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Coordination and decision making of regulation, operation, and market activities in power systems Nakashima, Tomoaki 2000

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Coordination and Decision making of Regulation, Operation, and Market Activities in Power Systems by Tomoaki Nakashima B.A.Sc, Kyoto University, Japan, 1991 M.A.Sc., Kyoto University, Japan, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 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 July 2000 ©Tomoaki Nakashima, 2000 ln 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 my written permission. Department of H(ec.-fV*icftl (A^A Gui^Kfur The University of British Columbia Vancouver, Canada Date Ayrr UM 2ooo DE-6 (2/88) Abstract Electric power has been traditionally supplied to customers at regulated rates by vertically integrated utilities (VIUs), which own generation, transmission, and distribution systems. However, the regulatory authorities of VIUs are promoting competition in their businesses to lower the price of electric energy. Consequently, in new deregulated circumstances, many suppliers and marketers compete in the generation market, and conflict of interest may often occur over transmission. Therefore, a neutral entity, called an independent system operator (ISO), which operates the power system independently, has been established to give market participants nondiscriminatory access to transmission sectors with a natural monopoly, and to facilitate competition in generation sectors. Several types of ISOs are established at present, with their respective regions and authorities. The ISO receives many requests from market participants to transfer power, and must evaluate the feasibility of their requests under the system's condition. In the near future, regulatory authorities may impose various objectives on the ISOs. Then, based on the regulators' policies, the ISO must determine the optimal schedules from feasible solutions, or change the market participants' requests. In a newly developed power market, market participants will conduct their transactions in order to maximize their profit. The most crucial information in conducting power transactions is price and demand. A direct transaction between suppliers and consumers may become attractive because of its stability of price, while in a power exchange market, gaming and speculation of participants may push up electricity prices considerably. To assist the consumers in making effective decisions, suitable methods for forecasting volatile market price are necessary. ii This research has been approached from three viewpoints: Firstly, from the system operator's point of view, desirable system operation and power market structure are explored. Two typical ISO models, centralized and decentralized, have been identified and compared. These ISO models have been simulated to observe the advantages and disadvantages of the different systems. If no powerful players exist, the centralized system would achieve the maximum market efficiency. However, in decentralized systems, freedom of trade protects market participants from strategic bidding caused by powerful players. Reduced market efficiency is the price markets have to pay to prevent strategic bidding. Secondly, from the regulator's point of view, the effects of different policies imposed by regulators on power transactions are examined. The optimal schedule could be affected greatly by the ideal goals and their allowable values. Therefore, when the ISO defines its objectives and their allowable ranges, an agreeable conclusion among market participants is required. Fuzzy multiobjective optimization methods can be suitably applied to the scheduling of the ISO, reflecting its objectives and their allowable ranges properly. Thirdly, from market participants' point of view, models to represent and forecast the price and demand of power are developed. Electricity consumption and price are forecasted based on possibility theory and fuzzy autoregression. The fuzzy model can represent highly volatile demand-price relations as a range, and gives the possibility distribution of prices. Based on the proposed model, a procedure to help consumers decide whether to accept a bilateral transaction contract or market-based purchases of electricity has been developed. The same procedure can also be used by an electricity supplier or broker to determine an offering price. iii Table of Contents Abstract ii Table of Contents iv List of Tables vi List of Figures vii List of Symbols viii List of Acronyms x Acknowledgements xi Chapter 1 Introduction 1 1.1 Synopsis of Work Accomplished 1 1.2 Background 5 1.3 Objectives 12 1.4 Methods 15 Chapter 2 ISO Functions for Different Market Structures 19 2.1 Deregulated ISO Functions and Market Structures 19 2.2 Mathematical Models for System Operators. : 23 2.3 ISO Functions 27 2.4 Existing ISOs 38 