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DISTRIBUTED ENERGY RESOURCE NETWORKS:

PLANNING, CONTROL AND MARKET DESIGN

A DISSERTATION

SUBMITTED TO THE INSTITUTE FOR

COMPUTATIONAL AND MATHEMATICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Junjie Qin

December 2017

This dissertation is online at: http://purl.stanford.edu/kp454xc7379

© 2017 by Junjie Qin. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

ii

http://purl.stanford.edu/kp454xc7379

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Ram Rajagopal, Primary Adviser


Abbas El Gamal


Ramesh Johari

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Distributed energy resources (DERs), such as energy storage and distributed generation,

are rapidly replacing centralized power generation due to their environmental, economic

and resiliency benefits. Integrating DERs into power network presents new challenges to

power system planning, control and market design.

In the first part of the thesis, we analyze a simple greedy strategy for the planning

problem of placing energy storage in a power network. Using structural characterizations

of the underlying power network control problem, we identify conditions under which the

placement value function is submodular so that the greedy strategy has a performance

guarantee. We then develop a computational procedure to certify these conditions for any

given problem instance based on multi-parametric programming.

In the second part of the thesis, we consider the stochastic control problem for operating

energy storage devices connected in a power network. As the exact solution of the problem

based on dynamic programming suffers from the curse of dimensionality, we propose a

simple online algorithm for the problem utilizing a stabilized greedy (myopic) controller.

For a rather general setting, we establish performance guarantees for the proposed method.

Finally, we study fundamental requirements for power network reliability in designing

novel power markets to integrate DERs. We demonstrate a transaction or trading based

market, on top of a system operator implementing these reliability requirements, could

achieve the same efficiency as centralized stochastic dispatch. We also obtain structural

results for radial networks which indicate efficient market outcomes can be reached with

bilateral trading for distribution networks.

iv

Acknowledgments

First and foremost, I am greatly indebted to my thesis advisor Professor Ram Rajagopal who

led me into this exciting area of power systems. He taught me how to conduct practically

relevant research, from reading the right paper/report and talking to the right person,

to formulating the right problem and presenting to the right audience. I am particularly

grateful to the freedom Ram has given me in taking courses and collaborating with faculty

and students in and out of Stanford campus. Together with Ram’s guidance, this unique

experience has defined the interdisciplinary pathway of my research, fusing control theory

and economics with practical power system problems. I could hardly imagine that I can go

this far on this path without the encouragement, support and intellectual stimulation from

Ram.

I am grateful to my thesis committee members, Professor Abbas El Gamal and Professor

Ramesh Johari. I enjoyed every smart grid algorithm meeting with Abbas and admire his

ability of resolving a difficult problem by asking simple questions. Ramesh’s suggestions on

modeling have been invaluable to me and will serve as the guide for my future research. I

would also like to thank my defense committee members, Professor Yinyu Ye and Professor

Walter Murray. Yinyu’s advice on looking at the simplest setting first is the major reason

that I could obtain the results in Chapter 2 of this thesis. Walter’s course has shaped my

understanding of numerical issues in solving large scale optimization problems, which has

been extremely helpful when implementing algorithms proposed in this thesis.

I have also benefited significantly from my collaborators during the past six years. In

particular, I would like to express my sincere appreciation to Professor Pravin Varaiya, Pro-

fessor Kameshwar Poolla, Professor Adam Wierman, Professor H. Vincent Poor, Professor

Andrea Goldsmith, Professor Amin Saberi, Professor Rahul Jain, Professor Baosen Zhang,

Professor Yue Zhao, Professor Rishee Jain, Proessor Insoon Yang, Professor Wenyuan Tang,

Han-I Su, Yinlam Chow, Jiyan Yang, Shai Vardi, Vahid Liaghat, Raffi Sevlian, Anthony

Kim, Jiafan Yu and Jonathan Mather. Working with and learning from them have enriched

not only my research but also my professional and personal life.

My sincere thanks also go out to my friends and colleagues in the Stanford Sustainable

Systems Lab. In no particular order, they are Chin-Woo Tan, Yang Yu, Xiao Chen, Yizheng

Liao, Gustavo Cezar, Matt Kiener, Sid Patel, Aaron Goldin, Camille Pache and Jianxiao

v

Wang.

Finally, I would like to thank my parents and my wife, for their support and unconditional

love, and for making my life as exciting as my work.

vi

Contents

Abstract iv

Acknowledgments v

1 Introduction 1

1.1 The Rise of Distributed Energy Resources . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Drivers and Trends for DERs . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Challenges in DER Integration . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Chapter 2: Planning of Energy Storage . . . . . . . . . . . . . . . . . 5

1.2.2 Chapter 3: Stochastic Control of Distributed Energy Storage . . . . . 6

1.2.3 Chapter 4: Flexible Market for Smart Grid . . . . . . . . . . . . . . . 6

2 Planning of Energy Storage 8

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Proposed Work and Its Contributions . . . . . . . . . . . . . . . . . . 10

2.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Power Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Energy Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.4 Economic Dispatch with Optimal Storage Control . . . . . . . . . . . 13

2.2.5 Storage Placement as Combinatorial Optimization . . . . . . . . . . . 14

2.2.6 Outline of Proposed Analyses . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Structures of Optimal Cost and Prices . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Dual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 Optimal Prices and Second Order Sensitivity . . . . . . . . . . . . . . 20

2.4 Submodularity of Placement Value Function . . . . . . . . . . . . . . . . . . . 22

2.4.1 Two-Bus Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

vii

2.4.2 Verification of Submodularity Using a Polyhedral Characterization of

Critical Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.3 Modified Greedy Algorithms . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.4 Risk-Aware Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.1 IEEE 14 Bus Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.2 Other Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Stochastic Control of Distributed Energy Storage 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Centralized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.3 Cluster based Distributed Control . . . . . . . . . . . . . . . . . . . . 40

3.3 Online Modified Greedy Algorithm for Networked Storage Control . . . . . . 41

3.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.2 Performance Guarantees . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Distributed Online Control Via Alternating Direction Method of Multipliers . 46

3.4.1 Node-Edge Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.2 Cluster-based Implementation . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5.1 Star Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5.2 IEEE 14 Bus Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.3 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 Conclusion and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Flexible Markets for Smart Grid 54

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.1 Contributions and Organization . . . . . . . . . . . . . . . . . . . . . . 56

4.1.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.2 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.3 Uncertainty Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.4 Participant Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.5 Efficiency Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Trading Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 Economic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Price Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

viii

4.6 Arrow-Debreu Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.8 Concluding remarks and open questions . . . . . . . . . . . . . . . . . . . . . 73

5 Conclusions 75

A Appendices of Chapter 2 76

A.1 Expression of the Shift Factor Matrix . . . . . . . . . . . . . . . . . . . . . . 77

A.2 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.3 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.4 Proof of Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.5 Proof of Theorem 2.1 and Corollary 2.1 . . . . . . . . . . . . . . . . . . . . . 79

A.6 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


A.8 Proof of Lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


B Appendices of Chapter 3 84

B.1 Definitions and Expressions for Section 3.3 . . . . . . . . . . . . . . . . . . . . 85

B.2 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

B.3 Derivation of the ADMM Algorithm . . . . . . . . . . . . . . . . . . . . . . . 91

C Appendices of Chapter 4 93

C.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.2 Proof of Lemma 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.3 Proof of Lemma 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C.4 Bilateral Trading in Tree Network . . . . . . . . . . . . . . . . . . . . . . . . 98

C.5 Trade Verification and Curtailment with Local Scenarios . . . . . . . . . . . . 101

Bibliography 103

ix

List of Tables

4.1 Power injection (unit: MW) of the initial trade proposed by the particpants. 72

4.2 Power injection (unit: MW) of the curtailed trade. . . . . . . . . . . . . . . . 73

4.3 Power injection (unit: MW) of the accumulated trade γp+∆p. . . . . . . . . 73

x

List of Figures

1.1 Generation types in U.S. annual capacity additions . . . . . . . . . . . . . . . 3

1.2 Installed price trends of distributed solar generation in the U.S. . . . . . . . . 4

1.3 Price trends of Lithium-Ion battery (cell and pack only) . . . . . . . . . . . . 4

2.1 Venn diagram for the set of all possible storage placement problem instances. 12

2.2 Critical regions for the two-bus examples. In the figure with a slight abuse of

notation, we use RAi and RB

i to denote the ith critical region for each case. . 25

2.3 Optimal flow for the case with negative Hessian entries. . . . . . . . . . . . . 26

2.4 Box plots of price and load data: (a) locational marginal price, and (b) load. 33

2.5 Average run-time comparison between the greedy algorithm and the MIQP-

based method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1 Percentage cost savings of a storage network operated for balancing. . . . . . 50

3.2 Bar plots for scaled hourly total load (upper panel) and wind data (lower

panel) used for the simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Convergence of ADMM and centralized subgradient method. Here ζk is the

step size of the subgradient algorithm at the kth iteration. . . . . . . . . . . . 52

3.4 Running time of distributed and centralized ADMM . . . . . . . . . . . . . . 53

4.1 Conceptual diagram for the trading process. . . . . . . . . . . . . . . . . . . . 61

4.2 Network diagram for the two-bus example. . . . . . . . . . . . . . . . . . . . . 72

xi

Chapter 1

Introduction

1

CHAPTER 1. INTRODUCTION 2

1.1 The Rise of Distributed Energy Resources

The architecture of today’s electric power grid is largely based on a top-down design: elec-

tricity flows from centralized generation through high voltage transmission system and then

medium and low voltage distribution networks to end users. This architecture is about

100 years old and goes back to the times of Tesla and Edison [133]. In the 21st century,

this paradigm is challenged due to a number of environmental, economic and policy trends,

which lead to the rise of distributed energy resources (DERs). Distributed energy resources,

in contrast to centralized energy resources, are energy resources that are scattered around

the grid. Common examples of DERs include distributed generation such as roof-top solar

generation, distributed energy storage, and demand management programs or technologies

such as those implemented by demand response aggregators [87]. In the rest of this section,

we examine the drivers for DERs and then briefly discuss challenges brought about by deep

penetration of DERs .

1.1.1 Drivers and Trends for DERs

The widening environmental concerns on greenhouse gas emissions, decreasing costs of

DERs, and favorable policies implemented in many countries and regions across the world

are major drivers for DERs.

Environmental Drivers

Centralized generation in the United States and many other countries is mainly based on

fossil fuels such as coal and nature gas. Such generation sources are often economical and

reliable, but produce greenhouse gases as byproducts of electricity. As combating global

warming becomes a wide-spread interest, alternative generation sources that have smaller

carbon footprints are receiving increasing amount of attention. In particular, renewable

generation sources such as wind and solar are considered as the primary candidates for

decarbonizing the electric grid [51]. In fact, solar power and wind power presented the

largest source and the third-largest source of U.S. electric-generating capacity additions in

2016, constituting 38% and 27% of all U.S. capacity additions, respectively [130]. Figure 1.1

depicts the relative contribution of generation types in annual capacity additions from year

1998 to year 2016 [130]. A large portion of the installed renewable generation capacities is

in the form of distributed generation.

These renewable generation sources are intermittent and uncontrollable, and have limited

availability during certain times of the day. Integrating these variable generation sources

presents a major challenge to the electric power grids, which typically are not designed to

handle variable generation sources. Issues arise from the need to align supply and demand

in order to avoid curtailing energy generated from renewable sources in a way that maintains

the reliability of the power grid in every time instance [127]. DERs like energy storage and


1998 2000 2002 2004 2006 2008 2010 2012 2014 2016Year

0

10

20

30

40

50

60

70

80

Ann

ual c

apac

ity a

dditi

ons

(GW

)Other non-renewableCoalGasOther renewableSolarWind

Figure 1.1: Generation types in U.S. annual capacity additions

demand management tools are well-suited to smoothing the variable generation from wind

and solar, so that energy supply can be shifted to be better aligned with demand.

Economic Drivers

Mass adoption of distributed generation and storage is not possible without the declining

costs for these technologies. Figure 1.2 depicts the cost trends of distributed solar generation

for various types of solar installation projects in the U.S. [8]. It is demonstrated that the

median installed prices fell by about $0.5/W per year on average for the last 16 years. This

represents an average annual percentage reduction of 7% per year for residential and small

non-residential systems (i.e., systems with DC capacity smaller than 500 kW), and 11% for

large non-residential systems (i.e., systems with DC capacity larger than 500 kW). Even

faster cost reduction is observed for utility scale solar systems (i.e., systems with capacity

larger than 5 MW) [16].

Energy storage costs are also declining in the recent years. The cost of Lithium-Ion

battery has declined by more than 60% since 2012. Figure 1.3 shows the price trends

of Lithium-Ion battery where only the prices for battery cells and packs are considered

[31]. The installed prices for Lithium-Ion battery systems tend to be higher due to other

hardware costs (e.g., inverter costs) and soft costs (e.g., installation costs). Comparing the

battery only prices reported in Figure 1.3 and California’s 2017 Self-Generation Incentive

Program (SGIP) statewide data suggests that the installed prices for Lithium-Ion battery

based storage system can be $650/kWh higher [1]. Nevertheless, given the steadily reducing

inverter costs and other soft costs, the overall costs for Lithium-Ion battery based energy


2000 2002 2004 2006 2008 2010 2012 2014 2016Year

0

2

4

6

8

10

12

14

Med

ian

inst

alle

d pr

ice

($/W

) ResidentialSmall non-residentialLarge non-residential

Figure 1.2: Installed price trends of distributed solar generation in the U.S.

storage system has been on a rapid reduction trend. Similar cost reductions trends have

also been observed or anticipated for other storage technologies [37, 107].

2010 2011 2012 2013 2014 2015 2016Year

0

200

400

600

800

1000

Lith

ium

-Ion

bat

tery

pric

e ($

/kW

h)

Figure 1.3: Price trends of Lithium-Ion battery (cell and pack only)

Policy Drivers

Given the environmental concerns, governments around the world have put forward plans

and/or legislation to reduce carbon emissions. The international Energy Agency estimates

that $13.5 trillion in additional investment will be needed to achieve the carbon reduction

goal set by the 2015 Paris agreement. It is expected that a part of such investment will

happen through the renewable energy industry, possibly stimulated by government mandates

or incentive programs.

For distributed solar, financial incentives provided through utility, state, and federal pro-

grams have been a driving force for photovoltaic (PV) markets in the United States. These

incentives typically include some combinations of cash incentives (rebate or performance-

based incentives) provided by state/utility’s PV program, federal/state investment tax cred-

its, and retail net metering programs. At their peak, these incentive programs have resulted

in savings for PV owners of $4-8/W [8]. These incentives are in the process of being phased


out in markets such as Arizona and California. Such reduction in cash incentives has to

some extent offset the reduction in the installed prices for distributed solar generation.

For energy storage, several states in the U.S. have established mandates or incentive

programs [1]. For instance, through legislation AB 2514, California requires the state’s three

investor owned utilities to procure 1.3 GW of energy storage by 2020. In 2016, Massachusetts

passed a law (H.4568) to direct the Massachusetts Department of Energy Resources (DOER)

to assess whether to set appropriate targets for energy storage procurements. As a result,

DOER has set an aspirational target of deploying 200 MWh of storage by 2020. Examples of

storage incentive programs include California’s SGIP program through 2020 and Maryland’s

30% tax credit for energy storage systems.

1.1.2 Challenges in DER Integration

The aforementioned drivers for DERs have resulted in a significant increase in DER pene-

trations in the electric power grid. This has led to many challenges in the planning, control

and market operation of the power grid as DERs differ from traditional centralized fossil

fuel based generation in two fundamental aspects. The first is that as DERs are scattered

around the power network, the interaction between the DERs and the power network be-

comes an important element to consider. The second is that as many distributed generation

sources are uncontrollable and intermittent renewable generation sources, it is important

to model them as stochastic resources. Incorporating these two aspects calls for a major

rethinking in the way that many power system studies are conducted, as a majority of the

prior studies ignore one or all of these aspects. In this thesis, we study three problems and

their solutions when the power network and uncertainty from DERs are both considered.

1.2 Outline and Contributions

1.2.1 Chapter 2: Planning of Energy Storage

Power system planning studies are the process of determining the capacities and locations to

build different types of energy resources, in order to fulfill the future load for the planning

horizon (ranging from a few years to 10 years). In Chapter 2, we consider the planning

problem of placing energy storage in power networks when all storage devices are optimally

controlled to minimize system-wide operation costs. We propose a discrete optimization

framework to accurately model heterogeneous storage capital and installation costs as these

fixed costs account for the largest cost component in most grid-scale storage projects. Iden-

tifying an optimal placement strategy is challenging due to (i) the combinatorial nature of

such placement problems, and (ii) the spatial and temporal transfer of energy via transmis-

sion lines and distributed storage devices. To develop a scalable near-optimal placement

strategy with a performance guarantee, we characterize a tight condition under which the


placement value function is submodular by exploiting our duality-based analytical charac-

terization of the optimal cost and prices. The proposed polyhedral analysis of a parametric

economic dispatch problem with optimal storage control also suggests a simple but rigorous

verification method for submodularity, and a novel insight that the spatio-temporal conges-

tion pattern of a power network is critical to submodularity. A modified greedy algorithm

provides a (1 − 1/e)-optimal placement solution and can be extended to obtain risk-aware

placement strategies when submodularity is verified. The results of this chapter have been

reported in [103]. An empirical study based on an extension of the algorithm analyzed in

this chapter to settings with both storage and distributed solar generation is reported in

[52].

1.2.2 Chapter 3: Stochastic Control of Distributed Energy Storage

After the DERs are built, a natural question is to determine how to operate them optimally.

Chapter 3 studies the problem of optimal control of energy storage in power networks

in stochastic environments, which is an important open problem. The key challenge is

that, even in small networks, the corresponding constrained stochastic control problems

on continuous spaces suffer from curses of dimensionality, and are intractable in general

settings. For large networks, no efficient algorithm is known to give optimal or provably near-

optimal performance for this problem. This chapter provides an efficient algorithm to solve

this problem with performance guarantees. We study the operation of storage networks,

i.e., a storage system interconnected via a power network. An online algorithm, termed

Online Modified Greedy algorithm, is developed for the corresponding constrained stochastic

control problem. A sub-optimality bound for the algorithm is derived, and a semidefinite

program is constructed to minimize the bound. In many cases, the bound approaches zero

so that the algorithm is near-optimal. A task-based distributed implementation of the

online algorithm relying only on local information and neighborhood communication is then

developed based on the alternating direction method of multipliers. Numerical examples

verify the established theoretical performance bounds, and demonstrate the scalability of

the algorithm. The results of this chapter have been reported in [96].

1.2.3 Chapter 4: Flexible Market for Smart Grid

As many DERs are owned and operated by individual users, we may need a market to

coordinate the benefits and needs of these users while ensuring the reliability of the power

network. In Chapter 4, a coordinated trading process is proposed as a design for an electric-

ity market with significant uncertainty, perhaps from renewables. In this process, groups of

agents propose to the system operator (SO) a contingent buy and sell trade that is balanced,

i.e.the sum of demand bids and the sum of supply bids are equal. The SO accepts the pro-

posed trade if no network constraint is violated or curtails it until no violation occurs. Each


proposed trade is accepted or curtailed as it is presented. The SO also provides guidance

to help future proposed trades meet network constraints. The SO does not set prices, and

there is no requirement that different trades occur simultaneously or clear at uniform prices.

Indeed, there is no price-setting mechanism. However, if participants exploit opportunities

for gain, the trading process will lead to an efficient allocation of energy and to the discovery

of locational marginal prices (LMPs). The great flexibility in the proposed trading process

and the low communication and control burden on the SO may make the process suitable

for coordinating producers and consumers in the distribution system. The results of this

section have been reported in [99].

Chapter 2

Planning of Energy Storage

8

CHAPTER 2. PLANNING OF ENERGY STORAGE 9

2.1 Introduction

Energy storage devices, ranging from batteries to hydropower plants, are considered to play

a key role in improving the reliability, efficiency and resilience of power systems. A strong

growth of energy storage installation has occurred around the world in recent years. For

example, the total storage deployment in the United States increased by 243% in power

capacity and 188% in energy capacity during 2014-2015 [42]. This was driven, in part,

by an increasing need for energy storage in modern power systems to compensate for the

variability of wind and solar energy sources. The value of storage in the power grid under

a large penetration of renewable energy sources has been quantified in several studies (e.g.,

[26, 92, 102]). It has also been claimed that energy storage can be used to shift load and

support frequency regulation to enhance system efficiency and reliability [54, 80]. Another

primary driving force has been the rapidly decreasing cost of storage devices, especially

batteries, as a consequence of growing public and commercial interest in electric vehicles [88].

The bulk of newly deployed storage devices has been front-of-meter deployment. In

2015, 85% of storage deployment in the United States was front-of-meter utility- or grid-

scale storage [79]. The value of such grid-scale storage depends critically on the location at

which it is installed due to the geographical heterogeneity of generation and load profiles and

the possibility of network congestion [19, 23]. Therefore, there is a strong need for efficient

strategies to place storage devices in power networks.

2.1.1 Related Work

A majority of prior studies have considered the energy storage placement problem as the

problem of sizing storage. This line of research has led to continuous optimization formula-

tions. For example, Thrampoulidis et al. [119] studied the allocation of a fixed total storage

capacity over a network to minimize the generation cost. By optimizing the capacity of each

storage device together with the decision variables in economic dispatch, they obtained a

structural characterization of the optimal allocation. This characterization eliminated the

need to place storage at certain generation-only buses. Pandzic et al. [91] and Wogrin and

Gayme [131] emphasized the multi-level nature of the placement problem. Their analyses

also provided useful insights on the effect of congestion, wind penetration and storage ser-

vice types. Sjodin et al. [110] employed chance constraints to limit the system operation

risk generated by variable renewable energy sources and jointly optimized generator dis-

patch and storage control and sizing. Kraining et al. [61] extended their convex model

predictive control based storage operation optimization to address the problem of allocat-

ing storage capacities over the network. Qin and Rajagopal [102] derived a constrained

linear-quadratic-Gaussian controller for distributed storage devices under uncertainty and

formulated a storage-sizing problem as a convex program. These studies all used linearal-

ized DC approximation of AC power flow in recognition of the complexity of the AC power


flow model. Castillo and Gayme [24] studied the storage allocation problem with line losses

considered. This led to a non-convex quadratic constrained quadratic program for which

exact convex relaxations based on semidefinite programs and second order cone programs

were developed. Bose et al. [20] developed a semidefinite relaxation approach to the

storage placement problem using the AC power flow model and demonstrated its effective-

ness through numerical simulations. With an AC power flow model, Castillo and Gayme

[25] considered the setup where storage is operated to maximize profit based on locational

marginal prices (LMPs) in the power network. Structural results between the storage deci-

sions and the LMPs were derived. Tang and Low [117] focused on distribution networks

by employing a branch flow model and derived the monotonicity properties of the optimal

placement solution under the assumption that all load profiles have the same shape.

Another line of research treats the energy storage placement problem as a form of fa-

cility location problem (cf. [109] and references therein). An example is Qi et al. [94],

which considers a planning problem for energy storage and transmission in the presence of

wind energy generation. Utilizing a simplified model for power flow, the authors formu-

late a mixed-integer second order conic program for uncapacitated storage and propose an

approximation scheme for capacitated storage.

Outside of energy storage placement, the concept of submodularity and optimization

techniques exploiting submodularity have been used in a number of power system applica-

tions. See [65], [67] and references therein.

2.1.2 Proposed Work and Its Contributions

Departing from the aforementioned continuous optimization approaches, we propose a dis-

crete optimization formulation for energy storage placement when all of the storage devices

are optimally controlled to minimize the total system-wide cost. This formulation is moti-

vated by the cost structure of storage deployment. The operating and maintenance costs

of storage are usually negligible compared to the fixed costs, which include installation and

capital costs. Depending on the storage technology used, the installation costs can be as

high as the capital costs. Therefore, the cost of deploying ten units of 1 MWh battery could

be dramatically different from the cost of deploying one unit of 10 MWh battery due to

differences in installation costs. Furthermore, additional fixed cost components, such as site

acquisition costs, could be sensitive to the installation location. Due to the discrete nature

of these heterogeneous cost factors, it is difficult to take into account all of them using a con-

tinuous optimization framework. However, discrete optimization with a budget constraint

limiting the total fixed cost offers a natural and accurate model of considering these cost

factors. Additionally, a discrete optimization framework is useful when handling practical

scenarios in which fixed-capacity storage devices are to be placed. These advantages are

elaborated in Section 2.2.5.


We formulate the placement problem as a maximization of the placement value func-

tion, a set function that represents the value of a storage placement decision, subject to a

knapsack constraint that models the budget constraint on the aforementioned fixed costs.

Unfortunately, this class of problems is NP-hard in general. To overcome this challenge,

we identify rich structures of the placement value function. In particular, we characterize

conditions under which the placement value function is submodular, suggesting that the

marginal benefit of adding a storage device decreases as more devices are installed. This

submodular structure allows us to employ a greedy algorithm that provides a near-optimal

solution with a provable suboptimality bound [55,85]. The submodularity of energy storage

placement is not unexpectable but characterizing conditions the under which it holds has

been recognized, e.g., in [35], as an unanswered question.

We summarize the contributions and main results of the proposed work as follows. First,

we provide a novel discrete optimization approach to energy storage placement that allows an

accurate modeling of fixed costs for storage deployment. Second, by analyzing an associated

multi-period economic dispatch problem with optimal storage control and its dual problem,

we analytically characterize several structural properties of the optimal system-wide cost,

energy prices and storage controls. In particular, we derive locational marginal prices and

network congestion prices as piecewise affine functions of the installed storage capacity

vector and a closed-form expression of the Hessian of the optimal objective function. Third,

exploiting these structural properties, we show that the submodularity of the placement

value function is not guaranteed (P 6= Psm in Fig. 2.1), although such situations are unlikely

to occur in practice. This examination provides a unique insight into the effect of spatio-

temporal (or network-storage) congestion patterns on submodularity, whereas such an effect

is not observed in other applications such as sensor placement [63,64] and actuator placement

[30, 90, 114, 122]. Fourth, by connecting the sign of Hessian entries to the submodularity of

the storage placement value function, we identify a tight condition under which the value

function is submodular through a polyhedral characterization of critical regions. Based on

this polyhedral analysis, we develop an efficient and rigorous computational procedure to

verify submodularity (i.e., testing whether a particular problem instance belongs to set Psm

in Fig. 2.1). Fifth, motivating by the fact that the total storage deployment over the network

is still small compared to the total hourly average load or generation, we define a small

storage condition (set Pss in Fig. 2.1) under which the verification procedure terminates in

one step. The small storage condition can be tested with the problem data via solving a

simple linear program. An extension to risk-aware placement strategies is also discussed for

cases with deep penetration of wind and solar energy sources.

