Fuzzy Set Based Decision Support System for Transactions of Electricity in a Deregulated Environment by Maninder Kaur Dhaliwal B.A.Sc., Punjab Technical University, India, 1998 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 2001 © Maninder Kaur Dhaliwal, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without rhy written permission. i Department of ~<£%Lt£xLrs> I ^ a ^ o / A & ^ j i j u t i ^ "^^^JteJ^A T U n i l _ : : * . . r > _ : * : _ U / - _ | u : _ U The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract In the restructured electricity market environment, the market participants conduct their power transactions with an aim to maximize their profits. Electricity is typically traded either through an auction at an open-access power exchange market or directly between supplier and retailer/consumer through bilateral and/or multilateral contracts. In i the power exchange, the balance of supply and demand determines the spot market price. These transactions are purely economic and are subject to the physical constraints of the transmission system. The transmission grid is controlled by an Independent System Operator (ISO), and information about the system operation is restricted and rarely available to the market entrants. The market players generally receive partial information about the system conditions, such as a forecast of the total demand. Hence, the pivotal information in conducting the spot market transactions of electricity is price and demand. In some cases, a direct bilateral contract between suppliers and consumers can provide an attractive alternative to the spot market pricing, where the price can be volatile due t o strategic behavior of market participants or tight demand-supply balance. The most likely measure of suitability of a bilateral contract is its comparison with the market price. However, the spot market price tends to be significantly volatile. Therefore, suitable methods for representing the volatile market price are needed. The traditional modeling methods are primarily based on statistical and probabilistic approaches, and it may not be entirely suitable to apply these stochastic methods to model the data generated by human activities such as power exchange markets. Besides, most of the 11 existing models are aimed at stock exchanges, and may not necessarily be applicable to the electricity markets. In this research work, a data representation model based on extended fuzzy regression is developed. The model represents the highly volatile demand-price relations as a range and estimates the possible distribution of the price range for a given value of a demand forecast. The highlight of the model is its capability to preserve the aggregate information contained in the original data set such as the uncertainty. The model was tested using actual data from California Power Exchange and results were found to be promising. Based on the proposed model, a procedure for evaluation of bilateral contracts i in an open market environment is developed. iii Table of Contents Abstract ii Table of Contents iv List of Tables vii i List of Figures viii List of Symbols x List of Acronyms xi Acknowledgements xii Chapter 1 Introduction 1 1.1 Background 1 1.2 Objectives 4 1.3 Related Work 5 1.4 Methods Used 6 1.4.1 Electricity Demand-Price Model 6 1.4.2 Consumer Decision Making Aid 7 1.5 Accomplishments 8 1.6 Outline of the Thesis 8 Chapter 2 Electricity Market Structure under Deregulation 10 2.1 Deregulation of the Electric Utility Industry - A Global Overview 10 2.2 Forces behind Electric Utility Restructuring 13 2.2.1 Conditions prior to Deregulation 13 2.2.2 The Public Utilities Regulatory Policies Act of 1978 14 2.2.3 The Development of Gas turbine Technology 15 2.3 Electric Utility Restructuring in California 15 2.3.1 The Deregulation Act of 1996 15 2.3.2 California Market Structure 17 2.3.2.1 The Power Exchange 19 2.3.2.2 California Independent System Operator 21 2.3.3 Revamping of California's Wholesale Electricity Markets 24 iv Chapter 3 Electricity Demand-Price Model 27 3.1 Relation between the Demand and Price of Electricity 27 3.1.1 Regression Analysis 28 3.1.2 Fuzzy Model to Combine Data Clusters 32 3.2 Fuzzy Regression to represent Demand-Price Relation 34 3.2.1 Fuzzy Regression Model 1 35 3.2.2 Optimization to Fit the Possible Range , 37 3.2.3 Data Pre-conditioning 40 3.2.4 Fuzzy Models on Pre-conditioned data 42 3.2.5 Fitness of the Composite Fuzzy Regression Model 44 3.2.6 Forecasting through Fuzzy Regression Model 45 3.3 Clustering on Demand-Price data ! 51 3.3.1 Clustering Algorithms 51 3.3.1.1 K-means Clustering • 51 3.3.1.2 Fuzzy C-means Clustering 54 3.3.1.3 K-means and FCM with User-defined Initial Points 57 3.3.2 Membership Function by FCM Clustering (60 3.3.3 Fuzzy Regression Model based on FCM Clustering 62 3.3.4 Forecasting through the FCM based Fuzzy Regression Model • 64 i 3.4 Summary 66 Chapter 4 Consumer Decision-Making Aid 68 4.1 Formulation of the Evaluation Process 69 4.1.1 Consumer Preference Index 70 i 4.1.2 Possible Range of Market Prices 72 4.1.3 Composite Evaluation Index 73 4.2 Evaluation of a Bilateral Contract 75 4.3 Summary 79 Chapter 5 Conclusion and Future Work 81 5.1 Conclusions 81 5.1.1 Electricity Demand-Price Model 81 5.1.2 Consumer Decision Making Support 82 5.2 FutureWork 82 5.2.1 Data Pre-conditioning 82 i 5.2.2 Focussing the Range of the Model 5.2.3 Congestion 5.2.4 Consumer Preference Indices 83 83 84 Chapter 6 Bibliography 85 Appendix I Fuzzy Set Theory 92 Appendix II Fuzzy Regression Model Area 95 Appendix III a-cuts of Fuzzy Sets 100 Appendix IV Transmission Congestion 103 vi List of Tables Table 1 Parameters for low demand model 42 Table 2 Parameters for high demand model 43 Table 3 Measure of Volatility 44 Table 4 Comparison of Fitness 45 Table 5 Parameters for low demand model 62 Table 6 Parameters for medium demand model 63 Table 7 Parameters for high demand model 63 Table 8 Fitness of the FCM based Model 64 Table 9 Consumer Preference 77 Table 10 Performance evaluation (y) 78 vii List of Figures Figure 1 Status of Electric Industry Restructuring in the US 11 Figure 2 California Electricity Market Structure 18 Figure 3 Clearing process of a PX market 20 Figure 4 ISO Service Area in California 22 Figure 5 Market Data of CalPX 29 Figure 6 Regression applied to California Demand-Price data 30 Figure 7 The Regression lines 31 Figure 8 Fuzzy Sets to represent Low and High Demand 32 Figure 9 Composite Regression Curve 34 Figure 10 Triangular Fuzzy Set 36 Figure 11 Fuzzy Demand-Price relation 39 Figure 12 Time-Series for Daily Demand-Price for California 41 Figure 13 The Demand Data Clusters 42 Figure 14 Composite Fuzzy Model on Pre-conditioned Data 43 Figure 15 Comparison of 1998 Model and 1999 Data 46 Figure 16 The Error Rate 46 Figure 17 Apportion of the Error Rate 47 Figure 18 Non-Preconditioned Data Sets 48 Figure 19 Time-Series Estimation 49 Figure 20 Time-Series of Price Estimation for 1999-2000 50 Figure 21 K-means Clustering on CalPX data set 53 viii Figure 22 Fuzzy C-means Clustering on CalPX data set 56 Figure 23 K-means with Cluster Center Initialization 58 Figure 24 FCM with Cluster Center Initialization 59 Figure 25 Membership Function for 3 Demand Clusters 61 Figure 26 Polynomial Approximation 61 Figure 27 Fuzzy Set Generation 62 Figure 28 Fuzzy Regression Model 63 Figure 29 Comparison of 1998 FCM Model and 1999 Data 65 Figure 30 The Error Rate 66 Figure 31 A Fuzzy Set to represent Consumer's Preference 71 Figure 32 Measurement of Price Differential against Preference 74 Figure 33 Estimated Range of Market Price (August 1999) 76 Figure 34 Performance Index for a Retail Supply Offer (August 1999)77 Figure 35 Estimated Range of Market Price 78 Figure 36 Performance Index for a Retail Supply Offer 79 I X List of Symbols p market clearing price m order of model membership of the fuzzy sets of demand D L lower demand threshold D H upper demand threshold M fuzzy number ai parameter of triangular fuzzy set (see Figure 10) + ci parameter of triangular fuzzy set (see Figure 10) ci parameter of triangular fuzzy set (see Figure 10) Y(x) fuzzy number representing degree of possibility Ci center of cluster i M membership of point j in cluster i U membership matrix for cluster groups d.. u Euclidean distance between point j and the cluster center c, middle demand threshold Ap price differential Pm market price at time t off p offered price for a bilateral contract G trapezoidal fuzzy set representing the consumer's price goals So parameter of trapezoidal fuzzy set (see Figure 31) s, parameter of trapezoidal fuzzy set (see Figure 31) membership of fuzzy set G D(k) fuzzy number representing price differential at time k P (k) fuzzy number representing the possible range of market price at time k membership of the overlapping of G and D Y overall performance evaluation for a bilateral contract X List of Acronyms CAISO California Independent System Operator CalPX California Power Exchange CPUC California Public Utilities Commission DWP Department of Water and Power, Los Angeles EPA The Energy Policy Act FERC Federal Electric Regulation Commission IOU Investor Owned utility ISO Independent System operator NUG Non-Utility Generator PG&E Pacific Gas and Electric Company PJM Pennsylvania-New Jersey-Maryland PURPA Public Utility Regulatory Act PX Power Exchange QF Qualifying Facility SC Scheduling Coordinator SCE Southern California Edison SDG&E San Diego Gas and Electric TSK Takagi-Sugeno-Kang VIU Vertically Integrated Utility FCM Fuzzy C-means Clustering xi Acknowledgements I am profoundly indebted to my supervisor, Dr. Tak Niimura, for his technical and financial support. This work would not have been possible without his able guidance. I would like to thank the past and present members of UBC Power Group for their friendship and assistance. I n particular, I w ould 1 ike t o e xpress m y gratitude t owards Tomoaki Nakashima for his earlier work on the model; Erica Roedern for her understanding and concern; Daniel Lindenmeyer, Khosro Kabiri, Mazana Lukic, Benidito Bonatto and Kenneth Wicks for their pragmatic and rewarding discussions. I have learnt a lot from my association with these people. I am also deeply obligated to Dr. Prabha Kundur for his moral support. Finally I would like to profess my gratefulness towards my family, especially my father who has always been a role model, and whose cheques make life easier. X l l Chapter 1 Introduction Electric utilities have traditionally operated as regional monopolies, with a varying degree of regulation from the government. But since the end of the 1980s, over 30 countries have either implemented or initiated the deregulation of their electric utility industry. The main idea behind the electric utility restructuring is to introduce an open market-based competition for liberalized electricity transactions. For this purpose, open-access 'power exchange' markets have been established, where electricity is traded by an auction as a commodity, and the prices are determined by the balance of supply a nd demand. However, the free market structure creates excessive fluctuation of electricity prices, as seen widely at stock exchange and currency exchange markets. Therefore, a new tool is strongly needed to grasp the uncertain situation in the new market structure. In this chapter, first the background concerning this research work is explained, followed by the objectives of the research work. Then the methods used for the study are discussed, subsequently followed by the accomplishments of the work and the outline of the thesis. 1.1 Background Conventionally, the electric power was supplied by vertically integrated utilities (VIUs) where all tasks were coordinated jointly under one umbrella with one common goal, that is, to minimize the total costs of operating the utility. In the process of restructuring the vertically integrated structures are divided into several autonomous 1 entities according to function (generation, transmission, distribution, retail and services). The production assets are fragmented and sold to independent parties in order to promote competition in the supply sector. Also the incumbent rights/obligations are removed to facilitate the entry of new participants into the industry. Australia, Chile, and Britain are recognized as pioneers in opening their electricity industries to competition. In the US, more than half of the states are in different stages of implementing electric utility restructuring, with California being one of the first states to break ground in the process. In Canada, the. provinces of Alberta and Ontario have adopted complete deregulation package, while in British Columbia minimal structural changes were made to ensure a participation in American electricity markets. Since the early 1990s, a series of legislations have led to the breaking up of the traditional utility structure and creation of open energy markets. Under the process of restructuring the electric utility industry, new institutions are created to coordinate commercial and technical activities in the new structure: an energy market and a power system operator. The energy market called the power exchange (PX) is a stock-exchange-like market for electricity and the price is decided through a bid and auction process. The technical operation of the entire system is independent of the economic interest of the parties involved and is controlled by a Independent System operator (ISO) who is also responsible for maintaining the physical integrity of the power system. The PX provides a spot market for electricity and is basically modeled on the commodity markets. Buyers and sellers participate in an auction where they provide information on the prices and quantities of electricity they are willing to buy or sell. The total demand is tallied up and matched with least expensive offers. The most expensive 2 offer to be retained determines the sale price for the entire lot being traded in that auction. Hence, in the PX the electricity prices are established as a function of supply, demand and production cost information provided by the market participants. There are many electricity markets in operation for different time frames relative to the real-time operation. During periods of high demand, the high-priced peaking generators supply the last unit of energy, and the price of electricity jumps. Hence, the spot price of electricity tends to be volatile. The bilateral and multilateral contracts provide an alternative to the volatile spot market pricing of the power exchange. They offer a stable way of buying or selling electricity for the risk-averse market participants. Under these contracts the customer can buy power for a period of time at a fixed price. Many different services are necessary to physically bring the retail power to the consumer including transmission and distribution; and to maintain its quality and reliability which constitute the ancillary services. Each of these services require a separate market, and some are deregulated while others are monopolies. The transmission grid is controlled by the ISO, while the distribution system is usually owned by the erstwhile utilities. The ISO may also be responsible for the ancillary services. In the restructured electricity markets, the market participants enter bids for electricity contracts in the power exchange, and the market prices are determined by an auction process. Hence, a proper assessment of the energy price is of prime importance to the deregulated electricity industry. Long-term forecast of the electricity price is crucial for decisions regarding transmission expansion, generation augmentation, distribution 3 and planning. While short-term price prediction is required to assist the market players in effective decision making. 