Modeling Electricity Prices for Generation Investment and Scheduling Analysis

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1 Modeling Electricity Prices for Generation Investment and Scheduling Analysis by Yang HE A thesis submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy at The University of Hong Kong. February 2010

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3 Abstract of thesis entitled Modeling Electricity Prices for Generation Investment and Scheduling Analysis Submitted by Yang HE for the degree of Doctor of Philosophy at The 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 analyzing electricity spot prices, which views electricity spot prices in three timeperspectives, 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 independent to each other, play in different timescales, and affect electricity spot prices in different time horizons, this work adopts the methodology Divide and Conquer to build the price model: it decomposes the historical electricity spot

4 price data into components that are driven by different and independent physical 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 methodology 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 companies for their making informed decisions in generation investment and scheduling analysis. Besides generation companies, the model could also be widely used by other players in electricity markets, like by power traders for pricing 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.

5 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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7 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 the guidance 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 student s work, and he shows to him, by correcting the student s 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 Fountainhead that 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 novel Atlas Shrugged I found the seven virtues she derived truly attractive, that is, Rationality, Independence, Honesty, Integrity, Justice, Productiveness, and

8 Pride. These virtues have since then become the guidance of my action, every day. 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 o clock in the morning till 6 o clock 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 persistent in 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. Deqiang 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. Wu s 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 attendance 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.

9 Contents List of Figures List of Tables xv xix 1 Introduction Background and Motivations Literature Review and the Gap to Fill Elaboration of the Problem The Requirements on the Price Model The Analytical and Mathematical Tools The Analysis Framework The Modeling Methodology Organization of the Thesis A Survey of Electricity Price Models Electricity Price Modeling Approaches Time Series Models The Autoregressive Model AR with Time-varying Mean AR with Exogenous Variables The Autoregressive Moving Average Model ARMA with Time-varying Mean ARMAX and Transfer Function Periodic Autoregressive Models ARIMA and its Extensions ix

10 CONTENTS Regime-switching Models Time-varying Volatility The GARCH Model GARCH with Asymmetric Effect Time-varying Volatility and Model Parameter Calibration Financial/Stochastic Process Models Modeling the Multi-seasonality Modeling the Mean-reverting Price Nature A Basic Electricity Price Model Modeling the Time-varying Volatility Modeling the Multi Risk-factors Modeling the Price Spikes Structural Models The Effect of Fuel Prices Decomposition Techniques Fourier Analysis Wavelet Analysis Principal Component Analysis Concluding Discussions Microeconomics applied to the Electricity Markets The Supply Curve of a Generation Company Heat Rate Curve of the Generation Company Fuel Price Curve of the Generation Company Marginal Cost Curve of the Generation Company Supply Curve of the Generation Company The Supply Curve of an Electricity Market The Demand Curve of an Electricity Market The Spot Price of an Electricity Market The Effect of Fuel Prices on Spot Prices The Effect of Generator Outage on Spot Prices Intra-day Electricity Spot Prices Time-varying Volatility of Intra-day Spot Prices x

11 CONTENTS 3.9 Electricity Spot Price Spikes A Multi-granularity View of Electricity Prices Intra-week Hourly Electricity Spot Prices Intra-year Weekly Prices Multi-year Yearly Prices The Effect of Fuel Prices on Yearly Prices A Multi-granularity View of Electricity Prices A Multi-granularity Electricity Spot Price Model Abstract Introduction The New-England Market Decompose the Data to Find Orders Fuel-Price Effect on Electricity Spot Prices Fuel-price Effect Adjustment Frequency Spectrum of the Data The High-frequency Component The Mid-frequency Component The Low-frequency Component Define the Orders by Models The Intra-week Model The Intra-year Model The Multi-year Model The Fuel Price Model The Overall Model Calibrate the Models The Intra-week Model Calibration The Intra-year Model Calibration The Multi-year Model Calibration The Sampling of the Data and the Time-Unit of Models Simulate Electricity Spot Prices with the Models Use Price Simulations to Valuate Generator Profit Discussion xi

