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Graduate Studies The Vault: Electronic Theses and Dissertations

2017

Optimal Operation Planning of Compressed Air

Energy Storage Plants in Competitive Electricity

Markets

Soroush, Shafiee

Soroush, S. (2017). Optimal Operation Planning of Compressed Air Energy Storage Plants in

Competitive Electricity Markets (Unpublished doctoral thesis). University of Calgary, Calgary, AB.

doi:10.11575/PRISM/27208

http://hdl.handle.net/11023/3837

doctoral thesis

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UNIVERSITY OF CALGARY

Optimal Operation Planning of Compressed Air Energy Storage Plants in Competitive Electricity

Markets

by

Soroush Shafiee

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN ELECTRICAL AND COMPUTER ENGINEERING

CALGARY, ALBERTA

MAY, 2017

c© Soroush Shafiee 2017

Abstract

This thesis focuses on the operation of a compressed air energy storage (CAES) facility in an

electricity market. CAES, a bulk energy storage technology, can provide time shifting due to its

capability of storing large amount of energy, as well as ancillary services including spinning and

non-spinning reserves due to its fast response. In order to participate effectively in electricity

markets and consequently quantify the economics of CAES technology, scheduling of the CAES

facility needs to be optimized considering market forecasts as well as efficiency of the components

and operational characteristics. Moreover, Due to inevitable price forecasting error, effective bid-

ding and offering strategy to purchase and sell electricity in the market is necessary to manage the

risk of forecasting error.

In this study, at first, a risk-constrained bidding/offering strategy for a merchant price-taker

CAES providing time shifting is proposed to manage the risk of price forecasting errors. Price-

taker utility refers to a utility, which is small enough compared to the market size that its operation

does not affect market price. Thereafter, since the operation of a large-scale facility in an elec-

tricity market could impact the prices, the scheduling of a merchant price-maker energy storage

facility, doing energy arbitrage is proposed. In this model, the impact of storage operation on mar-

ket clearing price are incorporated. As a continuation, this study proposes a bidding and offering

strategy for a price-maker ES facility taking the forecasting errors into account based on the robust

optimization to manage the associated risk when providing energy arbitrage. In order to maximize

revenue, the potential gain from providing ancillary services (including spinning and non-spinning

reserves) in addition to the energy arbitrage must be considered. In this regard, a scheduling model

of a merchant CAES facility participating in day-ahead energy and reserve markets is developed.

Meanwhile, the efficiency of a CAES facility deviates significantly from its nominal value de-

pending on its thermodynamics and operational conditions. Thus, in this study, these limitations

imposed on the facility are modelled as well when devising operations schedules.

ii

Acknowledgements

I would like to express my special appreciation and thanks to my supervisor Prof. Hamidreza

Zareipour for the continuous support of my Ph.D study and providing me with invaluable life and

career advice, and for providing a critical eye on my work. He has been a tremendous mentor for

me. I learned priceless lessons from his vision, personality, and professionalism.

I was also grateful to work with my co-supervisor, Prof. Andy Knight. I truly appreciate his

support and trust on me. He was always encouraging me and available when I needed his guidance.

Besides, I would like to thank the rest of my thesis committee, Dr. Yasser Abdel-Rady I.

Mohamed, Prof. Edward Roberts, Prof. Abu Sesay, Dr. Ed Nowicki, and Dr. majid Pahlevani, for

serving as my committee members even at hardship.

I am also greatly thankful of Mr. Payam Zamani-Dehkordi, Mr. Ehsan Nasrolahpour, Mr.

Hamid Shakerardakani, Mr. Hamed Chitsaz, Mr. Peyman Sindareh, Dr. Mostafa Kazemi, Mr.

Sepehr Tabatabaei, Mr. Mokhtar Tabari, Mr. Iman Erfan, Mr. Juan Arteaga, Mr. Babatunde

Odetayo, Dr. Behnam Mohammadi, Dr. Nima Amjady, and my other friends and colleagues who

encouraged, supported, and assisted me in my research and other aspects.

A special thanks to my parents. Words cannot express how grateful I am to my mother, and

father for their support, encouragement, and love during the long years of my studies.

My endless thanks also to my parents-in-law, to my grandparents, to my brother, and to my

sisters-in-law for all their support and kindness.

Last but not least, I would like express my deep appreciation to my beloved wife, Sahar, who

spent sleepless nights with and was always my support with her endless love and care. I truly

appreciate her support during the very stressful and challenging moments of my life.

iii

Dedicated to my Father, Zabihollah, my mother, Mahboubeh,

and

my beloved wife, Sahar

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Research Objectives and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Thesis Organization and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Risk-Constrained Bidding and Offering Strategy for a Merchant Compressed Air

Energy Storage Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 CAES self-scheduling Formulation . . . . . . . . . . . . . . . . . . . . . 152.2.2 Information-Gap Decision Theory . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Characteristics of the IGDT method . . . . . . . . . . . . . . . . . . . . . 19

2.3 The Proposed Methodology and Formulation . . . . . . . . . . . . . . . . . . . . 202.3.1 IGDT-Based Operation Scheduling Formulation for a CAES Plant . . . . . 212.3.2 The Equivalent Single-Level Optimization . . . . . . . . . . . . . . . . . . 22

2.4 The Proposed Method for Bidding and Offering Strategy . . . . . . . . . . . . . . 252.4.1 Sequential Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5.1 Risk-constrained Self-scheduling: A Demonstrative Case . . . . . . . . . . 302.5.2 Constructing Biding/Offering Curves Based on the Obtained Results from

IGDT-based Scheduling Cases . . . . . . . . . . . . . . . . . . . . . . . . 332.5.3 After-the-Fact Analysis Based on Constructed Bidding and Offering Cur-

ves and Simulated Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5.4 After-the-Fact Analysis Using Actual Market Data . . . . . . . . . . . . . 36

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Assessment of a Price-Maker Energy Storage Facility in the Alberta electricity

market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Methodology and Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.1 The Alberta Electricity Market Database . . . . . . . . . . . . . . . . . . 483.3.2 Construction of GPQCs and DPQCs of the Alberta Electricity Market . . . 513.3.3 Energy Storage self-scheduling Formulation . . . . . . . . . . . . . . . . . 523.3.4 Equivalent Linear Formulation . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.1 Base Case Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

v

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 Developing bidding and offering curves of a price-maker energy storage facility

based on robust optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.0.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Detereministic Scheduling of an ES Facility . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Equivalent linear formulation . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Proposed Robust-based Scheduling Model . . . . . . . . . . . . . . . . . . . . . . 794.4 Developing Bidding and Offering Curves . . . . . . . . . . . . . . . . . . . . . . 814.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5.1 Robust scheduling: a demonstrative case . . . . . . . . . . . . . . . . . . 854.5.2 The impact of uncertainties through a one-year analysis . . . . . . . . . . . 88

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 Considering Thermodynamic Characteristics of a CAES Facility in Self-scheduling

in Energy and Reserve Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2 CAES Self-scheduling Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.1 Objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2.2 Power Capacity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 985.2.3 Energy Capacity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3 Incorporating Thermodynamic Characteristics in Self-Scheduling Formulation . . . 995.3.1 Effect of SOC in Charging Process . . . . . . . . . . . . . . . . . . . . . . 1005.3.2 Effect of Generation level on SOC . . . . . . . . . . . . . . . . . . . . . . 1015.3.3 Effect of Discharging Rate on Heat Rate (HR) . . . . . . . . . . . . . . . . 104

5.4 Equivalent Linear Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.4.1 Linearizing the effect of SOC in Charging Process . . . . . . . . . . . . . 1055.4.2 Linearizing the Effect of Discharging Process on SOC . . . . . . . . . . . 1065.4.3 Linearizing the Cost of Natural Gas in Discharging Process . . . . . . . . 107

5.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.5.1 CAES Self-scheduling: a Demonstrative Case . . . . . . . . . . . . . . . . 1095.5.2 Participating in Energy Market: Five year Analysis . . . . . . . . . . . . . 1115.5.3 Participating in Energy and Reserve Markets: Five Year Analysis . . . . . 112

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A Copyright permission letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

vi

List of Tables

2.1 After-the-Fact Analysis using simulated prices . . . . . . . . . . . . . . . . . . . . 362.2 After-the-Fact Analysis Using Actual Market Prices . . . . . . . . . . . . . . . . . 38

3.1 Summaries of comparison with previous works . . . . . . . . . . . . . . . . . . . 483.2 Weekly Profit Analysis for a Price-Maker storage facility during 2010 to 2014

[Million $] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3 Price analysis without and with storage operation during discharging hours . . . . 633.4 Price analysis without and with storage operation during charging hours . . . . . . 633.5 Price analysis without and with storage operation for all hours . . . . . . . . . . . 64

4.1 Percentage of the potential profit captured by each strategy in each month of 2014[%] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1 Annual error of the estimated profit of CM and the profit improvement obtained byTBM, when participating in energy and reserve markets . . . . . . . . . . . . . . . 114

vii

List of Figures and Illustrations

2.1 The process of constructing a 4-step bidding curve . . . . . . . . . . . . . . . . . 262.2 The process of constructing a 4-step offering curve . . . . . . . . . . . . . . . . . 272.3 The sequences of defining steps of the bid and offer curves . . . . . . . . . . . . . 292.4 Forecasted price, the worst case of price for robust case and the best case of price

for opportunistic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Scheduling of storage for risk-neutral, robust and opportunistic cases . . . . . . . . 312.6 Bid curve for hour 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Offer curve for hour 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.8 Hourly forecasted and actual prices . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1 Hourly electricity price during 2013 in the Alberta electricity market . . . . . . . . 433.2 An example of a typical a) GPQC, b) DPQC . . . . . . . . . . . . . . . . . . . . . 443.3 Sample supply curve for hour ending 1, December 31, 2014 . . . . . . . . . . . . 493.4 Supply curves for each of the 24 hours on August 1, 2014 . . . . . . . . . . . . . . 503.5 Supply curves for hour ending 1 for the month of October 2014. . . . . . . . . . . 503.6 an example of price decrease due to a 150 MW new supply to the system . . . . . . 513.7 an example of price increase due to a new 100 MW demand to the system . . . . . 523.8 Generation PQC, the linearization process [1]. . . . . . . . . . . . . . . . . . . . . 553.9 Demand PQC, the linearization process. . . . . . . . . . . . . . . . . . . . . . . . 553.10 scheduling of storage plant and price of electricity during an arbitrary week in the

case of ignoring the impact of storage operation on electricity price . . . . . . . . . 573.11 scheduling, price of electricity before and after operation, for a price-maker storage

plant during an arbitrary week . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.12 Dispatch characteristic of a price-maker storage facility during 2010 to 2014 . . . . 593.13 Weekly profit of a price-maker energy storage facility during a) 2010, b) 2011, c)

2012, d) 2013, and e) 2014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.14 Price duration curve without and with operation of a price-maker storage facility

during 2013. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.15 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.16 Profit of storage facility as a function of charging capacity (discharging capacity is

fixed at 140 MW, storage capacity is fixed at 10-hr ) . . . . . . . . . . . . . . . . . 653.17 Profit of storage facility as a function of storage capacity (Discharging capacity is

fixed at 140 MW, charging capacity is fixed at 90 MW) . . . . . . . . . . . . . . . 663.18 Profit of storage facility vs. energy storage efficiency (Discharging capacity is

fixed at 140 MW, charging capacity is fixed at 90 MW, the storage capacity is 10hours of full discharging capacity) . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Linearization process for GPQC . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Linearization process for DPQC . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 An example of worst case for a confidence interval of a) GPQC, b) DPQC . . . . . 814.4 An example of GPQC intervals to build offering problems and the process of con-

structing offering curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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4.5 An example of a DPQC intervals to build bidding problems and the process ofconstructing bidding curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6 (a) DPQCs fpr hours 1 and 6, and (b) GPQCs for hour 17, 18, and 20 . . . . . . . . 854.7 The guaranteed profit versus the length of confidence interval . . . . . . . . . . . . 864.8 scheduling of CAES facility for each iteration corresponded to each subintervals . . 864.9 Obtained price profile for each iteration corresponded to each subintervals . . . . . 874.10 Resulting bidding curve for a) hour 1, b) hour 6 . . . . . . . . . . . . . . . . . . . 884.11 Resulting offering curve for a) hour 18, b) hour 20 . . . . . . . . . . . . . . . . . . 884.12 Resulting offering curve for a) hour 18, b) hour 20 . . . . . . . . . . . . . . . . . . 89

5.1 Schematic diagram of a CAES facility. . . . . . . . . . . . . . . . . . . . . . . . . 955.2 The variations of air flow rate and compressor power during charge process [2]. . . 1005.3 The level of air flow rate per MW of charging versus cavern SOC. . . . . . . . . . 1025.4 Variations of air flow rate under different generation levels [2] . . . . . . . . . . . 1025.5 The variations of air flow rate per unit of generated electricity under different ge-

neration levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.6 The variations of turbine efficiency under different generation levels [2]. . . . . . . 1035.7 The variations of HR under different generation levels [2]. . . . . . . . . . . . . . 1045.8 Linearization process for (a) Compression air flow rate versus cavern SOC, (b)

Dischrging air flow rate versus discharging rate, (c) heat rate versus discharging rate.1055.9 energy, spinning reserve, and non-spinning reserve price profiles. . . . . . . . . . . 1095.10 (a) CAES scheduling obtained from the simple model, (b) Actual CAES schedu-

ling when following simple model schedule considering thermodynamic charac-teristics, (c) CAES scheduling resulted from the developed thermodynamic-basedmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.11 The annual profit of the CAES facility providing energy arbitrage when using CMand TBM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.12 Dispatch characteristic of a CAES facility during 2011 providing energy arbitragewhen scheduling with CM and TBM. . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.13 The annual profit of the CAES facility providing energy arbitrage as well as reser-ves when using CM and TBM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.14 Dispatch characteristic of a CAES facility during 2011 providing energy arbitrageand reserves when scheduling with CM and TBM. . . . . . . . . . . . . . . . . . . 113

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List of Symbols, Abbreviations and Nomenclature

Abbreviation

AESO Alberta electric system operator.

AIL Alberta internal load.

CAES Compressed air energy storage system.

CM Conventional model.

DPQC Demand price quota curve.

ETS Energy trading system.

GAMS Generalized algebraic modeling systems.

GPQC Generation price quota curve.

HP High pressure.

HR Heat rate.

IGDT Information gap decision theory.

KKT Karush-Kuhn-Tucker.

MILP Mixed-integer linear programming.

MPEC Mathematical program with equilibrium constraints.

PHS Pumped hydroelectric storage.

SOC State of charge.

TBM Thermodynamic-based model.

x

Chapter 1

Introduction

Investor interest for grid-scale energy storage technology such as compressed air energy storage

(CAES), pumped hydroelectric storage (PHS), and large-scale battery storage has increased during

recent years. A market study by CitiGroup in 2015 estimated a global market of up to 240 GW

for energy storage by 2030 excluding pumped-storage hydroelectricity and car batteries [3]. The

capital cost of some technologies including batteries and CAES has been decreasing owing to

technology progress in energy storage systems at a pace unheard of 20 years ago [4]. Moreover,

such systems are able to provide load-shifting and peak capacity services to more effectively utilize

the existing capacity of the system, and potentially provide the necessary flexibility to deal with

uncertainty associated with the growing penetration of renewable resources [5, 6].

CAES technology stores large amount of electrical energy in the form of high-pressure air

in either underground (e.g. salt caverns) or aboveground reservoirs (pressure vessels) [7]. This

technology can serve grid-scale long term applications, on the order of 100’s of MW for tens of

hours. Huntrof and McIntosh CAES Plants have been operating for decades [7]. Moreover, there

are other CAES projects either announced, under construction or operating in recent years [8].

One of the best-comprehended and analyzed applications of CAES technology is time-shifting

due to its capability of storing large amount of energy [9]. Time-shifting is to store electricity at

low demand, low price periods, and inject it back to the grid at high demand periods. The high

price variations in electricity markets provide desirable opportunities for energy arbitrage [10].

The energy price in the Alberta electricity market experienced frequent price spikes during years

2011-2014, which brings several energy arbitrage opportunities. It is of great importance for the

investors to know the potential profitability of a large scale investment in bulk energy storage.

Thus, it is important to evaluate the potential profit earned by these storage devices through energy

1

arbitrage in a competitive electricity market.

When a CAES facility participates in an electricity market (e.g. a day ahead electricity mar-

ket), based on the market price forecasts, the operator has to submit bids to procure electricity and

store the air as well as submit offers to sell the electricity to the market at a relatively high price

to make energy arbitrage profitable. Since the price in electricity market has a high level of uncer-

tainty, the CAES operator needs an efficient participation strategy taking into account forecasting

uncertainties to capture energy arbitrage opportunities as much as possible and minimize the risk

of forecasting errors.

During energy arbitrage, purchasing the electricity to compress the air will increase the system

net demand and consequently lead to price increment. On the other hand, injecting the electricity

to the grid will increase system supply and cause price decline [11,12]. Accordingly, both charging

and discharging cases lead to a reduction in the off and on peak price differences and consequently

decrease energy arbitrage revenue, depending on the size of storage facility. As a result, the impacts

of CAES operation on market pool price should also be incorporated in economic analyses as

well as its strategy when participating in the market to avoid profit overestimation and develop an

acceptable participation strategy.

CAES is also well suited for balancing the fluctuations caused by intermittent renewable energy

output, due to its fast ramp response [7]. Thus, in addition to providing energy arbitrage in the

energy market, CAES is able to also provide ancillary services such as spinning and non-spinning

reserves. Therefore, there is another revenue opportunity for the CAES by participating in both

energy and reserve markets and exploiting different revenue streams. Stacking multiple revenue

streams improves the economics of energy storage, and thus, needs to be properly modeled [13–

15]. Consequently, from the investors’ point of view, it is important to assess the value which

could be added to the economics of the CAES technology by participating in the ancillary service

market.

Based on the aforementioned subjects, the studies on energy storage systems from economic

2

point of view, can be divided into two categories: 1) price-taker modeling, 2) price-maker mo-

deling. A price-taker facility refers to a utility whose operation does not impact market price.

A price-maker utility could alter market price by its activities in the market. In this study, the

self-scheduling problem of both price-taker and price-maker CAES facilities providing energy ar-

bitrage are addressed. Then, historical energy price data of electricity markets is used as the case

study to assess the economics of the CAES technology. Moreover, different bidding strategies for

price-taker and price-maker CAES facility providing energy arbitrage in an electricity market are

proposed. Furthermore, the CAES scheduling problem is extended to consider participation in an-

cillary service markets providing spinning and non-spinning reserve and consequently to explored

the additional revenue streams.

1.1 Research Objectives and Scope

The main objective of this research is to develop optimal scheduling frameworks for a privately-

owned CAES facility to participate effectively in an electricity market. At first, the focus is on the

operation of a CAES facility in an energy market providing energy arbitrage. Then, the participa-

tion in ancillary service market providing spinning and non-spinning reserve in addition to energy

arbitrage is addressed.

I first focus on how a price-taker CAES facility should effectively participate in an energy

market considering the price forecasting uncertainty. Since the price in an electricity market has a

high level of uncertainty, which could significantly affect the CAES revenue, the CAES operator

should take the price uncertainty into account in its decision making process to hedge the associated

risk. Hence, a merchant energy storage facility needs not only an appropriate offering strategy for

selling the electricity, but also a proper bidding strategy for purchasing energy from the market.

The stochastic programming, which is used wildly in the literature [16–18] to address the price

uncertainty, requires large computational burden as well as the necessity of knowing the probability

distribution function (PDF) of the uncertain parameters.

3

As an alternative method, the information gap decision theory (IGDT) [19,20] can be employed

to model the price uncerntainty in the problem of bidding strategy of a merchant CAES facility.

the main advantages of IGDT method are: no need for PDF of uncertain parameters, reduced

computational burden, and development of both robust and opportunistic functions.

I then focus on revenue estimation of a large-scale energy storage facility in energy market

in the Alberta electricity market while modeling its impact on pool prices. In order to optimize

the operation of a large size energy storage plant to participate in energy market providing energy

arbitrage, the impacts of the storage operation on the market pool price cannot be ignored. Be-

cause of its large charging or discharging capacity, its impacts on the supply or demand curve and

consequently on pool price become significant. Thus, a small shift in the supply curve during the

peak hours could lead to a significant change in pool price. Thus, neglecting the effect of CAES

operation on electricity market may lead to significant error and overestimating the potential gai-

ned revenue. The impact of energy storage charging and discharging operation should be more

accurately formulated to achieve a more efficient scheduling for the energy storage storage faci-

lity. As a result, this study aims to develop a self-scheduling formulation for a price-maker energy

storage facility and then, address the economic assessment of a large scale CAES facility in Alberta

electricity market.

The study conducted in the previous step provides the potential revenue, which could be gained

if the perfect forecast were available; thus, the uncertainty associated with the forecasts needs to

be incorporated to manage the risk. Thus, I develop a participation strategy for a price-maker

independent ES facility to participate in day-ahead energy market considering the uncertainties

related to the forecasts to manage the associated risk.

So far, the storage facility is assumed to only participate in the energy market to make profit

from energy arbitrage. When participating in competitive electricity markets, a merchant storage

facility may benefit from energy arbitrage. However, in addition to providing energy arbitrage,

the CAES technology can also provide spinning and non-spinning reserve services to the market

4

owing to its fast response [13,14,21]. The required reserve in ancillary service markets is growing

significantly due to substantial penetration of wind and solar energy resources. It is estimated

that in the U.S., in order to accommodate the variability due to additional installed capacity in

wind technology and load growth from 2011-2020, the additional balancing capacity of 18.57

GW is required with the total balancing capacity of 37.67 GW [22]. Thus, the revenue of the

storage facility can be improved by providing additional products in ancillary service market as

well, which is extremely important from the investors’ point of view. Moreover, it is shown that

the efficiency of the CAES facility depends on the operational condition of the facility [2, 23].

Hence, the varying efficiency of the facility needs to be incorporated in the scheduling problem.

Therefore, a self-scheduling model for a merchant CAES facility participating in day-ahead energy

and reserve markets needs to be developed incorporating the structure of both markets as well as

the practical limitations of a CAES facility.

1.2 Thesis Contributions

In this thesis, several contributions are added to the literature. The main contributions are described

in paragraphs below.

One of the main contributions of this thesis is to propose a non-probabilistic risk-constrained

operation scheduling for a merchant CAES plant based on IGDT method. The IGDT method is

used for decision making problems in an uncertain environment [19, 20]. The method enables

the decision maker to formulate optimistic and pessimistic self-scheduling problems without any

assumption on the probability distribution function of the uncertain parameter and with low com-

putational load. Moreover, an algorithm for constructing hourly offering and bidding curves are

presented based on the proposed IGDT-based optimistic and pessimistic scheduling problems in

order to incorporate different levels of risk of price forecasting errors, which is the next contribu-

tion of this thesis. This method enables the facility operator to not only makes its strategy robust

against unfavorable price deviation by including pessimistic bids and offers, but also take advan-

5

tages of favorable price spikes by considering optimistic bids and offers in the constructed bid and

offer curves.

Another contribution of this thesis is to conduct economic feasibility assessment of large-scale

energy storage systems providing energy arbitrage in the Alberta electricity market considering

impacts of storage operations on market price. To do so, the self scheduling problem of a merchant

price-maker energy storage plant is addressed. The impact of storage discharging operation on

market clearing price is modeled by means of generation price quota curves [1]. Demand price

quota curves are defined in order to incorporate the impacts of charging on market price as well.

The developed non-linear formulation is converted to its equivalent linear formulation to be able

to solve the problem by conventional solvers. Moreover, a detail analysis is conducted on the

historical data of the Alberta electricity market including actual hourly supply curves, pool prices,

and market equivalent demand to construct the hourly GPQCs and DPQCs of the Alberta electricity

market during years 2010 to 2014. The developed self-scheduling model is then applied to the

database to explore the potential revenue of providing energy arbitrage by a price-maker energy

storage facility. The impact of CAES component sizes on the economics of the facility are also

investigated through sensitivity analysis. The results demonstrates the importance of incorporating

the impact of energy storage operation on the electricity price in economic analysis and emphasize

on the fact that ignoring its impact causes high errors in results and overestimation in the potential

revenue of energy storage facility.

In the previous section, the actual historical DPQCs and GPQCs are used, since the focus is

on evaluating the economic feasibility of a storage facility based on historical market data. In

other words, I say what would have been the revenues if this facility was in operation and had a

perfect knowledge of the market. The outcomes are the upper bound of the economic feasibility,

and real-life uncertainties could affect the economics of the facility. Thus, as the other important

contribution of this thesis, a bidding and offering strategy for a large-scale price-maker independent

ES facility is developed to participate in day-ahead electricity market. The linear self-scheduling

6

formulation of an energy storage facility, developed in the previous section, is employed. Then, a

robust-based optimization platform is presented to model the uncertainty associated with GPQCs

and DPQCs. Robust optimization has been used in the literature to model the price uncertainty and

develop participation strategies of a price-taker generation company [24–26] as well as a consumer

[27]. In the case of a generation company (consumer), to solve the robust optimization, the worst

case of price deviation is easily determined as the lower (upper) bound of the confidence interval

[24,27]. Unlike for the case of a generator or a load in which determining the worst case scenario of

the uncertain parameter is straightforward, for the case of a price-maker ES facility, the worst case

scenario of the GPQCs and DPQCs depends on the charging and discharging status of the facility,

which needs to be properly modeled. Based on the developed robust scheduling formulation, I

present an algorithm to construct hourly multi-step bidding and offering curves for the storage

facility for participation in a day-ahead market. L. Baringo et al. [24] develops offer curves for a

price-taker thermal unit based on robust optimization using price subintervals. For the case of a

price-maker ES facility, this becomes more challenging; since, incorporating GPQCs and DPQCs

to model the price impacts of the ES facility, the developed algorithm in [24] needs to be modified

and extended further to construct not only offering curve for selling but also bidding curve for

purchasing electricity. The results shows significantly better performance of the proposed strategy

compared to the risk-neutral one.

The last main contribution that I develop is an optimization-based self-scheduling model of

a merchant CAES facility participating in day-ahead energy and reserve markets to investigate

the potential revenue gained from providing energy arbitrage, spinning and non-spinning reserves.

Moreover, in the CAES scheduling model presented in the literature, a constant efficiency para-

meters is considered for the CAES facility. However, it is shown that the efficiency of the CAES

facility is not constant. Conversely, it depends on the operational condition of the facility [2, 23].

Hence, the thermodynamic characteristics of the CAES technology is incorporated in the self-

scheduling problem to properly model the changes in the facility efficiency in different operational

7

conditions. The developed non-linear formulation is converted to its equivalent linear formulation

to be solved by conventional solvers. The results shows that the profit of the CAES facility could

be substantially improved by participating in both energy and reserve markets than that of only

energy market. Furthermore, the results demonstrates that in case of providing only energy arbi-

trage, the conventional model with nominal efficiency has acceptable accuracy. However, in case

of participating in energy and reserves markets, the results of the conventional model has signifi-

cant error, which illustrates the importance of considering CAES thermodynamic characteristics in

its scheduling.

1.3 Thesis Organization and Structure

The rest of this thesis is organized as follows: Chapter 2 develops a risk-constrained bidding and

offering strategy for a price-taker CAES facility in a day-ahead energy market. Chapter 3 con-

ducts a comprehensive economic feasibility study for a large-scale energy storage facility in the

Alberta electricity market incorporating the operation impacts on market prices. Chapter 4 deve-

lops a robust-based strategy for a price-maker energy storage facility to participate strategically

in the energy market considering the price forecasting errors. Chapter 5 addresses the scheduling

problem of a CAES facility when participating in energy and reserve markets.

This is a manuscript-based thesis. Chapters 2, 3, and 5 are published paper. Chapter 4 has been

submitted for publication. The articles have only been modified to fit the formatting requirements

for a thesis submission. Additional descriptions of the chapters are as follows.