1.2 Objectives The objective of this research work is to provide a model to represent the relation between t he electricity p rice a nd d emand in a p ower exchange market. In the power exchange, the market participants enter bids for electricity and the market prices are determined by the balance of supply and demand. The participant information is limited to the final spot price of the auction and system conditions such as a forecast of the total market demand. The first aim is to develop a price forecasting model with a requisite information of electricity demand, while accounting for the uncertainty related to other factors in the system which are either exclusive or indeterminate. A direct bilateral contract between suppliers and consumers can provide an attractive alternative to the volatile spot market pricing. The most probable measure of suitability of a bilateral contract is its comparison with the market price. But the price level acceptable to a consumer can be quite uncertain. The second section of the research aims at developing a procedure to aid the evaluation of a bilateral contract under uncertain market conditions. In the restructured electricity system, the study of the economic aspect is very important not only because of the prospect of tremendous profits, but also as revenue is a dictating factor in the development of new infrastructure such as power plants and transmission lines that could alter the whole picture of the power system operation. 4 1.3 Related Work Sophisticated models to simulate the system behavior and market operations for long-term price estimation have been developed in [1]. Short-term forecast using time-series and stochastic models is developed in [2], and using artificial neural networks is proposed in [ 3] a nd [ 4]. F uzzy regression a pproach ha s b een us ed for e lectrical 1 oad forecasting in [5], [6] and [7]. The application of TSK fuzzy system is explored in [8]. Although data mining is recognized as a key research topic in database system management, in the past its usage had been primarily limited to research fields of commerce and economics[9],[10]. However, with the electric utility restructuring, at present there is an increasing interest to apply data mining techniques to the domain of power s ystemsfl 1]. C lustering ha s b een e xplored in s ome p ower system data mining applications, such as, for load profiling in [12], for power system security in [13] and [14], for power system planning and operation in [15]. A generic method for the use of data mining to develop a fuzzy model for a system involving uncertainty has been explored in [16]. However there is no work specifically targeting the modeling of an electricity market through the demand-price relation. Previously reported studies in the electricity pricing area are mostly focused on the strategic decisions of suppliers in the spot market such as in [17]-[19]. Some strategies for bilateral contracts are analyzed[20],[21], but these early works are apparently targeting wholesale suppliers and distributors. Also, mathematical models representing the consumer "preference" are in an unexplored area, and conventional approaches based on stochastic models [22]are in many cases impractical. 5 1.4 Methods Used A fuzzy regression model is proposed to estimate the possible electricity price range for a given a demand value and later is used to develop a procedure for evaluating bilateral contracts[23],[24]. 1.4.1 Electricity Demand-Price Model First, a fuzzy regression method is introduced to determine the relationship between electricity demand and price. The conventional regression is applied in order to identify the structure of the model. As the application of regression analysis gives the most likely relation between given demand and price values, it aims at converging the functional relationship to a single line or curve. Different regression curves are used to represent different demand regions and then TSK-fuzzy model is constructed to join these demand these regression curves. Then, data c lustering t echniques a re us ed t o find t he na tural s ubgroups in t he market data. And the model is fine-tuned by replacing the arbitrary overlap demand regions by the ones derived from the inherent attribute characteristics. Next, the regression parameters are expanded to fuzzy numbers to encompass a wide range of possible data with varying degrees of possibility. The mean value of the fuzzy parameter is the one given by the conventional regression analysis and the spread is determined by linear optimization to encompass most of the actual demand-price data. 6 The resultant composite fuzzy regression model estimates the possible price range when a demand forecast is given. 1.4.2 Consumer Decision Making Aid Here, a procedure to help consumers evaluate an offered bilateral contract under uncertain market conditions is developed. The uncertain market prices are represented by the fuzzy regression model proposed in the earlier section. This extended regression model represents the market price by a fuzzy number with the most likely price as a mean value, and an interval encompassing the range of possible highest and lowest prices with varying degrees of possibility. The uncertain preference of the customer for a desirable bilateral price range with degrees of satisfaction is represented by a separate fuzzy number. The value of the bilateral transaction is then examined by a fuzzy decision involving the possible market price range, the bilateral price offered, and the consumer's preferred price range. The most likely and satisfactory performance of the bilateral contract transaction is measured by the intersection of the two fuzzy sets representing the market price and the consumer preference. By aggregating the composite fuzzy index for the entire time frame of the contract period, the most desirable bilateral contract for that consumer c ould b e found. This approach can be used for consumer decision making support when the consumer preference is uncertain. 7 1.5 Accomplishments In this research work a forecasting model based on fuzzy regression has been developed. This model can be used for predicting the electricity price based on power exchange market demand. The proposed fuzzy model is capable of representing the highly volatile relation between demand and price of electricity in the power exchange environment. Also, the rich information contained in the original data set, such as, uncertainty is preserved in the model. While in the conventional regression analysis the data set that is represented by a single curve, the model based on fuzzy regression can give the demand-price relations as a range with possibility distribution of prices, when a single value of demand is given. Based on the proposed model a procedure has been developed to assist an electricity customer to make decisions on bilateral power transaction under uncertain market conditions. This procedure can also be used by the electricity suppliers and/or brokers to determine an attractive price to offer to a target customer when the consumer preference is uncertain. 1.6 Outline of the Thesis The next chapter deals with the organization of restructured electricity markets and electric utility deregulation with special emphasis on California. 8 In Chapter 3, a electricity price forecasting model based on extended fuzzy regression is proposed and tested. In Chapter 4, the above-mentioned model is used to evaluate bilateral contracts in open market conditions. In Chapter 5, the conclusions of this study are discussed and possible future work is proposed. 9 Chapter 2 Electricity Market Structure under Deregulation Electric utility industry around the world is going through unprecedented restructuring. The traditionally vertically integrated structures are being broken down into separate t asks o f generation, transmission, and distribution, and these tasks are being opened to competition at different levels. 2.1 Deregulation of Electric Utility Industry - A Global Overview Deregulation of electricity utility intends to reduce the price of electricity, while maintaining reliability of service by introducing competitive marketing forces[25]. In the restructuring process, the major focus is on the deregulation of the price regulation of generation. Generation utilities will no longer have ownership or control of transmission and/or distribution facilities. This process is called 'unbundling'. In most cases of deregulation, transmission and distribution remain regulated monopolies while generation and marketing become competitive activities. Countries like Australia, Chile, and Britain are at the forefront of electricity deregulation. In U.K., the electricity industry reorganization and restructuring began in 1989, and entailed the privatization of the industry and setting up of a wholesale electricity market[26]. The European Commission introduced its proposal for a directive on common rules for the internal electricity market in 1992[27], which tends to phase in gradually into a single energy market covering all of the European Union. In Canada, 10 Alberta is leading the way into restructuring process, while the provinces of Ontario and British Columbia are actively considering the adoption of a restructuring plans. In U.S., twenty-four states and the District of Columbia have instituted electricity deregulation, as of April 2001 [28]. California became the first state to completely end utility monopolies and allow all customers to buy electric power from the cheapest supplier. Many others have either enacted utility deregulation laws or have them pending, a few have active pilot programs or direct access programs. Figure 1 shows a map of the U.S. with the states grouped into five different stages of electric utility restructuring[29]. Figure 1 : Status of Electric Industry Restructuring in the US. 11 The basis for deregulation was laid by the enactment of the Public Utility Regulatory Act (PURPA) in 1978, which focused on introducing competition at the generation stage of production. This act made provision for co-generation of energy by Non Utility Generated (NUG) power producers, also called qualifying facilities (QF)[25]. However at that time, these power producers were not allowed to sell the power directly to the consumers. The Energy Policy Act (EPAct) passed by the US Congress in 1992 took another major step by mandating the opening of the transmission system for wholesale generators[30]. Later on, the retail market deregulation was initiated by Open Access Order No. 888, issued by the Federal Electric Regulation Commission (FERC) in April 1996[31]. This required the electric utilities to open their transmission systems to the power generated by other companies. The idea behind this was four-fold : split up old power monopolies and open the market to healthy competition, create technology that would make electricity cheaper to generate, offer customer choice, and ultimately bring lower prices to consumers. Although the deregulation law was enacted by the FERC, states have discretion in regulatory policies. Deregulation has brought changes in the traditional organization of electricity markets. These changes involve the creation of power exchange (PX), stock-exchange-like markets for the sale of electricity, and the treatment of transmission and distribution lines as common carriers whose function is to ship the power determined to be cheapest by the activity of the stock-exchange like auction market. To maintain the physical integrity of the unbundled power system, there is an Independent System Operator (ISO). The ISO is a nonprofit organization set under the deregulation process to control the state's power grid. It is supervised by the FERC. 12 While the ISO tends to protect the market players, the legislations have provisions for consumer protection, for example, the rate cap. State of Pennsylvania has a provision called 'provider of last resort'[32]. This is to ensure that every customer has at least one electricity provider. 2.2 Forces behind Electric Utility Restructuring The transformation of electric utilities from regulation to competition has evolved over a period of time, with changes in government policies, advancements in technology, and belief in the benefits of competition creating a climate conducive to the process. 2.2.1 Conditions prior to Deregulation Before 1995, the electricity p rice in C alifornia and the No rtheastern U .S. w as higher than national average because of various reasons, such as expensive nuclear power plants. A big push for the creation of a 'deregulated generation market' came from large industrial customers in the Northeast and California, who did not want to pay vertically integrated traditional utilities for their expensive electricity. The big industries had electricity as a big portion of their total production cost. The high-tech, aerospace and defense factories in California found the high electricity prices a major competitive disadvantage[32]. In the past, five other major industries have been opened to competition: interstate trucking, railroads, long-distance telephone, natural gas, and airlines[33]. All these had 13 positive experiences when government-regulated monopolies were abolished, and the free market was allowed to work. As a result of market competition, prices decreased dramatically, reliability and customer service improved, jobs were created, and new technologies emerged. The striking success of these endeavors is offered as a reason for optimism about abolishing regulation in electric utility industry. 