and C-LOPF formulations are written without an

explicit voltage magnitude variable Voltage magnitude constraints are instead enforced

using the voltage sensitivity matrix and oset dened in (411b) and (411d)

v = Svqnw + S0

v (424a)

V le v le V (424b)

422 Generator Constraints

Generator limits ensure that the OPFs dispatch quantities does not exceed feasible gener-

ation levels For simplicity (41k) and (41l) are simple generator box constraints but they

could easily be replaced by more general D-curve constraints if desired and constructed

analogously to (420) and (422)

As described in Section 22 nonlinear convex generator cost functions are approximated

by a set of piece-wise linear constraints with breakpoint values P gml l isin Lpgm

cm ge C0m + C1

mPgml + C2

m(Pgml)

2 +(C1

m + 2C2mP

gml

) (pgm minus P g

ml


134



at least one l isin Lpgm as long as the cost function is convex and the linear approximation

cm can be made arbitrarily close to the nonlinear cost function by adding additional sample

points

423 LOPF Formulations

The S-LOPF D-LOPF C-LOPF and P-LOPF are dened below

bull Sparse LOPF (S-LOPF)

min z =summisinM

cm

st Constraints (45) (46) (49) (420) (425) (41j) (41k) (41l)

bull Dense LOPF (D-LOPF)

min z =summisinM

cm

st Constraints (45) (414) (415) (420) (424) (425) (41k) (41l)

bull Compact LOPF (C-LOPF)

min z =summisinM

cm

st Constraints (45) (414a) (414b) (417) (418)

(422) (424) (425) (41k) (41l)

135


bull Real Power LOPF (P-LOPF)

min z =summisinM

cm

st Constraints (45a) (414a) (417a) (418) (422) (425) (41k)

43 Simplication Techniques

This section describes simplications that can made to substantially reduce the size of the

DCP-LOPF model implementations First the lazy constraint algorithm reduces the

number of constraints included in the model and results in no approximation error Then

hybrid line loss constraints reduce the number of constraints needed for the D-LOPF models

line loss approximations which results in a hybrid model of the D-LOPF and C-LOPF and a

very small amount of approximation error Last a factor truncation procedure is described

that is commonly used in large-scale ISO applications Factor truncation usually results in

the largest power ow errors of the three simplications but the errors are typically small

431 Lazy Constraint Algorithm

An unappreciated advantage of distribution factor OPF models (such as the DCP-LOPF

models) is that non-binding network constraints can omitted from the formulation to reduce

the size of the model To implement this this section describes an active set method (Lu-

enberger and Ye 2008 Sec 123) called the lazy algorithm that initializes a small initial set

136


of transmission and voltage limit constraints and then adds binding or violated constraints

into the monitored constraint set in an iterative process The algorithm allows the model to

be solved with only a small fraction of the transmission constraints that are included in the

models explicit formulation Similar approaches are common in present ISO practices

Below Algorithm 2 implements the lazy algorithm for any of the distribution factor OPF

models Let Kmon and Nmon be input data for the initial set of constraints (414) and (424)

that are explicitly included in the model formulation Then each iteration solves the OPF

model and determines if any of the excluded constraints have been violated and adds them

to the monitored set Because the algorithm terminates only after all constraints have been

satised the resulting solution is both feasible and optimal in the extensive formulation

Algorithm 2 Lazy constraint algorithm for the D-LOPF and C-LOPF models

Input D-LOPF or C-LOPF model initial monitored sets KmonNmon

1 Kvio = N vio = empty2 repeat3 Kmon = Kmon cup Kvio and Nmon = Nmon cupN vio

4 Solve LOPF with monitored constraints Kmon and Nmon

5 pfα = Fpnwlowast + F0

6 qfα = Hqnwlowast +H0

7 v = Svqnwlowast + S0

v

8 Kvio = k isin K such that (pfαk )2 + (qfαk )2 gt T2

k9 N vio = k isin N such that vn lt V n or vn gt V n10 until Kvio sub Kmon and N vio sub Nmon

Output pnwlowastqnwlowast

A similar algorithm is not presented for the S-LOPF The S-LOPFs formulation consists

of K + N power ow and power balance constraints and K + N power ow and voltage

variables for both real and reactive power that must be simultaneously satised Dropping

137


any of the power ow constraints could therefore cause the power ow and voltage variables

to violate the Kirchhos laws introduced in Section 21

Algorithm 2 is implemented in Python using the Pyomo algebraic modeling language (Hart

et al 2011 2017) Pyomos persistent interface to the Gurobi solver (Gurobi Optimization

2020) was used to take advantage of LP warm-starting The algorithm is also implemented

for the P-LOPF and PTDF model formulations by dropping lines 6 7 and 9 as well as

the sets Nmon and N vio and the output variable qnwlowast Branch constraints were added to

the initial monitored set if the apparent power ows the base-point solution were within

25 of the limit and bus voltage constraints were added if the voltage magnitude in the

base-point solution was within 15 of either the upper or lower limit A maximum of 50

violated thermal constraints and 50 violated voltage magnitude constraints were added per

iteration which can often cause the model to add more constraints than necessary A more

ecient future implementation could use the implied constraint satisfaction technique by

Roald and Molzahn (2019) further reduce the number of added constraints by selecting the

added constraints in a more intelligent fashion

432 Hybrid Line Loss Constraints

The use of the lazy algorithm also suggests that it may also be possible to avoid calculating

the full sensitivity matrices F H L K and Sv Avoiding the full calculation can reduce

memory requirements and speed up pre-processing calculations in the LP software Many

138


attempts to solve the larger test cases described in Section 44 crashed due to memory

issues even before the OPF model could be passed to the Gurobi solver The following

section describes how to modify the D-LOPF formulation to include a residual loss function

for branch loss sensitivities that are not explicitly calculated

Suppose that sensitivities are only calculated for a specied set of branches called Kprime sub K

The partially solved matrix can be denoted F[kisinKprime] and is calculated by solving the system

below applying the implicit calculation method from Appendix A2

(A⊤F+

1

2|A|⊤L

)⊤

F⊤[kisinKprime] = minusF⊤

[kisinKprime] (426)

Analogous solves can also be performed to calculate H L K and Sv with the appropriate

substitutions for FHLK and I respectively per the sensitivity denitions in (411) and

(412) Sensitivity factors outside of Kprime and N prime are ignored

The complication that arises in the D-LOPF model is that a partial computation of

L and K results in underestimating line losses on the branches in K Kprime As explained

in Chapter 3 marginal line losses are a signicant component of marginal costs and can

therefore have a signicant eect on determining optimal dispatch and locational marginal

prices (LMPs) Accordingly the residual line losses are dened below to compensate for the

dierence between total losses and the branch losses in Kprime

pℓKprime= pℓ minus 1⊤pℓ

[kisinKprime] (427a)

qℓKprime= qℓ minus 1⊤qℓ

[kisinKprime] (427b)

139


Residual loss sensitivities can then be dened based on the denition above as the

dierence between the total loss sensitivity L and the sum of the modeled branch losses

LKprime

= Lminus 1⊤L[kisinKprime] L0Kprime= L0 minus 1⊤L0

[kisinKprime]

KKprime

= Kminus 1⊤K[kisinKprime] K0Kprime= K0 minus 1⊤K0

[kisinKprime]

(428)

Residual system losses are included in the D-LOPF with a modied balance constraint

and residual loss constraints analogous to (417)

1⊤pnw + 1⊤pℓ[kisinKprime] + pℓK

prime= 0 (429a)

pℓ = LKprimepnw + L0Kprime

(429b)

qℓ = KKprimeqnw + K0Kprime

(429c)

Implementing constraints (429) results in a hybrid of the D-LOPF and C-LOPF formu-

lations where each branchs line losses are either calculated individually or allocated to the

residual loss function Thus all line losses are accounted for in the system power balance

equation A promising avenue of future research may be to more intelligently select Kprime to

improve delity of the D-LOPFs line loss modeling in key parts of the network

433 Factor Truncation

Computational performance can also be improved by eliminating small sensitivity factors

from the dense constraint matrices In this factor truncation a tolerance ε is rst specied

Then any element of F H L or K that is less than ε is set to zero resulting in truncated

140


(or sparsied or trimmed) sensitivity matrices Fε Hε Lε or Kε This causes some error

when calculating power ows and losses in (414) so the constant vectors F0 H0 L0 and

K0 are also adjusted resulting in error-compensated osets F0ε H0ε L0ε and K0ε The

truncation and corrections are dened as follows using the indicator function 1x that is

equal to 1 if x is true or 0 if x is false and an absolute tolerance ε gt 0

F εik = Fik1Fikgtε forall(i k) isin N timesK (430a)

F 0εk = F 0

k +sumiisinN

Fikpξnwi 1Fikleε forallk isin K (430b)

If desired the absolute tolerance can set based on a relative tolerance level εrel gt 0

ε = εrel timesmaxik

Fik

The parameters of constraints (414b) (414c) and (414d) are modied similarly Factor

truncation reduces the number of nonzeros in the the power ow constraints and therefore

reduces memory requirements and improves computational performance of the D-LOPF and

C-LOPF models

Of course the truncation procedure also results in some amount of power ow error

While omitted from the present chapter a formal error analysis could be performed by an-

alyzing condition number properties of the Jacobian matrix to develop error bounds (see

Kincaid et al 2009 Sec 44) In lieu of formal analysis numerical results from computa-

tional experience are provided in Section 444 Like the lazy constraint algorithm factor

truncation is also common in many ISO software implementations

141


44 Computational Results

Computational testing was performed for the above models by rst solving the AC OPF (41)

to obtain a base-point solution First this section presents results to show that the LOPF

objective function values LMPs and power ow solutions are highly accurate compared

to solutions to the AC OPF Results from the modeling simplications in Section 43 are

presented and show that the simplications result in very little approximation error yet a

substantial decreases in solution times The use of an AC OPF solution is highly optimistic

for a base-point solution since real-world implementations would likely use the solution from

state estimator software so this section presents model results from solving the test cases

with varying levels of demand The results show that the accuracy of the LOPF models

does not substantially decline when the demand levels dier from the base-point Lastly a

comprehensive comparison of solution times is presented for all of the test cases and model

implementations

Results from the PTDF and B-theta implementations of the DC OPF are also provided

for comparison purposes These models assume a lossless network so they been implemented

by increasing all nodal demands by a factor proportional to the amount of line losses in the

base-point solution in the same manner as previously applied in Chapter 3

All problem instances were solved in a virtual Linux machine running Ubuntu 18041

with an allocation of 22 GB of RAM and 6 cores of an Intel i7-8650U 190 GHz processor

142


AC OPF problems were solved using IPOPT 31211 and linear problems were solved using

Gurobi 811 The software used for computational testing is a modied version of the open-

source EGRET software package (Knueven et al 2019) Table 41 shows the full suite of

test cases in which the solutions to the LOPF models were attempted where all cases are

sourced from Babaeinejadsarookolaee et al (2019) and reect typical operating conditions

The dense DCP-LOPF formulations required signicant time for preprocessing cal-

culating and loading the constraint sensitivity matrices into the Pyomo model could take

20-30 minutes on the larger test cases Reducing the computational time in these prepro-

cessing steps is certainly a useful area for future improvements However these steps are not

optimized in the implementation presented here Preprocessing time is not included in the

computational results and is outside the scope of the present chapter

In addition although the S-LOPF and D-LOPF (and the PTDF and B-theta models)

are isomorphic equivalents the default model implementations include a relative parameter

truncation tolerance of 10minus6 and partial calculation of branch sensitivities (ie from Equation

(426)) These dierences may cause some of the S-LOPF and D-LOPF (and PTDF and

B-theta) results to dier slightly

441 Objective Function Error

Objective function values for each test case are shown in Table 42 where each objective

function has been normalized by the locally optimal AC OPF solution resulting in a unitless

143


Table 41 OPF case study sources

Source IEEEa SDETb PEGASEc TAMUd

Cases case14_ieee

case30_ieee

case57_ieee

case118_ieee

case300_ieee

case588_sdet

case2316_sdet

case2853_sdet

case4661_sdet

case89_pegase

case1354_pegase

case2869_pegase

case9241_pegase

case13659_pegase

case200_tamu

case500_tamu

case2000_tamu

case10000_tamu

Source Polishe RTEc MISC

Cases case2383wp_k

case2736sp_k

case2737sop_k

case2746wop_k

case2746wp_k

case3012wp_k

case3120sp_k

case3375wp_k

case1888_rte

case1951_rte

case2848_rte

case2868_rte

case6468_rte

case6470_rte

case6495_rte

case6515_rte

case3_lmbdf

case5_pjmg

case24_ieee_rtsh

case30_asi

case30_fsri

case39_eprij

case73_ieee_rtsk

case162_ieee_dtcl

case179_gocm

case240_psercn

aIEEE Power Flow Test Cases (U of Washington 1999)bSustainable Data Evolution Technology (SDET) Test Cases (PNNL 2018)cPan European Grid Advanced Simulation and State Estimation (PEGASE) and Reacuteseau de TransportdEacutelectriciteacute (RTE) Test Cases (Josz et al 2016)

dTexas A amp M University (TAMU) Test Cases (Bircheld et al 2016)ePolish Test Cases (Zimmerman et al 2011)f3-Bus test Case (Lesieutre et al 2011)g5-Bus PJM Test Case (Li and Bo 2010)hRTS-79 (Albrecht et al 1979)i30 Bus-as and 30 Bus-fsr (Alsac and Stott 1974)jCase39 (Pai 2012)kRTS-96 (Grigg et al 1999)l17 Generator IEEE Dynamic Test Case (U of Washington 1999)m179 Bus Grid Optimization Competition Test Cases (Szechtman et al 1994)nWECC 240 Bus Power Systems Engineering Research Center (PSERC) Test Case (Price and Goodin 2011)

144


quantity The LOPF models are approximations rather than relaxations of the AC OPF and

therefore their optimal objective function may be higher or lower than the optimal AC OPF

objective Nevertheless most of the LOPF models fall within 1 of the AC OPF objective

Exceptions to this include case300_ieee2 case162_ieee_dtc3 case1888_rte case6495_rte

and case6515_rte

442 LMP Error

LMPs help to provide an economic signal that indicates how much power should be produced

at each location in the power network For example if there is a binding transmission

constraint then resources that reduce ow on the constraint (ie Fik lt 0 for node i and

binding constraint k) would receive a higher energy price than resources whose production

would increase the ow on the constraint (ie Fik gt 0 for node i and binding constraint k)

The following section compares the LMPs determined from the LOPF PTDF and B-theta

models to see how well each model is able to identify the same constraints and determine

similar prices as the AC OPF

A few caveats about LMP accuracy should rst be noted OPF formulation improvements

are intended to determine more ecient dispatch by improving the physical modeling of

the system In contrast LMPs are an economic signal that may have no objectively true

value to compare against When an OPF model (such as the LOPF PTDF or B-theta)

2The solution to case300_ieee is known to have unrealistic phase angle dierences3case162_ieee_dtc is intended for dynamic case studies

145


Table 42 Normalized objective function values default model implementations

Case S-LOPF D-LOPF C-LOPF P-LOPF PTDF B-theta

case14_ieee 1000 1000 1000 1000 1000 1000case30_ieee 1000 1000 1000 0992 0997 1001case57_ieee 0999 0999 0999 0999 0970 0969case118_ieee 0999 0999 0999 0999 0996 0998case300_ieee 0956 0957 0950 0957 0943 0943

case2383wp_k 0998 0998 0998 1000 1013 1009case2736sp_k 1000 1000 1000 1000 1000 1000case2737sop_k 1000 1000 1000 1000 1000 1000case2746wop_k 1000 1000 1000 1000 1000 1000case2746wp_k 1000 1000 1000 1000 1000 1000case3012wp_k 0999 0999 0999 1000 0998 1000case3120sp_k 1000 1000 1000 1000 1000 1001case3375wp_k 1000 1000 1000 1000 1000 1000

case588_sdet 1000 1000 1000 1000 1001 1002case2316_sdet 1000 1000 1000 1000 1006 1001case2853_sdet 1000 1000 1000 0999 1000 1002case4661_sdet 0999 0999 0999 1000 1002 1001

case1888_rte 0978 0978 0978 0979 0986 0986case1951_rte 1000 1000 1000 1000 1000 1000case2848_rte 1000 1000 1000 1000 0999 0999case2868_rte 1000 1000 1000 1000 1000 1000case6468_rte 1000 1000 1000 1002case6470_rte 0999 1000 1006 1006case6495_rte 0939 1017 0882 0971case6515_rte 0983 1003 0959 1002

case89_pegase 0999 0999 0999 0998 1000 case1354_pegase 0999 0999 0999 0992 0998 1000case2869_pegase 1000 1000 1000 0999 1000 0998case9241_pegase 1000 case13659_pegase 1000

case200_tamu 1000 1000 1000 1000 1000 1000case500_tamu 0999 0999 0999 1000 0999 1002case2000_tamu 1000 1000 1000 1000 1017 0999case10000_tamu 1000 1000

case3_lmbd 0990 0990 0990 0999 0999 0999case5_pjm 0997 0997 0997 0997 1006 1006case24_ieee_rts 1000 1000 1000 1000 1000 1000case30_as 1000 1000 1000 1000 0997 0997case30_fsr 0999 0999 0999 1000 0999 0999case39_epri 0998 0998 0998 1000 0998 0998case73_ieee_rts 1000 1000 1000 1000 1000 1000case162_ieee_dtc 0974 0974 0974 0990 0974 0987case179_goc 1000 1000 1000 1000 1000 1000case240_pserc 0995 0995 0995 0996 1000 1000

Lazy model default parameter tolerancesLazy model 10minus2 relative parameter tolerancesInfeasibleMemory crash

146


uses approximated power ows the dispatch solution may dier slightly from the AC OPF

solution and therefore the resulting LMPs could change signicantly although the dispatch is

still very close to the actual optimal solution Nonetheless the following LMP results indicate

that the models tend to correctly identify and price the systems binding constraints fairly

consistently with the AC OPF

Figure 41 shows a heatmap of LMPs in the IEEE 118-bus test case The gure shows the

higher delity of the SDCP-LOPF models compared to the PTDF and B-theta models

These DC OPF implementations correctly identify changes in LMP due to congestion but not

due to line losses in the network so the LMP heatmap for these simplied models appears

blurry compared to the other models with more accurate approximations especially the

eect of line loading on higher line losses LMPs from the SDCP-LOPF models are

almost identical to those from the AC OPF

Although the LOPF approximations tend to be more accurate than the PTDF and B-

theta models higher accuracy is not necessarily guaranteed4 Figure 42 displays LMP

heatmaps of four of the Polish test cases compared to the AC OPF solutions LMPs The

cases respectively represent network and demand conditions in winter peak (Fig 42a) winter

o-peak (Fig 42b) summer peak (Fig 42c) and summer o-peak (Fig 42d) periods

Figures 42a and 42c include price spikes that exceed $250MWh at some nodes indicated

by the white areas of the heatmap A few notes on these results are detailed below

4In other words the approximations cannot be said to be stronger in the same sense that some ACOPF relaxations are stronger than others (see introduction to Molzahn and Hiskens 2019)

147


Figure 41 LMP comparison in the 118-bus IEEE test case with nominal demand

First Fig 42a illustrates an important diculty in assessing the quality of OPF solutions

The diagram shows that the PTDF and B-theta models both correctly identify the location

of the highest LMP nodes and visually provide better matches to the AC OPFs LMPs

than the SDC-LOPF models However because each OPF solution may dier from the

AC OPF base-point the change in LMPs does not necessarily indicate a worse solution

According to Table 42 the solutions from the PTDF and B-theta models are about 1

more expensive than the AC OPF solution A more detailed look at the prices also shows

that highest LMPs in the DC OPF solutions are substantially higher than in the AC OPF

solution ($95752MWh in the PTDF $88748MWh in B-theta and $63483MWh in the

AC OPF) That is the PTDF and B-theta models have determined a more expensive dispatch

solution and the additional dispatch costs are also reected in higher LMPs In contrast

148


(a) case2383wp_k (b) case2746wop_k

(c) case3120sp_k (d) case2737sop_k

Figure 42 LMP errors in Polish test cases with nominal demand

149


Table 42 shows that the SDC-LOPF models are about 02 below the optimal objective

cost The SDC-LOPF models indeed violated a handful of thermal constraints in the

subsequent AC power ow but the violations are very small (40 40 and 38 MVAr on a

branch with 250 MVAr capacity) In the AC power ow solutions the PTDF and B-theta

generator dispatch resulted in constraint violations of 510 and 539 MVAr on a branch with

400 MVAr capacity The P-LOPF model also resulted in a 344 MVAr constraint violation on

the same 400 MVAr branch even though its LMPs match closely with the AC OPF solution

and its objective function is accurate to 001 In other words highly accurate LMP results

in Fig 42a do not necessarily correspond to higher quality OPF solutions since remedial

actions may be needed to resolve constraint violations that were not modeled accurately

Figures 42b and 42d show similar results as Figure 41 with the LOPF models providing

a higher delity model of the network and therefore more accurate LMPs Lastly Fig 42c

shows similar results as explained for Fig 42a As was the case for case2383wp_k the LOPF

solutions in case3120sp_k also appear to be less accurate than for the DC OPF solutions

However closer examination also reveals that the LOPF solutions only resulted in small

constraint violations in the AC power ow In comparison the PTDF and B-theta models

correctly identify the problematic constraints but the constraints have comparatively large

violations in the AC power ow because the PTDF and B-theta approximations are not as

accurate as those in the LOPF models Note however that the PTDF and B-theta models

are at start DC OPF implementations that are not typically used in the OPF software

used by ISOs

150


443 Power Flow Error

An AC power ow was solved after nding each optimal solution to assess power ow errors

The AC power ow is implemented in the standard fashion (Glover et al 2008 Sec 64) by

xing the real power output and voltage magnitude at each generator (PV) bus xing real

and reactive power demand at each load (PQ) bus and xing voltage angle and magnitude

at the slack or reference bus

Figure 43 shows the 50 largest real power ow errors in the each of the four Polish test

cases winter peak (Fig 43a) winter o-peak (Fig 43b) summer peak (Fig 43c) and

summer o-peak (Fig 43d) The proposed SDC-LOPF models result in very little power

ow error while the B-theta and PTDF models show signicant power ow error In each

test case the largest power ow errors occur at the branches connected to the reference bus

since it provides the additional power injections to resolve power ow infeasibilities Power

ows in the SDCP-LOPF models are nearly feasible therefore requiring less power from

the reference bus and resulting in less power ow error

Figure 44 summarizes the real power ow errors in terms of the median mean and

maximum absolute errors Note that the y-axis is shown in log scale In some cases such as

in Figures 44a and 44b real power ow errors are actually lower in the C-LOPF than for the

larger S-LOPF and D-LOPF models This underscores that the C-LOPFs simplications

maintain a very high degree of consistency with the underlying AC power ow physics

From the plots it can also be seen that the C-LOPF shows a higher degree of accuracy than

151




Figure 43 Real power ow errors in Polish test cases with nominal demand

152




Figure 44 Real power ow error statistics in Polish test cases with nominal demand

the P-LOPF even though the two formulations only dier in that the C-LOPFs reactive

power and voltage constraints are dropped in the P-LOPF formulation Since the real power

constraints are the same in both models it may be surprising that the two would result in

dierent power ow error This highlights the benets of modeling reactive power in OPF

formulations the C-LOPF is able to provide generator voltage set points vlowast that are more

consistent with each generators real power dispatch

153


(a) IEEE cases (b) Polish cases

Figure 45 Solution times in IEEE and Polish test cases with and without lazy algorithm

444 Simplication Results

Simplication techniques that are common in practice often do not appear to be very eective

when implemented on the standard set of IEEE test cases because the test cases are too small

to show improvements Figure 45 shows solution times with and without applying the lazy

algorithm Algorithm 2 Fig 45a shows that the lazy algorithm actually increases solution

times in the smaller IEEE cases However Fig 45 shows substantial speed improvements in

the set of larger Polish test cases These results are consistent with common ISO practices

and demonstrate that the lazy algorithms eectiveness is case dependent and tends to be

most favorable in larger test cases

As discussed in Section 433 another advantage of the dense OPF formulations is that

small distribution factors can be eliminated from transmission constraints to improve the

sparsity of the model

154



Figure 46 Solution times in IEEE and Polish test cases with factor truncation tolerances

Three relative tolerance levels εrel were tested 10minus6 10minus4 and 10minus2 and are denoted

by full e4 and e2 respectively in Figure 46 Like for the lazy algorithm the factor

truncation procedure is based on common ISO practices In Fig 46a the truncation pro-

cedure can be seen to improve the worst-case solution times in the IEEE test cases but has

no signicant eect on average or median solution times Fig 46b shows results from the

larger Polish test cases and the truncation procedure provides a very clear reduction in the

median and worst-case solution times such that there is almost no overlap between the box

plots with the smallest (full) and largest (e2) truncation tolerances Again the eectiveness

of this simplication technique may only be apparent in larger test cases

However factor truncation can distort power ows Figure 47 shows the eect of the

factor truncation procedure on power ow distortions in the D- and C-LOPF models As

shown in the gure only the e2 truncation threshold results in any signicant error How-

ever the errors are much smaller than the power ow errors of the P-LOPF model and

155


(a) case2383wp_k (b) case2737sop_k

(c) case2746wop_k (d) case3120sp_k

Figure 47 Real power ow error in Polish test cases with factor truncation tolerances

especially smaller than the power ow errors of the B-theta and PTDF models

Finally Figures 43 and 47 also provide conrmation that the hybrid line loss constraints

described in Section 432 does not result in signicant power ow errors In each Polish test

case the D-LOPF is implemented with hybrid line loss constraints and there is no noticeable

increase in power ow error compared to the S-LOPF errors

156



Although the above results indicate that the LOPF models perform well when the system

conditions are identical to the base-point solution it is also important to assess how sensitive

the models accuracy is to changes in system demand (Baldick et al 2005) Although power

ow accuracy is case dependent and is therefore aected by changes in demand the following

results show that the change tends to be small and within the same error as in the results

with nominal demand

The sensitivity analysis was performed by varying demand levels Pd by a multiplicative

factor This multiplier was initialized at 095 and increased to 105 in 001 increments Then

each LOPF model was solved at each demand using the sensitivity factors calculated from

the base-point solution (ie where the multiplier equals one) To prevent infeasible model

instances a screening step was performed which narrowed the range of the demand multiplier