2.1.3 Organization

The remainder of this chapter is organized as follows. In Section 2.2, we introduce a prob-

lem of jointly optimizing storage placement and control in a power network, and propose a


All problem instances (P)

Submodular

(Psm)

Small storage

(Pss)

Figure 2.1: Venn diagram for the set of all possible storage placement problem instances.

discrete optimization formulation. Section 2.3 contains several structural properties of the

optimal cost function and prices. In Section 2.4, we provide a computational tool to verify

the submodularity of the placement value function based on the identified structural prop-

erties and a polyhedral analysis. We demonstrate the effectiveness of the proposed approach

in Section 2.5.

2.2 Problem Formulation

2.2.1 Notation

For a transmission network with N buses and L lines, we use n ∈ N , 1, . . . , N to index

the buses, and ℓ ∈ L , 1, . . . , L to index the lines. We also use t ∈ T , 1, . . . , T to

index the time periods. For a matrix x ∈ RN×T with any given positive integers N and T ,

we use xn,t to denote its (n, t)-th entry, xt , (x1,t, · · · , xN,t)⊤ ∈ RN×1 to denote its tth

column, and x⊤n , (xn,1, · · · , xn,T ) ∈ R1×T to denote its nth row. For any real number z,

we use (z)+ , max(z, 0) to denote the positive part of z and (z)− , −min(z, 0) to denote

the negative part of z so that z = (z)+ − (z)−. For any Euclidean vector space RN , we use

1 ∈ RN to denote the all-one vector and 1k ∈ RN to denote the kth elementary vector, i.e.,

the vector with all zeros except for its kth element which is 1.

2.2.2 Power Flow Model

We begin by considering a connected power transmission network with N buses and L lines

operated over a finite horizon of T time periods. As common practice in the planning

studies, we use the classical linearized DC approximation to the steady-state AC power flow

[112], so that the power flow constraints can be compactly expressed as

1⊤pt = 0, (2.1a)

Hpt ≤ f , (2.1b)


for power injection pt ∈ RN , t ∈ T . Equation (2.1a) enforces net power balance in the

network, while (2.1b) limits the line flows induced by the power injection vector pt within

the line capacities f . The matrix H , which models the linear mapping from the nodal

injections to the line flows, is commonly referred to as the shift-factor matrix. Appendix A.1

provides a derivation of the structure of matrix H , relating it to matrices representing the

graph structure of the power network.

2.2.3 Energy Storage

We consider a stylized model of energy storage:1 for each bus n, the storage’s state of charge

(SOC) sn,t evolves as

sn,t+1 = sn,t − un,t, t = 1, . . . , T − 1, (2.2)

where un,t is the amount of energy discharged (if un,t > 0) or charged (if un,t < 0) in time

period t. The initial state of charge is assumed to be sn,1 = 0. Given the storage capacity

xn ≥ 0, the following constraints model the energy limit of the storage device:

0 ≤ sn,t ≤ xn, t ∈ T . (2.3)

Note that xn = 0 if there is no storage connected to bus n. Applying (2.2) recursively, we

can express constraint (2.3) as

0 ≤t∑

τ=1

−un,τ ≤ xn, t ∈ T ,

which can be compactly expressed in the following vector form:

0 ≤ Eun ≤ xn1,

where un ∈ RT is the vector of storage control over T periods, and E ∈ RT×T is a lower

triangular matrix with entries Eij = −1 for i ≥ j. In other words, the information about

the storage dynamics is embedded in the matrix E.

2.2.4 Economic Dispatch with Optimal Storage Control

The economic dispatch problem aims to identify an efficient generator dispatch to serve the

net demand, which is defined as load minus uncontrollable (renewable) generation. Let

the net demand for time t ∈ T be denoted by dt ∈ RN . When there are storage devices

connected to the network, a careful operation of storage could reduce the total generation

cost by moving energy across time periods. This can be achieved by linking T -single period

1Our analysis and results can be extended using a more detailed storage model with charging efficiencyand SOC decay. For the sake of simplicity, we use the idealized model.


economic dispatch problems, which results in the following multi-period economic dispatch

problem with storage dynamics :

J(x) , ming,u

T∑

t=1

Ct(gt) (2.4a)

s.t. βt : H(gt + ut − dt) ≤ f , t ∈ T , (2.4b)

γt : 1⊤(gt + ut − dt) = 0, t ∈ T , (2.4c)

µn : Eun ≤ xn1, n ∈ N , (2.4d)

νn : Eun ≥ 0, n ∈ N . (2.4e)

Here, gt ∈ RN is the vector of controllable power generation for each time period t ∈ T ,Ct(gt) ,

∑n∈N Cn,t(gn,t) is the system-wide cost for time period t and is taken to be

quadratic as is common in the literature [132], i.e.,

Ct(gt) ,1

2g⊤t Qtgt + r⊤t gt, t ∈ T ,

where Qt is a diagonal matrix whose diagonal entries are positive, which models the in-

creasing incremental (marginal) heat rate2, and rt ∈ RN is the linear cost coefficient for

generators over the network. The cost function mainly models the fuel cost of generating

gn,t MW of real power. The constraints (2.4b) and (2.4c) enforce power flow constraints (2.1)

with the nodal power injection pt = gt + ut − dt for each period t. The storage dynamics

and energy limit constraints are captured by (2.4d) and (2.4e). At buses with no storage

connection, we set xn = 0, and (2.4d) and (2.4e) reduce to un,t = 0 for all t ∈ T . Note

that we can obtain an optimal storage control schedule as well as an optimal generator dis-

patch schedule by solving the multi-period economic dispatch problem. The optimal value

of the multi-period economic dispatch problem, denoted by J , is a function of storage capac-

ities over the network x, as storage capacities affect the feasible region of the optimization

problem (2.4) via constraint (2.4d).

2.2.5 Storage Placement as Combinatorial Optimization

The optimal cost of this multi-period economic dispatch problem depends critically on the

storage capacity vector x ∈ RN over the network. If there is no storage connected to the

network (i.e., x = 0), the optimal cost of this multi-period problem reduces to the sum

of the optimal costs of T single-period economic dispatch problems. Conversely, if the

storage and line capacities are large enough for every node, the system cost for T periods

approaches a limit where, roughly speaking, the cheapest generators across the network and

2Heat rate is the unit amount of heat contained in fuel needed to produce one unit MW of power output.For each generator with a fixed type of fuel supply, an increasing marginal heat rate implies an increasingmarginal cost with a given fuel price.


over T periods are used. In this ideal case, marginal generation costs for all time periods are

equalized. When only a finite budget is available for installing storage devices, the location

at which a storage device is installed could have a large impact on its contribution to the

cost reduction due to line congestions that could isolate the benefits of storage.

In particular, given K different types of storage devices, each with some storage capacity

xk and capital and installation costs ck, k = 1, . . . ,K, we want to place the storage devices to

minimize the system operation cost with a total budget of b for total capital and installation

costs.

We proceed to formulate the placement problem as a combinatorial optimization prob-

lem. Consider the collection of all N ×K (bus, storage “type”) pairs3

Ω , (n, k) : n = 1, . . . , N, k = 1, . . . ,K.

Each subset X of Ω represents a valid placement decision, and all placement decisions can

be represented by a subset of Ω if we assume that only one storage device with each type

can be placed at each bus. For notational convenience, let I : 2Ω → 0, 1N×K be a set

indicator function such that In,k(X) , 0 if (n, k) /∈ X and In,k(X) , 1 if (n, k) ∈ X . Note

that the nth entry of the matrix-vector product I(X)x can be expressed as

(I(X)x)n =∑

k:(n,k)∈X

xk, (2.5)

which is equal to the total storage capacity at bus n. We now introduce a function, V :

2Ω → R, which we call the storage placement value function, defined as

V (X) , J(I(∅)x)− J(I(X)x).

For each placement decision X , the value V (X) represents the reduction in the minimum

T -period total generation cost caused by the optimal control of the storage devices given

the storage capacities induced by the (bus, storage type) pairs contained in X . The value

function V is normalized such that V (∅) = 0. An optimal placement solution can be obtained

by solving the following discrete optimization problem of maximizing the placement value

function:

maxX⊆Ω

V (X) (2.6a)

s.t.∑

k:(n,k)∈X

ck ≤ b. (2.6b)

3It can be the case that some buses should be ruled out a priori for certain systems. In this case, wecan define the set of possible storage placement decisions as Ω = (n, k) : n ∈ N , k = 1, . . . , K, whereN ⊆ 1, . . . , N is the set of buses where placing a storage is possible.


We claim that our problem formulation as a discrete optimization, has practical advan-

tages over continuous optimization formulations. First, our framework can handle practical

scenarios in which fixed-capacity storage devices are to be placed. Existing continuous

optimization formulations are valid under a strong assumption that the System Operator

can optimize the storage capacity at each bus. One can perform a post-processing, such

as thresholding, to convert the solutions of continuous optimization problems into discrete

solutions. However, such post-processing does not provide a performance guarantee in gen-

eral, whereas our method directly computes a discrete solution with a provable suboptimality

bound. Second, our problem formulation naturally incorporates storage devices’ capital and

installation costs through the knapsack constraint (2.6b), which accurately captures the to-

tal sum of the capital and installation costs as∑

k:(n,k)∈X ck and limits it by the budget

b. In contrast, it is difficult to expect such a precise regulation in continuous optimiza-

tion formulations as discussed in Section 2.1.2. Lastly, the proposed discrete optimization

formulation yields a very simple placement algorithm that only requires an input-output

(blackbox) model of a power system. Specifically, our greedy algorithm can use simulations

that capture electricity market input-output without using detailed information about the

network. This is a notable advantage over continuous optimization formulations, which often

require a full network model with complete information (e.g., parameters) about markets

to calculate the (sub)gradients of objective functions.

2.2.6 Outline of Proposed Analyses

We summarize the analyses conducted in this chapter as follows:

1. We characterize optimal locational marginal prices as affine functions of the storage

capacity x by examining the spatio-temporal (or network-storage) congestion patterns

of a power network via a dual quadratic program (Theorem 2.1).

2. Using the results of our dual analysis, we identify a closed-form expression of the

Hessian ∇2xxJ of the optimal cost function in each critical region (Theorem 2.2).

3. We connect the Hessian ∇2xxJ and the submodularity of the storage placement value

function V (Theorem 2.3). We also provide an insightful case in which V is not

submodular, although such a case is unlikely to occur in practice.

4. We investigate a polyhedral characterization of each critical region where the Hessian

∇2xxJ is invariant. We show that the spatio-temporal congestion pattern of a power

network defines the critical regions (Theorem 2.4).

5. The polyhedral characterization is then used to develop a computational tool for ver-

ifying the submodularity of V .

In Steps 1 and 2, we parametrize the multi-period economic dispatch problem with the

vector x of storage capacity by relaxing its domain as RN . Step 3 plays an important role in


connecting our analysis in the continuous domain to the discrete notion of submodularity. In

Step 4, the spatio-temporal congestion pattern of a power network is identified as an essential

factor that affects the submodularity of V . Verifying its submodularity via the method

proposed in Step 5, we are able to find a near-optimal solution with a provable suboptimality

bound via the polynomial-time modified greedy algorithm illustrated in Section 2.4.3.

2.3 Structures of Optimal Cost and Prices

2.3.1 Dual Analysis

In order to obtain efficient methods to solve the storage placement problem (2.6), which

is NP-hard, we establish the structural properties of the placement value function through

an analytical characterization of the optimal prices, i.e., the solution to the dual program

of (2.4).

Consider the standard dual quadratic program (QP) of (2.4):

maxλ,γ,β,µ,ν

φ(λ, γ, β, µ, ν) (2.7a)

s.t. λt = γt1−H⊤βt, t ∈ T , (2.7b)

λn = E⊤(µn − νn), n ∈ N , (2.7c)

β, µ, ν ≥ 0, (2.7d)

where the Lagrange dual function is given by

φ(λ, γ, β, µ, ν) ,

T∑

t=1

−1

2(λt − rt)⊤Q−1

t (λt − rt) + d⊤t λt − f⊤βt − x⊤µt.

Note that the variable λn,t represents the locational marginal price (LMP) at bus n in period

t since from the first order optimality condition of (2.4) we have

∇gtCt(gt) = λt, t ∈ T .

A more detailed derivation can be found in [73, 78].

Remark 2.1 (Economic interpretation). According to the spot pricing theory, the generator

(load) at each bus n of the network is paid (charged) at the locational marginal price at the

bus, i.e., λn,t. Therefore for each period t, regrouping terms in the dual objective function


reveals the economic meaning of each term:

φ(λ, γ, β, µ, ν) =

T∑

t=1

Ct(g⋆t (λt))︸︷︷︸

generator cost

+ λ⊤t dt︸︷︷︸load payment

− λ⊤t g⋆t (λt)︸︷︷︸

generator payment

− β⊤t f︸︷︷︸

line congestion charge

− µ⊤t x︸︷︷︸

storage congestion charge

.

Strong duality implies that we have

T∑

t=1

λ⋆⊤t dt − λ⋆⊤t g⋆t (λ⋆t ) =

T∑

t=1

β⋆⊤t f + µ⋆⊤

t x (2.8)

at the optimal solution (g⋆, u⋆, λ⋆, γ⋆, β⋆, µ⋆, ν⋆).4 The term on the left is the total amount

collected from the load less the total amount paid to the generator, which is often referred

to as the merchandising surplus of the system operator [135]. The terms on the right can

be interpreted as a form of economic rent paid to the owners of the transmission lines

(according to physical or financial transmission rights [48]) and storage devices.5 Thus,

(2.8) implies that the merchandising surplus of the system operator matches the total rent

paid to the transmission line and storage owners. This is a generalization of the result that

merchandising surplus is equal to the congestion rent when there is no storage in the network

[135].

We also note that β⋆t and µ⋆

t can be interpreted as the congestion prices for the trans-

mission lines and storage devices, respectively. If we treat the primal optimal value J as a

function parameterized by line capacity ℓ and storage capacity x, these dual variables are the

standard marginal values of increased line capacity and storage capacity ( cf. [135] for the

first identity and Theorem 2.2 for the second identity):

∇fJ = −T∑

t=1

β⋆t and ∇xJ = −

T∑

t=1

µ⋆t . (2.9)

The standard dual QP (2.7) can be further simplified. Observe that (2.4d) and (2.4e),

representing the lower and upper limits of state of charge, respectively, cannot bind simulta-

neously for any storage n and time period t. In other words, if the storage device connected

to bus n is empty in period t, i.e., sn,t = (Eun)t = 0, then it must be the case that

sn,t = (Eun)t < xn. Similarly, (Eun)t = xn signifies that (Eun)t > 0. By complementary

slackness, this implies that µn,tνn,t = 0 for all i and t, that is, at most one of µn,t and νn,t

can be positive at the optimal solution. Combining this with constraint (2.7c), which is

4By Slater’s condition, which is satisfied here because all constraints are linear and the domain of theobjective function is open, strong duality holds.

5Storage congestion charges are not common in current power system markets given the limited amountof storage connected to the system, but it has been proposed in recent studies [78, 118].


equivalent to (µn − νn) = E−⊤λn, we have

µn = (E−⊤λn)+ and νn = (E−⊤λn)

− ∀n ∈ N .

We can verify that, given the structure of matrix E, a more explicit display of the previous

relation is

µt = (λt+1 − λt)+ and νt = (λt+1 − λt)− ∀t ∈ T , (2.10)

where we define λT+1 , 0 ∈ RN for convenience. That is, the storage congestion price µn,t

is nonzero only when the LMP λn ramps up in the next time period, where its value equals

the LMP increment.

Substituting the expression of µt into the dual QP, we obtain the following reduced dual

program:

maxλ,γ,β

φ(λ, β) (2.11a)

s.t. λt = γt1−H⊤βt, t ∈ T , (2.11b)

β ≥ 0, (2.11c)

where φ is a piecewise quadratic function defined as

φ(λ, β) ,

T∑

t=1

−1

2(λt − rt)⊤Q−1

t (λt − rt) + d⊤t λt − f⊤βt − x⊤(λt+1 − λt)+.

By strong duality, we can characterize the function J(x) via a sensitivity analysis of the

primal-dual pair (2.4) and (2.11). Let (g⋆(x), u⋆(x), λ⋆(x), γ⋆(x), β⋆(x)) be a pair of primal

and dual solutions to (2.4) and (2.11) for a given capacity vector x. We focus on x values

which will induce nondegenerate solutions of (2.4). In particular, we assume the following

linear independence constraint qualification (LICQ) for the rest of this chapter.

Assumption 2.1 (Flow LICQ). For each t ∈ T , let Hnett be the collection of H’s rows

corresponding to the congested (oriented) lines for the flow induced by (g⋆t (x), u⋆t (x)), when

at least one congested line exists in period t.6 Then, Hnett is of full row rank for each t ∈ T .

We first show that the prices are uniquely defined in almost all practical scenarios:

Proposition 2.1 (Uniqueness of prices). For each fixed x ∈ Rn+, the optimal dual variables

λ⋆(x) and γ⋆(x) are unique. Furthermore, if Assumption 2.1 holds, then (λ⋆(x), γ⋆(x), β⋆(x))

is the unique solution to the dual problem (2.11).

Proof. See Appendix A.2.

In view of Proposition 2.1, for the rest of the chapter, we assume the constraint qual-

ification and take (λ⋆(x), γ⋆(x), β⋆(x)) as the unique dual solution. The following result

6The matrix Hnett is formally defined later in (2.14).


characterizes the locational marginal value of storage via the optimal LMP:

Lemma 2.1 (First order sensitivity). The optimal cost function J(x) is continuously dif-

ferentiable and its gradient is given by

∇xJ(x) = −T∑

t=1

(λ⋆t+1(x)− λ⋆t (x)

)+, (2.12)

where λ⋆T+1(x) , 0. Consequently, the optimal cost function J(x) is nonincreasing in xn

for each n ∈ N .


Coined in Bose and Bitar [19], the term locational marginal value of storage is used to

refer to the quantity −∇xJ(x), which characterizes the benefit of placing storage at different

locations of the network when the size of storage is infinitesimal. They also obtain the

expression (2.12) for the case where the marginal cost of generation and marginal benefit of

consumption are both constants (i.e., the cost function is a piecewise linear function with two

pieces). In fact, the expression (2.12) holds for any smooth convex cost function under mild

regularity assumptions as in the standard sensitivity theorem of nonlinear programming.

When the cost function is nonlinear and the size of the storage to be placed is far

from infinitesimal, the first-order approximation of the value function using the gradient

formula (2.12) may not be accurate.7 We now proceed to obtain a finer characterization of

the optimal cost J(x) by investigating its higher order derivatives. An immediate observation

is that J(x) is convex in x:

Lemma 2.2. The optimal cost function J(x) is convex in x.


2.3.2 Optimal Prices and Second Order Sensitivity

Given that the objective function is quadratic, we expect the curvature (or second-order)

information summarized by the Hessian matrix ∇2xxJ(x) together with the gradient infor-

mation discussed in Lemma 2.1 would provide a sufficient characterization of the optimal

system-wide cost function J(x). This is confirmed by the following result:

Lemma 2.3. The optimal cost function J(x) is a piecewise quadratic function with a finite

number of pieces, each of which is defined on a polytope in Rn+. In each polytope where J(x)

is a quadratic function, the optimal LMP vector λ⋆(x) is affine in x.

Proof. This is a standard multi-parametric quadratic programming result. See e.g.[9].

7The first-order approximation can be used for storage placement. Utilizing it with the approximationalgorithm proposed in [138], we find a solution with 0.6–0.7-a posteriori suboptimality bound.


Remark 2.2. The polytopes in Lemma 2.3 are referred to as critical regions in the litera-

ture of multi-parametric quadratic programming (e.g., [9,53]). In our context, each critical

region is defined as a set of x values such that the inequality constraints binding at the op-

timum remain unchanged. In a single-period economic dispatch problem, the set of binding

constraints conveys the network congestion pattern. When there are storage devices con-

nected to the system, the definition of critical regions also depends on whether the storage

constraints (2.4d) and (2.4e) bind at the optimum. See Theorem 2.4 for a detailed charac-

terization of the critical regions.

By considering each critical region, we can characterize the optimal LMPs based on the

network and storage congestion patterns at the optimum. For each (n, t) ∈ N × T , letzstn,t(x) = 1 if the constraint (Eun)t ≤ xn is binding at the optimum, zstn,t(x) = −1 if the

constraint (Eun)t ≥ 0 is binding at the optimum, and zstn,t(x) = 0 otherwise. In other

words, zst represents the storage congestion pattern. Under strict complementary slackness,

we use (2.10) to obtain

zstn,t , zstn,t(x) =

1, if λ⋆n,t+1(x)− λ⋆n,t(x) > 0,

−1, if λ⋆n,t+1(x)− λ⋆n,t(x) < 0,

0, otherwise.

(2.13)

We now let LCt ⊂ 1, . . . , 2L denote the set of transmission lines that are congested at the

solution in period t and Lt , |LCt | denote the number of congested lines. We define the

selection matrix znett ∈ RLt×2L as

(znett )i,j, (znett (x))i,j=

1, if the ith element in LCt is j,

0, otherwise,

for i = 1, . . . , Lt and j = 1, . . . , 2L, and the shift factor matrix for congested lines as

Hnett , znett H. (2.14)

Note that znett = 0 if all lines are uncongested in period t.

Theorem 2.1. In the critical region containing a given storage capacity vector x, where

the storage and network congestions are represented by zstt and znett , t ∈ T , the optimal

locational marginal prices are affine in x and can be expressed as

λ⋆t (x) =Wt(znet, zst)x+ λt(z

net, zst), t ∈ T , (2.15)

where Wt(znet, zst) and λt(z

net, zst) are defined in Appendix A.5.



As a useful byproduct of Theorem 2.1, we can obtain closed-form expressions of the

(reference) energy price γ⋆t = 1⊤1 λ

⋆t and the congestion price β⋆

t with respect to the capacity

vector x.

Corollary 2.1. Under the setting of Theorem 2.1, γ⋆t (x) and β⋆t (x) are affine functions of

x in the given critical region and can be expressed as

γ⋆t (x) = 1⊤1

(Wt(z

net, zst)x+ λt(znet, zst)

), (2.16)

β⋆t (x) = znett

⊤Bt(z

net, zst)x+ βt(znet, zst), (2.17)

where Bt(znet, zst) and βt(z

net, zst) are defined in Appendix A.5.


The Hessian of the optimal cost function plays a critical role in studying the submodu-

larity of the storage placement value function as we see in Section 2.4. Using Theorem 2.1

and Lemma 2.1, we can obtain a structural characterization (and a closed form expression)

for the Hessian of J(x) as follows:

Theorem 2.2. The optimal cost function J(x) is twice differentiable almost everywhere

with respect to the Lebesgue measure on Rn+. Furthermore, storage capacities x and x′ that

share the same congestion pattern, i.e., znet(x) = znet(x′) = znet and zst(x) = zst(x′) = zst,

have the same Hessian, i.e., (if both ∇2xxJ(x) and ∇2

xxJ(x′) exist),

∇2xxJ(x) = ∇2

xxJ(x′)

with expression given in Appendix A.6.


2.4 Submodularity of Placement Value Function

Equipped with the structural properties of the optimal cost function J(x), we now char-

acterize the storage placement function V (X) defined in Section 2.2.5. Recall that the set

function V (X) models the reduction of the optimal operational cost by employing the place-

ment decision X , which is defined as a subset of Ω that contains all admissible (bus, storage

type) pairs. In particular, we characterize the conditions under which the value function

belongs to the class of submodular functions, one of the most tractable classes in discrete

optimization.

Definition 2.1 (Submodularity and monotonicity). For a finite set Ω, a set function F :

2Ω → R is said to be submodular if, for any X ⊆ Y ⊆ Ω and e ∈ Ω \ Y ,

F (X ∪ e)− F (X) ≥ F (Y ∪ e)− F (Y ). (2.18)


The function is said to be monotonically nondecreasing if for any X ⊆ Ω and e ∈ Ω \X,

F (X ∪ e) ≥ F (X). (2.19)

In our case, (2.19) implies that the marginal benefit of installing a new storage device

is nonnegative and (2.18) states that this marginal benefit should diminish as more stor-

age devices are connected to the system. It is straightforward to check that any modular

function is submodular. Evidently, the nondecreasing property of V (X) follows from the

fact that J(x) is nonincreasing (Lemma 2.1). To check whether V (X) is submodular, it is

instrumental to consider an alternative characterization of submodularity based on discrete

derivatives defined for set functions.

Definition 2.2. For any set function F : 2Ω 7→ R, the discrete derivative of F in e ∈ Ω is

defined as

DeF (X) , F (X ∪ e)− F (X\e).

It is straightforward to check that the following lemma provides a necessary and sufficient

condition for submodularity [17].

Lemma 2.4. A set function F : 2Ω 7→ R is submodular if and only if

De (De′F (X)) ≤ 0, (2.20)

for all e, e′ ∈ Ω, e 6= e′ and X ⊆ Ω.

We relate the submodularity of V (X) to the sign of the Hessian entries of J(x) as follows:

Theorem 2.3 (Sufficient condition for submodularity). The storage placement value func-

tion V : 2Ω → R is submodular if

(∇2

xxJ(x))ij≥ 0, ∀i, j ∈ N ,

for all x ∈ X , [0, xmax]n, where xmax ,∑K

k=1 xk is the maximum storage capacity to be

achieved at each bus.


Theorem 2.3 provides a sufficient condition for the submodularity of V by just checking

the sign of the Hessian entries of the optimal cost function J(s), which can be computed

using Theorem 2.2. The characterization is tight in the following sense.

Corollary 2.2. If(∇2

xxJ(x))ij< 0 for some x ∈ Rn

+ and i, j ∈ N , then there exist a

storage capacity vector x ∈ Rn+ and a corresponding set, Ω, of (bus, storage type) pairs such

that V (X) is not submodular on 2Ω.