2.2.2 The Public Utilities Regulatory Policies Act of 1978 One of the policy responses in the US to the energy crises of the 1970s, was 'The Public Utilities Regulatory Policies Act of 1978', which opened the door for low cost alternatives[25]. The initial result of PURPA, however, was not low-cost alternatives but more high-cost obligations for the utilities. An important provision of the law required electric utilities to purchase power generated by independent producers at a price equal to what the regulated utility had to incur if it had generated the same amount of electricity. The purpose of that provision was to stimulate the development of alternatives to the traditional fossil-fuel steam electric generation, that was expected to become terribly expensive. But the fossil fuel prices dropped in the mid-1980s, and hence there was a mismatch between prices paid to independents and the market price for wholesale power. Therefore, the short-term effect of PURPA was to saddle the utilities, primarily in New York and California, with costly obligations that compounded their nuclear-power cost woes. 14 2.2.3 The Development of Gas turbine Technology The gas turbine is essentially a jet engine modified to produce electricity. These generating units have numerous advantages including smaller size and ease of installation. Hence, large users of electricity c ould self-generate and sell their surplus power under provisions of PURPA. Moreover the new advancements in small gas-turbine and co-generation technology, reached efficiency levels that exceeded those of conventional power stations. These developments in policy and technology turned the economics of traditional utilities upside down and led to the transformation of electricity from regulation to competition. 2.3 Electric Utility Restructuring in California California became the first state in the U.S. to completely end utility monopolies and allow all customers to buy electric power from the cheapest supplier, exposing the best and the worst of what electric utility deregulation has to offer. This is the reason why California has attracted such a lot of attention. 2.3.1 The Deregulation Act of 1996 Before the deregulation of the electric industry in C alifornia, t here w ere t hree major monopolies controlling the market : Pacific Gas and Electric Company (PG&E), 15 South C alifornia E dison (SCE), a nd S an D iego Ga s a nd Electricity (SDG&E). These companies gave big industrial consumers discounts in rates, while small and medium size consumers were paying 50 percent higher electricity rates than in other states[34]. The whole point of deregulation was to break up these monopolies and providing for competition in the market place, to give consumers a choice of an energy supplier, and hence reduce the cost of electricity. California's deregulation bill, passed in August 1996, which opened up the sale of electricity to competition and dismantled government-regulated monopoly, required the investor owned utilities (IOUs), such as PG&E, SCE, and SDG&E, to sell their generating plants and become buyers of wholesale power. The only e xception to this provision were the utilities owned by municipalities, such as, Department of Water and Power (DWP) of Los Angeles, which could still own their generation facilities[35]. The utilities were also required to transfer operational control of transmission lines and power grids to a private nonprofit organization - the Independent System Operator (ISO), that would manage the system. The utilities still retained control and ownership of distribution systems. The California Public Utilities Commission (CPUC) transferred pricing of electricity to the California Power Exchange (CalPX), which was overseen by the Federal Energy Commission (FERC)[36]. One of the key points in the legislation was the provision for stranded cost recovery, which granted the electric utilities an amount of $28 billion in stranded asset payments from the customers, to subsidize the regulation-induced cost purchases made by the utilities[37]. To achieve this objective, the electricity prices for consumers were frozen until March 2002 or when the utility firms pay off their power plant debts, 16 whichever happens first. Therefore, only after the debt to the state was repaid could the utilities begin to charge its customers the market prices. This state-mandated rate freeze protected the consumers from sudden price increases, and the utilities were ensured that their pre-regulation debt, which was incurred due to state-mandated outlays, was to be paid. 2.3.2 California Market Structure The. restructuring process brought about major changes in the operational structure of the California's electricity system. In order to separate market activity from the control of the power system, and hence to address concerns about market power and conflicts of interest, t wo inde pendent no n-profit e ntities w ere created : t he C alifornia Power Exchange (CalPX), a nd the California I ndependent System Operator (CAISO). Figure 2 shows the structure of California electricity market. The utilities are primarily in the business of distributing the wholesale power that they bought in the California Power Exchange (CalPX), and the ISO would operate the transmission system owned by these utilities as a single control area. Hence, the restructured industry had two main segments : a regulated transmission and distribution segment, and a competitive power sales segment[38]. Market players can buy or sell power directly through bilateral contracts as well as by entering bids in the auction process at the PX. For the risk-averse consumers who don't want to the play spot market in the PX, the bilateral or multilateral contracts provide a more stable way of buying or selling electricity. These contracts may include 17 Figure 2 : California Electricity Market Structure 'forward contracts' in which the customer can buy power at a fixed price (or tied to an index), for a fixed period of time, or 'futures contract', in which the right to take delivery of electricity can be bought or sold. After the transactions are decided upon, a schedule is prepared, which is primarily a statement of demand (including quantity, duration a nd take-out points), generation (including quantity, duration, location of generating unit and transmission losses), and ancillary services (if self provided)[39]. The energy schedule is considered balanced when scheduled demand equals scheduled supply. The schedules for bilateral contracts are aggregated by CAISO certified scheduling coordinators (SC). The initial schedule, also called the preferred schedule, is submitted to the CAISO by the PX and the SC, for the use of transmission system. 18 2.3.2.1 The Power Exchange Under the restructuring process, the California Power Exchange (CalPX), a state chartered, non-profit corporation, was created on April 1,1998, to run a California spot market for power[40]. It was created to be independent of the grid operations and other market participants, and was to provide a forward market for energy through bid and auction process. The PX was a financial market, whose participants included both generators and consumers. It handled power transactions based on economic dispatch, and ignoring physical constraints of the system. In the PX bid and auction process, the aim is to match the supply bids of the generators to the demand bids of the consumers. A bid in the PX indicate the quantity of energy the participant wishes to buy or sell and, if relevant, the maximum price that is buyer is willing to pay, or the minimum price that the supplier will accept. Each supplier bid is in the form of a price curve, a non-decreasing curve representing the suppliers' marginal cost. When these bids are aggregated, they form the supply curve as shown in Figure 3, which shows the quantity of power that would be sold at various market prices. On the other hand, each buyer bid is in the form a non-increasing curve, and when these buyer bids are aggregated, they form the demand curve as shown in Figure 3, which shows the quantity of power that would be purchased at various market prices[41]. The demand and the price curves intersect at point B, where the total demand and total supply are balanced. The market is considered to be in equilibrium at this point, and the corresponding price P, is considered to be the 'market clearing price'. Hence, at the market clearing price all demand prepared to pay at this 19 price has been satisfied and all supply prepared to operate at or below this price has been purchased. Price Supply Demand ~" — — _ N s s Market Clearing Price s \ \ s N N N < B p Volume of Electricity Figure 3 : Clearing process of a PX market. CalPX operated three electricity markets for different time frames relative to the real-time operation. In the 'Day-Ahead' market, energy was traded on an hourly basis, and prices and quantity of electricity for delivery during each hour of the following day was established. The 'Day-Of/Hour-Ahead' market operated in the similar way but offered trading closer to the delivery hour. This market originally began as the 'Hour-Ahead' market but was reconfigured to a 'Day-Of market to accommodate participants. 'Block Forwards Trading' offered standardized contracts for energy on forward basis up to six months in advance of delivery[3 9]. The P X model in California differs from the pool model found in U K , where the responsibilities of a power exchange are performed by the ISO. In the pool model, all 20 sellers and buyers interact with the pool, where the most efficient source is dispatched, subject to system constraints, based on the price and quantity information supplied by the sellers and buyers[43]. Whereas in the California PX system, the participants can interact with each other directly, outside of the PX. The transactions are decided purely on economic grounds completely ignoring any physical constraints of the transmission system. Then the schedules are coordinated by the ISO, a separate authority from the PX. The California Power Exchange ceased operations on January 31, 2001 [44]. Details are discussed in section 2.3.3. 2.3.2.2 California Independent System Operator In the California system, the ISO is responsible for neutrally administering the processes of scheduling and dispatching plants, and operating the transmission system in real time, subject to protocols established for system security and tariffs for the use of the system[45]. CAISO runs the electricity system for about 75 percent of the state. In the rest of California the transmission grid is not operationally controlled by the ISO, but is run by utilities owned by municipalities. Figure 4 shows the CAISO control area as of January 2001 [46]. CAISO m anages e lectricity flows o n t he b ulk t ransmission s ystem t aking int o account the transmission constraints and is responsible for system stability. CAISO also to ensure that there are sufficient ancillary services, such as, operating reserves (both spinning and non-spinning), black-start capability, and regulation[38]. Another of CAISO's responsibilities is the management of congestion in the power transmission 21 system. For fulfilling all these purposes, the CAISO has the authority to call on and call off generating units, regardless of the contractual commitments between buyers and sellers. Figure 4 : ISO Service Area in California. The scheduling process of the CAISO is divided into two time frames : the day-ahead schedule, to be completed approximately 5-10 hours before the beginning of the operating day (12 a.m.), and the hour-ahead schedule, to be completed approximately one 22 hour prior to each hour within the operating day. The CAISO scheduling process is discussed in detail in [47]. Near to the beginning of the day before the next operating day, the CAISO evaluates the balanced energy transaction schedules of the participants, and determines whether all of the submitted schedules can be accommodated on the transmission system. If there is congestion on the transmission system, the scheduling parties are informed, and they have the option of either rescheduling according to their adjustment bids, or using the congested lines by paying congestion fees based on difference between locational marginal costs. The day-ahead schedule is arrived at after one iteration between market participants and the CAISO. The CAISO is also responsible for the real-time balancing of system-wide generation with total load of the control area. If an imbalance occurs, it is considered a deviation in real time from the last schedule accepted, and it is adjusted for actual loss responsibility and real time congestion[38]. In other words, the erring parties have to pay the cost of purchasing the balancing energy from the real-time imbalance energy auction operated by the CAISO. In this process, the CAISO chooses from offers to buy and sell additional energy to maintain system balance. During each ten-minute interval, the CAISO pays the highest accepted sell bid to all generators dispatched for incremental energy, and charges the lowest accepted buy bid to all generators selected for decremental energy. The hourly price is the average of these ten-minute interval prices, weighed by the energy bought or sold each price. CAISO also reserves the right to impose a price cap in its real-time imbalance market, to limit the market power of the generators, during periods of high demand. 23 If a system emergency occurs, the CAISO has the authority to order on or off any generator or load connected to the grid and adjust any schedules into or out of the control area to maintain or restore reliable, stable system conditions. The CAISO is considered a de-centralized type of ISO and differs from the highly centralized ISO found in the Pennsylvania-New Jersey-Maryland (PJM) market, in its authority. PJMISO has full control o ver t he ne twork, a s it a lso f unctions a s a p ower exchange in addition to coordinating transmission services. In this system, market generators and customers submit their price information, without any preferred schedules to the ISO. Then the ISO optimizes both generation and transmission markets simultaneously subject to system constraints, and the market participants follow the schedules determined by the ISO[48]. Whereas in the case of CAISO the preferred schedules are determined by market participants, outside of the ISO, by free market action. The ISO tries to accommodate these preferred schedules, and manages only the operation of transmission grid. The ISO intervenes in energy markets only if congestion occurs and in this situation can readjust schedules only if necessary. 