(and decreased the increment size) as needed to ensure that feasible AC OPF solutions can

be obtained at each demand level A nominal demand multiplier equal to one was always

included bringing the maximum number of problem instances to 12

Figure 48 shows the sensitivity results for the IEEE 118-bus test case In Fig 48a the

total costs in each sensitivity remain well within 1 of the AC OPF cost The reference bus

slack shown in Fig 48b and power ow errors in Figures 48c and 48d also stay within a

close range of the error levels as the nominal demand case

Power ow errors are vector-valued so they are summarized by the 1-norm andinfin-norm

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(a) Normalized total cost (b) Reference bus real power slack

(c) Power ow error 1-norm (d) Power ow error infin-norm

Figure 48 Detailed error sensitivity analysis of the IEEE 118-bus test case

158


By denition the 1-norm is equivalent to a sum of the absolute power ow errors and the

infin-norm is equivalent to the maximum absolute error The general p-norm is denoted ∥ middot ∥p

and is dened for p ge 1 as

∥x∥p =

(sumk

|xi|p)(1p)

Figure 49 summarizes the error sensitivities of other IEEE test cases excluding case300

because it is known to include unrealistically large phase angle dierences The remaining

cases in the IEEE set all display less error than the 118-bus test case Errors of the SDC-

LOPF models are almost identical and are generally lower than for the P-LOPF model that

does not include reactive power and voltage constraints Additional sensitivity analyses are

not included for the other test case sets due to the memory and computational time required

to execute the larger test cases

446 Solution Times

Figure 410 displays the solution times of each model formulation variation and each set of

test cases The proposed model formulations tended to show the largest eect on solution

times in larger (gt 1 000-bus) test cases especially the Polish cases in Fig 410e In contrast

it is dicult to discern any signicant solution time reduction in sets with smaller test

cases like IEEE (Fig 410b) and MISC (Fig 410f) and as previously mentioned the lazy

algorithm appears to increase solution times in the small cases even though it reduces solution

159




Figure 49 Summary error sensitivity analysis of the IEEE test cases

160


(a) All Cases (b) IEEE Test Cases

(c) SDET Test Cases (d) RTE Test Cases

(e) Polish Test Cases (f) MISC Test Cases

Figure 410 Solution times in all test cases and model implementations

161


(g) PEGASE Test Cases (h) TAMU Test Cases

Figure 410 (cont) Solution times in all test cases and model implementations

times in the larger cases These smaller test cases are often used to demonstrate new OPF

solution algorithms yet our results suggest that small (lt 1 000-bus) test cases are almost

useless for assessing the performance of OPF algorithms and simplications

Lastly Table 43 displays the solution speedup of dierent model implementations com-

pared to the AC OPF Speedup is dened as the geometric mean of the AC OPF solution

times divided by the geometric mean of the specic models solution times so a speedup gt1

implies that the model solved faster than the AC OPF on average Geometric means are used

so that the model rankings are invariant to which model is used to dene the baseline (ie

AC OPF) and to limit the inuence of outliers Any cases that were not solved by all mod-

els are excluded from the mean data Default model implementations are marked D (for

default) and the implementations with the lazy algorithm and a relative factor truncation

tolerance of 10minus2 is marked LT (for lazytruncation) The default PTDF implementation

actually outperforms the B-theta models yet runs signicantly slower with the LT imple-

162


Table 43 Model speedup compared to AC OPF by implementation settings

Model S-LOPF D-LOPF C-LOPF P-LOPF PTDF B-theta

Implementation D D LT D LT D LT D LT D

IEEE 299 224 117 282 134 907 181 2446 108 875Polish 94 21 47 37 85 182 370 955 314 242SDET 162 32 31 48 54 304 66 965 20 609RTE 609 25 136 131 291 374 428 4441 390 1367PEGASE 91 21 20 41 40 119 120 751 53 618TAMU 222 35 69 52 39 463 707 2757 175 1206MISC 232 85 123 202 153 430 155 865 116 642

mentation settings However general conclusions avoided since the chapter omits detailed

explanations of the PTDF and B-theta implementations The P-LOPF almost always has

higher speedup than the S-LOPF and the S-LOPF generally has higher speedup than the

DC-LOPF models

Relative performance of each model is also highly case dependent For example the

S-LOPFs speedup is higher in the SDET cases than the Polish cases yet the C-LOPFs LT

implementation has higher speedup in the Polish cases than in the RTE cases Part of the

reasoning for this as previously alluded to in Section 444 is that the ecacy of the LT

simplications for the DC-LOPFs depends on the size of the test case Accordingly the

LT implementation only provides a D-LOPF speedup benet in the Polish RTE TAMU

and MISC test cases and it provides a C-LOPF speedup benet in the Polish SDET and

RTE test cases

Solution speeds in specic test cases is therefore nontrivial and should be thoroughly

investigated for each potential real-world application individually For example the sparse

formulation of the S-LOPF often outperforms the C-LOPF in small OPF test cases but

163


the compact formulation of the C-LOPF may have benets in larger test cases or when

embedded in more complex security-constrained or UC problems

45 Conclusion

This chapter contributes three novel linear OPF formulations that demonstrate substantially

better solution times than the standard AC OPF without substantially reducing the the

physical accuracy of the power ow solutions The rst linear OPF is a sparse linearization

of the AC power ow equations called the S-LOPF and two dense linearizations called

D-LOPF and C-LOPF are derived from this sparse model

Three simplication techniques have also been presented for implementing the dense

D-LOPF and C-LOPF models lazy constraints hybrid line loss constraints and factor

truncation Two of these simplications the lazy constraint algorithm and factor trunca-

tion are common practices in industry applications yet are not commonly implemented in

previously published comparisons of the PTDF and B-theta formulations of the DC OPF

and other sparse and dense OPF formulations They are presented here to highlight the

essential role of simplications in implementing large-scale OPF models

Future OPF studies should explore whether there are benets to implementing the C-

LOPF in more practically-focused problems such as SCUC and SCED Although the C-LOPF

is the most approximate of the proposed SDC-LOPF models computational experience

shows that it has no signicant reduction in accuracy compared to the other two models

164


and is sometimes the most accurate The SDC-LOPF solutions also approximate AC

power ows to a high degree of accuracy likely improving upon the OPF models that are

presently implemented in ISOs That is the proposed models are LPs that can be solved more

quickly than the nonlinear AC OPF and determine physical dispatch instructions that closely

approximate the AC OPF solution While all three models can be solved faster than the AC

OPF and provide similarly low power ow approximation errors the C-LOPFs dense and

compact formulation diers signicantly from state-of-the-art relaxed AC OPF formulations

It is instead more similar to the OPF models that are presently used by ISOs giving it better

scaling properties than other formulations and making it particularly interesting for future

implementation within more dicult problems such as SCUC and SCED

165

Chapter 5

Near-Optimal Scheduling in

Day-Ahead Markets Pricing Models

and Payment Redistribution Bounds

51 Introduction

Changes to traditional pricing methodologies in electricity markets continue to stir con-

troversy Wholesale electricity markets such as those coordinated by Independent System

Operators (ISOs) are often conceptualized as a uniform price auction where each participant

This chapter was previously published with co-authors Richard ONeill and Benjamin Hobbs Althoughco-authors include members of FERC sta the views expressed in the chapter do not necessarily representthe views of FERC or the US Government The previous publication has been edited for clarity andconsistency with the rest of the dissertation and can be cited as B Eldridge R ONeill and B F HobbsNear-optimal scheduling in day-ahead markets Pricing models and payment redistribution bounds IEEETransactions on Power Systems 35(3)16841694 2019

166

CHAPTER 5 NEAR-OPTIMAL SCHEDULING IN DAY-AHEAD MARKETS

submits a convex or linear supply curve to the market operator and the price is set by the

highest marginal cost of the accepted oers (Green and Newbery 1992 Baldick et al 2004)

However as previously discussed in Sections 23 and 243 ISOs must use unit commitment

(UC) software to model the non-convex production capabilities of many generating facili-

ties Convexity is an important assumption in classical economic analysis and a competitive

equilibrium and supporting prices are not guaranteed exist if the production technologies

are non-convex (Scarf 1994) Instead of implementing pure uniform price auctions ISOs

often include side-payments in market settlements to ensure that generators do not suer

nancial losses by following the socially ecient schedule (ONeill et al 2005) as well as

rules to discourage production from generators who are not part of the least-cost schedule

Thus the crux of the pricing controversy is whether to adhere to the usual marginal

pricing policy or if an alternative pricing scheme with somehow better incentives can be

formulated and adopted These pricing schemes are implemented by rst obtaining a physical

schedule (ie production quantities) and then executing a separate pricing model Most

ISOs now have implemented some version of this two-step procedure As pointed out by

Johnson et al (1997) and Sioshansi et al (2008a) ISOs use optimality tolerances to determine

UC schedules and many possible UC solutions can satisfy this tolerance yet result in very

dierent prices This chapter illustrates how dierent pricing models aect the market

settlements of sub- and near-optimal UC schedules

Price formation issues attracted interest from the Federal Energy Regulatory Commission

(FERC) following severe weather events in the winter of 2014-2015 Those events highlighted

167


the role of prices in aligning dispatch incentives maintaining reliability signaling ecient

investments and maximizing the market surplus (FERC 2014) A subsequent Notice of

Proposed Rulemaking (NOPR) highlighted the inclusion or exclusion of non-convexities in

pricing methodologies ie start-up and no-load operating costs minimum output levels

and minimum run times This NOPR proposed to create uniform fast-start pricing rules

for resources with quick response times that would be applied in the day-ahead and real-

time markets operated by ISOs (FERC 2016) Such resources are typically block-loaded

operated at full capacity or not at all and thus unable to set prices when the normal marginal

cost criterion is used All ISOs currently implement some form of fast-start pricing but

to varying degrees based on their resource mix and compatibility with existing ancillary

service markets (FERC 2016 NYISO 2016 Carey 2017 Patton 2018 PJM 2017 CAISO

2016 2017) Rather than pursuing uniform rules in all six ISOs and RTOs under FERCs

jurisdiction FERC concluded the NOPR by opening new dockets to examine specic pricing

rules for New York Independent System Operator (NYISO) PJM Interconnection (PJM)

and Southwest Power Pool (SPP) (FERC 2016)

While these issues are relevant in both the US and Europe US markets solve non-

convexities in a centralized fashion whereas European markets require participants to inter-

nalize non-convexities in their oer (see Reguant 2014 for example in the Spanish electricity

market) This chapter focuses on the treatment of non-convexities as now undertaken in US

markets

The main contribution of this chapter is to relate convex hull pricing (Gribik et al

168


2007) to an issue rst discussed by Johnson et al (1997) and later by Sioshansi et al

(2008a) This issue appears in markets with centralized UC and concerns how the nancial

settlements of near-optimal UC schedules may be considerably dierent than for an optimal

schedule despite little change in total cost This chapter denes a payment redistribution

quantity to measure the change in settlements dened by the sum of absolute deviations

of generator prots and consumer surplus compared to those in an optimal UC schedule

The redistribution quantity is then used to prove a previously unappreciated property that

convex hull pricing minimizes a bound on the change in market settlements and thus creates

a bound on incentives for generators to deviate from the ISOs commitment schedule

Since original publication a report by the Midcontinent Independent System Operator

(MISO 2020b) has shed light on potentially uneconomic levels of self-committed coal-red

power generation Self-commitment occurs when a generator decides it will operate in a given

period although it was not committed by the ISO Daniel et al (2020) estimates that self-

commitments potentially resulted in $350 million in extra costs to utility rate payers in MISO

in 2018 A conjecture in the original publication supposed that the payment redistribution

bounds proved in this chapter might reduce self-commitment incentives That supposition is

now made more explicit by Corollary 54 which did not appear in the original In addition

Appendix B solves Nash equilibrium self-commitment strategies to further demonstrate that

convex hull pricing approximations ie pricing models based on tight convex relaxations

of the UC problem can reduce incentives to self-commit This analysis is provided in the

appendices as it is less rigorous than the analysis included in this chapter However these

169


results have signicant implications in the ongoing electricity pricing debate and to my

knowledge have not been recognized previously in the convex hull pricing literature

This chapter is organized as follows Section 52 provides additional background about

the UC problem and then formulates a standard UC model and four pricing models Section

53 derives upper bounds on the payment redistribution when the pricing model is a convex

relaxation of the UC problem Section 54 illustrates these bounds in a simple example and

Section 55 demonstrates that the theoretical results are meaningful for a suite of larger test

cases Section 56 concludes the chapter and is followed by self-commitment equilibrium

examples in Appendix B

Notation

In this chapter theoretical analysis of the UC problem requires more abstraction than the

detailed optimal power ow formulations in previous chapters This chapters results are

instead presented using Cartesian mathematical conventions letters from the beginning of

the alphabet to represent model parameters the middle of the alphabet to represent set

indices and the end of the alphabet to represent model variables Uppercase letters will

represent matrices and lowercase letters will represent vector and scalar values

Sets

G Set of G generators i isin G

H Set of H generator types

h isin H

K Set of K system constraints

170


k isin K

L Set of L generator

constraints l isin L

M Set of M pricing models

m isinM = r pd td ld ch

S Set of S UC solutions

s isin S

χ Set of generator constraints

χ = cupiχi

Parameters

b0 isin RK System constraint limits

bi isin RL Generator constraint limits

c isin RN Marginal cost coecients

d isin RN Fixed (avoidable) cost

coecient

A0 isin RKtimesN System constraint

coecients

Ai isin RLtimesN Dispatch constraint

coecients

Bi isin RLtimesN Commitment constraint

coecients

Variables

δsopt isin R Optimality gap of solution s

δsmip isin R MIP gap δsmip ge δsopt

δmsi isin R Change in generator is

prot in near-optimal

schedule s

δmscs isin R Change in consumer surplus

in near-optimal schedule s

∆ms Payment redistribution

quantity in pricing model m

and solution s

λ isin RK System prices

x isin RN Dispatch decision variables

y isin ZN Commitment decision

variables


Functions

πsi (λ) Generator is linear prot

RK rarr R

πsi (λ) Generator is prot

including side-payments

RK rarr R

microsi (λ) Make-whole payment to

generator i RK rarr R

microsi (λ) Unpaid lost opportunity cost

to generator i RK rarr R

∆m(τ) Payment redistribution

quantity cdf Rrarr [0 1]

171


Σm(τ) Prot coecient of variance

cdf Rrarr [0 1]

L(λ) Lagrange function RK rarr R

U si (λ) Lost opportunity cost to


Additional Notation



value

conv(middotm) Convex relaxation model m


zs Solution s

zconv Solution of a convex

relaxation

zlb MIP lower bound

52 Unit Commitment and Pricing

As previously described in Section 243 there is generally no completely accepted method

for pricing UC schedules Diculties in resolving non-convex pricing issues stem from the

presence of lumpiness or indivisibilities in the production sets of electric generators (Scarf

1994) Examples of common instances include (1) a minimum output constraint such that

it cannot feasibly produce power at a level less than some threshold value unless it produces

exactly zero (2) xed costs that are required to begin producing power but are otherwise

independent of the amount of power produced or (3) requirements to remain on-line or

o-line for a specied amount of time before shutting o or coming back on-line Rather

than being rare or pathological examples the above features are common to most thermal

generating units A brief review of the discussion in Section 243 follows below

The standard market settlement method is formally presented in ONeill et al (2005) It

172


includes an energy payment based on the locational marginal price (LMP) and a make-whole

payment (MWP) that ensures recovery of as-bid costs for all scheduled participants One of

the objections to this approach is that it may result in large make-whole payments which are

believed to distort market entry incentives (Herrero et al 2015) Side-payments may also

create incentives to distort supply oers such as the well-known exercise of market power by

JP Morgan in California that resulted in a $410 million settlement (CAISO 2013) Various

alternatives to the ONeill et al (2005) pricing method have been proposed to reduce or

eliminate these side payments and can be loosely categorized into optimization models (see

Hogan et al 1996 Hogan and Ring 2003 Van Vyve 2011 ONeill et al 2016 Liberopoulos

and Andrianesis 2016) and equilibrium models (see Motto and Galiana 2002 Ruiz et al

2012 Huppmann and Siddiqui 2018)

Perhaps the most promising alternative is the convex hull pricing method proposed by

Gribik et al (2007) This method minimizes uplift payments a side-payment based on lost

opportunity costs (LOC) These LOC payments can are minimized by solving a Lagrangian

dual problem (see equation (229) and Denition 21) Schiro et al (2016) describes numer-

ous practical hurdles to implementation of convex hull pricing including the computational

diculty of the Lagrangian dual problem and concludes that the pricing method provides

no clear benet to market eciency

In contrast this chapter argues that computational diculty is not a major hurdle to

implementing convex hull pricing and that convex hull pricing may improve market eciency

by reducing incentives for inecient generators to self-commit into the market Instead of

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solving the Lagrangian dual directly convex hull prices can be approximated using tight and

computationally ecient relaxed UC formulations (Takriti et al 2000 Damc-Kurt et al

2016 Gentile et al 2017 Hua and Baldick 2017) Knueven et al (2017) discusses the

various UC formulations in more detail

However computational complexity of the UC problem often prevents ISOs from calcu-

lating a truly optimal UC schedule (Streiert et al 2005) Johnson et al (1997) and later

Sioshansi et al (2008a) investigate the consequences of this practical reality showing that

actual near-optimal UC scheduling decisions can result in signicantly dierent economic

consequences than if the ISO was able to select a truly optimal UC schedule In theory this

may undermine incentives for participation in the ISOs auction since a subset of market

participants may prefer that the ISO selects sub-optimal UC schedules (Sioshansi et al

2008a)

Good market design is multifaceted and requires careful analysis and balancing of a wider

range of issues than are discussed here Maximizing market eciency is often the primary

objective1 In addition to competitive behavior in the DAM and RTM other criteria such as

long-term incentives environmental externalities transparency simplicity fairness or other

stakeholder concerns are also relevant to good market design Although recent works (Herrero

et al 2015 Vazquez et al 2017 Mays et al 2018) have shown that convex hull pricing

may support better long-term investment incentives than the presently applied methods

Mays et al (2018) provides a simple example to show that this conclusion is not generally

1See Kreps (2013 Sec 86) on whether eciency is desirable as the only market design criteria

174


applicable Market design criteria are quite complex and this chapter neither proposes a new

market design nor explicitly endorses adoption of convex hull pricing by any ISO market

Rather the chapters analysis aims to add to the understanding of the properties of convex

hull pricing methods and the potential eect on self-commitment incentives

521 Models

The scheduling software used by ISOs uses mixed integer programming (MIP) techniques to

determine a near-optimal UC schedule Each day ISOs collect bids and oers that dene

consumer valuations and producer costs respectively and are used to calculate price and

quantity schedules The following formulation assumes that demand is xed in which case

maximizing the market surplus is equivalent to minimizing production cost but it is easily

generalized to include an active demand side in the market As previously formulated in

Section 23 the UC model is provided below

min z = c⊤x+ d⊤y (51a)

st A0x ge b0 (51b)


where the decision variables are the dispatch quantities x commitment decisions y total cost

z and xi and yi are the components of x and y associated with generator i the parameters

are marginal costs c xed costs d system constraint coecients A0 constraint limits b0

175


The system constraints are kept general so that all theoretical results in this chapter can

accommodate any linear equality (eg energy balance) or inequality (eg transmission and

ancillary service) system constraints All generator-level constraints are included in the

non-convex constraints (51c) where χi is dened below


where Ai and Bi are the generator constraint coecients and bi is the constraint limit With

a minor abuse of notation note that xi and yi need not be scalar values but typically will

be vectors describing generator is production quantities with elements that might reect

a stepped supply curve production in dierent time periods or dierent binary operating

status indicator variables

Feasible solutions to (51) are denoted by (xs ys) s isin S and s = lowast denotes an optimal

solution Let zconv le zlowast be the objective function of a convex relaxation m of (51) We

dene the optimality gap δsopt and integrality gap δsmip as follows

δsopt = zs minus zlowast le zs minus zconv = δsint (52)

In addition let zlb le zlowast be the lower bound on the optimal cost as determined by a MIP

algorithm and the MIP gap be dened as δsmip = zs minus zlb A solution s is optimal if δsmip

(ie zs = zlb) or near-optimal if zszlbminus 1 le α where α gt 0 is an optimality tolerance that

is usually something near 01 MIP algorithms will terminate after the rst near-optimal

solution is found Note that under these denitions a near-optimal solution may in fact be

176


optimal simply due to a poor lower bound Similarly the MIP gap may be larger than the

actual optimality gap

As previously discussed in Section 23 the UC model is an integer problem and therefore

does not have a standard dual problem that can be used to market clearing calculate prices

Instead most ISOs calculate prices using the method by ONeill et al (2005) that restricts

the binary variables to be equal to the UC solution y = ys where s is ideally an optimal

solution This pricing method will be called the restricted (r) model

The convex hull pricing method by Gribik et al (2007) has not been explicitly im-

plemented in any market and Gribik et al (2007) proposes an approximation called the

dispatchable model based on relaxing the integer constraints of the UC problem PJM and

MISO implement pricing methods which relax the integer constraints of some generators con-

ditioned on if the generator is selected by the ISOs UC software (MISO 2019 Shah 2019)

This pricing model will be called the partial dispatchable (pd) model In addition results

will also be presented for two convex hull pricing approximations called the tight dispatch-

able (td) and loose dispatchable (ld) models that relax all integer constraints regardless of

commitment status

Table 51 describes the main dierences between each pricing models formulation Al-

though we forego explicit UC formulations Formulation A applies tight constraints for

generator minimum up-time and down-time (Takriti et al 2000) two-period ramp inequal-

ities (Damc-Kurt et al 2016) variable upper bounds (Gentile et al 2017) and a convex

envelope of the cost function (Hua and Baldick 2017) Formulation B is a standard UC

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Table 51 Pricing model denitions

Model Description Formulation Binary Variable

r Restricted A y = ys

pd Partial Dispatchable A 0 le y le ys

td Tight Dispatchable A 0 le y le 1ld Loose Dispatchable B 0 le y le 1

formulation used in FERCs RTO Unit Commitment Test System (Krall et al 2012) It does

not aect the feasible solutions to (51) but relaxing the binary constraints usually results

in a larger integrality gap than Formulation A The r model results in the same prices for

either of Formulation A or B

After making the binary variable substitutions shown in Table 51 each pricing model is

a linear program and its shadow prices are given by the dual variable to (51b) λ isin RK As

described in Section 232 the LMP vector is given by A⊤0 λ isin RN In addition a generator

is energy payment is A⊤0iλxi and the total consumer charge for market product k is b0kλk2

Let ch denote the true convex hull price as proposed by Gribik et al (2007) For each UC

solution s denote the price vector derived from each model by λmsm isin rpdtdldch

respectively Even though the dual problem of each pricing model constrains positive prices

λ ge 0 the coecients in A0 and b0 could cause generator energy payments or consumer

charges to be either positive or negative

Each models pricing logic is as follows In the rmodel prices are set by the marginal cost

of any online resources that are dispatched at a level strictly between their maximum and

2Note that these consumer charges may include total energy payments ancillary service payments andpayouts to FTR holders

178


minimum output levels The pd model ignores the costs of all units that are not part of the

ISOs schedule Then it approximates convex hull pricing by relaxing the binary constraints

of the remaining generators The pdmodel roughly orders generators by total costs including

an amortization of xed costs and sets prices based on the marginal generators in the pd

models relaxed solution The td and ld models work the same way except all binary

variables are relaxed Since the pd td and ld relax generator binary constraints it is

possible and likely that the prices are based on dispatch solutions that are infeasible in (51)

Therefore the td and ld models may reect the costs of generators that are actually oine

Lastly the td model uses tighter constraint formulations than the ld model which will tend

to reduce the level of infeasibility in the pricing models dispatch

The rules of each ISO market also include many idiosyncrasies catalogued by Ela and

Helman (2016) Consequently the pricing models presented above are meant to be repre-

sentative but not perfect facsimiles of any ISOs specic pricing model Additional pricing

methods are reviewed in Section 243

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522 Side-Payment Policies

In the absence of side-payments generator i receives quasi -linear3 prots πsi (λ) also referred

to as linear prots

πsi (λ) = (A⊤

0iλminus ci)⊤xs

i minus d⊤i ysi (53)

Becasue UC is non-convex it often occurs that a generators socially optimal schedule does

not maximize its linear prot (Scarf 1994) That is given a UC solution s and a price vector

λ generator is lost opportunity cost (LOC) U si (λ) is dened as follows

U si (λ) = sup

(xiyi)isinχi

(A⊤0iλminus ci)

⊤xi minus d⊤i yi minus πsi (λ) (54)

Gribik et al (2007) derives convex hull pricing by minimizing uplift dened as the total

side-payments including LOC as well as an additional category called Product Revenue

Shortfall described in a few paragraphs As proposed LOC payments are an important

aspect of maintaining good market incentives these payments are conditioned on following

the ISOs dispatch signal to ensure that generators cannot protably deviate from the ISOs

schedule On the other hand consumers may have reasonable objections to being charged

for LOC payments LOCs could become very large in markets with large non-convexities

(Mays et al 2018) or if the convex hull price is poorly approximated (Schiro et al 2016)

and such payments could go to unscheduled generators (Schiro et al 2016) Unfortunately

3Quasi-linearity denotes that revenues (A⊤0iλ)

⊤xi are linear and and costs cixi + diyi are nonlinearboth with respect to production level xi

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a perfect resolution of all market participant desires may be unattainable in non-convex

markets (Scarf 1994)

The standard practice in ISOs is does not pay full lost opportunity costs but only the

portion of any scheduled generators as-bid costs that is in excess of its energy market

revenues The make-whole payment (MWP) microsi (λ) is dened as follows

microsi (λ) = max0minusπs

i (λ) (55)

Since producing nothing is assumed feasible in (52) 0 le microsi (λ) le U s

i (λ) The actual

prot including a possible make-whole payment is denoted by a tilde

πsi (λ) = πs

i (λ) + microsi (λ) (56)

Relaxed binary constraints in a pricing model can result in a special component of uplift

called Product Revenue Shortfall (PRS) Separate pricing and dispatch runs can results in

prices λ such that λ⊤(A0xs minus b0) gt 0 This quantity is the PRS and it results from the

fact that there may be dierent sets of binding inequality (eg transmission andor ancillary

service) constraints in the dispatch and pricing models (Schiro et al 2016 Cadwalader et al