This corollary is a partial converse of Theorem 2.3. Even when the point x that results

in negative Hessian entries is contained in X , the function V (X) could still be submodular

if the critical region with negative Hessian entries is relatively small (or the magnitude of

the negative Hessian entries is small) enough that its contribution to the discrete derivative

is outweighed by the contribution from other critical regions with positive Hessian entries.

Theorem 2.3 and Corollary 2.2 establish a strong connection between submodularity

of the storage value function V (X) and the sign of the Hessian entries of the optimal

cost function J(x). This allows us to understand the submodularity condition through an

economic interpretation of the Hessian entries:

Remark 2.3 (Submodularity and substitutability). Define a continuous version of the

storage value function as v(x) = J(0) − J(x). For any buses i, j ∈ N , the Hessian entry

(cross derivative) ∂2v(x)∂xj∂xi

= ∂∂xj

(∂v(x)∂xi

)is the rate of change of the locational marginal value

of storage at bus n when the storage capacity at bus j is changed. Thus, the condition in

Theorem 2.3 has the following economic interpretation:

• For i = j, storage at bus i has diminishing return;

• For i 6= j, storage at bus j substitutes storage at bus i.

The convexity of the optimal cost function J(x) (Lemma 2.2) establishes that the di-

agonal entries of the Hessian matrix ∇2xxJ(x) are always nonnegative. The conditions for

submodularity in Theorem 2.3 also require all off-diagonal entries of the Hessian matrix to

be nonnegative, which does not follow from properties that have already been established

for the optimal value function J(x).

2.4.1 Two-Bus Network

To gain insights into the sign of the off-diagonal entries of the Hessian matrix, we consider a

two-bus example with T = 3 together with synthetic load profiles. We demonstrate that for

the same network submodularity may hold for certain load profiles but fail to hold for certain

other load profiles. For simplicity, we use cost functions Ct(gt) =12g

⊤t gt, for t = 1, . . . , 3, i.e.,

Qt ≡ I ∈ R2×2 and rt ≡ 0. Given a time-varying demand profile over the network, if neither

storage nor line capacity is constraining, then the economic dispatch solution exhibits a form

of “water-filling” behavior where the optimal flows result in equalized generation from each

bus for all time periods. We also notice that g⋆t = λ⋆t for this cost function, by the first

order optimality condition of (2.4).

We now investigate the property of J(x) and the optimal primal and dual solutions

for the multi-period economic dispatch problem for all storage capacities x in the region

X = [0, 1]× [0, 1]. The line capacity is set to be 0.5. We consider the following two cases:

One is commonly observed in simulations where all of the critical regions inside of X have

J(x) with only nonnegative Hessian entries, and the other is specially designed so that one

of the critical regions has negative off-diagonal Hessian entries.


• Case A: dA =

[1 2 0

1 2 2

];

• Case B: dB =

[1 2 1

3 2 3

].

The critical regions for these cases are depicted in Fig. 2.2. For each critical region, we

x1

x2

10

1

RA1

RA2

(a) Case A

x1

x2

10

1

( 12,1

2)

1

3

1

3

RB1

RB2

RB3

RB4

(b) Case B

Figure 2.2: Critical regions for the two-bus examples. In the figure with a slight abuse ofnotation, we use RA

i and RBi to denote the ith critical region for each case.

obtain the expression of the optimal cost function J(x). In addition, for a set of points on

the mesh grid inside of each critical region, we solve the multi-period economic dispatch

problem and obtain the optimal primal dual solution. In all 6 critical regions across these

two cases, only the red region in case B, i.e., RB1 , has negative Hessian entries. This suggests

that submodularity holds for load profile dA but fails to hold for load profile dB. Therefore,

submodularity does not hold in general. We focus on this region for the remainder of this

subsection. The optimal cost function in the critical region is

J(x) =1

2x⊤

[1.5 −0.5−0.5 1.5

]x+

[−0.5 −0.5

]x+ 12.5.

Consider a particular storage capacity vector x = [0.2, 0.2]⊤ ∈ RB1 . The solution of

(2.4) for this given storage capacity is depicted in Fig. 2.3 by exploiting the observation

that storage can be thought of as an inter-temporal link that sends energy into the future


1, 1

2, 1

1, 2

2, 2

1, 3

2, 3

1.70

1.00

2.50

3.00

2.00

2.00

2.00

2.00

1.50

1.00

2.30

3.000.200.50 0.50

0.00

0.000.20

0.20

Figure 2.3: Optimal flow for the case with negative Hessian entries.

and that the multiperiod economic dispatch problem is a form of generalized network flow

problem on a time extended graph where storage edges connect the graph representations

of the power network for consecutive time periods [52, 137]. In Fig. 2.3, each node in the

graph represents a (bus, time period) pair. The vertical edges of the graph represent the

transmission line connecting the two buses, while the horizontal edges represent the power

stored for future use by each storage device. Around each node (n, t), the value associated

with an “inflow arrow” is the generation g⋆n,t, and the value associated with an “outflow

arrow” is the demand dn,t. The value on each vertical edge is the optimal flow sent through

the line; for each horizontal storage edge, the value on it represents the amount of energy

stored at the end of last time period. Red edges are congested at the optimal solution.

A key observation for this special case can be made: Given the load and network conges-

tion pattern, the usage of storage links are through the following spatial-temporal flow path

(1, 1)→ (1, 2)→ (2, 2)→ (2, 3). In this flow path, the storage capacity at bus 1 and storage

capacity at bus 2 complements, instead of substitutes (cf. Theorem 2.3 and Remark 2.3)

each other. We also notice that optimal prices λ⋆, as read from the generation values, follow

a low-high-low pattern on one bus and a high-low-high pattern on the other. This would be

unusual in practical settings, especially in planning scenarios, as the LMPs are often driven

by load profiles. If such a phenomenon were to occur in practice, it would indicate that (i)

the load profiles on these two buses complement each other in the sense that the load on bus

1 peaks when the load on bus 2 drops to its valley, and (ii) the transmission link between

the two buses is weak and congested so that the optimal/equilibrium prices still follow such

patterns. Given that each load bus in a transmission network often represents a collection

of smaller loads, the condition in (i) means that the aggregates of these small loads follow

very different temporal patterns at different locations in the network. Furthermore, if we

were concerned about determining which transmission lines to strengthen, conditions (i) and

(ii) are strong indicators for increasing the capacity of the line connecting these two buses.

In fact, for case B, doubling the line capacity eliminates the critical region with negative

Hessian entries.


2.4.2 Verification of Submodularity Using a Polyhedral Character-

ization of Critical Regions

Albeit the negative Hessian case above appears unlikely to occur in practice and we have yet

to observe negative Hessian entries in all our simulations as discussed in Section 2.5, there is

no a priori theoretical guarantee that V is submodular. In other words, its submodularity

depends on problem settings, particularly the load and network data. Thus, it is of interest

to develop an efficient computation procedure which verifies the submodularity of V .

This is generally a challenging task, as verifying the submodularity of V by definition

involves checking an exponential number of inequalities. Theorem 2.3 reduces this problem

to checking the sign of Hessian entries of a continuous function, J(x) on X . Theorem 2.2

provides a way to compute the Hessian for almost every x ∈ X . It also shows that the

Hessian is invariant in each critical region, and is therefore sufficient to evaluate the Hessian

once per critical region in X . We now address how we could iterate over the critical regions.

We begin by providing an explicit polyhedral characterization of the critical region that

contains almost every capacity vector x. When x is on the boundary of two critical regions,

strict complementary slackness fails to hold and, in general, one may obtain a degenerate

solution. However, the set of boundary points has a Lebesgue measure of 0 and therefore does

not contribute to our submodularity characterization as shown in the proof of Theorem 2.3.

Thus, we can ignore these points for the rest of the discussion. Upon solving the multi-

period economic dispatch problem at x, we can identify the set of binding constraints and

the associated zstt and znett for t ∈ T . The critical region containing x can then be expressed

as the set of storage capacity vectors where the storage and network congestion states are

not changed.

Theorem 2.4. Given zstt (x) and znett (x), t ∈ T evaluated at an arbitrary x ∈ X (except for

a set of measure zero), the critical region containing x, denoted as Rx, is an open convex

polytope defined by the set of x ∈ Rn+ satisfying the linear inequalities

λ⋆n,t+1(x)− λ⋆n,t(x) > 0 ∀(n, t) ∈ N × T s.t. zstn,t(x) = 1,

λ⋆n,t+1(x)− λ⋆n,t(x) < 0 ∀(n, t) ∈ N × T s.t. zstn,t(x) = −1,β⋆ℓ,t(x) > 0 ∀(ℓ, t) ∈ LCt × T .

In other words, for each x ∈ Rx, the associated storage congestion pattern zstt and network

congestion pattern znett satisfy zstt (x) = zstt (x) and znett (x) = znett (x), t ∈ T .


Given this polyhedral characterization of the critical regions, the iterative construction of

all critical regions and the complexity of this process both follow from the standard practice

of multi-parametric quadratic programming [9]. Here, we provide a brief description of this

procedure. To start, we select an initial point x ∈ X0 , X and compute the critical region


Rx that contains it by using Theorem 2.4. Focusing on the part of the critical region inside

of X0, i.e., Rx ∩ X0 while writing the inequality constraints that define this polytope as

Px ≤ q, we can partition the remaining region in X0 as

Xi , x ∈ X0 : P⊤i x ≥ qi, P⊤

j x ≤ qj , ∀j < i,

where P⊤i is the ith row of P . Recursively applying this process to Xi, we can obtain the

collection of all critical regions in X . This procedure is guaranteed to terminate in a finite

number of steps.

The rest of this subsection is devoted to a special case that garners a substantial amount

of practical interest as the amount of storage to be placed is usually small compared to the

total load and generation in the network. For example, the total power capacity of storage

installation in the United States was 221 MW in 2015 [42] while the average generation in

the same year was 467 GW.

Definition 2.3. A storage placement problem is said to satisfy the small storage condition,

if the capacity region X of interest is a subset of the closure of the critical region containing

x = 0, i.e., X ⊆ R0, with R0 defined in Theorem 2.4.

The immediate consequence of this condition for the purpose of verifying submodularity

is as follows [9].

Lemma 2.5. The multi-parametric programming procedure terminates in one step if small

storage condition holds.

Furthermore, if the small storage condition holds, the submodularity of V can be checked

using merely the LMP vector λ⋆t (0) and the network congestion pattern znett in the base case

where no storage has been installed. In other words, we can certify submodularity by simply

using the solutions of the single-period economic dispatch problems for time periods t ∈ T .

Corollary 2.3. Under the small storage condition, all x ∈ X share the same storage and

line congestion patterns as zstt (0) and znett (0), t ∈ T which can be obtained by solving T

single-period economic dispatch problems. Furthermore, if the network topology is a tree,

zstt (0) and znett (0), t ∈ T are uniquely determined using only the LMP data λ⋆(0).

To verify the small storage condition, we need to check whether the polytope R0 contains

the box X = [0, xmax]N . Instead of checking whether all 2N vertices of the box belong to

the polytope, we can simply compare the optimal value of the set containment problem

min ρ ∈ R+ : X ⊆ ρR0 with 1, where the scaled set ρR0 is x : (1/ρ)x ∈ R0. Write R0

as x : Px ≤ q. This set containment problem can be formulated as the following linear

program [36]:


minρ,Λ

ρ

s.t. Λ[I, −I]⊤ = P

Λ[xmax1⊤, 0⊤]⊤ ≤ ρqρ,Λ ≥ 0.

(2.22)

Finally, we note that the small storage condition is equivalent to requiring that in-

stalling the storage does not affect congestion patterns for the planning scenario considered.

This may not hold for some storage placement settings in which case the full-blown multi-

parametric programming procedure described above should be used for verifying submodu-

larity.

2.4.3 Modified Greedy Algorithms

If the submodularity of the value function is verified, we can employ a (modified) greedy

algorithm to obtain a near-optimal solution. In particular, Nemhauser et al. [85] show that

a standard greedy algorithm gives a (1 − 1e )-approximation of an optimal solution when

maximizing monotone submodular functions subject to a cardinality or matroid constraint.

This algorithm is applicable to our problem when there is only one type of storage devices,

i.e., K = 1.

When multi-type devices are used (K > 1), this standard greedy algorithm may not

fully use the diminishing return property as the knapsack (budget) constraint (2.6b) can

cause it become stuck at an unreasonable solution. However, a modification of the greedy

algorithm is shown to achieve the same performance guarantee [62, 115]. This algorithm

uses the partial enumeration heuristic proposed by Khuller et al. [55] which enumerates

all subsets of up to three elements. Its details are presented in Algorithm 1. The first

candidate X1 of the solution maximizes the benefit V among all feasible sets of cardinality

one or two as shown in Line 1. The second candidate X2 is constructed in a greedy way by

locally optimizing the incremental benefit-cost ratio [V (X ∪ (n, k)) − V (X)]/ck starting

from each set X of cardinality three as illustrated in Lines 2–15. Finally, the algorithm

generates an output by comparing the two candidates X1 and X2. This polynomial-time

algorithm can also be implemented in a distributed fashion using the paralellizable method

proposed in [76].

An interesting feature of the proposed algorithm is that its dependence on the operation

model (2.4) is only though the value function V . Therefore, we may substitute our simple

operation model with one that captures more detailed characteristics of the storage tech-

nologies and power flows or even with a black-box simulator. The same greedy algorithm

can be directly applied to the setting with the substituted model. However, our theoretical

performance guarantees need to be extended to be useful in that setting.


Algorithm 1: Modified greedy algorithm for energy storage placement

1 X1 ← argmaxV (X) : |X | ≤ 2,∑

(n,k)∈X ck ≤ b;2 X2 ← ∅;3 foreach X ⊆ Ω s.t. |X | = 3,

∑(n,k)∈X ck ≤ b do

4 Candidates← Ω \X ;5 while Candidates 6= ∅ do6 e← argmax(n,k)∈Candidates

V (X∪(n,k))−V (X)ck

;

7 if∑

k:(n,k)∈X∪e ck ≤ b then8 X ← X ∪ e;9 Candidates← Candidates \ e;

10 end

11 end12 if V (X) > V (X2) then13 X2 ← X ;14 end

15 end16 X⋆ ← argmaxX∈X1,X2 V (X);

2.4.4 Risk-Aware Placement

The uncertainty generated by the widespread penetration of variable renewable energy

sources (including wind and solar energy) is substantial in placement decision-making. In

this subsection, we explicitly consider the stochasticity of renewable generation or equiva-

lently net demand d. To this end, we now view the net demand vector d as a random variable

with a given density function, denoted as fd. The optimal cost function J(x) of economic

dispatch depends on the realization of d. To explicitly show this dependency, we rewrite

the cost function and its corresponding placement value function as J(x; d) and V (X ; d),

respectively. A standard way to account for the randomness of V (X ; d) is to extend the

storage placement (2.6) as a two-stage stochastic program that maximizes E[V (X ; d)] within

the same budget constraint. In the two-stage stochastic programming formulation8, the first

stage is for planning decisions determining the storage placement over the network; the sec-

ond stage is for operation decisions determining the generator and storage dispatch given

the realization of the net demand d. We consider a more general risk-aware formulation

than this mean-performance formulation as risks generated from renewable energy sources

are important factors in decision-making for power systems (e.g., [13,140]). Specifically, the

risk-aware placement problem can be formulated as the following combinatorial stochastic

8In principle, it is possible to utilize a multi-stage stochastic programming formulation featuring a moredetailed information update model for generator and storage operation. This is however less commonly thepractice in planning studies due to data availability and computational complexity concerns.


program featuring a mean-risk objective:

maxX⊆Ω

E[V (X ; d)]− κρ(V (X ; d))

s.t.∑

k:(n,k)∈X

ck ≤ b,(2.23)

where ρ is a convex risk measure and the weight κ ≥ 0 represents the importance of risk

relative to mean value. The following theorem suggests that the submodularity of the

placement value function is preserved through the risk-aware formulation with a convex risk

measure, which is monotonically nonincreasing.

Theorem 2.5. Suppose that X 7→ V (X ; d) is nondecreasing submodular for each d ∈ D,

where D is the support of the density function fd. If ρ is a convex risk measure, then the

objective function of the stochastic program (2.23)

E[V (X ; d)]− κρ(V (X ; d))

is nondecreasing submodular.


Theorem 2.5, which implies that submodularity is preserved through our risk-aware

formulation, is a strong result as it holds independent of the shape of the density function

fd. When we use an empirical distribution of d with a finite number of samples, we are

supposed to check the submodularity of X 7→ V (X ; d) for each sample of d.

We now discuss a solution method for the risk-aware placement problem through an

example. Conditional value-at-risk (CVaR) is one of the most popular convex risk measures,

which is also coherent in the sense of Artzner et al. [3]. It measures the expected value

conditional upon being within some percentage of the worst-case scenarios. Formally, the

CVaR of a random variable Z, representing a loss, is defined as9

CVaRα(Z) , E[Z | Z ≤ VaRα(Z)], α ∈ (0, 1),

where the value-at-risk (VaR) of Z (with the cumulative distribution function FZ) is given

by

VaRα(Z) , infz ∈ R | FZ(z) ≥ α.

In words, VaR measures (1 − α) worst-case quantile of a loss distribution, while CVaR is

equal to the conditional expectation of the loss within that quantile. The computation of

CVaR can be efficiently performed by using the following extremal representation [105]:

CVaRα(Z) = infy∈R

[y +

1

1− αE[(Z − y)+]

].

9This definition is valid when the distribution of Z has no probability atom.


Recalling that Z represents a loss or cost, we can compute the risk term in our optimization

problem as a convex program:

ρ(V (X ; d)) = CVaRα(−V (X ; d))

= infy∈R

[y +

1

1− αE[(−V (X ; d)− y)+]].

Since CVaR is a convex risk measure, if X 7→ V (X ; d) is nondecreasing submodular, The-

orem 2.5 allows us to use Algorithm 1 to solve the risk-aware placement problem (2.23).

This additional complexity comes from a one-dimensional convex optimization problem to

compute CVaRα(−V (X ; d)) at each greedy step. Other risk measures that result in such a

bilevel combinatorial-convex optimization problem can be found in [75].

2.5 Numerical Experiments

2.5.1 IEEE 14 Bus Case

Our placement algorithms are tested using the IEEE 14 bus test case. Hourly zonal aggre-

gated locational marginal price and load data are obtained from the PJM interconnection.

The data correspond to 14 zones inside PJM’s RTO for the year 2014. We consider the

hourly operation of storage over a representative day. The input data for each hour of the

representative day are obtained by averaging over all the 365 days of the year. The box

plots of the hourly price and load data over the 14 zones are plotted in Figure 2.4. The

hourly average load in the system is 80.5 GW.

The load and price time series for these 14 zones are assigned to the 14 buses of the

network, where the price data are used to specify the linear coefficient of generation cost.

We set the quadratic cost coefficients for all generators to be 0.01, which is the median value

of quadratic cost coefficients specified in the IEEE 14 bus test case in MATPOWER [141].

The capacity of each line is set to be the average load per bus over the 24 hours. We consider

a simple setting in which an exhaustive search is still feasible so that the performance of

the greedy placement can be compared to the exact optimal solution. To this end, we let

the type of storage be K = 1.

We now consider placing 5 storage devices in the 14 bus-network, with a total energy

capacity of 150 MWh. Using the optimal set containment optimization (2.22), we verify that

this setting satisfies the small storage assumption and that in the critical region the Hessian

condition in Theorem 2.3 holds. The greedy strategy in Algorithm 1 is implemented. We

also perform an exhaustive search over all feasible storage placements to verify the actual

performance of the algorithm. Instead of being (1 − 1/e) suboptimal as suggested by the

worst case performance bound, the greedy algorithm identifies the exact optimal placement

in this case, with buses 5, 11, 12, 1, 9 selected to place storage.


30

40

50

60

70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Hour

Pric

e ($

/MW

)(a)

5000

10000

15000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Hour

Load

(M

W)

(b)

Figure 2.4: Box plots of price and load data: (a) locational marginal price, and (b) load.

2.5.2 Other Test Cases

To further examine the performance of greedy placement, we test the algorithm with a

variety of other IEEE test cases together with a randomized assignment of the PJM load and

price data to the network buses.10 For larger networks, it is no longer feasible to benchmark

the greedy performance against the exact optimal placement which should be identified

through an exhaustive search. Therefore, we compare the greedy performance and run-time

against that of a mixed-integer quadratic programming (MIQP) solver from Gubori. As

the solver implements a branch-and-bound algorithm, the Gurobi solution comes with a

posterior performance bound on the optimal cost. For each of the test network topology, we

follow the setup in the previous subsection but vary the total storage capacity from 0.5% of

the average system-wide load to 5% of the system-wide load.

For each of the 40 problem instances (4 IEEE test cases and 10 total storage configura-

tions), the following observations hold:

• The small storage condition is valid and thus the placement value function is submod-

ular;

• The greedy algorithm achieves the same storage placement value (and the system-wide

cost) as the MIQP-based method.

• Per the error bound provided by the branch-and-bound procedure, the MIQP-based

10We use a random assignment due to a lack of real demand time series for the IEEE test cases. With therandom assignment, the demand (and price) time series for each node is a linear combination of the demand(and price) time series of the PJM zonal demand data with coefficients generated uniformly at random.


method finds the global optimal solution up to a numerical tolerance of 10−4 and

therefore so does the greedy algorithm.

14 30 57 118Bus numbers

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

MIQ

Pruntime

Greed

yruntime×

100%(%

)

7.65% 23.97%

248.77%

4948.53%

Figure 2.5: Average run-time comparison between the greedy algorithm and the MIQP-based method.

Figure 2.5 compares the run-time of these two methods, in which we have averaged the run-

times across the storage capacities as we have not observed significant or systematic variation

in run-time when the storage capacities are changed (the number of storage to be placed are

fixed to be 5 as in the previous subsection). The run-time comparison demonstrates that

the greedy method has superior scalability compared to the MIQP-based method, which is

consistent with the fact that the greedy approach is a polynomial time algorithm, whereas

branch-and-bound procedure takes exponential time in the worst case.

2.6 Conclusions

In this chapter, we have proposed a discrete optimization-based framework for placing en-

ergy storage devices in a power network when all storage resources are optimally controlled

to minimize system-wide cost. This approach is useful at explicitly accounting for hetero-

geneous storage installation and capital costs. To use a scalable modified greedy algorithm

to solve this NP-hard combinatorial optimization problem, we have investigated a tight

condition under which the placement value function is submodular. Based on our com-

prehensive analytical characterization of the optimal cost, prices, and critical regions in a

parametric economic dispatch problem with storage dynamics, we have also developed an

efficient computational method to verify submodularity, and gained the unique insight that

the spatio-temporal congestion pattern of a power network is a critical factor for submodu-

larity.

Chapter 3

Stochastic Control of

Distributed Energy Storage

35

CHAPTER 3. STOCHASTIC CONTROL OF DISTRIBUTED ENERGY STORAGE 36

3.1 Introduction

Deep penetration of renewable energy generation is essential to ensure a sustainable future.

Renewable energy resources, such as wind and solar, are intrinsically variable. Uncertainties

associated with these intermittent and volatile resources pose a significant challenge to their

integration into the existing grid infrastructure [81]. More flexibility, especially in shifting

energy supply and/or demand across time and network, is desired to cope with the increased

uncertainties.

Energy storage provides the functionality of shifting energy across time. A vast array

of technologies, such as batteries, flywheels, pumped-hydro, and compressed air energy

storages, are available for such a purpose [66, 80]. Furthermore, flexible or controllable

demand provides another ubiquitous source of storage. Deferrable loads – including many

thermal loads, loads of internet data-centers and loads corresponding to charging electric

vehicles (EVs) over certain time interval [33] – can be interpreted and controlled as storage

of demand [89]. Other controllable loads which can possibly be shifted to an earlier or later

time, such as thermostatically controlled loads (TCLs), may be modeled and controlled as a

storage with negative lower bound and positive upper bound on the storage level [46]. These

forms of storage enable inter-temporal shifting of excess energy supply and/or demand, and

significantly reduce the reserve requirement and thus system costs.

On the other hand, shifting energy across a network, i.e., moving excess energy supply

to meet unfulfilled demand among different geographical locations with transmission or

distribution lines, can achieve similar effects in reducing the reserve requirement for the

system. Thus in practice, it is natural to consider these two effects together. Yet, it remains

mathematically challenging to formulate a sound and tractable problem that accounts for

these effects in electric grid operations. Specifically, due to the power flow and network

constraints, control variables in connected buses are coupled. Due to the storage constraints,

control variables in different time periods are coupled as well. On top of that, uncertainties

associated with stochastic generation and demand dramatically complicate the problem, due

to the large number of recourse stages and the need to account for all probable realizations.

Two categories of approaches have been proposed in the literature. The first category

is based on exploiting structures of specific problem instances, usually using dynamic pro-

gramming. These structural results are valuable in providing insights about the system, and

often lead to analytical solution of these problem instances. However, such approaches rely

heavily on specific assumptions of the type of storage, the form of the cost function, and the

distribution of uncertain parameters. Generalizing these results to other specifications and

more complex settings is usually difficult, and consequently this approach is mostly used to

analyze single storage systems. For instance, analytical solutions to optimal storage arbi-

trage with stochastic price have been derived in [100] without storage ramping constraints,

and in [38] with ramping constraints. Problems of using energy storage to minimize energy

imbalance are studied in various contexts; see [98, 101, 102, 113] for reducing reserve energy


requirements in power system dispatch, [15,57] for operating storage co-located with a wind

farm, [124, 126] for operating storage co-located with end-user demands, and [50] for stor-

age with demand response. The other category relies on the use of heuristic algorithms,

such as Model Predictive Control (MPC) [136] and look-ahead policies [82], to identify sub-

optimal storage control rules. Usually based on deterministic (convex) optimization, these

approaches can be easily applied to general networks. The major drawback is that these

approaches usually do not have any performance guarantee. Consequently, it lacks theoret-

ical justification for their implementation in real systems. Examples of this category can be

found in [136] and references therein.