2.3.3 Revamping of California's Wholesale Electricity Markets Under the 1996 deregulation measure, the investor-owned utilities (IOUs) were required to sell most of their generation assets and become buyers of wholesale power. While these utilities bought their power in open market and paid market prices, t hey could not pass any price increases to their customers because of the retail rate-freeze. Moreover the IOUs were prohibited from entering any long-term contracts and had to 24 rely on the spot market, where the prices tend to be particularly volatile. Also, no significant generation ort ransmission w as a dded in C alifornia, while the demand for electricity grew steadily. This caused the balance between supply and demand to be tight during peak hours, and provided the sellers with the means to exercise their market power. Hence, the IOUs were buying wholesale electricity for much higher prices than that they sold for retail[49][50]. On December 15, 2000, FERC issued 'Docket Nos. EL00-95-000 et al', to change some of the wholesale market rules that arose from the original state restructuring[51]. The major remedial measure was to eliminate the requirement of CalPX being the mandatory exchange for the IOUs. In order to release the peak load from exposure to the spot market, t he IOUs w ere p ermitted torn ove t he p urchase oft heir p ower ne eds t o bilateral long-term contracts. The order also required setting up of $150 as a breakpoint ('price cap'), in the auction process of all the Californian electricity markets. The suppliers may submit bids above this level but cannot set the market-clearing price. Now the energy auction in real-time market of CAISO operates under a cap of $150[52]. Sellers bidding at or below this breakpoint receive the market clearing price but not more than $ 150 per MW. However, CalPX suspended trading in its Day-Ahead and Day-Of markets, citing its inability to satisfy the auction restrictions associated with the implementation of $150 breakpoint [44]. The last day of trading was January 31, 2001. The regulators consider Californian deregulation as a work in progress, and going through a transition period. They are trying to learn from successful deregulation efforts 25 in states like Pennsylvania and Maryland, where thriving, mature wholesale markets considered to be the main ingredient in the success of the restructuring process[53]. 26 Chapter 3 Electricity Demand-Price Model In the restructured electricity markets, the market participants conduct the power transactions with an aim to maximize their economic gain. In the power exchange, the suppliers and consumers enter bids for electricity contracts, and the market prices are determined by an auction process. Hence, price prediction can be vital for the participants competing in the power exchange. However, t he un regulated m arket s tructure c reates excessive fluctuation in electricity prices, as observed in stock-exchange and currency-exchange markets. Therefore, a meaningful representation of the electricity price can be crucial in assisting the market players in effective decision making. Stock-price models exist, but they may not necessarily be applicable to the electricity markets[20]. 3.1 Relation between the Demand and Price of Electricity In the conventional electricity industry structure, the utility companies were monopolies, and they determined the electricity prices largely based on their costs and certain tariffs. In the free market structure, however, the generators are free to set the price of electricity. Hence, in the power exchange markets electricity is traded as a commodity, and the spot price of the electricity in the market is determined by the balance of supply and demand. Keeping aside the strategic participant behavior factors like speculation and gaming which cause volatility in the electricity market; and transmission network elements like available transfer capacity, information about which is not readily 27 available, it is the demand of electricity at a given time, which acts as the key factor in determining the electricity price at that particular time. Hence, the relationship between the demand and price of electricity plays a major role in the method for prediction of the electricity price. 3.1.1 Regression Analysis Regression analysis can be used to represent the functional relationship between two sets of data, having some degree of correlation. In regression analysis, the relationship between parameters x and y is represented by the fol lowing polynomial equation: y = aQ +axx + a2x2 + + cimxm (1) Coefficients ao, ..., am are determined so that the distance (or error) between an observed data point and the corresponding point on the polynomial wi l l be minimal. Higher orders o f the parameter m can be used to represent nonlinear relationship. For the demand-price model development and analysis, market data from the California Power Exchange was used[54]. The data set for electricity demand and price in California for the period of Apr i l , 1998 to March, 1999 is shown in Figure 5. The employed data set consists of the hourly electricity demand in M W h and the unconstrained market clearing price in U S dollars per M W h of the demand. The unconstrained price denotes the price determined by the auction process of the P X . It is a 2 8 price based purely on economic decision making and is subject to physical constraints of the transmission system. Hence, we can assume that the unconstrained price represents the original intention of the market participants to maximize their economic gain. 200 _ 150 1 =- 100 8 '£ 50 0 10000 20000 30000 Demand [ M W h ] 40000 Figure 5 : Market Data of CalPX. Application of regression analysis on this data set gives the most likely relationship between any given demand and price values. Figure 6 shows the results of regression analysis using (a) linear and (b) nonlinear (second-order) polynomials applied to the California Power Exchange data. It is observable that fitting a polynomial to a given data set aims at converging to a single line or curve. In the given demand price data, there is a large cluster of data in the low demand area where the prices that shoot up out of the cluster does not heavily affect the determination of the regression curve. It can be interpreted that the linear regression line in Figure 6 (a), is representing the marginal cost of the system. However, as the demand 29 increases the trend to push up the price becomes apparent, and there are many points where prices reach several times more than the nominal marginal cost line. 200 , 150 100 10000 20000 30000 40000 Demand [MWh] (a) Linear 200 150 -10000 20000 30000 40000 Demand [MWh] (b) Non-linear (Quadratic) Figure 6 : Regression applied to California Demand-Price data. The reasons for this consistent trend could be explained with the fact that, as the demand goes higher, not only the chances of the high cost generators setting the market clearing price increase, but also as the supply-demand balance gets tighter, the generators get to dictate the terms in the bidding process and hence can inflate prices by exercising their market power. The non-linear regression line in Figure 6 (b), follows this upward trend to a certain extent, but is unable to completely represent the volatility of market prices under conditions of high demand. Figure 7 shows simultaneous linear and quadratic regression lines on the data set. 200 r — i 10000 20000 30000 40000 Demand [MWh] Figure 7 : The Regression lines. From Figure 7, it is evident that for an effective representation of this data set, linear curve should be applied to the low demand data while higher order curves would be more suitable for high demand data. However it is also apparent, that the division between high and low demand clusters is not distinct. Hence, conventional regression-based method cannot be used to model these overlapping demand data clusters. 31 3.1.2 Fuzzy Model to Combine Data Clusters These t wo o verlapping da ta s ets a long with their indistinct boundaries can be modeled using the TSK-fuzzy model[55],[56]. This model represents the relationship between multiple input and output data of a nonlinear system based on fuzzy reasoning. Fuzzy set theory is explained in Appendix I. The demand data clusters can be divided into two overlapping regions: low demand and high demand, as shown by the membership functions of fuzzy sets low and high in Figure 8. r H 1 H ^ T H >x D Demand D [MWh] Figure 8 : Fuzzy Sets to represent Low and High Demand. The membership of the fuzzy sets is denoted by p.D. The high and low fuzzy sets overlap between D L and D H as shown. Each demand value belongs to the high and/or low sets with a different amount of membership. For the demand values less than D L , the membership for low set is 1.0, and for high set is zero; for the demand values greater than D H , the membership for low set is zero, and for high set is 1.0; and the demand values lying between D L and D H , have non-zero membership in both the sets. 32 By this reasoning, we can represent overlapping the demand clusters such that the demand values less than D L are purely low; demand values more than D H are purely high; and demand values in between D L a nd D H lie in both the sets and h ence s ignify t he overlap. Therefore, the low and high demand data clusters are supposed to overlap between D L and D H , with the linear and quadratic regression curves applied to the respective regions. If the input x represents demand, and the output^ signifies electricity price, this fuzzy model can be represented by the following rules: IF x is low THEN yj = ao + x (linear regression) IF x is high THEN y2 = ao + ai x + 02 x~ (quadratic regression) And the nonlinear input-output relation is obtained by: y(x) = - ~——— \A) where /// is the degree of membership of the demand data belonging to either the low (i= 1) or high (i = 2) fuzzy sets. Using the same CalPX data set as shown in Figure 5, and with the lower overlap threshold D L set at 20,000 MWh and the upper overlap threshold D H at 30,000 MWh, we obtain the composite nonlinear curve as shown in Figure 9. 33 200 10000 20000 30000 40000 Demand [MWh] Figure 9 : Composite Regression Curve. It is observable that although this composite non-linear curve follows the overall trend of the data set, it does not represent other information present in the data such as the spread of the data. The conventional regression methods give the most likely relation between variables, and deviations from these mean values are considered to be errors. However, the lateral extent of the price data contains not only the degree of price volatility, but also the information about the extent of the dependence of electricity price on certain indeterminate elements in the system such as participant behavior, and restricted information such as physical constraints of the network. 3.2 Fuzzy Regression to represent Demand-Price Relation The conventional regression model can be extended to fuzzy regression, which takes into account the uncertainty information contained in the original data set, and gives the range of electricity prices including levels of possibility. Hence, the fuzzy regression 34 model will give more factual results by taking into account the indefiniteness of the system. H ere, t he d eviations f rom t he m ean a re a ssumed t o de pend on indeterminate factors in the system[57]. The conventional regression results are contained in this model, and it is considered to be a an extension of the conventional model with provision for the preservation of uncertainty contained in the original data set. 3.2.1 Fuzzy Regression Model In the fuzzy regression model the parameters in the polynomial of (1) are replaced with fuzzy numbers as shown in (3) to encompass a wider range of data[58],[59]. These fuzzy numbers are represented by triangular fuzzy sets with crisp (non-fuzzy) parameters. Y^A^+A.x + A^2 + + Amxm (3) The parameters Ao, A/, ... Am are determined so that the observed data y is encompassed by the fuzzy model. The resultant left-hand variable Y is also a fuzzy number with an interval of data covered in a varying degree of possibility. Figure 10 shows the triangular fuzzy set representing the fuzzy number A; with three crisp parameters, namely a[, c/+, c{ (c/ +, cf > 0), where a/ is the 'mean value' of the fuzzy parameter, and c/+ and cf show fuzziness of the parameter. 35 Figure 10 : Triangular Fuzzy Set. The mean value a/ of the fuzzy parameter is the one given by the conventional regression analysis and the spread parameters cf and c/+ are determined by linear optimization. The fuzzy regression model shows the possibility distribution or a possible interval of data at a point where each crisp input ('demand') is specified. The degree of possibility is represented by the fuzzy number Y(x), in which the conventional regression point is supposed to have the highest possibility of occurrence, while the degree of possibility declines linearly toward zero as the data y(x) goes further apart from the most possible point. The range of model represents the volatility information contained in the given set of data. The degree of volatility can be measured by the area of range encompassed by the fuzzy model. If we divide the model range by the conventional regression line/curve into two regions, say upper and lower, the difference in volatility indicates either the upward trend or the downward inclination of the observed data. 36 The fuzzy regression model represents the characteristics of the whole data set while the conventional model focuses on the most representative data and the deviations from that are discarded as errors. 3.2.2 Optimization to Fit the Possible Range While the mean value a[ o f t he f uzzy p arameter is g iven b y t he c onventional regression method, the spread parameters c/ and c/+ are determined by linear programming to fit t he m odel t o t he given d ata[57]. T he dim ension m is e mpirically determined by a human observation based on the data set or on the results of (conventional) regression analysis. The optimization model is formulated as follows : Minimize: (4) subject to: y\k)>y(k) for £ = 1,2,..., n (5) y-{k)<y{k) and 37 where y*(x) = (a0+c*) + (al+c*) y-(x) = (a0-c;)) + (a]-c;)x + --- + (am-c-)x' (6) x™ = Max[x(k)] xm'" =Min[x(k)] (7) and n is the total number o f observed data. The opt imizat ion is a imed at m i n im i z i n g the vagueness o f the f u z zy regression mode l b y f i t t ing the mode l w i th in as smal l area as possible, wh i l e cover ing al l the data considered to be w i th i n the range. Th i s phase o f mode l i ng does not need experience and data-specif ic knowledge. F igure 11 shows the demand-pr ice re lat ion mode l us ing l inear and quadratic curves on the raw data set. The fuzzy parameters i n (3) are determined as to encompass al l the pr ice-demand data i n the g iven data set. A s it is apparent i n the graph, the fuzzy regression mode l covers quite a w ide area because the parameters were determined such that a l l o f the g iven data is inc luded i n the range o f the f u z zy mode l . The lower out l ine curve seems reasonable but the upper out l ine is too far apart f r om the median. A l s o , it is evident that a few data points (unusual ly h igh prices) are mak i ng the range too wide-spread. Hence, some data pre-condi t ion ing wou l d assist i n obta in ing a reasonably fit mode l . 38 10000 20000 30000 40000 Demand [MWh] (a) Linear (b) Non-linear (Quadratic) Figure 11 : Fuzzy Demand-Price relation. 