modeling electricity prices for generation investment and scheduling analysis

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TitleModeling electricity prices for generation investment andscheduling analysis

Author(s) He, Yang; OU3

Citation

Issue Date 2010

URL http://hdl.handle.net/10722/130905

RightsThe author retains all proprietary rights, (such as patentrights) and the right to use in future works.


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Modeling Electricity Prices forGeneration Investment and

Scheduling Analysis

by

Yang HE

A thesis submitted in partial fulfillment of the requirements for

the Degree of Doctor of Philosophy

atThe University of Hong Kong.

February 2010
mailto:[email protected]://www.hku.hk/http://www.hku.hk/mailto:[email protected]


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Abstract of thesis entitled

Modeling Electricity Prices for GenerationInvestment and Scheduling Analysis

Submitted byYang HE

for the degree of Doctor of Philosophy

atThe University of Hong Kong

in February 2010

In the deregulated electric power industry, under the market environment,

electricity spot prices are highly volatile and uncertain. For private generation

companies, their profits are directly tied with and significantly affected by

these fluctuations of electricity spot prices. To make decisions on building new

power plants and scheduling the production thereafter, generation companies

desire an electricity spot price model to assist their making these decisions.

The system of electricity spot prices is of high complexity; it is driven by

various physical underlying forces that play in different timescales. In the

short time horizon of one week, the physical driving forces are intra-day and

intra-week variations of electricity load, generation forced outages, etc.; in

the mid-term of one year, it is the seasonal forces that are manifest, such

as seasonal weather and temperature, annual generation maintenance, etc.;

and in the time horizon of years and decades, the effecting physical forces

are economic development and economic cycles, generation investment and

retirement, fluctuations of fuel prices, etc.

This work develops a Multi-granularity Framework to facilitate analyz-

ing electricity spot prices, which views electricity spot prices in three time-

perspectives, that is, multi-year yearly, intra-year weekly, and intra-week hourly.

In each time-perspective, how the various underlying physical forces give rise

to the very peculiar behaviors of electricity spot prices is carefully discussed.

Because the physical forces that underlie electricity spot prices are indepen-

dent to each other, play in different timescales, and affect electricity spot prices

in different time horizons, this work adopts the methodology Divide and Con-

quer to build the price model: it decomposes the historical electricity spot
mailto:[email protected]://www.hku.hk/http://www.hku.hk/mailto:[email protected]


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price data into components that are driven by different and independent phys-ical underlying forces, then models each price component respectively, and

finally constructs a complete electricity spot price model out of the resulting

sub-models.

The overall price model explicitly considers the various physical forces that

drive electricity spot prices, the model extends on a time horizon of multiple

years and has a time unit of one hour, and its final result represents prices at

each hour by a probability density function of Lognormal distribution. The

model has been evaluated in the New-England and PJM electricity markets.Upon proper revisions, the same analysis framework and modeling methodol-

ogy probably could be applied to many other electricity markets in the world.

The proposed price model is physically grounded, mathematically simple,

and computationally fast. It provides an analytical tool to generation compa-

nies for their making informed decisions in generation investment and schedul-

ing analysis. Besides generation companies, the model could also be widely

used by other players in electricity markets, like by power traders for pric-

ing and trading electricity contracts, futures, options, and other electricity

derivatives, and by power retailers and large power consumers for their power

purchase and risk management.


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Declaration

I declare that the thesis and the research work thereof represents

my own work, except where due acknowledgment is made, and

that it has not been previously included in a thesis, dissertation or

report submitted to this University or to any other institution for

a degree, diploma or other qualifications.

Signed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Yang HE


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Acknowledgment

The saying Read Masters, Not Pupils, weighs heavily. I am fortunate in

the past four years having learned under a master. Prof. Wu is a master who

knows when to let the student loose, by allowing him to explore under theguidance of his own interest; and when to tighten him back, by pointing to

him the direction that is the most worth pursuing. He knows the importance

of good literature, and he recommends to his student the best works he ever

knows. He knows with exact in which stage the student is: he is the strictest

critics of his students work, and he shows to him, by correcting the students

work, how mathematically rigid and practically sound the final result could

be. He helps the student vision the ideal result of the work and helps him

setup the analysis framework, and asks him to work out a solution that is the

most mathematically concise and physically grounded. Prof. Wu has a good

habit in thinking, he starts with a problem, defines it clearly, and analyzes it

to a point by which he has got a thorough understanding. Prof. Wu is one

of the best lecturers I have ever seen. All these good examples Prof. Wu has

set, the pupil during the past four years has observed and mimicked, and the

student is determined to practice, in the rest of his life, these good fortunes he

has learned from his teacher.

Another person who makes this thesis work possible is the philosopher and

novelist Any Rand. At the end of the first year of this Ph.D. study, the author

was in great difficulty in seeking for the meaning of life. It was the novel The

Fountainheadthat taught me that man is a heroic rational being who pursues

his own happiness, that one achieves his happiness through productive work,

by productive work it is meant that one works by using his own mind to think,

and that reason is the absolute tool for the guidance of thought. In her another

novelAtlas ShruggedI found the seven virtues she derived truly attractive, that

is, Rationality, Independence, Honesty, Integrity, Justice, Productiveness, and


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Pride. These virtues have since then become the guidance of my action, everyday.

I grew up in a small town on the China eastern coast. My parents own

a small family business which buy, salt, dry, and sell fish. Their working

days start from 5 oclock in the morning till 6 oclock in the evening. They

work without holiday till the years ends. Since very early age I often went to

the factory watching them working. By observing my parents, I learned the

value of hard work and the responsibility of life: I learned that one has to be

responsible for his own work, and that one has to be diligent and persistentin pursuing his own convinced goals. I developed my working morals learning

from my parents, and these principles have guided me and proved valuable

throughout my time as a student.

I am grateful to my former supervisor in Zhejiang University, Prof. De-

qiang Gan, it was his reference that helped me be admitted to this Ph.D.

program; I am in debt to Dr. Yunhe Hou, during the early and late stages of

this research project, it was his discussions and advices that help me keep the

work progressing; I am grateful to Dr. Jin Zhong, who generously gave me two

chances to go to conferences; I am thankful to our laboratory technician Mr.

Peter Tam, who kept my computer out of trouble during this four years; I am

grateful to Prof. Wus secretary Ms. Clara Chung, who expertly steered me

through all the administrative requirements during the study; I am thankful

to my colleagues in the Center for Electrical Energy Systems, whose atten-

dance to my presentations and their following-up discussions gave me valuable

perspectives to criticize my work; and I am fortunate during this four years

having a few very intelligent and considerate friends, who made this journey

much more pleasant than it otherwise could be, they are Dr. Yanhui Geng

and his wife Ms. Qiong Sun.


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Contents

List of Figures xv

List of Tables xix

1 Introduction 1

1.1 Background and Motivations . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review and the Gap to Fill . . . . . . . . . . . . . . 4

1.3 Elaboration of the Problem . . . . . . . . . . . . . . . . . . . . 7

1.3.1 The Requirements on the Price Model . . . . . . . . . . 71.3.2 The Analytical and Mathematical Tools . . . . . . . . . 8

1.4 The Analysis Framework . . . . . . . . . . . . . . . . . . . . . . 9

1.5 The Modeling Methodology . . . . . . . . . . . . . . . . . . . . 10

1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . 11

2 A Survey of Electricity Price Models 13

2.1 Electricity Price Modeling Approaches . . . . . . . . . . . . . . 13

2.2 Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 The Autoregressive Model . . . . . . . . . . . . . . . . . 15

2.2.2 AR with Time-varying Mean . . . . . . . . . . . . . . . . 17

2.2.3 AR with Exogenous Variables . . . . . . . . . . . . . . . 18

2.2.4 The Autoregressive Moving Average Model . . . . . . . . 19

2.2.5 ARMA with Time-varying Mean . . . . . . . . . . . . . 20

2.2.6 ARMAX and Transfer Function . . . . . . . . . . . . . . 21

2.2.7 Periodic Autoregressive Models . . . . . . . . . . . . . . 21

2.2.8 ARIMA and its Extensions . . . . . . . . . . . . . . . . . 23

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CONTENTS

2.2.9 Regime-switching Models . . . . . . . . . . . . . . . . . . 252.2.10 Time-varying Volatility . . . . . . . . . . . . . . . . . . . 26