2010) Payments to ancillary service providers and Financial Transmission Rights (FTR)

holders may be underfunded if PRS is not somehow collected (Cadwalader et al 2010)

Next denitions and proofs are provided for the redistribution quantities of near-optimal

UC solutions It will be assumes that charges for all side-payments including PRS are paid

by consumers The proofs do not rely on the detailed make-whole payment denition but

181


only assume side-payments are microsi (λ) such that 0 le micros

i (λ) le U si (λ) In other words the

results do not implicate other important policy decisions regarding whether certain LOC

payments should be provided in addition to make-whole payments Numerical results in

Section 55 will reect side-payments assuming the make-whole denition (55) which is

consistent with the side-payment policies currently implemented in all ISOs

53 Theoretical Results

This section proves bounds on the payment redistribution quantity ie the aggregated

absolute change in individual market settlements when a near-optimal UC schedule is selected

in lieu of an optimal schedule Denitions are rst presented to dene the relations between

relevant settlement quantities Then a lemma a theorem and four corollaries are presented

to prove conditions for when the payment redistribution quantity is bounded

These theoretical results require that the price vector is the same for the near-optimal

solution s and for the optimal solution lowast Convex relaxations of the UC model such as the td

and ld models easily satisfy this criteria since their solutions are independent of any integer

constraints The bounds are not generally applicable to pricing models that depend upon

integer values in the UC solution such as the r model based on ONeill et al (2005) and

the pd based on the partially relaxed pricing methods currently implemented by PJM and

MISO (MISO 2019 Shah 2019) Nonetheless the presence of multiple identical optimal

solutions in the example problem in Section 54 allows a special case where the bounds can

182


be correctly applied to all four pricing models

It will be assumed that generator capabilities costs and system needs are accurately

portrayed in (51) that is that there is no exercise of market power the ISO procures

the correct amount of each product and there are no out-of-model adjustments to satisfy

uncertainty and reliability concerns (see Al-Abdullah et al 2014)

Next I will introduce the analytical framework used to dene the payment redistribution

quantity Changes in generator prots δmsi consumer surplus δms

cs and the optimality gap

δmsopt are related by a zero-sum balance equation

sumi

δmsi + δms

cs + δsopt = 0 (57)

where

δmsi = πs

i (λms)minus πlowast

i (λmlowast) (58a)

δmscs = (λmlowast)⊤A0x

lowast minus (λms)⊤A0xs +sumi

(microlowasti (λ

ms)minus microsi (λ

mlowast)) (58b)

δsopt = zs minus zlowast (58c)

Changes in consumer surplus δmscs reect the consumers energy payment PRS and

any make-whole payments This framework helps illustrate that market settlements depend

not only on prices but also the ISOs commitment and dispatch decisions Therefore mar-

ket settlements can be signicantly changed even if prices stay the same The payment

183


redistribution quantity ∆ms is dened below

∆ms =δmscs

+sumi

δmsi

(59)

Note that all consumers are aggregated in δmscs so the redistribution quantity appar-

ently does not reect transfers between consumers However the results still apply to such

transfers because the generic formulation of χi may include constraints of a consumer i

The Lagrange function (229) from Chapter 2 is copied below for convenience


c⊤x+ d⊤y + λ⊤(b0 minus A0x) (510)

Gribik et al (2007 pages 28-29) shows the Lagrange functions relation to the total lost

opportunity cost and PRS of any arbitrary integer UC solution as shown

sumi

U si (λ) + λ⊤(A0x

s minus b0) = sup(xy)isinχ

(A⊤0 λminus c)⊤xminus d⊤y

minus (A⊤0 λminus c)⊤xs + d⊤ys + λ⊤(A0x

s minus b0)

(511a)

= sup(xy)isinχ

(A⊤0 λminus c)⊤xminus d⊤y+ c⊤xs + d⊤ys minus λ⊤b0 (511b)

= minus inf(xy)isinχ

c⊤x+ d⊤y + λ⊤(b0 minus A0x)+ zs (511c)

= zs minus L(λ) (511d)

Convex hull prices are dened by λch = argmaxλ L(λ) which minimizes the sum of gen-

erator uplift and PRS (Gribik et al 2007) The resulting prices are inherently independent

of the UC solution The td and ld pricing models are also inherently independent of the UC

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solution as a consequence of being convex relaxations of (51) Independence between the

pricing model and the UC solution is the main necessary condition for the following lemma

theorem and corollaries4

Lemma 51 Let s and lowast denote a near optimal and optimal solution to (51) and let λ be

a price vector of appropriate dimension Suppose generator prots are πsi (λ) + U s

i (λ) and

πlowasti (λ) + Ulowast

i (λ) and the total consumer payments are λ⊤A0xs +

sumi U

si (λ) and λ⊤A0x

lowast +sumi U

lowasti (λ) Then

sumi |δms

i | = 0 and |δmscs | = δsopt

Proof From the denition of lost opportunity cost (54)

πsi (λ) + U s

i (λ) = sup(xiyi)isinχi

(A⊤0iλminus ci)

⊤xi minus d⊤i yi (512)

The right hand side is independent of the UC solution sosum

i |δmsi | = 0

Since δsopt ge 0 then the redistribution balance (57) implies the following

δmscs

= δsopt (513)

Theorem 51 Let s and lowast denote a near optimal and optimal solution to (51) and let λ

be a price vector of appropriate dimension Suppose generator prots are πsi (λ) + micros

i (λ) and

πlowasti (λ) + microlowast

i (λ) such that 0 le microsi (λ) le U s

i (λ) and 0 le microlowasti (λ) le Ulowast

i (λ) Let the total consumer

payments be λ⊤A0xs +

sumi micro


lowast +sum

i microlowasti (λ) Then the redistribution quantity

4The lemma theorem and corollaries dier from the published versions including a corrected proof andtighter bound for Theorem 51

185


∆ms is upper bounded by 3 (zs minus L(λ))

Proof Let microsi (λ) = U s

i (λ) minus microsi (λ) be the unpaid lost opportunity costs due to the revised

side-payment policy The net change in generator is prots δmsi can be expressed as

δmsi = |πs

s(λ)minus πlowasti (λ) + U s

i (λ)minus U si (λ)minus micros

i (λ) + microlowasti (λ)|

le |πss(λ)minus πlowast

i (λ) + U si (λ)minus U s

i (λ)|+ |microsi (λ)minus microlowast

i (λ)|(514)

From Lemma 51 the rst absolute value term is zero The remaining term is bounded

by uplift and PRS

|microsi (λ)minus microlowast

i (λ)| le U si (λ) + Ulowast

i (λ) + λ⊤(A0xs minus b0) + λ⊤(A0x

lowast minus b0) (515)

Next sum the above expressions over i and use equation (511) to simplify

sumi

δmsi le zs + zlowast minus 2L(λ) = δsopt + 2(zlowast minus L(λ)) (516)

Similarly to δmsi the net change in consumer surplus δms

cs can be expressed as

δmscs =

λ⊤A0(x

s minus xlowast) +sumi

(U si (λ)minus Ulowast

i (λ))minussumi

(microsi (λ)minus microlowast

i (λ))

le

λ⊤A0(x



i (λ))

+sum

i

microsi (λ)minus

sumi

microlowasti (λ)

(517)

From Lemma 51 the rst absolute value term is δsopt The terms in the remaining

absolute value are bounded below by zero and above by uplift and PRS

0 lesumi

microsi (λ) le

sumi


s minus b0) (518a)

186


0 lesumi

microlowasti (λ) le

sumi

Ulowasti (λ) + λ⊤(A0x

lowast minus b0) (518b)

From (511) and the fact that zs ge zlowast the upper bound in (518a) must be greater than

in (518b) Then again substituting from (511) we have the following boundsum

i

microsi (λ)minus

sumi

microlowasti (λ)

le zs minus L(λ) = δsopt + zlowast minus L(λ) (519)

Combining the above bounds completes the proof

∆ms le 3δsopt + 3(zlowast minus L(λ)) = 3(zs minus L(λ)) (520)

Before discussing the signicance of Theorem 51 note that the following four corollaries

are immediately apparent

Corollary 51 The redistribution quantity is ∆ms = δsopt for any pricing method that cal-

culates prices independently of the solution s and pays full LOC payments U si (λ)

Proof Direct consequence of Lemma 51

The rst corollary shows that any market price λ can satisfy the minimum possible

payment redistribution bounds if it is paired with a side-payment policy that compensates

all LOC payments However such a policy may be undesirable because it would require

consumers to pay generators that are not committed (ONeill et al 2016) and the total

uplift could saddle consumers with a very large bill if the price λ is a poor approximation of

187


the convex hull price (Schiro et al 2016)

A second corollary shows that accurate convex hull pricing approximations help to tighten

the Theorem 51 bounds

Corollary 52 Convex hull prices minimize the Theorem 51 bounds

Proof Direct consequence of convex hull price denition λch = argmaxλ L(λ)

The third corollary shows that the Theorem 51 bounds can be modied so that there is

no need to calculate L(λ) explicitly

Corollary 53 Let s and lowast denote a near optimal and optimal solution to (51) Let m

denote a convex relaxation of (51) with optimal objective function value zm and let λm be

the optimal dual variable to constraints (51b) in the convex relaxation Then the payment

redistribution quantity ∆ms is upper bounded by 3(zs minus zm)

Proof It suces to show that zm le L(λm) Let conv(χm) be model ms convex relaxation

of the generator constraint set χ Dene the convex relaxations Lagrangian function Lm(λ)

Lm(λ) = inf(xy)isinconv(χm)

c⊤x+ d⊤y + λ⊤(b0 minus A0x)

Since conv(χm) is convex and assumed to have a nonempty interior feasible region

strong duality implies that zm = Lm(λm) (Bertsimas and Tsitsiklis 1997 Sec 410) Because

conv(χm) is a relaxation of the constraint set χ it can also be seen that Lm(λ) le L(λ)

which completes the proof

188


A nal corollary relates the redistribution bound to self-commitment incentives

Corollary 54 Let the total cost of a sub-optimal UC solution be zs Let the price vector λ

be determined from the dual variables of a convex relaxation of the UC problem and let zm

be the optimal objective function value of this model Then no participant can benet from

the solutions suboptimality by more than 15times (zs minus zm)

Proof The proof is a relatively simple proof by contradiction Suppose that all conditions

of the above corollary are true except that δsi gt 15times (zs minus zm) for at least one i

Then include this δsi and all other settlement changes in equation (57) Rearrange

equation (57) so that all negative terms are moved to the right hand side and all positive

terms remain on the left hand side Notice that the left hand side includes δsi and other

nonnegative terms so the left hand side total must be strictly greater than 15times(zsminuszm) The

equation is balanced so the right hand side total is also strictly greater than 15times (zsminus zm)

Next calculate ∆ms which must be strictly greater than 3times (zsminus zm) This contradicts

Corollary 53 which conrms the proof

Before additional comment note that the above results can also be applied to any two

arbitrary UC solutions Redistribution quantities are dened in terms of a near-optimal

solution s and the optimal solution lowast based on the conventions from Johnson et al (1997)

and Sioshansi et al (2008a) However the optimal solution could be replaced by some other

solution sprime zsprime le zs with no major changes to the proofs The theoretical results are further

summarized as follows

189


Theorem 51 states the following If the side-payment policy proposed by Gribik et al

(2007) were amended to only pay uplift that meets certain criteria then this amended side-

payment policy will only increase the payment redistribution quantity by an amount no

larger than a multiple of the optimality gap plus the duality gap

Results from Corollaries 51 and 52 are rather straightforward Corollary 51 says that a

full uplift side-payment policy can result in the minimum possible payment redistributions

However it is not at all clear if this is a good thing the uplift payments could be very

costly and 100 of the payment redistribution quantity (ie the optimality gap δsopt) would

be paid by consumers Under the amended side-payment policy Corollary 52 shows that

convex hull pricing minimizes the upper bound on payment redistribution due to selecting a

near-optimal solution instead of an optimal solution That is the convex hull price ensures

that all participants receive approximately the same prots in the near-optimal UC schedule

as they would have if the UC schedule were optimal

Finally Corollaries 53 and 54 provide practical applications of Theorem 51 Corollary

53 shows that calculating the Lagrangian function L(λ) (which requires solving a MIP) can

be avoided and this value can be replaced with the objective function of the pricing model

This bound is much easier to calculate and does not signicantly reduce the quality of the

bound

Corollary 54 redenes the bound in terms of market incentives Its application is moti-

vated by the pervasive use of self-commitments by coal-red power plants in MISO (MISO

2020b) Nearly 88 of the coal-red power produced in MISO is produced by self-committed

190


or self-scheduled generators5 About 12 of MISOs coal powered generation is produced

uneconomically meaning that the markets prices are lower than the generators marginal

cost More than 12 may actually be ineciently committed without submitting actual

cost information to the ISO it is dicult to say how much of the self-committed and self-

scheduled coal power is actually part of the optimal UC schedule Current pricing methods

may provide incentives for generators to self-commit or self-schedule so Corollary 54 shows

that pricing methods based on tight convex relaxations of the UC problem may be able

to eliminate or reduce these incentives and therefore reduce uneconomic self-commitments

These incentives can increase the amount of self-commitments in a Nash equilibrium as

demonstrated in Appendix B

54 Example

This section presents a simple example to illustrate how scheduling changes with little or no

eect on total costs can disproportionately aect nancial outcomes of market participants

that is because there is a nonzero payment redistribution quantity (59) The example

consists of three types of generators that have each been replicated ve times shown in

Table 52 The demand quantity is 225 MWh plus a small perturbation ϵ gt 0 to prevent

degeneracy

Let G be the set of generators of each type h isin 1 2 3 and replication i isin 1 55Self-scheduling is similar to self-commitment and occurs when a generator species its output quantity

rather than its commitment status

191


Table 52 Generator attributes simple example

Gen i isin 1 5 Min P hi Max P hi Cost Chi

OldTech Output x1i (y1i times 25) MW (y1i times 25) MW $15MWhBaseload Output x2i 0 25 10Peaker Output x3i 0 25 25

The single-period UC problem that implements (51) is written below

min z =sum

(hi)isinG

Chixhi (521a)

stsum

(hi)isinG

xhi = 225 + ϵ (521b)

P hiyhi le xhi le P hiyhi forall(h i) isin G (521c)

yhi isin 0 1 forall(h i) isin G (521d)

The optimal UC is simple enough to solve by hand There are ve optimal integer

solutions6 In each optimal solution four of the ve OldTech generators are dispatched to

25 MW the remaining OldTech generator is dispatched to zero all ve Baseload generators

are dispatched to 25 MW and one Peaker generator is dispatched to ϵ

Prices can be calculated by xing or relaxing the appropriate binary constraints The r

and pd models set the price based on the Peaker generators marginal cost so λrs = λpds =

$25MWh The td and ld models set the price based on the OldTech generators marginal

cost so λtd = λld = $15MWh7 In each pricing model the prices are the same in all ve

6There are innite solutions with respect to the continuous variables but only 5 optimal integer solutions7Both the td and ld models calculate the exact convex hull price since the problem has strictly linear

costs and there are no intertemporal constraints Hua and Baldick (2017) so also λch = $15MWh

192


optimal integer solutions

For the $25MWh price (r and pd models) OldTech generators make a prot of $250 if

committed or $0 if left uncommitted All Baseload generators each make a prot of $375

and all Peaker generators either break even or are not dispatched As a result the r and

pd pricing models both result in a prot redistribution quantities ∆ms = $500 since each

alternative solution entails the the a $250 prot from one OldTech generator to another

On the other hand the $15MWh price (td and ld models) causes OldTech generators to

make $0 whether committed or not Baseload generators each make $125 prot and Peaker

generators either receive a make-whole payment or are not dispatched The cost of each

solution is the same and all market participants receive the same outcome regardless of

which OldTech generator is selected by the ISO

Since the r and pd models compute the same prices for each schedule Theorem 51

implies a $750 upper bound on the redistribution quantity If the price is instead set to

$15MWh as in the td or ld pricing models then the upper bound is $30ϵ ie arbitrarily

small Similarly Corollary 54 shows that the unscheduled OldTech generator can make no

more than $15ϵ by self-committing into the example market if the market operator is using

the td or ld pricing models In contrast the r and pd models create a large duality gap in

the market and this duality gap creates an incentive for the uncommitted OldTech generator

to self-commit

Small example problems like (521) can be helpful to illustrate concepts but can also be

misleading or deliver contrived results Accordingly the following section presents similar

193


results for a suite of more realistic test cases and Appendix B provides more discussion and

demonstration of the self-commitment incentives

55 Test Cases

UC pricing and market settlements were calculated for a suite of test cases listed in Ta-

ble 53 The rst set (RTS) from the IEEE 1996 reliability test system (Grigg et al

1999) consists of 96 generators and 24-hour load shapes for spring summer and win-

ter (sp-su-wi-) and weekdays and weekends (-d-e) It was solved with and without

transmission limits (txno) for a total of 12 RTS test cases The second set (PJM) made

available by Krall et al (2012) consists of two 24-hour snapshots of the PJM day-ahead

market from summer and winter of 2009 (suwi) each including about 1000 generators It

was also solved with and without transmission limits (txno) for a total of four PJM test

cases The model was implemented in GAMS with UC code from Tang and Ferris (2015)

Each test case was either solved to a 0 optimality tolerance or terminated after a 1000

second time limit All feasible integer solutions found during the MIP solvers algorithm

were saved if they met a 01 optimality tolerance at the end of the algorithms execution

resulting in 164 RTS solutions and 71 PJM solutions In the following results it will be

assumed that zlowast denotes the cost of the best known solution for test cases in which the MIP

solver terminated before an optimal solution could be veried

For computational eciency test cases with transmission limits were formulated us-

194


Table 53 Test case summary

Final MIP Gap Mean Binding Flow Limits

Test Case Solutions zszlb minus 1 r pd td ld

rtsspdno 21 0040 0 0 0 0rtsspeno 21 0050 0 0 0 0rtssudno 9 0029 0 0 0 0rtssueno 2 0014 0 0 0 0rtswidno 11 0030 0 0 0 0rtswieno 19 0046 0 0 0 0rtsspdtx 20 0046 40 40 0 0rtsspetx 12 0057 78 78 8 0rtssudtx 10 0029 33 38 1 0rtssuetx 6 0021 160 163 16 0rtswidtx 12 0033 03 03 0 0rtswietx 21 0061 60 60 5 0pjmsuno 23 0 0 0 0 0pjmwino 16 0 0 0 0 0pjmsutx 17 0 3996 3926 393 326pjmwitx 15 0 1460 1440 142 143

195


ing power transfer distribution factor (PTDF) transmission constraints (see Section 215)

Transmission limits in the RTS cases were reduced to 90 of their nominal values in order to

induce transmission congestion The last four columns of Table 53 show the average number

of binding transmission constraints in each test case and pricing model

551 Results Overview

Fig 51 shows load-weighted hourly prices in each of the four PJM cases The mean of those

prices across all solutions is shown for all four pricing models and bars for coecient of

variation (cv) are shown for the r and pd pricing models (cv is zero for the td and ld

models) The summer and winter price curves are both typical for each respective season

Price variation tends to be highest near peak periods in both the r and pd pricing models

However price variations can also persist throughout the day as in Fig 51a

The ld model tends to result in lower prices than the other three models despite including

xed costs in the price setting logic On the other hand r pd and td pricing models all

result in very prices on average especially in the summer cases Morning and evening peak

prices diverge more signicantly among the four pricing models but without an obvious

pattern In the PJM test cases the average energy payments by load were 176 179

176 and 171 times system cost for the r pd td and ld models respectively leading to

dierences short-run generator prots

Fig 52 shows the side-payment quantities for the RTS and PJM cases with make-whole

196


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005

(a) Summer no transmission limits

3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005

(b) Winter no transmission limits

3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005

(c) Summer with transmission limits

3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005

(d) Winter with transmission limits

Figure 51 Hourly price mean and coecient of variance in the PJM test case

payments (MWP=sum

i microsi (λ) from denition (55)) shown in dark as a component of the total

lost opportunity cost (LOC=sum

i Usi (λ) from denition (54)) The various pricing models

based on integer relaxation (pd td and ld) are often motivated by the desire to reduce

MWP and indeed the pd and td models result in much lower MWP than the other models

In both sets of test cases the td model lowers the total side-payments to be less than

the r modelss MWPs While there is a theoretical concern that the full uplift payments

proposed by Gribik et al (2007) could result in higher side-payments than status quo the

empirical results suggest that this is unlikely so long as the pricing model is a reasonably

tight approximation In contrast the ld model a (purposely) naive approximation has the

highest LOC in both sets of test cases

197


r pd td ld0

200

400

600

(a) RTS cases

r pd td ld0

50

100

(b) PJM cases

Figure 52 Make-whole payments and lost opportunity costs

552 Payment Redistribution Quantities

Because near-optimal solutions are a practical reality in ISO markets market designers may

prefer to adopt pricing models that accurately approximate the market settlements of the

optimal solution Put dierently an ecient market should ensure that market participants

only have small or minimal incentives to adopt strategies that reduce market eciency The

following numerical results show that the r pd and ld models often result in redistribution

quantities that are even higher than the td models theoretical bound from Corollary 53

Satisfying the bound is therefore nontrivial and shows that the tdmodel is the only one of the

included models whose settlements are not signicantly altered by the reality of near-optimal

scheduling decisions

The redistribution of payments between near-optimal solutions can become very complex

it is aected not only by changes to the price vector but also changes to generator schedules

side-payments and as shown in Section 53 the presence of a duality gap According to

198


Table 54 Mean payment redistribution quantities

Mean ∆mszlb ()Test Case Subset r pd td ld

RTS cases 101 251 012 228PJM cases 080 048 001 002

(57) payments may be redistributed from consumers to generators from generators to

consumers and from generators to other generators

Table 54 shows the average value of ∆ms (59) for all PJM and RTS test cases The

payment redistribution quantities in the RTS cases were much larger than in the PJM cases

on average However in both sets of test cases td pricing models Corollary 53 bounds are

relatively tight and thus the payment redistribution quantities are quite small compared to

the other pricing models

The td pricing models Corollary 53 bounds reproduced here can be used as a common

basis of comparison of the other pricing models

ˆ∆s = 3(zs minus ztd) (522)

While only the td pricing model will guarantee ∆ms le ˆ∆s comparing all pricing models

to the td models bound provides a basis for comparison that controls for the possibility

that the redistribution of payments may be larger in lower quality solutions The proportion

of solutions that satisfy some multiple of this bound τ ˆ∆s for some τ gt 0 is then used

to compare each pricing models relative eect on the redistribution of payments This

199


proportion will be called ∆m(τ) and is calculated as follows

∆m(τ) = (1S)sums

1∆msleτ ˆ∆s (523)

where S is the number of sampled solutions and 1middot is a counting operator

Fig 53 shows the proportion of solutions that satisfy the bound τ ˆ∆s As must be the

case td model satises the bound in all solutions (shown by the vertical line at τ = 1)

In fact the maximum td redistribution quantity is about 20 of the bound in the PJM

cases and 32 of the bound in the RTS cases In contrast the r and pd pricing models

which do not satisfy the conditions of Theorem 51 or Corollary 53 resulted in redistribution

quantities that typically did not satisfy the tdmodels worst case bound (at τ = 1) including

some redistribution quantities that were up to 140 times higher than the td models bound

Dierences between the convex relaxation models (ld and fd) are most apparent in the

redistribution quantities of the RTS cases In these test cases the td models integrality

gap was only 028 on average while the ld models average integrality gap was about 30

Applying Corollary 53 many of the ld models large redistribution quantities would not

have been possible but for this dierence in integrality gaps No analysis was performed to

determine which constraints (ie the polytopes proposed by Takriti et al (2000) Damc-

Kurt et al (2016) Gentile et al (2017) Hua and Baldick (2017)) were most eective at

reducing the size of the integrality gap but an implication remains that formulating a tight

relaxation of the UC problem (51) is a nontrivial task

200


00001 0001 001 01 1 10 100 1000 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 100 0

02

04

06

08

1

(b) PJM cases

Figure 53 Redistribution quantity cdf normalized by Corollary 53 bounds

201


553 Eects on Individual Market Participants

Results in this section show that the redistribution of payments does not aect all market

participants evenly but tends to have the largest eect on the prots of a small subset of

participants The coecient of variation (cv) of each generators prots is computed to

assess prot variability Sample mean and variance are computed from the pool of near-

optimal solutions s isin S Let πmi = 1

S

sums π

si (λ

ms) and (σmi )

2 = 1Sminus1

sums(π

si (λ

ms) minus πmi )

2 be

the mean and variance respectively of generator is prot when prices are determined by

pricing model m The prot cv is dened as Σmi = σm

i πmi and we dene the test case

sample cumulative distribution as follows

Σm(τ) =sumi

1Σmi leτ

sumi

1πmi gt0 (524)

Cumulative distributions of generator prot cv is shown in Fig 54 for each pricing

model Prot variation is consistently low for settlements determined by the td pricing

model The ld model resulted in consistently low prot variation in the PJM cases but

less so in the RTS cases The pd model produced high levels of prot variation in the RTS

cases some exceeding 1 (ie standard deviation greater than mean prots) Note that the

variation in prots in these test cases has nothing to do with variation or uncertainty in

market conditions the variability is wholly dependent on the UC schedules level of sub-

optimality

202


00001 0001 001 01 1 10 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 0

02

04

06

08

1

(b) PJM cases

Figure 54 Generator prot coecient of variance cdf

203


56 Conclusion

It has long been recognized that sub-optimal solutions can have signicant distributional

implications in markets with non-convexities and UC-based electricity markets in particular

(Johnson et al 1997 Sioshansi et al 2008a) This chapter is the rst work to explore

whether those implications are very dierent among alternative methods for determining

prices and settlements in such markets and as a result provides both a theoretical proof

and a numerical demonstration that the redistribution of payments due to solution sub-

optimality can be nearly eliminated by the application of UC pricing methods based on a

tight convex relaxation of the UC model

Results in this chapter demonstrate that indeed the magnitude of the redistribution of

payments is highly dependent on the pricing model being applied This was shown for a

suite of test cases showing that the redistribution of payments tends to be largest when

applying pricing models that require input data from a (possibly sub-optimal) UC schedule

The use of a sub-optimal UC solution as input to the pricing model can create substantial

settlement errors compared to what would have occurred if the UC schedule were optimal