In Chapter 2, the problem of CAES participation in a day-ahead energy market is addressed.

The market price uncertainty is modeled based on the IGDT method and then applied to develop

a risk-constrained self-scheduling model for a CAES facility. This chapter has been published in

IEEE Transaction of Power systems [28].

In Chapter 3, a linear optimization-based formulation for a price-maker energy storage facility

is proposed. GPQCs and DPQCs are utilized to model the impacts of discharging and charging on

8

market prices. Then, the historical hourly GPQCs and DPQCs of the Alberta electricity market are

constructed and applied to investigate the economic feasibility of energy storage systems in this

market.

Chapter 3 has been published in Energy [29]. Mr. Payam Zamani Dehkordi is a co-author of

this paper. He helped me in extracting relevant historical data of the Alberta electricity market

from the AESO website [30] and developing the database consists of hourly GPQCs and DPQCs.

I have developed the optimization formulation and done the simulations and related analyses.

Chapter 4 develops multi-step bidding and offering curves for a price-maker ES facility in

day-ahead energy market. GPQCs and DPQCs are utilized to model the impacts of discharging

and charging on market prices. Moreover, the uncertainty associated with GPQCs and DPQCs are

modeled based on robust optimization. This chapter has been submitted as a manuscript [31] to

IEEE Transaction on Smart Grid in April 2017.

In Chapter 5, I propose an optimization-base scheduling framework for a CAES facility pro-

viding energy arbitrage, spinning as well as non-spinning reserves in an electricity market. The

thermodynamic characteristics of the CAES technology is studied and applied to model the va-

rying efficiency of the facility in different operation conditions. This chapter has been published

in IEEE Transaction on Smart Grid [32].

I provide concluding remarks and future extensions to the research in Chapter 6.

9

Chapter 2

Risk-Constrained Bidding and Offering Strategy for a

Merchant Compressed Air Energy Storage Plant 1

Nomenclature

Index

t Index for operation intervals running from 1 to T .

Parameters

πEt Forecasted day-ahead electricity market price for interval t.

πNGt Natural gas price.

Emin minimum level of air storage.

Emax maximum level of air storage.

Eint Initial level of air storage.

ER Energy ratio.

HRd Required heat rate of CAES for discharging mode.

HRs Required heat rate of CAES for simple cycle mode.

Pexpmax maximum generation capacity of expander.

Pcmax maximum compression capacity of compressor.

Rr Robust profit level.

Rop Opportunistic profit level.

VOMexp Variable operation and maintenance cost of expander.

1 c© 2016 IEEE. Reprinted, with permission, from [28]: S. Shafiee, H. Zareipour, A. M. Knight, N. Amjady,and B. Mohammadi-Ivatloo, ”Risk-constrained bidding and offering strategy for a merchant compressed airenergy storage plant,” IEEE Transactions on Power Systems, vol. 32, no. 2, pp. 946957, March 2017.

10

VOMc Variable operation and maintenance cost of compressor.

Variables

αx Horizon of uncertain variable in robust (r) or opportunistic (op) cases.

πEt Actual day-ahead electricity market clearing price at interval t.

OCt Operation cost of the plant at time t.

Pi,dt Power generation in discharging mode in operation interval t.

Pi,st Power generation in simple cycle mode in operation interval t.

Pct Power consumption in charging mode in operation interval t.

uxt Unit status indicator in either modes x, i.e., discharging (d), simple cycle (s), or

charging modes (c) (1 is ON and 0 is OFF).

2.1 Introduction

There is currently significant interest in the use of electrical energy storage systems [33]. For

example, the total installed electricity storage capacity in the U.S. is forecasted to grow from 22

GW in 2014 to 103-152 GW by 2050 [34]. Bulk energy storage systems can serve load-shifting

and peak capacity services. Furthermore, high electricity price volatility in some markets provides

a business opportunity for energy arbitrage by these storage technologies [9]. Compressed air

energy storage (CAES) is one of the mature bulk energy storage technologies with the capability of

storing large amount of energy. In addition to the Huntrof and McIntosh CAES Plants, which have

been operating for decades, there are more CAES projects either announced, under construction

or operating in recent years [8]. For instance, a 317 MW CAES facility with 96 hours of storage

is announced in Texas, the USA, which is scheduled to be commissioned by summer 2019 [35].

CAES technology has the ability to operate as a gas turbine in case the air reservoir is depleted [36].

This mode makes CAES technology different from other types of energy storage, since unlike other

energy storage technologies, CAES is able to follow its scheduling plan and take advantage of price

spike in case of empty storage reservoir.

11

Several studies have been presented in the literature that focus on optimal self-scheduling

CAES facilities and estimating the energy arbitrage revenue in different electricity markets [13,

37, 38]. The feasibility of improving the economics of the CAES technology by distributing com-

pressors near heat loads is analyzed in [39]. In these studies, it is assumed that accurate electricity

market price forecasts are available. Then, the self-scheduling problem of the storage plant is

formulated as a profit maximization one. However, it is evident from the literature that price fo-

recasting errors could vary from 5% to 36% depending on the forecasting method and market

structure [40]. It could significantly affect the economics of energy storage. Hence, price uncer-

tainty should be considered when developing operation scheduling and bidding/offering strategies

for the CAES unit.

Previous studies have investigated the bidding strategies of generation companies considering

price uncertainty [25,41–45]. A risk-neutral approach for self-scheduling is used in [41,42], assu-

ming perfect forecast is available. Stochastic programming is also applied to model market price

uncertainties in self-scheduling of wind and thermal GenCos [43, 44]. Robust optimization is em-

ployed to hedge the risks associated with price uncertainty in [25, 45]. Furthermore, information

gap decision theory (IGDT) is applied in [46–48] to manage the risk of price forecast fluctuations

in self-scheduling and bidding strategies of generation units.

Compared to a generation company or a large load, the story is somewhat different for a mer-

chant CAES facility. The storage plant should decide when to purchase electricity from the mar-

ket and also when to sell it to the market to maximize its profit considering the operational cost

as well as the cost of purchasing electricity. In this decision making problem, price uncertainty

should be incorporated to manage the associated risks. Accordingly, the CAES facility needs not

only an appropriate offering strategy for selling the electricity, but also a proper bidding strategy

for purchasing energy from the market. Self-scheduling and bidding strategy of different energy

storage facilities has been reported in the previous literature [16–18,49–52]. In [16–18], stochastic

programming is applied to optimize the operation of an energy storage facility co-located with

12

a wind farm considering the uncertainties associated with the market price and wind generation.

Stochastic programming is also used in [49, 50] to investigate the optimal bidding strategy for an

independent battery storage in electricity market. Stochastic programming requires large computa-

tional burden as well as the necessity of knowing the probability distribution function (PDF) of the

uncertain parameters. Robust optimization is also applied for the bidding strategy of a wind farm

combined with energy storage in [51, 52]. Robust optimization does not incorporate opportunis-

tic actions in risk-constrained scheduling in order to take advantage of favorable price variations.

Moreover, in studies focusing on the bidding strategy of an energy storage facility in electricity

market [16–18, 49–52], only single block hourly bids and offers are constructed. In the day-ahead

electricity market, there is the possibility of submitting multi-step bids and offers. Thus, the storage

operator should be able to submit multi-step bids and offers to the market for purchasing and sel-

ling the electricity considering different level of risk to manage the risk of price uncertainty more

effectively. Thus, constructing multi-step bidding and offering curves for an energy storage facility

is important.

In this paper, an IGDT-based risk-constrained bidding/offering strategy is proposed for a mer-

chant CAES, which participates in the day-ahead energy markets, considering price uncertainty

based on IGDT. The IGDT method applies to decision making problems in an uncertain envi-

ronment. The method enables the decision maker to formulate optimistic and pessimistic self-

scheduling problems without any assumption on the probability distribution function of the un-

certain parameter and with low computational load. Instead of maximizing plant’s profit based

on some assumptions on uncertain price fluctuations, the proposed robust formulation maximizes

the horizon of price uncertainty around forecasted value and finds a scheduling solution that gua-

rantees a certain pre-determined revenue. Further, the proposed opportunistic method optimizes

the operation schedule to benefit from the favorable price fluctuations. In other words, for the

opportunistic case, the IGDT method investigates the minimum favorable price variation so that a

higher profit than that of the expected one could be achieved. The proposed bi-level IGDT based

13

method is converted to its equivalent single level optimization for both robust and opportunistic

formulation. Then, the proposed IGDT-based robust and opportunistic scheduling problems are

applied to construct hourly offering and bidding curves to submit to the market for each hour, in

order to take different levels of risk of price prediction into account. This approach enables the fa-

cility operator to not only act conservatively in the market by including pessimistic bids and offers,

but also take advantages of favorable price spikes by considering optimistic bids and offers in the

constructed bid and offer curves. The simple cycle mode of operation for the CAES facility is also

integrated into the proposed approach to illustrate its importance when providing energy arbitrage.

It should be noted that the proposed strategy is for bidding and offering into the day-ahead market.

Participation in the real-time market is not considered in this study.

As a conclusion, the contributions of this paper can be stated as follows:

• Proposing a non-probabilistic risk-constrained operation scheduling for a merchant

CAES plant based on IGDT method.

• Proposing a process for constructing hourly bidding and offering curves to hedge

the risk associated with the price uncertainty in a day-ahead market considering a

combination of risk-averse and risk seeking strategies.

• Converting the proposed bi-level IGDT-based optimization problem for the robust

and opportunistic functions separately to their equivalent single level formulations.

The background on the CAES technology, a generic formulation for the self-scheduling pro-

blem of a CAES and the IGDT method are presented in Section 2.2. In Section 2.3, the proposed

IGDT-based robust and opportunistic self-scheduling optimization problems and their equivalent

single level formulation are proposed. Section 2.4 explains the process to construct separate hourly

offering and bidding curves. Simulation results are presented and discussed in Section 2.5. The

paper is concluded in Section 2.6.

14

2.2 Background

2.2.1 CAES self-scheduling Formulation

A merchant CAES plant designed for energy arbitrage purchases electricity during low price pe-

riods to power large compressors to compress air into underground salt caverns. The stored air is

later used to power modified gas turbines (air expanders) during peak price hours. Unlike other

energy storage technology, the CAES technology considered in this paper requires natural gas as

the input fuel. The natural gas supply provides a fraction of the output power during discharge

mode and also enables the facility to operate as a simple cycle gas generator when the stored air

is depleted. Other types of CAES that do not require NG have been proposed [7] but these are not

considered in this formulation.

The efficiency of a CAES facility is expressed based on its heat rate and energy ratio. Heat rate

expresses the amount of fuel burned per unit of peak electricity generated by the expander. Energy

ratio indicates the amount of energy that the compressor of the plant consumes per unit of energy

that the expander generates during the peak hours [13].

In this section, the objective function for the self-scheduling of a merchant CAES and the

associated constraints are described. The goal of the CAES plant is to maximize profit through

energy arbitrage as a participant in the electricity market. The objective function and constraints

for the self-scheduling optimization are as follows.

maxT

∑t=1

[(Pi,dt +Pi,s

t −Pct )×π

Et −OCt ] (2.1)

Subject to:

OCt = [Pi,dt × (HRd×π

NGt +VOMexp)] (2.2)

+[Pi,st × (HRs×π

NGt +VOMexp +VOMc)]

+ [Pct ×VOMc] ∀t ∈ T

uct +ud

t +ust ≤ 1 ∀t ∈ T (2.3)

15

0≤ Pct ≤ Pc

max.uct ∀t ∈ T (2.4)

0≤ Pi,dt ≤ Pexp

max.udt ∀t ∈ T (2.5)

0≤ Pi,st ≤ Pexp

max.ust ∀t ∈ T (2.6)

Emin ≤ Et ≤ Emax ∀t ∈ T (2.7)

Et+1 = Et +Pct −Pi,d

t ×ER ∀t ∈ T (2.8)

E(0) = Eint (2.9)

The first term of objective function (2.1) is the revenue from electricity sales to the market from

discharging the stored air or purely using gas, i.e., the simple-cycle gas mode and also the cost of

purchasing the electricity from the market. The second term of objective function represents the

operating cost of the plant. The operating cost is expressed in three terms in (2.2). These terms

are respectively operating cost of generation in discharging mode, operating cost of generation in

simple cycle mode, and the variable cost of compressor in charging mode. Note that in simple-

cycle mode, fuel consumption of CAES would increase from the optimal design point (almost

twice) [53]. A CAES plant is not likely to operate at these high heat rates unless forced by the

circumstances (e.g., there are high prices in the market to take advantage of). The operational

constraint is expressed in (2.3) i.e., the CAES can operate in only one specific mode at a time. The

charging and discharging power and energy limits of the CAES are specified by (2.4)-(2.7). The

dynamic equation for the storage level is provided by (2.8). The initial level for the air storage

cavern is specified by (2.9).

2.2.2 Information-Gap Decision Theory

The IGDT method is a non-probabilistic interval optimization-based method that formulates robust

and opportunistic formulations under uncertainty [19, 20]. Since it makes no assumption on the

probability distribution of the uncertain variable, it makes it useful when high level of uncertainty

exists or no consistent probability distribution is available due to lack of information [19]. The

16

method has been used for various decision-making problems under uncertainty in different areas.

Such applications include reserve networks planning for biodiversity conservation [54], life cycle

engineering design problems [55] and water source planning [56]. In the area of power system,

IGDT method has been applied to various decision making problems such as scheduling of electric

vehicle aggregator [57], restoration decision-making model for distribution networks [58], and

also self-scheduling and bidding strategy of thermal generation companies [46–48]. The reasons

stated in those studies for choosing this method include severely deficient information, high level

of uncertainty, no need for knowing the probability distribution function of uncertain parameters,

significantly lower computational burden compared to probabilistic methods, and simply managing

financial risk without additional computational cost.

From the risk-aversion perspective, the IGDT method maximizes the horizon of uncertainty

and finds a solution that guarantees a certain expectation for the objective. It is referred to as

robustness function. In the context of self-scheduling, assume a set of price forecasts for the next

day is available. A pre-determined level of profit is guaranteed by the IGDT-based self-scheduling

solution, if the observed market prices fall into a maximized price band centered at the forecast

prices. Furthermore, from the risk-seeking viewpoint, the IGDT method determines a minimum

favorable price variation such that a higher profit than expected could be achieved. This is referred

to as opportunity function.

The IGDT method consists of three components, system model, uncertainty model and perfor-

mance parameters [19], which are described in the following:

System Model

System model represents the structure of input/output of the system. Consider P and Γ as decision

variable and uncertain parameter, respectively. R(P,Γ) is the system model which is objective to

be maximized taking uncertainty into account. In our problem, R(P,Γ) assesses the storage profit

with respect to P as storage charge/discharge schedule and Γ as day ahead price.

17

Uncertainty Model

Different models for uncertainties are presented in [19]. The uncertainty model (Γ(α, πt)) repre-

sents the information about the uncertain parameter, which here is the day-ahead price. It basically

shows the gap between the known parameter πEt and what needs to be known πE

t . Envelope bound

model with πEt as the surrounding function is used in this paper. It can be expressed as follows:

Γ(α, πEt ) = {πE

t : |πEt − π

Et | ≤ απ

Et } α ≥ 0,∀t ∈ T (2.10)

This model is a variant of the envelope-bound information-gap model. In this mode, maximal

variation is proportional to the forecasted value. In [19], α is the horizon of uncertain parameter.

The larger α is, the wider possible variation range for the uncertain parameter would be. The

reason for selecting this uncertainty model is that it represents the hourly relative absolute error.

In price forecasting literature, the mean of this error is calculated over an operation period (e.g.

24 hours) and commonly used as mean absolute percentage error (MAPE). Moreover, based on

(2.10), since the range of uncertainty is determined by απEt , wider error band is yield for high

price hours, i.e., higher error in price forecast could happen for higher price hours. Note that, the

function πEt determines the shape of the envelope. One may use another function for the shape of

the envelope, based on the nature of the price patterns and forecasting methods to capture price

volatility. For instance, instead of using the price forecasts πEt for all the hours, the forecasts can

be scaled by a controllable factor for expected more price volatile hours, which reflects the level

of fluctuation of those hours.

Performance Parameter

Based on the decision maker attitudes, two different performance functions could be developed in

an information-gap model i.e., robustness function corresponded to a risk-averse decision maker

or opportunity function corresponded to a risk-seeker decision maker.

A robustness function tries to find the maximum permissible deviation of price variation so that

18

the minimum pre-determined profit could be obtained. This can be expressed as (2.11).

αr(P,Rr) = maxP{α : min

πEt ∈Γ(αr,πE

t )R(P,Γ)≥ Rr} (2.11)

Equation (2.11) can be interpreted as determining the charging/discharging schedule of the

CAES by maximizing the possible range of unfavorable price variation ensuring the pre-determined

profit. Thus, following the obtained schedule could guaranteed the pre-defined profit if the uncer-

tain price falls into the maximized confidence interval defined by α .

An opportunity function investigates the minimum favorable price variation so that a higher

profit than that of expected could be achieved which formulated as (2.12).

β (P,Rop) = minP{αop : max

πEt ∈Γ(αop,πE

t )R(P,Γ)≥ Rop} (2.12)

If the uncertain parameter (e.g., price in this paper) favorably deviated from the forecasted

value by at least β , the higher profit Rop than what expected could be gained.

2.2.3 Characteristics of the IGDT method

A key feature of the IGDT method is that it can tie forecast error intervals to optimal scheduling. If

the user has a forecasting method with a reasonably consistent past performance, the IGDT method

enables the user to find schedules that are optimal considering the performance of the forecasting

system.

Furthermore, in the IGDT method, no assumption is made about the probability distribution

function of the uncertain parameters. This is important when the level of uncertainty is high and

finding a probability distribution for the uncertain variable is challenging. Given the high volatility

of electricity prices and existence of price spikes [40], determining a price probability distribution

over time may not always be possible. In probabilistic methods, on the other hand, it is necessary

to assume a probability distribution function for the uncertain parameters to be able to generate

scenarios. Compared to probabilistic methods where the computational burden could sometimes

19

be problematic [59,60], the IGDT method can handle high uncertainty levels with lower computa-

tional burden. Furthermore, unlike probabilistic methods where the risk is modeled using metrics

such as value at risk (Var) [61] and conditional value at risk (Cvar) [51] by adding some additional

constraints to the model, the risk is modeled in IGDT along with uncertainty by setting a guaran-

teed level of profit without adding computational cost to the problem. Thus, probabilistic methods

give a probabilistic risk metrics (e.g., Var or Cvar), whereas IGDT gives a confidence interval and

guarantees to achieve a predefined profit level if the uncertain parameter falls into the maximized

confidence interval.

Moreover, the IGDT method enables the decision maker to develop both optimistic and pessi-

mistic solutions that guarantee a certain value for the objective. Hence, the IGDT method covers

the decision making problems under uncertainty from risk-averse and also risk-seeking viewpoints.

Robust Optimization and minmax methods, on the other hand, only look at worst case scenarios

of the uncertain parameters. Further, IGDT could be seen as more understandable from a decision

making point of view. This is because in IGDT method, the user sets the level of expected pro-

fit, whereas in Robust and mimmax optimization methods, the user sets the uncertainty level; it

is perhaps easier for a high-level decision maker to deal with profit determination decisions than

uncertainty quantification decisions [47]. Thus, IGDT is a reasonable and suitable approach for

determining bidding and offering strategies taking into account price forecasting errors.

2.3 The Proposed Methodology and Formulation

In this section, an IGDT-based self-scheduling is presented for a risk-averse as well as a risk-seeker

operator of a merchant CAES. In the developed approach, the decision variables are the amount of

power to be purchased or generated at each interval, and the uncertain parameter is the wholesale

electricity price. It is assumed that natural gas prices are known in advance.

20

2.3.1 IGDT-Based Operation Scheduling Formulation for a CAES Plant

For a risk-averse CAES plant operator, the IGDT model objective is to maximize an uncertainty

parameter, referred to as α here onward, while a minimum pre-determined profit is guaranteed.

To develop the formulation, we start with the risk-neutral model of (2.1)-(2.9). Since price is the

source of uncertainty, a price deviation, say4πEt , will be added to the forecasted price values, i.e.,

πEt , as follows:

maxPi,d

t ,Pi,st ,Pc

t

αr (2.13)

subject to:

R≥ Rr = R0(1−σ) (2.14)

(2.2)− (2.9)

R = min4πE

t

T

∑t=1

[(Pi,dt +Pi,s

t −Pct )× (πE

t +4πEt )−OCt ]

(2.15)

subject to:

−απEt ≤4π

Et ≤ απ

Et ∀t ∈ T (2.16)

The above optimization consists of two levels, i.e., the upper level maximization and the lower

level minimization. The upper level of the proposed optimization is to find the maximum price

deviation that would satisfy the pre-specified profit. The lower level determines the worst case

price deviations. Observe that R0 is the risk-neutral profit obtained by solving the risk-neutral

self-scheduling of (2.1)-(2.9). Pre-determined profit Rr is the factor of R0 defined by σ . The

level of Rr shows the level of risk that the risk-averse operator would be willing to take. In other

words, it controls the risk level of the operator in the risk-averse case. Lower value for the Rr

means the decision maker is more conservative and does not want to take much risk and prefer

to guarantee a lower level of profit by considering more pessimistic price forecast. Changing

21

the defined parameter σ changes the level of predefined profit, which implies changing the risk-

aversion level.

For a risk-seeker plant operator, a favorable price deviation could lead to a higher profit than

what is expected. Thus, the IGDT model objective is to find the minimum favorable price fluctua-

tion, referred to as αop here, while hoping to earn a higher profit.

minPi,d

t ,Pi,st ,Pc

t

αop (2.17)

subject to:

R≥ Rop = R0(1+δ ) (2.18)

(2.2)− (2.9)

R = max4πE

t

T

∑t=1

[(Pi,dt +Pi,s

t −Pct )× (πE

t +4πEt )−OCt ]

(2.19)

subject to:

−απEt ≤4π

Et ≤ απ

Et ∀t ∈ T (2.20)

The upper level optimization is to find the minimum price deviation that could lead to the

targeted pre-defined profit. The lower level explores the best case of price deviations. Pre-defined

profit Rop is the factor of R0 defined by δ . The level of Rop shows the level of risk the risk-seeking

decision maker is willing to take. The higher the level of Rop is, the riskier the decision maker is

and more optimistic price forecasts he is expecting. The defined parameter δ is used to change the

level of risk-seeking.

2.3.2 The Equivalent Single-Level Optimization

The bi-level optimization problems described earlier are converted to their single-level equiva-

lents in order to be solved with conventional solvers. In the following, the proposed approach is

presented for robust and opportunistic cases.

22

1) Robustness function: As mentioned before, through (2.15)-(2.16), the objective is to find

the worst case scenario of price fluctuation that satisfies a minimum profit. Thus, in the lower

level, the decision variable is ∆πEt and the objective is to minimize the utility’s profit. Since the

variables Pi,dt , Pi,s

t , and Pct are decision variables at upper level optimization, they can be considered

as constant parameter at lower level [47]. Hence, the objective (2.15) is a linear optimization and

consequently, the minimum profit of the objective (2.15), i.e., the worst case scenario, occurs in

one of the bounds of the permitted variation horizon. Mathematically:

4πEt =

−αrπ

Et if Pi,d

t +Pi,st −Pc

t ≥ 0

αrπEt if Pi,d

t +Pi,st −Pc

t ≤ 0,∀t ∈ T (2.21)

In plain language, (2.21) states that the worst case for charging is when the price is higher than

the forecasts; the worst case for discharging is when the price declines from the forecasted value.

The two terms of (2.21) can be expressed as follows:

(Pi,st +Pi,g

t −Pct )(4π

Et +απ

Et )≤ 0 ∀t ∈ T (2.22)

(Pi,st +Pi,g

t −Pct )(4π

Et −απ

Et )≤ 0 ∀t ∈ T (2.23)

When the storage is in charging mode, the first term of (2.23) is negative. In addition, since

−απEt ≤ 4πE

t ≤ απEt , the second term of (2.23) is equal or less than zero. Thus, in order to

satisfy (2.23), the second term must be zero, which leads to 4πEt = απE

t . In this case, (2.22) is

neutral. Similarly, it can be proved that during discharging or simple cycle mode, (2.22) forces

4πEt =−απE

t .

Using (2.22) and (2.23), the single level optimization for the robustness function can be derived

23

as follows:

maxPi,d

t ,Pi,st ,Pc

t

αr (2.24)

subject to:

R≥ Rr = R0(1−σ) (2.25)

R =T

∑t=1

[(Pi,dt +Pi,s

t −Pct )× (πE

t +4πEt )−OCt ] (2.26)

(2.2)− (2.9), (2.22)− (2.23)

2) Opportunity function: through (2.19)-(2.20), the objective is to find the best scenario of

price fluctuation that could lead to a higher profit than expected. In the following, the approach to

convert the bi-level optimization of (2.17)-(2.20) to a single-level optimization is proposed.

Similar to what described for the robust case, the objective function (2.19) is linear with respect

to the only decision variable 4πEt and thus, the objective happens in one of the bounds of the

uncertainty horizon. However, in contrast with the previous case, since we are looking for the

maximum profit, i.e., the best case scenario, it occurs in the opposite price bound compared to that

of the robustness function. Mathematically:

4πEt =

αopπE

t if Pi,dt +Pi,s

t −Pct ≥ 0

−αopπEt if Pi,d

t +Pi,st −Pc

t ≤ 0,∀t ∈ T (2.27)

(2.27) shows that the best case for charging is when the price is lower than the forecasts. In

contrast, the best case for discharging is when the price is higher than the forecasts. The two terms

of (2.27) can be express as follows:

(Pi,st +Pi,g

t −Pct )(4π

Et +απ

Et )≥ 0 ∀t ∈ T (2.28)

(Pi,st +Pi,g

t −Pct )(4π

Et −απ

Et )≥ 0 ∀t ∈ T (2.29)

When charging, (2.28) is bounded which leads to4πEt =−απE

t . Similarly, when generating

electricity, (2.29) is bounded, i.e.,4πEt = απE

t .

24

Accordingly, the single level optimization for the risk-seeker case can be formulated as the

following:

minPi,d

t ,Pi,st ,Pc

t

αop (2.30)

subject to:

R≥ Rop = R0(1+δ ) (2.31)

R =T

∑t=1

[(Pi,dt +Pi,s

t −Pct )× (πE

t +4πEt )−OCt ] (2.32)

(2.2)− (2.9), (2.28)− (2.29)

The optimization problem of both robustness and opportunity functions are mixed integer non-

linear programming (MINLP) which can be solved using commercial MINLP solvers, such as

SBB [62]. For all the case studies in this work, the SBB solver was able to find the solution in

less than a few seconds using the GAMS platform [63]. It should be mentioned the nonlinear

IGDT-based proposed model with multiplication nonlinearity, known as bi-linear in the literature,

can be linearized using reformulation-linearization technique [64] or using the linear cutting plane

algorithms [65] with the cost of some over simplifications. However, the focus of this paper is not

linearizing the IGDT-based model.

2.4 The Proposed Method for Bidding and Offering Strategy

In order to sell and purchase energy, the storage plant needs to submit its hourly offers and bids to

the market. Thus, an appropriate bidding and offering strategy is required. The proposed IGDT-

based robust and opportunistic scheduling formulations are applied to build up the strategy, i.e.,

construct offering and bidding curves. The method for constructing the offering curve is in line

with the approach used in [47]. However, the approach in [47] is extended to incorporate opportu-

nistic actions described in Section 2.3. Moreover, an algorithm for constructing the bidding curves

for purchasing electricity is also presented. The approach not only guarantees a minimum level

25

Figure 2.1: The process of constructing a 4-step bidding curve

of profit, but also takes advantage of desirable price fluctuations. In the following, the process of

simultaneously building offering and bidding curves is presented.