2.2.11 The GARCH Model . . . . . . . . . . . . . . . . . . . . 26

2.2.12 GARCH with Asymmetric Effect . . . . . . . . . . . . . 28

2.2.13 Time-varying Volatility and Model Parameter Calibration 29

2.3 Financial/Stochastic Process Models . . . . . . . . . . . . . . . 30

2.3.1 Modeling the Multi-seasonality . . . . . . . . . . . . . . 30

2.3.2 Modeling the Mean-reverting Price Nature . . . . . . . . 33

2.3.3 A Basic Electricity Price Model . . . . . . . . . . . . . . 352.3.4 Modeling the Time-varying Volatility . . . . . . . . . . . 36

2.3.5 Modeling the Multi Risk-factors . . . . . . . . . . . . . . 38

2.3.6 Modeling the Price Spikes . . . . . . . . . . . . . . . . . 39

2.4 Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 The Effect of Fuel Prices . . . . . . . . . . . . . . . . . . . . . . 43

2.6 Decomposition Techniques . . . . . . . . . . . . . . . . . . . . . 44

2.6.1 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . 44

2.6.2 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . 452.6.3 Principal Component Analysis . . . . . . . . . . . . . . . 48

2.7 Concluding Discussions . . . . . . . . . . . . . . . . . . . . . . . 51

3 Microeconomics applied to the Electricity Markets 53

3.1 The Supply Curve of a Generation Company . . . . . . . . . . . 54

3.1.1 Heat Rate Curve of the Generation Company . . . . . . 54

3.1.2 Fuel Price Curve of the Generation Company . . . . . . 54

3.1.3 Marginal Cost Curve of the Generation Company . . . . 55

3.1.4 Supply Curve of the Generation Company . . . . . . . . 56

3.2 The Supply Curve of an Electricity Market . . . . . . . . . . . . 58

3.3 The Demand Curve of an Electricity Market . . . . . . . . . . . 58

3.4 The Spot Price of an Electricity Market . . . . . . . . . . . . . 59

3.5 The Effect of Fuel Prices on Spot Prices . . . . . . . . . . . . . 60

3.6 The Effect of Generator Outage on Spot Prices . . . . . . . . . 60

3.7 Intra-day Electricity Spot Prices . . . . . . . . . . . . . . . . . . 62

3.8 Time-varying Volatility of Intra-day Spot Prices . . . . . . . . . 63

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CONTENTS

3.9 Electricity Spot Price Spikes . . . . . . . . . . . . . . . . . . . . 64

4 A Multi-granularity View of Electricity Prices 67

4.1 Intra-week Hourly Electricity Spot Prices . . . . . . . . . . . . . 67

4.2 Intra-year Weekly Prices . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Multi-year Yearly Prices . . . . . . . . . . . . . . . . . . . . . . 71

4.4 The Effect of Fuel Prices on Yearly Prices . . . . . . . . . . . . 73

4.5 A Multi-granularity View of Electricity Prices . . . . . . . . . . 75

5 A Multi-granularity Electricity Spot Price Model 775.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 The New-England Market . . . . . . . . . . . . . . . . . . . . . 81

5.4 Decompose the Data to Find Orders . . . . . . . . . . . . . . . 82

5.4.1 Fuel-Price Effect on Electricity Spot Prices . . . . . . . . 82

5.4.2 Fuel-price Effect Adjustment . . . . . . . . . . . . . . . . 84

5.4.3 Frequency Spectrum of the Data . . . . . . . . . . . . . 85

5.4.3.1 The High-frequency Component . . . . . . . . . 855.4.3.2 The Mid-frequency Component . . . . . . . . . 87

5.4.3.3 The Low-frequency Component . . . . . . . . . 88

5.5 Define the Orders by Models . . . . . . . . . . . . . . . . . . . . 89

5.5.1 The Intra-week Model . . . . . . . . . . . . . . . . . . . 89

5.5.2 The Intra-year Model . . . . . . . . . . . . . . . . . . . . 90

5.5.3 The Multi-year Model . . . . . . . . . . . . . . . . . . . 91

5.5.4 The Fuel Price Model . . . . . . . . . . . . . . . . . . . . 92

5.5.5 The Overall Model . . . . . . . . . . . . . . . . . . . . . 93

5.6 Calibrate the Models . . . . . . . . . . . . . . . . . . . . . . . . 94

5.6.1 The Intra-week Model Calibration . . . . . . . . . . . . . 94

5.6.2 The Intra-year Model Calibration . . . . . . . . . . . . . 97

5.6.3 The Multi-year Model Calibration . . . . . . . . . . . . .100

5.6.4 The Sampling of the Data and the Time-Unit of Models 101

5.7 Simulate Electricity Spot Prices with the Models . . . . . . . . .102

5.8 Use Price Simulations to Valuate Generator Profit . . . . . . . .108

5.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111

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CONTENTS

5. 10 Concl usi on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6 The Multi-granularity Model applied to the PJM Market 113

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 The Fundamentals of the PJM Market . . . . . . . . . . . . . .114

6.3 The Electricity Prices and Fuel Price Index . . . . . . . . . . . .118

6.4 Data Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 120

6.5 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123

6.6 Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.6.1 The Intra-week Model Calibration . . . . . . . . . . . . .124

6.6.2 The Intra-year Model Calibration . . . . . . . . . . . . .124

6.6.3 The Multi-year Model Calibration . . . . . . . . . . . . .128

6.7 Price Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.8 Generator Profit Valuation . . . . . . . . . . . . . . . . . . . . .133

6.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7 A Structural Electricity Spot Price Model 135

7.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1357.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.3 The Effect of Generator Forced Outage on Spot Prices . . . . .138

7.4 The Structural Model . . . . . . . . . . . . . . . . . . . . . . . . 140

7.4.1 The Structural Model . . . . . . . . . . . . . . . . . . . .140

7.4.2 The Available Generation Capacity . . . . . . . . . . . .142

7.4.3 The Segment Slopes . . . . . . . . . . . . . . . . . . . . 142

7.4.4 The Expected Value of the Spot Prices . . . . . . . . . .144

7.5 The Simplified Model . . . . . . . . . . . . . . . . . . . . . . . . 146

7.5.1 The Simplified Model . . . . . . . . . . . . . . . . . . . .146

7.5.2 The Probability Density Function of the Spot Prices . .147

7.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 148

7.6.1 The Test Generation System . . . . . . . . . . . . . . . .148

7.6.2 The Structural Model . . . . . . . . . . . . . . . . . . . .148

7.6.3 The Simplified Model . . . . . . . . . . . . . . . . . . . .151

7.6.4 The Computation Burden of the Structural Models . . .151

7.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153

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CONTENTS

7.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8 Conclusion 157

8.1 Significance of the Work . . . . . . . . . . . . . . . . . . . . . . 157

8.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.3 Future Work on the Price Model . . . . . . . . . . . . . . . . . .158

8.4 Applications to Power System Operation and Planning . . . . .160

Appendix 162

A A Simple Structural Electricity Spot Price Model 163

B The Test Generation System 167

C Basic Stochastic Processes 169

C.1 Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . .169

C.2 Wiener Process with Drift . . . . . . . . . . . . . . . . . . . . .170

C.3 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . .170

C.4 Mean-reversion Process . . . . . . . . . . . . . . . . . . . . . . .171

C.5 Geometric Mean-reversion with Constant Mean . . . . . . . . .173

C.6 Geometric Mean-reversion with Time-varying Mean . . . . . . .174

D The Price Filters and the Filtering 177

D.1 Design of the Price Filters . . . . . . . . . . . . . . . . . . . . .177

D.2 Price Decomposition by Filtering . . . . . . . . . . . . . . . . .179

E The Mean-reversion Process in Discrete Time 181

F Build the Fuel Price Index 183

F.1 The Fuel Price Index Model . . . . . . . . . . . . . . . . . . . .183

F.2 Derivation of the Model for Parameter Estimation . . . . . . . .184

F.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . .186

References 189

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CONTENTS

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List of Figures

1.1 The Flow-chart of Generator Profit Valuation . . . . . . . . . . 2

1.2 Distribution of Literatures . . . . . . . . . . . . . . . . . . . . . 4

2.1 Index the Intra-day Hourly Spot Prices . . . . . . . . . . . . . . 22

2.2 A Two-regime Markov Chain . . . . . . . . . . . . . . . . . . . 25

2.3 Intra-day Electricity Price Pattern . . . . . . . . . . . . . . . . 31

2.4 Intra-week Electricity Price Pattern . . . . . . . . . . . . . . . . 32

2.5 Intra-year Seasonality of Electricity Prices . . . . . . . . . . . . 32

2.6 A Haar Wavelet . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1 The Heat-rate Curve of the Generation Company . . . . . . . . 54