Unlike all previously published analyses the work in this chapter proves a bound on the

redistribution of payments if convex hull pricing approximationsspecically tight convex

relaxations of the UC problemare adopted for calculating LMPs It is therefore possible for

the market settlements of a sub-optimal UC schedule to closely approximate the settlements

204


of an optimal UC schedule Importantly the redistribution bounds are valid even when

the optimal UC schedule is unknown which is the case in nearly all practical large-scale

UC instances solved by ISOs These results are dampened however by the fact that this

chapters theoretical results do not apply to pricing models that depend on integer values

from the UC solution such as the r and pd pricing models or any of the pricing models

currently implemented in any ISO

However there could be benets to adopting convex hull pricing approximations in an

ISO market The originally published version of this chapter posited that the payment re-

distribution bounds could be related to self-commitment incentives but no rigorous analysis

was provided Since then recent press articles (MISO 2020b Morehouse 2020) have shed

light on the pervasive use of self-commitments which Daniel et al (2020) argues may have

caused as much as $1 billion in additional production costs per year A corollary to the

payment redistribution bound is that convex hull pricing approximations create a bound

on the incentives for deviating from the optimal solution in other words bounding self-

commitment incentives Appendix B uses a Nash equilibrium framework to demonstrate

how self-commitment levels may be explained by the pricing models currently used in ISOs

market designs

205

Chapter 6

Conclusion

Advanced optimization modeling has led to vast eciency improvements in electricity pro-

duction over the past few decades and this dissertation aims to further improve the speed

accuracy and understanding of the optimization models that underpin todays electricity

markets Chapter 3 proposes a new OPF algorithm Chapter 4 proposes new model OPF

formulations and Chapter 5 proves a market settlement stability property for near-optimal

UC solutions These three contributions will soon be discussed in more detail but rst I will

describe some broader context in the remainder of this chapters introduction Although the

focus has been placed on the OPF and UC problems each of my contributions are aimed at

promoting consistency and compatibility with the many other interwoven models that ISOs

use to help ensure the safe ecient and reliable production of electricity

The broad context of this thesis is that ISOs are model pluraliststhey rely on a diverse

and increasingly sophisticated suite of software models that are each designed to perform

206

CHAPTER 6 CONCLUSION

a specic task Model pluralism helps avoid the use of gargantuan and cumbersome global

optimizers that might be time-consuming to solve prone to many errors and dicult or

impossible to debug Small purpose-built models allow piece-meal improvements to be made

without worrying too much whether all aspects of power systems optimization are being

addressed What model pluralism requires however is a certain amount of togetherness

between the various models Each model works with the others like a series of interlocked

gears so it is vital that each piece picks up the same information inputs from preceding

models and provides the same outputs to dependent models

Accordingly each of my contributions address a specic source of market ineciency

and the analysis can be limited to one of the ISOs small sub-problems rather than in a

full-edged electricity market simulation The SLP algorithm in Chapter 3 and linear OPF

formulations in Chapter 4 help improve the OPFs consistency with the physical power

system which reduces system costs and reduces the reliance on potentially expensive out-

of-model corrections to system dispatch The economic analysis in Chapter 5 shows that

it is possible to approximate the market settlements of an optimal UC schedule even if the

actual UC schedule is suboptimal and Appendix B shows that this property may promote

generator oer incentives that are consistent with the competitive market assumption that

all resources are oered at their actual cost Because of model pluralism the results in

Chapters 3 4 and 5 can all be used to improve market eciency even though the methods

of analysis are substantially dierent

207


61 Discussion

A key aspect of this dissertations contributions is their consistency and compatibility with

existing ISO processes For example the proposed OPF approaches do not need to provide

a strictly feasible AC power ow because ISOs already use models like the state estimator

and automatic generator control (AGC) that help maintain system feasibility as long as the

OPF model approximations are not too far o The proposed methods are also not much

more complicated than the models currently implemented by ISOs the proposed model

formulations can all be formulated as LPs

First I have shown in Chapter 3 that iterative procedures can provide highly accurate

line loss approximations without requiring more AC power ow solutions than are used in

current practice Solving one or two more LPs after the initial OPF solve only requires a

small amount of time but can eectively reduce line loss errors to less than 1 Not only

does this reduce dispatch costs it also improves LMPs by more accurately reecting actual

marginal costs Improved price accuracy helps signal ecient resource use in the short term

and ecient investments in the long term The proposed model also uses the same OPF

formulation that is currently used by ISOs so it can be implemented with relatively few

changes to the current OPF software

Chapter 4 the second main contribution proposes novel sparse dense and compact

linear OPF models with highly accurate AC power ow approximations of which the sparse

208


and dense formulations are isomorphically equivalent The third formulation called the

compact linear OPF or C-LOPF is a close approximation of the other two models and

probably shows the most potential for future study Most importantly the C-LOPFs model

size and solution accuracy occupy a nice middle area between DC OPF-based formulations

and state-of-the-art AC OPF approximations that are based on convex relaxation Unlike

these convex relaxation models the C-LOPF model uses a distribution factor formulation

similar to the PTDF formulation discussed in Section 215 that is currently implemented

in the OPF software at every ISO Not only is this formulation more compact using fewer

variables and constraints than other DC power ow formulations it also allows the use of

various simplication techniques such as the lazy constraint and factor truncation procedures

that were discussed in Section 43 and potentially any other proprietary methods that ISOs

have also developed to improve performance of PTDF-based models Implementing a new

linear OPF model for ISO dispatch would also likely be signicantly easier than implementing

nonlinear models there would be no need to change solver software vendors or to do a

complete software redesign to handle dierences in solution output solver errors and other

interactions between the ISOs OPF software and other models

Chapter 5 and Appendix B provide the dissertations last contributions There attention

turns away from computationally ecient AC power ow approximations and goes towards

analyzing the economic consequences of near-optimal UC schedules As background recall

that a variety of proposed ISO pricing methods are premised on the use of an optimal UC

schedule The unavoidable reality is that current UC scheduling software only provides near-

209


optimal solutions and under the pricing models currently adopted by all ISOs this causes

the actual market settlements to signicantly dier from the optimal schedules settlements

Many researchers will say that this is unavoidable or that the concerns are strictly

academic and have no real-world consequences In contrast my work shows that (1) convex

hull pricing accurately approximates the optimal schedules settlements (2) approximations

of convex hull pricing provide similar guarantees (3) the ISO can pay less than the full uplift

payments proposed by Gribik et al (2007) and still have similar settlement guarantees

(4) this settlement guarantee can be recast as a bound on incentives to self-commit and

(5) the pervasive use of self-commitments can be explained by the absence of this incentive

bound in the pricing methods currently implemented by all ISOs In other words convex hull

pricing approximations are a computationally simple method of approximating the market

settlements of an unknown optimal UC solution Simultaneously such pricing models may

also be a practical market design tool to disincentivize self-commitments that are argued to

increase production costs and reduce market eciency

There might be other reasons to believe that convex hull pricing should not be imple-

mented One of the primary concerns is that by not basing prices on the actual UC schedule

this may open the door to other gaming opportunities that would lead to inecient dispatch

To prevent this an ISO would need to implement a sensible side-payment policy and an eec-

tive means to monitor the physical output of each resource This dissertation also does not

fully address how rolling time horizons might aect pricing schemes since it may be necessary

to reect the cost of past commitment and dispatch decisions in future operating period It

210


also does not address how virtual bidders aect day-ahead and real-time market clearing

Some objections to convex hull pricing do not lend themselves to analytic or quantitative

answers For example many people believe that convex hull pricing (and its approximations)

is dicult to interpret since the prices are not set by a physically feasible schedule Another

reasonable point of view is that the ISO markets are already very ecient so something as

fundamental as the price-setting logic shouldnt be changed unless the benets are highly

certain

It will therefore suce to say that market design is multifaceted and requires a balance of

many objectives that are dicult to satisfy all at once Chapter 5 and Appendix B supply a

small piece to this puzzle by showing a new economic mechanism by which market eciency

can be reduced as well as an antidote to the possible ineciency More research is still needed

to determine whether implementing new pricing models would necessarily improve current

ISO market designs but the above contributions oer an exciting avenue in the search for

new eciencies

62 Looking Forward

Somebody has probably said it before a dissertation ends but its work is never nished

This dissertation ends with some concluding thoughts on how the contributions in Chapters

3 4 and 5 can be extended to further improve eciency in wholesale power markets

The numerical results in Chapter 3 show that the proposed SLP can converge to a very

211


accurate line loss approximation but it might not be the best possible approximation Re-

call that the SLP uses a three-parameter quadratic approximation of line losses taken from a

base-point solution but only two of those factors (the constant and linear terms) can be de-

termined from the base-point solution The proposed approach postulates that the quadratic

term will have a similar form to the common quadratic line loss approximation (originally

derived in Bohn et al 1984 Appendix) However this choice is somewhat arbitrary Al-

though the quadratic loss function minimizes the rst-order approximation errors it may be

possible to nd another parameterization that also minimizes second-order approximation

errors Perhaps the proposed approach is already close to doing this but perhaps not

Another obvious extension to Chapters 3 and 4 may be a synthesis of the two approaches

Since the SLP is already formulated for updating line losses it may make sense to formulate

analogous updates to the voltage and reactive power constraints that appear in the formula-

tions proposed in Chapter 4 and to evaluate the quality of the resulting reactive power and

voltage approximations My opinion however is that this might not provide very satisfac-

tory results The models in Chapter 4 solve much slower than the MW-only models used in

Chapter 3 Adding an iterative approach will slow the solution times of these models even

more and there is less benet since the power ows are already quite accurate

Other extensions to the modeling in Chapter 4 would likely be more fruitful Power ow

is typically nearly linear for real power ow then becomes progressively more nonlinear with

the inclusion of real power losses reactive power ow and reactive power losses respectively

The naive aspect of Chapter 4s formulation is that all of these aspects of power ow are

212


modeled with the same linearization routine but again the chapter does not show that these

are optimal approximations in the sense of minimizing average or maximum approximation

errors Other approaches should also be tried Relaxing reactive power equality constraints

into inequality constraints may help improve computational speeds Piece-wise lineariza-

tion may help improve the physical accuracy of the reactive power and voltage constraints

Perhaps the constraints of the SDR SOCR and QCR convex approximations of the AC

OPF could also be applied to calculating additional constraints A systematic evaluation of

possible formulations is needed

One extension of Chapter 5 is already underway as demonstrated by the preliminary

modeling and results included in Appendix B Future work should apply the same compu-

tational experiments in test cases with heterogeneous generator characteristics and multiple

time periods Heterogeneous generators pose no diculty to the use of reinforcement learning

algorithms since all decisions are made independently anyway However multiple time pe-

riods increase the number of possible self-commitment decisions exponentially so extending

the Appendix B results to a more realistic test case may require strong simplifying assump-

tions or perhaps a modeling approach that scales better with temporally-linked decisions

More research is also needed to determine the degree to which actual generator self-

commitments are caused by the incentives described in Chapter 5 and Appendix B There

are other possible explanations for the prevalence of self-commitments Without going into

detail take-or-pay fuel contracts cycling wear-and-tear the inability to oer a resources

actual xed costs in the ISO oer format cost recovery of state-regulated utilities and

213


possibly other situations may also explain the prevalence of self-commitments Questions

about causation are probably more suitable for statisticians than mathematical programmers

If it turns out that pricing models are signicant contributors to self-commitment incentives

thenunlike the alternative explanationsconvex hull pricing would be a relatively easy way

to eliminate the poor incentives Since the ISO is a model pluralist it can replace the current

pricing algorithms with very simple convex hull pricing approximations without also needing

to overhaul the rest of the ISOs processes

Lastly none of the contributions within this dissertation would necessarily be easy to

implement in an ISO Even the smallest changes in ISO procedures might require lengthy

discussions tari lings FERC hearing and possible appeals that will bring in perspectives

from various academics RampD experts software vendors policy makers consumer groups

and other stakeholders Change is expensive so it would be desirable for proposed changes to

undergo thorough simulation studies to ensure that the benets outweigh the costs When-

ever those studies occur the most likely implementation candidates will be whichever state-

of-the-art models require the fewest updates to other interconnected ISO processes and it

is this fact that motivates many of the contributions in this dissertation

214

Appendix A

Sensitivity Factor Calculations

The following appendices provide the detailed parameterizations and calculations that were

used to implement the OPF models formulated in Chapters 3 and 4

A1 Parametric Descriptions

The sparse sensitivity matrices (FHLK) and their osets (F0H0L0K0) are given

from the rst order Taylor series expansions of real and reactive power ow and line losses

(pfαqfαpℓqℓ respectively)

The rst-order Taylor series for real power mid-line ow pfαk assuming partpfαkpartv

= 0 is given

below

pfαk asymp Gk

((τkivξi)

2 minus vξ2j

)2minus Bkτkivξivξj sin(θξij minus ϕki)

minus Bkτkivξivξj cos(θξij minus ϕki)(θij minus θξij)

(A1)

215

APPENDIX A SENSITIVITY FACTOR CALCULATIONS

Summing together the respective linear and the constant terms denes F and F0

Fki =minus Bkτkivξivξj cos(θξij minus ϕki) (A2a)

Fkj = Bkτkivξivξj cos(θξij minus ϕki) (A2b)

F 0k = Gk

((τkivξi)

2 minus vξ2j


+Bkτkivξivξj cos(θξij minus ϕki)(θξij)

(A2c)

The rst-order Taylor series for reactive power mid-line ow qfαk assuming partqfαk

partθ= 0 is

given below

qfαk asympminus((Bk +Bs

ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j

)2minusGkτkivξivξj sin(θξij minus ϕki)

+(minus(Bk +Bs

ki)τ2kivξi minusGkτkivξj sin(θij minus ϕki)

)(vi minus vξi)

+((Bk +Bs

kj)vξj minusGkτkivξi sin(θij minus ϕki))(vj minus vξj)

(A3)

Summing together the respective linear and the constant terms denes H and H0

Hki =minus (Bk +Bski) τ

2kivξi minusGkτkivξj sin(θξij minus ϕki) (A4a)

Hkj = (Bk +Bskj)vξj minusGkτkivξi sin(θij minus ϕki) (A4b)

H0k =

((Bk +Bs

ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j

)2 +Gkτkivξivξj sin(θξij minus ϕki) (A4c)

The rst-order Taylor series for real power losses pℓk assumingpartpℓkpartv

= 0 is given below

pℓk asymp Gk

(τ 2kivξ

2i + vξ

2j

)minus 2Gkτkivξivξj cos(θξij minus ϕki)

+ 2Gkτkivξivξj sin(θξij minus ϕki)(θij minus θξij)

(A5)

216


Summing together the respective linear and the constant terms denes L and L0

Lki = 2Gkτkvξivξj sin(θξij minus ϕki) (A6a)

Lkj =minus 2Gkτkvξivξj sin(θξij minus ϕki) (A6b)

L0k = Gk

((τkivξi)

2 + vξ2j

)minus 2Gkτkvξivξj cos(θξij minus ϕki)

minus 2Gkτkvξivξj sin(θξij minus ϕki)(θξij)

(A6c)

The rst-order Taylor series for reactive power losses qℓk assumingpartqℓkpartθ

= 0 is given below

qℓk asympminus (Bk +Bski) τ

2kivξ

2i minus

(Bk +Bs

kj

)vξ

2j + 2Bkτkivξivξj cos(θξij minus ϕki)

+(minus2 (Bk +Bs

ki) τ2kivξi + 2Bkτkivξj cos(θξij minus ϕki)

)(vi minus vξi)

+(minus2(Bk +Bs

kj

)vξj + 2Bkτkivξi cos(θξij minus ϕki)

)(vj minus vξj)

(A7)

Summing together the respective linear and the constant terms denes K and K0

Kki =minus 2 (Bk +Bski) τ

2kivξi + 2Bkτkivξj cos(θξij minus ϕki) (A8a)

Kkj =minus 2(Bk +Bs

kj

)vξj + 2Bkτkivξi cos(θξij minus ϕki) (A8b)

K0k = (Bk +Bs

ki) τ2kivξ

2i +

(Bk +Bs

kj

)vξ

2j minus 2Bkτkivξivξj cos(θξij minus ϕki) (A8c)

The above denitions can then be used to calculate the other power ow coecients

described in Section A2

217


A2 Implicit Sensitivity Solutions

Equation (411) calculates the dense power ow sensitivities by solving a matrix inversion

This can be a computationally time-consuming process so an alternative is to calculate the

sensitivity matrices implicitly by solving the following linear systems1

minus(A⊤F+

1

2|A|⊤L

)⊤

F⊤= F⊤ (A9a)

minus(A⊤H+

1

2|A|⊤K

)⊤

H⊤= H⊤ (A9b)

minus(A⊤F+

1

2|A|⊤L

)⊤

L⊤= L⊤ (A9c)

minus(A⊤H+

1

2|A|⊤K

)⊤

K⊤= K⊤ (A9d)

The voltage magnitude sensitivity can also be computed this way

minus(A⊤H+

1

2|A|⊤K

)⊤

Sv⊤= I⊤ (A9e)

Denitions for the oset coecients F H L and K that do not depend on Sθ or Sv can

then be obtained from equations (411) (412) and (413)

F0 = F

(A⊤F+

1

2|A|⊤L

)+ F0 (A10a)

H0 = H

(A⊤H+

1

2|A|⊤K

)+H0 (A10b)

1Similarly to the PTDF calculation in Section 215 the calculation for F and L requires the referencebus modication from (218) The left hand side of the reactive power sensitivity equations have full rankand can be solved without modications

218


L0 = L

(A⊤F+

1

2|A|⊤L

)+ L0 (A10c)

K0 = K

(A⊤H+

1

2|A|⊤K

)+K0 (A10d)

The dense real power ow denition can be proven to be isomorphically equivalent to the

sparse denition as shown below

pfα = Fpnw + F0 (A11a)

= F(Sθp

nw + S0θ

)+ F0 (A11b)

= minusF(A⊤F+

1

2|A|⊤L

)minus1(pnw +A⊤F0 +

1

2|A|⊤L0

)+ F0 (A11c)

Note that equation (410a) can be rearranged

(A⊤F+

1

2|A|⊤L

)θ = minuspnw minusA⊤F0 minus 1

2|A|⊤L0

which simplies (A11c)

pfα = Fθ + F0 (A11d)

Similar equivalence can be drawn from the other power ow and loss denitions but are

omitted for brevity The dense power ow constraints (414) can therefore also be shown to

be equivalent to the sparse constraints (49)

The calculation of system loss sensitivity factors L and K from equation (416) depends

on rst computing the branch loss sensitivities L and K Instead L and K can be also

219


dened implicitly These factors are equal to as the power supplied as below

L =partpnwRpartpnwi

L0 = pℓξ minus Lpnw

K =partqnwRpartqnwi

K0 = qℓξ minus Kqnw

(A12)

As previously described in equation (38) in Chapter 3 the following linear systems dene

the marginal system line loss sensitivities and can be solved2 for L and K

(A⊤F+

1

2|A|⊤L

)⊤

L⊤=

(A⊤F+

1

2|A|⊤L

)⊤

1 (A13a)(A⊤H+

1

2|A|⊤K

)⊤

K⊤=

(A⊤H+

1

2|A|⊤K

)⊤

1 (A13b)

2Again the calculation for L requires the reference bus modication from (218) and K can be solvedwithout modication

220

Appendix B

Self-Commitment Equilibrium

Although Chapter 5 mentions that close approximations of convex hull pricing may have

substantial benets in terms of reducing the incentives of coal-red plants to self-commit

the chapter does not conclusively demonstrate that the incentives are strong enough to aect

generator oer behavior This appendix therefore provides a more thorough demonstration

rst using a small 15-generator example and then in a 1500-generator example

Self-committed generators do not submit their full costs to the ISO and consequently

are not optimized in the ISOs UC schedule This likely results in inecient commitment

schedules although it is dicult to say precisely how inecient without rerunning the mar-

ket clearing software with appropriate assumptions for the xed operations cost of self-

committed generators Another approach presented here is to calculate the equilibrium

The work in this appendix was completed during the 2019 Young Scientists Summer Program (YSSP)at the International Institute for Applied Systems Analysis in Laxenburg Austria

221

APPENDIX B SELF-COMMITMENT EQUILIBRIUM

self-commitment oers and then compare the equilibrium results with the market results

that assume no self-commitments

Supply function equilibrium is a classic method for computing Nash equilibrium strategies

in wholesale power markets (Green and Newbery 1992) where the generator strategies are

specied by a full supply schedule and this method can be a powerful method for analyzing

the eects of complex strategic interactions in ISOs (Baldick et al 2004) Another com-

mon approach is Cournot equilibrium in which generator strategies are specied by a just

a single production quantity (Hobbs et al 2000) However in both methods the standard

approaches assume that production costs are a continuous function so explicitly handling

the UC problems integer constraints adds signicant complexity to solving the equilibrium1

Many economic textbooks also avoid directly analyzing the eects of non-convexity by claim-

ing some variant of asymptotic convexity as the number of market participants grows to

innity If the ISO markets UC problem is approximately convex then a logical consequence

is that the market has no duality gap and therefore (remembering Section 232) there is no

diculty calculating supporting prices

Further study of the r pd td and ld pricing models is only interesting due to the

fact that none of the models can guarantee a market clearing solution when there is a

nonzero duality gap Chapter 5 has already shown that these four models calculate very

dierent prices from each other Now I will show that dierent pricing models also present

1For example Herrero et al (2015) nds that there typically is not an integer solution that satisesthe equilibrium conditions and therefore enumerates many integer solutions to nd the solution closest toequilibrium

222


substantially dierent oer incentives

The following analysis of mixed and pure strategy Nash equilibria uses the same replicated

market previously presented in Table 52 of Chapter 5 Replicating the market means that the

market only consists of multiples of the same three types of generators making it relatively

straightforward to compare results of a small 15-generator market with a 1500-generator

market The analysis is rather brief as all of the background and analytical heavy lifting

was already presented in Chapter 5 especially the incentive bound shown in Corollary 54

In the interest of brevity only the r and td pricing models are considered (due to being

the closest implementations of the ONeill et al (2005) and Gribik et al (2007) pricing

methods) Conveniently the r and pd pricing models produce identical results in these

examples as do the td and ld pricing models Section B1 solves the Nash equilibrium

explicitly in the small market and Section B2 presents a reinforcement learning algorithm

for solving the equilibrium in larger examples In both cases the r pricing model provides a

positive incentive for OldTech generators to self-commit whereas generators are indierent to

submitting self-schedules when the td model is used The modeled equilibrium behavior for

the r pricing model is very similar to the real-world self-commitments by coal-red power

plants documented by MISO (2020b) in that the self-committed resources are inexible

relatively expensive and mostly protable

223


B1 Nash Equilibrium in a Small Market

The following analysis of the optimal self-commitment decisions of non-convex generators

makes the assumptions below

bull The non-convex generator is block loaded so that if committed then its minimum

operating level is equal to its maximum operating level

bull There is no distinction between self-scheduling and self-committing

bull All generators follow the ISOs dispatch instruction

bull If needed generators receive make-whole payments to recover their costs-as-oered

bull No uplift or other side-payments are paid to oine or self-committed generators

bull Generators only behave strategically with respect to their decision to self-commit in

which case they oer their full production at zero cost

bull If a generator does not self-commit then it will be assumed that it oers its true costs

bull Each generator considers the equilibrium strategies of all other generators

bull All generators are owned separately and will therefore maximize with respect to its

individual prot and

bull The ISO uses the r model to calculate prices

224


Table B1 Optimal schedules given self-commit oers

OldTech Self-Commits N lt 5 N = 5

OldTech Start-upssum

k u1(k) 4 5OldTech Output

sumk x1(k) 100 MW 125 MW

Baseload Outputsum

k x2(k) 125 MW 100 + ϵ MWPeaker Output

sumk x3(k) ϵ MW 0 MW

OldTech Cost as Oered $375(4-N) $0Baseload Cost as Oered $1250 $1000+10ϵPeaker Cost as Oered $25ϵ $0

UC Objective Cost $(2750minus 375N + 25ϵ) $(1000+10ϵ)Actual Cost $(2750+25ϵ) $(2875+10ϵ)LMP $25MWh $10MWh

The replicated market includes three generator types and is the same as previously

presented in Table 52 Table B1 shows solution information based on the number of OldTech

generators that self-commit given by N

Like in Section 54 the optimal unit commitment is simple enough to solve by hand

Self-committed units are considered free to the ISOs scheduling software and are each

scheduled to their maximum output When N lt 5 four OldTech units are committed to

produce a total of 100 MWh all ve Baseload units are dispatched to produce 125 MWh and

the last ϵ demand is produced by a Peaker generator When N = 5 all ve OldTech units

are committed to produce 125 MWh and the Baseload units collectively produce 100 + ϵ

MWh

In the optimal schedule only four of the ve OldTech units can be committed Therefore

we will assume that an OldTech generator has probability of (4 minus N)(5 minus N) of being

committed if it does not self-commit given that N isin 0 1 2 3 4 other OldTech generators

225


decide to self-commit The r pricing model calculates LMPs of $25MWh if N lt 5 or

$10MWh if N = 5 There are no uplift payments so the uncommitted OldTech unit

has an unpaid LOC of $250 Instead the uncommitted OldTech generator has a $250

incentive to self-commit If it does self-commit then the additional commitment either

causes another OldTech generator to become uncommitted or it causes all ve OldTech

generators to become committed In the latter case the LMP drops down to $10MW so

each OldTech unit receives a loss of $125 It is therefore possible for the OldTech units to

protably self-commit so long as the probability of ve simultaneous self-commitments is

less than 23

This is indeed what happens in the mixed strategy Nash equilibrium which will now be

presented Each OldTech units expected prot depends on two things (1) its own decision to

self-commit (no-SC or SC) and (2) the total number of OldTech units that are self-committed

(N = 0 1 2 3 4 5) Dene an OldTech units strategy as the probability that it decides

to self-commit denoted α1 and assume that all other OldTech units choose the mixed

strategy α Note that a distinction is maintained between α1 and α to emphasize that the

OldTech units do not coordinate their strategies (ie collude) The expected prots for each

combination of events is given in Table B2 Note that no-SC and N = 5 are mutually

exclusive because N = 5 entails self-scheduling all OldTech units Joint probabilities are

given by the binomial distribution

226


Table B2 Expected prots given self-commit oers

Event ω isin Ω Pr(ω) E[π1(k)(λr)|ω]

no-SC N = 0 (1minus α1)times (1minus α)4 $20000no-SC N = 1 (1minus α1)times 4(1minus α)3α $18750no-SC N = 2 (1minus α1)times 6(1minus α)2α2 $16667no-SC N = 3 (1minus α1)times 4(1minus α)α3 $12500no-SC N = 4 (1minus α1)times α4 $000SC N lt 5 α1 times (1minus α4) $25000SC N = 5 α1 times α4 -$12500