This chapter aims at designing distributed online deterministic optimizations that solve

the stochastic control problem with provable guarantees. It contributes to the existing lit-

erature in the following ways. First, we formulate the problem of storage network operation

as a stochastic control problem with general cost functions, which encapsulates a variety

of problems with different types of storage as well as different uses of storage. Second, we

devise an online algorithm for the problem based on the theory of Lyapunov optimization,

and provide guarantees for its performance by proving a bound of its sub-optimality. This

converts the “intractable” stochastic control program to a sequence of tractable determin-

istic optimization programs. The bound is useful not only in assessing the performance of

our algorithm, but also in evaluating the performance of other sub-optimal algorithms when

the optimal costs are difficult to obtain. It can also be used to estimate the maximum cost

reduction that can be achieved by any storage operation, thus provides understanding for

the limit of a certain storage system. To the best of our knowledge, this is the first algorithm

with provable guarantees for the general networked storage operation problem. Finally, we

derive task-based distributed implementation of the online algorithm using the alternating

direction method of multipliers (ADMM).

The rest of the chapter is organized as follows. Section 3.2 formulates the problem of

operating a storage network under uncertainty. Section 3.3 gives the online algorithm and

states the performance guarantee. Section 3.4 discusses the distributed implementation of

the online program. Numerical examples are then given in Section 3.5. Section 3.6 concludes

the chapter.

3.2 Problem Formulation

3.2.1 Notation

Although we seek notational consistency across the thesis, definitions of variables should

be assumed to be independent across different chapters unless stated otherwise. For a

directed graph G(N ,L), with node set N , 1, . . . , N and edge set L , 1, . . . , L, defineN (ℓ) , n ∈ N : n ∼ ℓ, and L(n) , ℓ ∈ L : ℓ ∼ n, where n ∼ ℓ (or ℓ ∼ n) means

that edge ℓ and node N are incident. We assume that all these sets are equipped with


the natural order. For any vector v ∈ Rk and D ⊆ [k] , 1, . . . , k, vD ∈ R|D| is the

sub-vector containing entries of v indexed by set D. Similarly, for any matrix M ∈ Rd1×d2 ,

and D1 ⊆ [d1] and D2 ⊆ [d2], MD1,D2∈ R|D1|×|D2| is the sub-matrix containing rows and

columns of M indexed by sets D1 and D2, respectively. For any variable x ∈ RL that is

defined for each edge, if edge ℓ ∈ L is incident to nodes N and n′, we use the notations xℓ

and xnn′ interchangeably to refer to the ℓth element of x. For any x ∈ R, (x)+, max(x, 0)

and (x)−, (−x)+. An extended real valued function f(x) with domain dom f = C ⊆ Rk is

such that f(x) =∞ if x 6∈ C.

3.2.2 Centralized Problem

We model the power grid as a directed graph G(N ,L), with N = [N ] , 1, . . . , N, L =

[L] , 1, . . . , L, where N is the number of nodes and L is the number of edges. Here each

node models a bus and each edge models a line. To simplify the exposition, we assume each

bus n is connected to all of the following types of devices:1

• Uncontrollable net supply. A renewable generator and a load are connected to the bus,

with the net power supply, i.e., the generation minus the demand, at time period t

denoted by δn,t. As both demand and generation can be stochastic, δn,t is in general

stochastic.

• Energy storage2. A storage with storage capacity Smaxn , minimum storage level Smin

n ,

storage charging limit Umaxn , and storage discharging limit −Umin

n is connected to the

bus. The storage level (or state of charge) is denoted by sn,t and the storage control

is denoted by un,t with un,t > 0 representing charging and un,t < 0 representing

discharging. For each time period t, we have constraints

Sminn ≤ sn,t ≤ Smax

n

and

Uminn ≤ un,t ≤ Umax

n .

The storage dynamics is

sn,t+1 = λnsn,t + un,t,

where λn ∈ (0, 1] is the storage efficiency which models the energy loss over time

without storage operation. We denote the set of parameters for the storage at bus

1By setting the problem data properly, we can model buses which are only connected to a subset of thesedevices. For example, a generator bus with no renewables and no storage can be modeled by setting δn,t = 0and Smin

n = Smaxn = 0.

2The energy storage model differs from the energy storage model introduced in Section 2.2.3 in twoways. First, for the operation problem considered in this chapter, we utilize a more detailed energy storagemodel that captures storage characteristics via additional constraints and loss parameters. Second, the signconventions for charging and discharging in this chapter and in Section 2.2.3 are different. The choice ofsign convention is Section 2.2.3 is made for notational convenience in expressing certain price variables andfor consistency with prior literature on storage planning.


n by Sn , λn, Sminn , Smax

n , Uminn , Umax

n . Here the set of parameters for each stor-

age satisfies the feasibility and controllability assumptions (see Assumption B.1 in

Appendix B.2 and [97] for more discussions).

• Conventional generator. Its generation at time period t is denoted by g+n,t (≥ 0) and

its convex cost function is denoted by c+n (·). It is possible in certain scenarios to have

more supply than demand (e.g., when there is too much wind generation). In such

cases, let g−n,t (≥ 0) be the generation curtailment at time period t and c−n (·) be the

cost associated with the curtailment. Without loss of optimality, we can summarize

g+n,t and g−n,t by a single variable gn,t such that g+n,t = (gn,t)

+and g−n,t = (gn,t)

−. Then

the total cost at bus n and in time period t is

cn(gn,t) = c+n (g+n,t) + c−n (g

−n,t).

Optionally, the cost can depend on a stochastic price parameter pn,t ∈ [Pminn , Pmax

n ],

so that we write the cost as cn(gn,t; pn,t).

We use the classic DC approximation for AC power flow as in Section 2.2.2. For convenience,

in this chapter, we work with voltage phase angles denoted by θn,t for time period t on bus

n. The real power flow from bus n to bus n′ can be written as

fnn′,t = Ynn′(θn,t − θn′,t),

where Y ∈ RN×N is the imaginary part of the admittance matrix (Y-bus matrix) under

DC assumptions, and fnn′,t satisfies line flow constraints −Fmaxnn′ ≤ fnn′,t ≤ Fmax

nn′ , where

Fmaxnn′ = Fmax

n′n > 0 is the real power flow capacity of the line connecting bus n and bus n′.

We can now formulate the problem as a stochastic control problem as follows:

min (1/T )E

T∑

t=1

N∑

n=1

cn(gn,t; pn,t) (3.1a)

s.t. δn,t + gn,t = un,t +N∑

n′=1

fnn′,t, (3.1b)

sn,t+1 = λnsn,t + un,t, (3.1c)


n , (3.1d)


n , (3.1e)

Ynn′(θn,t − θn′,t) = fnn′,t, (3.1f)

− Fmaxnn′ ≤ fnn′,t ≤ Fmax

nn′ , (3.1g)

where T is the total number of time periods under consideration, the expectation is taken

over pn,t and δn,t, constraints (3.1b), (3.1c), (3.1d) and (3.1e) hold for all n and t, constraints


(3.1f) and (3.1g) hold for all n, n′ and t, and sn,1 ∈ [Sminn , Smax

n ] is given for each n. Here the

goal is to find an optimal control policy for each time period t which maps the information

available up to the time period to the optimal decisions (u⋆t , g⋆t , θ

⋆t , f

⋆t ).

Albeit the bulk of this chapter focuses on the formulation (3.1), we note that it can be

extended in various directions.

Remark 3.1 (Generalized storage Model). The storage model described above consider

primarily energy storage. But following the development in [97], it is easy to incorporate

other type of generalized storage such as deferrable loads modeled as storage of demand, and

collections of thermostatically controlled load. In addition, the energy loss during charg-

ing/discharging can be modeled with conversion functions. For example, a storage with

charging coefficient µC ∈ (0, 1] and discharging coefficient µD ∈ (0, 1] can be modeled us-

ing charging conversion function hC(u) = (1/µC)u and discharging conversion function

hD(u) = µDu, respectively. See [97] for more details.

Remark 3.2 (Nonconvex cost). The assumption that cn is convex for each n ∈ [N ] is not

strictly necessary. See [95] for generalization to general subdifferentiable functions.

Remark 3.3 (Other costs and constraints). Many other costs including operational cost

of storage due to charging and discharging, and other constraints including bounds on the

generation and phase angles can be added without altering our results and the proofs. In

fact, the cost can be a function of the form cn , cn(un,t, rn,t, θn,t, δn,t, pn,t).

Our prior work [97] can be viewed as the single bus special case of the problem formu-

lated here. Thus the examples for different use cases of the storage (e.g., balancing and

arbitrage) discussed in [97] can also be encapsulated into our current framework together

with a network. The incorporation of the network element allows our methodology to be

applied to a broader range of problems such as microgrid management and storage-based

real-time regulation for the bulk power grid.

3.2.3 Cluster based Distributed Control

Solving problem (3.1) in a centralized fashion may not be feasible due to concerns re-

garding privacy, communication, and computation. First of all, specifying the centralized

problem (3.1) requires information about the cost functions and parameters of the devices

connected to each of the buses, and the probability distributions of all local stochastic pa-

rameters. This process involves agents who own the generators, storages, as well as power

consumers who may not be willing to report such data. Even if the data reporting is granted,

gathering all these data from nodes of a large power network, and subsequently disseminat-

ing the optimal control signal obtained from the centralized solution in real time presents

a challenge on the communication system required. The large amount of data that have

to be sent to and from the centralized control center may lead to traffic congestions and


delays in the data delivery. Finally, granting an adequate communication infrastructure in

place, solving the stochastic control problem formulated in (3.1) over a large network is

not tractable due to a lack of practical algorithms, i.e., existing algorithms either do not

have any performance guarantee or do not scale gracefully with the number of buses of the

system.

A cluster-based control architecture for the future grid is envisioned in [4]. Here we

present a first step in achieving such an architecture. In particular, we consider solving the

centralized problem (3.1) with resource clusters. Suppose that the network is partitioned

into D clusters. Each cluster Cd consists of a subset of nodes Nd ⊂ N and a subset of lines

Ld ⊂ L, i.e., Cd , (Nd,Ld), and is controlled by a cluster controller (CC). The CC for each

cluster Cd

• possesses local static information including cn and Sn for all n ∈ Nd, and Yℓ and Fmaxℓ

for all ℓ ∈ Ld,

• senses local disturbances δn,t and pn,t for all n ∈ Nd and all t,

• controls local variables un,t, gn,t and θn,t for all n ∈ Nd, and fℓ,t for all ℓ ∈ Ld and all

t,

• and communicates with its neighbors GNdwhere GNd

is the collection of Gw’s for

which there exists ℓ ∈ Ld, n ∈ Nw such that ℓ ∼ n, or there exists ℓ ∈ Lw, n ∈Nd such that ℓ ∼ n.

Here we provide a bird-eye view of our approach for tackling the challenging distributed

stochastic control problem which we just formulated. Section 3.3 provides an online al-

gorithm that converts the centralized stochastic control program to a sequence of online

deterministic optimization. Section 3.4 then presents the decentralization of these online

deterministic optimization using the alternating direction method of multipliers (ADMM).

3.3 Online Modified Greedy Algorithm for Networked

Storage Control

3.3.1 Algorithm

We propose a very simple algorithm to solve the centralized problem (3.1) with performance

guarantees. The algorithm, termed the network online modified greedy (OMG) algorithm,

is composed of an offline and online phase. Next we describe the input data of the algorithm

and each phase.

Input Data. Similar to the single storage online modified greedy (OMG) algorithm [97],

for each bus n ∈ [N ], in addition to the storage parameters Sn and the cost functional form


cn, the algorithm requires two input parameters that are a lower bound, denoted by Dcn,

and an upper bound, denoted by Dcn, for the subdifferential of the objective function cn

with respect to un,t.3

Remark 3.4 (Distribution-free method). As in the single storage case [97], The OMG

algorithm is a distribution-free method in the sense that almost no information regarding

the joint probability distribution of the stochastic parameters δn,t and pn,t are required. The

only exception is when calculating Dcn and Dcn, the support of pn,t and δn,t may be needed.

Comparing to the entire distribution functions, it is much easier to estimate the supports of

the stochastic parameters from historical data.

Offline Phase. Before running the algorithm, each bus n ∈ [N ] needs to calculate two

algorithmic parameters, namely a shift parameter Γn and a weight parameterWn. Any pair

(Γn,Wn) satisfies the following conditions can be used:

Γminn ≤Γn ≤ Γmax

n , (3.2)

0 <Wn ≤Wmaxn , (3.3)

where Γminn , Γmax

n and Wmaxn are functions of the storage parameters Sn and subdifferential

bounds Dcn and Dcn whose definitions can be found in Appendix B.1.

It will be clear later that the sub-optimality bound depends on the choice of (Γn,Wn).

As in [97], we provide two approaches for selecting these parameters

• The maximum weight approach (maxW): Setting Wn = Wmaxn reduces the interval in

(3.2) to a singleton (Γminn = Γmax

n ) and hence determines a unique Γn.

• The minimum sub-optimality bound approach (minS): It turns out that the sub-

optimality bound of OMG, as a function of (Γn,Wn)’s for all n ∈ [N ], can be minimized

using a semidefinite program reformulation. This approach uses the set of (Γn,Wn)’s

minimizing the sub-optimality bound.

Online Phase. At the beginning of each time period t, the OMG algorithm solves a

deterministic optimization as follows

min

N∑

n=1

(λn/Wn)(sn + Γn)un + cn(gn; pn) (3.4a)

s.t. Uminn ≤ un ≤ Umax

n , (3.4b)

δn + gn = un +

N∑

n′=1

fnn′ , (3.4c)

Yn′n(θn′ − θn) = fn′n, (3.4d)

3Mathematical expressions for these parameters are relegated to Appendix B.1.


− Fmaxn′n ≤ fn′n ≤ Fmax

n′n . (3.4e)

where the decision variables are u, g, θ and f , and we have dropped the dependence on t to

simplify the notation. This treatment is justified by the fact that (3.4) does not involve the

charging and discharging constraints induced by the storage capacity and storage dynamics,

i.e., we have removed constraints (3.1c) and (3.1e), which can be alternatively summarized

as

Sminn ≤ λnsn + un ≤ Smax

n . (3.5)

It will be shown later in Appendix B.2 that (3.5) holds automatically given that the algo-

rithmic parameters of OMG satisfy conditions in (3.2) and (3.3).

The optimization is similar to the greedy heuristics which minimize the cost for each

period, i.e.,∑N

n=1 cn(gn; pn), subject to constraints of (3.4) together with constraint (3.5)

in each step. Instead of directly optimizing the cost at the current time period, for each

storage, the OMG algorithm optimizes a weighted combination of the stage-wise cost and a

linear term of un depending on the shifted storage level sn+Γn. Here the weight parameter

Wn decides the importance of the original cost in this weighted combination, while the shift

parameter Γn defines the shifted state given the original state sn. Roughly speaking, the

shifted state sn + Γn belongs to an interval [Sminn + Γn, S

maxn + Γn] which usually contains

0. For fixed Wn > 0, if the storage level is relatively high, the shifted state is greater than

0, such that the state-dependent term (i.e., (λn/Wn)(sn +Γn)un) encourages a negative un

(discharge) to minimize the weighted sum. As a result, the storage level in the next time

period will be brought down. On the other hand, if the storage level is relatively low, the

shifted state is smaller than 0, such that the state-dependent term encourages a positive

un (charge) and consequently the next stage storage level is increased. These two effects

together help to hedge against uncertainty by maintaining a storage level somewhere in

the middle of the feasible interval. More detailed discussion regarding the design of the

modification term in the objective can be found in [97].

3.3.2 Performance Guarantees

We provide a stylized analysis for the performance of OMG.

Assumption 3.1. The following assumptions are in force for the analysis in this section.

A1 Infinite horizon: The horizon length T approaches to infinity.

A2 IID disturbance: The disturbance process (δt, pt) ∈ R2N : t ≥ 1 is independent and

identically distributed (i.i.d.) across t and is supported on a compact set such that

δn,t ∈ [δminn , δmax

n ] and pn,t ∈ [Pminn , Pmax

n ] for all n ∈ [N ] and all t. Note that any

correlation structure is allowed for variables in the same time period.


A3 Frequent acting: The storage parameters satisfy Umaxn − Umin

n < Smaxn − Smin

n for all

n ∈ [N ].

Here A1 and A2 are technical assumptions introduced to simplify the exposition. Relax-

ingA1 leads to no change in our results except an extra term ofO(1/T ) in the sub-optimality

bound. For large T , this term is negligible. [97] discusses how to reduce A2. Under these

two assumptions, the storage operation problem can be cast as an infinite horizon average

cost stochastic optimal control problem in the following form

min limT→∞

(1/T )E[ T∑

t=1

N∑

n=1

cn(gn,t; pn,t)]

(3.6a)

s.t. (3.1b), (3.1c), (3.1d), (3.1e), (3.1f), (3.1g). (3.6b)

Assumption A3 states that the range of feasible storage control Umaxn − Umin

n is smaller

than the range of storage levels Smaxn − Smin

n , i.e., the ramping limits of the storage is

relatively small compared to the storage capacity. For any storage system, this assumption

is true as long as the length of each time period ∆t is made small enough; see [97] for more

details.

Define J(u, g, θ, f) as the total cost of problem (3.6) induced by the sequence of control

actions (ut, gt, θt, ft), t ≥ 1 and J⋆ = J(u⋆, g⋆, θ⋆, f⋆) as the minimum cost of the average

cost stochastic control problem with (u⋆t , g⋆t , θ⋆t , f⋆t ), t ≥ 1 being the corresponding optimal

control sequence. The main results regarding the performance of the OMG algorithm is

summarized as follows, whose proof is relegated to Appendix B.2.

Theorem 3.1 (Performance). The control sequence (uol, gol, θol, fol) , (uolt , golt , θolt , folt ), t ≥

1 generated by the OMG algorithm is feasible with respect to all constraints of problem (3.1)

and its sub-optimality is bounded by∑N

n=1Mn(Γn)/Wn, that is

J⋆ ≤ J(uol, gol, θol, fol) ≤ J⋆ +

N∑

n=1

Mn(Γn)/Wn, (3.7)

where

Mn(Γn) =Mun (Γn) + λn(1 − λn)M s

n(Γn),

Mun (Γn) =

1

2max

((Uminn +(1−λn)Γn

)2, (Umax

n +(1−λn)Γn)2),

M sn(Γn) = max

((Sminn + Γn

)2, (Smax

n + Γn)2).

The theorem above guarantees that the cost of the OMG algorithm is no greater than

J⋆ +∑N

n=1Mn(Γn)/Wn.

In many cases, we are interested to minimize the sub-optimality bound. This can be


cast as the following optimization

PO: min

N∑

n=1

Mn(Γn)/Wn

s.t. Γminn ≤ Γn ≤ Γmax

n , 0 < Wn ≤Wmaxn ,

where the optimization variables are (Γn,Wn), n ∈ [N ], and the constraints hold for all

n ∈ [N ]. Observing that the objective and constraints are separable across buses, we can

solve this program separately on each bus via a semidefinite program (SDP) as in the single

storage case [97]. Here the SDP is reproduced for completeness.

Lemma 3.1 (Semidefinite Reformulation of PO). For each n ∈ [N ], let symmetric positive

definite matrices Xmin,un , Xmax,u

n , Xmin,sn and Xmax,s

n be defined as follows

X(·),un =

[ηun U

(·)n + (1− λn)Γn

∗ 2Wn

], X(·),s

n =

[ηsn S

(·)n + Γn

∗ Wn

],

where (·) can be either max or min, and ηu and ηs are auxilliary variables. Then PO can

be solved via the following semidefinite program

min ηun + λn(1− λn)ηsn (3.8a)

s.t. Γminn ≤ Γn ≤ Γmax

n , 0 < Wn ≤Wmaxn , (3.8b)

Xmin,un , Xmax,u

n , Xmin,sn , Xmax,s

n 0, (3.8c)

where Γminn and Γmax

n are linear functions of Wn as defined in (B.2) and (B.3).

We close this section by summarizing some of the properties for the sub-optimality bound

at each bus n in the next remark; more detailed discussion and examples of the uses of the

sub-optimality bound can be found at [97].

Remark 3.5 (Properties ofMn(Γn)/Wn). For the per bus sub-optimality boundMn(Γn)/Wn,

let (Γ⋆n,W

⋆n) be the bound minimizing parameter choice. The following properties are true.

• For ideal storage (λn = 1), Mn(Γn)/Wn is minimized with maxW parameter specifica-

tion.

• The OMG algorithm is near-optimal for ideal storage with large storage capacity, i.e.,

with Umaxn − Umin

n being fixed, Mn(Γ⋆n)/W

⋆n → 0 when Smax

n − Sminn →∞.

• We also have Mn(Γ⋆n)/W

⋆n → 0 if Umax

n − Uminn → 0 and λn → 1, while Smax

n − Sminn

is fixed (which may be the case when the storage is controlled frequently such that the

length of each time period ∆t→ 0).


3.4 Distributed Online Control Via Alternating Direc-

tion Method of Multipliers

Results in previous section convert the stochastic control program (3.1) to a sequence of

online deterministic optimization programs. In this section, we take a bottom-up approach

in deriving a decentralized solution to (3.1). In particular, we first reformulate the online

program and then apply ADMM to obtain a fully distributed algorithm that specifies com-

putation and communication tasks for each bus and each line of the network. We then

associate the corresponding tasks to the CC’s to which these buses or lines belong. For a

survey of ADMM, see [21].

3.4.1 Node-Edge Reformulation

In order to obtain a fully distributed algorithm that uses only local computation and neigh-

borhood communication, it is necessary to ensure that all constraints of the optimization

program only couple variables controlled by pairs of neighboring node and edge so that all

communication can be implemented using simple pairwise messages. To this end, we refor-

mulate the online program (3.4) by creating local copies of certain variables. In particular, let

xn , (un, gn, θn, fn,L(n))⊤ be the local (primal) variables at node n, and zℓ , (fℓ, θℓ,N (ℓ))

⊤

be the local (primal) variables at edge ℓ, where fn,L(n) ∈ R|L(n)| is node n’s local auxiliary

copy of edge variable fL(n), and θℓ,N (ℓ) ∈ R2 is edge ℓ’s local auxiliary copy of node variable

θN (ℓ). Here we use the notation fn,ℓ for ℓ ∈ L(n) to refer to n’s local copy of variable fℓ;

the similar notation θℓ,n is also used. Then program (3.4) can be written as

min

N∑

n=1

qn(xn) +

L∑

ℓ=1

hℓ(zℓ) (3.9a)

s.t. fn,L(n) = fL(n), ∀n ∈ [N ], (3.9b)

θℓ,N (ℓ) = θN (ℓ), ∀ℓ ∈ [L], (3.9c)

where extended real valued functions qn and hℓ summarize the separable objective and

constraints at node n and edge ℓ, respectively, and are defined as follows

qn(xn) , qn(un, rn, θn, fn,L(n)) , (λn/Wn)(sn + Γn)un + cn(gn; pn),

with domain dom qn = xn : Uminn ≤ un ≤ Umax

n , δn + gn + An,L(n)fn,L(n) = un, θn ∈ R,and hℓ(zℓ) = 0 with domhℓ , zℓ : fℓ = BℓA

TN (ℓ),ℓθℓ,N (ℓ), −Fmax

ℓ ≤ fℓ ≤ Fmaxℓ . Here

constraints (3.9b) and (3.9c) ensure that at the solution, these local auxiliary variables must

be equal to the corresponding true variables. The (scaled) dual variables4 corresponding

to constraints (3.9b) and (3.9c) are denoted by ηn and ξℓ, respectively. We proceed to

4See Appendix B.3 for more details.


state the task-based distributed ADMM. The derivation of the algorithm is relegated to

Appendix B.3.

At each iterate, indexed by k, the following tasks (jobs) are issued and completed in

order:

• J NP,kn : Each node n ∈ [N ] performs node primal update:

xk+1n = argmin

xn

qn(xn) +ρ

2‖fn,L(n) − fk

L(n) + ηkn‖22 +∑

ℓ∈L(n)

ρ

2(θkℓ,n − θn + ξkℓ,n)

2,

and then passes a message containing θk+1n and fk+1

n,ℓ to each neighboring edge ℓ ∈ L(n).

• J EP,kℓ : Each edge ℓ ∈ [L] performs edge primal update:

zk+1ℓ = argmin

zℓ

hℓ(zℓ) +ρ

2‖θℓ,N (ℓ) − θk+1

N (ℓ) + ξkℓ ‖22 +∑

n∈N (ℓ)

ρ

2(fk+1

n,ℓ − fℓ + ηkn,ℓ)2,

and then passes a message containing fk+1ℓ and θℓ,n to each neighboring node n ∈ N (ℓ).

• J ND,kn : Each node n ∈ [N ] performs node dual update:

ηk+1n = ηkn + fk+1

n,L(n) − fk+1L(n),

and passes a message containing ηk+1n,ℓ to each neighboring edge ℓ ∈ L(n).

• J ED,kℓ : Each edge ℓ ∈ [L] performs edge dual update:

ξk+1ℓ = ξkℓ + θk+1

ℓ,N (ℓ) − θk+1N (ℓ),

and passes a message containing ξk+1ℓ,n to each neighboring node n ∈ N (ℓ).

We summarize the convergence property of the iterates specified above, whose proof is

relegated to Appendix B.3.

Lemma 3.2. The iterates (xk, zk) produced by jobs J k =J NP,k[N ] ,J EP,k

[L] ,J ND,k[N ] ,J ED,k

[L]

are convergent. Let x⋆ , limk→∞ xk and z⋆ , limk→∞ zk. Then (x⋆, z⋆) is primal feasible

and achieves the minimum cost of problem (3.9). Furthermore, the rate of convergence is

O(1/k).

Remark 3.6. Minimum amount of assumptions are required to obtain the convergence

results given in Lemma 3.2. In particular, we do not assume the objective function is

strongly convex which is a necessary assumption for standard distributed algorithms based

on primal or dual decomposition. Furthermore, the rate of convergence for our algorithm

is superior to primal or dual decomposition based algorithms, which usually have a rate of

convergence O(1/√k).


Remark 3.7 (Asynchronous variant). Based on the analysis in [129], one can easily ex-

tend the algorithm described above to its asynchronous counterpart with similar convergence

guarantees.