12 CONTENTS 5.10 Conclusion The Multi-granularity Model applied to the PJM Market Introduction The Fundamentals of the PJM Market The Electricity Prices and Fuel Price Index Data Decomposition Modeling Model Calibration The Intra-week Model Calibration The Intra-year Model Calibration The Multi-year Model Calibration Price Simulations Generator Profit Valuation Conclusion A Structural Electricity Spot Price Model Abstract Introduction The Effect of Generator Forced Outage on Spot Prices The Structural Model The Structural Model The Available Generation Capacity The Segment Slopes The Expected Value of the Spot Prices The Simplified Model The Simplified Model The Probability Density Function of the Spot Prices Numerical Examples The Test Generation System The Structural Model The Simplified Model The Computation Burden of the Structural Models Discussion xii

13 CONTENTS 7.8 Conclusion Conclusion Significance of the Work Contributions Future Work on the Price Model Applications to Power System Operation and Planning 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 C.2 Wiener Process with Drift C.3 Geometric Brownian Motion C.4 Mean-reversion Process C.5 Geometric Mean-reversion with Constant Mean C.6 Geometric Mean-reversion with Time-varying Mean D The Price Filters and the Filtering 177 D.1 Design of the Price Filters D.2 Price Decomposition by Filtering E The Mean-reversion Process in Discrete Time 181 F Build the Fuel Price Index 183 F.1 The Fuel Price Index Model F.2 Derivation of the Model for Parameter Estimation F.3 Parameter Estimation References 189 xiii

14 CONTENTS xiv

15 List of Figures 1.1 The Flow-chart of Generator Profit Valuation Distribution of Literatures Index the Intra-day Hourly Spot Prices A Two-regime Markov Chain Intra-day Electricity Price Pattern Intra-week Electricity Price Pattern Intra-year Seasonality of Electricity Prices A Haar Wavelet The Heat-rate Curve of the Generation Company The Fuel Price Curve of the Generation Company The Marginal Cost Curve of the Generation Company The Supply Curve of the Generation Company The Supply Curve of an Electricity Market The Demand Curve of an Electricity Market The Spot Price of an Electricity Market The Effect of Fuel Prices on Spot Prices The Effect of Generator Outage on Spot Prices Intraday Electricity Load Demand Intraday Electricity Spot Prices The Time-varying Volatility of Intra-day Electricity Spot Prices The Marginal Cost Curve with an additional Third Segment Intra-day Electricity Spot Price Spikes Intra-week Hourly Electricity Spot Prices xv

16 LIST OF FIGURES 4.2 Intra-year Weekly Prices Multi-year Yearly Prices The Prices of Fossil Fuels across Many Years The Yearly Prices Considering Fuel Price Fluctuations A Multi-granularity View of Electricity Prices New-England Market Fundamentals Electricity Prices and Natural Gas Prices Fuel-price Effect on Electricity Spot Prices Implied Marginal Generator Heat Rate Data The High-frequency Component Data The Mid-frequency Component Data The Low-frequency Component Data The Overall Electricity Spot Price Model The High-frequency Component and the Intra-week Pattern The Mid-frequency Component and the Intra-year Pattern Seasonality of the New-England Market Calibration of the Multi-year Model Construct the Electricity Spot Prices in the Year Day-ahead Electricity Spot Prices in the Year Empirical Probability Distributions of the Prices in Year The Mid-winter Week in The Mid-spring Week in The Mid-summer Week in The Mid-fall Week in One-year Generator Profit of the Coal Unit and the Gas Unit PJM Market Fundamentals Business-day Intra-day Pattern of Marginal Fuel Mixture Intra-year Seasonal Pattern of Marginal Fuel Mixture Multi-year Pattern of Marginal Fuel Mixture The Prices of the Three Marginal Fuels PJM Market Electricity Prices and Fuel Prices Fuel-price Effect on Electricity Spot Prices xvi