The rst order condition for E[π1|ω] with respect to α1 is

0 = partE[π1(λr)]partα1

=minus 200(1minus α)4 minus 750(1minus α)3αminus 1000(1minus α)2α2

minus 500(1minus α)α3 + 250(1minus α4)minus 125α4

A strategy α = 0831 satises the rst order condition The OldTech unit strategies are

assumed symmetrical so we also have α1 = 0831 The expected prot at equilibrium is

E[π1(λr)|α = 0831] = $7153 much less than the optimal coordinated (collusive) strategy

E[π1(λr)|α = 0] = $200 A consequence of the mixed strategy conditions is that the OldTech

generators receive the same expected prot whether self-committing or not

Considering the strategies α1 = 0 and α = 1 reveals an asymmetric Nash equilibrium in

pure strategies Since the generator with strategy α1 = 0 is arbitrary this represents ve

asymmetric equilibria in addition to the mixed strategy equilibrium It turns out that these

asymmetric equilibria are quite stable in the simulation results that follow in Appendix B2

227


Table B3 Pricing model eect on expected production cost and price

Price Model m Pr(N lt 5) Pr(N = 5) E[zs] E[λm]

r 060 040 $2790 $1907MWhtd 100 000 $2750 $1600MWh

Di () 144 2714

Next consider the market outcome if the ISO applies the td pricing model instead of

the r model The td price is $15MWh in the optimal solution and the integrality gap is

$10ϵ (ie the dierence in cost from producing ϵ in an OldTech generator instead of Peaker)

According to Corollary 54 the maximum that the uncommitted OldTech generator can earn

by self committing is $15ϵ ie essentially zero so there is no incentive to self-commit

The market eciency implications of the mixed strategy equilibrium are summarized in

Table B32 System costs are taken from Table B1 $2750 if the schedule is ecient (N lt 5)

and $2875 if all ve OldTech units self-commit (N = 5) Due to the OldTech generators

equilibrium self-commitment strategy there is an expected 144 increase in system costs

and 27 increase in market prices when the rmodel is used to calculate market prices When

the td model is used there is no incentive to self-commit and therefore no ineciency or

increase in expected prices

2In this case results for the pd and ld models are exactly the same as the r and td model resultsrespectively and are therefore not explicitly presented

228


B2 Simulating Equilibria Heuristically with

a Greedy Algorithm

One possible objection to the above equilibrium analysis is that in a real-world problem it

may be too complex for generators to nd their optimal self-commitment strategies In a

small market with 15 generators it might not be unreasonable to expect that each unit may

be able to reason about the strategies taken by other units and end up adopting a Nash

strategy But what if there are hundreds of other generators possibly with heterogeneous

production technologies In large and realistic markets some economists might say that the

generators are better o assuming that the market is competitive and therefore the only

reasonable strategy is to oer their true costs to the ISO

In this section I show that Nash equilibrium strategies can be found by very simple

heuristic methods Rather than solving for Nash strategies explicitly the following analysis

simulates each generators oer behavior by a greedy algorithm The greedy algorithm is

a reinforcement learning algorithm for solving the multi-armed bandit problem (Kuleshov

and Precup 2014) in which each agent (ie generator) only learns the protability of

each strategy (ie self-commit or not) based on its history of previous outcomes In the

greedy algorithm the agent chooses the strategy that it believes to be most protable with

probability 1 minus η or it chooses a random strategy with probability η where 0 lt η lt 1 is

called the exploration probability In the following simulation the outcomes of each strategy

229


jointly depend on the strategies selected by other generators so the simulation is called a

MAB game (Gummadi et al 2013)

The simulation is implemented as follows Each generator submits its oer to the ISO

selecting to either self-commit or to eco-commit (economic commitment meaning that the

submits its actual costs) The ISO optimizes based on the oered costs by solving the UC

problem (521) repeated below for convenience

min z =sum

(gk)isinG

Cgkxgk (B1a)

stsum

(gk)isinG

xgk = 225 + ϵ (B1b)

P gkugk le xgk le P gkugk forall(g k) isin G (B1c)

ugk isin 0 1 forall(g k) isin G (B1d)

Self-commits are implemented by changing the generators cost coecient to Cgk = 0

since the alternative method xing ugk = 1 can create infeasibilities

For comparison purposes one simulation subsequently uses r pricing model to calculate

prices and another simulation uses the td pricing model In both cases generators are

given make-whole payments if needed Each generators prots are calculated and logged

Then the process repeats with each generator independently deciding to self-commit or eco-

commit based on the greedy algorithm Using the framework proposed by Gummadi et al

(2013) the generators are randomly regenerated by deleting their previous oer history and

restarting the greedy algorithm which is a natural analog to new agents enteringleaving the

230


game It is also assumed that generators have a recency bias so the expectation of prots

is not calculated by the arithmetic mean of previous outcomes but using the exponential

smoothing method

microt+1i (α) = κπtα

i + (1minus κ)microti(α)

where microki (α) is generator is estimated prot of the oer strategy α at iteration t πtα

i is

the actual prot from oer strategy α at iteration t and κ isin (0 1) is the exponential

smoothing coecient If generator i does not choose oer strategy α then the update is

simply microt+1i = microt

i In the results to follow the greedy algorithm was implemented with the

random regeneration probability 00025 the exponential smoothing coecient κ = 005 and

random exploration probability η = 001

Figure B1 shows the average probability of OldTech generators submitting self-commitments

over the course of 1000 iterations Progressively larger markets are modeled by replicating

each generator type 5 15 50 150 and then 500 times As shown in the gure the self

commitment strategies converge to the same level regardless of market size One exception

is that the 5 replication market ends up with basically zero self-commitments in Fig B1b

while the larger replicated markets converge to 50 self-commitments This occurs be-

cause the probability of having gt80 OldTech self-commitments is large enough to avoid

self-commitments entirely in the small market (since this lowers the td market price from

$15MWh to $10MWh) In the larger markets the probability of gt80 OldTech self-

commitments is negligible under the td pricing model

231


0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100O

ldTe

ch S

elf-C

omm

itmen

ts(1

0-ite

ratio

n m

ovin

g av

erag

e)

5 replications15 replications50 replications150 replications500 replications

(a) r Pricing Model

0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100

Old

Tech

Sel

f-Com

mitm

ents

(10-

itera

tion

mov

ing

aver

age)


(b) td Pricing Model

Figure B1 Self-commitment strategies under dierent pricing models and market sizes

232


When the simulations are performed with the r pricing model shown in Fig B1a the

OldTech generators slowly learn to self-commit until about 80 of the OldTech generators

are self-committing This corresponds to the asymmetrical Nash equilibrium discussed in

Appendix B1 In contrast Fig B1b shows that the OldTech generators (mostly) converge

to 50 self-commitments because the OldTech generators are basically indierent to oering

a self-commitment or submitting an economic oer when using the td pricing model

The assumption that markets are competitivethat is no generator has market power

and therefore all oers reect actual costsis ubiquitous in power systems research yet the

simulation results show clear inconsistencies with competitive assumptions under the r pric-

ing model (ie similar pricing to what is implemented by ISOs today) Figure B2 compares

the competitive and simulated prots from the last 100 iterations of the 500 replication

market Shown in Fig B2a the r pricing model creates a clear incentive for the OldTech

generators to self-commit average prots were about $149 for self-commits but only $4 for

an economic oer In comparison the td pricing model shown in Fig B2a creates no incen-

tive to self-commit OldTech generators earned $0 whether self-committing or not which is

consistent with the competitive assumption

One might reasonably ask isnt it possible that the simulated results from the r pricing

model are inconsistent with the competitive results because the self commitment strategies

are found heuristically and therefore may be sub-optimal The answer is no that is not at

all what is going on Despite losing money 27 of the time in the market simulation3 the

3Since the OldTech generator either earns $250 with probability p or minus$125 with probability 1minus p then

233


Baseload OldTech PeakerGenerator Type

$0

$100

$200

$300

$400

Aver

age

Prof

it ($

) CompetitiveSimulatedEco-Commit (Sim)Self-Commit (Sim)

(a) r Pricing Model


$0

$100

$200

$300

$400

Aver

age

Prof

it ($



Figure B2 Comparison of competitive and simulated self-commitment strategies

self-committing generators actually make higher prots than those that oer economically

($149 to $4) Because the self-commitment oers are given priorty over economic oers

nearly 100 of committed OldTech generators are self-committed in the simulation MISO

(2020b) nds that about 88 of coal power generation comes from self-committed generators

which is basically consistent with the simulated results

Another criticism might be that MISOs ELMP pricing model is more similar to the pd

pricing model rather than the r pricing model so the results are correct for the simulation

but do not reect the real world However the r and pd pricing models produce identical

results in these examples so applying the pd model still provides the same results as shown

p is calculated by solving 149 = 250pminus (1minus p)125 so the probability of losing money is 1minus p asymp 027

234


in Figures B1a and B2a

More sophisticated analysis is still needed in order to empirically determine how much

of the self-commitment activity discussed in MISO (2020b) is caused by self-commitment

incentives of MISOs ELMP pricing model and how much is due to other reasons Still

more analysis is also needed to determine whether adopting a close convex hull pricing

approximation like the td model would remove the incentives for self-commitment and

what might be the benet in terms of improved market eciency if any These questions

are important to answer and have not been conclusively answered here This appendix

instead shows the following

bull ISOs currently use pricing methods that incentivize self-commitment in equilibrium

bull Self-commitment incentives can cause market outcomes to be inconsistent with com-

petitive assumptions

bull Equilibrium self-commitment strategies can be learned heuristically to a reasonable

degree of accuracy

In other words what this appendix shows is that the r and pd pricing models pro-

vide a mechanism which incentivizes inexible and costly generators to self-commit The

incentives are simple enough that market participants can respond to them without a so-

phisticated understanding of the mechanism that creates the self-commitment incentives

so it is reasonable to believe that generators in an actual market would be able to nd a

similar self-commitment equilibrium Indeed self-commitments are common and tend to

235


be protable among coal-red power plants (MISO 2020b) and this has been argued to

contribute to market ineciency (Daniel et al 2020) Implementing accurate convex hull

pricing approximations may be an eective way to remove the self-commitment incentives

Although more analysis is still needed to determine these things conclusively the simulation

results in this appendix oer a glimpse at what may be a productive avenue for future work

236

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252

Vita

Biography

Brent Eldridge received the BS degree in Industrial Engineering from Texas AampM University in 2011 andthe MS degree in Industrial Engineering and Operations Research from University of California Berkeleyin 2014 He began the PhD program at Johns Hopkins University in 2015 and he was a visiting studentat the Comillas Pontical University in Madrid Spain in 2016 and at the International Institute of AppliedSystems Analysis in Laxenburg Austria in 2019 Brent was awarded the Jack P Covan Award from TexasAampM University the Graduate Research Fellowship from UC Berkeley and the Young Scientists SummerProgram Fellowship from the National Academy of Sciences Brent previously worked at Entergy ServicesInc in The Woodlands TX and he currently works in the Oce of Energy Policy and Innovation at theFederal Energy Regulatory Commission in Washington DC where he uses optimization modeling to informnew electricity market policies

Education

PhD Environmental Health amp Engineering Aug 2015Oct 2019Johns Hopkins University Baltimore MD

Thesis Title Algorithms and Economic Analysis for the Use of Optimal Power Flow and Unit

Commitment in Wholesale Electricity MarketsAdvisor Benjamin Hobbs

MS Industrial Engineering amp Operations Research Aug 2013May 2014University of California-Berkeley Berkeley CA

BS Magna Cum Laude Industrial Engineering Aug 2007May 2011Texas AampM University College Station TX

Minors Economics and Math

Experience

Operations Research Analyst Oce of Energy Policy and Innovation Jun 2014PresentFederal Energy Regulatory Commission Washington DC

Formulation analysis and computational testing of optimal power ow algorithms and nonconvexpricing methodologies for wholesale electricity markets

Visiting Student Advanced Systems Analysis Program Jun 2019Aug 2019International Institute for Applied Systems Analysis Laxenburg Austria

253

VITA

Applied reinforcement learning algorithms to estimate ineciencies of various non-convex pricingmethodologies due to resource self-commitments

Grader Energy Policy and Planning Models Spr 2019Johns Hopkins University Baltimore MD

Teaching Assistant Energy Systems Analysis Spr 2017 Spr 2018Johns Hopkins University Baltimore MD

Visiting Student Institute for Research in Technology May 2016Jul 2016ICAI-School of Engineering Universidad Ponticia Comillas Madrid Spain

Development of optimal power ow linearizations to improve line loss reactive power and voltagemodeling

Forecast Analyst Planning Analysis Jun 2011Aug 2013Entergy Services Inc The Woodlands TX

Long term planning forecasts for electric power load and coal gas and other fuel prices for use inproduction cost modeling Retrospective load analyses to compare forecast and actual values

Student Researcher Department of Industrial amp Systems Engineering Apr 2010May 2011Texas AampM University College Station TX

Developed discrete event simulation model of US commercial border crossing to analyze eects ofnuclear material detection and interdiction policies

Papers

Brent Eldridge Richard ONeill and Benjamin Hobbs Near-Optimal Scheduling in Day-Ahead Mar-kets Pricing Models and Payment Redistribution Bounds IEEE Transactions on Power Systems35 no 3 (2019) 16841694

Brent Eldridge Richard ONeill and Benjamin Hobbs Pricing in Day-Ahead Markets with Near-Optimal Unit Commitment Electricity Policy Research Group Faculty of Economics University ofCambridge Working Paper No 1840 (2018)

Brent Eldridge Richard ONeill and Anya Castillo An Improved Method for the DCOPF with LossesIEEE Transactions on Power Systems 33 no 4 (2018) 37793788

Brent Eldridge and Richard ONeill Marginal Loss Calculations for the DCOPF FERC Sta ReportsJanuary 2017 [wwwfercgovlegalsta-reports2017marginallosscalculationspdf]

Richard ONeill Anya Castillo Brent Eldridge and Robin Broder Hytowitz Dual pricing algorithmin ISO markets IEEE Transactions on Power Systems 32 no 4 (2017) 33083310

Conference Presentations

Brent Eldridge and Richard ONeill Market design evaluation of oer incentives and eciency inelectricity markets INFORMS Annual Meeting Seattle WA October 2019

Brent Eldridge Richard ONeill and Benjamin Hobbs Resources in the MIP Gap of Near-OptimalRTO-scale Unit Commitment Solutions INFORMS Annual Meeting Phoenix AZ November 2018

Brent Eldridge Richard ONeill and Benjamin Hobbs Resources in the MIP Gap of Near-OptimalRTO-scale Unit Commitment Solutions Trans-Atlantic Infraday Conference Washington DCNovember 2018

Brent Eldridge and Richard ONeill Revisiting MIP Gaps and Pricing in RTO-scale Unit Commit-ment FERC Software Conference Washington DC June 2018

Brent Eldridge and Richard ONeill Linearized Reactive Power and Voltage Constraints for DCOPFFERC Software Conference Washington DC June 2017

254

VITA

Brent Eldridge and Richard ONeill Closed Loop Interface and Voltage Constraints with IntermittentWind Generation International Conference Windfarms2017 ICAI-School of Engineering UniversidadPonticia Comillas Madrid Spain May 2017

Brent Eldridge and Richard ONeill Extending the DCOPF to Include Reactive Power Trans-AtlanticInfraday Conference Washington DC November 2016

Brent Eldridge Anya Castillo and Richard ONeill First Order Line Loss Approximation for LMPCalculation INFORMS Annual Meeting Philadelphia PA November 2015

Honors amp Awards

Young Scientists Summer Program Fellowship Grant National Academy of Sciences 2019

WINDINSPIRE participant Universidad Ponticia Comillas ICAI-School of Engineering 2016

Graduate Study Fellowship University of California Berkeley IEOR Department 20132014

Jack P Covan Outstanding Senior Award Texas AampM University ISE Department 2011

Bonnie Hunt Scholarship Texas AampM University ISE Department 20092011

Deans Honor Award Texas AampM University Dwight Look College of Engineering 20082009

Professional Activities

Reviewer IEEE Transactions on Power Systems 20182020

Reviewer Energy Economics 2020

Session Chair INFORMS Annual Meeting 2019

Reviewer IEEE Transactions on Smart Grid 20182019

Reviewer Electricity Policy Research Group Cambridge University 2018

Reviewer Environmental Science amp Technology 2018

Reviewer Power Engineering Letters 2015

Affiliations

Student Member IEEE Power and Energy Society Jan 2017Present

Student Member INFORMS Jan 2016Present

President Institute of Industrial Engineers TAMU Chapter May 2010May 2011

Treasurer Alpha Pi Mu Industrial Engineering Honor Society TAMU Chapter Jan 2009Dec 2010

Technical Skills

Programming GAMS Python Pyomo Matlab Git LATEX

255

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures

Introduction

Brief Background

Research Questions

Contributions and Scope

Mathematical Preliminaries and Literature Review

Power Flow

Optimal Power Flow

Unit Commitment

State-of-the-Art and Current Gaps

An Improved Method for Solving the DC OPF with Losses

Introduction

Power Flow Derivations

Model

Proposed SLP Algorithm

Conclusion

Formulation and Computational Evaluation of Linear Approximations of the AC OPF

Introduction

Model Derivations

Simplification Techniques

Computational Results

Conclusion

Near-Optimal Scheduling in Day-Ahead Markets Pricing Models and Payment Redistribution Bounds

Introduction

Unit Commitment and Pricing

Theoretical Results

Example

Test Cases

Conclusion

Conclusion

Discussion

Looking Forward


Parametric Descriptions

Implicit Sensitivity Solutions


Nash Equilibrium in a Small Market

Simulating Equilibria Heuristically with a Greedy Algorithm

Vita

Abstract

Optimal power ow (OPF) and unit commitment (UC) are two of the most important op-

timization problems underlying both daily and minute-to-minute wholesale power market

operations However both problems are complex and require modeling simplications in

order to be used for market clearing purposes This dissertation provides three main con-

tributions to improve the delity of modeling simplications to the more dicult problems

that market operators would prefer to solve Formulating OPF with the physically correct

Kirchhos laws results in the alternating current (AC) OPF a non-convex and NP-hard

problem Market operators instead solve a simpler linear model called the direct current

(DC) OPF The UC problem includes binary variables and is also NP-hard Due to the UC

problems complexity market operators cannot solve UC to full optimality but only within

a tolerance of optimality

The rst contribution in this thesis is an iterative algorithm that improves the physical

accuracy of the DC OPF model The main advantage of the proposed algorithm is that it uses

the same DC OPF formulation that is used in current practices and does not substantially

increase the number of computations that must be performed by the market operator

ii

ABSTRACT












Readers




iii

Acknowledgments
















iv

ACKNOWLEDGMENTS



















v

Dedicated to my dad

vi

Contents

Abstract ii

Acknowledgments iv

Dedication vi

List of Tables ix

List of Figures x





vii

CONTENTS





Vita 253

viii

List of Tables





ix

List of Figures







x

Chapter 1

Introduction













1























2



these markets




















3












4




11 Brief Background

















5






(LP) software

















6























7






















8



















9





















10




















11























12

















13

Chapter 2


Literature Review





Chapters 3-5







14







21 Power Flow










15


211 AC Power Flow













(21)








16








voltages


(sumj


)⋆

(23)




pi = visumj


qi = visumj







17



pfk = Gkτ2kiv


ptk = Gkτ2kjv



2kiv



2kjv







summisinMi


iv2i minus

sumkisinKfr

i


i

ptk = 0 (26a)

summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i

qtk = 0 (26b)








18
















i (ie no additional

constraints)




19







213 DC Power Flow












20









qfk = qtk = 0 (212)



summisinMi


i minussumkisinKfr

i

pfk +sumkisinKto

i

pfk = 0 (213)







21



pnw +A⊤pf = 0 (214a)


θref = 0 (214c)









derivation







22











Solving for θ

θ =(A⊤BA

)minus1pnw (216)







(A⊤BA

)F

⊤= minus (BA)⊤ (217)


23




more generallysum



W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (218)





= minusBAθ







1⊤pnw = 0 (219)

24



















25



Cm(pgm) = C0

m + C1mp

gm + C2

m(pgm)

2 (220)


1m and C2

m are the coecients







cm ge C0m + C1

mpgml + C2

m(pgml)

2 +(C1

m + 2C2mp

gml

) (pgm minus pgml








26









max zAC =summisinM

Cm(pgm) (222a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj




2kiv

2i minus τkivivj



27



2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T 2


)2+(qtk

)2le T 2






θref = 0 (222n)











28





















29



2i = Gs

i


















30






















31








(226a)








λ = λ1+ F⊤micro




32





βi

(pgi minus P i

)= 0 (227b)











software





33







23 Unit Commitment














34



231 Formulation





min z = c⊤x

st Ax ge b

x ge 0








min z = c⊤x+ d⊤y (228a)

35


st A0x ge b0 (228b)








i χi where












36





















(229)

37











0iλminus ci)⊤

xi minus d⊤i yi






(xiyi)isinχi






38















prime) = 0 so that




Llowast = supλge0


supλge0



39






rearranged like so


= inf(xy)isinχ


= sup

(xy)isinχ

(A⊤

0 λlowast minus c


= sup

(xy)isinχ

sumi


= sup(xy)isinχ

sumi

πi(λlowast xi yi)

minus λlowast⊤b0

=sumi

sup

(xiyi)isinχi

πi(λlowast xi yi)

minus λlowast⊤b0








40



Llowast = supλge0

inf(xy)isinχ


supλge0





supλge0
















41











convex relaxation











42






















43






















44



















45





















46





















47

















48



infeasible













min z = tr(Cw)


w ⪰ 0



49


(where tr(Cw) =sum





















50




⎡⎢⎢⎣wii wij

wij wjj

⎤⎥⎥⎦ ⪰ 0

















51






















52






















53


Discussion




















54






















55
















56






















57






















58




losses









Diniz 2011)










59








AC OPF















60




















grid

61






















62























63











Discussion











64




power and voltage

















65





















66















periods11








67





















68























69






















70











Lagrangian dual












71























72




















73


Discussion




















74







in their infancy


75

Chapter 3


DC OPF with Losses

31 Introduction






76




















77














λ = λE + λL + λC (31)







78






















79

CHAPTER3

IMPROVEDMETHODFORSO

LVING

THEDCOPFWITHLOSSE

S


ISO(Source)

Used inSCED


CAISO(2020)




ERCOT(2016)



Seasonal

ISO-NE(2019)




MISO(Sutton2014)




NYISO(2020ab)




PJM(2010)




SPP(2020)




80






















81















313 Contributions






82



















83


Notation








Sets


k isin K



i

Kfri Kto


to bus i


i j n isin N



Parameters


n-side of branch k


branch k



length




j

and 0 otherwise



84






y isin 0 1 2



injections





matrix



vector


oset


susceptance matrix



real power output



matrix




Variables


elements θi



θi minus θj


from direction with

elements pfk




with elements pgn



with elements pnwi



elements vi



85


Dual Variables



limits






the AC OPF



constraints


function constraint

Additional Notation

Hadamard product



xAC AC OPF solution







pfk = Gkτ2kiv


ptk = Gkτ2kjv




86





from direction






equations are

pnwi +sumkisinKfr

i


i





pnw +A⊤pf = 0 (34)




87


321 DC Power Flow













) (35)







88






W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (36)










Mii =sumkisinKi

τkivξivξj


) foralli isin N

Mij = τkivξivξj




89



W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣∆θ

u

⎤⎥⎥⎦ =

⎡⎢⎢⎣∆pnw

0

⎤⎥⎥⎦ (37)









W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣ Sθ

U⊤

⎤⎥⎥⎦ =

⎡⎢⎢⎣ I

0⊤

⎤⎥⎥⎦ (38)



(32a) and (32b)

pℓk = Gk

(τ 2kiv

2i + τ 2kjv


) (39)

90






L = LSθ (310)














91



Lki =dpℓkdpnwn


dpnwn

=2Rk

R2k +X2

k


(312)




2Rk

R2k +X2

k


=2Rk

(R2k +X2

k)12

τkivivjpfk times

dθijdpnwn

(313)




k)minus12 dθij

dpnwn


n Rearranging



Li = 2sumk






pℓ asympsumk

Rk(pfk)

2 (315)

92


33 Model







)le T (316d)

P le pg le P (316e)











93




⊤Pd

)minus micro⊤

(T+ FPd

)minus micro⊤

(Tminus FPd


(317a)



)⊤FD = 0 (317c)







λE = λ1 (318a)

λL = σL (318b)


)⊤F (318c)

λ = λE + λL + λC (318d)



94




















95


332 LMP Accuracy




















96















Avg Dispatch Di =1

N

sumi


LMP MAPE =1

N

sumi


i |λACi

times 100 (320)


zACtimes 100 (321)


97


0 50 100 150 200 250 300-15

-10

-5

0

5

10



0 10 20 30 40 50 60-150

-100

-50

0

50

100




98














compared to DC OPF







99




A 1 3000 2950 10B 1 3000 2975 100C 2 3000 3000 100












2




100


















101









pℓ =sumk

pℓk =sumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (322)




pℓk = E(2)k (pfk)

2 +(2E

(2)k E

(1)k

)pfk +

(E

(2)k (E

(1)k )2 + E

(0)k

)(323)



fk + E

(1)k )pfk + E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (324)



fk +E

(1)k )) that can be


(1)k )2 minus (pξ

fk)

2) + E(0)k )

102












(1)k )Fkn (325a)

L0k = E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (325b)

Ln =sumk

Lkn (325c)

L0 =sumk

L0k (325d)




dpℓkdpnwn



k and E(0)k can

103




ℓξ τki

1 pfξ larr F(pg



3 E(1)k larr Lkn

(2E

(2)k Fkn

)minus pξ


4 E(0)k larr L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)





8 pξℓ larr

sumk E

(2)k (pξijk + E

(1)k )2 + E

(0)k

9 Ln larr 2sum

k

(E

(2)k (pξijk + E

(1)k )Fkn

)foralln isin N






E(1)k =

Lkn

2E(2)k Fkn

minus pξfk (327b)

E(0)k = L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)