3.4.2 Cluster-based Implementation

In a cluster-based distributed control environment, each CC is responsible for a subset of

resources in the grid. It is not necessary the case that there is a CC for each node and each

edge. However, issuing tasks defined for each node and edge to the associated CC would

implement our distributed algorithm in a cluster-based control environment. The iterates

now have the following form: in order, each CC d ∈ [D] (i) performs J NP,kn for all n ∈ Nd,

(ii) performs J EP,kℓ for all ℓ ∈ Ld, (iii) performs J ND,k

n for all n ∈ Nd, and (iv) performs

J ED,kℓ for all ℓ ∈ Ld. Note that if the source and destination of a message belong to different

CCs, instead of direct communications between the node-edge pair, the message is sent from

the CC containing the source to the CC containing the destination5; if a single CC controls

both the source and destination of a message, the corresponding messaging step may be

skipped.

3.5 Numerical Tests

In this section, we show three sets of numerical tests with different focuses. The first example

(Subsection 3.5.1) uses synthetic data that honor the i.i.d. assumption in Section 3.3.2 to

demonstrate the use of the online algorithm and to show how the sub-optimality bound scales

with storage parameters. The second example (Subsection 3.5.2) applies the algorithm on

IEEE 14 bus network together with real demand and wind data. The i.i.d. assumption no

longer holds in this setup. We also demonstrate the convergence of ADMM in this setting.

The last example (Subsection 3.5.3) is constructed in particular to show how the distributed

algorithm scales with the number of buses of the system. All examples are implemented

and tested using Matlab 2014a on a workstation with AMD Magny Cours 24-Core 2.1 GHz

CPU and 96GB RAM.

3.5.1 Star Network

Consider a star network, i.e., a tree with a root node and (N − 1) leaf nodes. Assuming a

homogeneous setting, all nodes are connected to identical power system components, and

thus we only provide specification for a single bus n. The storage network is operated for

the purpose of balancing the demand and supply residual due to forecast errors in the wind

5Recall the setup in Section 3.2.3: each CC d can communicate with its neighbors CNdwhere CNd

isthe collection of Cw’s for which there exists ℓ ∈ Ld, n ∈ Nw such that ℓ ∼ n, or there exists ℓ ∈ Lw,n ∈ Nd such that ℓ ∼ n. As all messaging tasks only involve incident node-edge pairs, the communicationbetween these CCs are possible.


power generation. The motivation of this setting in a single storage scenario is discussed in

detail in [113]. Let δn,t models the wind forecast error process for each bus n. We simulate

the δn,t processes by generating Laplace distributed random variables with zero mean and

standard deviation σδ = 0.149 p.u. as in [113], which are estimated empirically using the

NREL dataset. Two cases with different cost functions are considered. In the first case,

time homogeneous costs of the form

cn = (gn,t)−, (3.10)

are considered; in the second case, the cost function is modified to have a higher penalty

rate during the day

cn =

3 (gn,t)

−t ∈ T Day,

(gn,t)−, otherwise,

with T Day is the set of time points during the day (7am to 7pm in our tests). We consider

non-idealized storages which are operated frequently such that λn = 0.999 with conversion

coefficients being µCn = µD

n = 0.995 (cf., Remark 3.1) and −Uminn = Umax

n = (1/10)Smaxn .

We have N = 5 and Fmaxℓ = σδ for each line ℓ ∈ [L]. The time horizon for the simulation is

chosen to be T = 1000. Figure 3.1 shows the percentage cost savings compared to the no

storage scenario. Albeit the greedy heuristics have been proved to be the optimal solution

for single storage systems in the time homogeneous cost setting in [113], OMG outperforms

the greedy heuristics in the case with a network. The improvement over the greedy cost is

more significant for the time inhomogeneous case. For both cases, the costs of OMG are

close to the upper bounds estimated using the sub-optimality bounds of the algorithm.

3.5.2 IEEE 14 Bus Case

The network data from IEEE 14 bus test system [123] are used for this example, with

modifications described as follows. Three conventional power plants are connected to the

network, i.e., a coal power plant with capacity 500MW and (constant) marginal genera-

tion cost 50$/MWh connected to bus 1, a nuclear power plant with capacity 450MW and

marginal generation cost 25$/MWh connected to bus 2, and a natural gas power plant with

capacity 400MW and marginal generation cost 100$/MWh is connected to bus 8.6 A wind

power plant is connected to bus 3. Hourly data of wind power generation for January 2004

(Figure 3.2) are obtained from the NREL dataset [80], and are scaled to model a 30% pene-

tration scenario. The hourly load data are obtained from PJM interconnection for the same

period (Figure 3.2), and are scaled down and then factored out according to the portion of

different load buses. Three storages are connected to buses 6, 7 and 10. Their capacities are

Smax6 = 300MWh, Smax

7 = 240MWh, Smax10 = 300MWh, and charging/discharging power

6The labeling of the buses are consistent with [123]


0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

Total Smax in network

Percentagecost

savings(%

)

OMGGreedy

Upper bound

(a) Time homogeneous costs

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

Total Smax in network

Percentagecost

savings(%

)

OMGGreedy

Upper bound

(b) Time inhomogeneous costs

Figure 3.1: Percentage cost savings of a storage network operated for balancing.

ratings are Umax6 = Umax

7 = Umax10 = 10MW with Umin

n = −Umaxn for all n. For simplicity

(and in view of the fact that conversion to cluster based implementation is easy), we em-

ulate a complete distributed setting, where each node or each edge solves its own tasks in

the distributed ADMM algorithm.

The performance of OMG together with the greedy heuristic are simulated over T = 744

time periods (i.e., hourly for January 2004). We also compute the cost when there is no

storage in the system, and the offline clairvoyant optimal cost which corresponds to solving

the storage operation problem assuming the full knowledge of the future load and wind

ahead of time. For this example, the hourly average no storage cost is $51710. The costs

of the greedy heuristics, OMG, and offline optimal are 96.1%, 95.7% and 90.3% of the no

storage cost, respectively. Here the cost achieved by the offline optimal solution is a loose

lower bound as it requires information that is not available to the decision maker. The

stochastic lower bound, estimated by our algorithm under i.i.d. assumption is 94.6% of the

no storage cost. As the disturbances are not i.i.d., we expected the actual optimal cost to

lie between these two lower bounds.

The convergence of the fully distributed ADMM is shown in Figure 3.3. As a comparison,

we also plot the convergence of the projected subgradient method (SubGD). Figure 3.3a

shows the convergence of the objective values of the online program at a time period for

both algorithms with different algorithmic parameter choices, while Figure 3.3b depicts


the convergence of the norms of the primal residuals for the ADMM algorithm. For the

objective values, we observe that the convergence of ADMM is usually much faster compared

to SubGD. In fact, in all of our examples, SubGD does not converge after thousands of

iterations with the tolerance being 1 × 10−4. Comparing the performance of ADMM with

different parameter ρ’s, we note that smaller ρ leads to faster convergence of the objective

values but slower convergence of the primal residuals. Thus in practice, selecting a ρ that

properly trades off these two effects is necessary.

70

80

90

100

110

120

130

140

150

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour

Load(M

W)

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Hour

Wind(M

W)

Figure 3.2: Bar plots for scaled hourly total load (upper panel) and wind data (lower panel)used for the simulation.

3.5.3 Scalability

In this subsection, we give a preliminary account for the scalability of the distributed imple-

mentation using Matlab Distributed Computing Toolbox. Test results for larger networks

will be reported in the future. We consider star networks discussed in Subsection 3.5.1 with

the number of buses increasing from 2 to 16. We associate a processor to each of the buses,

and run the distributed ADMM using 2-16 processors. The running times of the distributed

ADMM implementation, together with the running times of solving the online programs

using the centralized ADMM algorithm, are shown in Figure 3.4. We note that while in

both scenarios, the running time increases approximately linearly with the number of buses,

the rate of the linear increase for distributed ADMM is significantly smaller. Loading the


0 100 200 300

0

0.5

1

1.5

2

2.5

3x 10

6

ADMM: ρ = 10

ADMM: ρ = 100

ADMM: ρ = 500

SubGD: ζk = 10−3/k

SubGD: ζk = 10−3/k0.5

SubGD: ζk = 10−3/k0.3

Objectivefunction

Iteration

(a) Objective value convergence

0 100 200 3000

20

40

60

80

100

120

ADMM: ρ = 10

ADMM: ρ = 100

ADMM: ρ = 500

Primalresidual

Iteration

(b) ADMM residual convergence

Figure 3.3: Convergence of ADMM and centralized subgradient method. Here ζk is the stepsize of the subgradient algorithm at the kth iteration.

data for problem specification and communication overheads may have contributed to the

linear running time increase for the distributed ADMM.

3.6 Conclusion and Future Directions

This chapter formulates the storage network operation problem as a stochastic control prob-

lem. An online algorithm is proposed to solve the problem efficiently. The performance of

the algorithm is analyzed and a sub-optimality bound is derived. The online programs

are then solved in a decentralized fashion with only local computation and neighborhood

communication with task-based ADMM iterations. Combining these elements, we obtain

an efficient task-based distributed online control strategy for operating distributed storage

systems under uncertainty with a guaranteed performance.

Many future directions are of interest for generalizing our results. (i) This chapter focuses

on the real power; incorporating the reactive power and a full AC power flow model may be

an important step towards a successful implementation in large-scale practical systems. As

the online optimization for each step becomes an AC optimal power flow (OPF) problem,

recent work on the convexification of such problems [68] [69] and on the distributed solution

of the convexified program [32] may be combined with the approach proposed in this chap-

ter. (ii) Our decentralized solution iroas based on the classical two block ADMM which has


ADMM with single processor

Fully distributed ADMM

Runningtime(second)

Number of nodes

15000

10000

5000

0

5 10 15

Figure 3.4: Running time of distributed and centralized ADMM

superior convergence properties compared to other popular methods for distributed opti-

mization such as primal or dual decomposition. Similar methods have been tested in much

larger networks for deterministic energy control problems [60]. However, the fact that such

an ADMM algorithm requires a two-block partition (corresponding to the node variables x

and edge variables z in Section 3.4.1) leads to the inconvenience that local copies of variables

controlled by the neighbors must be created. Multi-block variants of ADMM may eliminate

such need. However, the convergence is not guaranteed without additional assumptions

[28], [49], [45]. Validating these assumptions for specific storage control problem instances

may lead to simpler algorithm which has similar convergence properties. (iii) Utilizing the

sub-optimality bounds to assess the limit of the storage system for the purpose of storage

valuation and system design may also be of interest.

Chapter 4

Flexible Markets for Smart Grid

54

CHAPTER 4. FLEXIBLE MARKETS FOR SMART GRID 55

4.1 Introduction

The electric power grid increasingly relies on distributed and variable energy sources. In-

tegration of these new sources is helped by a market that facilitates matching intermittent

supply and flexible demand [104, 125]. Today the system operator (SO) achieves resource

adequacy, congestion management and efficiency through reserve requirements, day-ahead

(DA) and real-time (RT) markets, and centralized dispatch of standard energy commodities,

namely specified amount of energy delivered at specified nodes at fixed prices [59,111]. The

needs of important participants cannot be adequately expressed in terms of these standard

commodities, and so the SO allows bilateral contracts (e.g. Google [41], GM [2] and Ama-

zon [121]), with contractual arrangements that are not known to the SO. Over time, the

rigidity of the standard commodity was more broadly felt and fitful accommodations were

made by introducing new commodities, such as demand response, ramping, and capacity.

But given the legacy of the standard market this slow expansion of the SO’s responsibility

cannot unlock the full contractual flexibility that participants may wish. In particular, it

is challenging to repurpose today’s market design to serve the needs of distribution system

operators (DSOs) who must coordinate participants with small distributed generation and

controllable demand side devices, and who would benefit from differentiated micro-contracts

(i.e. contracts whose volumes are of the order of kWhs). Possible examples of such differen-

tiated contracts are (i) contracts for flexible amount of energy contingent on the realization

of uncertain supply or demand, (ii) contracts to serve deferrable loads that consume a fixed

amount of energy for (say) one hour but which could be scheduled for any hour of the day

[14,83], (iii) contracts that favorably price generation sources that are green or more flexible,

and (iv) contracts that encourage local sharing among prosumers with solar PV and storage

devices. Incorporating these differentiated contracts requires a significant deviation from an

electricity market with a small number of standard commodities.

In this chapter we propose a more flexible alternative to the current market design,

called coordinated multilateral trading. In this design, participants trade among themselves

according to terms and conditions fashioned to suit their own purposes like in today’s over

the counter markets (OTC), in contrast with exchanges for trading standardized commodi-

ties at transparent prices. These are contingent trades as the amount of energy delivered is

contingent on events or conditions specified in the contracts. Since the trades induce power

flow, they must be coordinated to ensure that network constraints are not violated. The SO

accomplishes this coordination task by curtailing trades if network constraints are violated,

and publishing information about the network state to guide participants regarding how

subsequent trades can avoid overloading congested lines. Thus the proposed market design

permits flexible contracts by allowing contingent trades while the SO maintains power sys-

tem security. In today’s design the SO computes an efficient dispatch that respects network

line constraints, but in the proposed design the SO is only concerned with reliability, and

the determination of an efficient dispatch is left to self-interested participants.


4.1.1 Contributions and Organization

Coordinated trading of contingent contracts (described in Section 4.3) is proposed as a flex-

ible market mechanism in the context of electric power transmission system operation. We

establish that the trading process is well-defined and during each step of the process, power

system reliability is guaranteed though the role of the SO is greatly simplified. Furthermore,

we show that the trading process converges to an efficient dispatch, which meets a bench-

mark defined using social welfare maximization as in the centralized stochastic economic

dispatch (Section 4.4). We also show that this trading process discovers the optimal loca-

tional marginal prices through the marginal costs of local participants (Section 4.5). Finally,

we prove that the dispatch and prices identified from the trading process support an Arrow-

Debreu equilibrium, a notion of competitive equilibrium under uncertainty (Section 4.6).

The trading process is illustrated with a simple two-bus example in Section 4.7.

4.1.2 Related Literature

In studies of the standard electricity market the basic framework is a one- or two-settlement

market (Day Ahead Market or DAM and Real Time Market or RTM) in a deterministic

setting [5,34,48,74]. In this framework, generators and load serving entities present supply

and demand function bids to the system operator (SO); the SO then calculates the equi-

librium as the generation and load schedule that maximizes social welfare (producer plus

consumer surplus), subject to the constraint that flows on transmission lines are limited by

their rated capacities. This centralized calculation has the form of a mathematical program-

ming problem called the optimum power flow problem. The dual variables at the optimum

solution are called locational or nodal marginal prices or LMPs. The LMP at a node is the

marginal cost of delivering additional power at that node. In a two-settlement market there

are day ahead and real time LMPs.

In a stochastic context, uncertainty is modeled by a probability distribution over a set of

scenarios. Each scenario has its specific supply and demand functions, and the SO finds the

schedule that maximizes expected social welfare. This schedule is contingent, since there is

a different schedule for each scenario [18,29,43,77,93,116]. The complexity of the stochastic

problem grows in three ways with the number of scenarios. First, each demand and supply

bid now is a function of prices and scenarios, so the number of decision variables and LMPs

will be multiplied by the number of scenarios, thereby increasing the SO’s communication

and computational burden. Second, there must be agreement among all participants about

the probability distribution over the scenarios, which precludes heterogeneous beliefs or

private information that can affect beliefs. Third, participants must work out in advance

the bids they will offer for each scenario and price vector. This complexity has precluded

real world implementation of the optimal stochastic power flow problem. In the absence

of contingent (stochastic) bids that permit risk mitigation and reduce volatility, stochastic


perturbations in demand and supply may lead to the large variations in LMPs that are

observed.

Two studies propose decentralized trading processes to replace the SO’s centralized cal-

culation. In [27], transmission rights are privately owned; the SO specifies “marginal loading

factors” that is, the amount of capacity on every transmission line that must be purchased

by every proposed bilateral transaction. Transmission prices are adjusted iteratively in

steps as follows. At any step nodal price differences adjust to eliminate arbitrage profits

from purchasing energy at one node and selling at another. Given nodal prices, transmission

prices then are adjusted to increase rents, subject to the competitiveness condition that the

transmission price for a line with excess capacity must be zero. The iteration converges in

the limit to the welfare maximizing solution, and the nodal prices converge to the LMPs.

Our proposed design is closer to the decentralized multilateral trading process in [134]

and generalizes their trading process developed for single period deterministic electricity

market into a setting with two periods and with uncertainty explicitly considered. In

the multilateral trading process, groups of buyers and sellers propose to the SO a balanced

trade, i.e. sum of buy bids equals sum of sell bids. The SO accepts the trade if (together

with previously accepted trades) no transmission line constraint is violated. Otherwise, the

SO curtails the proposed trade until the violation is eliminated. No price is announced.

The understanding is that the private terms and conditions of a trade (including monetary

payments) are acceptable to all parties. As in [27] the SO announces loading vectors to

guide participants towards trades that do not violate line constraints. It is shown that in

case generators are motivated by profit maximization and buyers by utility maximization,

the process will converge to a social welfare maximum.

Two important distinctions between these decentralized processes are worth noting.

First, in the language of mathematical programming, [27] describes a dual method, whereas

[134] gives a primal method. It is possible that at each step the iteration in [27] is infeasible

except in the limit, whereas each step in [134] is feasible and the process may be stopped

at any point. Second, even though it is decentralized, the process in [27] is synchronized:

trades in each step must occur at the same time; but in [134] trades are asynchronous.

4.2 Formulation

4.2.1 Notation

For a natural number N , [N ] denotes the set 1, . . . , N. Let x ∈ RI×J be a matrix, with

entries denoted by xi,j , i ∈ [I], j ∈ [J ]. We use xi ∈ RJ to denote the vector (xi,j)j∈[J] and

xj ∈ RI to denote the vector (xi,j)i∈[I].


4.2.2 Network Model

Consider a power network with N buses and L power lines with capacity constraints. We

utilize the same linearized DC approximation to AC power flow introduced in Section 2.2.2

so the region of feasible nodal injections can be written as

P :=p ∈ RN : Hp ≤ f , 1⊤p = 0

. (4.1)

The first inequality in (4.1) models the line capacity constraints, while the second equality

enforces power balance over the entire network. We will denote the rows of H , referred to

as loading vectors or shift factors, by h⊤ℓ ∈ R1×N . Throughout the chapter, we assume that

P has a non-empty interior.

4.2.3 Uncertainty Model

We consider the operation of the electricity market over two time periods, the DA market

and the RT market.

We explicitly model the RT uncertainty as a finite collection of S system scenarios, so

each scenario is indexed by s ∈ [S] with probability P(s) > 0. We assume that the set of

scenarios and the probabilities are known to all market participants and the SO and the

realization of a scenario is publicly verifiable by all of them. We could have the set of feasible

injections P depending on the scenario as well to model transmission line failures, in which

case H and f in (4.1) will be indexed by scenario s. We do not do this to simplify the

notation.

4.2.4 Participant Model

On each bus of the network n ∈ [N ], there resides a collection of electricity market partic-

ipants denoted by In, each of which is either an electricity producer or an electricity load.

We model each market participant by her (or his) RT power injection plan, denoted by

pi = (pi,s)s∈[S], her local feasible power injection sets, denoted by Pi,s such that pi,s ∈ Pi,s

for all i ∈ In and s ∈ [S], and her von NeumannMorgenstern utility function over such a

plan, denoted by Ui(pi) and taking the form of

Ui(pi) = E[ui,s(pi,s)] =∑

s∈[S]

P(s)ui,s(pi,s), (4.2)

where ui,s(pi,s) is the actual utility given scenario s. Throughout the chapter, we assume

a quasi-linear environment, so that the utility function is linear in the amount of monetary

payment of each market participant, i.e.,

ui,s(pi,s) = mi,s(pi,s) + ui,s(pi,s), (4.3)


where mi,s(pi,s) is the payment received by the participant in scenario s and ui,s(pi,s) is

the utility associated with power injection pi,s as discussed in detail below. We will assume

that the utility function ui,s(·) is concave for each i ∈ I and s ∈ [S].

For an electricity producer, the power injection is induced by the producer’s possibly

scenario-dependent electricity production so that pi ∈ RS+. The feasible power injection sets

model the generation limits, which could be scenario dependent in the case of renewable

generation. Thus, we have Pi,s = [0, pi,s], where pi,s is the maximum possible power output

in scenario s. The utility function is as defined in (4.2) and (4.3), with

ui,s(pi,s) = −ci,s(pi,s), (4.4)

where ci,s(·) is the cost function of the generation plant.

For an electricity load, the power injection is induced by the possibly scenario-dependent

electricity consumption so that pi ∈ RS−. Symmetrically with the producer, we have Pi,s =

[−pi,s, 0], where pi,s is the maximum possible power demand in scenario s. The utility

function is taken to be

ui,s(pi,s) = bi,s(pi,s), (4.5)

where bi,s(·) characterizes the benefit of using power by the particular load. For large

loads (e.g. resellers), the benefit corresponds to the profit made from the given power

consumption; for small loads such as individual consumers, the benefit function reflects the

monetary value of consuming electricity and is a widely used device for modeling how power

consumption varies with prices [58, 106, 108]. Allowing the benefit function to be scenario

dependent is useful for modeling e.g. demand response resources whose availability is not

known a priori.

We partition the set In as In = IDAn ∪ IRTn such that IDA

n ∩ IRTn = ∅ and denote

IDA = ∪n∈[N ]IDAn and IRT = ∪n∈[N ]IRTn , where IDA

n contains producers/loads connected

to bus n whose power injection has to be fixed in DA and cannot adapt to RT scenarios

and IRTn are those that can adapt to RT scenarios. We refer to participants in IDA as DA

participants and those in IRT as RT participants. Technically, the power injection plan of

DA participants must satisfy the non-anticipation constraint

pi,s = pi,s, for all s, s ∈ [S]. (4.6)

To simplify the notation, let

Pi = pi ∈ RS : pi,s ∈ Pi,s, s ∈ [S], (4.7)

and

Pi =

Pi ∩ pi ∈ RS : pi,s = pi,s, s, s ∈ [S], i ∈ IDA,

Pi, i ∈ IRT.(4.8)


Examples of DA participants include power plants that cannot ramp up or down following

the RT uncertainty, such as coal-based generation plants, and loads that contract a fixed

amount of consumption in each hour in DA. RT participants can either be variable generation

sources or demand modeling e.g. renewable generation or random power consumption, or

controllable generation or demand that can adapt to RT scenarios, such as fast-ramping gas

generation or demand response resources.

4.2.5 Efficiency Benchmark

A commonly used criteria for economic efficiency is Pareto optimality. In a quasi-linear

environment, it is equivalent to the following stochastic social welfare maximization problem

max U(p) :=∑

i∈I

Ui(pi) (4.9a)

s.t. pi ∈ Pi, i ∈ I, (4.9b)

xn,s =∑

i∈In

pi,s, n ∈ [N ], s ∈ [S], (4.9c)

xs ∈ P , s ∈ [S]. (4.9d)

Notice that when the system does not take money from outside sources, we must have ex

post budget adequacy: ∑

i∈I

mi,s(pi,s) ≤ 0. (4.10)

If ex post budget balance holds, i.e.,∑

i∈I mi,s(pi,s) = 0, then the social welfare maximization

program (4.9) is equivalent to the stochastic economic dispatch problem with the objective

of (4.9) replaced by ∑

i∈I

Ui(pi) =∑

i∈I

E [ui,s(pi,s)], (4.11)

where the summation is the net sum of ex ante generation costs and load benefits as discussed

in the previous subsection.

4.3 Trading Process

The simplest market mechanism is one based on meeting and trading among self-interested

agents. The electricity market is different in that centralized coordination has been com-

monly considered essential to ensure power system reliability constraints (4.1). Indeed,

completely decentralized trading without coordination could lead to line flows that violate

their capacity limits and compromise the reliability of the system. As such, the standard


power system market designs rely on a centralized clearing house (or market maker), re-

ferred to as system operator (SO), to solve an economic dispatch optimization in order to

determine the generator schedules and electricity prices. When uncertainty from renewable

generation is considered, the resulting stochastic economic dispatch problem is computa-

tionally more complex and leads to increased communication requirement between the SO

and market participants.

Wu and Varaiya [134] propose a remarkably simple fix to make the free-market style

meet-and-trade procedure respect the power system reliability constraints (4.1). The idea

is to inject minimal amount of coordination, implemented by the SO, into the free trades

so that the reliability (or feasibility) is guaranteed in every step of the trading process as

shown in Figure 4.1. They also establish that the trading process achieves economic effi-

ciency in the limit. We will generalize their coordinated trading framework developed for

single period deterministic electricity market into a setting with two periods and with un-

certainty explicitly considered. Although we consider only a two-period market (consisting

of a forward market, i.e. DA, and a delivery period, i.e. RT) below, the analysis readily

extends to settings with multiple delivery periods.

Participants propose

an admissible trade

SO checks

feasibility

SO accepts the trade

SO accepts the

curtailed trade

SO updates the system state and announces

trading requirements

Repeat until convergence

Yes

No

Figure 4.1: Conceptual diagram for the trading process.

In this chapter, we consider a setting where all market participants trade exclusively

in DA. This means that during the DA market, each DA participant i ∈ IDA trades and

determines her power injection while each RT participant i ∈ IRT trades and determines

her contingent power injection plan. Since under the current setting with a complete DA

forward market, there is no need for RT re-trading, i.e., additional RT trading can not


improve social welfare, we assume that there is no trading in real time.

We start with definitions for the trading process. Given the abstract nature of some of

the definitions, examples demonstrating them are provided in Section 4.7 and linked here

in footnotes.

The premise of our trading system is that self-interested market participants will meet

and propose trades for their own benefits, very much like how today’s bilateral power pur-

chase contracts are formed. Thus the fundamental building block of such a system is the

notion of trade:

Definition 4.1 (Contingent trade). A contingent multilateral trade (referred to as trade in

the sequel)1 among a group Ik ⊂ I of participants, is a collection of power injection plans

pk = (pki,s)i∈Ik, s∈[S], (4.12)

that are feasible with respect to participants’ local constraints, i.e. pi ∈ Pi, and ex post

balanced so that ∑

i∈Ik

pki,s = 0, s ∈ [S]. (4.13)

For convenience, we also define pki,s = 0 for i 6∈ Ik, so that given pk we can infer Ik via

Ik = i ∈ I : there exists a s ∈ [S] s.t. pki,s 6= 0. (4.14)

This definition is convenient from the point of view of the SO. In practice, a trade is

a transaction that exchanges power with money. We will touch upon the money side of

the trading process in Section 4.5 and 4.6. The power balance condition is natural: the

amounts of power supplied and consumed must be equal in each scenario. This definition

also stresses that the commodity for sale is scenario-contingent power. That is, 1 MWh in

different scenarios of RT are treated as different commodities.