39 3.2.3 Data Pre-conditioning To pre-condition the data, it is effective to apply some data-specific feature to distinguish irregularity, caused by, for example, social activity factors such as holidays. However, it largely depends on human observation and tends to be ad-hoc. In California, the weather is the dominant factor in deciding the electricity demand. California's yearly climate can be broadly divided into two seasons: summer and non-summer. Also, the state of California belongs to a warm climate zone. Hence, in the summer period, demand is high and supply-demand balance is tight, and therefore price volatility is prevalent. Whereas in the non-summer period, demand is low and electricity prices tend to settle down to the marginal cost of the generators. This is evident from Figure 12, which shows the time series data of daily maximum demand and the corresponding electricity price for years 1998 and 1999. In the process of pre-conditioning, w e have cropped s ome p eculiar days'da ta associated (presumably) with special events. In Figure 12 (a), there is a high price spike in mid-December, while the demand data does not indicate any irregularity. We assumed that there could have been some kind of speculative activities that took place in the market for the Christmas holiday season. This surmise is supported by the 1999 data as shown in Figure 12 (b), which does not show such a spike. Hence, this peculiarly high price data was truncated to an artificial limit of $50/MWh for this particular period. If we set the lower demand threshold at 30,000 MWh and upper demand threshold at 20,000 MWh, Figure 13 shows the data clustering into the low set, the high set and the irregular data which was cropped. 40 D e m a n d P r i c e (b) April 1998 to March 1999. D e m a n d P r i c e 4 0 0 0 0 3 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 M r ! * 2 0 0 150 100 50 0 A p r - 9 9 Jun -99 Sep-99 D e c - 9 9 M a r - 0 0 D a t e (b) April 1999 to March 2000. Figure 12 : Time-Series for Daily Demand-Price for California. 41 Low Demand data High Demand data • Irregular data 20000 30000 Demand [MWh] 40000 Figure 13 : The Demand Data Clusters. 3.2.4 Fuzzy Models on Pre-conditioned data The proposed fuzzy model was applied to the original data set after being preconditioned as described earlier. The demand data is fuzzily divided into low and high clusters overlapping between 20,000 and 30,000 MWh. Tables 1 and 2 list the parameters determined by the optimization on low demand and high demand clusters, respectively. Table 1 : Parameters for low demand model 1 st order constant a 2.119 xlO" J -22.799 + c 1.639 xl0"J 7.927x10"' c 0 19.246 42 Table 2 : Parameters for high demand model 2nd order 1 st order constant a 5.20 xlO" 7 -2.299 xlO" 2 277.36 + c 1.05 xlO'7 0 0 c 1.28x10"" 8.34 xlO"4 6.74x10"* Figure 14 shows the range of the composite model combined by the TSK-fuzzy modeling. As compared to Figure 11, the area of the model is less and hence the volatility of the model has been reduced by pre-conditioning. 10000 20000 30000 40000 Demand [ M W h ] Figure 14 : Composite Fuzzy Model on Pre-conditioned Data. Table 3 shows the area of the fuzzy regression model as an indicator of the volatility measure both on conditioned and unconditioned data sets. In both cases the area of the upper region (above the mean) is more than area of the lower region (below the mean) implying an upward trend in volatility of the price data. When we compare the areas of unconditioned and conditioned data sets, it is evident that in the case of the fuzzy 43 model on the conditioned data set, the area of the upper region is less than that of the model on unconditioned data. Hence, we can conclude that the fuzzy model on the pre-conditioned data set is more reasonable as we get a narrower range of possible relationships between demand and price. Table 3 : Measure of Volatility Model area [US$] Unconditioned Data Set Conditioned Data Set Total 4.8355 xlO6 3.0544 xlO6 Upper Region 3.9236 xlOb 2.1475 xlO6 Lower Region 9.1187 xlO5 9.0693x10' 3.2.5 Fitness of the Composite Fuzzy Regression Model While the original fuzzy regression model automatically determines the parameters of the fuzzy numbers by linear optimization, there is some room to tune fuzzy sets in the TSK-fiizzy model in this proposed approach. We can calculate the area between the upper and lower outline curves as a measure of 'fitness' of the model. The parameters D and D that define the overlap of low demand and high demand clusters can be determined so that the area is minimum. The calculation of the model area is explained in Appendix II. Table 4 compares the fitness by changing the overlapping region of low and high demand data for different cases. 44 Table 4 : Comparison of Fitness Cases D L D H Curve area [MWh] [MWh] [US$] 1 20,000 30,000 3.0543 xlO6 2 18,000 32,000 3.4500 xlO6 3 24,000 28,000 3.2058 xlO6 4 20,000 25,000 3.1581 xlO6 5 25,000 30,000 3.1035 xlO6 In terms of the fitness, it seems that the narrower the overlap is, the more fit the resultant model becomes. However, the model fitness also depends on the encompassed area of each low or high demand region. And in most of the cases there is a trade-off between a smaller area and coverage of more data points. 3.2.6 Forecasting through Fuzzy Regression Model Here, we apply the proposed fuzzy regression model for the estimation of possible ranges of electricity prices when an estimate of electricity demand is given. Figure 15 compares the performance of the obtained model built on the 1998 CalPX data against the actual record in 1999. The 1999 data set consists of raw data, that is, no preconditioning has been done. In this comparison of the model with the data set of 1999, the error rate (the number of data outside the outlines divided by the total number of data points) is approximately 0.67%. 45 200 150 100 "6 50 I 0 4-10000 20000 30000 40000 Demand [MWh] Figure 15 : Comparison of 1998 Model and 1999 Data. We can divide the net error of the model into error in two regions of the curve area: regions above and below the median (crisp regression curve). Figure 16 shows a bar chart comparing the error rate in upper and lower regions in the model area. From the figure we can deduce that the major part of the error lies in the upper region of the model curve and lower region's share in the error is negligible. 0.61 t °"3 I °-2 0.06 0.0 Upper Region Lower Region Figure 16 : The Error Rate. 46 Furthermore, both upper and lower regions consist of three demand areas: low demand area (with demand values less than D L ), high demand area (with demand values greater than D H ) , and overlap demand area (with demand values less than D H but greater than D L ) . Figure 17 shows the error rate in each of these six area segments. I L o w Demand • Overlap • High Demand 0.6 T -=- 0.4 u I o 0.2 C LU 0.0 I Upper Region Lower Region Figure 17 : Apportion of the Error Rate. From Figure 17, it is evident that the upper overlap demand segment accounts for a large portion of the total error, while the other five area segments have small share in the total area error. It is observable from Figures 16 and 17, that most of the error points lie in the upper region of the model curve, that is, the actual price was higher than that predicted by the model. This could be explained by the fact that there seems to be a general trend of market prices going upwards in 1999 as shown in Figure 18. It compares two raw data sets: 1998 data on which the model was developed and 1999 data which was used for comparison. 47 • 1999 Price Data 1998 Price Data 200 20000 30000 Demand [MWh] 40000 Figure 18 : Non-Preconditioned Data Sets. Despite the uncertain factors involved, the error rate in electricity price prediction using the fuzzy regression model is less than 1%, showing the potential of using this model for estimation of future price ranges. To demonstrate the effectiveness of the model in the estimation of electricity price range, we compare a time series of the fuzzy regression model and the actual price data for that period. First, for clarity, the comparison was done for a seven day period. A week in the month of August was chosen when electricity prices are the most volatile. Figure 19 shows a time series of the fuzzy regression model and the actual price data for that period. The median value (denoted by the thin real curve) is determined by conventional regression. Then it is extended by fuzzy regression to give the range of the fuzzy model (denoted by broken curves). The actual price curve (denoted by the thick real curve) is also shown. We find that the fuzzy range encompasses the actual price with a possible 48 span on both the upper and the lower side of the price curve, with one exception when a price spike goes out of the model range. The conventional model on the other hand largely misses the price value. Actual Crisp - Fuzzy (upper) Fuzzy (lower) M 100 ( X 50 0 - - '- r.,.S > / : t * ' t : ^JKA 23-Aug 24-Aug 25-Aug 26-Aug 27-Aug 28-Aug 29-Aug Figure 19 : Time-Series Estimation. This approach can be expanded to predict the time series for the entire year. Figure 20 shows the time-series estimation for the daily maximum electricity price for the period of April 1999 to March 2000. With the exception of a few price spikes, most of the actual price curve is well within the range predicted by the fuzzy regression model. 49 Figure 20 : Time-Series of Price Estimation for 1999-2000. In the proposed model, fuzzy regression is applied to the analysis and estimation of electricity demand-price data. First, the conventional regression is applied in order to identify the structure of the model. Then, the regression parameters are expanded to fuzzy numbers to encompass a wide range of possible data with varying degrees of possibility. The spread of these fuzzy numbers is determined by linear programming. Finally, other supplementary provisions such as data pre-conditioning and measures for model fitness are done in order to minimize the vagueness of the model while covering the data as broadly as possible. The fuzzy regression model so obtained is applied to predict electricity price data when the electricity demand is given, and the results are compared with the actual price data. 50 3.3 Clustering on Demand-Price data Data mining can be used to find patterns and correlations in the given demand-price data set. And based on the recognized patterns in the data set, the arbitrarily defined demand data clusters can be replaced by the ones derived from the inherent data structure, and the proposed model can be further fine tuned. 3.3.1 Clustering algorithms Clustering algorithms partition the data set into groups such that the similarity within a group is larger than that among groups, and hence, can help arrive at some rules about the structure of the data. K-means and the fuzzy C-means are two important clustering techniques. 3.3.1.1 K-means Clustering The K-means clustering algorithm partitions a collection of n vector Xj,j = 1, n, into c groups G„ i = 1, c, and finds the cluster center in each group such that an objective function of dissimilarity (or distance) measure is minimized[60]. When the Euclidean distance is chosen as the dissimilarity measure between a vector Xk in group G„ and the cluster center c„ the objective function can be defined by: • / - Z - ^ Z z \xk-c, (8) 51 where Ji = ^ \\xk -ci ||2 is the cost function within group G,. The partitioned groups k,x^Gf are typically defined by an c x n binary membership matrix U, where the element jUy is 1 if the yth data point Xj belongs to group 1, and 0 otherwise. Once the cluster centers c, are fixed, the minimizing /uy for Equation (8) can be derived as follows: ^ _ | l , i f || Xj -c, ||2<|| Xj-ck ||2,foreachA:^ /, 0, otherwise Since a given data point can only be in a group, the membership matrix U has the following properties: c C II _>,,.=LV. =!,...,» and Z2>//=" (10> /=i j=\ On the other hand, i f i s fixed then the optimal center c, that minimize Equation (8) is the mean of all vectors in group G,: where I G, 1= V" w... 52 The K-means algorithm is presented with a data set x, = 1, the algorithm determines the membership matrix U and the cluster centers c, iteratively using the following steps: Step 1 : Initialize the membership matrix U randomly such that conditions in Equation (10) are satisfied. Step 2 : Determine the cluster centers by Equation (11). Step 3 : Compute the objective function according to Equation (8). Stop i f its improvement over previous iteration is below a certain threshold. Step 4 : Update the membership matrix according to Equation (9). Go to Step 2. Figure 21 shows K-means clustering being applied to normalized CalPX demand-price data set. The data set is partitioned into 3 clusters. Figure 21 : K-means Clustering on CalPX data set. 53 3.3.1.2 Fuzzy C-means Clustering In the Fuzzy C-means clustering (FCM), each data point belongs to each of the clusters to a degree specified by the membership grade. FCM partitions a collection of n vector i = 1, n, into c fuzzy groups, and finds a cluster center in each group such that a cost function of dissimilarity is minimized[60]. The major difference between FCM and k-means is that FCM employs fuzzy partitioning such that a given data point can belong to several groups with the degree of belonging specified by membership grades between 0 and 1. To take into account the fuzzy partitioning, the membership matrix U is allowed to have elements with values between 0 and 1, however the summation of degrees of belonging for a data set is equal to unity: where //// is between 0 and 1; c, is the cluster center of fuzzy group i; dy = ||c, - x7|| is the Euclidean dis tance b etween it h c luster c enter a nd j th da ta p oint, a nd m e [ 1 ,oo) is a HUj = 1 > v , - = 1 , . . . , " (12) The cost function (or objective function) for FCM is then given by: (13) 54 weighing component. The necessary conditions for Equation (13) to reach a minimum are found by forming a new objective function j as follows: /(iv,Cl,...,cc,/i],...^ j=/(^ c1,...,cj+x;1^ 4ZL -^1) (i4) where XjJ =1 to n, are the Lagrange multipliers for the n constraints in Equation (12). By differentiating j(U,C\,...,cc,A\,...,A,n)with respect to all its input arguments[60], the necessary conditions for Equation (13) to reach its minimum are: = J ~ l (15) k=\ 4 For a given data set, the FCM determines the cluster centers c, and membership matrix Uusing the following steps: Step 1 : Initialize the membership matrix U with random values between 0 and 1 such that the constraints in Equation (12) are satisfied. Step 2 : Calculate c fuzzy cluster centers c,-, i = 1, ..., c, using Equation (15). 55 Step 3 : Compute the objective function according to Equation (13). Stop if its improvement over previous iteration is below a certain threshold. Step 4 : Update the membership matrix according to Equation (16). Go to Step 2. Figure 22 shows Fuzzy C-means clustering being applied to normalized CalPX demand-price data set. The data set is partitioned into 3 clusters. Dem and Figure 22 : Fuzzy C-means Clustering on CalPX data set. As it is apparent from Figures 21 and 22, for the given data set both k-means and F C M give very similar cluster shapes and centers. 