3.2 The Fuel Price Curve of the Generation Company . . . . . . . . 55

3.3 The Marginal Cost Curve of the Generation Company . . . . . . 56

3.4 The Supply Curve of the Generation Company . . . . . . . . . . 57

3.5 The Supply Curve of an Electricity Market . . . . . . . . . . . . 58

3.6 The Demand Curve of an Electricity Market . . . . . . . . . . . 59

3.7 The Spot Price of an Electricity Market . . . . . . . . . . . . . 60

3.8 The Effect of Fuel Prices on Spot Prices . . . . . . . . . . . . . 613.9 The Effect of Generator Outage on Spot Prices . . . . . . . . . 62

3.10 Intraday Electricity Load Demand . . . . . . . . . . . . . . . . . 62

3.11 Intraday Electricity Spot Prices . . . . . . . . . . . . . . . . . . 63

3.12 The Time-varying Volatility of Intra-day Electricity Spot Prices 64

3.13 The Marginal Cost Curve with an additional Third Segment . . 65

3.14 Intra-day Electricity Spot Price Spikes . . . . . . . . . . . . . . 66

4.1 Intra-week Hourly Electricity Spot Prices . . . . . . . . . . . . . 68

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LIST OF FIGURES

4.2 Intra-year Weekly Prices . . . . . . . . . . . . . . . . . . . . . . 694.3 Multi-year Yearly Prices . . . . . . . . . . . . . . . . . . . . . . 72

4.4 The Prices of Fossil Fuels across Many Years . . . . . . . . . . . 74

4.5 The Yearly Prices Considering Fuel Price Fluctuations . . . . . 74

4.6 A Multi-granularity View of Electricity Prices . . . . . . . . . . 75

5.1 New-England Market Fundamentals . . . . . . . . . . . . . . . . 81

5.2 Electricity Prices and Natural Gas Prices . . . . . . . . . . . . . 82

5.3 Fuel-price Effect on Electricity Spot Prices . . . . . . . . . . . . 83

5.4 Implied Marginal Generator Heat Rate Data . . . . . . . . . . . 84

5.5 The High-frequency Component Data . . . . . . . . . . . . . . . 86

5.6 The Mid-frequency Component Data . . . . . . . . . . . . . . . 88

5.7 The Low-frequency Component Data . . . . . . . . . . . . . . . 89

5.8 The Overall Electricity Spot Price Model . . . . . . . . . . . . . 93

5.9 The High-frequency Component and the Intra-week Pattern . . 96

5.10 The Mid-frequency Component and the Intra-year Pattern . . . 98

5.11 Seasonality of the New-England Market . . . . . . . . . . . . . . 99

5.12 Calibration of the Multi-year Model . . . . . . . . . . . . . . . . 101

5.13 Construct the Electricity Spot Prices in the Year 2005 . . . . . . 103

5.14 Day-ahead Electricity Spot Prices in the Year 2005 . . . . . . .104

5.15 Empirical Probability Distributions of the Prices in Year 2005 .105

5.16 The Mid-winter Week in 2005 . . . . . . . . . . . . . . . . . . .106

5.17 The Mid-spring Week in 2005 . . . . . . . . . . . . . . . . . . .107

5.18 The Mid-summer Week in 2005 . . . . . . . . . . . . . . . . . .107

5.19 The Mid-fall Week in 2005 . . . . . . . . . . . . . . . . . . . . . 108

5.20 One-year Generator Profit of the Coal Unit and the Gas Unit .110

6.1 PJM Market Fundamentals . . . . . . . . . . . . . . . . . . . . 115

6.2 Business-day Intra-day Pattern of Marginal Fuel Mixture . . . .116

6.3 Intra-year Seasonal Pattern of Marginal Fuel Mixture . . . . . .116

6.4 Multi-year Pattern of Marginal Fuel Mixture . . . . . . . . . . .117

6.5 The Prices of the Three Marginal Fuels . . . . . . . . . . . . . .118

6.6 PJM Market Electricity Prices and Fuel Prices . . . . . . . . . .119

6.7 Fuel-price Effect on Electricity Spot Prices . . . . . . . . . . . .119

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LIST OF FIGURES

6.8 Implied Marginal Generator Heat Rate Data . . . . . . . . . . .1216.9 The High-frequency Component Data . . . . . . . . . . . . . . .122

6.10 The Mid-frequency Component Data . . . . . . . . . . . . . . .122

6.11 The Low-frequency Component Data . . . . . . . . . . . . . . .123

6.12 The High-frequency Data and the Intra-week Pattern . . . . . .125

6.13 The Mid-frequency Data and the Seasonal Pattern . . . . . . . . 126

6.14 Seasonality of the PJM Market . . . . . . . . . . . . . . . . . .127

6.15 Calibration of the Multi-year Model . . . . . . . . . . . . . . . . 128

6.16 Day-ahead Hourly Electricity Spot Prices in Year 2008 . . . . .1306.17 Empirical Probability Distributions of the Prices in Year 2008 .130

6.18 Mid-winter Week in 2008 . . . . . . . . . . . . . . . . . . . . . . 131

6.19 Mid-spring Week in 2008 . . . . . . . . . . . . . . . . . . . . . . 131

6.20 Mid-summer Week in 2008 . . . . . . . . . . . . . . . . . . . . . 132

6.21 Mid-fall Week in 2008 . . . . . . . . . . . . . . . . . . . . . . .132


7.1 The Generation Supply Stack of an Electricity Market . . . . .139

7.2 The Effect of Generator Forced Outage on Spot Prices . . . . .1397.3 The Structural Electricity Spot Price Model . . . . . . . . . . .140

7.4 The Integration Regions of the Three Cases . . . . . . . . . . .145

7.5 The Generation Supply Stack of the Test Generation System . .149

7.6 The Expected Value of the Spot Prices . . . . . . . . . . . . . .150

7.7 The Standard Deviation and Skewness of the Spot Prices . . . .150

7.8 The Errors of the Model . . . . . . . . . . . . . . . . . . . . . . 151

7.9 The Expected Value of the Spot Prices . . . . . . . . . . . . . .152

7.10 The Standard Deviation and Skewness of the Spot Prices . . . .1527.11 The Errors of the Model . . . . . . . . . . . . . . . . . . . . . .153

A.1 A Simple Structural Electricity Spot Price Model . . . . . . . .165

F.1 Parameter Estimation of the Fuel Price Index Model . . . . . .187

F.2 The Monthly Intra-year Seasonal Pattern . . . . . . . . . . . . .187

xvii


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LIST OF FIGURES

xviii


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List of Tables

2.1 Separate the Intra-day Hourly Spot Prices into 24 Time Series . 22

5.1 Parameters of the Intra-week Model . . . . . . . . . . . . . . . . 95

5.2 Parameters of the Intra-year Model . . . . . . . . . . . . . . . . 97

5.3 Parameters of the Multi-year Model . . . . . . . . . . . . . . . .100

5.4 Statistics of the Empirical Probability Distributions . . . . . . .105

5.5 Characteristics of the Coal Unit and the Gas Unit . . . . . . . .108


6.1 Parameters of the Intra-week Model . . . . . . . . . . . . . . . .124

6.2 Parameters of the Intra-year Model . . . . . . . . . . . . . . . .127

6.3 Parameters of the Multi-year Model . . . . . . . . . . . . . . . .129

6.4 Statistics of the Empirical Probability Distributions . . . . . . .129


7.1 The Computation Burden of the Structural Models . . . . . . .153

B.1 Characteristics of the Generators in the Test System . . . . . .168

D.1 The Price Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 179

D.2 Electricity Spot Price Decomposition by Filtering . . . . . . . .180

F.1 Parameters of the Fuel Price Index Model(2002-2009) . . . . . .186

F.2 Parameters of the Fuel Price Index Model(2005-2009) . . . . . .186

xix


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LIST OF TABLES

xx


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Chapter 1

Introduction

1.1 Background and Motivations

Large scale electric power systems have a history of less than one hundred

years. It was only in 1882 that Thomas A. Edison brought his power station

in Pearl Street on-line to supply electricity to the financial district in New

York City, see Wasik(2006). Since early 20th

century power systems gaineddramatic development first in the Western world and thereafter spread to the

rest of the world. Up to early 1990s, partly due to historical reasons and partly

due to the nature of the electric power industry, most power systems around

the world operated under strict government regulation, in which government

granted the rights of building power plants to generation companies, fixed

the electricity tariff, and guaranteed a certain rate of return for the invested

capital.