The CAES operator tends to submit descending bidding for purchasing electricity and ascen-

ding offering curves for selling electricity. The lower the price is, the more power the operator is

willing to purchase. The operator is also willing to sell more power for higher prices. Both robust

and opportunistic actions are taken into account when forming the bidding and offering curves. As

an example, Figs. 2.1 and 2.2 illustrate the process of constructing a 4-step bidding and a 4-step

offering curve. To do so, we select four levels of profit below and above the expected profit (Rex),

i.e., Rr1 ≤ Rr2 ≤ Rex ≤ Rop1 ≤ Rop2. Then, the proposed IGDT-based scheduling is applied se-

quentially. For each level of profit, it determines the optimum confidence level, the corresponding

price profile, and the corresponding hourly charging and discharging quantity. Using the obtained

results, the hourly bidding and offering curves are constructed. In the following, the procedure for

constructing each step of the bidding curve for a specific hour of charging period is described.

Step 1

According to the results of the first profit level, for each hour of charging period, the obtained price

level is considered as the bid price. This level of price is actually the upper bound of the confidence

26

Figure 2.2: The process of constructing a 4-step offering curve

interval, i.e., the worst case. The corresponding charging quantity, i.e., Pct,1, is also considered as

the charging power. If the market price is lower than the bid price, the CAES would purchase the

submitted amount of electricity.

Step 2

After defining the first step, the results of the second profit level is employed. Higher level of profit

compared to the first level leads to a lower value of αr2 ≤ αr1. The market price needs to be lower

to justify purchasing more power. The difference between the charging quantity of this step and

the previous one, i.e., Pct,2−Pc

t,1 is considered as the bidding quantity. The corresponding prices is

chosen as the bid price. This level of price is actually higher than the forecasted value by term of

αr2.

Step 3

For the third level of profit, the proposed IGDT-based scheduling determines the minimum favo-

rable price deviation, i.e., αop1, the corresponding scheduling and price profile. Thereafter, for

charging hour, the defined price level is the bid price. It is the lower bound of the confidence inter-

val, i.e., the best case as determined by the proposed method. Furthermore, the difference between

27

the quantity of this step and the previous one, Pct,3−Pc

t,2, is considered as the bid quantity for this

step of hourly curves. The next block of hourly bidding curves is determined in a similar way.

Similar process is employed to construct the hourly offering curves. The difference is that,

during discharging hours, for the robust cases, i.e., Rr1 and Rr2, the IGDT-based optimization

determines the price level as the lower bound of the confidence interval, which is the worst case

scenario. For the opportunistic part, i.e., Rop1 and Rop2, the price level is the higher bound of

confidence interval, i.e., the best case scenario of discharging.

2.4.1 Sequential Constraints

As depicted in Figs. 2.1 and 2.2, since the quantity submitted to the market for either purchasing or

selling at each step must be greater or equal than that of submitted in the previous step, constraints

(2.33)-(2.34) are added to both robust as well as opportunistic problems.

Pct (R1)≤ Pc

t (R2), ∀t ∈ T,R1 ≤ R2 (2.33)

[Pi,dt +Pi,s

t ](R1)≤ [Pi,dt +Pi,s

t ](R2),∀t ∈ T,R1 ≤ R2 (2.34)

Starting from the lowest profit level, constraints (2.33)-(2.34) are updated sequentially for hig-

her level of profit. Fig. 2.3 shows the flowchart summarizing the process of sequentially con-

structing the steps of bid and offer curves.

It should be mentioned that with the proposed bidding/offering strategy, the storage plant might

face an infeasible schedule in the case when charging bids are accepted while storage reservoir is

full. In such case, the storage facility would be in a situation where it is committed to buy for

the market but has no room to store the energy. Such situations could happen for other markets

participants too. For example, a wind producer participating in the market may forecast a level

of available power that is less than what is actually produced. Or a retailer may bid in day-ahead

market to buy power but the real-time demand turns out to be less than expected. In such cases,

the market participants need to adjust their schedules in the real-time market buy selling or buying

the extra/deficit energy.

28

Figure 2.3: The sequences of defining steps of the bid and offer curves

29

2.5 Numerical Example

Numerical simulations are performed for a CAES facility with 150 MW of discharging power, 100

MW of charging power, and 20 hours of full discharging capability as the storage capacity. Heat

rate, energy ratio, and VOM of expander and compressor are taken from [39]. The required heat

rate for the simple cycle mode is assumed to be twice as that of discharging mode [39]. The price

of gas is assumed to be 3.5 $/GJ. A 24-hour scheduling period is considered and the initial storage

level is set to zero. Fig. 2.4 depicts the forecast prices for a typical 24 hours period.

2.5.1 Risk-constrained Self-scheduling: A Demonstrative Case

This section demonstrates the application of the proposed robust and opportunistic self-scheduling

approach proposed in section 2.3 for one typical day. At first, risk-neutral case is considered as

the reference, in which the forecasted price is applied to schedule the plant, i.e., deterministic

scheduling. Then, the CAES self-scheduling problem is solved from both risk-averse and risk-

seeking perspectives by applying the robust and opportunistic IGDT-based scheduling methods,

respectively. Solving the risk-neutral self-scheduling leads to $31,905 profit, which is considered

as the expected profit, i.e., R0. Fig. 2.5 shows the corresponding risk-neutral scheduling. In this

case, the robustness and opportunity parameter are set to 0.25, i.e., σ = 0.25 and δ = 0.25. In

other words, 25% of risk aversion and 25% of risk-seeking is chosen as the risk levels for the

robust and opportunistic cases, respectively. The robust case finds the maximum unfavourable

deviation in price that limits the reduction in operating profit to within 25% of the the risk neutral

case. The opportunistic case finds the minimum favourable deviation in price required to return

a 25% increase in profit. Fig. 2.4 depicts the price forecast, the worst case and the best case

of price fluctuation for those two pre-determined robust and opportunistic profit level. Fig. 2.5

also illustrates the corresponding hourly scheduling plans of the CAES plant for two robust and

opportunistic cases. In the following, the results of each case are discussed in more details.

30

Figure 2.4: Forecasted price, the worst case of price for robust case and the best case of price foropportunistic case

Figure 2.5: Scheduling of storage for risk-neutral, robust and opportunistic cases

Robust scheduling

The robust optimization is solved with the predetermined profit level Rr1 =R0(1−0.25)=$23,929.0.

Fig. 2.4 shows the worst case of price deviations that guarantees Rr1. It indicates that Rr1 could

be achieved if the unfavourable hourly price deviation is less than αr = 8.9% for charging and dis-

charging hours. In another word, if the hourly cleared prices during charging (discharging) hours

deviate unfavorably no more than 8.9% from forecasted value, i.e., go above (below) the forecasts

no more than 8.9%, the predefined level of profit could be achieved. Fig. 2.5 shows the correspon-

ding robust scheduling. The comparison of these two figures show that during charging hours, i.e.,

1-7, and 9, higher price than the forecast is considered as the worst case. Conversely, during the

generation hours, i.e., 12-14, and 21-24, lower price than the forecasts is considered as the worst

case scenario. Since, from a risk-averse decision maker’s view point, the worst case of price devi-

ation is considered as a higher and lower price than forecasts for charging and discharging periods,

respectively. The results in the figure verify that the proposed method is able to effectively find the

31

worst case of price deviation and then determine the optimal scheduling that could guarantee the

predetermined profit level.

Figure 2.5 shows that for either cases, the scheduling plan follows the price pattern. In other

words, it purchases electricity to store the compressed air during off peak when the prices are low

and in contrast, sells the energy during peak hours which coincides high price periods. However,

comparing the risk-averse and risk-neutral scheduling indicates that taking lower level of risks, i.e.,

σ = 0.25, would lead to lower hours of charging and discharging or lower charging and discharging

power values compared to that of the risk-neutral case, and consequently gaining lower profits. In

contrast, in the risk-neutral case, the storage plant is committed to charge and discharge for more

hours than that of σ = 0.25; since in order to earn a higher profit, higher level of risk should be

taken. Thus, as depicted in Figs. 2.4 and 2.5, the operator would not consider any unfavourable

price deviations and consequently decides to charge more during off peak hours in order to generate

more power during peak period.

Opportunistic scheduling

In this case, δ is considered to be 25% as the level of risk-seeking, which means the plant is

looking to make 25% more profit by taking risk and looking for some favourable deviations in

the actual prices with respect to the price forecasts. Fig. 2.4 and Fig. 2.5 show the best cases of

price deviation and the corresponding opportunistic scheduling, respectively. The results show that

the favourable price deviation to gain additional 25% profit should be at least 7.95% for charging

and discharging hours. In another word, if the hourly cleared prices during charging (discharging)

hours fluctuate favorably at least by 7.95% from forecasted value, i.e., go below (above) the fore-

casts at least by 7.95%, the predefined level of profit could be gained. Moreover, the comparison

of these two figures shows that during charging hours, lower prices, and during discharging hours,

higher prices than the forecasts are of interest. It can also be observed from Fig. 2.5 that in the

opportunistic scheduling, the charging or discharging power of the storage unit is higher than that

of risk-neutral or risk-averse. Since, in order to gain higher profit, the schedule should be optimis-

32

tic to price deviations. Thus, as a general observation, increasing the profit level would require an

increase in the charging and discharging power of the unit compared to the risk-neutral one (e.g.,

at hour 10 for charging, and at hour 16 for discharging) or at least, no change (e.g., at hours 1-9 for

charging, and hours 12-14, and 21-24 for discharging).

2.5.2 Constructing Biding/Offering Curves Based on the Obtained Results from IGDT-based

Scheduling Cases

The results obtained in Section 2.5.1 are employed to construct the steps of the bidding and offering

curves. Without loss of generality, three step curves are selected for this study. To construct

the curves, at first, three levels of profit are selected. For this case study, one robust level i.e.,

σ = 0.25, one risk-neutral level, and one opportunistic level i.e., δ = 0.25 are chosen. Then,

the proposed IGDT-based scheduling approach is applied for each of the profit levels, which are

presented and discussed in the previous subsection through Figs. 2.4-2.5. Thereafter, based on the

obtained results, the steps of the bidding and offering curves, i.e., the amount of power and the

corresponding price, are determined.

The algorithm proposed in section 2.4 is employed to construct the bidding and offering curves.

According to Fig. 2.4, charging periods is between hours 1 to 10, i.e., the off peak hours. As an

example, Fig. 2.6 shows the constructed bid curve for hour 7. Based on Fig. 2.5, in this hour,

the storage unit is committed to compress the air with 87.5 MW of power corresponded to the first

profit level. Thus, this amount of power is submitted for purchasing with the corresponding price

of the first profit level i.e., $32.6/MWh; since in this hour, for second profit levels , the unit charges

with 100 MW, the difference i.e., 100−87.5 = 12.5 MW, is submitted with the price determined

for the second profit level, which can be found in Fig. 2.4, i.e., $29.9/MWh. The third step of the

curve is zero, since, based on Fig. 2.5, for the opportunistic case, the unit also charges with 100

MW in this hour. Thus, the opportunistic step of Fig. 2.6 is aligned with risk neutral step.

Similarly, for offering curves, according to Fig. 2.4, discharging periods is between hours 12

to 24. For instance, Fig. 2.7 illustrates the constructed offer curve for hour 15. In this hour, 133.5

33

Figure 2.6: Bid curve for hour 7.

Figure 2.7: Offer curve for hour 15.

and 16.5 MW are submitted to the market for selling with the prices determined for the second and

third profit levels, respectively. This is because 0, 133.5 and 150 MW of discharging is considered

for the mentioned hour, as can be seen in Fig. 2.5, for the first, second, and third profit levels,

respectively. Note that, the offer curve has two step instead of three step, since as shown in Fig.

2.5, in robust case, the unit does not discharge. Hence, in Fig. 2.7 , the robust step has zero value.

Similarly, the bidding/offering steps for other hours are created.

2.5.3 After-the-Fact Analysis Based on Constructed Bidding and Offering Curves and Simulated

Prices

In this section, different simulated market prices are generated. The simulated prices are distri-

buted randomly around the forecasted values with uniform density function by different range of

variation. Then, the submitted offers and bids, constructed in Section 2.5.2, are applied to inves-

tigate how the bidding and offering strategy, proposed in Section 2.4, would work. The effects of

34

price fluctuation on the operation of the CAES unit and the gained profit are also explored.

As a reminder, the guaranteed level of profit is $23,929.0 considering the risk level σ = 0.25

for the risk averse case. The would-be profit of the merchant CAES facility is calculated assuming

that the facility participate strategically and the simulated prices are observed in the market. In

each scenario, based on the bid and offer curves constructed in section 2.5.2 and the simulated

prices, the accepted bids and offers are determined. The gained profits are calculated accordingly,

which are presented in table 2.1. According to this table, as the price forecasting e,

For the sake of comparison, another case is also considered, in which the CAES facility ope-

rates as a self-scheduling plant considering forecasted price for its scheduling. A self-scheduling

generator submits its schedule (hourly generation quantity) to the market and then, follows it in

real time regardless of market price [66]. The market operator does not send this kind of ge-

nerators dispatch instructions [66]. In our case, it is assumed that the self-scheduling of CAES

facility is defined in a deterministic way based on the forecasted price depicted in Fig. 2.4. In this

case, the expected profit based on the forecast prices is $31,905.0. The resulted deterministic self-

scheduling is the same as risk-neutral self-scheduling shown in Fig. 2.5. Hence, the CAES unit

operates based on the resulted scheduling in this figure, regardless of market price. Then, based

on the charge/discharge scheduling and the simulated market price in each scenario, the would-be

gained profit in that scenario is calculated, which are shown in Table 2.1. Observe from the table

that the gained profit with deterministic self-scheduling is lower than that of IGDT-based strategy

in the three scenarios. This is because of taking the risk of forecasting error into account in the

developed strategy. Thus, in one hand, it makes the strategy robust against undesirable forecasting

errors and prevents unprofitable actions. On the other hand, considering optimistic actions in the

proposed strategy enables the plant to take advantages of desirable forecasting errors.

According to Table 2.1, as the forecasting error gets higher, the level of gained profit decreases.

However, comparing the obtained profit with the guaranteed profit shows that the gained profits in

scenario 1 and 2 are higher than the guaranteed one. It shows the robustness of the proposed

35

Table 2.1: After-the-Fact Analysis using simulated prices

Scenario Range ofprice variation

Strategicgained profit

Self-schedulinggained profit

1 ±5% $26,600 $25,9262 ±10% $26064 $25,6733 ±20% $22,860 $ 22,354

0 24 48 720

50

100

150

200

250

300

350

Time (Hour)

Pric

e($/

MW

h)

Actual PriceForecasted Price

Figure 2.8: Hourly forecasted and actual prices

bidding and offering strategy. In scenario 1, price forecasting error, 5%, is less than the maximum

allowable price deviation of first profit level , 8.9%. As a result, the gained profit is higher than

the corresponding guaranteed level of profit, presented in Table 2.1. For the second scenario,

the variation slightly exceeds the allowable interval. However, due to considering opportunistic

actions, the gained profit is higher than the guaranteed level. For the third scenario, due to high

forecasting error, the guaranteed profit level is not achieved. However, the performance of the

developed bidding and offering strategy is better than the deterministic self-scheduling.

2.5.4 After-the-Fact Analysis Using Actual Market Data

Three-Day Analysis

In this section, the same analysis as the previous section is applied sequentially using the actual

and forecasted price of three-day period in the New England electricity market. Fig. 2.8 depicts

the actual and generated forecasted prices for the three days.

In this analysis, the arbitrary values of σ = 0.25, σ = 0, i.e., risk-neutral, and δ = 0.25 are used

36

as the risk levels to construct hourly offering and bidding curves. Thus, the profit level 25% lower

than the deterministic risk-neutral profit is selected as the guaranteed profit, i.e., Rr = R0(1−0.25).

In this way robust actions are taken into account in the strategy. The two others profit levels are also

chosen to incorporate more optimistic actions in the bidding and offering strategy. The developed

strategy is applied for each day based on the price forecasts. Then, according to the actual market

price, the accepted offers and bids are determined. Accordingly, other outcomes, such as the daily

profit values, the state of charge, are calculated. The process is repeated sequentially for all days

of the week. In order to investigate the performance of the proposed strategy, Table 2.2 reports the

results of IGDT-based scheduling for the three days.

For day 1, according to Fig. 2.8, the price is mostly overestimated. Thus, all the bids for

purchasing the electricity are accepted, as presented in Table 2.2. On the other hand, none of the

10 supply offers are cleared due to prices lower than the forecast during most of the peak hours,

as shown in Fig. 2.8. Thus, as provided in Table 2.2, the unit does not make any profit and only

purchases energy, at a total cost of $29,805. Through this process it charges the air reservoir to the

level of 1,062.0 MWh at low prices. Consequently in the second day, the proposed strategy leads

to significantly high profit. As shown in Fig. 2.8, due to lower prices than forecast during charging

periods and high price spikes during discharging periods, 4 out of 5 demand bids and 9 out of 14

supply offers are accepted. As a result of accepted bids and offers, the storage facility not only

makes high level of profit, $125,130, but also retains 450 MWh of compressed air at the end of the

day. During the third day, the market prices are mainly underestimated, as seen in Fig. 2.8. Thus,

as presented in Table 2.2, none of the charging bids are accepted. However, during nine out of the

ten hours which discharging offers are submitted, market prices are higher than forecast, thus nine

supply offers are accepted. The accepted offers lead to $32807 profit which exceeds the guaranteed

level of profit. Note that since the initial storage level is 450 MW and none of the charging bids

are accepted in this day, after four hours of discharging, the storage reservoir is depleted. Hence

the unit switches to pure gas mode for the rest of the discharging hours, in spite of its inefficiency,

37

Table 2.2: After-the-Fact Analysis Using Actual Market Prices

Day Guarant.Profit [$]

αr(%)

No.Acc.Bids

No.Acc.Offer

SCHr.

GainedProfit

[$]

FSOC[MWh]

1 21383 6.55 10/10 0/10 0 -29805 10622 29011 15.91 4/5 9/14 0 125130 4503 11326 14.71 0/6 9/10 5 3807 0

Guarant.: Guaranteed, Acc.: Accepted, SC: Simple Cycle, FSOC: Final SOC

in order to follow the schedule.

Four-week Analysis

In this section, in order to show the performance of the developed bidding and offering strategy,

the strategy is applied daily to a period of four arbitrary weeks with the same risk levels as those

of the previous section. Then, the four-week simulation is repeated for several times with different

generated daily price profiles; in each run, the daily actual prices are distributed randomly around

the forecasted values with uniform density function by different range of variation which is chosen

randomly between %5 to %15. The cumulative guaranteed profit level for this period is $700,061±

23,503 whereas the total profit achieved using the proposed strategy is $816,243± 35,979. For

the sake of comparison, profit obtained by deterministic self-scheduling is also investigated and is

$45,379±22,164. As can be seen, the gained profit by the proposed strategy is significantly higher

than that of deterministic scheduling. This comparison demonstrates better performance of the

proposed strategy compared to the deterministic scheduling when high level of price uncertainty

exists. Furthermore, the results show that overall, the proposed strategy is capable of guaranteeing

a minimum level of profit by taking pessimistic actions into account and preventing uneconomical

charging or discharging actions. By incorporating optimistic actions, the decision maker is also

able to benefit from unforeseen price drops or spikes. These factors combine to provide a higher

profit than that obtained by deterministic self-scheduling.

38

2.6 Conclusion

This paper develops an IGDT-based risk-constrained bidding/offering strategy for a merchant

CAES facility taking price uncertainty into account. Robust actions in the proposed bidding/offering

strategy guarantees a minimum critical profit if the future prices fall within a maximized robust-

ness region. The opportunistic actions enable the plant to benefit from favorable price deviation

and potentially earn a higher profit. The numerical results verify the applicability of the develo-

ped strategic scheduling approach. In the robust scheduling case, the IGDT-based optimization

method is able to find the worst case of price deviation and then, determine the corresponding op-

timal scheduling which guarantees the predefined profit. For the opportunistic case, the desirable

price variation and the corresponding optimal scheduling are effectively defined for any level of

profit by the proposed opportunistic formulation. The obtained results from robust, risk-neutral,

and opportunistic cases are employed to construct the hourly offering and bidding curves. The de-

monstration clearly shows that in different price scenarios, the proposed strategic scheduling leads

to a more profitable scheduling than that of deterministic one. In other words, the strategy avoids

uneconomic actions. At the same time, it benefits from desirable price deviation and thus, higher

profits are achieved.

It should be noted that, although constant efficiency parameters are considered in the CAES

self-scheduling model, the efficiency of the CAES system somehow depends on its operation con-

dition. Considering this issue is not in the scope of this paper, and is left to future work.

39

Chapter 3

Economic Assessment of a Price-Maker Energy Storage Facility

in the Alberta electricity market 1

Nomenclature

Indices

s Index for the steps of generation price quota curves from 1 to ndt .

s′ Index for the steps of demand price quota curves from 1 to nct .

t Index for operation intervals running from 1 to T .

Parameters

µ Roundtrip storage efficiency.

πdt,s Price corresponding to step number s of the GPQC at hour t.

πct,s′ Price corresponding to step number s′ of the DPQC at hour t.

bd,maxt,s Size of step s of the GPQC at hour t.

bc,maxt,s′ Size of step s′ of the DPQC at hour t.

Emin Minimum level of energy storage.

Emax Maximum level of energy storage.

Eint Initial level of energy storage.

Pdmax Maximum discharging capacity.

Pcmax Maximum charging capacity.

qd,mint,s Is the summation of power blocks from step 1 to step s−1 of GPQC for hour t.

1 c© 2016 Elsevier Ltd. Reprinted, with permission, from [29]: S. Shafiee, P. Zamani-Dehkordi, H. Zareipour,and A. M. Knight, ”Economic assessment of a price-maker energy storage facility in the Alberta electricitymarket,” Energy, vol. 111, pp. 537 - 547, 2016.

40

qc,mint,s′ Is the summation of power blocks from step 1 to step s′−1 of DPQC for hour t.

VOMd Variable operation and maintenance cost of discharging.

VOMc Variable operation and maintenance cost of charging.

Variables

bdt,s The fractional value of the power block corresponding to step s of the GPQC to

obtain discharging quota Pdt in hour t.

bct,s′ The fractional value of the power block corresponding to step s′ of the QPQC to

obtain charging quota Pct in hour t.

Est Level of energy storage at time t.

OCt Operation cost of the plant at time t.

Pdt Discharging power of the storage unit at hour t.

Pct Charging power of the storage unit at hour t.

uxt Unit status indicator in either modes x, i.e., discharging (d) or charging modes (c)

(1 is ON and 0 is OFF).

xdt,s Binary variable that is equal to 1 if step s of GPQC is the last step to obtain dischar-

ging quota Pdt in hour t and 0 otherwise.

xct,s′ Binary variable that is equal to 1 if step s′ of DPQC is the last step to obtain charging

quota Pct in hour t and 0 otherwise.

Functions

πdt (P

dt ) Stepwise decreasing function that indicates the market price as a function of the

price-maker discharge quantity at time t.

πct (P

ct ) Stepwise increasing function that indicates the market price as a function of the

price-maker charge quantity at time t.

41

3.1 Introduction

The implementation of large-scale energy storage systems has been shown to be technically fe-

asible in the province of Alberta [67]. Such systems are able to provide load-shifting [68] and

potentially provide the necessary flexibility to deal with uncertainties associated with the growing

penetration of renewable resources [69–71]. Load shifting is one of the best-comprehended and

analyzed applications of energy storage , i.e., to buy and store electricity at low demand, low price

periods, and sell at high demand, high price periods [9]. This is referred to as energy arbitrage. It

has been shown that the dynamics of the Alberta electricity market and relatively high price variati-

ons provide desirable opportunities for energy arbitrage [10]. As an example, the hourly electricity

prices in this market for 2013 are shown in Fig. 3.1. Over the year, electricity prices averaged

$80.20/MWh. For 3208 hours, the price was below $25/MWh and for the remaining hours, the

average price was over $115/MWh with 204 hours settling between $800/MWh and $1,000/MWh,

the market price cap. As a result of this variation, energy storage systems have attracted the at-

tention of investors; in 2014, a 160 MW compressed air energy storage (CAES) plant was filed

with the Alberta Electric System Operator (AESO) in 2014 [72]. It is important for the investors

to know the potential profitability of a large-scale investment in bulk energy storage; economic

feasibility is the deciding factor for developing new energy storage facilities. Projects must be

attractive to capital from the investors’ viewpoint and it is crucial to evaluate the potential profit

available to be earned through energy arbitrage in the Alberta electricity market.

The profitability of providing energy arbitrage by energy storage systems in various electricity

markets are shown in [9,13,37–39,73,74]. These studies assume that the energy storage facility is

a ”price-taker”, i.e, storage operation in the market does not affect the pool price [75,76]. However,

in the case of a large-scale energy storage facility it can be assumed that charging and discharging

operations change the net demand and supply. As a consequence, a large-scale energy storage

facility can be expected to be a price-maker, i.e., its actions could affect the market price. A few

studies have modeled the impacts of energy storage operation on market price. The operation of

42

0 1000 2000 3000 4000 5000 6000 7000 8000 87600

200

400

600

800

1000

Hours

Pri

ce (

$/M

Wh

)

Figure 3.1: Hourly electricity price during 2013 in the Alberta electricity market

large-scale price-maker energy storage systems is optimized in [77]. The profitability of energy

arbitrage for a price-maker energy storage in the PJM [9], the Iberian Electricity Market [78, 79]

and the Alberta electricity market [80] is investigated. In [80], one representative supply curve is

considered for all the hours. The impact of energy storage charging and discharging operation on

market prices should be accurately formulated and historical hourly data should be employed to

achieve a better understanding of the energy storage profitability in the Alberta electricity market.

Several efforts have been devoted in modeling of price-maker generation companies (Gen-

cos). The developed modeling methods can be divided into two categories: game based and non-

game based. Game based methods aim to calculate the Nash Equilibrium in a market with a sin-

gle or multiple price-maker Gencos using the mathematical program with equilibrium constraints

(MPEC) approach and binary expansion techniques [81–84]. In [81, 82], the bidding strategy pro-

blem of a price-maker Genco is initially formulated as a bi-level optimization problem, consisting

of bidding strategy and market clearing problems in the upper and lower levels, respectively. Then,

using Karush-Kuhn-Tucker (KKT) optimality conditions, the problem is converted to its equiva-

lent single nonlinear MPEC problem. Binary expansion is used in [82] to transform the nonlinear

MPEC problem to a mixed-integer linear programming (MILP) form and then solve the bidding

strategy for one price-maker thermal generator in an electricity market. In [83], this work is further

extended to find the Nash equilibrium for a market with multiple price-maker firms. Bakirtzis et

al. [84] apply the approach in [82] to construct multi-step price-quantity offer curves for a single

43

Figure 3.2: An example of a typical a) GPQC, b) DPQC

price-maker producer.

In non-game based methods, the impact of a participant’s operation on the market price is mo-

deled by generation price quota curves (GPQCs) [85]. The GPQC for a given hour, is a stepwise

decreasing curve that indicates the market price as a function of the total accepted production of

the price-maker generator. Figure 3.2-(a) shows an example of a GPQC with steps of 10 MW up

to 100 MW. The use of GPQCs enables self-scheduling of price-maker producers to be formulated

efficiently [1, 86–88]. In [1], the self-scheduling problem of a price-maker thermal producer is

addressed using a MILP approach with PQCs. This work illustrates the efficient and proper functi-

oning of the proposed formulation. PQCs are used to address the short term operation planning of a

price-maker hydro producer in a day-ahead electricity market [86,87]. A mid-term self-scheduling

model for a price-maker hydro producer is developed in [88], in which PQCs are used to model

the producer’s interaction with other market participants.