2.5 Challenges of ISO 41 2.6 Comparison of ISO Models 42 2.7 Summary - ISO Functions for Different Market Structures 54 Chapter 3 Different Policies Imposed by Regulators 56 3.1 Possible Obj ectives of Regulators 57 3.2 Methods 59 3.3 Assumptions 62 3.4 Results, 64 3.5 Comparison of AC and DC Models 70 3.6 Graphical User Interface for Power Transaction Simulation 77 3.7 Summary - Different Policies Imposed by Regulators 79 Chapter 4 Decision-Making Aid for Market Participants 83 4.1 Electricity Markets 84 IV 4.2 Market Observation 88 4.3 Forecasting Method for Market Participants 97 4.4 Electricity Consumption Forecasting 106 4.5 Electricity Price Forecasting Based on Demand Data 114 4.6 Consumer Decision-Making Aid , 125 4.7 Summary - Decision-Making Aid for Market Participants 133 Chapter 5 Conclusion 135 Chapter 6 Future work 138 Chapter 7 Bibliography 140 Appendix I Sequential Quadratic Programming 150 Appendix II IEEE 30 Bus System 152 Appendix III Multiobjective Optimization 154 Appendix IV Market Observations 157 Appendix V Fuzzy Set and Possibility Theory 159 v List of Tables Table 1 List of symbols for Chapter 2 and Chapter 3 viii Table 2 List of symbols for Chapter 4 ix Table 3 Price function coefficients of generators... 46 Table 4 Benefit function coefficients of customers 47 Table 5 Power exchange market combination 48 Table 6 Parameters of generator at bus 2 after adjustment for strategic bidding ...48 Table 7 Total social welfare of plural PX 49 Table 8 Total social welfare after strategic bidding 50 Table 9 Relationship of plural power exchange market and strategic bidding 51 Table 10 Lists of congested lines 53 Table 11 Generator characteristics (active power) 63 Table 12 Generator characteristics (reactive power) : 63 Table 13 Customer characteristic 64 Table 14 Extreme and operating points in Figure 12 65 Table 15 Transmission losses in AC system 67 Table 16 Extreme and operating points in Figure 15 69 Table 17 Social welfare and emission 71 Table 18 Active constraints for DC and AC models 72 Table 19 Marginal social welfare of reactive power ($) 73 Table 20 Social welfare and emission under Condition 1 74 Table 21 Social welfare and emission under Condition 2 74 Table 22 Market size of PJM and California..... - 90 Table 23 Daily volatility of NYMEX Futures at PJM Western Hub 96 Table 24 Daily volatility of NYMEX Futures at COB 96 Table 25 Parameters of fuzzy numbers •••••• 109 Table 26 Vagueness of models 112 Table 27 Vagueness of forecasted data 112 Table 28 Parameters for first-order model 119 Table 29 Parameters for second-order model 119 Table 30 Parameters for first-order model 122 Table 31 Parameters for second-order model 122 Table 32 Performance evaluation (y) 133 VI List of Figures Figure 1 Independent System Operator 8 Figure 2 Clearing process of power exchange market : 20 Figure 3 Max-ISO model . 30 Figure 4 Min-ISO model : • 33 Figure 5 Mix-ISO model 37 Figure 6 PJM market structure : 39 Figure 7 California market structure 40 Figure 8 Network configuration (30 bus system) 43 Figure 9 Relationship between strategic bidding and plural-PX condition 51 Figure 10 Relationship between strategic bidding and plural-PX condition without network constraints 52 Figure 11 Linear membership function 61 Figure 12 Tradeoff relationships in AC and DC systems 64 Figure 13 Generator outputs and customer demands in AC model 66 Figure 14 Generator outputs and customer demands in DC model 66 Figure 15 Three-dimensional relationship 68 Figure 16 Generator output at operating points 69 Figure 17 Customer demand at operating points 69 Figure 18 Generator output at operating point A 71 Figure 19 Customer demands at operating point A 72 Figure 20 File structure 78 Figure 21 Graphical user interface of PSOS 79 Figure 22 Geographical area of PJM market 90 Figure 23 Locational prices of PJM market in 1999 91 Figure 24 Geographical area of California market 92 Figure 25 Locational prices of California market in 1999 92 Figure 26 NYMEX Futures at PJM Western Hub 94 Figure 27 NYMEX Futures at COB 95 Figure 28 Symmetrical triangular fuzzy set 99 Figure 29 Unsymmetrical fuzzy set , 101 Figure 30 Recorded electricity consumption in California 107 Figure 31 Autocorrelation function values of the California data 108 Figure 32 Time-series data of electricity consumption by fuzzy autoregression models 111 Figure 33 Forecast by fuzzy AR models 114 Figure 34 Market data of CalPX in 1998 116 Figure 35 Regression applied to California demand-price data 117 Figure 36 Fuzzy relationship between demand and price at CalPX 119 Figure 37 Data clusters 121 Figure 38 Fuzzy models on preconditioned data 123 Figure 39 Error rates versus a-cut 124 Figure 40 Fuzzy demand-price model with a = 0.5 124 Figure 41 An example of price time-series estimated by the fuzzy model 125 Figure 42 A fuzzy set to represent the consumer's preference 129 Figure 43 Measurement of price differential 130 