Some further remarks are in order for Definition 4.1.

Remark 4.1 (Need for multilateral trades). As indicated in Definition 4.1, a trade may

involve more than two market participants. Although multilateral trades are less common in

practice compared with bilateral trades, for our purpose, it is necessary to consider multilat-

eral trades so that the trading process is guaranteed to converge to an efficient dispatch. See

[134] for an example in which bilateral trading fails to converge to the optimal dispatch due

to loop externality [48]. When the network does not have cycles, it is possible to show that

bilateral trades suffice under certain conditions. See Appendix C.4.

Remark 4.2 (SO’s sufficient statistics). While market participants must keep track of their

own power injection plans, from the power system’s perspective, contingent (network) nodal

1See Table 4.1 in Section 4.7 for an example.


injection vectors, calculated from

qkn,s =∑

i∈In

pki,s, n ∈ [N ], s ∈ [S], (4.15)

carry all the necessary information for checking the reliability constraints in (4.1). In par-

ticular, a trade among participants at the same node of network makes zero contribution to

the actual nodal injection and thus is not of concern to the SO.

Trades motivated by participants interests do not take into account power system re-

liability constraints. So it is necessary to have the SO verify that trades meet the power

system constraints, and in case of violation to curtail trades so that compliance is achieved.

Throughout the chapter, we consider a simple curtailment scheme:

Definition 4.2 (Uniform curtailment). A trade pk is said to be curtailed if only a portion of

the proposed power injection, γkpk, is accepted by the SO, where γk ∈ [0, 1) is the curtailment

factor and

(γkpk)i,s = γkpki,s, i ∈ Ik, s ∈ [S]. (4.16)

For notational convenience, we also define γk = 1 when a trade is accepted without curtail-

ment.2

Remark 4.3 (Scenario-dependent curtailment). The uniform curtailment scheme is the

simplest curtailment scheme that ensures local feasibility of curtailed trades given that the

initial trades satisfy local constraints. That is, given a trade pk such that pki ∈ Pi, i ∈ I,the curtailed trade always satisfies γkpki ∈ Pi, i ∈ I. It is possible to make the curtailment

scenario-dependent, i.e., for each scenario s ∈ [S], we can pick a different curtailment

factor γks ∈ [0, 1]. This curtailment scheme no longer has the local feasibility property if

DA participants are involved in the initial trade. In particular, the curtailed trade will not

satisfy non-anticipative constraints of DA participants if the curtailment factors for different

scenarios are taken to be different values. A hybrid of the uniform curtailment and scenario-

dependent curtailment is to use the former when a trade involves DA participants and to

use the latter when it does not. One can verify that all our results hold for this curtailment

scheme as well.

During the DA market time window, a sequence of trades will come up for SO’s approval.

Thus the notion of power system reliability and the calculation of curtailment depend on

trades that are already accepted into the system. We define a notion of system trading state

as follows:

Definition 4.3 (Trading state). Given a sequence of trades pκ and their curtailment factor

γκ, κ = 0, . . . , k − 1, the global trading state is the accumulated participants’ contingent

2Table 4.2 in Section 4.7 provides an example of a curtailed trade.


power injection

yki,s =

k−1∑

κ=0

γκpκi,s, i ∈ I, s ∈ [S], (4.17)

and the network state for the SO is the accumulated network power injection

xkn,s =

k−1∑

κ=0

γκqκn,s =

k−1∑

κ=0

∑

i∈In

γκpκi,s, n ∈ [N ], s ∈ [S]. (4.18)

The network and trading states relate as

xkn,s =∑

i∈In

yki,s. (4.19)

Given the current system state xk, a characterization for a trade pk to be feasible for

network constraints (4.1) is that its corresponding network injection vector qk as defined in

(4.15) satisfies

xks + qks ∈ P , s ∈ [S]. (4.20)

Define the scenario-contingent feasible set of network injection as

Qs(xs) = P − xs = qs ∈ RN : xs + qs ∈ P, s ∈ [S], (4.21)

and Q(x) = Qs(x1)× . . .×QS(xS). Then (4.20) is equivalent to qk ∈ Q(xk).A potential issue of the trading process, in view of Definition 4.2, is that γk may have to

be 0 to bring many trades back to feasible. Indeed, if the market participants are proposing

trades without any information regarding the current network state xk, then it is likely that

many trades overburdening lines which are already congested at xk will be proposed. To

forestall such a possibility, the SO requires participants to only submit trades that are in

the feasible direction of the network given the current state.

Definition 4.4 (Feasible direction trade). Given a network state xk, let Ls(xks ) be the set

of active (binding) line constraints in scenario s, that is,

Ls(xks ) = ℓ ∈ [L] : h⊤ℓ xks = fℓ, s ∈ [S]. (4.22)

Then a trade pk is a feasible-direction (FD) trade at xk if its corresponding network injection

qk as defined in (4.15) satisfies

h⊤ℓ qks ≤ 0, ℓ ∈ Ls(xks ), s ∈ [S]. (4.23)

If market participants are constrained to propose only FD trades, then it is guaranteed

that γk > 0 so that every trade updates the network state. At this moment, it is unclear

whether such update is favorable in any sense. Formalizing the notion of “self-interested”


participants, we have the following definition.

Definition 4.5 (Worthwhile trade). We call a trade pk an ǫ-worthy trade at trading state

yk if it leads to welfare improvement no smaller than ǫ, i.e.,

∑

i∈Ik

Ui(yki + pki )− Ui(y

ki ) ≥ ǫ, (4.24)

and an ǫ-unworthy trade if (4.24) does not hold. A profitable trade is an ǫ-worthy trade with

ǫ = 0.

Notice that if an ǫ-worthy trade is proposed by some participants and accepted by the SO,

then it improves the social welfare by at least ǫ as the power injection plans of participants

not involved in the trade are not changed.

We can now formalize the coordinated trading process.

Step 1. Initialization. The SO initializes the system state xk corresponding to some initial

feasible trade pk, k = 0.

Step 2. Announcement. The SO checks the congestion state of the system at xk, identifies

Ls(xks ) for s ∈ [S] and announces the network loading vectors hℓ, ℓ ∈ Ls(xks ) for

each s ∈ [S].

Step 3. Trading. If a profitable trade3 in the feasible direction pk is identified, participants

arrange it. If no profitable trade is found, go to Step 6.

Step 4. Curtailment. If pk is not feasible, i.e., the corresponding network injection qk is

such that qk 6∈ Q(xk), the SO curtails the trade with

γk = maxγ : γqk ∈ Q(xk)

∈ (0, 1). (4.25)

If pk is feasible, set γk = 1.

Step 5. Update. The SO updates the network state as xk+1 ← xk + γkqk, k ← k+1. Go to

Step 2.

Step 6. Termination.

It is evident from the description of the trading process that the SO only has the following

responsibilities. (i) SO checks whether the trade newly submitted by participants is feasible

with respect to network constraints. If not, it curtails the trade so that the resulting trade is

feasible. (ii) In case there are congested lines, the SO computes and broadcasts the loading

vectors to the participants. Note that in our framework, the SO does not carry out any

3We refer to the resulting trading process as an ǫ-trading process when the requirement of profitabletrade is replaced by ǫ-worthy trade.


optimization. Instead, market participants seek to optimize their own profit during the

trading process.

Remark 4.4 (Feasibility). An important feature of the trading process is that the proposed

system state xk for any k is feasible with respect to the power network constraints. Thus

even if the trading process is stopped at any stage before termination, the trades still result

in a safe power flow solution.

Remark 4.5 (Pay-as-bid settlement). The trading process allows a pay-as-bid payment

settlement approach. Immediately after submitting the trade, the market participants are

informed whether their trades will be scheduled (or partially scheduled if curtailed); this

information can then be used to calculate and settle the payment among these participants.

Comparing to the locational pricing used in the standard market, such a payment settlement

process could limit the price risk faced by market participants which are expected to increase

when the system integrates more renewables. This is also part of the reason why bilateral

long-term contracts are widely used by large utility companies and power producers.

4.4 Economic Efficiency

Similar to arguments in [134], one can verify that trading process described in the previous

subsection is well-defined, and whenever a ǫ-worthy trade is identified (ǫ > 0), the social

welfare is strictly increased (even if the trade is to be curtailed). Thus when the trading

process terminates, that is, when there exists no additional profitable trade that is not yet

arranged, one may expect that the resulting power injection plan matches the economic

efficiency benchmark defined by stochastic optimization problem (4.9).

Theorem 4.1 (Efficiency). Suppose the following assumptions are in force:

(i) for any fixed ǫ > 0, any ǫ-unworthy trade in the feasible direction will not be arranged

and any ǫ-worthy trade will eventually be identified and arranged, and

(ii) once a worthy profitable trade is identified, the market participants involved are willing

to carry it out.

Then the ǫ-trading process is well-defined and the accumulated global trading state yk con-

verges in the sense that

U⋆ − limk→∞

U(yk) ≤ ǫ, (4.26)

where U⋆ is the optimal value of (4.9).

Proof. See Appendix C.1.

When ǫ is sufficiently small, Theorem 4.1 states that the trading process will converge

to a dispatch that is practically optimal for any desired accuracy. The key message of

Theorem 4.1 is that the extremely simple feedback procedure of the SO based on curtailment


and loading vector announcement suffices in providing coordination for the trades so that

efficiency is achieved while network reliability is guaranteed in every iteration of the trading

process.

Remark 4.6 (Trade formation). Like in [134], we purposely leave the details of trading

group formation open. Theorem 4.1 is powerful in that it is agnostic to the actual underlying

mechanism dictating which subset of participants meet and propose trade k. For instance,

a conceptually simple mechanism is that in every iteration k, a subset of I is picked at

random such that there is a positive probability for picking every subset4. If it is possible

for this group of participants to identify a profitable trade in the feasible direction, they will

propose it, as in Step 3 of the trading process. If not, we can simply continue this process by

generating another random group of participants. Since there is a finite number of subsets of

I, Theorem 4.1 guarantees that this process converges to an efficient dispatch with probability

one. In practice, trading group formation processes depend on a lot of factors that we do

not model in this chapter. As a result, it could be the case that each participant i ∈ Imay only have access to a small subset of other participants in the market. An important

future research direction is to design information platforms that facilitate trade discovery

and reduce search cost.

Remark 4.7 (Profit allocation). Similarly, we do not specify how profit is allocated among

the participants if a profitable trade is proposed and accepted by the SO. One can verify (or

cf. [134]) that for every profitable trade, there is a profit allocation that makes all involving

participants better off.

Remark 4.8 (Merchandising surplus). In the standard market, the total payment collected

from loads is larger than that paid to generators when there are line congestions. This

merchandising surplus [135] is paid to transmission owners. In our setting, as the SO does

not collect money from participants and all trades are budget balanced, separate payment

streams might be needed to cover the costs of the transmission owners. Possible ways include

charging a fee for using the transmission or requiring participants to acquire transmission

rights for making trades across the network.

Remark 4.9 (Algorithmic interpretation). The trading process may be thought of as a

projected line search algorithm for solving (4.9), whose iteration k performs update

yk+1 = yk + γkpk, (4.27)

where pk is the search direction and γk is the step length introduced to project the step into

the feasible region. The algorithm is distributed, in that the search direction is identified

based on information (and objective functions) of a subset of participants. The algorithm is

4We can e.g. first sample a random group size Ik from [|I|] and then randomly sample a group from Iof size Ik.


special as its search direction pk is identified from a profitable trade, which is an economic

construct, rather than based on gradient or Hessian of the objective function.

Remark 4.10 (Subjective probability). In general, different market participants may have

their own subjective assessment of the probabilities of the scenarios. Denote the subjective

probabilities of participant i ∈ I by Pi(s), s ∈ [S]. Then yk converges to an optimal solution

of (4.9) with the ex ante utility function replaced by

Ui(pi) = Ei[ui,s(pi,s)] =∑

s∈[S]

Pi(s)ui,s(pi,s). (4.28)

In this case, the resulting dispatch is Pareto optimal but may not maximize the ex ante social

welfare as the latter notion is defined upon the unknown true probability distribution of the

uncertainty.

Remark 4.11 (Distribution system operator). The trading process also offers a way to

design a lightweight or minimal distribution system operator (minDSO) for coordinating

distributed generation (DG), flexible loads and other distribution level resources. With

minDSO, the DG owners and demand side flexibility providers do not need to report cost

and benefit data to the minDSO; so long as they can determine profitable trades among

themselves, social welfare will improve. To adapt our formulation to the distribution system

setting, the linearized DistFlow model[6,7] provides an accurate model of the real power flow

on the distribution network. Line capacity and transformer limits can be modeled similarly

as transmission line limits. The tree network topology offers potential simplification to the

trading process (see Appendix C.4). Voltage constraints can be modeled as additional lin-

ear constraints [39]. Distribution topology switching can be accommodated by updating the

network constraint set P according to the current switch states.

4.5 Price Discovery

In standard markets, the SO solves an economic dispatch problem that determines both

the dispatch and the locational marginal prices of power at all buses. When uncertainty is

considered, the computationally demanding stochastic economic dispatch must be solved by

the SO.

The trading process, on the contrary, does not require the SO to solve any optimization

problem. Theorem 4.1 suggests that an efficient dispatch is achieved in the limit; here

we show that the optimal locational marginal prices also emerge when the trading process

converges5. The idea is simple. Suppose that in the last few minutes of the DA trading

window, when the trading process has already converged, a new load comes into the system

5An alternative treatment, involving setting up trade-based prices and characterizing the convergence ofthe price process, is also possible (cf. [27, 71]). However, this requires a detailed specification of paymentsassociated with each trade which we avoid in this chapter.


and demands a ǫ → 0 unit of power at bus n for scenario s. Producers who can still

generate additional power could each quote a price based on their marginal cost evaluated

at the current trading state. We thus discover the locational marginal price at bus n for

scenario s by finding the minimum price announced by those generators that can indeed

send power to bus n given the congestion state in the scenario.

To formalize this idea, denote the optimal dual variable associated with constraint (4.9c)

by λ⋆n,s, n ∈ [N ], s ∈ [S]. Furthermore, as constraint (4.9b) for RT participants is a box

constraint, denote the optimal dual variables associated with the lower and upper bounds

by η⋆i,s

and η⋆i,s, respectively. Notice that the trading process has a balanced budget by

construction, as the system operator is not involved in any financial aspect of the system.

Therefore, problem (4.9) is equivalent to the stochastic economic dispatch problem.

Lemma 4.1 (Price discovery). For each bus n ∈ [N ] and s ∈ [S], if there exists a participant

i ∈ IRTn whose utility function is differentiable and whose optimal contingent power injection

p⋆i,s is in the interior of her local feasible set, i.e., p⋆i,s ∈ Pi,s, then we have6

λ⋆n,s = −P(s)∂ui,s(p

⋆i,s)

∂pi,s. (4.29)

In general, suppose the utility function of some participant i ∈ IRTn is differentiable, then

λ⋆n,s = −P(s)∂ui,s(p

⋆i,s)

∂pi,s+ (η⋆i,s − η⋆i,s). (4.30)


While the price calculation based on (4.29) is intuitive and only requires local informa-

tion, that based on (4.30) may require solving the dual program of (4.9) to identify the

values of the optimal dual variables η⋆i,s and η⋆i,s. Fortunately, solving for the dual program

is greatly simplified when the optimal primal solution p⋆ is known (cf. [22]).

4.6 Arrow-Debreu Equilibrium

Section 4.4 established that the trading process converges to a stationary contingent power

injection plan p⋆. Section 4.5 then showed that there is a well-defined notion of price λ⋆

that emerges alongside with the stationary injection plan. Here we connect the pair (p⋆, λ⋆)

to the suitable economic concept of general equilibrium under uncertainty. Taken together,

this will formally establish that the contingent trading process converges to a (contingent

plan, price) equilibrium, which respects the power system reliability constraints and achieves

economic efficiency.

6The sign convention is that pi,s > 0 represents power injection (supply) into the network. Thus if ui,s

is a utility function that in the usual sense is increasing with demand, λ⋆n,s as computed below will be

nonnegative.


To start, we need to define an electricity market economy, similar to that done in [18]

(also see [128]). The commodities of the economy is contingent power at each node n ∈ [N ]

and in each scenario s ∈ [S]. Buying (selling) a unit contingent power (n, s) in DA leads

to the right to consume (responsibility to generate) a unit of power at node n if scenario s

occurs in RT.

The market participants are those in I as defined in Section 4.2.4 and a traditional

system operator7 who may convert power at one node into that at another node using the

network.

For each participant i ∈ I, given prices for contingent power λ, the following optimization

is solved to determine the participant’s contingent power injection plan

max∑

s∈[S]

λn,spi,s + P(s)ui,s(pi,s) (4.31a)

s.t. pi ∈ Pi, (4.31b)

where λn,s is the price at node n faced by i ∈ In. Here the objective function is the same

as Ui(pi) as defined in (4.2) with linear payment scheme

mi,s(pi,s) = λn,spi,s, (4.32)

where λn,s = λn,s/P(s). Notice that the first term in the summation in (4.31a) is the

monetary payment that clears in DA; the second term is the expected utility derived from

the power injection in RT.

The SO is modeled as a firm that uses technology (i.e. power network) to convert one

type of commodity (i.e. contingent power on one node) to other types of commodities (i.e.

contingent power on other nodes), in order to maximize its profit. Formally, the SO solves

the following optimization to determine the contingent network power injection x given

prices for contingent power λ:

max∑

s∈[S]

∑

n∈[N ]

−λn,sxn,s (4.33a)

s.t. xs ∈ P , s ∈ [S], (4.33b)

where the entire profit of SO in (4.33a) is cleared in DA.

The suitable notion of competitive equilibrium of such a market for contingent claims

is that of Arrow-Debreu, which generalizes the Walrasian concept of general equilibrium to

settings with uncertainty [72]. Here we state the definition of Arrow-Debreu equilibrium for

the electricity market economy:

7This notion of SO is consistent with that in the literature and different from the SO described inSection 4.3.


Definition 4.6 (Arrow-Debreu equilibrium). A collection of contingent power injection

plans (p⋆, x⋆), with p⋆ ∈ R|I|×S and x⋆ ∈ RN×S and a system of prices for contingent power

λ⋆ ∈ RN×S constitute an Arrow-Debreu equilibrium if:

(i) For every i ∈ I, p⋆i solves (4.31) given prices λ⋆.

(ii) For the SO, x⋆ solves (4.33) given prices λ⋆.

(iii) The market for each contingent power commodity clears:

x⋆n,s =∑

i∈In

p⋆i,s, n ∈ [N ], s ∈ [S]. (4.34)

By the first fundamental theorem of welfare economics and in a quasi-linear environment,

we expect that a dispatch-price tuple (p⋆, x⋆, λ⋆) at an Arrow-Debreu equilibrium achieves

economic efficiency defined by (4.9) (cf. [10, 72]). Our previous result suggests that the

dispatch at the limit of the trading process together with the emerged prices matches the

solution of (4.9). Our next result establishes that the dispatch-price tuple obtained from

the trading process indeed constitutes an Arrow-Debreu equilibrium.

Lemma 4.2. Suppose that an Arrow-Debreu equilibrium exists and that the utility functions

are differentiable. Then the contingent power injection plan p⋆ obtained from the trading

process, the corresponding network injection plan x⋆ calculated from (4.9c), and the prices

computed from (4.29) or (4.30) constitute an Arrow-Debreu equilibrium.


Remark 4.12 (Tatonnement process). In light of Lemma 4.2, the trading process can

be thought of as a way to drive an out-of-equilibrium market into its equilibrium. Such

processes, characterizing the dynamic laws of out-of-equilibrium movement of the market

state, is in general referred to as a tatonnement process; see [44].

4.7 Examples

We provide an illustrative example for the trading process in this section.

Consider a two bus network depicted in Figure 4.2. There are 3 generators and 1 load

connected to the system. We list the relevant data for the participants as follows

• G1 is a coal power plant that can generate up to 200 MW at a constant marginal cost

50 $/MW. It can only be scheduled in the DA stage due to its lead time.

• G2 is a wind farm that generates 100 MW in the first scenario (windy scenario) and

50 MW in the second scenario (breezy scenario) at no operational cost. Suppose there

are only these two scenarios for the system and one of them is realized at the delivery

time. The underlying probabilities for these two scenarios are 0.6 and 0.4, respectively.


• G3 is a gas power plant that can ramp up rapidly at real time with 100 MW capacity

and constant marginal cost of 80 $/MW.

• Load represents an inelastic power consumption of 150 MW.

G1

Load

Limit = 120

G2

G3

p1

p2

p3

p4

Figure 4.2: Network diagram for the two-bus example.

For the purpose of illustrating the interaction between market participants and the PSO,

in this example we assume that in each iteration all four participants meet and propose a

trade with no knowledge of the network constraint. In DA, as an example, the participants

could solve the following optimization problem to identify the cost minimization trade

minimize 50p1 + E [80p3,s] = 50p1 + 48p3,1 + 32p3,2

subject to p1 + p2,s + p3,s + p4 = 0, s = 1, 2,

p4 = −150,0 ≤ p1 ≤ 200,

0 ≤ p3,s ≤ 100, s = 1, 2,

0 ≤ p2,1 ≤ 100, 0 ≤ p2,2 ≤ 50,

where the optimization variables are the day-ahead scheduled coal power generation p1, the

real-time gas power generation p3,s corresponding to the two scenarios, and wind power

generation corresponding to the two scenarios p2,s (which is controllable up to curtailment).

Upon solving this linear program, the participants propose its solution as their initial trade

to the SO, which is shown in Table 4.1.

Table 4.1: Power injection (unit: MW) of the initial trade proposed by the particpants.

Scenario G1 G2 G3 Load

Windy 50 100 0 -150Breezy 50 50 50 -150

This trade is not feasible with respect to the line limit in the windy scenario. As such,

the SO curtails the trade to the one shown in Table 4.2 with γ = 0.8. The SO also announces

the loading vector such that the constraints (4.23) can be expressed as ∆p1+∆p2,s−∆p3,s−∆p4 ≤ 0, where ∆p’s are the corresponding changes in the power injections, and s = 1, 2


as the line limit constraint is binding for both scenarios. The participants then solve the

following program to identify a profitable trade in the feasible direction:

Table 4.2: Power injection (unit: MW) of the curtailed trade.


Windy 40 80 0 -120Breezy 40 40 40 -120

minimize 50(p1+∆p1)+48(p3,1+∆p3,1)+32(p3,2+∆p3,2)

subject to ∆p1 +∆p2,s +∆p3,s +∆p4 = 0, s = 1, 2,

∆p1 +∆p2,s −∆p3,s −∆p4 ≤ 0, s = 1, 2,

p4 +∆p4 = −150,0 ≤ p1 +∆p1 ≤ 200,

0 ≤ p3,s +∆p3,s ≤ 100, s = 1, 2,

0 ≤ p2,1+∆p2,1 ≤ 100, 0 ≤ p2,2+∆p2,2 ≤ 50,

where the p’s are the curtailed trade given in Table 4.2. The resulting accumulated trade

γp + ∆p is shown in Table 4.3. The trading process would terminate now as there is no

profitable feasible direction trade can be further identified. One can easily verify that the

accumulated trade coincides with the solution to (4.9) for this example, and therefore the

trading process indeed achieves efficiency.

Table 4.3: Power injection (unit: MW) of the accumulated trade γp+∆p.


Windy 20 100 30 -150Breezy 20 50 80 -150

4.8 Concluding remarks and open questions

Contingent coordinated trading is proposed as a market framework for power system re-

source allocation under uncertainty. Within the framework, the economic efficiency is

achieved via coordinated trades proposed by any groups of market participants for their

own benefit. The trading process also discovers the optimal contingent locational marginal

prices, and supports an Arrow-Debreu equilibrium of the market. Allowing the trades to be

contingent on properly defined system scenarios greatly enhances the flexibility of the trades

and could result in an improvement of social welfare compared to standard deterministic

dispatch based market clearing. The role of the SO is minimal in our framework as the SO


only monitors the trades, curtails them if necessary, and does not collect any cost data or

directly intervene in economic decisions. As such, all suppliers and consumers have open

access to the power network, which promotes competition and expedites the processes of

new generation and consumer-side technology adoption.

We envision that the proposed framework could help address many challenges in design-

ing new DSOs for distribution systems with deep distributed energy resource penetration.

Given the novelty of the proposed framework, it is natural that this chapter leaves a variety

of fundamental questions open.

• Uncertainty model. In practice, it is unlikely that we can obtain an exact charac-

terization of all possible scenarios for the entire system in DA. Thus extending our

ideal uncertainty model by incorporating information updates could make the trading

framework more realistic. Under such settings, it may become advantageous to allow

RT re-trading as the realized RT scenario may not be exactly one of the pre-scribed

scenarios in DA. Additionally, even if it is possible to characterize the set of all pos-

sible scenarios, the total number of scenarios may be very large due to the fact that

many scenarios are local (see Appendix C.5 for a model where all scenarios are local).

Thus in practice, suitable factorization (decomposing the scenario tree into system

wide scenarios and local scenarios) or scenario reduction is necessary for successful

market design based on the proposed trading framework.

• Trade implementation. As all trades happen before real time, in real time, the partic-

ipants need to supply and consume according to the scheduled trades. To ensure this

indeed happens, advance metering infrastructure (AMI) systems and suitable financial

incentive (or penalty) scheme have to be in place. Thus an open question is how to

design such financial schemes that encourage consistent participant behaviors while

limits potential gaming activities.

• Trade formation. For distribution system applications, requiring participants to meet

and trade seems overwhelming. A more likely setting is to rely on one or many third-

party marketplaces to identify profitable trades on behalf of (subsets of) the partic-

ipants. Our analysis also applies to such settings thanks to our general assumption

on the trade formation process. In this context, our results are better understood as

a form of separation principle, which ensures lossless separation of network reliability

from market efficiency considerations with our trading framework. Under such sepa-

ration, third party marketplaces can fill the role of trade identification and formation

without any explicit knowledge of the power network, as long as it follows the rules

set by the SO. Designing and implementing such third-party marketplaces to unlock

potentials from distributed energy resources and flexible loads thus is an important

future direction to explore.