This paper addresses the economic assessment of energy arbitrage for a large-scale energy

storage facility in the Alberta electricity market, considering its impact on pool prices. Self-

scheduling of a merchant price-maker storage plant is proposed, using an approach which incor-

porates the impact of storage operation on market clearing price by means of price quota curves.

The impact of large-scale energy storage discharging activities in the market is modeled by hourly

GPQCs. However, the storage plant must decide not only when to sell electricity to the market, but

also when to purchase the electricity from the market for charging with the lowest cost. Purchasing

electricity from the market will increase the demand and consequently may negatively impact mar-

ket prices. Thus, the impact of energy storage charging activities on market prices should also be

44

modeled in the self-scheduling problem in order to achieve an optimal scheduling solution. In so

doing, in addition to the GPQCs for discharging operations, an hourly demand price quota curve

(DPQC) is also defined here for discharging operations. The DPQC states how electricity price

changes as the demand quantity of the load changes. The DPQC is a stepwise non-decreasing

curve; the more power absorbed from the grid, the more the electricity price will increase. Hourly

DPQCs help a price-responsive load to participate efficiently in the market in order to meet de-

mand at the lowest cost. Figure 3.2-(b) shows an example of a DPQC with steps of 10 MW up to

100 MW.

The formulation, presented in this paper, is non-linear and is therefore converted to its equi-

valent linear formulation to be enable solution by conventional solvers. Thereafter, the historical

hourly supply curves and hourly pool prices of the Alberta electricity market for years 2010 to

2014 are extracted to construct the hourly GPQCs and DPQCs in this period. The developed

self-scheduling model is then applied to the historical hourly GPQCs and DPQCs of the Alberta

electricity market to investigate the economic feasibility of a price-maker energy storage during

these years. The model is used to explore the sensitivity of the storage plant profit to a range of

design and performance parameters.

The main contributions of this paper can be stated as follows:

• To develop a linear self-scheduling formulation for a price-maker energy storage

facility using hourly GPQCs and DPQCs

• To construct the hourly GPQCs and DPQCs of the Alberta electricity market during

years 2010 to 2014 using actual hourly supply curves.

• To assess the economic feasibility of large-scale energy storage systems providing

energy arbitrage in the Alberta electricity market considering impacts of storage

operations on market price.

The remainder of the paper is outlined as follows. Section 3.2 reviews the literature on the

economic assessment of energy storage systems in different electricity markets. In section 3.3, the

45

process of constructing GPQCs and DPQCs for the Alberta electricity market is described, and the

non-linear self-scheduling problem of a merchant price-maker storage and, its equivalent linear

formulation are developed. In Section 3.4, the developed model is employed. A base case of the

potential revenue gained by energy arbitrage is presented along with sensitivity analyses. Finally,

the paper is concluded in Section 3.5

3.2 Literature Review

This section reviews the literature on scheduling of energy storage systems and estimation of the

energy arbitrage value. The study presented in this paper is compared with the literature.

In the price-taker approach for self-scheduling of energy storage systems ( [9, 13, 37–39, 73,

74]), it is assumed that storage operation in the market does not affect the pool price. The arbitrage

value of a storage device in the PJM and New York are explored in [9] and [38], respectively. It is

shown that the value of arbitrage for an 80% efficient storage device has a range from $60/MW-

year to $110/MW-year in PJM market. The effects of natural gas and electricity price fluctuations

on the energy arbitrage revenue of a pure storage device and a CAES facility are investigated

in [37]. It is concluded that the annual arbitrage value of a CAES facility is lower than that of an

80% efficient pure storage device due to the effect of the cost of burning natural gas in the CAES

system. It is shown that the energy arbitrage revenue of a CAES facility could be improved by

distributing compressors near heat loads in the Alberta electricity market [39, 73]. The value of

providing energy arbitrage as well as operating reserves for energy storage systems in different

electricity markets in US and UK are evaluated in [13, 74]. The limitation of all all these studies

is the assumption that storage operations have no impact on price. However, purchasing and sel-

ling activities will change the net demand and supply, and consequently, the market price. Thus,

storage operation could reduce the peak price differential and consequently decrease the net energy

arbitrage revenue for storage operators.

An optimization framework for the optimal operation of a price-maker energy storage system

46

is developed in [77]. In this paper, MPEC approach is applied to model the price impacts of the

energy storage facility. The impact of large-scale storage on arbitrage value in PJM is investigated

in [9]. In that paper, in order to model the price impacts, only one linear supply curve is assumed

for each month based on the historical price and load data. The self-scheduling of a price-maker

PHS is developed in [78, 79]. In these papers, the impact on price is modeled by residual demand

curve, which is defined by an approximated sigmoid function. This leads to a mixed-integer non-

linear formulation. In [80], profitability of energy arbitrage by different storage technologies are

evaluated in the Alberta electricity market in year 2012 considering the price impacts. In this

report, one representative supply curve is considered for all the hours in this market.

Compared to [9, 13, 37–39, 73, 74], in this paper, the impacts of energy storage operation on

market price are taken into account, i.e., we consider the facility to be price-maker. This is im-

portant in the case of large storage facilities. Compared to [77], instead of MPEC approach, a

non-game based method using PQCs is used to model the impacts of energy storage operation on

market prices. The proposed methodology in the present paper makes it possible to use real-life

market data with high volume at a reasonable computational cost. Compared to [9], instead of

using a monthly supply curve, which is estimated using load and price data, actual hourly supply

curves and pool prices are extracted to model the impacts of energy storage operation on market

price for each hour. Compared to the model presented in [78, 79], instead of using an approxima-

ted function to model the residual demand curves, the GPQCs and DPQCs are constructed using

actual hourly supply curves. Moreover, compared to the non-linear model presented in [9, 78, 79],

a mixed-integer linear programming approach is developed in this study, the global optimality of

which is guaranteed. Compared to [80], instead of using one representative supply curve for all

hours in the Alberta electricity market, the hourly supply curves of this market for years 2010 to

2014 are extracted and used to investigate the economic feasibility of energy storage systems in

this market during recent years. The importance of considering hourly supply curves is going to

be explained later in Section 3.3.1. As a summary, the comparison of our work with the related

47

Table 3.1: Summaries of comparison with previous worksReference Price-taker/maker Modeling Approach for Impact on Price[9, 37–39][13, 73, 74] Price-taker N/A

[9] Price-maker Monthly linear supply curve[77] Price-maker MPEC-Game based

[78, 79] Price-makerapproximate residual demand curve(RDC) and nonlinear optimization

[80] Price-makerOne representative supply

curve for all hours

This Study price-makerHourly price quota curves

and non-game based linear optimization

literature are presented in table 3.1.

Tables

3.3 Methodology and Formulation

The methodology, used in this paper, requires pre-process of significant data in order to develop

the required DPQCs and GPQCs. These curves are the input to the nonlinear formulation, which

can then be linearized

3.3.1 The Alberta Electricity Market Database

In order to model the potential impacts of an energy storage facility, a first necessary step is to

build a historical database of market operations over the period of interest. In Alberta, the data

required is publicly available, published by the Alberta Electric System Operator (AESO) through

its online data-publishing portal [30]. The database for the study includes hourly generator offers

and pool price data for the period of January 1, 2010 to December 31, 2014.

In the Alberta electricity market, generators submit their offers in the form of quantity and

price pairs to sell energy in the market to the AESO through the Energy Trading System (ETS).

Sorting the price-quantity offers from the lowest-priced to the highest-priced, a supply curve is

48

Figure 3.3: Sample supply curve for hour ending 1, December 31, 2014

constructed for each hour. The sensitivity of market price to changes in supply or demand depends

on the market structure and particularly, the system supply curve. As an example, the supply curve

of the Alberta electricity market for the hour 1 on December 31, 2014 is shown in Fig. 3.3. As

can be seen, the supply curve is very steep towards the right side of the curve. This could cause

significant price fluctuations as a result of a relatively small change to supply offers. Moreover,

the steep supply curve could lead to a significant change in pool price if the demand in the market

varies. As a result, the impacts of storage operation on market pool prices in Alberta should be

incorporated in economic analyses in order to prevent profit overestimation.

As stated earlier, this study uses the actual supply curve data for each of the 43,824 hours in the

study period. The importance of using actual hourly data, rather than a ”representative” curve, can

be seen by considering Fig. 3.4 and Fig. 3.5. Figure 3.4 plots the supply curves for each of the 24

hours in a single day, August 1 2014; Figure 3.5 plots the supply curves for a given hour of all the

days in a specific month, hour 1 for each day of October 2014. Both plots demonstrate significant

variability in the supply curves. This is an indication that market participants submit their offers

strategically depending on market conditions. Thus, using a single supply curve for all hours

of a long study period, as that of considered in [80], may not fully capture the realities of market

participant’ offering strategies. Moreover, with lower resolution data (e.g., using one representative

supply curve per day instead of hourly supply curves) we might overestimate the energy arbitrage

opportunities that exist during a period of say a week or a year. Thus, it could cause overestimation

49

Figure 3.4: Supply curves for each of the 24 hours on August 1, 2014

Figure 3.5: Supply curves for hour ending 1 for the month of October 2014.

of the potential profit gained through energy arbitrage. Conversely, low resolution data could also

cause underestimation of the arbitrage opportunity and consequently failure to capture an arbitrage

opportunity. Therefore, use of high resolution data leads to more effective scheduling and revenue

prediction.

The Alberta Internal Load (AIL) is reported by the AESO on a hourly basis. However, the AIL

data does not provide sufficient information to determine price from a supply curve such as that in

Fig.3.3. The supply curves in the database only consider dispatchable generation above 5MW. To

determine equivalent system load on a supply curve such as that developed in the study, the pool

price is cross-referenced against the supply curve, with the intercept providing the load supplied

by the dispatchable generation in the merit order. We refer to this value as the ”Market Equivalent

Demand”.

50

Figure 3.6: an example of price decrease due to a 150 MW new supply to the system

3.3.2 Construction of GPQCs and DPQCs of the Alberta Electricity Market

In order to investigate the economic performance of a price-maker energy storage system in the

Alberta electricity market, the hourly GPQCs and DPQCs are created. Construction of the PQCs

requires the database of hourly supply curves, pool prices and market equivalent demand. For each

hour of years 2010-2014, the impact of additional generation or additional demand have on on

market pool prices is explored. The impact of [rice is determined by incrementally adding either

demand or generation, in 10 MW steps, up to 200 MW. Additional generation is added to the merit

order at an offer price of $0/MWh. Figure 3.6 illustrates the case of a new supply of 150 MW;

as a result of the new supply, the supply curve is extended to the right. The impact of the new

supply offer can be seen as the difference between the original and modified supply curves, at the

equivalent market demand. At 10,800 MW of demand, the original price is $500/MWh, the new

price is $259.12/MWh; a $240.88/MWh decrease in price due to the new supply in the system.

Figure 3.7 shows the impact of an additional 100 MW demand on market price. The original

price is $32.28/MWh and the market equivalent demand is 9110 MW. As can be seen in the figure,

the 100 MW additional demand cause the price to increase to $39.4/MWh.

Two sets of data are created, one consists of hourly GPQCs and the other hourly DPQCs. Each

set of data has 43824 rows and 20 columns. The curves shown in Figs. 3.2-(a) and 3.2-(b) are

GPQC of hour 1043 and DPQC of hour 39 in 2010 up to 100 MW, respectively. Note that, the

described process is based on the assumption that during the analyzed period, historical supply

51

Figure 3.7: an example of price increase due to a new 100 MW demand to the system

offers and demand bids remain unchanged.

3.3.3 Energy Storage self-scheduling Formulation

A merchant storage plant, designed for energy arbitrage, purchases electricity during low price

periods to charge the plant. The stored energy is later used to discharge the power and sell it to

the market during peak price hours. The storage device characteristics consist of charging power

capacity, discharging power capacity, energy capacity, and efficiency. The efficiency of a storage

facility is expressed as the amount of output energy per unit of energy consumed for charging

during the off-peak hours.

In this section, a general optimization-based formulation for the self-scheduling of a merchant

price-maker energy storage plant is presented. The developed model can be modified depending

on the characteristics of the storage technology such as battery [89–91] or CAES [37]. The goal

of the storage plant is to maximize profit through energy arbitrage as a participant in the electricity

market. To formulate the operation of the facility, it is assumed that the storage operator has the

forecasts of the GPQCs and DPQCs for the upcoming hours. Publicly available data of electricity

markets can be used to forecast the DPQCs and GPQCs. For instance, in this paper, the historical

hourly supply curves and pool prices of the Alberta electricity market for five years, which are

available online at [30], are extracted to build the hourly PQCs for five years. Different forecasting

methods can be used and the historical PQCs can be fed to those forecasting engines to forecast

PQCs for the upcoming day or week. Hourly GPQCs and DPQCs allow the self-scheduling profit

52

maximization problem to be precisely formulated. The objective function and constraints for the

self-scheduling optimization are as follows.

maxT

∑t=1

[Pdt ×π

dt (P

dt )−Pc

t ×πct (P

ct )−OCt ] (3.1)

Subject to:

OCt = Pdt ×VOMd +Pc

t ×VOMc ∀t ∈ T (3.2)

udt +uc

t ≤ 1 ∀t ∈ T (3.3)

0≤ Pct ≤ Pc

max.uct ∀t ∈ T (3.4)

0≤ Pdt ≤ Pd

max.udt ∀t ∈ T (3.5)

Esmin ≤ Es

t ≤ Esmax ∀t ∈ T (3.6)

Est+1 = Es

t +Pct ×µ−Pd

t ∀t ∈ T (3.7)

Es(0) = Es

int (3.8)

The objective function (3.1) consists of three terms. The first term is the revenue from elec-

tricity sales to the market from discharging the stored energy. The second term is the cost of

purchasing the electricity from the market. The third term of the objective function represents the

operating cost of the plant. It is expressed as the variable operation and maintenance (VOM) cost

of the storage plant during charging and discharging hours in (3.2). The operational constraint is

expressed in (3.3) i.e., the storage can operate in only one of charging or discharging modes at a

time. The charging and discharging power and energy limits of the storage are specified by (3.4) -

(3.6). The dynamic equation for the storage level is provided by (3.7). The initial level for the air

storage cavern is specified by (3.8).

3.3.4 Equivalent Linear Formulation

In the objective function, the market clearing price is not an input parameter. The price is a varia-

ble, which is a function of charging or discharging quantities. The relation between the price and

53

charging and discharging quantities is expressed through hourly GPQCs and DPQCs. Due to the

products between these variables, i.e., hourly charging and discharging power and hourly market

price, the formulation is nonlinear. A mixed-integer linear programming approach is presented

in [1] to convert the non-linear problem of a price-maker generation company to its linear equi-

valent. The optimization problem developed above can be also converted to its equivalent linear

formulation in a similar manner to that described in [1]. However, since the charging side should

be also scheduled, the proposed approach in [1] is modified and extended further for energy storage

charge/discharge scheduling.

Figs. 3.8 and 3.9 demonstrate the linearization process for a sample five step GPQC and a

sample four step DPQC, respectively. Based on this approach, the linearization process may be

written as follows:

maxT

∑t=1

[ ndt

∑s=1

πdt,s(b

dt,s + xd

t,sqd,mint,s )−

nct

∑s′=1

πct,s′(b

ct,s′+ xc

t,s′qc,mint,s′ )−OCt

](3.9)

Subject to:

(3.2)− (3.8)

Pdt =

ndt

∑s=1

(bdt,s + xd

t,sqd,mint,s ) ∀t ∈ T (3.10)

0≤ bdt,s ≤ xd

t,sbd,maxt,s ∀t ∈ T (3.11)

ndt

∑s=1

xdt,s = ud

t ∀t ∈ T (3.12)

Pct =

nct

∑s′=1

(bct,s′+ xc

t,s′qc,mint,s′ ) ∀t ∈ T (3.13)

0≤ bct,s′ ≤ xc

t,s′bc,maxt,s′ ∀t ∈ T (3.14)

nct

∑s′=1

xct,s′ = uc

t ∀t ∈ T (3.15)

The objective function (3.9) states the profit of the price-maker energy storage during the sche-

duling horizon. As seen in 3.9, the profit consists of three terms, which are respectively discharging

54

Figure 3.8: Generation PQC, the linearization process [1].

Figure 3.9: Demand PQC, the linearization process.

55

revenue, charging cost, and operating cost. Figure 3.8 illustrates the variables, i.e., bdt,s,x

dt,s, and

parameters, i.e., πdt,s,q

d,mint,s ,bd,max

t,s , used to linearize the revenue of the storage plant as a function

of its hourly discharging power. The shaded area in this figure represents the revenue, which is

the discharging power multiplied to the market price at that discharging level. It is mathematically

expressed in the first term of objective function. In (3.10), the discharging power is linearly ex-

pressed as a function of variables bdt,s,x

dt,s, shown in Fig. 3.8. Equation (3.11) expresses the limit

on the block of the GPQC at each hour, which is between zero and the size of that step. Equation

(3.12) states that at each hour of discharging period, only one instance of the variable xdt,s is non-

zero, which shows the corresponding step of GPQC the storage is operating at that hour. Based on

(3.12), all instances of the variable xdt,s are zero at time t if storage is not in discharging mode at

that hour. Based on (3.11) and (3.12), during a discharging hour, only one instance of the variable

bdt,s could vary between zero and the size of selected step of GPQC of that hour. All the others are

forced to be zero. Based on above discussion, for each hour, The revenue is the sum of the product

of the corresponding price of each step of GPQC with the corresponding term of the discharging

power constraint (3.10). If the storage is not in discharging mode, all terms of (3.10) are zero,

which means no revenue at that hour.

The linearization process for charging is similar to that used for discharging. The second

term in the objective function (3.9) indicates the charging cost of storage. Figure 3.9 shows the

variables, i.e., bct,s′,x

ct,s′ , and parameters, i.e., πc

t,s′,qc,mint,s′ ,bc,max

t,s′ , used to linearize the charging cost

of storage plant as a function of its hourly charging power. In (3.13), the charging power is linearly

formulated as a function of variables bct,s′,x

ct,s′ shown in Fig. 3.9. The charging cost is shadowed in

this figure, which is formulated in second term of the objective function (3.9).

3.4 Results and Discussions

The model developed in Section 3.3 is used to assess the potential operating profit gained by a

merchant price-maker storage plant participating in the Alberta electricity market. A period of

56

Figure 3.10: scheduling of storage plant and price of electricity during an arbitrary week in thecase of ignoring the impact of storage operation on electricity price

seven days is selected as the scheduling horizon in order to take advantage of hourly and daily

fluctuations in electricity prices. It is assumed that the perfect forecast of hourly GPQCs and

DPQCs are available. The hourly PQCs of the Alberta electricity market from 2010 to 2014 are

used as the input to the storage self-scheduling problem. A base case and sensitivity analyses are

presented

3.4.1 Base Case Analysis

The base case evaluates a storage facility with 140 MW discharging power, 90 MW charging

capacity, 1400 MWh energy capacity, i.e, 10 hours generation at full discharging capacity, and

70% roundtrip efficiency. $1/MWh VOM cost [4] is considered for charging and discharging

modes.

Single Week Example: To demonstrate the importance of the price-maker formulation, a single

week of operation is scheduled under each of price-taker and price-maker assumptions. Figure

3.10 plots the scheduling plan when the impact of storage operations on the electricity price is

neglected. It is clear that with this price-taker assumption, the storage facility charges during

low price hours and discharges during high price hours. It is also clear that the majority of these

operations are conducted at maximum power capacity.

Figure 3.11 plots the storage scheduling when the impact on market price is considered. The

hourly price without and with the storage operation are also plotted. As with the price-maker as-

57

Figure 3.11: scheduling, price of electricity before and after operation, for a price-maker storageplant during an arbitrary week

sumption, the overall operational trend of the storage facility with responsive price is to charge

when prices are low and discharge when prices are high. However, the price curves indicate that

even a 140/90 MW storage facility has a significant impact on the electricity price specially du-

ring peak hours. The peak generation capacity of Alberta’s electric system for the period in the

study is 12.35 GW, with peak net demand of 10.51 GW. Comparison of Fig. 3.10 and Fig. 3.11

demonstrates that with responsive price optimization, charging and discharging operations are so-

metimes curtailed when their impacts on price are significant, making energy arbitrage less profi-

table. Under the price-maker assumption, the price profile becomes smoother with higher prices

during charging and lower prices during discharging. The proposed self-scheduling solution pre-

vents excessive price impacts during operation hours. This leads to a more profitable solution and

higher net operating profit than would be obtained if scheduling was carried out using price-taker

formulation. Figure 3.11 demonstrates that the impact of energy storage operation on the electri-

city price should be taken into account, since ignoring its impact causes high errors in results and

overestimation in the potential revenue of energy storage facility.

Dispatch Characteristics of the Storage Facility During Five Years: The percentage of time the

storage facility is charging, discharging or idle for the five-year period from 2010 to 2014 is plotted

in Fig. 3.12. Considering the asymmetric nature of the storage facility, with 140MW discharge

capacity and only 90MW charge capacity and with 70% roundtrip efficiency, 100% utilization

corresponds to 69% charge time and 31% discharge time. From Fig. 3.12, one can see that the

58

Figure 3.12: Dispatch characteristic of a price-maker storage facility during 2010 to 2014

facility is idle for between 41% and 54% of the time, and that generation occurs for less than 24%

of the available hours, with charging occurring between 27% to 35% of the hours.

Considering the charging operations first, between 37% to 45% of the charging hours, the

storage facility charges at the rate lower than it charging capacity. It shows that considering char-

ging impacts on market price, the charging power is limited to lower rates to reduce price incre-

ment. Instead, it charges for more hours to store sufficient amount of energy. moreover, it can be

seen that the majority of the charging occurs at full capacity. This implies that during majority of

charging hours, which is likely to coincide with low price periods, the market is relatively insensi-

tive to an additional load; charging is not curtailed by increasing prices. Conversely, much of the

generation is conducted at a rate below maximum power. This result is unlike that which may be

expected from [13]. It is reported in [13] that the energy storage mostly operates at full discharge

when providing energy arbitrage. The result of our study shows that at high price periods, the Al-

berta market price is sensitive to additional supply, and that during most of the hours discharging

should be curtailed below maximum power in order to balance price and generated volume.

Based on Fig. 3.12, during years 2010 and 2014 the number of operating hours is lower than

other years. This implies that there are fewer arbitrage opportunities during those years (e.g., lower

price volatility) whereas during 2013, the plant operates for more hours than that of other years,

59

Figure 3.13: Weekly profit of a price-maker energy storage facility during a) 2010, b) 2011, c)2012, d) 2013, and e) 2014.

59% hours. This result implies that among these five years, the highest price volatility occurs

during 2013, which brings profitable opportunities for energy arbitrage.

Weekly Profit Analysis During Five years:

Figure 3.13 plots the weekly operating profit earned through arbitrage in years 2010 to 2014.

Table 3.2 also provides the statistics of weekly profit during these years. Based on the total profit

reported in this table, the level of gained profit is much lower compared to the cases presented in

price-taker studies [39, 73]. This shows the fact that storage operation in the market noticeably

impacts market price, which leads to significant overestimation of potential profit in the case of

price-taking assumption. As a result, the price impact should be necessarily considered in the

economic analysis.

It can be observed from Fig. 3.13 that the profit varies significantly week by week and also

year by year. For example, for year 2014, the weekly profit could be as low as $ 0.006 million or

as high as $2.55 million. This can be also concluded from Table 3.2; the standard deviation of the

60

Table 3.2: Weekly Profit Analysis for a Price-Maker storage facility during 2010 to 2014 [Million$]

Year Min. Max. Mean Median StandardDeviation Total

2010 0.006 2.55 0.194 0.053 0.418 10.102011 0.015 2.69 0.578 0.247 0.682 30.052012 0.002 3.20 0.494 0.285 0.635 25.672013 0.021 3.39 0.615 0.198 0.895 31.992014 0.020 2.50 0.269 0.060 0.494 14.00Total - - - - - 111.81

weekly profit is higher than the average profit in all these years, implying significant variation in

the number and level of arbitrage opportunities in different weeks of the year. The comparison of

weekly profit for these five years and also the results provided in Table 3.2 show that in year 2013,

the average weekly profit is higher than in the other years. The data in Table 3.2 and Fig. 3.13

indicate that in the years 2010 and 2014, although the price varied hourly and daily, the energy

arbitrage opportunities rarely happened and are not highly profitable. Thus, the weekly profit is

mostly low compared to the other years. As provided in the fifth column of Table 3.2, i.e., median

column, for half of the weeks in years 2011, 2012, and 2013, the obtained profit is higher than

$0.247 M, $0.258 M, and $0.198 M, respectively. This level is as low as $0.053 M and $0.060 for

years 2010 and 2014, respectively. Moreover, the comparison of Figs. 3.12 and 3.13 demonstrates

that although the number of charging and discharging hours in years 2011 and 2012 are slightly

higher than those of years 2010 and 2014, the final annual profit in years 2011 and 2012 is almost

two to three times as that of 2010 and 2014. This implies that there are a few highly profitable

energy arbitrage opportunities in 2011 and 2012, which makes a noticeable difference in total

profit between these two years, and years 2010 and 2014. Furthermore, based on this table, the

median of profit for the year 2013 is lower than that of years 2011 and 2012 in spite of total higher

operating profit of year 2013 than that of years 2011 and 2012. As shown in Fig. 3.13, there are

a few weeks with significantly high profit, which makes the total profit of year 2013 higher than

the other years. All these descriptions show the fact that the energy storage should be able to take

advantage of high price spikes and price drops as much as possible to get the most out of energy

61

0 1000 2000 3000 4000 5000 6000 7000 80000

200

400

600

800

1000

Time (Hour)

Price($

/MW

h)

Without storage operationWith storage operation

Figure 3.14: Price duration curve without and with operation of a price-maker storage facilityduring 2013.

arbitrage.

Overall Impact on Market Price: The price duration curves for the year 2013 without and with

the operation of storage unit are presented in Fig. 3.14. According to Fig. 3.14, the operation of

storage facility has a significant impact on the electricity price during peak hours, since the energy

storage is only likely to discharge during peak hours. At this time, the market supply curve is very

steep and the storage discharging reduces the electricity price significantly.

Table 3.3 presents an analysis of the impact of energy storage discharging activities on the

electricity price in each year. During discharging operation, the energy storage facility causes a

noticeable decline in the average price. For instance, the average price during discharging periods

in year 2013 decreases by $35.3/MWh, from $217.87/MWh to $182.58/MWh. Furthermore, even

though the percentage price changes are similar for the first three years and about 3% higher than

that of 2013, the overall impact is obviously more significant when discharging at higher prices

(e.g., the average price decrease is higher in years 2011, 2012, and 2013 than the other years).

Table 3.4 presents the impacts of charging operation on market price. Based on the data plotted

in Fig. 3.14 and presented in table 3.4, the charging of energy storage has a smaller impact on price

than discharging. This is because storage facility mostly charges during low price periods during

which both the demand and the gradient of the supply curve are low. As a result, a small increase

in demand does not significantly impact the price. The lowest increase in price is for year 2014,

62

Table 3.3: Price analysis without and with storage operation during discharging hours

Year No.Hours

Price average[$/MWh] Price change

Without With [$/MWh] %2010 1597 128.32 103.69 -24.62 -19.22011 2089 216.17 174.41 -41.75 -19.32012 1974 198.05 158.60 -39.45 -19.92013 2110 217.87 182.58 -35.30 -16.22014 1810 122.81 106.24 -16.58 -13.5

Table 3.4: Price analysis without and with storage operation during charging hours

Year No.Hours

Price average[$/MWh] Price change

Without With [$/MWh] %2010 2408 24.52 27.23 2.71 11.12011 2696 21.78 23.90 2.12 9.72012 2976 18.69 21.70 3.01 16.12013 3023 24.26 27.53 3.27 13.52014 2879 21.13 22.88 1.75 8.3

which is on average $1.75/MWh. Charging operations have the highest impact on market prices

during year 2012 and 2013.