Figure 44 Price range determination from January and February 1999 131 Figure 45 Forecasted price range for March 1999 132 Figure 46 An example of a-cuts 159 vii List of Symbols Table 1 List of symbols for Chapter 2 and Chapter 3 active power output of generator (MW) n min Gi minimum active power output of generator (MW) rf max maximum active power output of generator (MW) PC, preferred schedule of active power for generator (MW) QGl reactive power output of generator (MVar) minimum reactive power output of generator (MVar) QT maximum reactive power output of generator (MVar) coefficient of generator price function ($/MW2) bi coefficient of generator price function ($/MW) Ci coefficient of generator price function ($) fs(Pc) marginal price function of generator active power demand of generator (MW) pmin A minimum active power demand of customer (MW) nmax A maximum active power demand of customer (MW) Qo, reactive power demand of generator (MW) preferred schedule of active power for customer (MW) di coefficient of customer benefit function ($/MW ) e-i coefficient of customer benefit function ($/MW) fi coefficient of customer benefit function ($) MPD) marginal benefit function of customer ps active power injection at bus i (MW) Qf reactive power injection at bus i (MVar) 0 , bus angle at bus i (per unit) v, bus voltage at bus i (per unit) ~y min minimum bus voltage at bus i (per unit) T / " max maximum bus voltage at bus i (per unit) G, conductance of admittance matrix element i, j Bij susceptance of admittance matrix element i, j pL total active power loss (MW) a off-nominal tap ratio of a transformer PV active power flow between bus i and bus j (MW) rjniax U active power flow limit between bus i and bus j (MW) X vector of real numbers vector of Lagrange multiplier for equality and inequality constraints viii Table 2 List of symbols for Chapter 4 X(t) estimated value (fuzzy number) at time t t t is discrete time of an equal interval m order of model description xi observed variable Ai fuzzy coefficient ai crisp parameter of triangular fuzzy set (see Figure 28 or Figure 29) crisp parameter of triangular fuzzy set (see Figure 28) + upper crisp parameter of triangular fuzzy set (see Figure 29) c'i lower crisp parameter of triangular fuzzy set (see Figure 29) estimated price (fuzzy number) at time t ^ o f f offered price for bilateral contract Pmif) market price at time t G trapezoidal fuzzy set defining preference of consumer membership value of fuzzy set G gi crisp parameter of trapezoidal fuzzy set (see Figure 42) So crisp parameter of trapezoidal fuzzy set (see Figure 42) D(i) fuzzy set defining price differential Ho membership value of fuzzy set D MDG membership value of overlapping G and D Y overall evaluation of offered price for bilateral contract IX List of Acronyms AC Alternating Current AR AutoRegression ATC Available Transfer Capability CalPX California Power Exchange CBOT Chicago Board Of Trade DC Direct Current FAR Fuzzy AutoRegression FERC Federal Energy Regulatory Commission IPP Independent Power Producer ISO Independent System Operator NERC North American Electric Reliability Council NGC National Grid Company OPF Optimal Power Flow problem PJM Pennsylvania-New Jersey-Maryland PSOS Power System Operation simulator Support environment PURPA Public Utility Regulatory Policy Act PX Power eXchange QP Quadratic Programming problem SC Scheduling Coordinator SQP Sequential Quadratic Programming problem TAC Total Adjustment Cost TRM Transmission Reliability Margin TSC Total Supply Cost TSW Total Social Welfare TTC Total Transfer Capability UMCP Unconstrained Market-Clearing Price VIU Vertically Integrated Utility X Acknowledgements I owe my gratitude to all the people who helped me accomplish this work. First and foremost, I would like to express my deepest gratitude to my supervisor, Professor Takahide Niimura for his patience and professional guidance during my studies. I also feel grateful towards Professor Hermann Dommel for his support in my research. Conducting this work would not have been possible without him. I would also like to thank Professor Masao Nakamura for introducing me to the field of economics, which was new to me, and guiding me through it. Much valuable advice from Professor Kazuhiro Ozawa, Hosei University, was greatly appreciated. The financial assistance of the Natural Science and Engineering Research Council of Canada, and of BC Hydro & Power Authority, through funding provided for the NSERC-BC Hydro Industrial Chair in Advanced Techniques for Electric Power Systems Analysis, Simulation and Control is gratefully acknowledged. The financial assistance of the Centre for Japanese Research, UBC, is also greatly appreciated. I would like to thank all of the Power Group members at the University of British Columbia for their friendship, and I am grateful for receiving continuous and friendly encouragement. This work would hardly have been complete without them. Last, but not least, I would like to thank my parents for their unconditional love, help, and understanding. Tomoaki Nakashima The University of British Columbia July 10, 2000 xi Chapter 1 Introduction In this chapter, the summary of this research work is presented first. Following the summary, the background concerning the research work is explained. Lastly, the objectives and methods of the research work are defined. 