Chapter 5

Conclusions

In this thesis, we first analyze a simple greedy strategy for the planning problem of placing

energy storage in power network. Using structural characterizations of the underlying power

network control problem, we identify conditions under which the placement value function

is submodular so that the greedy strategy has a performance guarantee. We then develop

a computational procedure to certify these conditions for any given problem instance based

on multi-parametric programming.

In the second part of the thesis, we consider the stochastic control problem for operating

energy storage devices connected in a power network. As the exact solution of the problem

based on dynamic programming suffers from the curse of dimensionality, we propose a

simple online algorithm for the problem utilizing a stabilized greedy (myopic) controller.

For a rather general setting, we establish performance guarantees for the proposed method.

Finally, we study fundamental requirements for power network reliability in designing

novel power markets to integrate DERs. We demonstrate a transaction or trading based

market, on top of a system operator implementing these reliability requirements, could

achieve the same efficiency as centralized dispatch. We also obtain structural results for

radial networks which indicate efficient market outcome can be reached with bilateral trading

for distribution networks.

75

Appendix A

Appendices of Chapter 2

76

APPENDIX A. APPENDICES OF CHAPTER 2 77

A.1 Expression of the Shift Factor Matrix

For the directed graph G(N ,L) representing the power network, the node-edge incidence

matrix A ∈ RN×L is defined as

An,ℓ =

1 if ℓ→ n,

−1 if ℓ← n,

0 otherwise,

where ℓ → n denotes that n is the head of ℓ, and ℓ ← n denotes that n is the tail of ℓ.

Under the classical DC approximation to the steady-state AC power flow [112], the lines are

characterized by their reactance and real power flow capacity f ∈ RL. Let Y ∈ RN×N be

the network admittance matrix, which can be represented as

Y = A∆yA⊤,

where ∆y ∈ RL×L is the diagonal matrix with the ℓth diagonal element being yℓ > 0 which

is the reciprocal of the reactances of the line. Note that rank(Y ) = N − 1. Taking bus 1 to

be the reference bus, we let Y ∈ R(N−1)×(N−1) be the sub-matrix of Y which contains all

the entries of Y except its first row and first column. Let

Y † ,

[0 0

0 Y −1

]

be the constrained generalized inverse of Y . For each time period t = 1, . . . , T , we can then

relate the line flows ft ∈ RL with the nodal power injection pt ∈ RN using a linear map

H ∈ RL×N :

ft = Hpt, with H , ∆yA⊤Y †.

Then we have

H ,

[I

−I

]H ∈ R2L×N , f ,

[f

f

]∈ R2L,

and constraint Hpt ≤ f captures the flow constraints for the power network.

A.2 Proof of Proposition 2.1

Proof. As the objective function of (2.11) is strongly convex in λ, we know that λ⋆(x)

must be unique. Suppose that the first bus is the reference bus of the network, then by

the definition of the shift-factor matrix, H11 = 0. Thus constraint (2.11b) implies that

γ⋆t (x) = λ⋆1,t(x), and therefore γ⋆(x) is also unique. Under Assumption 2.1, the set of

primal flow constraints (2.4b) that are binding at the optimal solution is given by those


corresponding to Ht. That is, βt can be partitioned into βt for the binding constraints and

β′t for the slack constraints for which we know that β′

t = 0 by complementary slackness. In

fact, using this decomposition, the dual constraint (2.11b) can be written as

λt = γt1−Hnett

⊤βt, t ∈ T .

Now as Hnett is a full row rank matrix, βt is uniquely determined by the equation above

given fixed λt and γt. This implies that βt is also unique.

A.3 Proof of Lemma 2.1

Proof. Consider the primal program (2.4), which has an infinitely differentiable objective

function and linear constraints. Under the non-degeneracy condition, we can apply standard

sensitivity theorem of nonlinear programming [70], which suggests the differentiability of

J(x) and that

∂J(x)

∂xn= −

T∑

t=1

µ⋆n,t = −

T∑

t=1

(λ⋆n,t+1(x) − λ⋆n,t(x)

)+,

for any i ∈ N . To show that ∂J(x)/∂xn itself is again a continuous function, we observe that

λ⋆(x) is the unique solution of the dual QP (2.7). By the smoothness of the objective and

constraints of (2.7), we know that the parameter to solution mapping λ⋆(x) is continuous

in x. Furthermore, the positive part function (·)+ is a continuous function from R to R+.

Therefore, we conclude that ∂J(x)/∂xn is continuous and J(x) is continuously differentiable.

As ∂J(x)/∂xn ≤ 0, the function J(x) is nonincreasing in xn for each i ∈ N .


Proof. We write the primal problem (2.4) as

J(x) = ming

T∑

t=1

Ct(gt) + ω(g, x),

where extended real-valued function ω(g, x) is defined to be 0 if, given (g, x), there exists a

control u satisfying all the constraints of (2.4), and +∞ otherwise. Let x1, x2 ∈ Rn+ be two

arbitrary vectors of storage capacities, and let (g1, u1) and (g2, u2) be the optimal primal

solutions associated with x1 and x2 respectively. We claim that the function ω(g, x) is convex

in (g, x). Indeed, it is easy to verify that, for ρ ∈ [0, 1], ω(ρg1+(1−ρ)g2, ρx1+(1−ρ)x2) = 0

if ω(gi, xi) = 0 for i = 1, 2, as ρu1 +(1− ρ)u2 is a feasible solution for the set of constraints

given (ρg1 + (1 − ρ)g2, ρx1 + (1 − ρ)x2). Therefore, J(x) is convex as it is the minimum

value of another convex function optimized in g over a convex set.


A.5 Proof of Theorem 2.1 and Corollary 2.1

Proof. By strict complementary slackness,

βℓ,t = 0 if ℓ 6∈ LCt and βℓ,t > 0 if ℓ ∈ LCt .

Thus, we can focus on the reduced dual variable

βt , znett βt ∈ RLt , t = 1, . . . , T.

For convenience, we denote

ξn,t =(zstn,t

)+, ηn,t = 1zstn,t = 0, n ∈ N , t ∈ T .

Then we can consider the following reduced form of dual program (2.7) in a neighborhood

of the given storage capacity vector x

maxλ,γ,β

ψ(λ, γ, β) (A.1)

s.t. λt = γt1−Hnett

⊤βt, t ∈ T , (A.2)

F (ηt)(λt+1 − λt) = 0, t ∈ T , (A.3)

where the objective function is

ψ(λ, γ, β) ,T∑

t=1

− 1

2(λt − rt)⊤Q−1

t (λt − rt) + d⊤t λt − f⊤t βt − x⊤∆ξt(λt+1 − λt),

ft , znett f , and matrix F (ηt) ∈ Rmt×N , (mt =∑

n∈N ηn,t), is formed by removing zero

rows from diagonal matrix ∆ηt. We can vectorize the variables and coefficients by denoting

λ =

λ1...

λT

, β =

β1...

βT

, γ =

γ1...

γT

, r =

r1...

rT

, d =

d1...

dT

, f =

f1...

fT

,

and define matrices

Q = diag(Q1, . . . , QT ) ∈ RNT×NT ,

G = diag(1, . . . ,1) ∈ RNT×T ,

H = diag(Hnet1 , . . . , Hnet

T ) ∈ R(∑

t∈TLt)×NT ,


F =

−F (η1) F (η1)

−F (η2) F (η2)

. . .. . .

−F (ηT−1) F (ηT−1)

−F (ηT )

∈ R(∑

t∈Tmt)×NT .

Then the optimization can be written as

maxλ,γ,β

− 1

2(λ− r)⊤Q−1(λ − r) + κ⊤λ − f⊤β

s.t. λ = Gγ −H⊤β,

Fλ = 0,

where

κ = d− Y x,

and

Y =

−∆ξ1

∆ξ1 −∆ξ2

...

∆ξT−1 −∆ξT

.

We can reduce the variable λ using constraint λ = Gγ −H⊤β, as a result, we get

maxγ,β

− 1

2(Gγ −H⊤β − r)⊤Q−1(Gγ −H⊤β − r) + κ⊤(Gγ −H⊤β)− f⊤β

s.t. F (Gγ −H⊤β) = 0.

Converting this to the standard form of QP, we have

minγ,β

1

2

[γ

β

]⊤ [G⊤Q−1G G⊤Q−1H⊤

HQ−1G HQ−1H⊤

]

︸︷︷︸,A

[γ

β

]−[

G⊤κ+G⊤Q−1r

−Hκ−HQ−1r − f

]

︸︷︷︸,B

⊤ [γ

β

]+

1

2r⊤Q−1r

s.t.[FG −FH⊤

]

︸︷︷︸,C

[γ

β

]= 0.

This is a standard equality constrained QP. Given the uniqueness of the prices (Proposi-

tion 2.1), we can use the formula for the solution of standard equality constrained QP and

conclude that [γ

β

]=

(A−1C⊤(CA−1C⊤)−1)CA−1 −A−1

)B.


Substituting this expression into λ = Gγ −H⊤β, we get

λ =[G −H⊤

] (A−1C⊤(CA−1C⊤)−1)CA−1 −A−1

)B.

Notice that as B is affine in κ, which is an affine function of the storage capacities x, the

resulting prices γ, β and λ are affine functions of the storage capacities. Substituting the

definition of B and κ, we obtain expression for Wt(znet, zst), λt(z

net, zst), Bt(znet, zst) and

βt(znet, zst), for t ∈ T .

A.6 Proof of Theorem 2.2

Proof. We can compute the optimal value J by substituting the expression of the optimal

γ and β, as a result

J(x) =1

2B⊤

(A−1 −A−1C⊤(CA−1C⊤)−1)CA−1

)B − 1

2r⊤Q−1r.

Recognizing that

B =

[G⊤

−H

]κ+

[G⊤Q−1r

−HQ−1r − f

]=

[G⊤

−H

](d − Y x) +

[G⊤Q−1r

−HQ−1r − f

],

we obtain the Hessian for critical region containing x being

∇xxJ(x) = Y ⊤[G −H⊤

] (A−1 −A−1C⊤(CA−1C⊤)−1)CA−1

)[G⊤

−H

]Y .


Proof. For any X ⊆ Ω, and without loss of generality ei , (i, ki) 6∈ X , ej , (j, kj) 6∈ X , we

have DeiV (X) = V (X ∪ (i, ki))− V (X) and

Dej(Dei

V (X)) = [V (X ∪ (i, ki), (j, kj))− V (X ∪ (j, kj))]− [V (X ∪ (i, ki))− V (X)].

Let x0 , I(X)x. Using the definition of V , we have

Dej(Dei

V (X)) =[J(x0 + xj1j)− J(x0 + xi1i + xj1j)

]− [J(x0)− J(x0 + xi1i)].

Since J(x) is continuously differentiable, the following integral expression is well-defined:

Dej(Dei

V (X)) =

∫ xi

0

[∂J

∂xi(x0 + ξ1i)−

∂J

∂xi(x0 + xj1j + ξ1i)

]dξ.


Meanwhile, given that ∂J/∂xi is differentiable almost everywhere with respect to Lebesgue

measure, we have

Dej(Dei

V (X)) = −∫ xi

0

∫ xj

0

∂2J

∂xj∂xi(x0 + ξ1i + ζ1j) dζ dξ.

As(∇2

xxJ(x))ij≥ 0, we have Dej

(DeiV (X)) ≤ 0 for any i, j ∈ N and any ki and kj . Thus,

using (2.20), we conclude that the set function V is submodular.


Proof. By the proof of Theorem 2.1 and Corollary 2.1, we know that (λ⋆(x), γ⋆(x), β⋆(x)) is

a stationary point of the objective of the dual QP (A.1). In other words, (λ⋆(x), γ⋆(x), β⋆(x))

is the unconstrained local maximizer of (A.1) in an affine subspace defined by the set of

equality and inequality constraints in (A.1) which are binding at x. Recall that, given the

storage congestion state zst ∈ RN×T , the objective function of (A.1) can be written as

φ(λ, β) =

T∑

t=1

−1

2(λt − rt)⊤Q−1

t (λt − rt) + d⊤t λt − f⊤βt − x⊤∆ξt(λt+1 − λt).

Now consider a vector x that satisfies the inequality conditions in Theorem 2.4. The first two

inequality conditions ensure that, at x, the storage congestion state zst(x) given by (2.13) is

unchanged from zst(x). Therefore, (λ⋆(x), γ⋆(x), β⋆(x)) is still the unconstrained local max-

imizer of (A.1) in the same affine subspace when x is replaced with x in the above expression

of the objective function φ. The last inequality condition in Theorem 2.4 guarantees that

(λ⋆(x), γ⋆(x), β⋆(x)) is feasible for (A.1). Indeed, we observe from the expression of β⋆t (x)

in (2.17) that modifying x does not affect β⋆ℓ,t(x) for a line ℓ that is uncongested and hence

β⋆ℓ,t(x) is 0 if line ℓ is uncongested, while the second inequality condition in Theorem 2.4

ensures that β⋆ℓ,t(x) remains positive if line ℓ is congested. Therefore, (λ⋆(x), γ⋆(x), β⋆(x))

must be a global maximizer of the dual QP (A.1) because the problem is concave. By strict

complementary slackness, the conditions defining Rx ensure that the set of binding inequal-

ity constraints is unaltered with x. This implies that Rx is the critical region containing x.


Proof. Since submodularity is preserved through addition, E[V (X ; d)] is submodular. On

the other hand, ρ is nonincreasing and convex since it is a convex risk measure. We now

use the fact that the composition of a nonincreasing convex function and a nondecreas-

ing submodular function is nonincreasing supermodular [120]. Therefore, we observe that

ρ(V (X ; d)) is nonincreasing supermodular. Finally, we again use the fact that submodular-

ity is preserved through addition to conclude that E[V (X ; d)]−κρ(V (X ; d)) is nondecreasing


submodular for any κ ≥ 0.

Appendix B


84

APPENDIX B. APPENDICES OF CHAPTER 3 85

B.1 Definitions and Expressions for Section 3.3

Here we provide the define mathematically the parameters used by the algorithm, including

Dcn, Dcn, Γminn , Γmax

n and Wmaxn . We start by defining Dcn and Dcn for each n ∈ [N ].

Definition B.1. Let yn , (f, δn, pn). For function φn(un, yn) , cn(un−δn+∑N

n′=1 fnn′ , pn)

that is convex (but not necessarily differentiable) in un, a real number αn is called a

(partial) subgradient of φn with respect to argument un at given (un, yn) if φn(u′n, yn) ≥

φn(un, yn) + αn(u′n − un) for all u′n ∈ [Umin

n , Umaxn ]. The set of all subgradients at (un, yn),

denoted by ∂unφn(un, yn), is called the (partial) subdifferential of φn(un, yn) with respect

to un at (un, yn). Denote Un , [Uminn , Umax

n ], Yn , F × [δminn , δmax

n ] × [Pminn , Pmax

n ] where

F =f : −Fmax

nn′ ≤ fnn′ ≤ Fmaxnn′ , ∀n, n′ ∈ [N ]

. Define the set

Dcn ,⋃

(un,yn)∈×Un×Yn

∂unφn(un, yn),

and let real numbers Dgn and Dgn be defined such that

Dcn ≤ inf Dcn ≤ supDcn ≤ Dcn. (B.1)

That is, Dcn and Dcn are a lower bound and an upper bound of the sub-gradient of φn over

its (compact) domain, respectively.

More details and examples regarding how to calculate Dcn and Dcn can be found in our

previous work [97]. The bounds for the algorithmic parameters are

Γminn ,

1

λn(−WnDcn + Umax

n − Smaxn ) , (B.2)

Γmaxn ,

1

λn

(−WnDcn − Smin

n + Uminn

), (B.3)

and

Wmaxn ,

(Smaxn − Smin

n )− (Umaxn − Umin

n )

Dcn −Dcn. (B.4)

B.2 Proof of Theorem 3.1

Similar to the analysis in [97] for the single bus storage case, we will prove Theorem 3.1 via

the following steps:

1. Reformulate problem (3.6) and link it to the sequence of OMG online optimizations

(3.4).

2. Prove that the control policy obtained from OMG is feasible to problem (3.6).

3. Derive the performance bound in Theorem 3.1.


First, we proceed by reformulating problem (3.6). For n = 1, . . . , N , define

un , limT→∞

1

TE

[T∑

t=1

un,t

], sn , lim

T→∞

1

TE

[T∑

t=1

sn,t

].

Note that for sn,1 ∈ [Sminn , Smax

n ],

un = limT→∞

1

TE

[T∑

t=1

sn,t+1 − λnsn,t]= (1 − λn)sn.

As sn,t ∈ [Sminn , Smax

n ] for all t ≥ 0, the above expression implies

(1− λn)Sminn ≤ un ≤ (1 − λn)Smax

n .

Problem (3.6) can be equivalently written as follows

P1: min limT→∞

1

TE

T∑

t=1

N∑

n=1

cn(gn,t; pn,t) (B.5a)

s.t. δn,t + gn,t = un,t +

N∑

n′=1

fnn′,t, (B.5b)

sn,t+1 = λnsn,t + un,t, (B.5c)

Sminn − λnsn,t ≤ un,t ≤ Smax

n − λnsn,t, (B.5d)


n , (B.5e)

(1− λn)Smin ≤ un ≤ (1− λn)Smaxn (B.5f)

Ynn′(θn,t − θn′,t) = fnn′,t, (B.5g)


nn′ , (B.5h)

where bounds on sn,t are replaced by (B.5d), and (B.5f) is added without loss of optimality.

Here we use JP1(u, g, θ, f) to denote the objective value of P1 with operation sequence

(u, g, θ, f) (as an abbreviation of ut, gt, θt, ft : t ≥ 1), Λ⋆(P1) = (u⋆(P1), g⋆(P1), θ⋆(P1),

f⋆(P1)) to denote the optimal control sequence for P1, J⋆P1 , JP1(Λ

⋆(P1)), and we define

similar quantities for P2. Here P2 is an auxilliary problem we construct to bridge the

infinite horizon storage control problem P1 to online optimization problems (3.4). It has

the following form

P2: min limT→∞

1

TE

T∑

t=1

N∑

n=1

cn(gn,t; pn,t) (B.6a)

s.t. δn,t + gn,t = un,t +N∑

n′=1

fnn′,t, (B.6b)



n , (B.6c)

(1− λn)Sminn ≤ un ≤ (1− λn)Smax

n (B.6d)

Ynn′(θn,t − θn′,t) = fnn′,t, (B.6e)


nn′ . (B.6f)

Notice that it has the same objective as P1, and evidently it is a relaxation of P1. This

implies that u⋆(P2) may not be feasible for P1, and

J⋆P2 = JP1(Λ

⋆(P2)) ≤ J⋆P1. (B.7)

The reason for the removal of state-dependent constraints (B.5d) (and hence (B.5c) as the

sequence st : t ≥ 1 becomes irrelevant to the optimization of ut : t ≥ 1) in P2 is that the

state-independent problem P2 has easy-to-characterize optimal stationary control policies.

In particular, from the theory of stochastic network optimization [84], the following result

holds.

Lemma B.1 (Stationary disturbance-only policies). Under Assumption 3.1 there exists a

stationary disturbance-only policy Λstat,t = (ustatt , gstatt , θstatt , f statt ) satisfying the constraints

in P2 and providing the following guarantees ∀t:

(1− λn)Sminn ≤ E[ustatn,t ] ≤ (1 − λn)Smax

n , ∀n ∈ [N ]

E

[N∑

n=1

cn(gn,t; pn,t)

∣∣∣∣∣Λstat,t

]= J⋆

P2,

where the expectation is taken over the randomization of δn,t, pn,t, and possibly Λstat,t in

case the policy is randomized.

Recall the online optimization solved by OMG:

P3: min

N∑

n=1

(λn/Wn)(sn + Γn)un + cn(gn; pn) (B.8a)

s.t. Uminn ≤ un ≤ Umax

n , (B.8b)

δn + gn = un +

N∑

n′=1

fnn′ , (B.8c)

Yn′n(θn′ − θn) = fn′n, (B.8d)

− Fmaxn′n ≤ fn′n ≤ Fmax

n′n . (B.8e)

We use Λol,t = (uolt , golt , θ

olt , f

olt ) to denote the solution of P3 at time step t, Λ⋆(P3) =

(u⋆(P3), g⋆(P3), θ⋆(P3), f⋆(P3)) to denote the sequence Λol,t : t ≥ 1, JP3,t(Λt) to

denote the objective function of P3 at time period t using policy Λt, and J⋆P3,t to denote


the corresponding optimal cost.

Now, we turn to the feasibility analysis of Λ⋆(P3) with respect to P1. Following as-

sumption holds for any storage system that is controllable.

Assumption B.1 (Feasibility and Controllability). Each storage n ∈ [N ] is feasible and

controllable:

• (feasibility) starting from any feasible storage level, there exists a feasible storage opera-

tion such that the storage level in the next time period is feasible, i.e., λnSminn +Umax

n ≥Sminn and λnS

maxn + Umin

n ≤ Smaxn .

• (controllability) starting from any feasible storage level, there exists a sequence of fea-

sible storage operations to reach any feasible storage level in a finite number of steps,

i.e., λnSmaxn + Umax

n ≥ Smaxn and λnS

minn + Umin

n ≤ Sminn .

In order to prove that the solution of P3 is feasible to P1, we have the following technical

lemma.

Lemma B.2. At each time period t, the optimal storage operation of P3 at node n, uoln,t,

for n = 1, . . . , N , satisfies

1. uoln,t = Uminn whenever λnsn,t ≥ −WnDcn,

2. uoln,t = Umaxn whenever λnsn,t ≤ −WnDcn,

where

sn,t = sn,t + Γn.

Proof. The proof follows from similar arguments used to prove Lemma 3 of [97]. Details are

omitted for brevity.

We are ready to prove the feasibility of the control sequence generated by the algorithm.

Proof of Theorem 3.1, feasibility. For any n = 1, . . . , N , we first validate that the intervals

of Γn and Wn are non-empty. By A3 of Assumption 3.1, one concludes Wmaxn > 0, thus it

remains to show Γmaxn ≥ Γmin

n . Based on (B.4), Wn ≥ 0, and Dcn ≥ Dcn, one obtains

Wn(Dcn −Dcn) ≤ [(Smaxn − Smin

n )− (Umaxn − Umin

n )].

Re-arranging terms results in

−WnDcn + Umaxn − Smax

n ≤ −WnDgn − Sminn + Umin

n ,

which further implies Γmaxn ≥ Γmin

n .

We proceed to show that


n , (B.9)


for t = 1, 2, . . . and any n ∈ [N ], when Λ⋆(P3) is implemented. The base case holds by

assumption. Let the inductive hypothesis be that (B.9) holds at time t. The storage level

at t + 1 is then sn,t+1 = λnsn,t + uoln,t. We show (B.9) holds at t + 1 by considering the

following three cases.

Case 1. −WnDcn ≤ λnsn,t ≤ λn(Smaxn + Γn).

First, it is easy to verify that the above interval for λnsn,t is non-empty using (B.2) and

Γn ≥ Γminn . Next, based on Lemma B.2, one obtains uoln,t = Umin

n ≤ 0 in this case. Therefore

sn,t+1 = λnsn,t + Uminn ≤ λnSmax

n + Uminn ≤ Smax

n ,

where the last inequality follows from Assumption B.1. On the other hand,

sn,t+1 = λnsn,t + Uminn ≥ −WnDcn − λnΓn + Umin

n

≥−WnDcn − λnΓmaxn + Umin

n ≥ Sminn ,

where the third inequality used Dcn ≥ Dcn.Case 2. λn(S

minn + Γn) ≤ λnsn,t ≤ −WnDcn.

The above interval for λnsn,t is non-empty by (B.3) and Γn ≤ Γmaxn . Lemma B.2 implies

uoln,t = Umaxn ≥ 0 in this case. Therefore, again using Assumption B.1,

sn,t+1 = λnsn,t + Umaxn ≥ λnSmin

n + Umaxn ≥ Smin

n .

On the other hand,

sn,t+1 = λnsn,t + Umaxn ≤ −WnDcn − λnΓn + Umax

n

≤−WnDcn − λnΓminn + Umax

n ≤ Smaxn ,

where the third inequality used Dcn ≥ Dcn.Case 3. −WnDcn < λnsn,t < −WnDcn.

By Uminn ≤ uoln,t ≤ Umax

n , one obtains

sn,t+1 = λnsn,t + uoln,t ≤ λnsn,t + Umaxn

<−WnDcn − λnΓn + Umaxn

≤−WnDcn − λnΓminn + Umax

n ≤ Smaxn .

On the other hand,

sn,t+1 = λnsn,t + uoln,t ≥ λnsn,t + Uminn

>−WnDcn − λnΓn + Uminn

≥−WnDcn − λnΓmaxn + Umax

n ≥ Sminn .


Combining these three cases, and by mathematical induction, we conclude (B.9) holds

for all t = 1, 2, . . . .

It remains to show that the sub-optimality bounds claimed in Theorem 3.1 indeed hold.

Proof of Theorem 3.1, performance. Consider a quadratic Lyapunov function Ln(sn) = s2n/2.

Let the corresponding Lyapunov drift be

∆n(sn,t) = E [Ln(sn,t+1)− Ln(sn,t)|sn,t] .

Recall that sn,t+1 = sn,t+1 + Γn = λsn,t + un,t + (1− λn)Γn, and so

∆n(sn,t)

= E

[(1/2)(un,t + (1− λn)Γn)

2 − (1/2)(1− λ2n)s2n,t + λnsn,tun,t + λn(1− λn)sn,tΓn|sn,t]

≤Mun (Γn)− (1/2)(1− λ2n)s2n,t +E

[λnsn,tun,t + λn(1− λn)sn,tΓn|sn,t

]

≤Mun (Γn) +E [λnsn,t(un,t + (1− λn)Γn)|sn,t] .

It follows that, with arbitrary Λt = (ut, gt, θt, ft),

∆n(sn,t)

Wn+E[cn(gn,t; pn,t)|sn,t]

≤ Mun (Γn)

Wn+λn(1 − λn)sn,tΓn

Wn+E

[λnsn,tun,t

Wn+cn(gn,t; pn,t)|sn,t

].