Considering both charging and discharging operations, table 3.5 presents the net impact on

average electricity price for each year in the study. The table indicates that the operation of energy

storage leads to a decrease in the annual mean price as the price reduction due to discharging

operation is more substantial than the price increase due to charging operation. The highest price

decrease is in 2011 by $9.26/MWh, following by the years 2012 and 2013, which are respectively,

$7.89/MWh and $7.39/MWh. During years 2010 and 2014, the storage operation does not have as

significant effect on prices as that of the other years.

3.4.2 Sensitivity Analysis

The sensitivity of the annual profit of the storage plant to different design changes is explored to

provide information to economically optimize storage energy, charging and discharging capacities.

In the sensitivity analysis, only one characteristic is varied at a time, all other characteristics are

63

Table 3.5: Price analysis without and with storage operation for all hours

YearPrice average

[$/MWh] Price change

Without With [$/MWh] %2010 50.95 47.20 -3.75 -7.42011 76.27 67.01 -9.26 -12.12012 64.64 56.75 -7.89 -12.22013 80.22 72.83 -7.39 -9.22014 49.66 46.81 -5.75 -5.7

Figure 3.15: Profit of storage facility as a function of discharging capacity (Charging capacity is fixed at 90

MW, storage capacity is fixed at 10-hr )

assumed to be the same as that of base case.

Figure 3.15 shows the impact of the discharging capacity on the annual and total profit of

the energy storage facility. It can be seen that larger discharging capacity leads to higher profit.

Expanding the discharging capacity from 60 MW to 100 MW, or from 100 MW to 180 MW,

increase the total profit by 40.7% and 36.5%, respectively. This is because with larger discharging

capacity, the unit is able to generate more power. Consequently, more power could be sold during

the hours with the highest prices. This increases sales profit as well as the number of hours when

energy arbitrage is profitable. The increment rate differs from year to year. This rate is higher for

years 2011, 2012, and 2013 compared to that of year 2010 and 2014. This is due to more frequent

price spikes during years 2011-2013. Thus, a storage with larger discharging capacity could better

exploit these opportunities.

According to Fig. 3.15, the marginal incremental profit declines as discharging power increa-

64

Figure 3.16: Profit of storage facility as a function of charging capacity (discharging capacity isfixed at 140 MW, storage capacity is fixed at 10-hr )

ses. The higher the level of injected power, the higher the (negative) impact on price. Based on the

developed formulation, when the discharging power capacity is high, the operator sometimes de-

cides to limit its production level and not to sell high level of power to the market in order to avoid

high level of price drop. Thus, the storage would not benefit much from higher level of discharging

capacity. However, in studies in which the storage facility is assumed to be price-taker [13, 39],

larger discharging power would almost linearly lead to profit increment, since the impact of energy

storage operation on market price is ignored. Hence, even with larger discharging power, the price

is assumed not to change and consequently higher revenue is obtained.

Figs. 3.16 presents the effect of charging capacity, ranged between 10 MW to 190 MW, on the

storage profit in different years. Figs. 3.16 shows that increasing the charging power improves the

operating profit. For instance, increasing charging capacity from 50 MW to 90 MW improves total

profit by 12.8%. However, Fig. 3.16 demonstrates that the incremental rate of profit improvement

is lower than that of discharging capacity. Additionally, the incremental increase in profit declines

with charging capacity such that there is negligible improvement for charging power larger than

130 MW. This is due to the fact that in the Alberta electricity market, the hours of low price

happens frequently and thus, with not necessarily large charging capacity, there is enough time

for charging sufficient amount of energy and taking advantage of peak prices. Therefore, a larger

charging power does not noticeably improve system total operating profit.

65

Figure 3.17: Profit of storage facility as a function of storage capacity (Discharging capacity isfixed at 140 MW, charging capacity is fixed at 90 MW)

Figure 3.17 shows the sensitivity of the annual profit to the energy capacity. Storage size larger

than 18 hours does not provide significant incremental arbitrage opportunity. This results implies

that the additional storage is not used by the unit. Due to the price pattern, the unit does not need

to charge large amount of energy and at most 18 hours of storage is sufficient to take advantage of

all hourly and daily energy arbitrage opportunities. Moreover, Fig. 3.17 indicates more than 72%

of the total potential value comes from the first 4 h of storage, i.e., intra-day arbitrage. Additional

operating profit is achieved by longer-term storage; 10 h of storage captures about 91.6% of the

potential profit, while 20 h of storage captures about 99% of potential profit.

Figure 3.18 represents the relationship between the operating profit gained by the storage faci-

lity and the unit efficiency. The efficiency of the unit impacts the annual profit as a higher efficient

unit needs less energy to purchase to generate one MWh of energy. Thus, a lower gap between

the purchasing and selling price is required to make profit out of energy arbitrage. Figure 3.18

illustrates that improving the storage efficiency from 60% to 70% would lead to around $7.35 M

additional revenue in five years. The additional gained profit can be used to compare with the

additional capital cost required to improve system efficiency and determine profitability of the

investment.

The results presented in this paper are based on a general modeling of an energy storage system.

It is not the purpose of this paper to conduct financial analysis for every storage technology. Ho-

66

Figure 3.18: Profit of storage facility vs. energy storage efficiency (Discharging capacity is fixed at140 MW, charging capacity is fixed at 90 MW, the storage capacity is 10 hours of full dischargingcapacity)

wever, for a specific storage technology, depending on technology and planned application, fixed

and variable cost of the device, as well as efficiency, the reported results can be useful for optimal

sizing of that storage device. There is no universal optimal size of storage and the marginal cost of

the next incremental MW of charging or discharging or hour of storage is wildly ranged depending

on the technology.

3.5 Conclusion

This paper conducts a comprehensive study on the economic evaluation of a large-scale energy

storage facility in the Alberta electricity market, incorporating the impacts of energy storage acti-

vities on market price. Hourly GPQCs and DPQCs are utilized to precisely formulate the self-

scheduling of a price-maker energy storage in an electricity market. Then, the developed model is

applied using the historical hourly GPQCs and DPQCs of the Alberta electricity market to explore

the economics of energy storage in this market. The results show that energy storage operation

significantly affect market price, especially during high price hours. During high price hours the

supply curves are steep and a relatively small change in supply may change the price substantially.

As a result, the proposed formulation mat curtail charging and discharging operations relative to

price-taking self-scheduling, when their impacts on price are high and energy arbitrage becomes

67

unprofitable. In this way, the predicted gained profit is lower than that reported in price-taker

studies, demonstrates the necessity of incorporating price impacts during economic studies.

Sensitivity analyses are performed to investigate the impacts of different storage characteristics

on its profit. The results illustrate that larger discharging capacity leads to a higher level of profit

due to the ability to sell more power during peak price periods. However, the incremental return

declines due to the impact of discharging power on the market price. The sensitivity analysis on

charging power shows that, larger charging capacity up to 180 MW can return higher profit. Howe-

ver, the profit increment saturates quickly as low prices occur frequently in the Alberta electricity

market. Even with a low charging power capacity, there is time to store sufficient energy. It is

shown that 4 and 10 hours of energy storage capacity are able to capture 72% and 91.6% of po-

tential profit profit, respectively. The higher storage efficiency decreases the cost of purchasing

the electricity and consequently increase the annual profit. The presented results can be used to

optimize the size of storage device depending of the storage technology.

In the developed self-scheduling model, forecasts of DPQCs and GPQCs are required inputs

to the problem. In our study, the actual historical DPQCs and GPQCs are used, since we focus on

evaluating the economic feasibility of a storage facility based on historical market data. In other

words, we say what would have been the revenues if this facility was in operation and had a perfect

knowledge of the market. The outcomes are the upper bound of the economic feasibility, and real-

life uncertainties could make the economics less attractive depending how the actual curves deviate

from forecast curves. However, the focus of our paper in not designing bidding strategies under

forecast uncertainty of these parameters. The authors are currently working on bidding strategies

for such a facility under various sources of uncertainty and revenue streams.

Moreover, the database of GPQCs and DPQCs and the developed linearized scheduling model

can be used to formulate an optimization framework to find the optimal size for an specific energy

storage technology. This is an ongoing study in our research group.

The important assumptions that are made for this study are (i) market participants do not react

68

to the presence of an energy storage facility and act the way they did without it, and (ii) no demand

charges are considered when calculating profit values. As for assumption (i), it is hard to model

how market participants would have reacted if the the facility was in service in each of those years.

The authors are investigating alternative methodologies that could be used for doing just that. For

the second assumption, demand charges are sometimes high and could significantly impact the

profitability of a facility. These two assumptions need to be kept in mind when interpreting the

findings of this research.

69

Chapter 4

Developing Bidding and Offering Curves of a Price-maker

CAES Facility in Day-Ahead Energy Market Based on Robust

Optimization 1

Nomenclature

Superscripts

c Charging mode.

d Discharging mode.

indices

t Index for operation intervals running from 1 to T .

s Index for the steps of generation price quota curves from 1 to ndt for hour t.

s′ Index for the steps of demand price quota curves from 1 to nct for hour t.

Parameters

πNGt Natural gas price.

πdt,s Forecasted price corresponding to step number s of the GPQC at hour t.

πct,s′ Forecasted price corresponding to step number s′ of the DPQC at hour t.

πdt,s/πd

t,s Upper/Lower bound of price corresponding to step number s of the GPQC at hour

t.

πct,s′/πc

t,s′ Upper/Lower bound of Price corresponding to step number s′ of the DPQC at hour

t.

1Submitted to IEEE. Transaction on Smart Grid [31], under review: S. Shafiee, H. Zareipour, and A. Knight,”Developing bidding and offering curves of a price-maker energy storage facility based on robust optimization,”IEEE Transactions on Smart Grid, under review, April 2017.

70

bd,maxt,s Size of step s of the GPQC at hour t.

bc,maxt,s′ Size of step s′ of the DPQC at hour t.

Esmax maximum energy level of air storage.

Pdmax maximum generation capacity of expander.

Pcmax maximum compression capacity of compressor.

qd,mint,s Is the summation of power blocks from step 1 to step s−1 of GPQC for hour t.

qc,mint,s′ Is the summation of power blocks from step 1 to step s′−1 of DPQC for hour t.

SOCmin minimum state of charge (SOC) of air storage.

SOCmax maximum SOC of air storage.

SOCinit Initial SOC level of air storage.

VOMexp Variable operation and maintenance cost of expander.

VOMc Variable operation and maintenance cost of compressor.

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Variables

πEt Electricity market clearing price at interval t.

πdt,s/πc

t,s price corresponding to step number s of the GPQC/DPQCs at hour t within the

specified confidence interval.

bdt,s The fractional value of the power block corresponding to step s of the GPQC to

obtain discharging quota Pdt in hour t.

bct,s′ The fractional value of the power block corresponding to step s′ of the QPQC to

obtain charging quota Pct in hour t.

OCt Operation cost of the plant at time t.

Pdt Discharging power at time t.

Pct Charging power at time t.

SOCt Cavern state of charge at time t.

uxt Unit status indicator in either modes x, i.e., discharging (d) or charging modes (c)

(1 is ON and 0 is OFF).

xdt,s Binary variable that is equal to 1 if step s of GPQC is the last step to obtain dischar-

ging quota Pdt in hour t and 0 otherwise.

xct,s′ Binary variable that is equal to 1 if step s′ of DPQC is the last step to obtain charging

quota Pct in hour t and 0 otherwise.

4.0.1 Functions

πdt (P

dt ) GPQCs, i.e, stepwise decreasing function that determines the market price as a

function of the discharge quantity at time t.

πct (P

ct ) DPQCs, i.e., stepwise increasing function that determines the market price as a

function of the charge quantity at time t.

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4.1 Introduction

Energy storage systems are rapidly integrated into power systems worldwide due to their various

applications [8, 33]. A market study by CitiGroup in 2015 estimated a global market of up to 240

GW by 2030 excluding pumped-storage hydroelectricity and car batteries [3]. From the system

perspective, large scale ES systems such as batteries, compressed air energy storage (CAES), and

pumped hydro storage systems are able to provide time shifting and peak shaving, facilitate the fast

growing penetration of the renewable resources into the power system, and in conclusion, improve

the efficiency and reliability of the system [14, 92]. Several studies have investigated the impacts

of ES systems on the operation of power systems from a centralized system viewpoint [69,93–95].

From another perspective, the ES facility can operate as a merchant investor-owned entity. From

the investor’s point of view, the capability of storing considerable amount of energy brings business

opportunities for energy arbitrage by an ES facilities [37].

A large scale ES operator needs to optimize its participation strategy in an electricity market to

gain higher operation profit. Thus, the operator needs a strategy for an ES facility, which consists

of an offering strategy for selling electricity to maximize the revenue and a bidding strategy for

purchasing electricity to minimize the cost. Over the past years, several studies have concentrated

on developing optimal bidding and offering strategy of a privately-owned ES facility in competitive

electricity markets as described below.

The existing methods can be divided into two separate groups: price-taker storage facilities and

price-maker ones. A price-taker storage facility refers to a facility, which cannot change the market

price by its charge/discharge actions. Thus, a forecast of day-ahead prices are used to optimize the

charge/discharge operation of the facility accordingly [13, 38, 39]. In this group, the electricity

market is modeled by the forecast of day-ahead prices. In the second group, a price-maker ES

facility is large enough that its action can alter market prices.

To optimize the scheduling of a price-maker market player, one method is to estimate the rivals’

marginal cost. Solving the problem by predicting the rivals’ marginal cost leads to a mathematical

73

programming with equilibrium constraints (MPEC) problem. MPEC approach is used to address

strategic bidding and offering of a storage facility [77, 96, 97]. MPEC approach imposes high

computational burden due to its complicated mathematical formulation, which makes it difficult to

be solved to the optimality specially for large systems [98]. Another method to solve the scheduling

of a price-maker participant in an electricity market is to forecast the price quota curves (PQCs) [1].

For a generation unit, the generation PQC (GPQCs) for an hour, is a stepwise decreasing curve

that shows the market price as a function of the total generation of the unit [86–88]. GPQC models

the producers interaction with other market participants. This method has significantly simpler

formulation and consequently less computational burden compare to the MPEC approach [98]. In

the present paper, the concept of price quota curves is implemented to model the price impacts a

price-maker ES facility. In addition to the GPQCs, demand PQCs (DPQCs) are employed here

to model the impacts of charging on market price to optimize the scheduling of a price-maker ES

facility. This issue is not addressed in the literature.

Due to inevitable error in the forecasts of the GPQCs and DPQCs, the associated uncertainty

needs to be incorporated in the scheduling problem to minimize the risk. Therefore, an appro-

priate offering strategy for selling the electricity as well as a bidding strategy for purchasing the

electricity is required to maximize the arbitrage profit while managing the risk of forecasting un-

certainty. Some studies do not consider the uncertainty associated with the input data and develop

a risk-neutral scheduling [1]. In order to deal with forecasting uncertainties when participating in

electricity market, different methods are applied such as stochastic programming [77, 87, 88, 97]

or information Gap decision theory (IGDT) [28, 99, 100]. As an alternative, robust optimization

is used to develop self-scheduling and bidding strategy of market participants such as thermal

and hydro thermal generation companies [24–26, 101], virtual power plants [102], and a wind

farm combined with ES [51, 52]. Robust optimization is an interval based optimization method

in which, instead of scenarios, the uncertainty is specified via intervals [101]. Thus, this method

is applicable when high level of uncertainty exists. It also has less computational burden than

74

stochastic programming. Moreover, robust optimization makes no assumption on the probability

density function (PDF) of the uncertain parameter. This feature of the robust optimization is highly

desirable for the case of PQCs, since providing PDF for PQCs is not straightforward. Moreover, in

the day-ahead electricity market, the storage operator can submit multi-step bids and offers to the

market for purchasing and selling electricity to manage the risk of price uncertainty more effecti-

vely. Thus, constructing multi-step bidding and offering curves for an ES facility is important.

In this paper, we develop a bidding and offering strategy for a large-scale price-maker inde-

pendent ES facility to participate in day-ahead electricity market. While such facility could stack

multiple revenue sources by providing other services to the grid, in this work, we focus only on

energy arbitrage; developing bidding strategies for such cases is the focus of the authors’ future

work. The impact of ES discharging and charging operation on market price is modeled by means

of GPQCs and DPQCs. Then, we develop a robust-based optimization to model the uncertainty as-

sociated with GPQCs and DPQCs. In order to solve the min-max problem, it needs to be converted

to a linear maximization problem. Robust optimization has been used in the literature to model the

price uncertainty and develop participation strategies of a price-taker generation company [24–26]

as well as a consumer [27]. In the case of a generation company and consumer, to solve the ro-

bust optimization, the worst cases of price deviation are easily determined as the lower and upper

bound of the confidence interval, respectively [24,27]. Unlike for the case of a generator or a load

in which determining the worst case scenario of the uncertain parameter is straightforward, for the

case of a price-maker ES facility, the worst case scenario of the GPQCs and DPQCs depends on

the charging and discharging status of the facility, which needs to be properly modeled. Based on

the developed robust scheduling formulation, we presents an algorithm to construct hourly multi-

step bidding and offering curves for the storage facility for participation in a day-ahead market.

L. Baringo et al. [24] develops offer curves for a price-taker thermal unit based on robust optimi-

zation using price subintervals [24]. For the case of a price-maker ES facility, this becomes more

challenging; since, incorporating GPQCs and DPQCs to model the price impacts of the ES facility,

75

the developed algorithm in [24] needs to be modified and extended further to construct not only

offering curve for selling but also bidding curve for purchasing electricity. To do so, the predefi-

ned confidence intervals of GPQCs and DPQCs are divided into subintervals, where the process

of defining subintervals for GPQCs differs from that of DPQCs. Then, the robust formulation is

solved sequentially for each set of subintervals. Afterward, the obtained results are employed to

construct the hourly multi-step bidding and offering curves in order to take different levels of risk

of forecast error into account. It enables the operator to immune its decision against unfavorable

price deviations during charging or discharging. Meanwhile, it exploits favorable price fluctuation

by incorporating less conservative actions.

4.2 Detereministic Scheduling of an ES Facility

In this section, a deterministic scheduling problem of an ES facility participating in a day-ahead

energy market is developed. The storage facility aims to maximize its profit by purchasing and

storing electricity during low price hours. The stored energy is used to generate electricity during

peak price hours. The ES facility is assumed to be a price-maker and can alter the market price by

its operation. Compared to [77, 96, 97], where MPEC model is applied to schedule a price-maker

ES facility, GPQCs and DPQCs are used in this paper to schedule a price-maker ES facility, which

has less computational burden for a large problem. The GPQCs are used in Ref. [1] to address

the self-scheduling problem of a price-maker thermal producer. The method is extended in our

previous work [29] to develop a self-scheduling framework for a price-maker ES facility. The

formulation in this paper is in line with that of proposed in [29]. In the deterministic scheduling, it

is assumed that the hourly GPQCs and DPQCs are known parameters.

76

Figure 4.1: Linearization process for GPQC

maxT

∑t=1

[Pdt ×π

Et (P

dt )−Pc

t ×πEt (P

ct )−OCt ] (4.1)

Subject to:

Ψt(Pdt ,P

ct ) ∀t ∈ T (4.2)

The objective function (5.1) consists of three terms. The first term is the revenue from selling

the electricity to the market. The second term is the cost of purchasing the electricity from the

market to charge the facility. The third term represents the operating cost of the plant. Ψt represents

the associated operation constraints of the ES facility related to the charging and discharging power

capacity and ES capacity. The set of constraints Ψt as well as the operation cost OCt are presented

in Appendix. This is a general formulation for an energy storage facility. Depending on the storage

technology, the objective function and the constraints Ψt could be adjusted.

4.2.1 Equivalent linear formulation

The developed self-scheduling formulation is non-linear due to multiplication of two variables in

the objective function. Considering stepwise GPQCs and DPQCs, as shown in Figs. 4.1 and 4.2,

the equivalent linear model is mathematically formulated as below:

77

Figure 4.2: Linearization process for DPQC

maxT

∑t=1

[ ndt

∑s=1

πdt,s(b

dt,s + xd

t,sqd,mint,s )

−nc

t

∑s′=1

πct,s′(b

ct,s′+ xc

t,s′qc,mint,s′ )−OCt

](4.3)

Subject to:

(4.2)

Pdt =

ndt

∑s=1

(bdt,s + xd

t,sqd,mint,s ) ∀t ∈ T (4.4)

0≤ bdt,s ≤ xd

t,sbd,maxt,s ∀t ∈ T (4.5)

ndt

∑s=1

xdt,s = ud

t ∀t ∈ T (4.6)

Pct =

nct

∑s′=1

(bct,s′+ xc

t,s′qc,mint,s′ ) ∀t ∈ T (4.7)

0≤ bct,s′ ≤ xc

t,s′bc,maxt,s′ ∀t ∈ T (4.8)

nct

∑s′=1

xct,s′ = uc

t ∀t ∈ T (4.9)

πEt =

ndt

∑s=1

πdt,s× xd

t,s +nc

t

∑s′=1

πct,s× xc

t,s′ ∀t ∈ T (4.10)

78

The objective function (4.3) states the operating profit of the storage facility. The first and

second term is the linearized discharging and charging revenue, respectively. Using the binary and

continuous variables as shown in Figs. 4.1 and 4.2, the discharging revenue and charging cost are

linearly expressed in (??). The discharging power and charging power are linearly formulated in

(4.4) and (4.8), respectively. These equations are used to calculate the discharging revenue and

charging cost in the objective function. The market price, which is a function of discharging or

charging power, is calculated in (4.10) based on the defined binary variables. Based on which steps

of GPQCs during discharging or DPQCs during charging, the storage is operating at, the market

price is determined. More details of the linearization process can be found in [29]

4.3 Proposed Robust-based Scheduling Model

In the previous section, a deterministic scheduling optimization method for a price-maker ES plant

was developed. It is assumed that the perfect forecasts of hourly GPQCs and DPQCs are available

for all hours of the scheduling horizon. However, it is evident from the forecasting errors could

exist depending on the forecasting method, market structure, and forecasting time horizon. It could

significantly affect the economics of ES. It is difficult to accurately forecast PQCs, as predicting

the market price based on the accepted amount of units bids or offers is complicated. Hence,

forecasting error should be taken into account in ES facility’s operation bidding/offering strategies

to manage the risks.

Robust optimization has been used widely in power system scheduling problems [24–26,51,52,

101,102]. In robust optimization, the variation ranges of uncertain parameter, known as confidence

interval [103], is defined assuming that after the fact parameter falls into this interval. Then, the

worst case of uncertainties occurrence inside of the interval is evaluated; and, the decision variables

are determined base on this worst case analysis [25].

In this paper, we propose a robust-based self-scheduling model for a price-maker ES facility, in

which the uncertainty associated with the PQCs is modeled using robust optimization. In the case

79

of modeling uncertainty of hourly PQCs in (??), a confidence interval is considered for the curve.

Publicly available data of electricity markets along with different forecasting method can be used

to estimate the confidence interval of the PQCs for upcoming hours. Compared to [87, 88], where

scenario-based methods are used to model the uncertainty of GPQCs for a generation company,

robust optimization, which is used in this paper to model the uncertainty of GPQCs and DPQCs

for a ES facility, has less computational burden.

The daily optimization scheduling problem of (??) can be rewritten as (4.11), by including a

robust-based model of the PQCs.

max minπd

t,s,πct,s

T

∑t=1

[ ndt

∑s=1

πdt,s(b

dt,s + xd

t,sqd,mint,s )

−nc

t

∑s′=1

πct,s′(b

ct,s′+ xc

t,s′qc,mint,s′ )−OCt

](4.11)

Subject to:

(4.2) and (4.4)− (4.9)

πdt,s ≤ π

dt,s ≤ π

dt,s (4.12)

πct,s ≤ π

ct,s ≤ π

ct,s (4.13)

πEt =

ndt

∑s=1

πdt,s× xd

t,s +nc

t

∑s′=1

πct,s× xc

t,s′ ∀t ∈ T (4.14)

This is a max-min optimization problem. The profit is maximized with respect to the decision

variable to find the optimal scheduling. The profit is minimized with respect to the uncertain pa-

rameters, πdt,s, π

ct,s to find the worse case of πd

t,s, πct,s within the interval of [πd

t,s,πdt,s] and [πc

t,s,πct,s].

Note that, the intervals are defined such that the decreasing (increasing) nature of the GPQCs

(DPQCs) are retained. For example, one way is the upper and lower bound of the interval to be

proportional to the forecast (e.g. 20% above and below the forecasted PQCs).

80

Figure 4.3: An example of worst case for a confidence interval of a) GPQC, b) DPQC

In order to solve the max-min optimization problem of (4.11) with commercial solvers, it

should be converted to a maximization problem. In (4.11), the optimization problem is linear

with respect to the uncertain variables. Hence, the worst case of these variables would occur at

the beginning or end of the intervals. During discharging, the worst case of price happens in the

lower bound of GPQCs interval, i.e., πdt,s = πd

t,s. Conversely, the worst case of price during char-

ging is the upper bound of the DPQCs interval, i.e., πct,s = π

ct,s. For example, figure 4.3 shows the

worst case for a GPQC and DPQC confidence interval. Therefore, the final formulation of robust

optimization in our problem could be expressed as follows:

maxT

∑t=1

[ ndt

∑s=1

πdt,s(b

dt,s + xd

t,sqd,mint,s )

−nc

t

∑s′=1

πct,s′(b

ct,s′+ xc

t,s′qc,mint,s′ )−OCt

](4.15)

Subject to:

(4.2) and (4.4)− (4.9)

πEt =

ndt

∑s=1

πdt,s× xd

t,s +nc

t

∑s′=1

πct,s× xc

t,s′ ∀t ∈ T (4.16)

4.4 Developing Bidding and Offering Curves

The operator of a ES facility needs to submit multi-step hourly offers and bids for selling and

purchasing electricity in order to manage the risk of forecasting errors. Compared to [51, 52, 102],

in which the facility is assumed to be a price-taker, we proposed a bidding strategy for a price-

81

maker ES facility. Based on the proposed robust scheduling, we develop a bidding and offering

strategy for a price-maker ES facility. The proposed method is in line with the approach used in

[24] to construct offering curve for a price-taker generation unit. However, compare to [24], where

robust optimization is use to construct offering curve for a price-taker thermal generation company,

in the present paper, the algorithm is modified and extended further to develop a framework to

construct not only offering curve, but also bidding curve for purchasing electricity for a price-

maker ES facility. Moreover, the bidding and offering curves are interrelated due to the time

dependency of the charging and discharging of the ES facility and cannot be built independently.

To construct the bidding and offering curves, a set of robust optimization problem are solved

sequentially as explained below:

1. The confidence interval for the hourly GPQCs and DPQCs are divided into several

subintervals, based on the desired number of steps the operator wants to submit to

the market. For example, Figs. 4.4 and 4.5 show the confidence interval and the

subintervals for a sample GPQC and DPQC with four subintervals.

2. Starting from the first subinterval, i.e., subinterval (a), the robust optimization sche-

duling is solved for each subinterval sequentially to obtain the hourly pairs of char-

ging and discharging level and the corresponding prices for each subinterval k, i.e,

(Pc,kt ,πE,k

t ),(Pd,kt ,πE,k

t ). In order to have a increasing offering curve and decrea-

sing bidding curve, the following constraints are added to the robust scheduling

when solving the problem for step k:

Pd,k−1t ≤ Pd,k

t ∀k (4.17)

Pc,k−1t ≤ Pc,k

t ∀k (4.18)

πE,kt ≥ π

E,k−1t i f Pd,k−1

t ≥ 0 (4.19)

πE,kt ≤ π

E,k−1t i f Pc,k−1

t ≥ 0 (4.20)

82

Figure 4.4: An example of GPQC intervals to build offering problems and the process of con-structing offering curve

Equations (4.17) and (4.18) force the charge/discharge level of the current step be

higher or equal than the previous step. Since the price is a function of charging

or discharging level based on the PQCs, equations (4.19) and (4.20) result in an

increasing offered price for offering curve and decreasing bid price for bidding

curves.