56 3.3.1.3 K-means and FCM with User-defined Initial Points As both k-means and FCM rely on least square minimization of the distances from cluster centers to various data points, the effects of initial points of the algorithm can be measured by specifying different initial centers. The initial set of cluster centers can either be user defined, or determined by the algorithm itself by randomly picking up data points to serve as initial centers. In the beginning of the iteration process, instead of starting off by determining the membership matrix U, we can initialize a set of cluster centers and then compute U accordingly. Figure 23 (a) and 23 (b) show k-means with random initial cluster centers, and user defined initial cluster centers respectively. While Figure 24 (a) a nd 24 (b) show FCM with random initial cluster centers, and user defined initial cluster centers respectively. Despite starting off from distinctly different centers, the difference between resultant clusters is not significant and both the algorithms manage to reach the same optimum cluster centers. However, the k-means has a slight bias on the position of initial cluster centers, whereas FCM produced consistently the same results by starting from different points. Therefore, FCM may be considered to be more robust for data sets having non-linear characteristics among its attributes. 57 O Initial Centers • Final Centers 0.4 0.6 Demand (a) Random initial centers. 1 0.8 0.6 0.4 0.2 0 0 Initial Centers • Final Centers 0.2 ,« i p. i « • » * o) 0.4 0.6 Demand 0.8 (b) User defined initial centers. Figure 23 : K-means with Cluster Center Initialization. 58 o Initial Centers • Final Centers 0.2 0.4 0.6 0.8 Demand 1 0.8 0.6 o £ 0.4 0.2 0 (a) Random initial centers. O Initial Centers • Final Centers »2 * i .* 0.2 0.4 0.6 0.: Demand O o (b) User defined initial centers. Figure 24 : FCM with Cluster Center Initialization. 59 3.3.2 Membership Function by FCM Clustering After the natural subgroups in the data set are identified through clustering, this analysis of the data structure, can be used to build a model for the system represented by that data set. In FCM, the measure of data points belonging to a certain cluster is available as a grade of membership, rather than the 0-1 measure (belong or not) measure of k-means. By mapping the three clusters obtained by FCM on to the demand a xis, three distinct fuzzy sets are observed as shown in Figure 25. This set of figures can be interpreted such that demand data can be partitioned into three clusters each representing low demand, medium demand, and high demand. And this information on demand characteristic can be used to build a more precise demand-price relation. The different demand clusters membership functions can be approximated by polynomials as illustrated in Figure 26. The low and high clusters can be represented by linear approximation; while the medium cluster is best depicted by sixth-order approximation. These taxonomic cluster representations can be used to generate fuzzy sets as shown in Figure 27, where cluster partition is represented by overlap demand points DL, D M and D H . The medium cluster is portrayed by a second-order approximation for simplification. 60 Clusterl • Cluster2 Cluster3 0 0.2 0.4 0.6 0.8 1 Demand Figure 25 : Membership Function for 3 Demand Clusters. • Low • Medium High Demand Figure 26 : Polynomial Approximation. 61 Demand Figure 27 : Fuzzy Set Generation. 3.3.3 Fuzzy Regression Model based on FCM Clustering Based on the fuzzy sets for demand, generated through clustering, the new model is obtained by using the fuzzy regression and optimization principles proposed in Sections 3.2.1 and 3.2.2, respectively. The model has low and high demand represented by linear regression curves, and medium demand represented by a quadratic regression curve as specified by the generated fuzzy sets in Figure 27. Tables 5, 6 and 7 list the parameters determined by optimization on low, medium and high demand clusters, respectively. Table 5 : Parameters for low demand model 1 st order constant a 2.242 xlO"j -25.165 + c 1.682x10"' 0 c 9.474 xlO"4 0 62 Table 6 : Parameters for medium demand model 2nd order 1 st order constant a -1.611 xlO" 8 2.686 xlO"' -27.253 + c 1.224x10"* 1.260 xlO"' 3.076 c -2.014 xlO"4 1 0 19.627 Table 7 : Parameters for high demand model 1 st order constant a 5.254x10"' -99.453 + c 3.672x10"' 0 c 1.068x10"' 0 The composite model is obtained by combining the three fuzzy demand clusters with TSK-fuzzy model as discussed in Section 3.1.2. The resulting model is shown in Figure 28. Figure 28 : Fuzzy Regression Model. 63 As defined in Section 3.2.5, the 'fitness' of the fuzzy regression model depends on the area between upper and lower outline curves of the model range. The 'fitness' of the new model along with the overlap demand points D L , D M and D H is outlined in Table 8. The overlap demand points are determined through membership function approximation as shown in Figure 26. Table 8 : Fitness of the FCM based Model D L [MWh] D M [MWh] D H [MWh] Curve area [US$] 17,981.6 22,686.1 29,386.9 2.9492 xlO6 In the fuzzy regression model obtained through the membership function approximation has a lower area than all the overlap combinations of the other model as shown in Table 4. Hence, we can conclude that the overlap points based on the inherent data structure give better 'fitness' than the ones that are arbitrarily selected. 3.3.4 Forecasting through the FCM based Fuzzy Regression Model Now, we apply the new fine tuned fuzzy regression model for the estimation of possible r anges o f electricity p rices w hen an e stimate o f electricity demand is given. Figure 29 compares the performance of obtained model built on the 1998 CalPX data against the actual record in 1999. The 1999 data set consists of raw data, that is, no preconditioning has been done. 64 10000 20000 30000 40000 Demand [MWh] Figure 29 : Comparison of 1998 FCM Model and 1999 Data. In this comparison of the model with the data set of 1999, the error rate (the number of data outside the outlines divided by the total number of data points) is approximately 0.54% which is lower than the earlier model. We can divide the net error margin of the model into error in two regions of the curve area: regions above and below the median (crisp regression curve). Figure 30 shows a bar chart comparing the error rate in upper and lower regions in the model area. From the figure we can deduce that the major part of the error lies in the upper region of the model curve and lower region's share in the error is negligible. Also it is observable that the total error and the error in upper region is lower than the earlier model but the error in the lower region is slightly greater than the earlier model. 65 0.5 ^ 0.4 S 0.3 Z 0.2 0.0 0.46 -0.08 1 Upper Region Lower Region Figure 30 : The Error Rate. In this section the proposed fuzzy regression model is refined by using data clustering techniques. The clustering of the given demand-price data set divides the data into natural subgroups and hence the intrinsic structure of the data is identified. This is used to derive the underlying relationships between the variables involved, and based on the membership function the fuzzy regression model is obtained. This model is applied to predict electricity price data when the electricity demand is given, and the results are compared with the actual price data to find the accuracy of the prediction. 3.4 Summary In this Chapter, a fuzzy regression model is developed for the estimation of electricity de mand and p rice. F irst, t he conventional r egression is a pplied in o rder to identify the structure of the model. Then, the obtained regression parameters are expanded to fuzzy numbers to encompass the range of possible data with varying degrees of possibility. The spread of these fuzzy numbers is determined by optimization. The 66 obtained fuzzy regression model is applied to predict electricity price data when the demand is given, and the results are compared to the actual price data. Next, the model is fine tuned b y applying data mining techniques to the given data set, and the natural subgroups in the data are used to improve the model. Finally the fine tuned fiizzy regression model is applied to predict electricity price data when the electricity demand is given, and the results are compared with the actual price data and to the ones obtained from the earlier model. 67 Chapter 4 Consumer Decision-Making Aid In the deregulated electricity system, the consumers have the option buying their electricity needs either through the auction process of the power exchange or by entering long-term multilateral/bilateral contracts with the suppliers. The major objective behind the deregulation of electricity utilities is to reduce the price of electricity by introducing competitive market forces. However, under the free market conditions there is no effective means o f c ontrolling t he p rices, and t he e lectricity p rices t raded at t he s pot market can be quite volatile due to the strategic behaviors of market participants. In such a situation, a bilateral contract from a supplier, with a fixed price over a period of time, can be very attractive to the consumer. As the long-term supply contracts tend to act as an shield against wildly fluctuating spot market prices, they are offered at a prices generally higher than the market prices. Hence, the consumer might have to pay some premium for the added security of the long term contract. When different electricity rates are offered by different competing sources, the consumer will select the offer that will best match his preference, most probably a low price. Hence, when opting for a bilateral contract the consumer will probably compare the offered price to the market price, and if the offered price is too high as compared to the market price, the offer might be unappealing. However, there are two major difficulties in m aking s uch d ecisions: t he c onsumer's p referred p rice is us ually no t a clear-cut boundary but there may be some margins to accept a less satisfactory offer, and the market prices are fluctuating so much that a simple average does not substantially support decisions. 68 Previously reported studies in the electricity pricing area are mostly focused on the strategic decisions of suppliers in the spot market (for example, [17]-[19]). Some strategies for bilateral contracts are analyzed in [20] and [21 ] but these early works are apparently targeting wholesale suppliers and distributors. Also, mathematical models representing the consumer "preference" are an unexplored area, and conventional approaches based on stochastic models [22] are in many cases impractical. In this chapter, the market price is estimated by the fuzzy regression model developed in Chapter 3, and the consumer's preference is represented by a separate fuzzy number. The value of the bilateral transaction is then examined by a fuzzy decision involving the possible market price range, the bilateral price offered, and the consumer's preferred price range. 4.1 Formulation of the Evaluation Process In this approach, the uncertain market prices are represented by fuzzy regression models with fuzzy parameters. The extended regression model represents a market price by a fuzzy number which is composed of the most likely price as a mean value and the interval encompassing the range of possible highest and lowest prices with varying degrees of possibility. The consumer's preference for a desirable bilateral price range with degrees of satisfaction is represented by a separate fuzzy number. The most likely and satisfactory performance of the transaction is measured by the intersection of these two fuzzy sets. 69 4.1.1 Consumer Preference Index The preeminent factor in the evaluation process of an electricity supply contract is the preference of the consumer for a price offered by the supplier. It is desirable to get a price lower than the electricity rate available from the market, but the market price is unpredictable and can be several times higher or occasionally lower than an offered contract price because of market competition. On the other hand, the attraction of a long-term contract is the stability of price. Therefore, the consumer could settle for a slightly higher price than what the market indicates. Additionally, the offer would be unacceptable if the price for the supply contract is too high compared with the indicated market price. Hence, the preference for the (possibly low) price can be a range of acceptable prices with a varying degree of satisfaction for each price in the range. Let price differential expressed by Ap such that: Ap = poff-pm(t) (17) off where pm is the market price which changes over time t, while p is the offered price for the bilateral contract and has a fixed value throughout the contract period. The varying degree of consumer satisfaction for a range of offered prices can be expressed by a piecewise linear function. Figure 31 shows a trapezoidal fuzzy set representing the consumer's preference index. 70 [$/MWh] Figure 31 : A Fuzzy Set to represent Consumer's Preference. The trapezoidal fuzzy set G is defined by two parameters, gl and gQ (with gt < g0y representing the threshold values for the consumer's price goals. The membership of set G is denoted by p.G which gives the level of consumer's satisfaction. If the price differential is smaller than g the offer is completely satisfactory (p.G = 1). The degree declines a s t he dif ferential w idens, a nd a t g 0 the price offered becomes unacceptable (uG = 0). In other words, the value g ; denotes the extra amount the consumer is willing to pay with confidence for the security of the bilateral contract, while the value gQ denotes the limit of the extra amount which the consumer is willing to pay for the indemnity to volatility of spot market prices. Assuming that the preference of the consumer does not change during the contract period, the preference index can be defined by a piecewise linear function such that: JUQ = Min 8o-g\ . (18) 71 4.1.2 Possible Range of Market Prices While the preference index is arbitrarily defined by consumers, the market reality must be somehow reflected in the evaluation process. The prices at the wholesale market, although wildly fluctuating at times, show the intention of wholesale suppliers, and they will be eventually reflected in the rate that retailers can offer. Therefore, it is important to know the possible range of market prices for a reasonable evaluation of a supply contract. To represent relations of electricity demand and possible price range of the spot market, we use the fuzzy regression model developed in Chapter 3. The forecast of electricity demand is an extensively studied area, and in many cases, demand information released by authorities is far more reliable than the estimate of spot market prices. Therefore, given the demand forecast, we apply the method based on an extended regression to obtain the possible price ranges for the contract period. Given a set of history data of electricity demand and market prices, we first divide the data set into tw,o overlapping clusters, low demand and high demand, with threshold values of demand: D L and D H (DL < DH). Data sets with demand values falling between L H D and D n are allowed to belong to both clusters in a varying degree and are smoothly connected by the TSK fuzzy model. Next, we apply regression analysis on the low and high demand clusters independently and obtain the most likely demand-price relation for each cluster. This is based on the fact that spot prices are quite volatile in the high demand region. Then, the regression parameters are extended as fuzzy numbers so that the observed demand-price data is encompassed by the fuzzy model. 