In the regulated electric power industry, because government policies guar-

antee that generation companies are receiving a fixed tariff for generating a

certain amount of electricity, the concern of these generation companies thus

is how to build a particular power plant and then how to dispatch the gen-

erators in a manner so that to minimize the cost of generating that amount

of electricity. The cost of generating electricity is a function of electric load,

efficiency and availability of generators, prices of fuel, and various operating

constraints of generators and transmission networks, etc. Power system re-

searchers have developed sophisticated methodologies and models to estimate

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1. INTRODUCTION

the cost of generating electricity up to a time horizon of several decades, whichare conventionally categorized as Production Cost Models, refer to Wood &

Wollenberg(1996).

Since early 1990s, deregulating the electric power industry in the purpose of

introducing competition became a trend around the world. The deregulation

first started on the generation side, generation assets were sold to a few private

generation companies. These generation companies then compete with each

other to supply electricity to electricity markets, and the generators with lower

bids have the priority to sell to the market. Electricity prices are no longer fixedby the government, but determined by the supply and demand of electricity

markets.

In the deregulated electric power industry, private generation companies, to

make their decisions on building power plants and scheduling their production,

concern the profit they are going to make if they build a particular power

plant and if they dispatch their generators in a certain manner. The profit

of a generator is a function of future electricity prices, the availability of the

generator, the cost of generating electricity, and the discount-rate for futureprofit, refer toHou & Wu(2008), and see Figure1.1.

Electricity Spot Prices

0t1T 2T

t

t

1T

2T

Q2

Generator

Availability

( )A t

Total

Profit

max ( ) ( ),0S t C t

Hourly Profit

Q3

Present Value

Rte

Q4Summation

Q5Q1

( )S t

Figure 1.1: The Flow-chart of Generator Profit Valuation

Namely, at the present time t0, the total profit for generating electricity

during a future time period [T1, T2] could be calculated as follows: at each

future hour t, if the generator is available and not in outage, namely A(t) =

1(otherwise A(t) = 0), the generator will watch the market and take chances

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1.1 Background and Motivations

to generate electricity; if the electricity spot pricesS(t) is greater than the costof generating electricityC(t),S(t)> C(t), the generate will produce and earn

a profit (t) = S(t) C(t); and if the electricity spot price is less than thecost of generating electricity,S(t)< C(t), the generator does not operate and

earn a zero profit(t) = 0; the future profit at hourt,(t), is then discounted

to obtain its present value (t) =eRt (t); finally, the profits at each futurehour t is summed over the whole period [T1, T2] to arrive at the total profit

during the period, = t[T1,T2](t).

Due to the limitations of human knowledge and the inherent uncertainties

of Nature, the forecasts on future profits are always uncertain: future electricity

prices, as they have been, will be highly volatile, generators could be forced to

outage due to equipment failure, the cost of generating electricity is subject to

fluctuations of fuel prices, and the discount-rate for future profit is further a

function of interest rate and the extent of the uncertainty associated with the

profit in concern. Therefore, on the downside, generation companies very much

concern the possibility of not earning enough profit to cover their initial capital

investment; on the upside, these generation companies also care the possibility

of reaping unusual spectacular profit if the markets go in their favor.

Therefore, generation companies, in order to make informed decisions on

generation investment and scheduling, have to calibrate carefully the key fac-

tors which will significantly affect their future earnings, that is, future electric-

ity prices, generation forced outage rate, future fuel prices, and future interest

rate. This thesis work will only focus on understanding, interpreting, and then

modeling the first key factor, namely,electricity prices.

Even compared with the prices of other energy commodities like oil and

natural gas, which have been famously volatile, electricity, the most important

direct source of power for mankind, has prices that are notoriously volatile.

During a day, electricity spot prices are higher during the day-time and lower

in the night-time; during a week, electricity prices are higher during weekdays

and lower on weekends; during one year, electricity prices are high and volatile

in high demand seasons, and usually low and milder in low demand seasons;

across years, electricity prices are high when the economy is active and low

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1. INTRODUCTION

when the economy falls to recessions. Meanwhile, short-life extreme-high pricespikes, which could be as high as tens of times of usual prices, are frequently

encountered in many electricity markets around the world.

Generation companies, whose profits are directly tied with and significantly

affected by these movements of electricity prices, crave for a deeper understand-

ing on electricity prices and at best a physically well-grounded, simple, and

fast electricity price model to empower them to make informed decisions even

on a day-to-day basis.

1.2 Literature Review and the Gap to Fill

Modeling Approach

Statistical / Financial

Structural / Hybrid

Fundamental / Simulation

Time-horizonLong-term

/Multi-year

Mid-term

/Intra-year

Short-term

/Intra-week

Figure 1.2: Distribution of Literatures- according to their approaches and

the time-horizons in their concern, and the darkness of the cells represents the

number of publications

Many researchers have worked on electricity price modeling. In terms of

time-horizon, the price modeling problems can be categorized into short-term,

mid-term, and long-term, see Figure 1.2. In terms of modeling approach,

there are Statistical/Financial Approach, Structural/Hybrid Approach, and

Fundamental/Simulation Approach: many researchers have taken the statis-

tical/financial approach to model either the short-term or mid-term prices,

using Time Series Models or Financial Models; a few authors have taken the

structural/hybrid approach to model either the short-term, mid-term, or the

long-term prices; and some researchers have taken the fundamental/simulation

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1.2 Literature Review and the Gap to Fill

approach to study power markets. (A detailed discussion on the various mod-eling methods could be found in Chapter2.)

Looking at this landscape of literature, one observes that few people have

taken the statistical/financial approach to model the long-term prices. The

reason for this observation is mostly due to the limitations of the mathemat-

ical tools that are available for the statistical/financial approach, which are

Time-Series/Financial Models. According to the conventional wisdom, Time-

Series/Financial Models are only good for modeling short-term prices, and at

most for mid-term prices, but not for modeling long-term prices, because in thelong-run of many years, the fundamental system is undergoing so significant a

change that previous calibrations of models from historical data are no longer

valid.

Another observation on this literature landscape is that, comparing with

the large number of publications on the statistical approach, only a few authors

have addressed the structural approach. The main reason for this observation

is due to the complexity of the structural approach. Structural approach mod-

els electricity spot prices indirectly: it first models the constituent physicalunderlying forces such as generation supply stack, electricity load, generation

forced outages, fuel prices, etc., and then constructs these constituent models

into an electricity spot price model. In order to build a structural price model

that isrealisticand thus has its potential use inpractice, the perquisite is that

one has to first model the constituent physical forces satisfactorily. However,

the problems of modeling these constituent physical forcesthemselvesare chal-

lenging and so far havent received definitive solutions. Further discussions on

the structural approach could be found in Section2.4.

In this landscape of literature, we discuss the position of our work. Our

primary purpose is to build a long-term electricity spot price model, and this

model will be used by generation companies for generation investment and

scheduling analysis. The approach we take is the statistical/financial approach.

The reason for taking the statistical/financial approach is mostly because sta-

tistical/financial models provide a final result that is mathematically simple

and elegant, and this mathematically simple and elegant final result the other

modeling approaches usually are not able to serve. This simple final result is

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1. INTRODUCTION

of particular importance for us because in generation investment and schedul-

ing analysis prices is only the input variable for much further computation.

Another reason for taking the statistical/financial approach is that few people

have ever taken the statistical/financial approach for modeling the long-term

prices, and there might be much space worthing further exploration.

Even though themodeling approachwe will take is statistical/financial, the

analysis approachwe will take is both structural and fundamental. Namely, in

the process of this work, we will extensively employ structural approach and

fundamental approach to understand the fundamental operation of engineering

power systems and power markets, and based on this understanding, we then

move on to devise our modeling approach and design the price models.