The process is repeated until the whole confidence intervals are covered.

3. The robust scheduling is solved considering the upper bound of GPQCs and lower

bound of DPQCs as the last iteration to find the maximum charging and discharging

levels and the corresponding price profile.

4. Construct the offering and bidding curves using the obtained results. In each step k,

a charge/discharge schedule and the corresponding price profile are determined. For

each hour of discharging, the set of prices for all steps and the discharge schedule

result in an increasing offering curve for that hour. For each hour of charging, the

charge schedule and the corresponding price result in a decreasing bidding curve.

Figures 4.4 and 4.5 illustrate the process of constructing offering and bidding curves

based on the subintervals of the GPQCs and DPQCs, and the obtained results.

83

Figure 4.5: An example of a DPQC intervals to build bidding problems and the process of con-structing bidding curve

4.5 Numerical Results

For the case study, the compressed air energy storage technology is considered as the storage

technology, since this technology can serve grid-scale applications, on the order of 100’s of MW

for tens of hours [7]. Thus, due to its large power capacity, it is reasonable to be considered as

a price-maker facility. Note that, the formulation presented in this paper is general and can be

modified based on the storage technology. The scheduling formulation in this paper is modified

for a CAES facility based on the CAES scheduling formulation proposed in our previous study

[32]. Numerical simulations are performed for a CAES facility with rated capacities of 100 MW

discharging, 60 MW charging and 10 hours of full discharge storage capacity. The details of the

scheduling formulation, associated operation constraints and efficiency parameters can be found

in [32]. The proposed mixed integer linear optimization problem is solved using CPLEX in GAMS.

GAMS and MATLAB c© are used to solve the model sequentially to construct the offering and

bidding curves and also to perform after-the-fact analysis for sequential days. For the case studies,

the historical GPQCs and DPQCs of the Alberta electricity market in year 2014 are used. The

process of constricting GPQCs and DPQCs based on hourly supply curves and system demand can

be found in [29, 104]

84

Figure 4.6: (a) DPQCs fpr hours 1 and 6, and (b) GPQCs for hour 17, 18, and 20

4.5.1 Robust scheduling: a demonstrative case

The hourly GPQCs and DPQCs of the Alberta electricity market at 18 February 2014 are used

as the forecasts to build the confidence interval for the PQCs. For the illustration purposes, the

GPQCs for the hours 17 to 21 are modified to obtain more distinguishable results than those of

original ones. As some examples, figure 4.6 shows the DPQCs for hour 1 and 6, and the modified

GPQCs for hours 17, 18, and 20.

The robust self-scheduling, proposed in section 4.3 is solved for different length of confidence

intervals from 5% to 50% for the GPQCs and DPQCs. For instance, for 20% confidence interval,

the intervals are considered to be 20% below and 20% above the hourly PQCs of this day. The

guaranteed level of profit versus length of the confidence interval is plotted in figure 4.7. Observe

from this figure that the guaranteed level of profit decreases for wider intervals. Wider confidence

interval corresponds to a more conservative case in which higher range of forecasts error is consi-

dered at the cost of lower profit expectation. For example, based on this figure, for 10% confidence

interval, the guaranteed profit is $7,946.0; the forecasting error must be less that 10% in order to

gain this level of profit. The profit level is $4,633.0 for 20% interval; In this case, higher forecasting

error up to 20% guarantees a lower profit of $4,633.0.

For the case of 20% confidence interval, without loss of generality, the intervals are divided

into four subintervals, similar to that shown in Figs. 4.4 and 4.5. Then, the proposed method to

construct the bidding and offering curve, solving sequential robust scheduling, are applied. Figure

4.8 shows the obtained scheduling for each iteration from one to five. The scheduling of the first

iteration has the lowest charge/discharge level, as it is corresponded to the worst subinterval, i.e.,

the largest subinterval in which the lower bound of the hourly GPQCs is the lowest among all the

85

Figure 4.7: The guaranteed profit versus the length of confidence interval

Figure 4.8: scheduling of CAES facility for each iteration corresponded to each subintervals

subintervals and the upper bound of the hourly DPQCs is the highest among all the subintervals.

Thus, the price differences between on peak and off peak hours are already the low without storage

operation and these become even lower as storage charge and discharge with higher power due to

the impacts on market price based on the GPQCs and DPQCs. Therefore, the robust scheduling

model decides to charge and discharge partially in most of the operation hours and retain some

of the capacity to prevent high impacts on market prices as defined by the PQCs and make the

arbitrage profitable. For the higher iterations, the charge/discharge levels increases; since they are a

better case compared to the previous iteration, i.e, a higher iteration has smaller confidence interval

with higher GPQCs bound and lower DPQCs bound, which leads to a higher price difference

between on peak and off peak hours. Thus, the facility participate with higher level of power in the

market and retain less charge/discharge capacity in spite of higher impact on market prices, since

the arbitrage is still profitable.

86

Figure 4.9: Obtained price profile for each iteration corresponded to each subintervals

Figure 4.9 also shows the corresponding resulting price profiles for each iteration. As can be

seen in this figure, higher iteration leads to a higher prices during discharging hours and lower

price during charging hours; since the corresponding subinterval for higher iteration is a better

case compared to the previous one, i.e., smaller confidence interval with higher GPQCs bound

and lower DPQCs bound, which leads to a higher price difference between on peak and off peak

hours than previous iterations. Moreover, the sequential constraints (4.17)-(4.20) ensure to obtain

higher discharging prices and lower charging prices for higher iterations to keep the increasing

and decreasing nature of the offering and bidding curves, respectively. In another word, the lower

iteration corresponds to a more conservative case where flatter price profile is expected, while

higher iteration leads to a more volatile price profile.

The proposed method in section 4.4 is applied to construct the bidding and offering curves.

Basically, to construct the curves, the results, presented in Figs. 4.8 and 4.9 are used; the pairs of

charging power and prices as well as discharging power and prices are sorted to build the bidding

and offering curves. As an example, figure 4.10 and 4.11 shows the obtained bidding curves for

hours 1 and 6, and offering curves for hours 18 and 20, respectively. In hour 1, as shown in

figure 4.8, in the first iteration, storage unit is charging with 5.6 MW of power and the resulted

price is $44.8/MWh. Thus, these values are submitted as the first step of the bidding curve. The

pair of charge power and price for the second and third iteration is (18.3MWh,$41.4/MWh) and

(60MWh,$38.1/MWh), respectively. These pairs are corresponded to second and third steps of

87

Figure 4.10: Resulting bidding curve for a) hour 1, b) hour 6

Figure 4.11: Resulting offering curve for a) hour 18, b) hour 20

curve. Since the charging power for the fourth and fifth iteration is also 60 MW, these two steps

are zero. Bidding and offering curves for the other hours are build in the same way.

4.5.2 The impact of uncertainties through a one-year analysis

In order to show the performance of the proposed bidding and offering strategy in managing the

risk of forecasting uncertainties, the proposed robust optimization based strategy is compared with

the risk neutral strategy. The historical PQCs of the Alberta electricity market during 2014 is

used. The proposed bidding and offering strategy is applied sequentially in a daily basis for the

year using the historical GPQCs and DPQCs of the Alberta electricity market as the forecasts

considering 20% confidence interval below and above the forecasts. In each day, after the fact

analysis is conducted to show the impacts of considering uncertainties in the actual gained profit.

To do so, the after the fact GPQCs and DPQCs are artificially generated by adding up to 30%

level of random error to the forecasted curves. In other words, based on the forecasts, simulated

PQCs are generated, which has 30% error. Then, based on the constructed bidding and offering

curves and the simulated GPQCs and DPQCs, the accepted bids and offers, the market price, and

consequently the gained profit is calculated. For the risk neutral scheduling, the deterministic

88

Figure 4.12: Resulting offering curve for a) hour 18, b) hour 20

Table 4.1: Percentage of the potential profit captured by each strategy in each month of 2014 [%]Strategy Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec. TotalProposedStrategy 83.8 84.1 76.0 83.7 83.1 84.4 81.4 76.8 95.1 82.0 84.6 83.5 81.7

Risk-neutralStrategy 74.1 67.6 57.5 69.5 58.2 70.8 55.7 56.9 71.7 59.7 73.3 64.5 61.8

strategy based on the forecasts is applied and the after the fact gain profit is calculated similarly.

The process is repeated sequentially for all days of the year 2014. Furthermore, the potential profit,

which could be gained if the perfect forecasts were available, are also calculated to explore how

much of the potential profit can be captured by each strategy.

Figure 4.12 presents the monthly gained profit for risk-neutral, the proposed strategies, and

the potential profit. It illustrates that for all months the gained profit of the proposed robust-

based strategy is higher than that of risk neutral strategy. This fact shows a significantly better

performance of the robust strategy compare to the risk neutral method; since the proposed multi-

step strategy makes the scheduling decisions immune against undesirable price variations. At the

same time, dividing the confidence interval into subintervals and including multiple steps enables

the strategy to take advantage of high (low) prices during discharging (charging). This comparison

shows the effect of forecast uncertainties on the company’s profit, and also the importance of

incorporating uncertainties when participating in the electricity market.

Moreover, based on potential profit and the actual profit gained by each strategy illustrated in

figure 4.12, table 4.1 shows the percentage of the potential profit captured by each strategy in each

month. Based on this table, the proposed robust-based strategy is able to capture a 76% up to 95%

89

of the monthly potential profit. Overall, the strategy captures 81.7% of the total potential profit

during the year. However, the capture rate is significantly lower for the risk-neutral strategy. This

rate ranges between 55.7% and 74.1%. Overall, 61.8% of the potential profit could be gained by

the risk-neutral strategy, which is 20% lower than that of proposed strategy.

4.6 Conclusion

This paper develops a robust-based bidding and offering strategy for a price-maker ES facility in a

day-ahead electricity market. The impact of ES operation on market prices is modeled by means

of GPQCs and DPQCs. The uncertainty in the GPQCs and DPQCs is also modeled by robust

optimization in this work. Then, a sequential algorithm is proposed to develop multi-step bidding

and offering curves for the facility to participate in the market. The applicability of the developed

strategy is verified in the numerical results. The one-year analysis shows significantly better per-

formance of the proposed strategy compared to the risk-neutral one. The proposed strategy has the

capability of capturing 81.7% of the potential profit, while this rate is 61.8% for the risk-neutral

strategy.

90

Chapter 5

Considering Thermodynamic Characteristics of a CAES

Facility in Self-scheduling in Energy and Reserve Markets 1

Nomenclature

Indices

s Steps of the curve representing the compression air flow rate versus cavern state of

charge (CAFRC) from 1 to nc.

s′ Steps of the curve representing the turbine air flow rate versus discharging rate

(TARFC) from 1 to nd .

s′′ Steps of the heat rate curve (HRC) from 1 to nh.

t Operation intervals running from 1 to T .

k Scenario index from 1 to K.

Parameters

πEt,k Day-ahead energy price for hour t in scenario k.

πsrt,k Day-ahead spinning reserve price for hour t in scenario k.

πnrt,k Day-ahead non-spinning reserve price for hour t in scenario k.

πNG Natural gas price.

γk Probability of scenario k.

ARFcs Charge air flow rate corresponding to step number s of the CARFC

ARFds′ Discharge air flow rate corresponding to step number s of the TARFC

1 c© 2016 IEEE. Reprinted, with permission, from [32]: S. Shafiee, H. Zareipour, and A. Knight, Conside-ring thermodynamic characteristics of a CAES facility in self-scheduling in energy and reserve markets, IEEETransactions on Smart Grid, vol. PP, no. 99, pp. 11, 2016.

91

bc,maxs size of step s of the CARFC.

bd,maxs′ size of step s′ of the TARFC.

bh,maxs′′ size of step s′′ of the HARFC.

CAmax Total mass of cushion air in cavern in kg

depsr/nrt,k Status of spinning/non-spinning reserve deployment at time t in scenario k (1 is

deployed and 0 is not deployed).

Emax maximum stored energy capacity of air storage cavern in MWh.

ER Nominal energy ratio of CAES facility.

HRhs′′ Heat rate corresponding to step number s of the HRC

HRnom Heat rate of the CAES facility at 100% discharging rate.

Pexpmax maximum generation capacity of the expander.

Pcmax maximum compression capacity of the compressor.

Pexpmin minimum generation capacity of expander.

Pcmin minimum compression capacity of compressor.

QSC Quick start capacity of the CAES facility.

qd,mins′ Summation of power blocks from step 1 to step s′−1 of TARFC.

qh,mins′′ Summation of power blocks from step 1 to step s′′−1 of HRC.

SOCmin minimum state of charge (SOC) of air storage cavern.

SOCmax maximum SOC of air storage cavern.

SOCinit Initial SOC of air storage cavern.

SOC f inal SOC of air storage cavern at the end of the day.

Sc,mins Summation of SOC blocks from step 1 to step s−1 of CARFC.

VOMexp Variable operation and maintenance cost of expander.

VOMc Variable operation and maintenance cost of compressor.

92

Variables

airch/dt Total amount of air compressed to/ extracted from the air cavern at time t.

bc/d/ht,s/s′/s′′ The fractional value of the SOC/power/power block corresponding to step s/s′/s′′

of the CAFRC/TARFC/HRC to obtain SOCt/Pdt /Pd

t at time t.

CONGt Cost of natural gas consumption at time t.

OCt Operation cost of the plant at time t.

Pdt Discharging power at time t.

Pct Charging power at time t.

Psr,xt Spinning reserve power at time t in either modes x , i.e, discharging (d), or charging

(c).

Pnrt Non-spinning reserve power at time t.

SOCt,k Cavern state of charge at time t in scenario k in percent.

uc/d/ht,s/s′/s′′ Binary variable that is equal to 1 if step s/s′/s′′ of CARFC/DAFRC/HRC is the last

step to obtain SOCt/Pdt /Pd

t and 0 otherwise.

xdt Unit status indicator in discharging mode at time t (1 is ON and 0 is OFF).

xct Unit status indicator in charging mode at time t.

Functions

Γ(Pct ,SOCt) The amount of air stored in the cavern in term of kg as a function of charging power,

Pct , the SOC of the cavern at time t.

AFRc(SOCt) Stepwise decreasing function that indicates the charging air flow rate as a function

of the SOC of the cavern at time t.

AFRd(Pdt ) Stepwise decreasing function that indicates the required discharging air flow rate as

a function of discharging rate, Pdt , at time t.

HR(Pdt ) Stepwise decreasing function that indicates the heat rate as a function of the dis-

charging rate, Pdt , at time t.

93

5.1 Introduction

The total installed electricity storage capacity worldwide is estimated to grow from around 85 GW

in 2011 to 460 GW with 27% renewable energy share in annual power generation by 2050 [34].

Compressed air energy storage (CAES), as one of bulk energy storage technologies, has a variety

of potential applications due to its capability of storing large amount of energy as well as its fast

response. These applications includes energy time-shifting, facilitating the large-scale integration

of renewable energy resources, and enhancing power system reliability [51, 92, 97, 105].

Figure 1 illustrates the schematic diagram of a conventional CAES plant with a two-stage high

pressure and low pressure compressors and turbines. Large compressors use electricity to compress

and store air into a reservoir, typically an underground salt cavern. The high pressure air is later

heated in a combustor using natural gas fuel and then used to power gas turbines to generate

electricity. In order to quantify the economics of CAES technology in an electricity market, an

appropriate scheduling model for the CAES facility needs to be developed considering efficiency

of the components and operational characteristics. Several studies develop self-scheduling models

of a CAES facility to estimate the energy arbitrage revenue of the CAES technology in different

electricity markets [28,37,39,96]. The self-scheduling of generation company with a CAES facility

as well as thermal units and renewable resources are addressed in [106, 107].

When participating in competitive electricity markets, a large merchant storage facility may

benefit from energy arbitrage. In addition to providing energy arbitrage, the CAES technology

can also provide spinning and non-spinning reserves services to the market. Stacking multiple

revenue streams improves the economics of energy storage, and thus, needs to be properly modeled

[13, 14]. Previous studies have modeled participation of energy storage systems in energy and

reserve markets [13, 14, 49, 50]. The additional revenue of a CAES facility gained by providing

ancillary services in different U.S. electricity markets are explored in [13]. The benefits of a CAES

facility providing arbitrage and reserves in a power system are studied in [69, 93, 95].

The efficiency of a CAES facility is expressed based on its heat rate and energy ratio [23]. Heat

94

Figure 5.1: Schematic diagram of a CAES facility.

rate expresses the amount of fuel burned per unit of electricity generated by the turbine. Energy

ratio indicates the amount of energy that the compressor of the plant consumes per unit of energy

that the expander generates [13]. The energy ratio is calculated based on the air flow rate the com-

pressor compresses and stores as well as the required discharging air flow rate. In the developed

CAES scheduling model in [13, 14, 28, 37, 39, 50, 69, 93, 95, 96, 106, 107], the nominal heat rate

and energy ratio, i.e., required heat and air flow rate at full discharging capacity, is considered

for the facility. However, it has been shown that the efficiency of a CAES facility depends on its

operational status [2, 23]. For instance, the heat rate increases for lower discharging powers. Se-

veral studies have concentrated on thermodynamic analysis of the CAES technology [2,108,109].

It is demonstrated that the air flow rate during charging depends on the cavern SOC. Moreover,

the heat and air flow rate during discharging vary significantly for different discharging rates [2].

Thus, the varying efficiency of the CAES facility based on the system thermodynamics should be

taken into account in the facility scheduling plan to avoid costly and unprofitable operations, and

consequently prevent overestimation of the facility’s revenues.

This paper proposes an optimization-based self-scheduling model of a merchant CAES facility

participating in day-ahead energy and reserve markets incorporating the thermodynamic characte-

ristics of the CAES technology. Thus, the proposed formulation properly models the changes in

the facility efficiency in different operational conditions and optimize its scheduling accordingly.

In doing so, the self-scheduling of a CAES facility providing energy arbitrage, spinning and non-

spinning reserves with nominal heat rate and energy ratio is initially developed. The thermodyna-

mic characteristics of the CAES facility during charge and discharge processes are then taken into

95

consideration and the model is modified accordingly. Since the formulation is non-linear, binary

techniques are used to convert it to its equivalent linear formulation to be solved by conventional

solvers. The main contribution of this paper is to incorporate practical limitations of a CAES fa-

cility in operation scheduling model. The significance of this contribution is that the prescribed

schedules revenue estimations are more realistic compared to when the thermodynamic charac-

teristics are ignored. Note that in his paper, it is assumed that the CAES facility is a merchant

privately-owned unit, which operates independently in the electricity market. We do not consider

co-operation of the facility with a wind/solar farm.

5.2 CAES Self-scheduling Formulation

In this section, the self-scheduling of a CAES facility participating in day-ahead energy and reserve

markets is developed. The storage facility is assumed to be a price-taker and cannot alter the

market price by its operation. This assumption is only valid when the CAES capacity is very small

compared to the size of the supply-side in the market. Modeling the behavior of a price-maker

CAES facility considering the thermodynamic characteristics is the subject of the authors’ future

work.

5.2.1 Objective function

The goal of the CAES plant is to maximize its profit through energy arbitrage as well as offering

spinning and non-spinning reserves as a participant in the day-ahead energy and reserves markets.

In order to take price uncertainty into account, we generate different price scenarios on historical

price data. We also generate different spinning and non-spinning deployment scenarios to consider

the uncertainty of their deployment. Then, the expected value of the profit for the day is calculated,

in line with [50]. The objective function is expressed as follows

96

maxK

∑k=1

γk

T

∑t=1

[(Pdt −Pc

t )×πEt,k +(Psr,d

t +Psr,ct )×π

srt,k

+Pnrt ×π

nrt,k +[(Psr,d

t +Psr,ct )×depsr

t,k

+Pnrt ×depnr

t,k]×πEt,k−OCt,k] (5.1)

The objective functions (5.1) consists of five terms. The first term represents the energy ar-

bitrage revenue, i.e. the profit of selling electricity to the market minus the cost of purchasing

the electricity from the market to power the compressor. The second term is the spinning reserve

revenue determined by the spinning reserve price and the spinning reserve capacity offered during

either charging or discharging modes. Note that a responsive load can offer spinning reserve ser-

vice in electricity markets such as ERCOT and NYISO [110]. The third term is the non-spinning

reserve revenue. The forth term is the revenue comes from the real-time spinning and non-spinning

reserves deployment respectively assuming a probability of deployment depsrt,k and depnr

t,k in sce-

nario k. In this study, based on the deployment probability for each reserve, depsrt,k and depnr

t,k are

vectors consist of 0 and 1 elements, in which 1 at time t states total deployment of offered spinning

or non-spinning reserve at that time in scenario k. It is assumed the storage facility is paid by

the energy price (πEt,k) in case it is deployed. The fifth term in the objective functions shows the

operation cost of the CAES facility at time t in scenario k. It is stated as follows:

OCt,k = [(Pdt +Psr,d

t ×depsrt,k +Pnr

t ×depnrt,k)

× (HRnom×πNG +VOMexp)]

+ [(Pct −Psr,c

t ×depsrt,k)×VOMc] ∀t ∈ T, ∀k ∈ K (5.2)

The operation cost is expressed in two terms in (5.2). The first term shows the operation

cost during discharging. This is the cost of burning natural gas and the variable operation and

maintenance cost of expander to provide energy offered in the energy market Pdt plus the spinning

or non-spinning reserves during discharging if deployed at time t. The second term in (5.2) is the

total variable operation and maintenance cost during compression.

97

5.2.2 Power Capacity Constraints

Equation (5.3) states the CAES facility can operate in only one specific mode at a time. The limits

on the charging and discharging power and the spinning reserve during charging and discharging

are presented in (5.4)-(5.7) based on the minimum and maximum capacity of compressor and

expander. The non-spinning reserve capacity is limited by the quick start capacity of the CAES

facility, as expressed in (5.8).

xct + xd

t ≤ 1 ∀t ∈ T (5.3)

Pct ≤ Pc

max.xct ∀t ∈ T (5.4)

Pcmin.x

ct ≤ Pc

t −Psr,ct ∀t ∈ T (5.5)

Pdt +Psr,d

t ≤ Pexpmax.x

dt ∀t ∈ T (5.6)

Pexpmin .x

dt ≤ Pd

t ∀t ∈ T (5.7)

0≤ Pnrt ≤ QSC× [1− (xc

t + xdt )] ∀t ∈ T (5.8)

5.2.3 Energy Capacity Constraints

Participation in energy and reserve markets creates operational constraints based on cavern capa-

city. These constraints must be taken into account when scheduling operation in multiple markets.

In [13, 93], this issue is not addressed and only the effect of charging and discharging power is

considered in energy capacity constraints.

During discharging, the CAES facility needs to store sufficient compressed air in the cavern

to not only follow its discharge schedule in the energy market, but also provide ancillary service

in response to the system operator’s deployment dispatch, as specified in (5.9). The effect of

discharging on the cavern SOC is a function of the energy ratio. Additionally, the storage cavern

must have sufficient available capacity to be able to follow the charge schedule in case the spinning

reserve during charging is not deployed by the system operator. This constraints is defined in

(5.10). Equation (5.11) calculates the state of charge for the next hour (SOCt+1,k) based on the

current SOC, the level of charging and discharging power (Pct and Pd

t ), as well as the amount of

98

spinning and non-spinning reserves deployed in that hour. The deployed spinning reserve during

charging is calculated as (Psr,ct × depsr

t,k) in scenario k. If depsrt,k = 1 at time t, i.e, the spinning

reserve is deployed, the CAES facility must decrease its charging level by Psr,ct , as stated in (5.11).

Similarly, if the spinning reserve is deployed during discharging, the facility must increase its

discharging level by Psr,dt . In a similar way, the effect of non-spinning reserve deployment on

SOC is calculated. The initial level for the air storage cavern is specified by (5.12). In order

to have sufficient energy at the end of the day to be able to take advantages of energy arbitrage

opportunities in the next day, a minimum level for SOC is considered for the end of the day, as

stated in (5.13).

SOCmin ≤ SOCt,k−(Pd

t +Psr,dt +Pnr

t )×EREmax

∀t ∈ T, ∀k ∈ K (5.9)

SOCt,k +Pc

tEmax ≤ SOCmax ∀t ∈ T, ∀k ∈ K (5.10)

SOCt+1,k = SOCt,k +(Pc

t −Psr,ct ×depsr

t,k)

Emax

−(Pd

t +Psr,dt ×depsr

t,k +Pnrt ×depnr

t,k)×ER

Emax

∀t ∈ T, ∀k ∈ K (5.11)

SOC1,k = SOCinit ∀t ∈ T, ∀k ∈ K (5.12)

SOC f inal ≤ SOCT+1,k ∀t ∈ T, ∀k ∈ K (5.13)

5.3 Incorporating Thermodynamic Characteristics in Self-Scheduling Formula-

tion

In Section 5.2, constant heat rate and energy ratio are assumed for the facility, disregarding the

facility operational conditions. This is the case for previous studies focusing on self-scheduling of

the CAES technology in an electricity market. However, the efficiency depends on its operational

99

Figure 5.2: The variations of air flow rate and compressor power during charge process [2].

status [2,23]. In [2], the charge/discharge process analysis of a conventional compressed air energy

storage system is conducted. In that study, the air storage cavern operates between the pressure

range of 7.2 MPa and 4.2 MPa and the rated powers of the two-stage compressor and two-stage

turbine are 60 MW and 290 MW, respectively. Developing the thermodynamic equations for dif-

ferent components, the thermodynamic analysis is carried out for charging process and also for a

range of discharging rate, i.e., 30% to 100%. It is shown that the cavern SOC affects the air flow

rate during charging. Moreover, the heat rate and required air flow rate during discharging varies

significantly for different discharging rates.

5.3.1 Effect of SOC in Charging Process

Reference [2] investigates the variation of air flow rate and compressor power during charging time

with rated compression power when the compressor fully charges the storage cavern with initial

minimum SOC. It is shown in figure 5.2. As shown in figure 5.2, the air flow rate of the compressor

drops as the SOC and consequently cavern pressure increase. This is because as more air is stored

in the cavern, it gets more difficult to compress air in a higher pressure. As seen in figure 5.2,

compressor power increases before decreasing. This is because of the increased compressor outlet

temperature and decreased air flow rate [2].

Based on the above discussion, the impact of charging on the cavern SOC is not constant. De-

pending on the level of SOC, the amount of air which could be stored in the cavern with the power

Pct directly depends on the level of SOC. In other words, the more amount of air is compressed and

100

stored, the lower level of air flow could be stored in the cavern due to the higher pressure level of

the cavern. Thus, if at time t, the compressor is operating with the power level of Pct , the mass of

air stored in the cavern in kg depends on Pct as well as the current cavern SOC. This is presented as

a function of Pct and SOCt : say Γ(Pc

t ,SOCt). Thus, the SOC equation can be expressed as follows:

SOCt+1 = SOCt +Γ(Pc

t ,SOCt)

CAmax −Φ(Pdt ) ∀t ∈ T (5.14)

Φ(Pdt ) represents the effect of discharging on the cavern SOC. It will be explained later in

this section. The function Γ(Pct ,SOCt) is defined based on the information presented in figure

5.2. Based on this figure, the air mass flow rate of compressors per MW of energy consumed by

the compressor, AFRc, is calculated by dividing the air flow rates by the compressor power. The

cavern SOC at time t of charging is also calculated, which is the cumulative mass of air stored by

the time t, i.e., integral of airflow rate curve up to t. By means of this information, AFRc versus

SOC is illustrated in figure 5.3. Note that, based on the air flow rate curve shown in figure 5.2, the

total amount of compressed air is 6.4 million kg. Total mass of cushion air in cavern is reported to

be 9.48 million kg [2]. Thus, it can be concluded that, in order to maintain the minimum required

pressure for the cavern, the minimum amount of 3.08 million kg of compressed air must remain in

the cavern. In other words, the minimum cavern SOC mus be 33%.