2) (327c)




fk are available If


(1)k = E

(0)k = 0 and




104




k is



k lt ε


of the algorithm

pξgh+1n = ωpξ


pξfh+1k = ωpξ








pℓ gesumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (329)





105
















in the algorithm





106


0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 5 10 15 201e-09

1e-06

0001

1










107

















108












300-bus networks)




109



001

01

1

10

100

1000LMP

MAPE


-03

-02

-01

0

01

RelativeLoss

Error













110




35 Conclusion

















111










voltage levels

112

Chapter 4



of the AC OPF

41 Introduction





113













physical accuracy








114









less accurate














115
















412 Contribution





116























117







413 Outline





chapter

Notation






118




Sets


k isin K



i

Kfri Kto


to bus i




m isinM



i j n isin N

Parameters


i-side of branch k


branch k


of ones




j

and 0 otherwise


of branch k





sensitivities




sensitivities


I Identity matrix


sensitivities



loss sensitivity


loss oset


119




sensitivity


oset


demand



output







matrix

S0θ S




limit


Variables





branch line losses










withdrawals


loss



Additional Notation



value




ξ isin Ξ

120














max zAC =summisinM

Cm(pgm) (41a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj



121



2kiv

2i minus τkivivj




2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T

2


)2+(qtk

)2le T

2





θref = 0 (41m)










122









2i minus αGkv

2i


) (42a)


2kiv

2i + α(Bk +Bs

kj)v2j


) (42b)




pfαk = Gk

(τ 2kiv

2i minus v2j



ki)τ2kiv

2i minus (Bk +Bs

kj)v2j





123







pℓk = Gk

(τ 2kiv

2i + v2j



2kiv

2i minus

(Bk +Bs

kj








4211 Power Balance






124



pnwi = P di +Gs


summisinMi


qnwi = Qdi minus Bs


summisinMi




tk = minusp

fαk + 1

2pℓk q

fk = qfαk + 1


fαk + 1




pnw +A⊤pfα +1

2|A|⊤pℓ = 0 (46a)

qnw +A⊤qfα +1

2|A|⊤qℓ = 0 (46b)





Appendix A1



(47)



ξ pℓξ

125




ξ minusHvξ


ξ minusKvξ

(48)



pfα = Fθ + F0 (49a)

qfα = Hv +H0 (49b)

pℓ = Lθ + L0 (49c)

qℓ = Kv +K0 (49d)









126














pnw +A⊤(Fθ + F0

)+

1

2|A|⊤

(Lθ + L0

)= 0 (410a)

qnw +A⊤ (Hv +H0)+

1

2|A|⊤

(Kv +K0

)= 0 (410b)


Sθ = minus(A⊤F+

1

2|A|⊤L

)minus1

(411a)

Sv = minus(A⊤H+

1

2|A|⊤K

)minus1

(411b)


127


S0θ = Sθ

(A⊤F0 +

1

2|A|⊤L0

)(411c)

S0v = Sv

(A⊤H0 +

1

2|A|⊤K0

)(411d)


θ

and v = Sθvq



F = FSθ H = HSv

L = LSθ K = KSv

(412)


F0 = FS0θ + F0 H0 = HS0

v +H0

L0 = LS0θ + L0 K0 = KS0

v +K0

(413)







pℓ = Lpnw + L0 (414c)

qℓ = Kqnw + K0 (414d)

128














balance constraint

1⊤pnw + 1⊤pℓ = 0 (415)







129




summation A1 = 0

















130




L = 1⊤L L0 = 1⊤L0

K = 1⊤K K0 = 1⊤K0

(416)



pℓ = Lpnw + L0 (417a)

qℓ = Kqnw + K0 (417b)


1⊤pnw + pℓ = 0 (418)






131



1

2pℓk

)2

+

(qfαk +

1

2qℓk

)2

le T2

k if(pfξ k

)2+(qfξ k

)2gt(ptξk

)2+(qtξk

)2(pfαk minus

1

2pℓk

)2

+

(qfαk minus

1

2qℓk

)2

le T2






2pfξ kl

(pfαk +

1

2pℓk

)+ 2qfξ kl(q

fαk +

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2pℓk

)+ 2qfξ kl(q

fαk minus

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl


(420)




Ldk = pℓξk

sumκisinK


sumκisinK

qℓξκ (421)


132



2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk +

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk minus

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl


(422)





2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl

)2

if(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl


(423)



is if η = Tk

(pξfkl)

2+(qξfkl)


fkl =

radicηpξ

fkl and

qξfkl =

radicηqξ

fkl

133


4216 Voltage Limits



reformulated




v = Svqnw + S0

v (424a)

V le v le V (424b)








cm ge C0m + C1

mPgml + C2

m(Pgml)

2 +(C1

m + 2C2mP

gml

) (pgm minus P g

ml


134





points




min z =summisinM

cm



min z =summisinM

cm



min z =summisinM

cm


(422) (424) (425) (41k) (41l)

135



min z =summisinM

cm
















136



















v







137





















138









(A⊤F+

1

2|A|⊤L

)⊤


[kisinKprime] (426)














139




LKprime


[kisinKprime]

KKprime


[kisinKprime]

(428)




prime= 0 (429a)


(429b)


(429c)










140








F 0εk = F 0

k +sumiisinN




Fik




C-LOPF models







141














implementations







142





















143




Cases case14_ieee

case30_ieee

case57_ieee

case118_ieee

case300_ieee

case588_sdet

case2316_sdet

case2853_sdet

case4661_sdet

case89_pegase

case1354_pegase

case2869_pegase

case9241_pegase

case13659_pegase

case200_tamu

case500_tamu

case2000_tamu

case10000_tamu


Cases case2383wp_k

case2736sp_k

case2737sop_k

case2746wop_k

case2746wp_k

case3012wp_k

case3120sp_k

case3375wp_k

case1888_rte

case1951_rte

case2848_rte

case2868_rte

case6468_rte

case6470_rte

case6495_rte

case6515_rte

case3_lmbdf

case5_pjmg

case24_ieee_rtsh

case30_asi

case30_fsri

case39_eprij

case73_ieee_rtsk

case162_ieee_dtcl

case179_gocm

case240_psercn



144






and case6515_rte

442 LMP Error














145












146






















147














148





149






















used by ISOs

150






















151





152











153
















154

















155










156








with nominal demand














157





158





∥x∥p =

(sumk

|xi|p)(1p)








446 Solution Times







159





160






161

















162









DC-LOPF models








RTE test cases




163




45 Conclusion

















164












165

Chapter 5




51 Introduction





166























167





















markets


168























169











Notation







Sets



h isin H


170


k isin K






s isin S


χ = cupiχi

Parameters





coecient


coecients


coecients


coecients

Variables





schedule s





and solution s




variables


Functions


RK rarr R



RK rarr R







171



cdf Rrarr [0 1]




Additional Notation



value



zs Solution s


relaxation

zlb MIP lower bound













172























173













2008a)










174






521 Models









st A0x ge b0 (51b)





175





















176

















commitment status






177























178
















179




to as linear prots

πsi (λ) = (A⊤

0iλminus ci)⊤xs





U si (λ) = sup

(xiyi)isinχi

(A⊤0iλminus ci)













180








i (λ) (55)


i (λ) The actual


πsi (λ) = πs












181






















182











sumi

δmsi + δms

cs + δsopt = 0 (57)

where

δmsi = πs


i (λmlowast) (58a)



(microlowasti (λ


mlowast)) (58b)






183



∆ms =δmscs

+sumi

δmsi

(59)






c⊤x+ d⊤y + λ⊤(b0 minus A0x) (510)



sumi





s minus b0)

(511a)

= sup(xy)isinχ








184







i (λ) and



sumi U


lowast +sumi U

lowasti (λ) Then

sumi |δms



πsi (λ) + U s


(A⊤0iλminus ci)



i |δmsi | = 0


δmscs

= δsopt (513)



i (λ) and






sumi micro


lowast +sum



185






δmsi = |πs







i (λ)|(514)


by uplift and PRS






sumi




δmscs =

λ⊤A0(x



i (λ))minussumi


i (λ))

le

λ⊤A0(x



i (λ))

+sum

i

microsi (λ)minus

sumi

microlowasti (λ)

(517)



0 lesumi

microsi (λ) le

sumi


s minus b0) (518a)

186


0 lesumi


sumi





i

microsi (λ)minus

sumi

microlowasti (λ)














187





















188






















189



















bound




190












54 Example






degeneracy



191






min z =sum

(hi)isinG

Chixhi (521a)

stsum

(hi)isinG

xhi = 225 + ϵ (521b)













192




















to self-commit



193




55 Test Cases

















194






195





















196


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005



payments (MWP=sum












197


r pd td ld0

200

400

600

(a) RTS cases

r pd td ld0

50

100

(b) PJM cases















198




















199



∆m(τ) = (1S)sums



















200


00001 0001 001 01 1 10 100 1000 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 100 0

02

04

06

08

1

(b) PJM cases


201








S

sums π

si (λ

ms) and (σmi )

2 = 1Sminus1

sums(π

si (λ

ms) minus πmi )

2 be





Σm(τ) =sumi

1Σmi leτ

sumi

1πmi gt0 (524)








optimality

202


00001 0001 001 01 1 10 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 0

02

04

06

08

1

(b) PJM cases


203


56 Conclusion



















204


















market designs

205

Chapter 6

Conclusion













206






















207


61 Discussion



















208























209























210








certain







new eciencies

62 Looking Forward





211























212























213


















214

Appendix A









= 0 is given

below

pfαk asymp Gk

((τkivξi)

2 minus vξ2j



(A1)

215





F 0k = Gk

((τkivξi)

2 minus vξ2j



(A2c)


partθ= 0 is

given below


ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j


+(minus(Bk +Bs


)(vi minus vξi)

+((Bk +Bs


(A3)





H0k =

((Bk +Bs

ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j



= 0 is given below

pℓk asymp Gk

(τ 2kivξ

2i + vξ

2j



(A5)

216





L0k = Gk

((τkivξi)

2 + vξ2j



(A6c)


= 0 is given below


2kivξ

2i minus

(Bk +Bs

kj

)vξ


+(minus2 (Bk +Bs


)(vi minus vξi)

+(minus2(Bk +Bs

kj


)(vj minus vξj)

(A7)




Kkj =minus 2(Bk +Bs

kj


K0k = (Bk +Bs

ki) τ2kivξ

2i +

(Bk +Bs

kj

)vξ




217






minus(A⊤F+

1

2|A|⊤L

)⊤

F⊤= F⊤ (A9a)

minus(A⊤H+

1

2|A|⊤K

)⊤

H⊤= H⊤ (A9b)

minus(A⊤F+

1

2|A|⊤L

)⊤

L⊤= L⊤ (A9c)

minus(A⊤H+

1

2|A|⊤K

)⊤

K⊤= K⊤ (A9d)


minus(A⊤H+

1

2|A|⊤K

)⊤

Sv⊤= I⊤ (A9e)



F0 = F

(A⊤F+

1

2|A|⊤L

)+ F0 (A10a)

H0 = H

(A⊤H+

1

2|A|⊤K

)+H0 (A10b)


218


L0 = L

(A⊤F+

1

2|A|⊤L

)+ L0 (A10c)

K0 = K

(A⊤H+

1

2|A|⊤K

)+K0 (A10d)




= F(Sθp

nw + S0θ

)+ F0 (A11b)

= minusF(A⊤F+

1

2|A|⊤L


1

2|A|⊤L0

)+ F0 (A11c)


(A⊤F+

1

2|A|⊤L


2|A|⊤L0








219



L =partpnwRpartpnwi


K =partqnwRpartqnwi


(A12)



(A⊤F+

1

2|A|⊤L

)⊤

L⊤=

(A⊤F+

1

2|A|⊤L

)⊤

1 (A13a)(A⊤H+

1

2|A|⊤K

)⊤

K⊤=

(A⊤H+

1

2|A|⊤K

)⊤

1 (A13b)


220

Appendix B













221






















222




















223
















individual prot and


224







Baseload Outputsum














MWh




225










less than 23











226


















227




r 060 040 $2790 $1907MWhtd 100 000 $2750 $1600MWh

Di () 144 2714














228



a Greedy Algorithm


















229








min z =sum

(gk)isinG

Cgkxgk (B1a)

stsum

(gk)isinG

xgk = 225 + ϵ (B1b)












230




smoothing method




i is

















231


0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100O

ldTe

ch S

elf-C

omm

itmen

ts(1

0-ite

ratio

n m

ovin

g av

erag

e)


(a) r Pricing Model

0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100

Old

Tech

Sel

f-Com

mitm

ents

(10-

itera

tion

mov

ing

aver

age)




232























233



$0

$100

$200

$300

$400

Aver

age

Prof

it ($


(a) r Pricing Model


$0

$100

$200

$300

$400

Aver

age

Prof

it ($














234















degree of accuracy







235







236

Bibliography










237

BIBLIOGRAPHY















238

BIBLIOGRAPHY

















239

BIBLIOGRAPHY















240

BIBLIOGRAPHY















241

BIBLIOGRAPHY












gurobicom



242

BIBLIOGRAPHY















243

BIBLIOGRAPHY















244

BIBLIOGRAPHY















245

BIBLIOGRAPHY


















246

BIBLIOGRAPHY






577508











247

BIBLIOGRAPHY















248

BIBLIOGRAPHY














249

BIBLIOGRAPHY

















250

BIBLIOGRAPHY
















251

BIBLIOGRAPHY






252

Vita

Biography


Education







Experience




253

VITA










Papers












254

VITA




Honors amp Awards















Affiliations





Technical Skills


255

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures

Introduction

Brief Background

Research Questions



Power Flow

Optimal Power Flow

Unit Commitment



Introduction


Model


Conclusion


Introduction

Model Derivations



Conclusion


Introduction


Theoretical Results

Example

Test Cases

Conclusion

Conclusion

Discussion

Looking Forward







Vita

ABSTRACT












Readers




iii

Acknowledgments
















iv

ACKNOWLEDGMENTS



















v

Dedicated to my dad

vi

Contents

Abstract ii

Acknowledgments iv

Dedication vi

List of Tables ix

List of Figures x





vii

CONTENTS





Vita 253

viii

List of Tables





ix

List of Figures







x

Chapter 1

Introduction













1























2



these markets




















3












4




11 Brief Background

















5






(LP) software

















6























7






















8



















9





















10




















11























12

















13

Chapter 2


Literature Review





Chapters 3-5







14







21 Power Flow










15


211 AC Power Flow













(21)








16








voltages


(sumj


)⋆

(23)




pi = visumj


qi = visumj







17



pfk = Gkτ2kiv


ptk = Gkτ2kjv



2kiv



2kjv







summisinMi


iv2i minus

sumkisinKfr

i


i

ptk = 0 (26a)

summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i

qtk = 0 (26b)








18
















i (ie no additional

constraints)




19







213 DC Power Flow












20









qfk = qtk = 0 (212)



summisinMi


i minussumkisinKfr

i

pfk +sumkisinKto

i

pfk = 0 (213)







21



pnw +A⊤pf = 0 (214a)


θref = 0 (214c)









derivation







22











Solving for θ

θ =(A⊤BA

)minus1pnw (216)







(A⊤BA

)F

⊤= minus (BA)⊤ (217)


23




more generallysum



W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (218)





= minusBAθ







1⊤pnw = 0 (219)

24



















25



Cm(pgm) = C0

m + C1mp

gm + C2

m(pgm)

2 (220)


1m and C2

m are the coecients







cm ge C0m + C1

mpgml + C2

m(pgml)

2 +(C1

m + 2C2mp

gml

) (pgm minus pgml








26









max zAC =summisinM

Cm(pgm) (222a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj




2kiv

2i minus τkivivj



27



2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T 2


)2+(qtk

)2le T 2






θref = 0 (222n)











28





















29



2i = Gs

i


















30






















31








(226a)








λ = λ1+ F⊤micro




32





βi

(pgi minus P i

)= 0 (227b)











software





33







23 Unit Commitment














34



231 Formulation





min z = c⊤x

st Ax ge b

x ge 0








min z = c⊤x+ d⊤y (228a)

35


st A0x ge b0 (228b)








i χi where












36





















(229)

37











0iλminus ci)⊤

xi minus d⊤i yi






(xiyi)isinχi






38















prime) = 0 so that




Llowast = supλge0


supλge0



39






rearranged like so


= inf(xy)isinχ


= sup

(xy)isinχ

(A⊤

0 λlowast minus c


= sup

(xy)isinχ

sumi


= sup(xy)isinχ

sumi

πi(λlowast xi yi)

minus λlowast⊤b0

=sumi

sup

(xiyi)isinχi

πi(λlowast xi yi)

minus λlowast⊤b0








40



Llowast = supλge0

inf(xy)isinχ


supλge0





supλge0
















41











convex relaxation











42






















43






















44



















45





















46





















47

















48



infeasible













min z = tr(Cw)


w ⪰ 0



49


(where tr(Cw) =sum





















50




⎡⎢⎢⎣wii wij

wij wjj

⎤⎥⎥⎦ ⪰ 0

















51






















52






















53


Discussion




















54






















55
















56






















57






















58




losses









Diniz 2011)










59








AC OPF















60




















grid

61






















62























63











Discussion











64




power and voltage

















65





















66















periods11








67





















68























69






















70











Lagrangian dual












71























72




















73


Discussion




















74







in their infancy


75

Chapter 3


DC OPF with Losses

31 Introduction






76




















77














λ = λE + λL + λC (31)







78






















79

CHAPTER3

IMPROVEDMETHODFORSO

LVING

THEDCOPFWITHLOSSE

S


ISO(Source)

Used inSCED


CAISO(2020)




ERCOT(2016)



Seasonal

ISO-NE(2019)




MISO(Sutton2014)




NYISO(2020ab)




PJM(2010)




SPP(2020)




80






















81















313 Contributions






82



















83


Notation








Sets


k isin K



i

Kfri Kto


to bus i


i j n isin N



Parameters


n-side of branch k


branch k



length




j

and 0 otherwise



84






y isin 0 1 2



injections





matrix



vector


oset


susceptance matrix



real power output



matrix




Variables


elements θi



θi minus θj


from direction with

elements pfk




with elements pgn



with elements pnwi



elements vi



85


Dual Variables



limits






the AC OPF



constraints


function constraint

Additional Notation

Hadamard product



xAC AC OPF solution







pfk = Gkτ2kiv


ptk = Gkτ2kjv




86





from direction






equations are

pnwi +sumkisinKfr

i


i





pnw +A⊤pf = 0 (34)




87


321 DC Power Flow













) (35)







88






W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (36)










Mii =sumkisinKi

τkivξivξj


) foralli isin N

Mij = τkivξivξj




89



W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣∆θ

u

⎤⎥⎥⎦ =

⎡⎢⎢⎣∆pnw

0

⎤⎥⎥⎦ (37)









W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣ Sθ

U⊤

⎤⎥⎥⎦ =

⎡⎢⎢⎣ I

0⊤

⎤⎥⎥⎦ (38)



(32a) and (32b)

pℓk = Gk

(τ 2kiv

2i + τ 2kjv


) (39)

90






L = LSθ (310)














91



Lki =dpℓkdpnwn


dpnwn

=2Rk

R2k +X2

k


(312)




2Rk

R2k +X2

k


=2Rk

(R2k +X2

k)12

τkivivjpfk times

dθijdpnwn

(313)




k)minus12 dθij

dpnwn


n Rearranging



Li = 2sumk






pℓ asympsumk

Rk(pfk)

2 (315)

92


33 Model







)le T (316d)

P le pg le P (316e)











93




⊤Pd

)minus micro⊤

(T+ FPd

)minus micro⊤

(Tminus FPd


(317a)



)⊤FD = 0 (317c)







λE = λ1 (318a)

λL = σL (318b)


)⊤F (318c)

λ = λE + λL + λC (318d)



94




















95


332 LMP Accuracy




















96















Avg Dispatch Di =1

N

sumi


LMP MAPE =1

N

sumi


i |λACi

times 100 (320)


zACtimes 100 (321)


97


0 50 100 150 200 250 300-15

-10

-5

0

5

10



0 10 20 30 40 50 60-150

-100

-50

0

50

100




98














compared to DC OPF







99




A 1 3000 2950 10B 1 3000 2975 100C 2 3000 3000 100












2




100


















101









pℓ =sumk

pℓk =sumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (322)




pℓk = E(2)k (pfk)

2 +(2E

(2)k E

(1)k

)pfk +

(E

(2)k (E

(1)k )2 + E

(0)k

)(323)



fk + E

(1)k )pfk + E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (324)



fk +E

(1)k )) that can be


(1)k )2 minus (pξ

fk)

2) + E(0)k )

102












(1)k )Fkn (325a)

L0k = E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (325b)

Ln =sumk

Lkn (325c)

L0 =sumk

L0k (325d)




dpℓkdpnwn



k and E(0)k can

103




ℓξ τki

1 pfξ larr F(pg



3 E(1)k larr Lkn

(2E

(2)k Fkn

)minus pξ


4 E(0)k larr L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)





8 pξℓ larr

sumk E

(2)k (pξijk + E

(1)k )2 + E

(0)k

9 Ln larr 2sum

k

(E

(2)k (pξijk + E

(1)k )Fkn

)foralln isin N






E(1)k =

Lkn

2E(2)k Fkn

minus pξfk (327b)

E(0)k = L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)

2) (327c)




fk are available If


(1)k = E

(0)k = 0 and




104




k is



k lt ε


of the algorithm

pξgh+1n = ωpξ


pξfh+1k = ωpξ








pℓ gesumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (329)





105
















in the algorithm





106


0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 5 10 15 201e-09

1e-06

0001

1










107

















108












300-bus networks)




109



001

01

1

10

100

1000LMP

MAPE


-03

-02

-01

0

01

RelativeLoss

Error













110




35 Conclusion

















111










voltage levels

112

Chapter 4



of the AC OPF

41 Introduction





113













physical accuracy








114









less accurate














115
















412 Contribution





116























117







413 Outline





chapter

Notation






118




Sets


k isin K



i

Kfri Kto


to bus i




m isinM



i j n isin N

Parameters


i-side of branch k


branch k


of ones




j

and 0 otherwise


of branch k





sensitivities




sensitivities


I Identity matrix


sensitivities



loss sensitivity


loss oset


119




sensitivity


oset


demand



output







matrix

S0θ S




limit


Variables





branch line losses










withdrawals


loss



Additional Notation



value




ξ isin Ξ

120














max zAC =summisinM

Cm(pgm) (41a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj



121



2kiv

2i minus τkivivj




2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T

2


)2+(qtk

)2le T

2





θref = 0 (41m)










122









2i minus αGkv

2i


) (42a)


2kiv

2i + α(Bk +Bs

kj)v2j


) (42b)




pfαk = Gk

(τ 2kiv

2i minus v2j



ki)τ2kiv

2i minus (Bk +Bs

kj)v2j





123







pℓk = Gk

(τ 2kiv

2i + v2j



2kiv

2i minus

(Bk +Bs

kj








4211 Power Balance






124



pnwi = P di +Gs


summisinMi


qnwi = Qdi minus Bs


summisinMi




tk = minusp

fαk + 1

2pℓk q

fk = qfαk + 1


fαk + 1




pnw +A⊤pfα +1

2|A|⊤pℓ = 0 (46a)

qnw +A⊤qfα +1

2|A|⊤qℓ = 0 (46b)





Appendix A1



(47)



ξ pℓξ

125




ξ minusHvξ


ξ minusKvξ

(48)



pfα = Fθ + F0 (49a)

qfα = Hv +H0 (49b)

pℓ = Lθ + L0 (49c)

qℓ = Kv +K0 (49d)









126














pnw +A⊤(Fθ + F0

)+

1

2|A|⊤

(Lθ + L0

)= 0 (410a)

qnw +A⊤ (Hv +H0)+

1

2|A|⊤

(Kv +K0

)= 0 (410b)


Sθ = minus(A⊤F+

1

2|A|⊤L

)minus1

(411a)

Sv = minus(A⊤H+

1

2|A|⊤K

)minus1

(411b)


127


S0θ = Sθ

(A⊤F0 +

1

2|A|⊤L0

)(411c)

S0v = Sv

(A⊤H0 +

1

2|A|⊤K0

)(411d)


θ

and v = Sθvq



F = FSθ H = HSv

L = LSθ K = KSv

(412)


F0 = FS0θ + F0 H0 = HS0

v +H0

L0 = LS0θ + L0 K0 = KS0

v +K0

(413)







pℓ = Lpnw + L0 (414c)

qℓ = Kqnw + K0 (414d)

128














balance constraint

1⊤pnw + 1⊤pℓ = 0 (415)







129




summation A1 = 0

















130




L = 1⊤L L0 = 1⊤L0

K = 1⊤K K0 = 1⊤K0

(416)



pℓ = Lpnw + L0 (417a)

qℓ = Kqnw + K0 (417b)


1⊤pnw + pℓ = 0 (418)






131



1

2pℓk

)2

+

(qfαk +

1

2qℓk

)2

le T2

k if(pfξ k

)2+(qfξ k

)2gt(ptξk

)2+(qtξk

)2(pfαk minus

1

2pℓk

)2

+

(qfαk minus

1

2qℓk

)2

le T2






2pfξ kl

(pfαk +

1

2pℓk

)+ 2qfξ kl(q

fαk +

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2pℓk

)+ 2qfξ kl(q

fαk minus

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl


(420)




Ldk = pℓξk

sumκisinK


sumκisinK

qℓξκ (421)


132



2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk +

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk minus

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl


(422)





2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl

)2

if(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl


(423)



is if η = Tk

(pξfkl)

2+(qξfkl)


fkl =

radicηpξ

fkl and

qξfkl =

radicηqξ

fkl

133


4216 Voltage Limits



reformulated




v = Svqnw + S0

v (424a)

V le v le V (424b)








cm ge C0m + C1

mPgml + C2

m(Pgml)

2 +(C1

m + 2C2mP

gml

) (pgm minus P g

ml


134





points




min z =summisinM

cm



min z =summisinM

cm



min z =summisinM

cm


(422) (424) (425) (41k) (41l)

135



min z =summisinM

cm
















136



















v







137





















138









(A⊤F+

1

2|A|⊤L

)⊤


[kisinKprime] (426)














139




LKprime


[kisinKprime]

KKprime


[kisinKprime]

(428)




prime= 0 (429a)


(429b)


(429c)