By summing the above expression over n = 1, . . . , N ,

N∑

n=1

∆n(sn,t)

Wn+E[cn(gn,t; pn,t)|sn,t] ≤

N∑

n=1

Mun (Γn)

Wn+λn(1− λn)sn,tΓn

Wn+E

[JP3,t(Λt)|st].

where it is clear that minimizing the right hand side of the above inequality over Λt is

equivalent to minimizing the objective of P3. Since Λstat,t, the disturbance-only stationary

policy of P2 described in Lemma B.1, is feasible for P3, then the above inequality implies

N∑

n=1

∆n(sn,t)

Wn+E[cn(gn,t; pn,t)|sn,t,Λol,t]

≤N∑

n=1

Mun (Γn)

Wn+λn(1− λn)sn,tΓn

Wn+E

[J⋆P3,t|st]

≤N∑

n=1

Mun (Γn)

Wn+λn(1−λn)sn,tΓn

Wn+E

[JP3,t(Λ

stat,t)|st]

(a)=

N∑

n=1

Mun (Γn)

Wn+λnsn,tE

[ustatn,t + (1− λn)Γn

]

Wn+

N∑

n=1

E[cn(gn,t; pn,t)|Λstat,t)]


(b)

≤N∑

n=1

Mn(Γn)

Wn+E[cn(gn,t; pn,t)|Λstat,t]

(c)

≤N∑

n=1

Mn(Γn)

Wn+ J⋆

P1.

Here (a) uses the fact that ustatt is induced by a disturbance-only stationary policy; (b)

follows from inequalities

|sn,t| ≤(max

((Smax

n + Γn)2, (Smin

n + Γn)2))1/2

and ∣∣E

[ustatn,t

]+ (1− λn)Γn

∣∣ ≤ (1 − λn)(max((Smaxn + Γn)

2, (Sminn + Γn)

2))1/2;

and (c) used E[∑N

n=1 cn(gn,t; pn,t)|Λstat,t] = J⋆P2 from Lemma B.1 and J⋆

P2 ≤ J⋆P1. Taking

expectation over st on both sides gives

E

[N∑

n=1

cn(gn,t; pn,t)|Λol,t

]+

N∑

n=1

E [Ln(sn,t+1)− Ln(sn,t)]

Wn≤

N∑

n=1

Mn(Γn)

Wn+ J⋆

P1. (B.10)

Summing expression (B.10) over t from 1 to T , dividing both sides by T , and taking the

limit T →∞, we obtain the performance bound in expression (3.7).

B.3 Derivation of the ADMM Algorithm

The first step in deriving the ADMM iterations for the reformulated problem (3.9) is to

form the augmented Lagrangian function as follows:

Lρ(x, z, µ, ν)

=

n∑

n=1

qn(xn) + µ⊤n (fn,L(n) − fL(n)) +

ρ

2‖fn,L(n) − fL(n)‖22

+m∑

ℓ=1

hℓ(zℓ) + ν⊤ℓ (θℓ,N (ℓ) − θN (ℓ)) +ρ

2‖θℓ,N (ℓ) − θN (ℓ)‖22,

where µn ∈ R|L(n)| and νℓ ∈ R|N (ℓ)| are dual variables for constraints (3.9b) and (3.9c),

respectively, and ρ > 0 is a parameter. The centralized ADMM iterates are then

xk+1 = argminx

Lρ(x, zk, µk, νk), (B.11a)

zk+1 = argminz

Lρ(xk+1, z, µk, νk), (B.11b)

µk+1n = µk

n + ρ(fk+1n,L(n) − fk+1

L(n)), ∀n ∈ [N ], (B.11c)

νk+1ℓ = νkℓ + ρ(θk+1

ℓ,N (ℓ) − θk+1N (ℓ)), ∀ℓ ∈ [L], (B.11d)


where k is the iteration count. Let ηn = µn/ρ for all n and ξℓ = νℓ/ρ for all ℓ be the scaled

dual variables. Then upon recognizing that updates (B.11a) and (B.11c) are separable across

all nodes, and that updates (B.11b) and (B.11d) are separable across all edges, we obtain

the following distributed ADMM iterates:

xk+1n = argmin

xn

qn(xn) +ρ

2‖fn,L(n) − fk

L(n) + ηkn‖22 +∑

ℓ∈L(n)

ρ

2(θkℓ,n − θn + ξkℓ,n)

2, ∀n ∈ [N ],

zk+1ℓ = argmin

zℓ

hℓ(zℓ) +ρ

2‖θℓ,N (ℓ) − θk+1

N (ℓ) + ξkℓ ‖22 +∑

n∈N (ℓ)

ρ

2(fk+1

n,ℓ − fℓ + ηkn,ℓ)2, ∀ℓ ∈ [L],

ηk+1n = ηkn + fk+1

n,L(n) − fk+1L(n), ∀n ∈ [N ],

ξk+1ℓ = ξkℓ + θk+1

ℓ,N (ℓ) − θk+1N (ℓ), ∀ℓ ∈ [L].

The observation that the message passing scheme proposed indeed facilitates the local com-

putation completes this derivation.

Proof of Lemma 3.2. Based on the derivation above, it is easy to check that the iterations

given above implement the standard two block ADMM with x and z being the (two-block)

primal variables, and (µ, ν) be the dual variable for the linear equality constraints. The

convergence analysis of [21] applies directly. The linear convergence rate follows from e.g.

[47] and [49].

Appendix C


93

APPENDIX C. APPENDICES OF CHAPTER 4 94

C.1 Proof of Theorem 4.1

We prove Theorem 4.1 by first establishing a more general version of the result for opti-

mization of the form

maximize U(p) (C.1a)

subject to Ep = 0, (C.1b)

Dp ≤ d, (C.1c)

pi ∈ Pi, i ∈ [m], (C.1d)

and then show that Theorem 4.1 is a special case. Here the decision variables are pi ∈ Rn,

i ∈ [m] so that p ∈ Rmn, constraint matrices E ∈ Rr1×mn and D ∈ Rr2×mn are such that

p : Ep = 0, Dp ≤ d is a closed compact subset of Rmn, local feasible sets Pi, i ∈ [m] are

closed, compact and convex subsets of Rn, where n,m, r1 and r2 are positive integers.

For each ǫ > 0, we consider an iterative algorithm for solving (C.1) of the following form:

given y0 ∈ Rmn feasible for (C.1), the iterations are generated by a point-to-set mapping

Aǫ : Rmn 7→ 2R

mn

. That is,

yk+1 ∈ Aǫ(yk), k ∈ Z+. (C.2)

We consider a specific algorithmic mapping that is a composition of two mappings: Aǫ =

CFǫ. Here the point-to-set mapping Fǫ : Rmn 7→ Rmn × 2R

mn

is defined such that (y′, p) ∈Fǫ(y) if y

′ = y and

p ∈ Gǫ(y) =

p ∈ Rmn :

Ik ⊂ [m], pi = 0, i 6∈ Ik,yi + pi ∈ Pi, i ∈ Ik,Ep = 0, D(y)p ≤ 0,

U(y + p)− U(y) ≥ ǫ

, (C.3)

where D(y) is the matrix containing rows of D corresponding to binding constraints at y.

The point-to-point mapping C : Rmn × Rmn → Rmn is defined such that C(y, p) = y + γp

where

γ = maxγ ∈ (0, 1] : D(y + γp) ≤ d. (C.4)

The convergence theorem that we wish to establish is formally stated as follows.

Lemma C.1. Suppose U is concave and problem (C.1) has a solution and its feasible set

has a nonempty interior. Denote the optimal value of (C.1) by U⋆. Then for any given

feasible y0, any process ykk∈Z+generated by algorithmic mapping Aǫ in the sense of (C.2)

has objective values U(yk)k∈Z+such that

U⋆ − limk→∞

U(yk) ≤ ǫ. (C.5)


Notice that in (C.2), any point in the set Aǫ(yk) can be picked as yk+1, and thus

Lemma C.1 is asserting a form of convergence for a family of an infinite numbers of processes

ykk∈Z+. A classical result concerning with this type of convergence is Zangwill’s global

convergence theorem [139] (also see [70]):

Theorem C.1. Let A be an algorithm on set X , and suppose that starting from x0 the

sequence xkk∈Z+is generated satisfying xk+1 ∈ A(xk). Let a solution set X ⋆ ⊂ X be

given, and suppose

1. all points xk are contained in a compact set S ⊂ X ,

2. there is a continuous function U on X such that (i) if x 6∈ X ⋆, then U(y) > U(x) for

all y ∈ A(x), and (ii) if x ∈ X ⋆, then U(y) ≥ U(x) for all y ∈ A(x),

3. the mapping A is closed at points outside of X ⋆.

Then the limit of any convergent subsequence of xkk∈Z+is in X ⋆.

The proof of Lemma C.1 amounts to checking conditions in the theorem above for

solution set P⋆ǫ which are feasible points p for (C.1) such that U⋆ − U(p) ≤ ǫ.

Proof of Lemma C.1. The objective function U is concave, hence continuous. The feasible

set of (C.1) is an intersection of closed, compact, and convex sets and is also closed, compact

and convex. It remains to show that the sequence ykk∈Z+is feasible and ascent, and the

mapping Aǫ is closed.

(a) The sequence ykk∈Z+is feasible for constraints of (C.1).

The initial point y0 is feasible by assumption. Suppose that yk is feasible. As pk ∈Gǫ(y

k), we have Epk = 0 and so E(yk + γpk) = 0 for any γ. Therefore Eyk+1 = 0. We

also have Dyk+1 ≤ d by the definition of γ in (C.4). (Notice that a nonzero γ exists as the

feasible set of (C.1) has nonempty interior.) Finally, yk+1i = yki + γpki ∈ Pi since y

ki ∈ Pi,

yki + pki ∈ Pi and Pi is convex. Thus yk is feasible for all k by induciton.

(b) The sequence U(yk)k∈Z+is nondecreasing for yk 6∈ P⋆

ǫ .

By the last condition in the definition of set Gǫ, we have U(yk + pk) − U(yk) ≥ ǫ > 0.

As γ ∈ (0, 1] and U is concave, we have U(yk + γpk)− U(yk) > 0.

(c) The mapping Aǫ is closed.

As the feasible set is closed, it suffices to prove that the graph of Aǫ is closed. That is,

for any sequence (zk, yk)k∈Z+with limit (z, y) such that zk ∈ Aǫ(y

k), we need to show

that z ∈ Aǫ(y). It is easy to check that zi ∈ Pi as the set Pi is closed for each i. We

thus proceed to show that the search direction defined by p = z − y always satisfies linear

constraints Ep = 0 and D(y)p ≤ 0. Indeed, the equality constraint holds by continuity of

the linear mapping. Suppose that the inequality constraint D(y)p ≤ 0 does not hold. Then

there exists a constraint, say the ℓth constraint, binding at y, i.e., D⊤ℓ y = dℓ such that


D⊤ℓ p = δ > 0, where Dℓ is the ℓth row of matrix D. By yk → y and D⊤

ℓ y = dℓ, there exists

a natural number K1 such that for all k ≥ K1,

D⊤ℓ y

k ≥ dℓ − δ/4. (C.6)

Meanwhile, by yk → y and zk → z, there exists a natural number K2 such that for all

k ≥ K2,

D⊤ℓ (z

k − yk) ≥ δ/2 > 0. (C.7)

It follows that for all k ≥ max(K1,K2), we have

D⊤ℓ z

k ≥ D⊤ℓ y

k + δ/2 ≥ dℓ + δ/4 > dℓ, (C.8)

contradicting to the fact zk is feasible as proved in item (a), as zk ∈ Aǫ(yk).

It remains to show that there exists a η ∈ [1,∞) such that the search step before

applying the mapping C (curtailment), p = η(z − y) is ǫ-worthy. Suppose otherwise, then

for all η ∈ [1,∞),

U(y + η(z − y))− U(y) ≤ ǫ1 < ǫ. (C.9)

As U is continuous, yk → y and zk → z, there exists a K3 such that for all k ≥ K3,

U(yk + η(zk − yk))− U(yk) ≤ ǫ2, ǫ2 ∈ (ǫ1, ǫ), (C.10)

contradicting to zk ∈ Aǫ(yk). Therefore z ∈ Aǫ(y) and so Aǫ is closed.

Proof of Theorem 4.1. We first recognize that (4.9) is a special case of (C.1), with n = S,

m = |I|, Ep = 0 modeling the power balance constraints, Dp ≤ d modeling the line capacity

constraints. We further notice that the ǫ-trading process (see footnote 3) is algorithm (C.2)

with matrices E and D suitably defined. Invoking Lemma C.1 suggests that for any ǫ > 0,

the trading state process ykk∈Z+is such that U⋆ − limk→∞ U(yk) ≤ ǫ and therefore the

claim in Theorem 4.1 follows.

Remark C.1 (Trading process for distributed optimization). Given the general form of

optimization (C.1) and per Lemma C.1, the trading process and its algorithmic correspon-

dence (C.2) define a framework for solving distributed optimization problems. Different from

popular algorithms such as coordinate descent [86] and ADMM [21], which iterate among

coordinates with well-defined update order, the trading process would converge by updating

some subsets of the coordinates (pairs of coordinates in the case of tree network or for net-

work flow problems as discussle ed in Appendix C.4) according to any order, as long as

conditions in the algorithmic mapping (C.2) are met. Another distinct feature of the trad-

ing process is that it does not specify a search direction for each update; it permits any

search direction corresponding to a trade with suitable economic incentives (ǫ-worthy trade)


thus allows a great flexibility for designing platforms or systems in which agents could trade

freely for their own benefits with an essential amount of coordination in place to ensure

global constraints are satisfied.

C.2 Proof of Lemma 4.1

Writing the Lagrangian of (4.9) and taking derivative with respect to pi,s for some RT

participant i gives

Pi(s)∂ui,s(p

⋆)

∂pi,s+ λn,s − (ηi,s − ηi,s) = 0. (C.11)

Thus (4.30) follows from this first order condition. When it is known that p⋆i,s ∈ Pi,s, we

have η⋆i,s = η⋆i,s

= 0 by complementary slackness, and so equation (4.29) holds in this case.

C.3 Proof of Lemma 4.2

Consider the optimization of participant i at bus n. The local constraint pi ∈ Pi can be

expressed as

pi≤ pi ≤ pi,

pi =1

S11⊤pi, if i ∈ IDA, (C.12)

where the second constraint is the vector form of pi,s = 1S

∑Ss=1 pi,s, a convenient way

to express the non-anticipation constraint. Let the dual variables for these constraints be

denoted ηi∈ RS , ηi,s ∈ RS and ζi ∈ RS , respectively. Then the optimality condition for

(4.31) is

λn,spi,s + Pi(s)∂ui,s(pi,s)

∂pi,s− (ηi,s − ηi,s)

− (1s −1

S1)⊤ζi1i∈IDA = 0, (C.13a)

pi≤ pi ≤ pi, (C.13b)

pi =1

S11⊤pi, if i ∈ IDA, (C.13c)

ηi, ηi ≥ 0, (C.13d)

where 1i∈IDA = 1 if i ∈ IDA and 0 otherwise.

Now consider the optimization for SO. Denote the dual variable for power balance con-

straint by γs ∈ R and the dual variable for flow constraint by βs ∈ RL, s ∈ [S]. Then the


optimality condition is

λn,s + γs + (H⊤βs)n = 0, n ∈ [N ], s ∈ [S], (C.14a)

1⊤xs = 0, s ∈ [S], (C.14b)

Hxs ≤ f, s ∈ [S], (C.14c)

βs ≥ 0, s ∈ [S]. (C.14d)

Collecting optimality conditions (C.13) for all i ∈ I and that for SO (C.14), together with

(4.34), we recover the optimality condition for (4.9), where λn,s is the dual for constraint.

The claim in Lemma 4.2 thus follows.

C.4 Bilateral Trading in Tree Network

In an example, [134] demonstrates that multilateral trades involving more than two par-

ticipants could be necessary when the network has cycles. The goal of this section is to

complement that result by showing that when the network has no cycle, bilateral trades are

sufficient for the trading process to converge to the solution of centralized dispatch.

For simplicity, we consider the deterministic case so that S = 1 and I = IRT. Without

loss of generality, we further compress the notation by assuming there is only one participant

connected to each node, so that In is a singleton for each n and we use network index n

and participant index i whichever is more convenient. These assumptions reduce our model

to that of [134]. The network is radial so there is no cycle. With the DC approximation,

the power flow model is equivalent to the standard network flow model [12] as shown in e.g.

[56].

We establish two decomposition results. We will state these results assuming that the

accumulated trade of the network is zero (x = 0) and we will work with the general line

capacity constraints instead of constraints specified by the loading vector requirements. In

a tree network, incorporating a nonzero accumulated trade x amounts to modifying the line

capacities and the utility functions; the loading vector requirements are special cases of the

general line capacity constraints as they simply require the induced line flows on congested

lines have nonpositive contribution to the congested direction. Thus our treatment leads to

no loss of generality.

We first consider decomposing feasible multilateral trades into feasible bilateral trades.

Proposition C.1. Suppose that the line capacities are rational numbers, i.e., f ∈ QL. For

any given feasible multilateral trade p ∈ QN involving more than two participants, i.e., p ∈ Pand ‖p‖0 := |n ∈ [N ] : pn 6= 0| ≥ 3, there exists a finite number K ∈ Z+ of bilateral trades

pk ∈ QN with ‖pk‖0 = 2, k = 1, . . . ,K, that are sequentially feasible such that∑K′

k=1 pk ∈ P

for any K ′ ≤ K and satisfies∑K

k=1 pk = p.


Furthermore, under the same assumptions, there exists a finite number K ∈ Z+ of

bilateral trades pk ∈ QN with ‖pk‖0 = 2, k = 1, . . . , K, that are sequentially feasible for

any ordering satisfying∑K

k=1 pk = p. That is, let σ : [K] 7→ [K] be any permutation of the

indices 1, . . . , K, then∑K′

k=1 pσ(k) ∈ P, K ′ < K, and in particular pk ∈ P for any k ∈ [K].

Proof. Let (V , E) be the graph underlying the radial power network, with the node set

V = [N ] and the edge set containing the N − 1 lines of the network. Define the edge

capacity to the capacity of the corresponding line. A flow on the graph is a (N − 1)-

vector f that assigns a flow on each edge of the network satisfying flow conservation (for

each node, in-flow equals the out-flow) and edge capacity constraints. Given a multilateral

trade p, we denote the set of supply nodes by V+ = n ∈ V : pn > 0 and the set of

demand nodes by V− = n ∈ V : pn < 0. We then extend the graph by adding a

source node vs connecting to all the supply nodes and adding a sink node vt connecting

to all the demand node, so that the extended graph is (V , E) with V = V ∪ vs, vt and

E = E ∪ (vs, v) : v ∈ V+ ∪ (v, vt) : v ∈ V−. For edges (vs, v), v ∈ V+, we assign edge

capacity to be pv > 0. Similarly for edges (v, vt), v ∈ V−, the edge capacity is −pv > 0.

Now we make the observation that the multilateral trade p and its induced power flow

on the radial network is equivalent to the max flow from vs to vt on the flow network (V , E).In particular, the max flow solution would assign flows on the additional edges in E\E equal

to the capacities. Together with the power flow induced by p, we obtain a feasible flow

that maximizes the flow value from vs to vt. Therefore, the problem of identifying feasible

bilateral trades representing the multilateral trades is equivalent to finding feasible simple

flows representing the max flow on the flow network, where a simple flow from vs to vt is a

flow on a simple path from vs to vt, which must contain exactly one supply node and one

demand node.

The Ford-Fulkerson algorithm solves the max flow problem by iteratively identifying a

feasible simple flow on the residual graph. By the definition of the residual graph, this se-

quence of simple flows is sequentially feasible. Furthermore, it is known that Ford-Fulkerson

terminates in a finite number of steps for rational inputs. We thus conclude that the finite

collection of simple flows found by Ford-Fulkerson represents the finite collection of bilateral

trades satisfying the requirements of the first part of the proposition.

For the second part of the proposition, by Conformal Realization Theorem[11, Proposi-

tion 1.1] we know that there is a decomposition of the flow induced by p on network (V , E)into simple flows that are conformal in the sense that the flow direction of the simple flows

on each edge in E is the same as that of the flow induced by p. It follows that this finite col-

lection of simple flows is sequentially feasible for any ordering as for each edge if we choose

the positive direction to be that of the flow direction induced by p, then for any e ∈ E ,fke ≥ 0 and

∑Kk=1 fe = fe, where f is the flow induced by p and fk is the flow induced by

pk. The fact that each of the simple flows (and the corresponding bilateral trade) is feasible

with respect to the original network constraint follows from choosing the kth simple flow as


the first simple flow in the sequence.

We proceed to show that any non-redundant profitable multilateral trades can be de-

composed into a collection of profitable bilateral trades on a tree network.

Definition C.1. We say a profitable (and feasible) multilateral trade p ∈ RN contains

redundancy if there exists a curtailment γ ∈ [0, 1]N such that p ∈ RN defined by pn = γnpn,

n ∈ [N ], is a feasible trade that achieves at least the same amount of profit as p. A profitable

multilateral trade is deemed non-redundant if it does not contain redundancy.

Without loss of practicality, we focus on and first state the result for the linear utility

case so that Ui(pi) = αipi for some αi ∈ R, i ∈ [N ]. The extension to nonlinear case is

discussed after that.

Proposition C.2. Under the same assumptions of Proposition C.1 and supposing that the

utility function U is linear, any non-redundant profitable multilateral trades can be decom-

posed into a finite collection of profitable bilateral trades. Formally, given a non-redundant

profitable trade p ∈ Qn, there exists a finite number K of bilateral trades pk that are sequen-

tially feasible for any ordering and profitable, and satisfies∑K

k=1 pk = p.

Proof. Consider the decomposition for the second part of Proposition C.1. If all the bilateral

trades in the decomposition are profitable, there is nothing to prove. Suppose there exists a

bilateral trade in the decomposition that is not profitable, denoted by p′. We claim that the

remaining trade p′′ = p−p′ is a feasible profitable trade that yields at least the same amount

of profit as p and therefore p has redundancy. Indeed, by the proof of Proposition C.1, p′′

is feasible. Furthermore,

U(p′′)− U(0) =

N∑

i=1

αip′′i − 0 =

N∑

i=1

αi(pi − p′i) > U(p)− U(0), (C.15)

as U(p′) =∑N

i=1 αi < U(0) = 0 as p′ is not profitable.

In general, when Ui(pi) is nonlinear but differentiable, we can decompose any given

profitable trade p intoM ∈ Z+ copies of trades p/M , each of which is profitable by concavity

of U . For pi ∈ [m∆pi, (m + 1)∆pi], m = 0, . . . ,M − 1, Taylor series offers a good linear

approximation

Ui(pi) = Ui(m∆pi) + U ′(m∆pi)pi + o(pi/M), (C.16)

where the last term denotes the approximation error which is of order higher than pi/M and

is negligible for practical purposes when M is sufficiently large. Applying Proposition C.2

gives a finite collection of profitable bilateral trades for each multilateral trade p/M with

mp/M already scheduled into the system. Pooling these collections of bilateral trades gives

a collection of bilateral trades that approximately represents the original multilateral trade.


C.5 Trade Verification and Curtailment with Local Sce-

narios

The bulk of the chapter assumes that there is a commonly known set of global scenarios

[S], which may not be practically available when participants are distributed over a large

geographical area. In this section, we briefly discuss the other extreme setting where no

global scenario is known a priori and participants are still allowed to submit contingent

trades pks , where s ∈ Sk with Skdenoting the set of local scenarios that is known to the

participants involved in the kth trade. Motivated by common sources of uncertainty in power

systems (e.g. renewable generation level), we consider the setting that pks ∈ R|I| : s ∈ Skis an interval1 in R|I|. That is, the participants submit a lower bound pk and an upper bound

pk to the SO, so that given any realization of the local uncertainty s ∈ Sk, the resulting

contingent trade satisfies pks ∈ [pk, pk]. Since the SO may not be capable to identify the

exact correlation among these local scenarios, the verification of power network constraints

and curtailment has to be robust with respect to any combinations of the local scenarios.

That is, the accumulated network power injection defined by

xkn,s =

k−1∑

κ=0

∑

i∈In

γκpκi,s, n ∈ [N ], (C.17)

must be feasible, i.e.,

xks ∈ P , (C.18)

for every global scenario s = (s0, . . . , sk−1) generated by local scenarios sκ ∈ Sκ, κ =

0, . . . , k − 1.

We show that checking and curtailing a new trade pks ∈ [pk, pk] with its corresponding

network injection qks ∈ [qk, qk] can be done in an efficient way so that this process can

be carried out inductively. Under our assumption that the set of contingent trades is an

interval, the verification of xks + qks ∈ P , for all s = (s0, . . . , sk), sκ ∈ Sκ, is equivalent to

checking

H(xks + qks ) ≤ f , (C.19)

where xks is defined as in (C.17) with pκs ∈ [pκ, pκ], κ = 0, . . . , k− 1, and qks ∈ [qk, qk]. Given

the intervals for previous trades pκs ∈ [pκ, pκ], κ = 0, . . . , k−1, and their curtailment factors,

we can form the corresponding interval xks ∈ [xk, xk]. Then verifying the condition above

can be done by solving the following robust linear program:

maxγ∈[0,1]

γ (C.20a)

1When this does not hold, we can form the interval by finding the point-wise extremum of vector pks .


s.t. xkn,s =k−1∑

κ=0

∑

i∈In

γκpκi,s, n ∈ [N ], (C.20b)

H(xks + γqks ) ≤ f , (C.20c)

where the last constraint must hold for all xks ∈ [xk, xk] and qks ∈ [qk, qk]. In particular, if

the optimal value of this program is γ⋆ = 1, then the new trade is feasible; if the optimal

value is less than 1, the new trade needs to be curtailed with curtailment factor γk = γ⋆ < 1.

Finally, we note that this optimization can be solved efficiently by a bisection process (as

the optimization variable is a scaler) equipped with a constraint checking sub-routine, which

verifies the strong solvability of a set of interval linear inequalities in polynomial time (cf.

[40, Section 2.13]).

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