As seen in figure 5.3, AFRc is a function of cavern SOC. Thus, the function Γ(Pct ,SOCt) and

(5.14) can be expressed as follows:

Γ(Pct ,SOCt) = Pc

t ×AFRc(SOCt)×3600 ∀t ∈ T (5.15)

SOCt+1 = SOCt +Pc

t ×AFRc(SOCt)×3600CAmax −Φ(Pd

t ) (5.16)

5.3.2 Effect of Generation level on SOC

During discharging, the required air flow rate of a CAES facility is a function of generation power

- higher generation power requires higher air flow rates through the expander to meet the energy

101

Figure 5.3: The level of air flow rate per MW of charging versus cavern SOC.

Figure 5.4: Variations of air flow rate under different generation levels [2]

requirement. Based on the data in [2], this relationship can be plotted, as shown in figure 5.4. It

can be seen that generation level and air flow rate are linearly related.

An interesting point can be found from the data presented in figure 5.4. Dividing the air flow

rates by the generation power for different generation power, the required air flow rate per unit

of generated electricity at different generation levels is derived, which is depicted in figure 5.5.

According to this figure, the air flow rates per MW of generated electricity increases with for

lower generation levels. For instance, when generating at 30% generation level, the required air

flow rate per MW is 2.30 kg/s.MW , which is 42% higher than that of generating with full capacity,

1.38 kg/s.MW . This shows that the impacts of generating electricity during discharging mode

on cavern SOC is not constant. Conversely, it significantly depends on the generation level. The

lower the generation level is, the higher air flow rates per unit of electricity is required. Based

on data in [2], the reason behind this issue is implied in figure 5.6. This figure shows the turbine

efficiency under different generation level conditions. As seen in this figure, the efficiency of the

high pressure (HP) turbine decreases significantly with the decrease in generation level. Hence, in

102

Figure 5.5: The variations of air flow rate per unit of generated electricity under different genera-tion levels

Figure 5.6: The variations of turbine efficiency under different generation levels [2].

order to compensate the decrease in turbine efficiency for lower generation level, higher air flow

rate is required to generate one unit of electricity.

Based on the above discussion, the required mass of air per MW of generated electricity, re-

leased from the cavern, to generate a certain level of power is not constant and directly depends

on the discharging level as shown in 5.5. These issues must be incorporated in the self-scheduling

model of a CAES system. The mass of air released from the cavern at time t to generate Pdt is

Pdt ×AFRd(Pd

t ). AFRd is the discharging air flow rate per MW, which is a function of generation

power, as shown in figure 5.5. Thus, the cavern SOC equation (5.16) is updated as follows:

SOCt+1 =SOCt +Pc

t ×AFRc(SOCt)×3600CAmax

− Pdt ×AFRd(Pd

t )×3600CAmax ∀t ∈ T (5.17)

103

Figure 5.7: The variations of HR under different generation levels [2].

5.3.3 Effect of Discharging Rate on Heat Rate (HR)

During discharging, the required heat rate is a function of generation level. Based on the data

presented in [2], figure 5.7 plots the variations of HR for different generation levels. As seen in

this figure, the heat rate increases noticeably for lower generation levels. For instance, the heat rate

increases by 26% from the rate value when operating at 30% generation level. The reason for this

increment in HR is implied in figure 5.6. Due to decrease in turbine efficiency for lower generation

level, higher air flow rate and consequently higher fuel flow rate is required to generated one unit

of electricity.

According to figure 5.7, HR is a function of generation level and thus, the cost of natural gas

when discharging at Pdt can be expressed as follows:

CONGt = [Pd

t ×HR(Pdt )×π

NGt ] ∀t ∈ T (5.18)

5.4 Equivalent Linear Formulation

The developed equation for the cavern SOC in (5.17) and the cost of natural gas in (5.18) are non-

linear due to the products between the variables. In this section, their equivalent linear formulations

are presented.

104

Figure 5.8: Linearization process for (a) Compression air flow rate versus cavern SOC, (b) Dischr-ging air flow rate versus discharging rate, (c) heat rate versus discharging rate.

5.4.1 Linearizing the effect of SOC in Charging Process

In (5.17), the term Pct × AFRc(SOCt) shows the amount of air compressed and stored at time

t, which causes non-linearity. In the following, the equivalent linear constraints of the term is

developed.

As an example, the curve in figure 5.3 is represented by a four step decreasing curve as shown in

figure 5.8-(a). A set of binary variables uct,s is defined for each hour. Then, based on the variables

bct,s,u

ct,s and parameters Sc,min

s ,bc,maxs , shown in figure 5.8-(a), the set of following equations are

developed to find the corresponding step the level of cavern SOC is at.

SOCt =nc

∑s=1

(bct,s +uc

t,sSc,mins ) ∀t ∈ T (5.19)

0≤ bct,s ≤ uc

t,sbc,maxs ∀t ∈ T,∀s ∈ nc (5.20)

nc

∑s=1

uct,s = 1 ∀t ∈ T (5.21)

In (5.19), SOCt is linearly expressed as a function of variables bct,s and uc

t,s, shown in figure

5.8-(a). Equation (5.20) expresses the limit on the block of the curve shown in figure 5.8-(a) in

every hour, which is between zero and the size of that step. Equation (5.21) states that in every

105

hour, only one instance of the variable uct,s is nonzero, which shows the corresponding step of the

curve the cavern SOC is at that hour.

Using the defined parameters and variables, the total amount of air compressed when compres-

sor is operating at the power Pct can be expressed as follows:

aircht = Pc

t ×[ nc

∑s=1

uct,sAFRc

s]

(5.22)

Based on the fact that at time t only one instance of the variable uct,s is one, the summation in

(5.22) shows the compressor air flow rate. The equation (5.22) is still non-linear due to the products

between the variables Pct and uc

t,s. The following constraints are the equivalent linear constraints

of (5.22). The method used here to resolve the non-linearity of (5.22) is the extension of big M

method presented in [101, 111]. The details of this method can be found in [111].

aircht +M ≥ AFRc

s.Pct +uc

t,s×M,∀t ∈ T,∀s ∈ nc (5.23)

aircht −M ≤ AFRc

s.Pct −uc

t,s×M,∀t ∈ T,∀s ∈ nc (5.24)

where M is a positive big enough number.

5.4.2 Linearizing the Effect of Discharging Process on SOC

In (5.17), the term Pdt ×AFRd(Pd

t ) is the required amount of air released from the cavern to gene-

rate Pdt . In the following, the equivalent linear constraints of the term is presented. The process is

in line with the approach used in [29].

Figure 5.8-(b) illustrates the linearization process for a sample four step discharging air low

106

rate curve. Based on this approach, the linearization process may be written as follows:

Pdt =

nd

∑s′=1

(bdt,s′+ud

t,s′qd,mins′ ) (5.25)

0≤ bdt,s′ ≤ ud

t,s′bd,maxs′ (5.26)

nd

∑s′=1

udt,s′ = xd

t (5.27)

airdt =

nd

∑s′=1

AFRds′× (bd

t,s′+udt,s′q

d,mins′ ) (5.28)

Figure 5.8-(a) shows the variables, i.e., bdt,s′,u

dt,s′ , and parameters, i.e., AFRd

s′,qd,mins′ ,bd,max

s′ ,

used to linearize the amount of air as a function of hourly discharging power. The shaded area in

this figure represents the total required amount of air, which is the discharging power multiplied

to the air flow rate per MW. In (5.25), the discharging power is linearly expressed as a function of

variables bdt,s′,u

dt,s′ , shown in 5.8-(b). Equation (5.26) expresses the limit on the block of the curve,

which is between zero and the size of that step. Equation (5.27) specifies that in every hour of

discharging, only one instance of the variable udt,s′ is nonzero, which shows the corresponding step

of the discharging air flow rate the storage is operating at that hour. Based on (5.27), all instance

of the variable udt,s′ are zero at time t if storage is not in discharging mode at that hour. Based

on (5.26) and (5.27), during a discharging hour, only one instance of the variable bdt,s′ could vary

between zero and the size of selected step of the curve. All the others are forced to be zero. Based

on above discussion, for each hour, the total amount of air required to discharge at Pdt is linearly

expressed as (5.28). Therefore, the term Pdt ×AFRd(Pd

t ) in (5.17) is replaced by the variable airdt

and the constraints (5.25)-(5.28) are added to the optimization problem.

5.4.3 Linearizing the Cost of Natural Gas in Discharging Process

The cost of natural gas in (5.18), which is used in the operation cost constraint, is non-linear. A

similar approach, described in Section 5.4.2, is used to develop the equivalent linear constraints of

(5.18). Figure 5.8-(c) illustrates the linearization process for a sample four step heat rate curve.

107

The linearization process is expressed as follows:

Pdt =

nh

∑s′′=1

(bht,s′′+uh

t,s′′qh,mins′′ ) (5.29)

0≤ bht,s′′ ≤ uh

t,s′′bh,maxs′′ (5.30)

nh

∑s′′=1

uht,s′′ = xd

t (5.31)

CONGt =

nh

∑s′′=1

HRhs′′× (bh

t,s′′+uht,s′′q

h,mins′′ ) (5.32)

The process of incorporating thermodynamic characteristics of the conventional CAES techno-

logy in self-scheduling formulation and the the linearization processes, developed in sections 5.3

and 5.4, respectively, are used to update the CAES self-scheduling optimization problem presented

in section 5.2.

5.5 Numerical Results

Daily and yearly numerical simulations are performed for a CAES facility with 100 MW of dis-

charging power, 60 MW of charging power, and 8 hours of full discharging capability as the storage

capacity. Minimum discharging, charging and SOC levels are respectively, 30 MW, 10 MW, and

33%. The charging and discharging air flow rate and the heat rate curves presented in Section 5.3

are also used for this case study. The minimum level of SOC at the end of the day is assumed to be

at least 60% to have enough compressed air in the cavern to take advantage of the opportunities in

the next day. The energy and reserve prices of the ERCOT market for a five year period from 2011

to 2015 are used for the yearly analysis. Based on the available installed generation capacity of the

ERCOT market, which is 77,000 MW [112], considering a 100 MW storage facility as price-taker

is a reasonable assumption. The proposed mixed integer linear model is implemented in generali-

zed algebraic modeling systems (GAMS) software package and solved using CPLEX solver. The

solution time for daily scheduling on a PC with an Intel Core 7 CPU (2.8 GHz) and 8.0 GB RAM

is in the order of few seconds. GAMS c© and MATLAB c© are used to solve the model for the five

108

Figure 5.9: energy, spinning reserve, and non-spinning reserve price profiles.

year period on a daily basis.

In the next subsections, the CAES scheduling model with constant efficiency parameters and

the proposed thermodynamic-based model are respectively referred to as CM and TBM (conventi-

onal model and thermodynamic-based model).

5.5.1 CAES Self-scheduling: a Demonstrative Case

The energy and reserves price profiles for a typical 24 hours period is depicted in figure 5.9. The

resulted schedule for energy and reserve markets from CM is shown in figure 5.10-(a). As seen

in this figure, the facility charges at low price hours and also participates in the spinning reserve

market during charging periods. Moreover, the facility decides to discharge partially with the

minimum capacity in all discharging periods and offers the remaining capacity in the spinning

reserve market to maximize its profit. The estimated profit in this case is $9,800.

The schedule obtained from the CM, shown in figure 5.10-(a), is imported to TBM to see

whether the storage facility is capable of following the schedule and what the actual profit is based

on this schedule considering the thermodynamic characteristics. The actual schedule is depicted

in figure 5.10-(b). According to this figure, the CAES facility should also charge with 20 MW

at hour 4 to compressed more air because of two reasons. First, due to the effect of SOC on the

compression air flow rate, which is not considered in CM, not enough air is stored. Secondly, due

to partial discharge during discharging periods, the required air flow rate increases due to drop

in turbine efficiency, which means more amount of air than estimated is required to follow the

109

Figure 5.10: (a) CAES scheduling obtained from the simple model, (b) Actual CAES schedu-ling when following simple model schedule considering thermodynamic characteristics, (c) CAESscheduling resulted from the developed thermodynamic-based model.

discharge schedule. As shown in figure 5.10-(a), the storage facility fails to follow the schedule

in hours 14, 22, and 24 due to faster depletion of the air storage cavern than what estimated and

accordingly lack of compressed air in the air storage cavern. Moreover, due to partial discharging,

the heat rate also increases, which imposes higher cost of burning natural gas. Therefore, the actual

revenue gained from this schedule is $5,600, which is $4,200 lower than what is estimated.

The optimal scheduling of the CAES facility resulted from the TBM is shown in figure 5.10-

(c). Comparison of figure 5.10-(a) and figure 5.10-(c) demonstrates that with thermodynamic-

based scheduling, the partial discharging operations in some hours are curtailed. Furthermore,

the facility discharges with higher level in hours 18, 20, and 21; since due to decrease in turbine

efficiency and higher required air and fuel flow rates in partial operation, it is not profitable to

operate partially. This schedule leads to $6,940 operation profit, which is $1,340 higher than the

actual profit gained from the CM. Therefore, the results show that

taking into account the CAES thermodynamic characteristics would lead to a more efficient

scheduling, with lower operation hours and higher operating profit.

110

Figure 5.11: The annual profit of the CAES facility providing energy arbitrage when using CMand TBM.

Figure 5.12: Dispatch characteristic of a CAES facility during 2011 providing energy arbitragewhen scheduling with CM and TBM.

5.5.2 Participating in Energy Market: Five year Analysis

In this section, it is assumed that the CAES facility only participates in the energy market providing

energy arbitrage. The CM and TBM are applied sequentially on a daily basis to the energy price

of the ERCOT market during years 2011-2015.

Figure 5.11 illustrates the annual estimated profit gained from the CM, the actual gained profit

from the CM when the resulted schedules in CM are imported to the TBM, and the TBM profit

during year 2011 and 2015. As shown in this figure, the actual annual profit decreases slightly from

the estimated profit when using CM. Moreover, the improvement in profit obtained from the TBM

compared to the actual profit is small. This implies that in case of providing only energy arbitrage

in the market, the CM, widely used in the literature, has acceptable accuracy. The dispatch charac-

teristics of the storage facility during the year shows why this happens. As an example, figure 5.12

shows the percentage of time the storage facility is charging, discharging or idle during 2011. As

shown in this figure, in case of using CM (left bar), the CAES facility mostly operates at full dis-

charge when providing energy arbitrage, in which the facility operates with the nominal efficiency.

111

Thus, as shown in the middle bar of figure 5.12, the CAES facility is able to follow all discharges

scheduling. It should only charge for 2% more hours to be able to follow the schedule due to a

few percentage of the hours it discharges partially and also not considering the effect of SOC in

charging process. This causes a small decrease in the actual profit compared to the estimated one.

Moreover, as shown in the right bar of figure 5.12, the dispatch characteristic of the storage when

providing energy arbitrage using TBM is similar to that of CM. Therefore, the gained profit out of

CM and the TBM are close to each other.

5.5.3 Participating in Energy and Reserve Markets: Five Year Analysis

In this section, it is assumed that the CAES facility participates in both energy and reserve market

The developed thermodynamic-based model and the conventional model are applied to investigate

the effect of CAES thermodynamic characteristics on its annual operating profit obtained from

energy and reserve markets.

The annual estimated profit resulted from the CM, the actual gained profit from the CM, and

the TBM profit comes from providing energy arbitrage, spinning and non-spinning reserves during

year 2011 and 2015 are depicted in figure 5.13. The numbers above the bars shows the total profit

in each case. As seen in this figure, the actual annual profit for energy arbitrage and spinning

reserve decreases noticeably from the profit estimated by the CM. Table 5.1 presents the error of

the estimated profit resulted from the CM compared to the actual profit for the year 2011 to 2015.

The results demonstrate in case of scheduling for both energy and reserves markets, the CM leads

to significant overestimation, which obviously affects the economics of the facility. This shows

the importance of considering the thermodynamics of the facility when scheduling for the energy

and reserve markets. Furthermore, the comparison of the actual profit and the TBM profit, shown

in figure 5.13, states that by considering the thermodynamics of the facility in the scheduling,

proposed in the TBM, the profit of the CAES facility is improved, as shown in table 5.1.

Investigating the dispatch characteristics of the storage facility during the year for each model

shows the reason of such significant error in the profit estimated by the CM. As an instance, figure

112

Figure 5.13: The annual profit of the CAES facility providing energy arbitrage as well as reserveswhen using CM and TBM.

Figure 5.14: Dispatch characteristic of a CAES facility during 2011 providing energy arbitrageand reserves when scheduling with CM and TBM.

5.14 shows the dispatch characteristics of the facility during the year 2011. As illustrated in this

figure, in the case of using CM (left bar), the CAES facility mostly discharge partially to offer

the remaining capacity as the spinning reserve while ignoring the decreasing efficiency for lower

discharging rate; This is similar to that of reported in [13]. Since, when discharging partially, the

required air flow rate increase significantly, the facility needs to charge more to store more amount

of air in the cavern, as shown in the middle bar of figure 5.14. However, in spite of compressing

more air, the CAES facility fails to follow all partial discharge schedules due to fast depletion of

the cavern. Moreover, higher heat rate, required for those hours of partial discharging, imposes

higher operation cost to the facility. Therefore, the actual revenue comes from the energy and

reserve market drops significantly, as depicted in figure 5.13.

113

Table 5.1: Annual error of the estimated profit of CM and the profit improvement obtained byTBM, when participating in energy and reserve markets

Year 2011 2012 2013 2014 2015Error of estimated profit

compared to actual in CM 10.6% 19.5% 32.5% 23.1% 18.6%

Improvement of TBMcompared to CM 2.50% 2.90% 5.47% 3.18% 3.73%

As depicted in the right bar of figure 5.14, the dispatch characteristic of the storage using

TBM is different from that of the CM. It discharges with full capacity for more number of hours.

Moreover, the number of hours it discharges partially is much lower than those of the CM, since

the lower efficiency of the facility for lower discharging rates are considered in the TBM. Thus,

it only discharge partially in those hours that the revenue gained by participating in the reserve

market offsets the high operation cost of partial discharge. Therefore, although in TBM, the facility

discharge for less number of hours compared to those of the CM, due to the efficient scheduling

considering the thermodynamics of the system and preventing unprofitable actions, the obtained

profit is higher than the actual profit gained by the CM, as shown in figure 5.13.

5.6 Conclusion

In this paper, a self-scheduling approach for a merchant CAES facility participating in energy and

reserve markets is developed incorporating the thermodynamic characteristics of the facility. The

developed model is applied to the energy and reserve prices of the ERCOT market to analyze the

effect of the system thermodynamics on the economics of energy storage and compare it with

the case of conventional scheduling with nominal constant efficiency parameters. The results de-

monstrates that in case of providing only energy arbitrage, the conventional model with nominal

efficiency has acceptable accuracy. However, in case of participating in energy and reserves mar-

kets, the results of the conventional model has significant error, which illustrates the importance of

considering CAES thermodynamic characteristics in its scheduling.

In this paper, it is assumed that the CAES facility is a price-taker and does not change the mar-

114

ket price by its operation. Modeling the self-scheduling of a price-maker CAES facility in energy

and reserve markets considering the thermodynamic characteristics is the subject of the authors’

future work. Moreover, considering the fact that a CAES plant could affect the power flow in the

grid, the CAES operation in a power grid could have impact on transmission congestion. Incor-

porating the transmission network and the impacts of CAES operation limitations on transmission

congestion is left to future work. Furthermore, the proposed thermodynamic-based CAES self-

scheduling model can be expanded to the scheduling of an energy storage facility in the electricity

market co-located with a wind or solar farm to investigate what role the thermodynamic limitations

play when co-operation with a wind or solar farm is of interest.

115

Chapter 6

Conclusion

In this thesis, the optimal operation of a CAES facility in a competitive electricity market is addres-

sed. An energy storage facility optimizes its operation schedules, referred to as self-scheduling,

in order to maximize its profit in an electricity market. In this thesis, the self-scheduling of a

price-taker as well as price-maker CAES facility in energy market is developed. Then, uncertainty

modeling techniques are applied to the self-scheduling models to develop risk-contained bidding

strategies for both cases of price-taker and price-maker to enable the facility operator to incorpo-

rate the risk of forecasting uncertainties in its scheduling problem in energy market. Afterward,

the participation of the CAES facility in ancillary service market in addition to the energy market

is addressed. Overall, this thesis is significant to the literature because it focuses its efforts on the

practical approaches for scheduling a CAES facility in energy and ancillary service markets and

quantifying its economics. The detailed conclusions for each of the Chapters 3 to 6 are summarized

next.

In chapter 2, an efficient and applicable bidding and offerings strategy for a CAES facility is

developed based on IGDT method. The developed IGDT-based strategy incorporates both robust

and opportunistic actions in strategic scheduling process. This approach enables the plant to be-

nefit from desirable price fluctuations in addition to guaranteeing a minimum level of profit. The

results show that the proposed scheduling strategies lead to a more profitable result than that of

deterministic one when the forecasting uncertainties are ignored.

For a comparatively large scale energy storage facility in an electricity market, the assumption

of being price-taker is not valid anymore. Hence, their impacts on market price should be accu-

rately modeled to prevent profitability overestimation. Thus, in chapter 3, an optimization-based

self-scheduling is developed for an energy storage facility incorporating its impacts on market

116

price using GPQCs and DPQCs. Then, using the developed model, a comprehensive economic

assessment is conducted to investigate the profitability of energy storage systems in Alberta. To

do so, the historical data of Alberta electricity market during 2010 to 2014 are utilized to construct

GPQCs and DPQCs. Thereafter, the curves are used as the input of the proposed model to evalu-

ate the economics of a price-maker energy storage in this market. The results illustrate there are

plenty of energy arbitrage opportunities in the Alberta electricity market. However, energy storage

operation significantly affects market price, especially during high price hours.

Because of the presence of inevitable errors in forecasted GPQCs and DPQCs, an applicable

bidding and offering strategy for participation of a price-maker CAES facility in energy market is

important in order to capture the arbitrage opportunities as much as possible. Hence, Chapter 4

develops a robust-based bidding and offering strategy for a price-maker CAES facility in a day-

ahead electricity market. The self-scheduling model presented in chapter 3 is extended by applying

robust optimization to model the uncertainty in the GPQCs and DPQCs. Then, a sequential algo-

rithm is proposed to develop multi-step bidding and offering curves for the facility to participate

in the market. The applicability of the developed strategy is verified in the numerical results. The

one-year analysis shows significantly better performance of the proposed strategy compared to the

risk-neutral one.

Chapter 5 develops a scheduling model for the operation of a price-taker CAES facility taking

into account participation in both energy and ancillary service markets providing energy arbitrage

as well as ancillary services such as spinning and non-spinning reserves. Moreover, the efficiency

of the facility in different operation condition is modeled since the efficiency is not constant and it

somehow depends on its operation condition. The results show noticeable improvement in the re-

venue of the facility when participating in both markets compare to the case of only energy market.

Furthermore, it is demonstrated that in case of providing only energy arbitrage, the conventional

model with nominal efficiency has acceptable accuracy. However, in case of participating in energy

and reserves markets, the results of the conventional model has significant error, which illustrates

117

the importance of considering CAES thermodynamic characteristics in its scheduling.

6.1 Future Work

The following is a list of potential extensions to the present work:

1. Chapters 2 and 4 could be extended for considering real time market in addition

to the day-ahead market. Since some of the submitted bids or offers might not be

accepted in the day-ahead market, the storage facility has the opportunity to parti-

cipate in the real-time market and adjust its bids and offer to increase its revenue.

2. The developed general scheduling framework in Chapter 3 can be modified for

different energy storage technologies (e.g. Battery, PHS) to investigate and compare

the economic feasibility of these technologies in the Alberta electricity market.

3. Chapter 3 could be extended to the other electricity markets. Historical data of other

markets can be extracted and analyzed to construct the PQCs. Then, the database

and the scheduling model is used to explore energy arbitrage opportunities in those

markets.

4. The database of GPQCs and DPQCs and the developed linearized scheduling model

in Chapter 3 can be used to formulate an optimization framework to find the optimal

size for an specific energy storage technology.

5. The work in Chapter 5 could be extended to model the self-scheduling of a price-

maker CAES facility in energy and reserve markets using the concept of PQCs.

Historical data of the reserve market should be extracted and analyzed to construct

the hourly price quota curves for spinning and non-spinning reserve markets. Then,

the potential revenue of the CAES facility in energy and reserve market is investi-

gated considering its operation impacts on energy and reserve prices. It will give a

more accurate estimation of the economics of the technology.

118

6. Considering the fact that a CAES plant could affect the power flow in the grid, the

CAES operation in a power grid could have impact on transmission congestion.

Therefore, the transmission network can be incorporated in the studies conducted

in this thesis.

7. As a future work, the CAES self-scheduling model proposed in Chapter 5 can be

expanded to the scheduling of an energy storage facility in the electricity market

co-located with a wind or solar farm to investigate the role of a CAES facility when

co-operation with a wind or solar farm in energy and reserve markets.

6.2 Summary

In this thesis, the optimal operation of a CAES facility in energy and reserve markets is investiga-

ted. The case of provision of only energy arbitrage and also energy arbitrage as well as spinning

and non-spinning reserves are addressed. These improvements, the ability of a CAES facility to

effectively participate in a restructured electricity market, may help the integration of storage in

electricity markets. Future works are also identified to further improve the operation of storage fa-

cilities. In summary, this thesis will help the integration of new merchant energy storage facilities

into the market.

119

Bibliography

[1] S. de la Torre, J. Arroyo, A. Conejo, and J. Contreras, “Price maker self-scheduling in

a pool-based electricity market: a mixed-integer lp approach,” IEEE Trans. Power Syst.,

vol. 17, no. 4, pp. 1037–1042, Nov 2002.

[2] P. Zhao, L. Gao, J. Wang, and Y. Dai, “Energy efficiency analysis and off-design analy-

sis of two different discharge modes for compressed air energy storage system using axial

turbines,” Renewable Energy, vol. 85, pp. 1164 – 1177, 2016.

[3] CITI Group, “Investment Themes in 2015: Dealing with Divergence,” 2015. [On-

line]. Available: https://ir.citi.com/20AykGw9ptuHn0MbsxZVgmFyyp%20puQUUt3HV\

hTrcjz4ibR%2Bx79LajBxIyoHIoSD%20J3S%2BWRSMg8WOc%3D

[4] V Viswanathan, M Kintner-Meyer, P Balducci, C Jin, “National Assessment of Energy

Storage for Grid Balancing and Arbitrage, Phase II, Volume 2: Cost and Performance Cha-

racterization,” Pacific Northwest National Laboratory, September 2013.

[5] The Legislative Assembly Of Alberta, “Renewable Electricity Act,” THE MINISTER

OF ENERGY, 2016. [Online]. Available: http://www.assembly.ab.ca/ISYS/LADDAR files/

docs/bills/bill/legislature 29/session 2/20160308 bill-027.pdf

[6] “Renewable Roadmap,” SaskPower. [Online]. Available: http://www.saskpower.com/

our-power-future/renewables-roadmap/

[7] Electric Power Research Institute, “Compressed air energy storage scoping study for califor-

nia,” Prepared for the California Energy Commission. Report number CEC-500-2008-069.

EPRI: Palo Alto, CA, November 2008.

[8] Office of Electricity Delivery & Energy Reliability, “Global Energy Storage Database,”

Department of Energy. [Online]. Available: http://www.energystorageexchange.org/projects

120

[9] R. Sioshansi, P. Denholm, T. Jenkin, and J. Weiss, “Estimating the value of electricity

storage in PJM: Arbitrage and some welfare effects,” Energy Economics, vol. 31, no. 2,

pp. 269 – 277, 2009.