140








F 0εk = F 0

k +sumiisinN




Fik




C-LOPF models







141














implementations







142





















143




Cases case14_ieee

case30_ieee

case57_ieee

case118_ieee

case300_ieee

case588_sdet

case2316_sdet

case2853_sdet

case4661_sdet

case89_pegase

case1354_pegase

case2869_pegase

case9241_pegase

case13659_pegase

case200_tamu

case500_tamu

case2000_tamu

case10000_tamu


Cases case2383wp_k

case2736sp_k

case2737sop_k

case2746wop_k

case2746wp_k

case3012wp_k

case3120sp_k

case3375wp_k

case1888_rte

case1951_rte

case2848_rte

case2868_rte

case6468_rte

case6470_rte

case6495_rte

case6515_rte

case3_lmbdf

case5_pjmg

case24_ieee_rtsh

case30_asi

case30_fsri

case39_eprij

case73_ieee_rtsk

case162_ieee_dtcl

case179_gocm

case240_psercn



144






and case6515_rte

442 LMP Error














145












146






















147














148





149






















used by ISOs

150






















151





152











153
















154

















155










156








with nominal demand














157





158





∥x∥p =

(sumk

|xi|p)(1p)








446 Solution Times







159





160






161

















162









DC-LOPF models








RTE test cases




163




45 Conclusion

















164












165

Chapter 5




51 Introduction





166























167





















markets


168























169











Notation







Sets



h isin H


170


k isin K






s isin S


χ = cupiχi

Parameters





coecient


coecients


coecients


coecients

Variables





schedule s





and solution s




variables


Functions


RK rarr R



RK rarr R







171



cdf Rrarr [0 1]




Additional Notation



value



zs Solution s


relaxation

zlb MIP lower bound













172























173













2008a)










174






521 Models









st A0x ge b0 (51b)





175





















176

















commitment status






177























178
















179




to as linear prots

πsi (λ) = (A⊤

0iλminus ci)⊤xs





U si (λ) = sup

(xiyi)isinχi

(A⊤0iλminus ci)













180








i (λ) (55)


i (λ) The actual


πsi (λ) = πs












181






















182











sumi

δmsi + δms

cs + δsopt = 0 (57)

where

δmsi = πs


i (λmlowast) (58a)



(microlowasti (λ


mlowast)) (58b)






183



∆ms =δmscs

+sumi

δmsi

(59)






c⊤x+ d⊤y + λ⊤(b0 minus A0x) (510)



sumi





s minus b0)

(511a)

= sup(xy)isinχ








184







i (λ) and



sumi U


lowast +sumi U

lowasti (λ) Then

sumi |δms



πsi (λ) + U s


(A⊤0iλminus ci)



i |δmsi | = 0


δmscs

= δsopt (513)



i (λ) and






sumi micro


lowast +sum



185






δmsi = |πs







i (λ)|(514)


by uplift and PRS






sumi




δmscs =

λ⊤A0(x



i (λ))minussumi


i (λ))

le

λ⊤A0(x



i (λ))

+sum

i

microsi (λ)minus

sumi

microlowasti (λ)

(517)



0 lesumi

microsi (λ) le

sumi


s minus b0) (518a)

186


0 lesumi


sumi





i

microsi (λ)minus

sumi

microlowasti (λ)














187





















188






















189



















bound




190












54 Example






degeneracy



191






min z =sum

(hi)isinG

Chixhi (521a)

stsum

(hi)isinG

xhi = 225 + ϵ (521b)













192




















to self-commit



193




55 Test Cases

















194






195





















196


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005



payments (MWP=sum












197


r pd td ld0

200

400

600

(a) RTS cases

r pd td ld0

50

100

(b) PJM cases















198




















199



∆m(τ) = (1S)sums



















200


00001 0001 001 01 1 10 100 1000 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 100 0

02

04

06

08

1

(b) PJM cases


201








S

sums π

si (λ

ms) and (σmi )

2 = 1Sminus1

sums(π

si (λ

ms) minus πmi )

2 be





Σm(τ) =sumi

1Σmi leτ

sumi

1πmi gt0 (524)








optimality

202


00001 0001 001 01 1 10 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 0

02

04

06

08

1

(b) PJM cases


203


56 Conclusion



















204


















market designs

205

Chapter 6

Conclusion













206






















207


61 Discussion



















208























209























210








certain







new eciencies

62 Looking Forward





211























212























213


















214

Appendix A









= 0 is given

below

pfαk asymp Gk

((τkivξi)

2 minus vξ2j



(A1)

215





F 0k = Gk

((τkivξi)

2 minus vξ2j



(A2c)


partθ= 0 is

given below


ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j


+(minus(Bk +Bs


)(vi minus vξi)

+((Bk +Bs


(A3)





H0k =

((Bk +Bs

ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j



= 0 is given below

pℓk asymp Gk

(τ 2kivξ

2i + vξ

2j



(A5)

216





L0k = Gk

((τkivξi)

2 + vξ2j



(A6c)


= 0 is given below


2kivξ

2i minus

(Bk +Bs

kj

)vξ


+(minus2 (Bk +Bs


)(vi minus vξi)

+(minus2(Bk +Bs

kj


)(vj minus vξj)

(A7)




Kkj =minus 2(Bk +Bs

kj


K0k = (Bk +Bs

ki) τ2kivξ

2i +

(Bk +Bs

kj

)vξ




217






minus(A⊤F+

1

2|A|⊤L

)⊤

F⊤= F⊤ (A9a)

minus(A⊤H+

1

2|A|⊤K

)⊤

H⊤= H⊤ (A9b)

minus(A⊤F+

1

2|A|⊤L

)⊤

L⊤= L⊤ (A9c)

minus(A⊤H+

1

2|A|⊤K

)⊤

K⊤= K⊤ (A9d)


minus(A⊤H+

1

2|A|⊤K

)⊤

Sv⊤= I⊤ (A9e)



F0 = F

(A⊤F+

1

2|A|⊤L

)+ F0 (A10a)

H0 = H

(A⊤H+

1

2|A|⊤K

)+H0 (A10b)


218


L0 = L

(A⊤F+

1

2|A|⊤L

)+ L0 (A10c)

K0 = K

(A⊤H+

1

2|A|⊤K

)+K0 (A10d)




= F(Sθp

nw + S0θ

)+ F0 (A11b)

= minusF(A⊤F+

1

2|A|⊤L


1

2|A|⊤L0

)+ F0 (A11c)


(A⊤F+

1

2|A|⊤L


2|A|⊤L0








219



L =partpnwRpartpnwi


K =partqnwRpartqnwi


(A12)



(A⊤F+

1

2|A|⊤L

)⊤

L⊤=

(A⊤F+

1

2|A|⊤L

)⊤

1 (A13a)(A⊤H+

1

2|A|⊤K

)⊤

K⊤=

(A⊤H+

1

2|A|⊤K

)⊤

1 (A13b)


220

Appendix B













221






















222




















223
















individual prot and


224







Baseload Outputsum














MWh




225










less than 23











226


















227




r 060 040 $2790 $1907MWhtd 100 000 $2750 $1600MWh

Di () 144 2714














228



a Greedy Algorithm


















229








min z =sum

(gk)isinG

Cgkxgk (B1a)

stsum

(gk)isinG

xgk = 225 + ϵ (B1b)












230




smoothing method




i is

















231


0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100O

ldTe

ch S

elf-C

omm

itmen

ts(1

0-ite

ratio

n m

ovin

g av

erag

e)


(a) r Pricing Model

0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100

Old

Tech

Sel

f-Com

mitm

ents

(10-

itera

tion

mov

ing

aver

age)




232























233



$0

$100

$200

$300

$400

Aver

age

Prof

it ($


(a) r Pricing Model


$0

$100

$200

$300

$400

Aver

age

Prof

it ($














234















degree of accuracy







235







236

Bibliography










237

BIBLIOGRAPHY















238

BIBLIOGRAPHY

















239

BIBLIOGRAPHY















240

BIBLIOGRAPHY















241

BIBLIOGRAPHY












gurobicom



242

BIBLIOGRAPHY















243

BIBLIOGRAPHY















244

BIBLIOGRAPHY















245

BIBLIOGRAPHY


















246

BIBLIOGRAPHY






577508











247

BIBLIOGRAPHY















248

BIBLIOGRAPHY














249

BIBLIOGRAPHY

















250

BIBLIOGRAPHY
















251

BIBLIOGRAPHY






252

Vita

Biography


Education







Experience




253

VITA










Papers












254

VITA




Honors amp Awards















Affiliations





Technical Skills


255

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures

Introduction

Brief Background

Research Questions



Power Flow

Optimal Power Flow

Unit Commitment



Introduction


Model


Conclusion


Introduction

Model Derivations



Conclusion


Introduction


Theoretical Results

Example

Test Cases

Conclusion

Conclusion

Discussion

Looking Forward







Vita

Acknowledgments
















iv

ACKNOWLEDGMENTS



















v

Dedicated to my dad

vi

Contents

Abstract ii

Acknowledgments iv

Dedication vi

List of Tables ix

List of Figures x





vii

CONTENTS





Vita 253

viii

List of Tables





ix

List of Figures







x

Chapter 1

Introduction













1























2



these markets




















3












4




11 Brief Background

















5






(LP) software

















6























7






















8



















9





















10




















11























12

















13

Chapter 2


Literature Review





Chapters 3-5







14







21 Power Flow










15


211 AC Power Flow













(21)








16








voltages


(sumj


)⋆

(23)




pi = visumj


qi = visumj







17



pfk = Gkτ2kiv


ptk = Gkτ2kjv



2kiv



2kjv







summisinMi


iv2i minus

sumkisinKfr

i


i

ptk = 0 (26a)

summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i

qtk = 0 (26b)








18
















i (ie no additional

constraints)




19







213 DC Power Flow












20









qfk = qtk = 0 (212)



summisinMi


i minussumkisinKfr

i

pfk +sumkisinKto

i

pfk = 0 (213)







21



pnw +A⊤pf = 0 (214a)


θref = 0 (214c)









derivation







22











Solving for θ

θ =(A⊤BA

)minus1pnw (216)







(A⊤BA

)F

⊤= minus (BA)⊤ (217)


23




more generallysum



W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (218)





= minusBAθ







1⊤pnw = 0 (219)

24



















25



Cm(pgm) = C0

m + C1mp

gm + C2

m(pgm)

2 (220)


1m and C2

m are the coecients







cm ge C0m + C1

mpgml + C2

m(pgml)

2 +(C1

m + 2C2mp

gml

) (pgm minus pgml








26









max zAC =summisinM

Cm(pgm) (222a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj




2kiv

2i minus τkivivj



27



2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T 2


)2+(qtk

)2le T 2






θref = 0 (222n)











28





















29



2i = Gs

i


















30






















31








(226a)








λ = λ1+ F⊤micro




32





βi

(pgi minus P i

)= 0 (227b)











software





33







23 Unit Commitment














34



231 Formulation





min z = c⊤x

st Ax ge b

x ge 0








min z = c⊤x+ d⊤y (228a)

35


st A0x ge b0 (228b)








i χi where












36





















(229)

37











0iλminus ci)⊤

xi minus d⊤i yi






(xiyi)isinχi






38















prime) = 0 so that




Llowast = supλge0


supλge0



39






rearranged like so


= inf(xy)isinχ


= sup

(xy)isinχ

(A⊤

0 λlowast minus c


= sup

(xy)isinχ

sumi


= sup(xy)isinχ

sumi

πi(λlowast xi yi)

minus λlowast⊤b0

=sumi

sup

(xiyi)isinχi

πi(λlowast xi yi)

minus λlowast⊤b0








40



Llowast = supλge0

inf(xy)isinχ


supλge0





supλge0
















41











convex relaxation











42






















43






















44



















45





















46





















47

















48



infeasible













min z = tr(Cw)


w ⪰ 0



49


(where tr(Cw) =sum





















50




⎡⎢⎢⎣wii wij

wij wjj

⎤⎥⎥⎦ ⪰ 0

















51






















52






















53


Discussion




















54






















55
















56






















57






















58




losses









Diniz 2011)










59








AC OPF















60




















grid

61






















62























63











Discussion











64




power and voltage

















65





















66















periods11








67





















68























69






















70











Lagrangian dual












71























72




















73


Discussion




















74







in their infancy


75

Chapter 3


DC OPF with Losses

31 Introduction






76




















77














λ = λE + λL + λC (31)







78






















79

CHAPTER3

IMPROVEDMETHODFORSO

LVING

THEDCOPFWITHLOSSE

S


ISO(Source)

Used inSCED


CAISO(2020)




ERCOT(2016)



Seasonal

ISO-NE(2019)




MISO(Sutton2014)




NYISO(2020ab)




PJM(2010)




SPP(2020)




80






















81















313 Contributions






82



















83


Notation








Sets


k isin K



i

Kfri Kto


to bus i


i j n isin N



Parameters


n-side of branch k


branch k



length




j

and 0 otherwise



84






y isin 0 1 2



injections





matrix



vector


oset


susceptance matrix



real power output



matrix




Variables


elements θi



θi minus θj


from direction with

elements pfk




with elements pgn



with elements pnwi



elements vi



85


Dual Variables



limits






the AC OPF



constraints


function constraint

Additional Notation

Hadamard product



xAC AC OPF solution







pfk = Gkτ2kiv


ptk = Gkτ2kjv




86





from direction






equations are

pnwi +sumkisinKfr

i


i





pnw +A⊤pf = 0 (34)




87


321 DC Power Flow













) (35)







88






W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (36)










Mii =sumkisinKi

τkivξivξj


) foralli isin N

Mij = τkivξivξj




89



W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣∆θ

u

⎤⎥⎥⎦ =

⎡⎢⎢⎣∆pnw

0

⎤⎥⎥⎦ (37)









W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣ Sθ

U⊤

⎤⎥⎥⎦ =

⎡⎢⎢⎣ I

0⊤

⎤⎥⎥⎦ (38)



(32a) and (32b)

pℓk = Gk

(τ 2kiv

2i + τ 2kjv


) (39)

90






L = LSθ (310)














91



Lki =dpℓkdpnwn


dpnwn

=2Rk

R2k +X2

k


(312)




2Rk

R2k +X2

k


=2Rk

(R2k +X2

k)12

τkivivjpfk times

dθijdpnwn

(313)




k)minus12 dθij

dpnwn


n Rearranging



Li = 2sumk






pℓ asympsumk

Rk(pfk)

2 (315)

92


33 Model







)le T (316d)

P le pg le P (316e)











93




⊤Pd

)minus micro⊤

(T+ FPd

)minus micro⊤

(Tminus FPd


(317a)



)⊤FD = 0 (317c)







λE = λ1 (318a)

λL = σL (318b)


)⊤F (318c)

λ = λE + λL + λC (318d)



94




















95


332 LMP Accuracy




















96















Avg Dispatch Di =1

N

sumi


LMP MAPE =1

N

sumi


i |λACi

times 100 (320)


zACtimes 100 (321)


97


0 50 100 150 200 250 300-15

-10

-5

0

5

10



0 10 20 30 40 50 60-150

-100

-50

0

50

100




98














compared to DC OPF







99




A 1 3000 2950 10B 1 3000 2975 100C 2 3000 3000 100












2




100


















101









pℓ =sumk

pℓk =sumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (322)




pℓk = E(2)k (pfk)

2 +(2E

(2)k E

(1)k

)pfk +

(E

(2)k (E

(1)k )2 + E

(0)k

)(323)



fk + E

(1)k )pfk + E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (324)



fk +E

(1)k )) that can be


(1)k )2 minus (pξ

fk)

2) + E(0)k )

102












(1)k )Fkn (325a)

L0k = E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (325b)

Ln =sumk

Lkn (325c)

L0 =sumk

L0k (325d)




dpℓkdpnwn



k and E(0)k can

103




ℓξ τki

1 pfξ larr F(pg



3 E(1)k larr Lkn

(2E

(2)k Fkn

)minus pξ


4 E(0)k larr L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)





8 pξℓ larr

sumk E

(2)k (pξijk + E

(1)k )2 + E

(0)k

9 Ln larr 2sum

k

(E

(2)k (pξijk + E

(1)k )Fkn

)foralln isin N






E(1)k =

Lkn

2E(2)k Fkn

minus pξfk (327b)

E(0)k = L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)

2) (327c)




fk are available If


(1)k = E

(0)k = 0 and




104




k is



k lt ε


of the algorithm

pξgh+1n = ωpξ


pξfh+1k = ωpξ








pℓ gesumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (329)





105
















in the algorithm





106


0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 5 10 15 201e-09

1e-06

0001

1










107

















108












300-bus networks)




109



001

01

1

10

100

1000LMP

MAPE


-03

-02

-01

0

01

RelativeLoss

Error













110




35 Conclusion

















111










voltage levels

112

Chapter 4



of the AC OPF

41 Introduction





113













physical accuracy








114









less accurate














115
















412 Contribution





116























117







413 Outline





chapter

Notation






118




Sets


k isin K



i

Kfri Kto


to bus i




m isinM



i j n isin N

Parameters


i-side of branch k


branch k


of ones




j

and 0 otherwise


of branch k





sensitivities




sensitivities


I Identity matrix


sensitivities



loss sensitivity


loss oset


119




sensitivity


oset


demand



output







matrix

S0θ S




limit


Variables





branch line losses










withdrawals


loss



Additional Notation



value




ξ isin Ξ

120














max zAC =summisinM

Cm(pgm) (41a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj



121



2kiv

2i minus τkivivj




2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T

2


)2+(qtk

)2le T

2





θref = 0 (41m)










122









2i minus αGkv

2i


) (42a)


2kiv

2i + α(Bk +Bs

kj)v2j


) (42b)




pfαk = Gk

(τ 2kiv

2i minus v2j



ki)τ2kiv

2i minus (Bk +Bs

kj)v2j





123







pℓk = Gk

(τ 2kiv

2i + v2j



2kiv

2i minus

(Bk +Bs

kj








4211 Power Balance






124



pnwi = P di +Gs


summisinMi


qnwi = Qdi minus Bs


summisinMi




tk = minusp

fαk + 1

2pℓk q

fk = qfαk + 1


fαk + 1




pnw +A⊤pfα +1

2|A|⊤pℓ = 0 (46a)

qnw +A⊤qfα +1

2|A|⊤qℓ = 0 (46b)





Appendix A1



(47)



ξ pℓξ

125




ξ minusHvξ


ξ minusKvξ

(48)



pfα = Fθ + F0 (49a)

qfα = Hv +H0 (49b)

pℓ = Lθ + L0 (49c)

qℓ = Kv +K0 (49d)









126














pnw +A⊤(Fθ + F0

)+

1

2|A|⊤

(Lθ + L0

)= 0 (410a)

qnw +A⊤ (Hv +H0)+

1

2|A|⊤

(Kv +K0

)= 0 (410b)


Sθ = minus(A⊤F+

1

2|A|⊤L

)minus1

(411a)

Sv = minus(A⊤H+

1

2|A|⊤K

)minus1

(411b)


127


S0θ = Sθ

(A⊤F0 +

1

2|A|⊤L0

)(411c)

S0v = Sv

(A⊤H0 +

1

2|A|⊤K0

)(411d)


θ

and v = Sθvq



F = FSθ H = HSv

L = LSθ K = KSv

(412)


F0 = FS0θ + F0 H0 = HS0

v +H0

L0 = LS0θ + L0 K0 = KS0

v +K0

(413)







pℓ = Lpnw + L0 (414c)

qℓ = Kqnw + K0 (414d)

128














balance constraint

1⊤pnw + 1⊤pℓ = 0 (415)







129




summation A1 = 0

















130




L = 1⊤L L0 = 1⊤L0

K = 1⊤K K0 = 1⊤K0

(416)



pℓ = Lpnw + L0 (417a)

qℓ = Kqnw + K0 (417b)


1⊤pnw + pℓ = 0 (418)






131



1

2pℓk

)2

+

(qfαk +

1

2qℓk

)2

le T2

k if(pfξ k

)2+(qfξ k

)2gt(ptξk

)2+(qtξk

)2(pfαk minus

1

2pℓk

)2

+

(qfαk minus

1

2qℓk

)2

le T2






2pfξ kl

(pfαk +

1

2pℓk

)+ 2qfξ kl(q

fαk +

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2pℓk

)+ 2qfξ kl(q

fαk minus

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl


(420)




Ldk = pℓξk

sumκisinK


sumκisinK

qℓξκ (421)


132



2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk +

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk minus

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl


(422)





2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl

)2

if(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl


(423)



is if η = Tk

(pξfkl)

2+(qξfkl)


fkl =

radicηpξ

fkl and

qξfkl =

radicηqξ

fkl

133


4216 Voltage Limits



reformulated




v = Svqnw + S0

v (424a)

V le v le V (424b)








cm ge C0m + C1

mPgml + C2

m(Pgml)

2 +(C1

m + 2C2mP

gml

) (pgm minus P g

ml


134





points




min z =summisinM

cm



min z =summisinM

cm



min z =summisinM

cm


(422) (424) (425) (41k) (41l)

135



min z =summisinM

cm
















136



















v







137





















138









(A⊤F+

1

2|A|⊤L

)⊤


[kisinKprime] (426)














139




LKprime


[kisinKprime]

KKprime


[kisinKprime]

(428)




prime= 0 (429a)


(429b)


(429c)










140








F 0εk = F 0

k +sumiisinN




Fik




C-LOPF models







141














implementations







142





















143




Cases case14_ieee

case30_ieee

case57_ieee

case118_ieee

case300_ieee

case588_sdet

case2316_sdet

case2853_sdet

case4661_sdet

case89_pegase

case1354_pegase

case2869_pegase

case9241_pegase

case13659_pegase

case200_tamu

case500_tamu

case2000_tamu

case10000_tamu


Cases case2383wp_k

case2736sp_k

case2737sop_k

case2746wop_k

case2746wp_k

case3012wp_k

case3120sp_k

case3375wp_k

case1888_rte

case1951_rte

case2848_rte

case2868_rte

case6468_rte

case6470_rte

case6495_rte

case6515_rte

case3_lmbdf

case5_pjmg

case24_ieee_rtsh

case30_asi

case30_fsri

case39_eprij

case73_ieee_rtsk

case162_ieee_dtcl

case179_gocm

case240_psercn



144






and case6515_rte

442 LMP Error














145












146






















147














148





149






















used by ISOs

150






















151





152











153
















154

















155










156








with nominal demand














157





158





∥x∥p =

(sumk

|xi|p)(1p)








446 Solution Times







159





160






161

















162









DC-LOPF models








RTE test cases




163




45 Conclusion

















164












165

Chapter 5




51 Introduction





166























167





















markets


168























169











Notation







Sets



h isin H


170


k isin K






s isin S


χ = cupiχi

Parameters





coecient


coecients


coecients


coecients

Variables





schedule s





and solution s




variables


Functions


RK rarr R



RK rarr R







171



cdf Rrarr [0 1]




Additional Notation



value



zs Solution s


relaxation

zlb MIP lower bound













172























173













2008a)










174






521 Models









st A0x ge b0 (51b)





175





















176

















commitment status






177























178
















179




to as linear prots

πsi (λ) = (A⊤

0iλminus ci)⊤xs





U si (λ) = sup

(xiyi)isinχi

(A⊤0iλminus ci)













180








i (λ) (55)


i (λ) The actual


πsi (λ) = πs












181






















182











sumi

δmsi + δms

cs + δsopt = 0 (57)

where

δmsi = πs


i (λmlowast) (58a)



(microlowasti (λ


mlowast)) (58b)






183



∆ms =δmscs

+sumi

δmsi

(59)






c⊤x+ d⊤y + λ⊤(b0 minus A0x) (510)



sumi





s minus b0)

(511a)

= sup(xy)isinχ








184







i (λ) and



sumi U


lowast +sumi U

lowasti (λ) Then

sumi |δms



πsi (λ) + U s


(A⊤0iλminus ci)



i |δmsi | = 0


δmscs

= δsopt (513)



i (λ) and






sumi micro


lowast +sum



185






δmsi = |πs







i (λ)|(514)


by uplift and PRS






sumi




δmscs =

λ⊤A0(x



i (λ))minussumi


i (λ))

le

λ⊤A0(x



i (λ))

+sum

i

microsi (λ)minus

sumi

microlowasti (λ)

(517)



0 lesumi

microsi (λ) le

sumi


s minus b0) (518a)

186


0 lesumi


sumi





i

microsi (λ)minus

sumi

microlowasti (λ)














187





















188






















189



















bound




190












54 Example






degeneracy



191






min z =sum

(hi)isinG

Chixhi (521a)

stsum

(hi)isinG

xhi = 225 + ϵ (521b)













192




















to self-commit



193




55 Test Cases

















194






195





















196


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005



payments (MWP=sum












197


r pd td ld0

200

400

600

(a) RTS cases

r pd td ld0

50

100

(b) PJM cases















198




















199



∆m(τ) = (1S)sums



















200


00001 0001 001 01 1 10 100 1000 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 100 0

02

04

06

08

1

(b) PJM cases


201








S

sums π

si (λ

ms) and (σmi )

2 = 1Sminus1

sums(π

si (λ

ms) minus πmi )

2 be





Σm(τ) =sumi

1Σmi leτ

sumi

1πmi gt0 (524)








optimality

202


00001 0001 001 01 1 10 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 0

02

04

06

08

1

(b) PJM cases


203


56 Conclusion



















204


















market designs

205

Chapter 6

Conclusion













206






















207


61 Discussion



















208























209























210








certain







new eciencies

62 Looking Forward





211























212























213


















214

Appendix A









= 0 is given

below

pfαk asymp Gk

((τkivξi)

2 minus vξ2j



(A1)

215





F 0k = Gk

((τkivξi)

2 minus vξ2j



(A2c)


partθ= 0 is

given below


ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j


+(minus(Bk +Bs


)(vi minus vξi)

+((Bk +Bs


(A3)





H0k =

((Bk +Bs

ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j



= 0 is given below

pℓk asymp Gk

(τ 2kivξ

2i + vξ

2j



(A5)

216





L0k = Gk

((τkivξi)

2 + vξ2j



(A6c)


= 0 is given below


2kivξ

2i minus

(Bk +Bs

kj

)vξ


+(minus2 (Bk +Bs


)(vi minus vξi)

+(minus2(Bk +Bs

kj


)(vj minus vξj)

(A7)