Once we decided to pursue the statistical/financial approach for modeling

the long-termprices, it immediately implies that we have to resolve its limita-

tion of vulnerability for system fundamental changes. In order to make statis-

tical models robust for the underlying changes of systems, statistical models

have to be designed to somehow contain these system changes:

a) Intuitively, one way to achieve that is to take the Hybrid Approach that

combines the Statistical Approach and the Structural Approach: namely,

one derives the parameters of the statistical models from the more fun-

damental structural models that are more capable of containing system

fundamental changes;

b) The other way is to decompose systematically the electricity spot price

data into a few components that are driven by different and independent

physical underlying forces, and then model each price component respec-

tively by one statistical model. Because each statistical model captures

only one type of driving forces and each type of physical forces behaves

consistently along its own time horizon, each statistical model proba-

bly is capable of capturing that one particular type of physical forces

satisfactorily. Finally a complete electricity spot price model might be

somehow constructed with the resulting statistical sub-models.

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1.3 Elaboration of the Problem

Our research work thus has adventured into both paths, and the result

turned out that the second approach of decomposition is more feasible and

produces a desirable model. Therefore, a significant portion of this thesis is

devoted to discussing the second approach of decomposition; and only the

very last chapter introduces an adventure of modeling the prices by Structural

Approach, which only achieves limited success.

1.3 Elaboration of the ProblemThis section defines the problem by visioning the ideal result of the model and

enlists the analytical and mathematical tools that are available for solving the

problem.

1.3.1 The Requirements on the Price Model

Modeling the Physical Forces

Generation investment and scheduling analysis concerns future generator profit

from as short as a few months to as long as years and decades. Electricity spot

prices up to many years and even decades is a complex system, for that in this

long time-horizon electricity prices are driven by many different and indepen-

dent physical forces from both the supply and demand sides, and these physical

forces play in different timescales: some forces affect prices in short-term(intra-

week) and have ignorable effect beyond a week, such as regular generation

dispatches, generation forced outage, and intra-day and intra-week variations

of electricity load; the seasonal forces play at the mid-term(intra-year) level,

such as seasonal weather and temperature, seasonal hydro-generation capacity,

annual generation planned outage, etc., and these forces could influence elec-

tricity spot prices up to a few weeks, but have negligible effect beyond a year;

in the longer run of many years and up to a few decades, the physical forces

that are manifest are economic development and economic cycles, generation

investment and retirement, and fluctuations of fuel prices, etc. A model in-

tended for generation investment and scheduling analysis ought to define and

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1. INTRODUCTION

capture properly these various physical forces that underlie electricity spot

prices.

Time Horizon

The electricity spot price model for generation investment and scheduling anal-

ysis, therefore, ought to capture the dynamics of electricity spot prices in a

future time horizon from as short as a few months to as long as many years.

Time Unit

In terms of time unit, as between day-time and night-time electricity spot

prices change a lot, thus particularly for valuating the profit of peaking gen-

erators, which are usually only profitable to operate during peak hours, an

hourly based model is necessary.

Final Result

Besides satisfying all the above requirements, the model ought to have a final

result that is mathematically simple, for that electricity spot prices is the input

variable for generator profit valuation and further the generation investment

and scheduling analysis. Ideally, the prices at each hour ought to be captured

by a simple probability density function.

1.3.2 The Analytical and Mathematical Tools

Time-Series Models and Financial Models

The basic building blocks for the price model are borrowed from the estab-

lished field of Time-Series Analysis and Financial Engineering. The reasons

for choosing the Time-Series/Financial Models as the basic tools are due to

that they are capable of modeling one very important feature of electricity

prices, that is, its mean-reverting nature, and that they provide a simple and

elegant final representation for electricity spot prices as a continuous proba-

bility density function.

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1.4 The Analysis Framework

Microeconomics Price TheoryAfter hundreds of years efforts of many generations of economists, the Price

Theory in Microeconomics has become a solid foundation for the science of

Economics. It provides the basic analytical tool for understanding and inter-

preting the outcomes of almost any markets that feature people who are free to

buy and to sell. Electricity markets, despite the complexity of its underlying

engineering systems and so far the inability for large-scale storage of electric-

ity, are no exception. As we will see, Microeconomics Price Theory provides a

powerful tool for analyzing and understanding electricity spot prices.

Electric Power Engineering

Underpinning any electricity market, there is a huge and complicated engineer-

ing power system, which consists of a large number of generators, a network

of transmission lines, substations, distribution networks, load centers, various

control and communication devices, and control centers. This system is oper-

ating at real-time, at any moment, power engineers have to ensure that power

generation and electricity load are always balanced. This power generation

and electricity load, if interpreted into the terms familiar to economists, power

generation corresponds to market supplyand electricity load stands for mar-

ket demand. No surprise, power generation and electricity load have been the

subjects of study by power engineers for long time, and this rich expertise on

engineering power systems no doubt is a valuable source for understanding

electricity markets and hence electricity prices.

1.4 The Analysis FrameworkThe system of electricity prices up to a few decades is a complex system; there

are many different and independent physical forces under playing. To put these

various underlying physical forces in perspective, electricity spot prices will be

studied carefully in a Multi-granularity Framework, that is, electricity spot

prices will be analyzed into three time-perspectives of different granularity:

multi-year yearly, intra-year weekly, and intra-week hourly. In each of the

three time-perspectives, by applying the analytical tools like Microeconomics

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1. INTRODUCTION

Price Theory and Knowledge on operations of fundamental engineering powersystems, how these physical underlying forces affect electricity spot prices will

be carefully investigated:

a) The multi-year perspective looks at the long-term trend of electricity

prices, which is under driven by the physical forces like economic devel-

opment and economic cycles, generation investment and retirement, and

fluctuations of fuel prices.

b) The intra-year perspective examines the seasonal behavior of electricity

prices, which is driven by the mid-term forces such as seasonal variations

of electricity load due to seasonal weather and temperature, seasonal

hydro-generation capacity, annual generation planned outage, etc.

c) The intra-week perspective investigates the intra-day and weekday-weekend

variations of electricity spot prices, which are driven by short-term forces,

such as intra-day and weekday-weekend variations of electricity load, reg-

ular generation dispatches, generation forced outages, etc.

1.5 The Modeling Methodology

Electricity prices are driven by various physical underlying forces, and these

underlying forces are playing in different timescales. In other words, these

physical forces are of different frequencies, that is, the physical forces in the

multi-year perspective are of the lowest frequency, the intra-year forces are of

mid-frequency, and the intra-week forces the highest frequency; these physical

forces probably locate at different and separate bands in their frequency spec-

trum. Taking advantage of this insight, if somehow one could decompose the

electricity spot price data into a few components, of which each is driven by

a different and independent type of physical forces, one could thus divide the

original price modeling problem into a few sub-problems that are by nature

much simpler. This is the thought we went through when devising our plan

for solving the problem. This modeling methodology will be referred as Divide

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1.6 Organization of the Thesis

and Conquer, which is a general strategy for solving complex problems of anykind.

Therefore, the plan for our adventure goes as follows: we will begin with

historical electricity spot price data; the historical electricity spot price data

will then be decomposed into a few components that are driven by different

and independent physical underlying forces; then these price components will

be defined respectively by separate models; finally the resulting sub-models

will be constructed into a complete electricity spot price model.

1.6 Organization of the Thesis

The forthcoming chapters of this thesis are organized as follows. Chapter2

surveys the existing literatures on electricity price modeling, namely, the par-

ticular price modeling problems, the usual approaches, popular mathematical

tools, and some special techniques. Chapter 3 applies the Microeconomics

Price Theory to electricity markets and develops the basic analytical tools for

analyzing electricity spot prices. Making use of the analytical tools that aredeveloped in Chapter3, Chapter4introduces theMulti-granularity Framework

for analyzing electricity spot prices in three time-perspectives; in each perspec-

tive, it endeavors to understand how the operations of fundamental engineer-

ing power systems give rise to the very peculiar behaviors of electricity spot

prices. Taking advantage of the understanding on electricity spot prices that

has been gained in previous chapters, Chapter 5proposes an electricity spot

price model that is based on the idea of decomposition, and demonstrates the

modeling methodology in the New-England electricity market. Upon some mi-

nor revisions on the model, Chapter 6 brings the same modeling methodology

to the Pennsylvania-New Jersey-Maryland(PJM) electricity market. Chap-

ter7records an early adventure of us trying to model electricity spot prices by

Structural Approach. Chapter8summarizes and concludes this thesis work.