[10] Carlson, C., “Wind and energy storage: Potential market opportunities in Alberta,” Alberta

Innovates Technology Futures, 2012.

[11] P. Zamani-Dehkordi, L. Rakai, and H. Zareipour, “Estimating the price impact of proposed

wind farms in competitive electricity markets,” IEEE Transactions on Sustainable Energy,

vol. 8, no. 1, pp. 291–303, Jan 2017.

[12] ——, “Deciding on the support schemes for upcoming wind farms in competitive electricity

markets,” Energy, vol. 116, Part 1, pp. 8 – 19, 2016.

[13] E. Drury, P. Denholm, and R. Sioshansi, “The value of compressed air energy storage in

energy and reserve markets,” Energy, vol. 36, no. 8, pp. 4959 – 4973, 2011, PRES 2010.

[14] B. Kaun and S. Chen, “Cost-effectiveness of energy storage in california,” Electric Power

Research Institute (EPRI), 2013.

[15] P. Zamani-Dehkordi, S. Shafiee, L. Rakai, A. M. Knight, and H. Zareipour, “Price impact

assessment for large-scale merchant energy storage facilities,” Energy, pp. –, (2017) , doi:

10.1016/j.energy.2017.02.107.

[16] J. Garcia-Gonzalez, R. de la Muela, L. Santos, and A. Gonzalez, “Stochastic joint optimi-

zation of wind generation and pumped-storage units in an electricity market,” IEEE Trans.

Power Syst., vol. 23, no. 2, pp. 460–468, May 2008.

[17] M. Khodayar and M. Shahidehpour, “Stochastic price-based coordination of intrahour wind

energy and storage in a generation company,” IEEE Trans. Sustainable Energy, vol. 4, no. 3,

pp. 554–562, July 2013.

121

[18] Y. Yuan, Q. Li, and W. Wang, “Optimal operation strategy of energy storage unit in wind

power integration based on stochastic programming,” IET Renewable Pwr. Gen., vol. 5,

no. 2, pp. 194–201, Mar. 2011.

[19] Y. Ben-Haim, “Information gap decision theory, designs under severe uncertainty,” 2nd ed.

San Diego, CA: Academic, Feb. 2006.

[20] Y. Ben-Haim”, “Uncertainty, probability and information-gaps,” Reliability Engineering &

System Safety, vol. 85, no. 13, pp. 249 – 266, 2004.

[21] H. Daneshi and A. Srivastava, “Security-constrained unit commitment with wind generation

and compressed air energy storage,” IET Gen. Trans. Dist., vol. 6, no. 2, pp. 167–175,

February 2012.

[22] M. Kintner-Meyer, P. Balducci, W. Colella, M. Elizondo, C. Jin, T. Nguyen, V. Viswana-

than, and Y. Zhang, “National assessment of energy storage for grid balancing and arbitrage

phase: Ii: Wecc, ercot, eic volume 1: Technical analysis,” Pacific Northwest National Labo-

ratories, September, 2013.

[23] S. Succar, R. H. Williams et al., “Compressed air energy storage: theory, resources, and

applications for wind power,” Princeton environmental institute report, vol. 8, 2008.

[24] L. Baringo and A. Conejo, “Offering strategy via robust optimization,” IEEE Trans. Power

Syst., vol. 26, no. 3, pp. 1418–1425, Aug 2011.

[25] A. Soroudi, “Robust optimization based self scheduling of hydro-thermal genco in smart

grids,” Energy, vol. 61, no. 0, pp. 262 – 271, 2013.

[26] L. Fan, J. Wang, R. Jiang, and Y. Guan, “Min-max regret bidding strategy for thermal ge-

nerator considering price uncertainty,” IEEE Transactions on Power Systems, vol. 29, no. 5,

pp. 2169–2179, Sept 2014.

122

[27] S. Nojavan, H. Ghesmati, and K. Zare, “Robust optimal offering strategy of

large consumer using {IGDT} considering demand response programs,” Electric

Power Systems Research, vol. 130, pp. 46 – 58, 2016. [Online]. Available:

http://www.sciencedirect.com/science/article/pii/S0378779615002540

[28] S. Shafiee, H. Zareipour, A. M. Knight, N. Amjady, and B. Mohammadi-Ivatloo, “Risk-

constrained bidding and offering strategy for a merchant compressed air energy storage

plant,” IEEE Transactions on Power Systems, vol. 32, no. 2, pp. 946–957, March 2017.

[29] S. Shafiee, P. Zamani-Dehkordi, H. Zareipour, and A. M. Knight, “Economic assessment of

a price-maker energy storage facility in the alberta electricity market,” Energy, vol. 111, pp.

537 – 547, 2016.

[30] Alberta Electric System Operator (AESO), “Energy trading system.” [Online]. Available:

ets.aeso.ca

[31] S. Shafiee, H. Zareipour, and A. Knight, “Developing bidding and offering curves of a price-

maker energy storage facility based on robust optimization,” IEEE Transactions on Smart

Grid, under review, April 2017.

[32] ——, “Considering thermodynamic characteristics of a caes facility in self-scheduling in

energy and reserve markets,” IEEE Transactions on Smart Grid, vol. PP, no. 99, pp. 1–1,

2016.

[33] P. Denholm, E. Ela, B. Kirby, and M. Milligan, “The role of energy storage with renewa-

ble electricity generation,” National Renewable Energy Laboratory, NREL/TP-6A2-47187,

2010.

[34] “Technology roadmap energy storage,” International Energy Agency, 2014.

[Online]. Available: https://www.iea.org/publications/freepublications/publication/$\

$TechnologyRoadmapEnergystorage.pdf

123

[35] “Bethel Energy Center,” Apex.CAES. [Online]. Available: http://www.apexcaes.com/

project

[36] E. Fertig and J. Apt, “Economics of compressed air energy storage to integrate wind power:

A case study in ERCOT,” Energy Policy, vol. 39, no. 5, pp. 2330 – 2342, 2011.

[37] R. Sioshansi, P. Denholm, and T. Jenkin, “A comparative analysis of the value of pure and

hybrid electricity storage,” Energy Economics, vol. 33, no. 1, pp. 56 – 66, 2011.

[38] R. Walawalkar, J. Apt, and R. Mancini, “Economics of electric energy storage for energy

arbitrage and regulation in New york,” Energy Policy, vol. 35, no. 4, pp. 2558 – 2568, 2007.

[39] H. Safaei and D. W. Keith, “Compressed air energy storage with waste heat export: An

Alberta case study,” Energy Conversion and Management, vol. 78, no. 0, pp. 114 – 124,

2014.

[40] H. Zareipour, A. Janjani, H. Leung, A. Motamedi, and A. Schellenberg, “Classification of

future electricity market prices,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 165–173, Feb

2011.

[41] J. Arroyo and A. Conejo, “Optimal response of a thermal unit to an electricity spot market,”

IEEE Trans. Power Syst., vol. 15, no. 3, pp. 1098–1104, Aug 2000.

[42] H. Yamin and S. Shahidehpour, “Self-scheduling and energy bidding in competitive electri-

city markets,” Elec. Pwr Syst. Res., vol. 71, no. 3, pp. 203 – 209, 2004.

[43] A. Al-Awami and M. El-Sharkawi, “Coordinated trading of wind and thermal energy,” IEEE

Trans. Sustainable Energy, vol. 2, no. 3, pp. 277–287, July 2011.

[44] T. Dai and W. Qiao, “Trading wind power in a competitive electricity market using stochastic

programing and game theory,” IEEE Trans. Sustainable Energy, vol. 4, no. 3, pp. 805–815,

July 2013.

124

[45] R. Jabr, “Robust self-scheduling under price uncertainty using conditional value-at-risk,”

IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1852–1858, Nov 2005.

[46] B. Mohammadi-ivatloo, H. Zareipour, N. Amjady, and M. Ehsan, “Application of

information-gap decision theory to risk-constrained self-scheduling of gencos,” IEEE Trans.

Power Syst., vol. 28, no. 2, pp. 1093–1102, May 2013.

[47] M. Kazemi, B. Mohammadi-ivatloo, and M. Ehsan, “Risk-constrained strategic bidding of

gencos considering demand response,” IEEE Trans. Power Syst., vol. 30, no. 1, pp. 376–384,

Jan 2015.

[48] M. Kazemi, B. Mohammadi-Ivatloo, and M. Ehsan, “Risk-based bidding of large electric

utilities using information gap decision theory considering demand response,” Electric Po-

wer Systems Research, vol. 114, pp. 86 – 92, 2014.

[49] H. Akhavan-Hejazi and H. Mohsenian-Rad, “Optimal operation of independent storage sy-

stems in energy and reserve markets with high wind penetration,” IEEE Trans. Smart Grid,

vol. 5, no. 2, pp. 1088–1097, March 2014.

[50] G. He, Q. Chen, C. Kang, P. Pinson, and Q. Xia, “Optimal bidding strategy of battery

storage in power markets considering performance-based regulation and battery cycle life,”

IEEE Trans. Smart Grid, vol. PP, no. 99, pp. 1–1, 2015.

[51] A. Thatte, L. Xie, D. Viassolo, and S. Singh, “Risk measure based robust bidding strategy

for arbitrage using a wind farm and energy storage,” IEEE Trans. Smart Grid, vol. 4, no. 4,

pp. 2191–2199, Dec 2013.

[52] A. Thatte, D. Viassolo, and L. Xie, “Robust bidding strategy for wind power plants and

energy storage in electricity markets,” in 2012 IEEE Power and Energy Society General

Meeting, July 2012, pp. 1–7.

125

[53] S. Knoke, “Compressed air energy storage (CAES),” S. Eckroad, Editor, Handbook of

energy storage for transmission or distribution applications,, The Electric Power Research

Institute (EPRI); 2002.

[54] A. Moilanen, M. C. Runge, J. Elith, A. Tyre, Y. Carmel, E. Fegraus, B. A. Wintle, M. Burg-

man, and Y. Ben-Haim, “Planning for robust reserve networks using uncertainty analysis,”

Ecological Modelling, vol. 199, no. 1, pp. 115 – 124, 2006.

[55] S. J. Duncan, B. Bras, and C. J. Paredis, “An approach to robust decision making under

severe uncertainty in life cycle design,” International Journal of Sustainable Design, vol. 1,

no. 1, pp. 45–59, 2008.

[56] E. S. Matrosov, A. M. Woods, and J. J. Harou, “Robust decision making and info-gap deci-

sion theory for water resource system planning,” Journal of Hydrology, vol. 494, pp. 43 –

58, 2013.

[57] J. Zhao, C. Wan, Z. Xu, and J. Wang, “Risk-based day-ahead scheduling of electric vehicle

aggregator using information gap decision theory,” IEEE Trans. Smart Grid, vol. PP, no. 99,

pp. 1–10, 2015.

[58] K. Chen, W. Wu, B. Zhang, and H. Sun, “Robust restoration decision-making model for

distribution networks based on information gap decision theory,” IEEE Trans Smart Grid,

vol. 6, no. 2, pp. 587–597, March 2015.

[59] A. Soroudi and T. Amraee, “Decision making under uncertainty in energy systems: state of

the art,” Renewable and Sustainable Energy Reviews, vol. 28, pp. 376–384, 2013.

[60] M. Aien, A. Hajebrahimi, and M. Fotuhi-Firuzabad, “A comprehensive review on uncer-

tainty modeling techniques in power system studies,” Renewable and Sustainable Energy

Reviews, vol. 57, pp. 1077–1089, 2016.

126

[61] J. Garcıa-Gonzalez, R. Moraga, and A. Mateo, “Risk-constrained strategic bidding of a

hydro producer under price uncertainty,” in Power Engineering Society General Meeting,

2007. IEEE. IEEE, 2007, pp. 1–4.

[62] M. R. Bussieck and A. Drud, “SBB: A new solver for mixed integer nonlinear program-

ming,” Talk, OR, 2001.

[63] A. Brooke, D. Kendrick, A. Meeraus, R. Raman, and R. Rosenthal, “Gams a user’s guide

(gams development corporation, washington dc),” 1998.

[64] H. D. Sherali and A. Alameddine, “A new reformulation-linearization technique for bilinear

programming problems,” Journal of Global optimization, vol. 2, no. 4, pp. 379–410, 1992.

[65] X. Ding and F. Al-Khayyal, “Accelerating convergence of cutting plane algorithms for

disjoint bilinear programming,” Journal of Global Optimization, vol. 38, no. 3, pp. 421–

436, 2007.

[66] “Introduction to ontarios physical markets,” Ontario Independent Electricity Sy-

stem Operator; Febuaray 2014. [Online]. Available: www.ieso.ca/Documents/training/

IntroOntarioPhysicalMarkets.pdf

[67] Chambers, A., “Electric Energy Storage Technology. Alberta,” Alberta Innovates Techno-

logy Futures, 2010.

[68] R. Barzin, J. J. Chen, B. R. Young, and M. M. Farid, “Peak load shifting with energy storage

and price-based control system,” Energy, vol. 92, Part 3, pp. 505 – 514, 2015.

[69] N. Li and K. Hedman, “Economic assessment of energy storage in systems with high levels

of renewable resources,” IEEE Trans. Sustainable Energy, vol. 6, no. 3, pp. 1103–1111, July

2015.

127

[70] E. Rodrigues, R. Godina, S. Santos, A. Bizuayehu, J. Contreras, and J. Catalo, “Energy

storage systems supporting increased penetration of renewables in islanded systems,”

Energy, vol. 75, pp. 265 – 280, 2014.

[71] R. Edmunds, T. Cockerill, T. Foxon, D. Ingham, and M. Pourkashanian, “Technical benefits

of energy storage and electricity interconnections in future british power systems,” Energy,

vol. 70, pp. 577 – 587, 2014.

[72] Alberta Electric System Operator (AESO), “Project list of AESO by dec. 2014,” Dec. 2014.

[73] H. Safaei, D. W. Keith, and R. J. Hugo, “Compressed air energy storage (CAES) with com-

pressors distributed at heat loads to enable waste heat utilization,” Applied Energy, vol. 103,

no. 0, pp. 165 – 179, 2013.

[74] G. Locatelli, E. Palerma, and M. Mancini, “Assessing the economics of large energy storage

plants with an optimisation methodology,” Energy, vol. 83, no. 0, pp. 15 – 28, 2015.

[75] M. Kazemi, B. Mohammadi-Ivatloo, and M. Ehsan, “Igdt based risk-constrained strategic

bidding of gencos considering bilateral contracts,” in Electrical Engineering (ICEE), 2013

21st Iranian Conference on. IEEE, 2013, pp. 1–6.

[76] ——, “Risk-based bidding of large electric utilities using information gap decision theory

considering demand response,” Electric Power Systems Research, vol. 114, pp. 86–92, 2014.

[77] H. Mohsenian-Rad, “Coordinated price-maker operation of large energy storage units in

nodal energy markets,” IEEE Trans. Power Syst., vol. 31, no. 1, pp. 786–797, Jan 2016.

[78] F. Teixeira, J. de Sousa, and S. Faias, “How market power affects the behavior of a pumped

storage hydro unit in the day-ahead electricity market?” in European Energy Market (EEM),

2012 9th International Conference on the, May 2012, pp. 1–6.

[79] “Impact of a price-maker pumped storage hydro unit on the integration of wind energy in

power systems,” Energy, vol. 69, no. 0, pp. 3 – 11, 2014.

128

[80] P. Mannan, G. Baden, L. Olein, C. Brandon, B. Scorfield, N. Naini, and J. Cheng, “Techno-

economics of energy storage,” Alberta Innovates Technology Futures, BECL and Associa-

tes Ltd, March 2014.

[81] B. Hobbs, C. Metzler, and J.-S. Pang, “Strategic gaming analysis for electric power systems:

an mpec approach,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 638–645, May 2000.

[82] M. Pereira, S. Granville, M. Fampa, R. Dix, and L. Barroso, “Strategic bidding under uncer-

tainty: a binary expansion approach,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 180–188,

Feb 2005.

[83] L. Barroso, R. Carneiro, S. Granville, M. Pereira, and M. Fampa, “Nash equilibrium in

strategic bidding: a binary expansion approach,” IEEE Trans. Power Syst., vol. 21, no. 2,

pp. 629–638, May 2006.

[84] A. Bakirtzis, N. Ziogos, A. Tellidou, and G. Bakirtzis, “Electricity producer offering strate-

gies in day-ahead energy market with step-wise offers,” IEEE Trans. Power Syst., vol. 22,

no. 4, pp. 1804–1818, Nov 2007.

[85] G. B. Sheble, Computational auction mechanisms for restructured power industry opera-

tion. Springer Science & Business Media, 2012.

[86] H. Pousinho, J. Contreras, and J. Catalao, “Operations planning of a hydro producer acting

as a price-maker in an electricity market,” in 2012 IEEE Power and Energy Society General

Meeting, July 2012, pp. 1–7.

[87] H. Pousinho, J. Contreras, A. Bakirtzis, and J. Catalao, “Risk-constrained scheduling and

offering strategies of a price-maker hydro producer under uncertainty,” IEEE Trans. Power

Syst., vol. 28, no. 2, pp. 1879–1887, May 2013.

[88] C. Baslis and A. Bakirtzis, “Mid-term stochastic scheduling of a price-maker hydro producer

with pumped storage,” IEEE Trans. Power Syst., vol. 26, no. 4, pp. 1856–1865, Nov 2011.

129

[89] M. Fotuhi-Firuzabad, S. Shafiee, and M. Rastegar, “Optimal in-home charge scheduling of

plug-in electric vehicles incorporating customers payment and inconvenience costs,” in Plug

In Electric Vehicles in Smart Grids. Springer, 2015, pp. 301–326.

[90] S. Shafiee, M. Fotuhi-Firuzabad, and M. Rastegar, “Impacts of controlled and uncontrolled

phev charging on distribution systems,” 2012.

[91] ——, “Investigating the impacts of plug-in hybrid electric vehicles on distribution conges-

tion,” 2013.

[92] “Energy Storage Integration,” Alberta Electrici System Operator, Recommendation

Paper, June 2015. [Online]. Available: www.aeso.ca/downloads/Energy$ $Storage$

$Integration$\ $Recommendation$ $Paper.pdf

[93] H. Daneshi and A. Srivastava, “Security-constrained unit commitment with wind generation

and compressed air energy storage,” Generation, Transmission Distribution, IET, vol. 6,

no. 2, pp. 167–175, February 2012.

[94] C. Suazo-Martinez, E. Pereira-Bonvallet, R. Palma-Behnke, and X.-P. Zhang, “Impacts of

energy storage on short term operation planning under centralized spot markets,” Smart

Grid, IEEE Transactions on, vol. 5, no. 2, pp. 1110–1118, March 2014.

[95] B. Cleary, A. Duffy, A. OConnor, M. Conlon, and V. Fthenakis, “Assessing the economic

benefits of compressed air energy storage for mitigating wind curtailment,” IEEE Transacti-

ons on Sustainable Energy, vol. 6, no. 3, pp. 1021–1028, July 2015.

[96] E. Nasrolahpour, S. J. Kazempour, H. Zareipour, and W. D. Rosehart, “Impacts of ramping

inflexibility of conventional generators on strategic operation of energy storage facilities,”

IEEE Trans. Smart Grid, Accepted for publication, 2016.

[97] ——, “Strategic sizing of energy storage facilities in electricity markets,” IEEE Transactions

on Sustainable Energy, vol. PP, no. 99, pp. 1–1, 2016.

130

[98] G. Steeger, L. Barroso, and S. Rebennack, “Optimal bidding strategies for hydro-electric

producers: A literature survey,” IEEE Trans. Power Syst., vol. 29, no. 4, pp. 1758–1766,

July 2014.

[99] M. Kazemi, B. Mohammadi-Ivatloo, and M. Ehsan, “Igdt based risk-constrained strategic

bidding of gencos considering bilateral contracts,” in Electrical Engineering (ICEE), 2013

21st Iranian Conference on. IEEE, 2013, pp. 1–6.

[100] ——, “Risk-based bidding of large electric utilities using information gap decision theory

considering demand response,” Electric Power Systems Research, vol. 114, pp. 86–92, 2014.

[101] M. Kazemi, H. Zareipour, M. Ehsan, and W. D. Rosehart, “A robust linear approach for of-

fering strategy of a hybrid electric energy company,” IEEE Transactions on Power Systems,

vol. PP, no. 99, pp. 1–1, 2016.

[102] M. Rahimiyan and L. Baringo, “Strategic bidding for a virtual power plant in the day-ahead

and real-time markets: A price-taker robust optimization approach,” IEEE Transactions on

Power Systems, vol. 31, no. 4, pp. 2676–2687, July 2016.

[103] J. M. Mulvey, R. J. Vanderbei, and S. A. Zenios, “Robust optimization of large-scale sys-

tems,” Operations research, vol. 43, no. 2, pp. 264–281, 1995.

[104] P. Zamani-Dehkordi, S. Shafiee, L. Rakai, A. M. Knight, and H. Zareipour, “Price impact

assessment for large-scale merchant energy storage facilities,” Energy, vol. 125, pp. 27–43,

2017.

[105] H. Khani and M. R. D. Zadeh, “Real-time optimal dispatch and economic viability of

cryogenic energy storage exploiting arbitrage opportunities in an electricity market,” IEEE

Trans. on Smart Grid, vol. 6, no. 1, pp. 391–401, Jan 2015.

[106] A. Soroudi, “Smart self-scheduling of gencos with thermal and energy storage units under

price uncertainty,” International Transactions on Electrical Energy Systems, vol. 24, no. 10,

131

pp. 1401–1418, 2014.

[107] A. N. Ghalelou, A. P. Fakhri, S. Nojavan, M. Majidi, and H. Hatami, “A stochastic self-

scheduling program for compressed air energy storage (caes) of renewable energy sources

(ress) based on a demand response mechanism,” Energy Conversion and Management, vol.

120, pp. 388 – 396, 2016.

[108] N. Hartmann, O. Vohringer, C. Kruck, and L. Eltrop, “Simulation and analysis of different

adiabatic compressed air energy storage plant configurations,” Applied Energy, vol. 93, pp.

541–548, 2012.

[109] J.-L. Liu and J.-H. Wang, “A comparative research of two adiabatic compressed air energy

storage systems,” Energy Conversion and Management, vol. 108, pp. 566–578, 2016.

[110] “Ancillary Services Matrix,” North American Electric Reliability Corporation (NERC),

2014. [Online]. Available: http://www.nerc.com/docs

[111] B. Alizadeh, S. Dehghan, N. Amjady, S. Jadid, and A. Kazemi, “Robust transmission system

expansion considering planning uncertainties,” Generation, Transmission Distribution, IET,

vol. 7, no. 11, pp. 1318–1331, November 2013.

[112] “ERCOT Quick Facts,” Electric Reliability Council of Texas, 2015. [Online].

Available: http://www.ercot.com/content/news/presentations/2015/$\$ERCOT$ $Quick$

$Facts$ $12715.pdf

132

Appendix A

Copyright permission letters

To Whom It May Concern:

I, Payam Zamani-Dehkordi, hereby grant permission to Mr. Soroush Shafiee to reuse the below

article in his thesis titled ”Optimal Operation planning of Compressed Air Energy Storage Plants

in Competitive Electricity Markets”.

S. Shafiee, P. Zamani-Dehkordi, H. Zareipour, and A. M. Knight, Economic assessment of a price-

maker energy storage facility in the alberta electricity market, Energy, vol. 111, pp. 537 547,

2016.

I agree to the terms outlined in the University of Calgary Non-Exclusive Distribution License. I am

aware that all University of Calgary Theses are also achieved by the Library and Archives Canada

(LAC) and the University of Calgary Theses may be submitted to ProQuest.

Date:

Signature:

133

To Whom It May Concern:

I, Hamidreza Zareipour, hereby grant permission to Mr. Soroush Shafiee to reuse the below three

articles in his thesis titled ”Optimal Operation planning of Compressed Air Energy Storage Plants

in Competitive Electricity Markets”.

1- S. Shafiee, H. Zareipour, A. M. Knight, N. Amjady, and B. Mohammadi-Ivatloo, ”Risk-constrained

bidding and offering strategy for a merchant compressed air energy storage plant,” IEEE Tran-

sactions on Power Systems, vol. 32, no. 2, pp. 946957, March 2017.

2- S. Shafiee, P. Zamani-Dehkordi, H. Zareipour, and A. M. Knight, ”Economic assessment of a

price-maker energy storage facility in the Alberta electricity market,” Energy, vol. 111, pp. 537

547, 2016.

3- S. Shafiee, H. Zareipour, and A. Knight, ”Considering thermodynamic characteristics of a caes

facility in self-scheduling in energy and reserve markets,” IEEE Transactions on Smart Grid, vol.

PP, no. 99, pp. 11, 2016.

I agree to the terms outlined in the University of Calgary Non-Exclusive Distribution License. I am

aware that all University of Calgary Theses are also achieved by the Library and Archives Canada

(LAC) and the University of Calgary Theses may be submitted to ProQuest.

Date:

Signature:

134

To Whom It May Concern:

I, Andrew M. Knight, hereby grant permission to Mr. Soroush Shafiee to reuse the below three

articles in his thesis titled ”Optimal Operation planning of Compressed Air Energy Storage Plants

in Competitive Electricity Markets”.

1- S. Shafiee, H. Zareipour, A. M. Knight, N. Amjady, and B. Mohammadi-Ivatloo, ”Risk-constrained

bidding and offering strategy for a merchant compressed air energy storage plant,” IEEE Tran-

sactions on Power Systems, vol. 32, no. 2, pp. 946957, March 2017.

2- S. Shafiee, P. Zamani-Dehkordi, H. Zareipour, and A. M. Knight, ”Economic assessment of a

price-maker energy storage facility in the Alberta electricity market,” Energy, vol. 111, pp. 537

547, 2016.

3- S. Shafiee, H. Zareipour, and A. Knight, ”Considering thermodynamic characteristics of a caes

facility in self-scheduling in energy and reserve markets,” IEEE Transactions on Smart Grid, vol.

PP, no. 99, pp. 11, 2016.

I agree to the terms outlined in the University of Calgary Non-Exclusive Distribution License. I am

aware that all University of Calgary Theses are also achieved by the Library and Archives Canada

(LAC) and the University of Calgary Theses may be submitted to ProQuest.

Date:

Signature:

135

To Whom It May Concern:

I, Nima Amjady, hereby grant permission to Mr. Soroush Shafiee to reuse the below article in his

thesis titled ”Optimal Operation planning of Compressed Air Energy Storage Plants in Competitive

Electricity Markets”.

S. Shafiee, H. Zareipour, A. M. Knight, N. Amjady, and B. Mohammadi-Ivatloo, ”Risk-constrained

bidding and offering strategy for a merchant compressed air energy storage plant,” IEEE Tran-

sactions on Power Systems, vol. 32, no. 2, pp. 946957, March 2017.

I agree to the terms outlined in the University of Calgary Non-Exclusive Distribution License. I am

aware that all University of Calgary Theses are also achieved by the Library and Archives Canada

(LAC) and the University of Calgary Theses may be submitted to ProQuest.

Date:

Signature:

136

To Whom It May Concern:

I, Behnam Mohammadi-Ivatloo, hereby grant permission to Mr. Soroush Shafiee to reuse the

below article in his thesis titled ”Optimal Operation planning of Compressed Air Energy Storage

Plants in Competitive Electricity Markets”.

S. Shafiee, H. Zareipour, A. M. Knight, N. Amjady, and B. Mohammadi-Ivatloo, ”Risk-constrained

bidding and offering strategy for a merchant compressed air energy storage plant,” IEEE Tran-

sactions on Power Systems, vol. 32, no. 2, pp. 946957, March 2017.

I agree to the terms outlined in the University of Calgary Non-Exclusive Distribution License. I am

aware that all University of Calgary Theses are also achieved by the Library and Archives Canada

(LAC) and the University of Calgary Theses may be submitted to ProQuest.

Date:

Signature:

137

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