72 The estimated price so obtained is also a fuzzy number, which has an interval of data with a varying degree of possibility. Its possibility is the highest (1.0) at the mean value. The possibility declines if the price goes apart from the mean value, and at the edges of the interval, the possibility becomes zero. The mean value of this fuzzy number is determined by conventional regression, while the spread is determined by linear optimization as described in section 3.2.2. Hence, for the given estimated demand values, the fuzzy regression model is capable of producing a possible range of prices. 4.1.3 Composite Evaluation Index With consumer preference defined by the price differential between the offered price for the bilateral contract and the market price, and the market price range predicted by fuzzy regression from an estimate of demand as a time series over a contract period, we evaluate an offered price in fuzzy number domain as follows: 1) For each time stage, calculate the price differential D(k). Equation (17) is rewritten using fuzzy numbers such as: D(k)=poff-Pm{k) (19) 73 where PJk) is the fuzzy number representing the possible range of market price at a discrete time k as determined by the fuzzy regression model. The market price is most likely at the mean value and there is no possibility assumed outside of the interval. In (19) the non-fuzzy offered pricepff is mapped into a fuzzy number {a -pff and c+ = c = 0) and the resultant price differential D(k) will also be a fuzzy number. 2) Compare the preference fuzzy number G and the price differential fuzzy number D(k), and measure the maximum degree pDG(k) of the overlapping two fuzzy sets as: MDG =Max{Min[/JG,fiD(k)§ (20) This operation, illustrated in Figure 32, assures the consumer's preference index to be feasible. If there is no overlap, the degree is 0 (The price differential is unacceptable or infeasible). On the other hand, if D(k) is completely enclosed by G, the degree is 1.0 (completely satisfactory and feasible). DG < 7\ 1 k g l [$/MWh] Figure 32 : Measurement of Price Differential against Preference. 74 3) Repeat steps 1) and 2) for all the time stages k = 1 , n . 4) The overall evaluation of the offered price is given by the averaging process as: 4.2 Evaluation of a Bilateral Contract To demonstrate the proposed procedure we present examples using California Power Exchange data[54]. To reflect the reality of a supply contract for an industrial consumer, only the data set for working hours (7am to 11pm) on weekdays is used. First, the de mand-price da ta s et a s s hown in Figure 5 (total a rea de mand and unconstrained market clearing price for the period of April 1998 to March 1999) is used to determine the demand-price relation model as discussed in Chapter 3. Then, the weekday peak-hours demand data for the month of August 1999 is used to produce the estimated price range of the same period. Figure 33 compares the estimated price range and the actual market clearing price for the same period. Next, we set the preference parameters are set arbitrarily as gl = $2/MWh and gQ = $5/MWh. It could be interpreted as 'the bilateral contract is fully satisfactory for price differential up to $2/MWh; moderately satisfactory for price differential between $2/MWh ~ $5/MWh; and unacceptable for price differential more than $5/MWh'. (21) 75 Crisp Fuzzy (upper) Fuzzy (lower) •Actual i — i — i -2-Aug 5-Aug 10-Aug 16-Aug 19-Aug 25-Aug 30-Aug Figure 33 : Estimated Range of Market Price (August 1999). The average market price for the same period was $40.19/MWh. For an offered price of $50/MWh, overall performance evaluation (y) was obtained to be 0.948. Hence, for the month of August (weekday, peak-hours), a bilateral contract at $50/MWh seems reasonable. Figure 34 shows the fluctuation of the performance index for this case. The chart reflects the high satisfaction level for the particular contract offer, as the performance index stays close to the complete satisfaction level (1.0) for most of the time. We can extend this method to evaluate a bilateral contract for the one year period from April 1999 to March 2000 period. In order to simulate different attitudes of the consumers, three different sets of preference parameters are used: moderate (gl = $2/MWh and gn = $5/MWh) aggressive (g, = $0/MWh and gQ = $2/MWh) and conservative (g/ = $5/MWh and gn = $10/MWh). These preference parameters are tabulated in Table 9. 76 0 -J , r , , r - S , r—^ , , , , ^ 2-Aug 5-Aug 10-Aug 16-Aug 19-Aug 25-Aug 30-Aug Figure 34 : Performance Index for a Retail Supply Offer (August 1999). Table 9 : Consumer Preference Parameters Case 1 Case 2 Case 3 ($/MWh) (Moderate) (Aggressive) (Conservative §n 5 2 ' 10 g, 2 0 5 The weekday peak-hours demand data for the period April 1999 to March 2000 was used to produce the estimated price range of that time period. Figure 35 compares the estimated price range and the actual market clearing price for the same period. The average market price for the same period was $36.73/MWh. The offered price was varied to three different levels: $30/MWh, $40/MWh and $50/MWh. Table 10 summarizes the overall performance of these offered prices for each type of consumer. 77 250 Crisp - • - Fuzzy (upper) - - - Fuzzy (lower) Actual 200 4/1/99 5/1/99 6/1/99 7/1/99 9/1/99 10/1/99 11/1/99 12/1/99 2/1/00 3/1/00 Figure 35 : Estimated Range of Market Price. Table 10 : Performance evaluation (y) Offered Price ($/MWh) Case 1 Case 2 Case 3 30.00 0.950 0.921 0.984 40.00 0.798 0.759 0.854 50.00 0.627 0.583 0.693 Figure 36 shows the fluctuation of the performance index for Case 1 for an offered price of $50/MWh. The variation of the performance index over time gives us information about the difference between the market price and the price of the bilateral contract being studied. Hence, we can see the variation of market price in California, relative to the offered price ($50/MWh) in Figure 36. In the months of April to June the performance index lies on the lower side of the scale as the market price was low in these non-summer months. The performance index is on the upper side of the scale in the months of July and 78 August because of higher temperatures of the summer. The negative peaks in this time period can be attributed to a summer which was cooler than average. In the winter months, the performance index comes down as the falling temperatures brought down the market prices. 0 I- r '—i , , , r , , r 4/1/99 5/26/99 7/21/99 9/14/99 11/9/99 1/3/00 2/28/00 Figure 36 : Performance Index for a Retail Supply Offer. 4.3 Summary In this chapter, a procedure to help consumers decide between a bilateral contract and spot market purchases of electricity is developed. The electricity price was predicted by the fuzzy regression model proposed in Chapter 3. Then bilateral contracts were evaluated by fuzzy decision, considering that both the market price and the consumer preference have some level of uncertainty. A typical target consumer for this process is, for example, a small to medium scale industrial user of electricity where the electricity usage is not very flexible and the lowering of the electricity cost is of interest. This approach can be used not only for consumer decision making support, but also for 79 electricity suppliers and/or brokers to determine a price to offer when the consumer preference is uncertain. 80 Chapter 5 Conclusion and Future W o r k The achievements of this study include a realization of a prototype m odel f or electricity price prediction under the restructured electric utility environment. Further augmentations can aid in enhancing the model performance. 5.1 Conclusions A model for representing electricity price and demand is proposed in this research work and subsequently is used for consumer decision making support under the competitive market conditions. 5.1.1 Electricity Demand-Price Model A model for forecasting electricity price, based on fuzzy regression is introduced in this work. This fuzzy model can represent highly volatile demand-price relations as a range with a possibility distribution of prices for a given single value of demand forecast. The model construction is based on techniques of data clustering, regression analysis and linear programming, and it preserves the rich information contained in the original data set including uncertainty. 81 5.1.2 Consumer Decision Making Support A procedure to assist electricity consumers to make decisions on bilateral power transaction under uncertain market conditions has been developed. The uncertain market prices are represented by the fuzzy regression model mentioned above. This procedure can also be used by the electricity suppliers and/or brokers to determine an attractive price to offer to a target customer when the consumer preference is uncertain. 5.2 Future Work This work is the first step in determining the feasibility of using the proposed technique for electricity price forecasting in a power exchange frame of reference. There are a few significant issues that need to be addressed in order to increase the effectiveness of the model. 5.2.1 Data Pre-conditioning It is observed that some data pre-conditioning is required to obtain a reasonably 'fit' model. For the purpose of pre-conditioning of the data set, it is effective to apply some target-specific heuristics such as seasonal differences. Also, some data-specific features are required to distinguish irregularity in the observed data caused by, for example, social activity factors such as holidays. However, such heuristics largely depend 82 on hum an o bservation a nd t end t o b e a d-hoc. H ence, t heir generalization is difficult. Improving the pre-conditioning technique could a topic for further study. 5.2.2 Focussing the Range of the Model In the proposed method, the parameters of the fuzzy regression model were determined so that all the given data is included in the range of the model. Hence, the resultant model covered quite a wide area. In some cases, it would be desirable to focus on the fuzzy regions of higher possibility. For this purpose, the possibility distribution of the data can be focussed by using a-cuts of fuzzy parameters. The details of a-cuts of fuzzy parameters are explained in Appendix III. 5.2.3 Congestion In the proposed method, the unconstrained price data has been used. The unconstrained power transactions are based solely on economic decision making and are subject to physical constraints of the transmission system. When the available transfer capacity is insufficient to simultaneously dispatch all the preferred schedules, the scheduling parties either pay a congestion charge to the system operator or have to revise their proposed schedules[41]. Hence, the actual price of the electricity transaction can be very different from the one based purely on economic optimization. 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[31] Website http://www.ferc.fed.us/newsl/rules/pages/order888.htm: Promoting Wholesale Competition Through Open Access Non-discriminatory Transmission Services by Public Utilities; Recovery of Stranded Costs by Public Utilities and Transmitting Utilities, FERC Order No. 888, April 1996. [32] Website http://www.alleghenyinstitute.org: Transcript of a speech by John Hanger, former Pennsylvania PUC Commissioner. [33] Website http://www.cato.org/pubs/regulation/regl8n2f.html: Article: "Clearing the Track: The Remaining Transportation Regulations", By Thomas Gale Moore. [34] Website http://www.energy.ca.gov/BR96: California State Energy Plan-"Critical Changes: California's Energy Future", Biennial Report, adopted by California Energy Commission on December 17, 1997. [35] Website http://www.cnn.com./SPECIALS/2001/power.crisis: CNN In-Depth Specials- Power Crisis. 88 [36] Website http://www.cpuc.ca. gov: Home Page of California Public Utilities Commission. [37] Website http://www.emagazine.com/november-december 1997/1197featl sbl.html: Magazine Article: "Will Utility Deregulation Finally Unplug 'Dirty' Electricity?", By Jim Motavalli. [38] B. Barkovich, "Charting a New Course in California", IEEE Spectrum, Vol. 33, No. 7, 1996, pp. 26-31. [39] ISO Tariff Original Sheet No.300-357, "Master definitions Supplement", ISO Tariff Appendix A, is sued b y C alifornia Independent System Operator Corporation on October 13, 2000. [40] Website http://www.calpx.com: Home Page of California Power Exchange Corporation. [41] N. Fuller, Principles of Micro Economics, Tudor Publishers, Great Britain, 1997. [42] Website http ://www. calpx. com/prices/index .htm: Analysis of California Power Exchange Electricity Markets. [43] R. Green, "The Political Economy of The Pool", in M . Illic, F. Galiana, L. Fink, (ed.) Power Systems Restructuring, Kluwer Academic Publishers, Boston, 1998. [44] Website http://208.152.75.64/index.asp: Notice by The California Power Exchange Corporation to the Secretary of FERC, January, 30,2001. [45] Website http://www.caiso.com: Home Page of California Independent System Operator Corporation. [46] Website http:// www, energy, ca. gov/maps/index .html: California On-Line Energy Maps. 89 [47] ISO Tariff Original Sheet No.363-366, "ISO Scheduling Process", ISO Tariff Appendix C, issued by California Independent System Operator Corporation on October 13,2000. [48] Website http://www.pim.com: Home Page of PJM Interconnection L.L.C. [49] Website http://www.caiso.com/docs/09003a6080/07/40/09003a6080074029.pdf: Report on California Energy Market prepared by the California ISO, August 10, 2000. [50] E. K ahn, e t a 1., " Pricing in t he C alifornia Power Exchange Electricity Market: Should California Switch from Uniform Pricing to Pay-as-Bid Pricing?", A study commissioned by the California Power Exchange, January 2001. [51] Website http://www.caiso.eom/docs/2000/l 1/01/2000110112414127623.html: Statement of FERC Order: Remedies for California Wholesale Electric Structure. [52] Website http://www.eia.doe.gov/cneaf/electricitv/wholesale/wholesalelinks.html: Energy Information Administration website providing Official Wholesale Energy Statistics from the U.S. government. [53] D. Eisenberg, "Which State is Next?", Time, Vol. 157, Issue 4, pp. 45-48, January 29, 2001. [54] Website http://www.calpx.com/prices/index_prices_dayahead_trading.html: Posting of California Power Exchange Market Price Data. [55] T. Takagi and M. Sugeno, "Fuzzy Identification of Systems and its Applications to Modeling and Control," IEEE Transactions on Systems, Man, and Cybernetics, Vol. 15, No. 1, 1985, pp. 116-132. [56] M. Sugeno and G. T. Kang, "Structure Identification of Fuzzy Model," Fuzzy Sets and Systems, Vol. 28, 1988, pp. 15-33. [57] H. Tanaka, S. Uejima, and K. Asai, "Linear Regression Analysis with Fuzzy Model," IEEE Transactions on Systems, Man, and Cybernetics, Vol.12, No.6, 1982, pp. 903-907. [58] K. Tanaka, An Introduction to Fuzzy Logic for Practical Applications, Springer, New York, 1996. [59] D. Dubois and H. Prade, Fuzzy Sets and Systems: Therory and Applications, Academic Press, New York, 1980. [60] J. Jang, C. Sun, E. Mizutani, Neuro Fuzzy and Soft Computing, Prentice Hall, New Jersey, 1997. [61] T. J. Hammons, et al. "Market Forces Propelling Renewable Energy Technologies", IEEE Power Engineering Review, January 1999, pp. 12-22. 