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1. INTRODUCTION

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Chapter 2

A Survey of Electricity PriceModels

2.1 Electricity Price Modeling Approaches

From the point of view of a generation company, modeling and forecasting

electricity prices are either for formulating bidding strategies, scheduling gen-eration production, pricing electricity derivatives, or generation investment

analysis. For formulating bidding strategies one concerns the day-ahead 24-

hour spot prices; for scheduling generation and pricing electricity derivatives

one cares the spot prices from as short as the next week to as long as the

next year; for generation investment analysis, one concerns the electricity spot

prices during the life time of a generator, that is, years or even decades. In

order to make these pricing, scheduling, and investment decisions, generation

companies need a thorough understanding on electricity spot prices, at best,

they want a physically well-grounded, simple, and fast electricity spot price

model to assist their making these decisions.

A price model suitable for a particular application has to at least satisfy

five requirements: it has to capture properly the movements of electricity spot

prices, it must give the expected value of the foreseen spot prices, it must

quantify the uncertainty of the spot prices, it has to define the prices of a time

unit that is suitable for a particular application, and it has to cover the time

horizon that a particular application concerns. There are a few mathematical

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2. A SURVEY OF ELECTRICITY PRICE MODELS

tools, modeling approaches, or techniques that have been applied to analyzeand model electricity prices, i.e., Time Series Models, Financial/Stochastic

Process Models, Structural Models, Decomposition Techniques, etc.

In terms of Time Series Models, they start with the price data and define

a model to capture the correlations between the prices at one time period

and the prices at the previous periods. Time series models are usually good

options for short-term applications, like forecasting next-day 24-hr and next

week 168-hr prices. We will, start with the simplest time series model and

end with some rather involved ones, step by step, discuss the nature of timeseries models and their possible applications in electricity price modeling and

forecasting.

Financial/Stochastic Process Models have seen their successful applica-

tions in modeling the financial, commodity, and energy markets. The financial

models are defined by stochastic differential equations. The financial models,

though complicated at first glance, give final results that are mathematically

simple. These mathematically elegant final results enable financial models to

be used in applications that desire the least computation and the fastest speed,such as pricing electricity futures, contracts, and other derivatives. We will

introduce, carefully, the development of financial models in modeling electric-

ity prices, the problems that have been satisfactorily solved, and the problems

that are still remain.

Time series models and financial models belong to Statistical Models. Both

of them use the historical data to induce, calibrate, and experiment the mod-

els. Structural Approach, on the other hand, models electricity spot prices in-

directly. It first models the physical forces that underlie electricity spot prices

by a few constituent models, then constructs these constituent sub-models into

a complete electricity spot price model. For that Structural Approach utilizes

the more fundamental knowledge on electricity markets, it could model elec-

tricity spot prices of a longer time-horizon than that the statistical approach

could usually do. We will briefly introduce the idea of structural models, their

advantages and disadvantages, and the literatures along to this line.

Besides the statistical approach and the structural approach in modeling

electricity prices, there is another method for studying electricity markets,

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2.2 Time Series Models

which will be referred as Fundamental Approach. Compared with the struc-tural approach, fundamental approach is of much further detail. It may model

each generator unit, behavior of each market player, seasonal generation main-

tenance and available generation capacity, the dynamics of electricity load

demand, the transmission networks, the rules of the electricity market, the

investment cycles of the electric power sector, etc. It then uses these detailed

models to construct electricity spot prices. Due to these further details, prob-

ably there is no analytical result for a fundamental model. In this survey,

we will not give any further discussions on the fundamental approach, inter-ested readers please refer to the literatures, such as, Angelus(2001);Bastian

et al.(1999);Baughman & Lee(1992);Bessembinder & Lemmon(Jun. 2002);

Olsina et al. (2006);Ruibal & Mazumdar(2008);Wang & Mazumdar(2007).

Finally, decomposition techniques probably have huge potential in analyz-

ing and modeling electricity prices. Electricity prices has a complex nature:

intra-week hourly spot prices has an intra-day pattern and a weekday-weekend

pattern; intra-year weekly prices has a seasonal pattern, and in different sea-

sons prices behave distinctly; and short-life price spikes prevail in electricitymarkets all over the world. This complex nature of electricity prices is due

to the various physical driving forces that underlie electricity spot prices. If

the electricity spot prices could somehow be decomposed into a few price com-

ponents that are driven by the different and independent physical underlying

forces, the original problem then is divided into several sub-problems that

are by nature much simpler and thus much easier to solve. We will introduce

briefly the various decomposition techniques, such as Fourier Analysis, Wavelet

Analysis, and Principal Component Analysis, and discuss their possible appli-

cations in electricity price analysis and modeling.


2.2.1 The Autoregressive Model

The discussions on the time series electricity price models are based onFranses

(1998); Hamilton (1994); Wei (2006); Weron (2006). We will start with the

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simplest Autoregressive model of order One, namely the AR(1) model. Denote

a time series as xk,k = 1,....,T, where the tilde defines a random variable.An AR(1) model defines xk as a function of its value in the previous time

period xk1, as

xk =c + xk1+ k 0<


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wherea(L) = 11L12L2 ...mLm. The unconditional mean of AR(m)is found by taking the expectation on both sides of Equation (2.1),

E {xk 1xk1 2xk2 ... mxkm} = E {c + k}E {xk} 1E {xk1} 2E {xk2} ... mE {xkm} =c + E {k}

1 2 ... m= c (1 1 2 ... m) =c

Thus,

= c1 1 2 ... m

The unconditional mean is a constant, namely, given the initial value x0,

when time goes to infinite k , the expected value of xk is a constant.

AR and Electricity Prices

AR models assume a constant unconditional mean, while electricity spot prices

have a time-varying mean, and its mean varies by the time of day, week and

year. AR models alone thus are not suitable for modeling electricity prices,

but they are ideal candidates for capturing the mean-reverting stochastic com-

ponent of electricity prices, therefore, AR models, as we will see, will serve as

an vitalcomponent in building an electricity price model.

2.2.2 AR with Time-varying Mean

Lets introduce a time-varying function f(k) into the simplest AR(1) model

xk, the resulting model is named AR model with a Time-varying Mean(ARV),

denoted as yk,yk =f(k) + xk

Here xk is simplified to have the constant c = 0, thus it is xk = xk1+ k.

Write yk into the form of AR(1),

yk = [f(k) f(k 1)] + yk1+ k=g(k) + yk1+ k

where g(k)=f(k)

f(k

1).

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ARV and Electricity Prices

An ARV model is the sum of a deterministic time-varying function f(k) and

an AR process xk. Electricity prices are observed to have a time-varying

mean and a stochastic component that is mean-reverting. The deterministic

function f(k) could model the time-varying mean of electricity prices, and

the AR process xk could model the mean-reverting stochastic component of

electricity prices. The design of the deterministic function f(k) depends on

the specific modeling problems, it could include the intra-day and weekday-

weekend pattern of spot prices, and it may also include the annual seasonal

pattern. The specification of the AR process xk, say, how many time lags to

include in the model, may depend on data and particular applications. ARV

models have been empirically applied to model electricity prices by Bhanot

(2000); Escribano et al. (2002); Hamm & Borison (2006); Knittel & Roberts

(2005); Misiorek et al. (2006); Rambharat et al. (2005); Serna & Villaplana

(2007);Weron & Misiorek(2008).

2.2.3 AR with Exogenous Variables

Another way to model the time-varying function f(k) is to substitute f(k)

with a function of an exogenous variable. The exogenous variable is denoted

as zk. AR models with eXogenous variables are named ARX models. An ARX

model yk with the time lags of the exogenous variable zk is,

yk = (c + 0zk+ 1zk1+ ... + qzkq)+(1yk1+ 2yk2+ ... + mykm)+ k

Move the zk terms to the left and write it in a more concise form,

yk 1yk1 2yk2 ... mykm= c + 0zk+ 1zk1+ ... + q zkq+ k1 1L1 2L2 ... mLm

yk =c +

0+ 1L

1 + ... + qLq

zk+ k

a(L)yk =c + c(L)zk+ k

where a(L) = 1

1L

1

2L

2

...

mL

m, andc(L) =0 + 1L1 + ... + qL

q.