17 LIST OF FIGURES 6.8 Implied Marginal Generator Heat Rate Data The High-frequency Component Data The Mid-frequency Component Data The Low-frequency Component Data The High-frequency Data and the Intra-week Pattern The Mid-frequency Data and the Seasonal Pattern Seasonality of the PJM Market Calibration of the Multi-year Model Day-ahead Hourly Electricity Spot Prices in Year Empirical Probability Distributions of the Prices in Year Mid-winter Week in Mid-spring Week in Mid-summer Week in Mid-fall Week in One-year Generator Profit of the Coal Unit and the Gas Unit The Generation Supply Stack of an Electricity Market The Effect of Generator Forced Outage on Spot Prices The Structural Electricity Spot Price Model The Integration Regions of the Three Cases The Generation Supply Stack of the Test Generation System The Expected Value of the Spot Prices The Standard Deviation and Skewness of the Spot Prices The Errors of the Model The Expected Value of the Spot Prices The Standard Deviation and Skewness of the Spot Prices The Errors of the Model A.1 A Simple Structural Electricity Spot Price Model F.1 Parameter Estimation of the Fuel Price Index Model F.2 The Monthly Intra-year Seasonal Pattern xvii

18 LIST OF FIGURES xviii

19 List of Tables 2.1 Separate the Intra-day Hourly Spot Prices into 24 Time Series Parameters of the Intra-week Model Parameters of the Intra-year Model Parameters of the Multi-year Model Statistics of the Empirical Probability Distributions Characteristics of the Coal Unit and the Gas Unit One-year Generator Profit of the Coal Unit and the Gas Unit Parameters of the Intra-week Model Parameters of the Intra-year Model Parameters of the Multi-year Model Statistics of the Empirical Probability Distributions One-year Generator Profit of the Coal Unit and the Gas Unit The Computation Burden of the Structural Models B.1 Characteristics of the Generators in the Test System D.1 The Price Filters D.2 Electricity Spot Price Decomposition by Filtering F.1 Parameters of the Fuel Price Index Model( ) F.2 Parameters of the Fuel Price Index Model( ) xix

20 LIST OF TABLES xx

21 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 20 th century power systems gained dramatic 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 guarantee 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 generators 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 researchers have developed sophisticated methodologies and models to estimate 1

22 1. INTRODUCTION the cost of generating electricity up to a time horizon of several decades, which are 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 fixed by 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 future profit, refer to Hou & Wu (2008), and see Figure 1.1. Q1 Q5 T 1 Electricity Spot Prices St () t 0 t T1 T2 t Q2 Generator Availability At () Q3 Hourly Profit max St ( ) Ct ( ),0 Q4 Present Value e Rt Summation Total Profit Π T 2 Figure 1.1: The Flow-chart of Generator Profit Valuation Namely, at the present time t 0, the total profit Π for generating electricity during a future time period [T 1, T 2 ] 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 2

23 1.1 Background and Motivations to generate electricity; if the electricity spot prices S(t) is greater than the cost of generating electricity C(t), S(t) > C(t), the generate will produce and earn a profit π(t) = S(t) C(t); and if the electricity spot price is less than the cost of generating electricity, S(t) < C(t), the generator does not operate and earn a zero profit π(t) = 0; the future profit at hour t, π(t), is then discounted to obtain its present value π (t) = e Rt π(t); finally, the profits at each future hour t is summed over the whole period [T 1, T 2 ] to arrive at the total profit during the period, Π = π (t). t [T 1,T 2 ] 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 factors which will significantly affect their future earnings, that is, future electricity 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 3

24 1. INTRODUCTION when the economy falls to recessions. Meanwhile, short-life extreme-high price spikes, 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 understanding 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 Short-term Mid-term Long-term Time-horizon /Intra-week /Intra-year /Multi-year 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 statistical/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 4

25 1.2 Literature Review and the Gap to Fill approach to study power markets. (A detailed discussion on the various modeling methods could be found in Chapter 2.) 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 mathematical 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 the long-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 models electricity spot prices indirectly: it first models the constituent physical underlying 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 is realistic and thus has its potential use in practice, the perquisite is that one has to first model the constituent physical forces satisfactorily. However, the problems of modeling these constituent physical forces themselves are challenging and so far haven t received definitive solutions. Further discussions on the structural approach could be found in Section 2.4. 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 statistical/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 5

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