91 Appendix I Fuzzy Set Theory Appendix A fuzzy set A in the universe X is defined by a membership function U A representing a mapping: U A : X - > [ 0 , 1 ] (22) Here the value of U A ( X ) of the fuzzy set A is called the membership or the grade of membership of x e X. The membership value represents the degree of x belonging to the fuzzy set A. Fuzzy logic provides a methodical means of dealing with uncertainty and ambiguity. The membership functions can be used to map natural language syntax onto a numerical value. In the characteristic function of crisp sets, the degree of belonging to a particular description or class is either 0 or 1, whereas in the fuzzy domain, the membership function can have a arbitrary real value between 0 and 1. Figure 37 shows characteristic functions (crisp) and membership functions (fuzzy) for 'low', 'middle' and 'high' stature. According to the crisp 'conventional' set theory in Figure 37(a), a person with height of 168 cm belongs to the 'low' set and a person with height of 171 cm belongs to the 'middle' set, even though the difference in the two heights is only 3 cm. According to the fuzzy set theory in Figure 37(b), the person with height of 168 cm belongs to the 'low' set with a greater degree and to the 'middle' set with a smaller degree, and a person with height of 171 cm belongs to the 92 Appendix 'middle' set with a greater degree and to the 'low' set with a smaller degree. Hence the second person would be defined as 'lower middle' and the first person would be defined as 'relatively low', which agree with the natural language descriptions. Expressing the height in fuzzy sets did not result in a compelling difference and the fuzzy set division is much closer to human feeling than the crisp sets. Low Middle High 0 Height [cm] 170 180 (a) Crisp Sets Low Middle High 0 Height 170 [cm] .180 (b) Fuzzy Sets Figure 37 : Crisp and Fuzzy Sets of Height 93 Appendix The membership functions of fuzzy sets can be viewed as possibility distributions hence, the fuzzy sets can systematically treat uncertainty which is not random in nature. Triangular fuzzy numbers are popularly used to express possibility distributions. Figure 38 shows a symmetrical triangular fuzzy set. Figure 38 : Triangular Fuzzy Set . The fuzzy set A has a base width of 2c and a peak at x = 0. The membership U A is 1 at x = 0, and goes on decreasing at either side until x = ± c, after which pA = 0. The set has infinite number of elements with different memberships values. The triangular fuzzy sets can also be unsymmetrical. 94 Appendix Appendix II Fuzzy Regression Model Area In the fuzzy regression model, the overlapping demand data sets are represented using t he T SK-fuzzy m odel w ith t he 1 ow and hig h d emand da ta c lusters overlapping between the demand thresholds of D L and D H . The area covered by the model gives a measure of its 'fitness' and the parameters D L and D H that define the overlap can be determined so as to make the area minimum. The net TSK model area is comprised of three demand regions defined by different set of regression curves as shown in Figure 39. 0 Low \ Overlap , High Linear Quadratic Regression Regression 0 D L D H D M Demand Figure 39 : The Demand Regions Following Figure 39, the regression curves split the model area into three regions: linear curve for the low demand, quadratic curve for the high demand, and nonlinear heterogeneous curve for the overlap demand region. The total model area is the sum of these three regions areas. 95 J Appendix Low Demand Area The low demand region is represented by linear regression curve with parameters ao, a/, co+, c;+, co and cf. If the demand is denoted by JC , the upper and the lower linear relations to represent possible range are given by : y]+(x) = (a0+cZ)+(al+c?)x and yf(x) = (a0-c-)+(al-c-)x (23) If we denote AQ=c 0 +-c 0- and /\Q=c;-c; (24), The area for linear regression region is given by: AreaL = A&D1 + Afx (DL )2 (25) High Demand Area The high demand region is represented quadratic regression curve with parameters ao, a/, a2, co+, c;+, C2+, Co, cf and cf. The value of maximum demand in the data set is represented by D M . With the demand denoted by x, the upper and the lower envelope of quadratic fuzzy curves is given by: 96 Appendix y2+ =(a0 +c0+)+(a, +c+x)x+(a2 +c2+)x2 y{ ={ao - c o)+(«i ~^)x+(a2-c-)x2 and (26) If we denote 4 C 0 = c 0 + - c - , A C ^ c ^ - c r and 4 C 2 = c 2 + - C - (27) The area for quadratic regression region is given by: AreaQ = A^C^ + Z^C,^ + Z^C^ (28) where S i , S 2 and S 3 are the demand differentials with: SX=DM -DH S2JD"ff"f and , ] = K ) > " ) ( » , Overlap Demand Area The demand data with a value greater than D L but less than D H belongs to the overlap demand region and is represented by a non-linear set of curves obtained from linear and quadratic regression parameters by fuzzy reasoning. Each demand point in the overlap region has a non-zero membership in both low and high demand sets. If yi denotes the linear curve and ^ represents the quadratic curve, the non-linear composite fuzzy curve is given by : 97 Appendix where L7/ o h , and Uhigh are the memberships of low and high fuzzy sets given by : U, = X ~ D " and u - x ~ D l (31) The upper and lower set of composite fuzzy curves can be obtained from equations (23), (26) and (30), and the area is composed of two parts : _ A c 0 ( z ) ^ - g 2 ) + A c 1 ( ^ g 2 - g 3 ) a n d sireuNX — Area N2 AO) ( a - p'a)+AC. ( a - ^ a ) + ( a - s " a ) ( 3 2 ) 0 where Oj, Q 2 , Q 3 and Q 4 are the second set of demand differentials with: a . l £ M and a . K M (33, The total area for non-linear composite regression region given by: AreaN = Aream + AreaN2 (34) The total model area is the sum of areas of the linear, the non-linear composite and quadratic regression regions. AreaModei = A r e a L + A r e a N + Area0 (35) 98 Appendix Figure 40 shows a three-dimensional plot of the total model area versus the demand thresholds D L and D H . x10 6 3 .5 . High Demand Threshold (MWh) 1 1 . . . . u Low Demand Threshold (MWh) Figure 40 : The Model Area Note that model area is inversely proportional to both of the demand thresholds. As the area goes on decreasing as the threshold values are increased, the optimum area is achieved when D L and D H have the maximum values permitted by the data set constraints. 99 Appendix Appendix III a-cuts of Fuzzy Sets In the fuzzy regression model, the a-cuts of fuzzy parameters can be used to evaluate the performance of the model in encompassing the possibility distribution of the obtained range. If for a fuzzy set A, UA (X) is the membership of x e X (universe), the a-cuts of fuzzy set A can be defined as: • K = {x|//A(x)>a},ae[0,l] (36) where a-cut A a is an interval projected to the X domain. Figure 41 shows an exponential fuzzy set A with two a-cuts aj and 0.2 (ai > a2), giving rise to two subsets of A: Aai and Aa2. HA 1 ai 0:2 0 Figure 41 : The a-cuts During the optimization stage of the model development, the spread of fuzzy parameters was determined to include all the observed data, and as a consequence the 100 Appendix resultant range of the model might appear too broad. The band of possibility intervals can be narrowed down by using a-cuts of fuzzy numbers such that the higher the value of a, the more focussed is the model on high possibility region as it is observable in Figure 41. With application of a-cuts, the fuzzy numbers in Equation (3) will be modified as: A> =<a0,Co(l-a),c~(\-a)> A=<a],c;(\-a),c;(\-a)> An =<am,c+m(\-a),cm(\-a)> where 0 < a < 1. Figure 42 shows the fuzzy regression model with the conventional regression curve enveloped by the original fuzzy range (a = 0) and the model range with an a-cut of 0.5. 10000 20000 30000 40000 Demand [MWh] Figure 42 : Fuzzy Regression Model with an a-cut 101 Appendix It is observable from Figure 42, that with a = 0.5, the fuzzy regression envelope is more focussed towards the data points of highest possibility (the conventional regression curve), and some lower possibility data points are outside the range of model. With a = 0, the model is same as the original fuzzy model and as the value of a increases towards 1, the range encompassed by the fuzzy envelope narrows and eventually shrinks to the conventional regression model at a = 1. For a > 0, some of the data points are outside the range of fuzzy model, and the number of missed points will increase as a approaches to 1. The a-cut levels can be considered to indicate the willingness to accept the risk of missing data points by focussing on a narrower scope and an acceptable level of a-cut depends on human judgment. 102 Appendix Appendix IV Transmission Congestion Congestion is a condition that occurs when the volume of scheduled transfer of electrical power exceeds the physical and operational limits of the transmission facilities. Figure 43 shows a two bus electrical network with two different power flow conditions. There is a high capacity generator and load located at each of these buses. The buses are connected with a power line of definite transfer capacity. Nodel Capacity 100 MW Node2 $12/MW Flow 50 MW Loadl 200 MW Load2 50 MW r !L $18/MW (a) Non-Congested Transmission Line Nodel $12/MW Loadl 200 MW Capacity 100 MW Flow 100 MW Node2 Load2 150 MW r $18/MW (b) Congested Transmission Line Figure 43 : Electrical Network with different Power Flows 103 Appendix From Figure 43, it is observable that the two generators have different variable costs with the generation of node-2 being more expensive than that of node-1. Hence, it is economical for load at node-2 to import power from node-1 rather than utilizing its local generation. In Figure 43 (a), the entire load at node-2 can be fed by the remote power without overloading the transmission system. In such a situation the price of power will be competed down to marginal cost of the generation at node-1. In Figure 43 (b), the load at node-2 has increased and an attempt to feed the total load with imported power that will overload the connecting power line. The actual power flow is limited by the capacity. In such a situation the transmission line is congested and even though both supply and demand exist on the system, the equilibrium market price of power will no longer be equal to the marginal cost as the marginal costs of node-1 to node-2 will be different. When the market participants want to buy or sell more than the sufficient available transfer capacity, they have to pay a congestion charge which is determined in an ISO auction for transfer capacity. The other option is to re-dispatch their schedules through non-congested transmission paths. 104
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Fuzzy set based decision support system for transactions of electricity in a deregulated environment Dhaliwal, Maninder Kaur 2001
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Title | Fuzzy set based decision support system for transactions of electricity in a deregulated environment |
Creator |
Dhaliwal, Maninder Kaur |
Date Issued | 2001 |
Description | In the restructured electricity market environment, the market participants conduct their power transactions with an aim to maximize their profits. Electricity is typically traded either through an auction at an open-access power exchange market or directly between supplier and retailer/consumer through bilateral and/or multilateral contracts. In the power exchange, the balance of supply and demand determines the spot market price. These transactions are purely economic and are subject to the physical constraints of the transmission system. The transmission grid is controlled by an Independent System Operator (ISO), and information about the system operation is restricted and rarely available to the market entrants. The market players generally receive partial information about the system conditions, such as a forecast of the total demand. Hence, the pivotal information in conducting the spot market transactions of electricity is price and demand. In some cases, a direct bilateral contract between suppliers and consumers can provide an attractive alternative to the spot market pricing, where the price can be volatile due to strategic behavior of market participants or tight demand-supply balance. The most likely measure of suitability of a bilateral contract is its comparison with the market price. However, the spot market price tends to be significantly volatile. Therefore, suitable methods for representing the volatile market price are needed. The traditional modeling methods are primarily based on statistical and probabilistic approaches, and it may not be entirely suitable to apply these stochastic methods to model the data generated by human activities such as power exchange markets. Besides, most of the existing models are aimed at stock exchanges, and may not necessarily be applicable to the electricity markets. In this research work, a data representation model based on extended fuzzy regression is developed. The model represents the highly volatile demand-price relations as a range and estimates the possible distribution of the price range for a given value of a demand forecast. The highlight of the model is its capability to preserve the aggregate information contained in the original data set such as the uncertainty. The model was tested using actual data from California Power Exchange and results were found to be promising. Based on the proposed model, a procedure for evaluation of bilateral contracts in an open market environment is developed. |
Extent | 9935512 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-08-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065391 |
URI | http://hdl.handle.net/2429/12059 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2002-05 |
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UBCV |
Scholarly Level | Graduate |
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