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ARX and Electricity PricesDriven by the intra-day, intra-week, and seasonal variations of electricity load

demand, electricity spot prices have a time-varying mean, and its mean varies

by the time of day, week, and year. Besides electricity load, electricity spot

prices are also affected by generation capacity, fuel prices, etc. Capable of in-

cluding these exogenous variables into consideration, ARX models are proba-

bly able to improve the accuracy of price forecasting, seeKarakatsani & Bunn

(2004); Misiorek et al. (2006); Rambharat et al. (2005); Weron & Misiorek

(2008).

2.2.4 The Autoregressive Moving Average Model

The simplest Moving Average Model of order one is MA(1) Model,

xk =c + 0k+ 1k1

It says that xk equals to a constant c plus the weighted average of time-lagged

random noises k and k1. In other words, if xk is subtracted by the constant

c, the residuals rk

= x

k c =

0k

+ 1k1

are highly correlated. A general

Moving Average Model of time lag n, MA(n), is,

xk =c + (0k+ 1k1+ ... + nkn)

=c + b(L)k

where b(L) =0+ 1L + ... + nLn.

ARMA stands for Autoregressive-Moving Average. The purpose for ex-

tending an AR model to an ARMA model is to model the autocorrelation

of residuals. An ARMA model is a combination of an AR model and a MA

model. The simplest ARMA model is a combination of AR(1) and MA(1),namely, ARMA(1,1),

(1 1L)xk =c + (0+ 1L)ka(L)xk =c + b(L)k

A general ARMA model, ARMA(m,n) is:

a(L)xk =c + b(L)k

where a(L) = 1 1L1 2L2 ... mLm, andb(L) =0 + 1L + ... + nLn.The unconditional mean of the an ARMA(m,n) model is = c

1

1

2

...

m

.

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ARMA and Electricity PricesThe same as an AR model, an ARMA model assumes a constant uncondi-

tional mean. Thus it only models the mean-reverting stochastic component of

electricity prices. An ARMA model is of more complexity than an AR model.

Whether an AR model or an ARMA model is a better option for modeling the

mean-reverting stochastic component of electricity prices probably depends on

data and particular applications. Whether the extra complexity of an ARMA

model could be justified has to be carefully judged, for that a model is not at

all merely verified by its fitting of the historical data, but more importantly itought to be measured by its ability in forecasting. A model of more complexity

is usually more vulnerable to fundamental changes of systems. Literatures on

modeling electricity prices by ARMA models areGarciaet al.(2005);Huurman

et al. (2008);Nogales & Conejo(April 2006);Swider & Weber(2007).

2.2.5 ARMA with Time-varying Mean

In analogy to the extension from AR model to an ARV model, an ARMA model

could be extended to an ARMA model with Time-varying Mean, denoted as

ARMAV model. An ARMAV model, denoted as yk, is the summation of a

deterministic time-varying function f(k) and an ARMA process a(L)xk =

c + b(L)k, as

yk =f(k) + xk

Written in terms of yk, it is,

a(L)yk =g(k) + b(L)k

ARMAV and Electricity Prices

The difference between an ARMAV model and an ARV model is the way they

model the stochastic mean-reverting component of electricity prices. Whether

an ARMAV model or an ARV model is a better option for modeling electricity

prices probably depends on specific empirical cases. Empirical evidences from

Crespo Cuaresma et al. (2004);Weron & Misiorek(2005) show that ARMAV

models do not outperform the simpler ARV models. The moral lesson from

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these empirical studies is that, when a model of higher complexity is not welljustified by the particular modeling problem in concern, one must be cautious

to move beyond a simpler model and prefer to a model of higher complexity,

even though the model of higher complexity usually gives a better fitting of

historical data.

2.2.6 ARMAX and Transfer Function

If one replaces the deterministic functiong(k) by a function of exogenous vari-

ables,c+c(L)zk, the ARMAV model becomes an ARMA model with eXogenous

variables, referred as an ARMAX model,

a(L)yk =c + c(L)zk+ b(L)k

where c(L) =0+ 1L1 + ... + qL

q. Divide a(L) from both sides, one arrives

at the ARMAX model written in the form of a Transform Function,

yk =

c

a(L)+

c(L)

a(L) zk+

b(L)

a(L) k

ARMAX Model and Electricity Prices

The difference between an ARMAX model and an ARX model is again the

way they model the mean-reverting stochastic component of electricity prices.

Though probably a ARMAX model of higher complexity fits the historical data

better, it does not necessarily outperform a simpler ARX model in forecasting.

Empirical studies on ARMAX models areConejo et al. (2005a);Garcia et al.

(2005); Huurman et al. (2008); Knittel & Roberts(2005); Nogales & Conejo(April 2006);Nogaleset al.(2002);Swider & Weber(2007);Weron & Misiorek

(2005).

2.2.7 Periodic Autoregressive Models

In previous sections, we model the time-varying mean of electricity prices with

a deterministic function. In this section, we will introduce another way of mod-

eling the time-varying mean, which is Periodic Autoregressive Model(PAR).

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Lets look at the hourly electricity spot prices across several days, for ex-ample, there are Ndays, each day has 24 hours, thus there are 24 N spotprices. We are going to separate the hourly spot prices into 24 time series

according to their hour index, which is the order of a particular hour in the 24

hours in a day.

Hour 1 Hour 2 . . . Hour 24

Day 1 x1,1 x2,1 . . . x24,1

Day 2 x1,2 x2,2 . . . x24,2... . . . . . . . . . ...

Day N x1,N x2,N . . . x24,N

Table 2.1: Separate the Intra-day Hourly Spot Prices into 24 Time Series

Day 1k Day k

Hourth

i

Hourth1i 1,i kx

,i kx, 1i kx

1, 1i kx

Figure 2.1: Index the Intra-day Hourly Spot Prices

The spot prices are organized in Table 2.1: the columns expand the 24

hours, and the rows spread the days. Instead of considering the 24 N pricesas a whole, we group them according to their hour index, namely, the index

of the columns. Take the firstcolumn for example, we took the price at the

1st hour of each day, order them by day 1, day 2, ..., till day N, we thus

obtained a new segment time series consisting ofx1,1, x1,2, . . ., x1,N, and put

them in the first column. Continue the same procedure for the rest of the 24

hours, we obtain 24 segment time series. This way of separating the original

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time series according to their hour index is the key concept underlying PARModels, Figure2.1.

If one models each segment time series with one AR model, there will be

24 AR models. In the simplest case, each AR model is an AR(1) model, the

model for hour i, thus is,

xi,k =ci+ ixi,k1+ i,k i= 1,...24

The above PAR model could be extended to include the spot price at the

immediate previous hour xi1,k

, one then arrives at a more general PAR model,

xi,k =ci+ ixi,k1+ ixi1,k+ i,k i= 1,...24

A general PAR model could have more price lags. The purpose of PAR model

is to model prices at different hours by different AR models.

PAR and Electricity Prices

For hourly electricity spot prices, on-peak prices are more volatile than the

off-peak prices. It is reasonable to model on-peak prices and off-peak prices

by different AR models. Hourly spot prices have three orders of seasonality:intra-day pattern, weekday-weekend pattern, and intra-year seasonality. A

PAR model that treats each hour respectively avoids modeling the intra-day

pattern. For daily average prices, prices on weekdays are usually more volatile

than these on weekends, it is reasonable to model weekday and weekend prices

with different AR models. Empirical Studies on PAR models could be found

in Crespo Cuaresma et al. (2004); Garcia-Martos et al. (2007); Guthrie &

Videbeck(2007);Huisman et al. (2007).

2.2.8 ARIMA and its Extensions

The simplest ARIMA model is an ARIMA(0,1,0), which is a Random Walk

model,

xk xk1 = xk = kIt describes a process whose increase from time k 1 to k is random and thenoise k is characterized by a random variable of Standard Normal Distribu-

tion. In some cases, the first order difference, namely the increase xk, may

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require more specifications. One may describe it with an AR(1) Model. Letsdenote xk

=k,

k =c + k1+ k

Reorganize it as,

(1 L)k =c + kWe call it an ARI(1,1) model. The ARI(1,1) could be extended into a more

complicated ARIMA(1,1,1) model, where the residuals are assumed to be cor-

related as a MA(1) process,

(1 L)k =c + (1 + L)k

A general ARIMA(m,1,n) model thus is:

a(L)k =c + b(L)k

In some cases one may need to describe the second order differences,

2

xk =k k1 = xk xk1

we then arrive at an ARIMA(m,2,n) model,

a(L)2xk =c + b(L)k

Higher order of differences is denoted as dxk, thus a general ARIMA(m,d,n)

model describing the dth order difference is,

a(L)dxk =c + b(L)k

ARIMA and Electricity Prices

The nature of ARIMA model is to model the price differences. Based on our

understanding on electricity spot prices, which are the intersections of marginal

cost curve and electricity load demand, we think it is the original prices, rather

than the differences of prices, that probably have a systematic structure. Em-

pirical studies concerning the ARIMA models could be found inConejo et al.