Kkj =minus 2(Bk +Bs

kj


K0k = (Bk +Bs

ki) τ2kivξ

2i +

(Bk +Bs

kj

)vξ




217






minus(A⊤F+

1

2|A|⊤L

)⊤

F⊤= F⊤ (A9a)

minus(A⊤H+

1

2|A|⊤K

)⊤

H⊤= H⊤ (A9b)

minus(A⊤F+

1

2|A|⊤L

)⊤

L⊤= L⊤ (A9c)

minus(A⊤H+

1

2|A|⊤K

)⊤

K⊤= K⊤ (A9d)


minus(A⊤H+

1

2|A|⊤K

)⊤

Sv⊤= I⊤ (A9e)



F0 = F

(A⊤F+

1

2|A|⊤L

)+ F0 (A10a)

H0 = H

(A⊤H+

1

2|A|⊤K

)+H0 (A10b)


218


L0 = L

(A⊤F+

1

2|A|⊤L

)+ L0 (A10c)

K0 = K

(A⊤H+

1

2|A|⊤K

)+K0 (A10d)




= F(Sθp

nw + S0θ

)+ F0 (A11b)

= minusF(A⊤F+

1

2|A|⊤L


1

2|A|⊤L0

)+ F0 (A11c)


(A⊤F+

1

2|A|⊤L


2|A|⊤L0








219



L =partpnwRpartpnwi


K =partqnwRpartqnwi


(A12)



(A⊤F+

1

2|A|⊤L

)⊤

L⊤=

(A⊤F+

1

2|A|⊤L

)⊤

1 (A13a)(A⊤H+

1

2|A|⊤K

)⊤

K⊤=

(A⊤H+

1

2|A|⊤K

)⊤

1 (A13b)


220

Appendix B













221






















222




















223
















individual prot and


224







Baseload Outputsum














MWh




225










less than 23











226


















227




r 060 040 $2790 $1907MWhtd 100 000 $2750 $1600MWh

Di () 144 2714














228



a Greedy Algorithm


















229








min z =sum

(gk)isinG

Cgkxgk (B1a)

stsum

(gk)isinG

xgk = 225 + ϵ (B1b)












230




smoothing method




i is

















231


0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100O

ldTe

ch S

elf-C

omm

itmen

ts(1

0-ite

ratio

n m

ovin

g av

erag

e)


(a) r Pricing Model

0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100

Old

Tech

Sel

f-Com

mitm

ents

(10-

itera

tion

mov

ing

aver

age)




232























233



$0

$100

$200

$300

$400

Aver

age

Prof

it ($


(a) r Pricing Model


$0

$100

$200

$300

$400

Aver

age

Prof

it ($














234















degree of accuracy







235







236

Bibliography










237

BIBLIOGRAPHY















238

BIBLIOGRAPHY

















239

BIBLIOGRAPHY















240

BIBLIOGRAPHY















241

BIBLIOGRAPHY












gurobicom



242

BIBLIOGRAPHY















243

BIBLIOGRAPHY















244

BIBLIOGRAPHY















245

BIBLIOGRAPHY


















246

BIBLIOGRAPHY






577508











247

BIBLIOGRAPHY















248

BIBLIOGRAPHY














249

BIBLIOGRAPHY

















250

BIBLIOGRAPHY
















251

BIBLIOGRAPHY






252

Vita

Biography


Education







Experience




253

VITA










Papers












254

VITA




Honors amp Awards















Affiliations





Technical Skills


255

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures

Introduction

Brief Background

Research Questions



Power Flow

Optimal Power Flow

Unit Commitment



Introduction


Model


Conclusion


Introduction

Model Derivations



Conclusion


Introduction


Theoretical Results

Example

Test Cases

Conclusion

Conclusion

Discussion

Looking Forward







Vita

ACKNOWLEDGMENTS



















v

Dedicated to my dad

vi

Contents

Abstract ii

Acknowledgments iv

Dedication vi

List of Tables ix

List of Figures x





vii

CONTENTS





Vita 253

viii

List of Tables





ix

List of Figures







x

Chapter 1

Introduction













1























2



these markets




















3












4




11 Brief Background

















5






(LP) software

















6























7






















8



















9





















10




















11























12

















13

Chapter 2


Literature Review





Chapters 3-5







14







21 Power Flow










15


211 AC Power Flow













(21)








16








voltages


(sumj


)⋆

(23)




pi = visumj


qi = visumj







17



pfk = Gkτ2kiv


ptk = Gkτ2kjv



2kiv



2kjv







summisinMi


iv2i minus

sumkisinKfr

i


i

ptk = 0 (26a)

summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i

qtk = 0 (26b)








18
















i (ie no additional

constraints)




19







213 DC Power Flow












20









qfk = qtk = 0 (212)



summisinMi


i minussumkisinKfr

i

pfk +sumkisinKto

i

pfk = 0 (213)







21



pnw +A⊤pf = 0 (214a)


θref = 0 (214c)









derivation







22











Solving for θ

θ =(A⊤BA

)minus1pnw (216)







(A⊤BA

)F

⊤= minus (BA)⊤ (217)


23




more generallysum



W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (218)





= minusBAθ







1⊤pnw = 0 (219)

24



















25



Cm(pgm) = C0

m + C1mp

gm + C2

m(pgm)

2 (220)


1m and C2

m are the coecients







cm ge C0m + C1

mpgml + C2

m(pgml)

2 +(C1

m + 2C2mp

gml

) (pgm minus pgml








26









max zAC =summisinM

Cm(pgm) (222a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj




2kiv

2i minus τkivivj



27



2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T 2


)2+(qtk

)2le T 2






θref = 0 (222n)











28





















29



2i = Gs

i


















30






















31








(226a)








λ = λ1+ F⊤micro




32





βi

(pgi minus P i

)= 0 (227b)











software





33







23 Unit Commitment














34



231 Formulation





min z = c⊤x

st Ax ge b

x ge 0








min z = c⊤x+ d⊤y (228a)

35


st A0x ge b0 (228b)








i χi where












36





















(229)

37











0iλminus ci)⊤

xi minus d⊤i yi






(xiyi)isinχi






38















prime) = 0 so that




Llowast = supλge0


supλge0



39






rearranged like so


= inf(xy)isinχ


= sup

(xy)isinχ

(A⊤

0 λlowast minus c


= sup

(xy)isinχ

sumi


= sup(xy)isinχ

sumi

πi(λlowast xi yi)

minus λlowast⊤b0

=sumi

sup

(xiyi)isinχi

πi(λlowast xi yi)

minus λlowast⊤b0








40



Llowast = supλge0

inf(xy)isinχ


supλge0





supλge0
















41











convex relaxation











42






















43






















44



















45





















46





















47

















48



infeasible













min z = tr(Cw)


w ⪰ 0



49


(where tr(Cw) =sum





















50




⎡⎢⎢⎣wii wij

wij wjj

⎤⎥⎥⎦ ⪰ 0

















51






















52






















53


Discussion




















54






















55
















56






















57






















58




losses









Diniz 2011)










59








AC OPF















60




















grid

61






















62























63











Discussion











64




power and voltage

















65





















66















periods11








67





















68























69






















70











Lagrangian dual












71























72




















73


Discussion




















74







in their infancy


75

Chapter 3


DC OPF with Losses

31 Introduction






76




















77














λ = λE + λL + λC (31)







78






















79

CHAPTER3

IMPROVEDMETHODFORSO

LVING

THEDCOPFWITHLOSSE

S


ISO(Source)

Used inSCED


CAISO(2020)




ERCOT(2016)



Seasonal

ISO-NE(2019)




MISO(Sutton2014)




NYISO(2020ab)




PJM(2010)




SPP(2020)




80






















81















313 Contributions






82



















83


Notation








Sets


k isin K



i

Kfri Kto


to bus i


i j n isin N



Parameters


n-side of branch k


branch k



length




j

and 0 otherwise



84






y isin 0 1 2



injections





matrix



vector


oset


susceptance matrix



real power output



matrix




Variables


elements θi



θi minus θj


from direction with

elements pfk




with elements pgn



with elements pnwi



elements vi



85


Dual Variables



limits






the AC OPF



constraints


function constraint

Additional Notation

Hadamard product



xAC AC OPF solution







pfk = Gkτ2kiv


ptk = Gkτ2kjv




86





from direction






equations are

pnwi +sumkisinKfr

i


i





pnw +A⊤pf = 0 (34)




87


321 DC Power Flow













) (35)







88






W⊤

⎤⎥⎥⎦ F⊤=

⎡⎢⎢⎣(BA)⊤

0

⎤⎥⎥⎦ (36)










Mii =sumkisinKi

τkivξivξj


) foralli isin N

Mij = τkivξivξj




89



W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣∆θ

u

⎤⎥⎥⎦ =

⎡⎢⎢⎣∆pnw

0

⎤⎥⎥⎦ (37)









W⊤ 0

⎤⎥⎥⎦⎡⎢⎢⎣ Sθ

U⊤

⎤⎥⎥⎦ =

⎡⎢⎢⎣ I

0⊤

⎤⎥⎥⎦ (38)



(32a) and (32b)

pℓk = Gk

(τ 2kiv

2i + τ 2kjv


) (39)

90






L = LSθ (310)














91



Lki =dpℓkdpnwn


dpnwn

=2Rk

R2k +X2

k


(312)




2Rk

R2k +X2

k


=2Rk

(R2k +X2

k)12

τkivivjpfk times

dθijdpnwn

(313)




k)minus12 dθij

dpnwn


n Rearranging



Li = 2sumk






pℓ asympsumk

Rk(pfk)

2 (315)

92


33 Model







)le T (316d)

P le pg le P (316e)











93




⊤Pd

)minus micro⊤

(T+ FPd

)minus micro⊤

(Tminus FPd


(317a)



)⊤FD = 0 (317c)







λE = λ1 (318a)

λL = σL (318b)


)⊤F (318c)

λ = λE + λL + λC (318d)



94




















95


332 LMP Accuracy




















96















Avg Dispatch Di =1

N

sumi


LMP MAPE =1

N

sumi


i |λACi

times 100 (320)


zACtimes 100 (321)


97


0 50 100 150 200 250 300-15

-10

-5

0

5

10



0 10 20 30 40 50 60-150

-100

-50

0

50

100




98














compared to DC OPF







99




A 1 3000 2950 10B 1 3000 2975 100C 2 3000 3000 100












2




100


















101









pℓ =sumk

pℓk =sumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (322)




pℓk = E(2)k (pfk)

2 +(2E

(2)k E

(1)k

)pfk +

(E

(2)k (E

(1)k )2 + E

(0)k

)(323)



fk + E

(1)k )pfk + E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (324)



fk +E

(1)k )) that can be


(1)k )2 minus (pξ

fk)

2) + E(0)k )

102












(1)k )Fkn (325a)

L0k = E

(2)k ((E

(1)k )2 minus (pξ

fk)

2) + E(0)k (325b)

Ln =sumk

Lkn (325c)

L0 =sumk

L0k (325d)




dpℓkdpnwn



k and E(0)k can

103




ℓξ τki

1 pfξ larr F(pg



3 E(1)k larr Lkn

(2E

(2)k Fkn

)minus pξ


4 E(0)k larr L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)





8 pξℓ larr

sumk E

(2)k (pξijk + E

(1)k )2 + E

(0)k

9 Ln larr 2sum

k

(E

(2)k (pξijk + E

(1)k )Fkn

)foralln isin N






E(1)k =

Lkn

2E(2)k Fkn

minus pξfk (327b)

E(0)k = L0

k minus E(2)k ((E

(1)k )2 minus (pξ

fk)

2) (327c)




fk are available If


(1)k = E

(0)k = 0 and




104




k is



k lt ε


of the algorithm

pξgh+1n = ωpξ


pξfh+1k = ωpξ








pℓ gesumk

(E

(2)k (pfk + E

(1)k )2 + E

(0)k

) (329)





105
















in the algorithm





106


0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 2 4 6 8 10 12 14 16 18 2010-15

10-12

10-9

10-6

10-3

100

103



0 5 10 15 201e-09

1e-06

0001

1










107

















108












300-bus networks)




109



001

01

1

10

100

1000LMP

MAPE


-03

-02

-01

0

01

RelativeLoss

Error













110




35 Conclusion

















111










voltage levels

112

Chapter 4



of the AC OPF

41 Introduction





113













physical accuracy








114









less accurate














115
















412 Contribution





116























117







413 Outline





chapter

Notation






118




Sets


k isin K



i

Kfri Kto


to bus i




m isinM



i j n isin N

Parameters


i-side of branch k


branch k


of ones




j

and 0 otherwise


of branch k





sensitivities




sensitivities


I Identity matrix


sensitivities



loss sensitivity


loss oset


119




sensitivity


oset


demand



output







matrix

S0θ S




limit


Variables





branch line losses










withdrawals


loss



Additional Notation



value




ξ isin Ξ

120














max zAC =summisinM

Cm(pgm) (41a)

stsum

misinMi


iv2i minus

sumkisinKfr

i


i


summisinMi

qgm minusQdi +Bs

i v2i minus

sumkisinKfr

i


i


pfk = Gkτ2kiv

2i minus τkivivj



ptk = Gkτ2kjv

2j minus τkjvivj



121



2kiv

2i minus τkivivj




2kjv

2j + τkjvivj



(pfk

)2+(qfk

)2le T

2


)2+(qtk

)2le T

2





θref = 0 (41m)










122









2i minus αGkv

2i


) (42a)


2kiv

2i + α(Bk +Bs

kj)v2j


) (42b)




pfαk = Gk

(τ 2kiv

2i minus v2j



ki)τ2kiv

2i minus (Bk +Bs

kj)v2j





123







pℓk = Gk

(τ 2kiv

2i + v2j



2kiv

2i minus

(Bk +Bs

kj








4211 Power Balance






124



pnwi = P di +Gs


summisinMi


qnwi = Qdi minus Bs


summisinMi




tk = minusp

fαk + 1

2pℓk q

fk = qfαk + 1


fαk + 1




pnw +A⊤pfα +1

2|A|⊤pℓ = 0 (46a)

qnw +A⊤qfα +1

2|A|⊤qℓ = 0 (46b)





Appendix A1



(47)



ξ pℓξ

125




ξ minusHvξ


ξ minusKvξ

(48)



pfα = Fθ + F0 (49a)

qfα = Hv +H0 (49b)

pℓ = Lθ + L0 (49c)

qℓ = Kv +K0 (49d)









126














pnw +A⊤(Fθ + F0

)+

1

2|A|⊤

(Lθ + L0

)= 0 (410a)

qnw +A⊤ (Hv +H0)+

1

2|A|⊤

(Kv +K0

)= 0 (410b)


Sθ = minus(A⊤F+

1

2|A|⊤L

)minus1

(411a)

Sv = minus(A⊤H+

1

2|A|⊤K

)minus1

(411b)


127


S0θ = Sθ

(A⊤F0 +

1

2|A|⊤L0

)(411c)

S0v = Sv

(A⊤H0 +

1

2|A|⊤K0

)(411d)


θ

and v = Sθvq



F = FSθ H = HSv

L = LSθ K = KSv

(412)


F0 = FS0θ + F0 H0 = HS0

v +H0

L0 = LS0θ + L0 K0 = KS0

v +K0

(413)







pℓ = Lpnw + L0 (414c)

qℓ = Kqnw + K0 (414d)

128














balance constraint

1⊤pnw + 1⊤pℓ = 0 (415)







129




summation A1 = 0

















130




L = 1⊤L L0 = 1⊤L0

K = 1⊤K K0 = 1⊤K0

(416)



pℓ = Lpnw + L0 (417a)

qℓ = Kqnw + K0 (417b)


1⊤pnw + pℓ = 0 (418)






131



1

2pℓk

)2

+

(qfαk +

1

2qℓk

)2

le T2

k if(pfξ k

)2+(qfξ k

)2gt(ptξk

)2+(qtξk

)2(pfαk minus

1

2pℓk

)2

+

(qfαk minus

1

2qℓk

)2

le T2






2pfξ kl

(pfαk +

1

2pℓk

)+ 2qfξ kl(q

fαk +

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2pℓk

)+ 2qfξ kl(q

fαk minus

1

2qℓk)

le T2

k +(pfξ kl

)2+(qfξ kl


(420)




Ldk = pℓξk

sumκisinK


sumκisinK

qℓξκ (421)


132



2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk +

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl

)2 if

(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)+ 2qfξ kl

(qfαk minus

1

2Kd

kqℓ

)le T

2

k +(pfξ kl

)2+(qfξ kl


(422)





2pfξ kl

(pfαk +

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl

)2

if(pfξ kl

)2+(qfξ kl

)2gt(ptξkl

)2+(qtξkl

)2 or

2pfξ kl

(pfαk minus

1

2Ldkp

ℓ

)le T

2

k +(pfξ kl

)2minus(qfξ kl


(423)



is if η = Tk

(pξfkl)

2+(qξfkl)


fkl =

radicηpξ

fkl and

qξfkl =

radicηqξ

fkl

133


4216 Voltage Limits



reformulated




v = Svqnw + S0

v (424a)

V le v le V (424b)








cm ge C0m + C1

mPgml + C2

m(Pgml)

2 +(C1

m + 2C2mP

gml

) (pgm minus P g

ml


134





points




min z =summisinM

cm



min z =summisinM

cm



min z =summisinM

cm


(422) (424) (425) (41k) (41l)

135



min z =summisinM

cm
















136



















v







137





















138









(A⊤F+

1

2|A|⊤L

)⊤


[kisinKprime] (426)














139




LKprime


[kisinKprime]

KKprime


[kisinKprime]

(428)




prime= 0 (429a)


(429b)


(429c)










140








F 0εk = F 0

k +sumiisinN




Fik




C-LOPF models







141














implementations







142





















143




Cases case14_ieee

case30_ieee

case57_ieee

case118_ieee

case300_ieee

case588_sdet

case2316_sdet

case2853_sdet

case4661_sdet

case89_pegase

case1354_pegase

case2869_pegase

case9241_pegase

case13659_pegase

case200_tamu

case500_tamu

case2000_tamu

case10000_tamu


Cases case2383wp_k

case2736sp_k

case2737sop_k

case2746wop_k

case2746wp_k

case3012wp_k

case3120sp_k

case3375wp_k

case1888_rte

case1951_rte

case2848_rte

case2868_rte

case6468_rte

case6470_rte

case6495_rte

case6515_rte

case3_lmbdf

case5_pjmg

case24_ieee_rtsh

case30_asi

case30_fsri

case39_eprij

case73_ieee_rtsk

case162_ieee_dtcl

case179_gocm

case240_psercn



144






and case6515_rte

442 LMP Error














145












146






















147














148





149






















used by ISOs

150






















151





152











153
















154

















155










156








with nominal demand














157





158





∥x∥p =

(sumk

|xi|p)(1p)








446 Solution Times







159





160






161

















162









DC-LOPF models








RTE test cases




163




45 Conclusion

















164












165

Chapter 5




51 Introduction





166























167





















markets


168























169











Notation







Sets



h isin H


170


k isin K






s isin S


χ = cupiχi

Parameters





coecient


coecients


coecients


coecients

Variables





schedule s





and solution s




variables


Functions


RK rarr R



RK rarr R







171



cdf Rrarr [0 1]




Additional Notation



value



zs Solution s


relaxation

zlb MIP lower bound













172























173













2008a)










174






521 Models









st A0x ge b0 (51b)





175





















176

















commitment status






177























178
















179




to as linear prots

πsi (λ) = (A⊤

0iλminus ci)⊤xs





U si (λ) = sup

(xiyi)isinχi

(A⊤0iλminus ci)













180








i (λ) (55)


i (λ) The actual


πsi (λ) = πs












181






















182











sumi

δmsi + δms

cs + δsopt = 0 (57)

where

δmsi = πs


i (λmlowast) (58a)



(microlowasti (λ


mlowast)) (58b)






183



∆ms =δmscs

+sumi

δmsi

(59)






c⊤x+ d⊤y + λ⊤(b0 minus A0x) (510)



sumi





s minus b0)

(511a)

= sup(xy)isinχ








184







i (λ) and



sumi U


lowast +sumi U

lowasti (λ) Then

sumi |δms



πsi (λ) + U s


(A⊤0iλminus ci)



i |δmsi | = 0


δmscs

= δsopt (513)



i (λ) and






sumi micro


lowast +sum



185






δmsi = |πs







i (λ)|(514)


by uplift and PRS






sumi




δmscs =

λ⊤A0(x



i (λ))minussumi


i (λ))

le

λ⊤A0(x



i (λ))

+sum

i

microsi (λ)minus

sumi

microlowasti (λ)

(517)



0 lesumi

microsi (λ) le

sumi


s minus b0) (518a)

186


0 lesumi


sumi





i

microsi (λ)minus

sumi

microlowasti (λ)














187





















188






















189



















bound




190












54 Example






degeneracy



191






min z =sum

(hi)isinG

Chixhi (521a)

stsum

(hi)isinG

xhi = 225 + ϵ (521b)













192




















to self-commit



193




55 Test Cases

















194






195





















196


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005


3 6 9 12 15 18 21 24Hour

20

30

40

50

0

001

002

003

004

005



payments (MWP=sum












197


r pd td ld0

200

400

600

(a) RTS cases

r pd td ld0

50

100

(b) PJM cases















198




















199



∆m(τ) = (1S)sums



















200


00001 0001 001 01 1 10 100 1000 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 100 0

02

04

06

08

1

(b) PJM cases


201








S

sums π

si (λ

ms) and (σmi )

2 = 1Sminus1

sums(π

si (λ

ms) minus πmi )

2 be





Σm(τ) =sumi

1Σmi leτ

sumi

1πmi gt0 (524)








optimality

202


00001 0001 001 01 1 10 0

02

04

06

08

1

(a) RTS cases

00001 0001 001 01 1 10 0

02

04

06

08

1

(b) PJM cases


203


56 Conclusion



















204


















market designs

205

Chapter 6

Conclusion













206






















207


61 Discussion



















208























209























210








certain







new eciencies

62 Looking Forward





211























212























213


















214

Appendix A









= 0 is given

below

pfαk asymp Gk

((τkivξi)

2 minus vξ2j



(A1)

215





F 0k = Gk

((τkivξi)

2 minus vξ2j



(A2c)


partθ= 0 is

given below


ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j


+(minus(Bk +Bs


)(vi minus vξi)

+((Bk +Bs


(A3)





H0k =

((Bk +Bs

ki)τ2kivξ

2i minus (Bk +Bs

kj)vξ2j



= 0 is given below

pℓk asymp Gk

(τ 2kivξ

2i + vξ

2j



(A5)

216





L0k = Gk

((τkivξi)

2 + vξ2j



(A6c)


= 0 is given below


2kivξ

2i minus

(Bk +Bs

kj

)vξ


+(minus2 (Bk +Bs


)(vi minus vξi)

+(minus2(Bk +Bs

kj


)(vj minus vξj)

(A7)




Kkj =minus 2(Bk +Bs

kj


K0k = (Bk +Bs

ki) τ2kivξ

2i +

(Bk +Bs

kj

)vξ




217






minus(A⊤F+

1

2|A|⊤L

)⊤

F⊤= F⊤ (A9a)

minus(A⊤H+

1

2|A|⊤K

)⊤

H⊤= H⊤ (A9b)

minus(A⊤F+

1

2|A|⊤L

)⊤

L⊤= L⊤ (A9c)

minus(A⊤H+

1

2|A|⊤K

)⊤

K⊤= K⊤ (A9d)


minus(A⊤H+

1

2|A|⊤K

)⊤

Sv⊤= I⊤ (A9e)



F0 = F

(A⊤F+

1

2|A|⊤L

)+ F0 (A10a)

H0 = H

(A⊤H+

1

2|A|⊤K

)+H0 (A10b)


218


L0 = L

(A⊤F+

1

2|A|⊤L

)+ L0 (A10c)

K0 = K

(A⊤H+

1

2|A|⊤K

)+K0 (A10d)




= F(Sθp

nw + S0θ

)+ F0 (A11b)

= minusF(A⊤F+

1

2|A|⊤L


1

2|A|⊤L0

)+ F0 (A11c)


(A⊤F+

1

2|A|⊤L


2|A|⊤L0








219



L =partpnwRpartpnwi


K =partqnwRpartqnwi


(A12)



(A⊤F+

1

2|A|⊤L

)⊤

L⊤=

(A⊤F+

1

2|A|⊤L

)⊤

1 (A13a)(A⊤H+

1

2|A|⊤K

)⊤

K⊤=

(A⊤H+

1

2|A|⊤K

)⊤

1 (A13b)


220

Appendix B













221






















222




















223
















individual prot and


224







Baseload Outputsum














MWh




225










less than 23











226


















227




r 060 040 $2790 $1907MWhtd 100 000 $2750 $1600MWh

Di () 144 2714














228



a Greedy Algorithm


















229








min z =sum

(gk)isinG

Cgkxgk (B1a)

stsum

(gk)isinG

xgk = 225 + ϵ (B1b)












230




smoothing method




i is

















231


0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100O

ldTe

ch S

elf-C

omm

itmen

ts(1

0-ite

ratio

n m

ovin

g av

erag

e)


(a) r Pricing Model

0 200 400 600 800 1000Iteration

0

10

20

30

40

50

60

70

80

90

100

Old

Tech

Sel

f-Com

mitm

ents

(10-

itera

tion

mov

ing

aver

age)




232























233



$0

$100

$200

$300

$400

Aver

age

Prof

it ($


(a) r Pricing Model


$0

$100

$200

$300

$400

Aver

age

Prof

it ($














234















degree of accuracy







235







236

Bibliography










237

BIBLIOGRAPHY















238

BIBLIOGRAPHY

















239

BIBLIOGRAPHY















240

BIBLIOGRAPHY















241

BIBLIOGRAPHY












gurobicom



242

BIBLIOGRAPHY















243

BIBLIOGRAPHY















244

BIBLIOGRAPHY















245

BIBLIOGRAPHY


















246

BIBLIOGRAPHY






577508











247

BIBLIOGRAPHY















248

BIBLIOGRAPHY














249

BIBLIOGRAPHY

















250

BIBLIOGRAPHY
















251

BIBLIOGRAPHY






252

Vita

Biography


Education







Experience




253

VITA










Papers












254

VITA




Honors amp Awards















Affiliations





Technical Skills


255

Abstract

Acknowledgments

Dedication

List of Tables

List of Figures

Introduction

Brief Background

Research Questions



Power Flow

Optimal Power Flow

Unit Commitment



Introduction


Model


Conclusion


Introduction

Model Derivations



Conclusion


Introduction


Theoretical Results

Example

Test Cases

Conclusion

Conclusion

Discussion

Looking Forward







Vita