(2005a,b);Contreraset al.(2003);Garciaet al.(2005);Huurmanet al.(2008).

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2.2.9 Regime-switching Models

A regime-switching model assumes two regimes, R1 and R2, the switching

between the two regimes is governed by a Markov Chain, Figure 2.2, and the

transient probability between the two regimes are,

11= Pr

R(t) =R1

R(t dt) =R1

12= Pr

R(t) =R2

R(t dt) =R1

21

= Pr

R(t) =R1

R(t dt) =R2 22 = PrR(t) =R2 R(t dt) =R2

12

11 22

21

2R

1R

Figure 2.2: A Two-regime Markov Chain

A regime-switching process pt has two sub-processes, p1t and p2t: when the

Markov Chain is in the first regime R(t) =R1, pt= p1t; and when it is in the

second regime R(t) =R2, pt= p2t. Namely,

pt=

p1t if R(t) =R1p2t if R(t) =R2

The two processes p1t and p2t each could be an AR process or extensions of an

AR process. The parameters of the model p1tand p2tare different, by this way

they are designed to capture the different price behaviors in the two different

regimes.

Regime-switching Models and Electricity Prices

During one day, on-peak prices are more volatile than the off-peak prices;

during one year, prices are usually higher and more volatile during the high

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demand seasons than these during the low demand seasons. Driven by someextreme weather, when electricity load demand reaches some very high levels,

price spikes tend to occur. Overall, when prices are low, high, or in spike,

they behave differently. Regime-switching price models are designed to cap-

ture these different behaviors of electricity prices in different time. Empirical

studies applying the regime switching models to electricity prices are Huisman

& Mahieu(2003);Mount et al. (2006); Swider & Weber(2007);Weron et al.

(2004).

2.2.10 Time-varying Volatility

Lets first recall the AR(1) with Time-varying Mean model, see Section 2.2.2,

yk =g(k) + yk1+ k k N(0, 1)

where the volatility , measured by the standard deviation of the noise k, is

constant. Now the constant volatility is to be relaxed as time-varying, namely,

the standard deviation becomes a function of time k, ask. The ARV model

with time-varying volatility thus is,

yk =g(k) + yk1+ kk k N(0, 1)

2.2.11 The GARCH Model

GARCH stands for Generalized Autoregressive Conditional Heteroscedasticity.

GARCH models are designed to capture the stochastic time-varying volatility.

The simplest GARCH(1,1) model is,

2k =c + 2k1+ (k1k1)

2

where +


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(k1k1)2 models the effect of the innovation noises: a large noise at timek 1 will cause the variance 2k in the time k to be large, and a small noise attime k 1 contributes little to the variance 2k.

Given the initial variance at time 0,20, the variance at timek, 2k, is derived

as,

2k =20

ki=1

+ 2ki

+ c

1 +

k1j=1

ji=1

+ 2ki

The expected value of the variance 2k thus is,

E

2k

= 20

ki=1

+ E

2ki

+ c

1 +

k1j=1

ji=1

+ E

2ki

=20( + )k + c

1 +

k1j=1

( + )j

=20( + )k + c

1 +

( + ) + ( + )k

1 ( + )

As time k goes to infinite, k , its expected value is limk E{2k} =

c

1 + (+)1(+)

. Namely, the unconditional mean of 2k is a constant.

GARCH and Electricity Price Volatility

The volatility of electricity spot prices is time-varying, see Section3.8, specif-

ically, the volatility is price-dependent: high prices are of high volatility; low

prices are of low volatility. The volatility of hourly electricity spot prices thus

is cyclic and highly predictable: price volatility during the day-time is higher

than that during the night-time, and the price volatility during the high de-

mand seasons is higher than that during the low demand seasons.

GARCH model, by its nature, is a stochastic process. The unconditional

mean of a GARCH process is a constant, it means that the variance has a

tendency to revert to a constant level. And in the GARCH model, that the

variance is time-varying is due to the random return noises. The GARCH

Model, therefore, is only capable of modeling the stochastic componentof the

time-varying volatility of electricity prices, and it is not able to model the

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deterministic partof the time-varying volatility of electricity prices. The de-terministic part of the time-varying volatility of electricity prices, probably, is

more significantthan the stochastic counterpart. Empirical studies on apply-

ing GARCH models to electricity prices modeling are Escribano et al.(2002);

Garcia et al. (2005);Hadsell & Shawky(2006);Hadsell et al. (2004);Higgs &

Worthington(2005); Karakatsani & Bunn(2004); Knittel & Roberts(2005);

Serna & Villaplana(2007);Swider & Weber(2007);Worthingtonet al.(2005).

And general discussions on the volatility of electricity prices are Li & Flynn

(2004b);Nakamuraet al.(2006);Robinson & Baniak(2002);Simonsen(2005);Zareipouret al. (2007).

2.2.12 GARCH with Asymmetric Effect

The standard GARCH model could be extended to model the asymmetric

effect of noises on the volatility. For example, the positive noises, k1 > 0,

may have larger effect on the volatility than the negative ones, k1 < 0.

The GARCH model with Asymmetric Effect is referred as Threshold-GARCH

model. The model is written as,

2k =c + 2k1+ (k1k1)

2 + (k1k1)2 dk1

where dk1 =

1 if k1 > 00 if k1 < 0

.

The additional term (k1k1)2 dk1 differentiates the asymmetric effect

of the positive and negative noises. If > 0, the model implies that positive

noises, k1 > 0, have more significant effects on the volatility k than the

negative noises; if


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follows will be less volatile electricity prices. The TGARCH model, designedfor modeling the asymmetric effect of the noises, therefore, may match some

realities in electricity prices. One empirical study on the Threshold-GARCH

is Knittel & Roberts (2005). It observes that positive price shocks have a

larger effect on the volatility than the negative shocks.

2.2.13 Time-varying Volatility and Model Parameter Cal-

ibration

Lets use a simple linear least-square regression example to illustrate the effect

of time-varying volatility on parameter calibration. We will start with constant

volatility and than add time-varying volatility.

a) In the case of constant volatility, a regression model of three time periods

is,

y1y2y3

=

x1x2x3

+

123

Write it in vectors as, y= x +. The parameter calibration is to find

the optimalthat minimizes, as

min

f()= (y x)T (y x)

Take the first order derivative and the optimal solution satisfies the

condition,df()

d

=

= 0

one thus has xTy xTx= 0, thus the estimate is,

=

xTx1

xTy=

3i=1

x2i

1 3i=1

xiyi

b) If the volatility is relaxed to be time-varying, namely, the model becomes,

y1y2y3

=

x1x2x3

+

112233

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where1,2, and3 are different, written into vectors as, y = x + .The optimization problem becomes,

min

f()= (y / x /)T (y / x /)

denote y =y /, and x =x /, one has,

min

f()= (y x)T (y x)

Take derivative and let df()d

=

= 0, namely, xTy xTx= 0, thusthe estimate is,

=

xT

x1

xT

y =

3i=1

x2i2i

1 3i=1

xiyi2i

The above equation tells that idetermines the weight of the point (xi,yi).

Ifi ,the effect of (xi,yi) on is canceled out; ifi 0, solely depends

on (xi, yi). If1 =2 =3, each point (xi, yi), i = 1, 2, 3 is weighed equallyin calculating . If1 = 2 = 3 = 1, it becomes the example of constant

volatility.

Up to this point, it is natural to infer that if a time series is generated by a

process by nature of time-varying volatility, and then one calibrates it with a

model which assumes a constant volatility, surely the estimated parameters are

biased. The extent of the bias depends on the difference between the assumed

volatility and the true volatility.

2.3 Financial/Stochastic Process Models

2.3.1 Modeling the Multi-seasonality

Electricity spot prices, driven by the temporal

modeling electricity prices for generation investment and scheduling analysis

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