1.1 Synopsis of Work Accomplished Electric power has been traditionally supplied to customers at regulated rates by vertically integrated utilities (VIUs), which own generation, transmission, and distribution systems. However, the regulatory authorities of VIUs are promoting competition in their businesses to lower the price of electric energy. Consequently, in new deregulated circumstances, many suppliers and marketers compete in the generation market, and conflict of interest may often occur over transmission. Therefore, a neutral entity, called an independent system operator (ISO), which operates the power system independently, has been established to give market participants nondiscriminatory access to transmission sectors with a natural monopoly, and to facilitate competition in generation sectors. Several types of ISOs are established at present, with their respective regions and authorities. The ISO receives many requests from market participants to transfer power, and must evaluate the feasibility of their requests under the system's condition. In the near future, regulatory authorities may impose various objectives on the ISOs, which are not restricted to economic benefits, but, for example, to the reduction of environmental impact or network security, which are often conflicting. Then, based on the regulators' policies, which have negatively correlated, often conflicting, and tradeoff relationships, the ISO has to determine the optimal schedules from feasible solutions, or change requests in an agreeable and reasonable way amongst market 1 participants. In a newly developed power market, market participants will conduct their transactions to maximize their profits. The most crucial information in conducting power transactions is supply and demand, which determine the market price of electricity. In a power exchange market, power suppliers and power consumers enter sealed bids for short-period contracts (typically 1/2 or 1 hour). Market prices are then determined by an auction process, subject to feasibility constraints imposed by the ISO's objective function. Market participants observe the equilibrium prices, but not the bids of their competitors. By comparison, information is completely private when market participants negotiate contracts directly. Such direct transactions can be attractive if market participants are risk-averse, and wish to enter long-term contracts, but they also provide an opportunity for power suppliers to price-discriminate amongst their contract partners. I have approached the problems arising from the new environment surrounding the deregulated power industry from three viewpoints: Firstly, I have explored desirable system operation and power market structure from the system operator's point of view [1]. Two typical ISO models, centralized and decentralized, have been identified and compared based on actual ISO models. In the centralized system, all generators and customers receive orders from the ISO based on the price information they give to the ISO. In the decentralized system, all participants are allowed to trade power freely by keeping their supply-demand balance. These ISO models have been simulated to observe the advantages and disadvantages of the 2 different systems. The effect of reduced market efficiency due to strategic bidding in the market was examined. If no powerful players exist, the centralized system will achieve the maximum market efficiency. However, in the decentralized systems, freedom of trade protects market participants from strategic bidding caused by powerful players. Reduced market efficiency is the price markets have to pay to prevent strategic bidding. Secondly, I have examined the effect of different policies imposed by regulators on power transactions [2] [3] [4]. Since the ISO is a monopoly in the transmission sector, often formed as a non-profit organization, it is still under the control of the regulatory authority. Therefore, the regulator may find it practical to impose several policies directly on the ISO operation. The policies of the regulators can be various, and are not restricted to economic efficiency. For example, the total amount of environmental emission in the ISO service area can be constrained. Then, the ISO will have to coordinate between the outputs of economically efficient but environmentally polluting generators, and the outputs of less efficient but cleaner generators. Therefore, the ISO needs to solve the multiobjective optimizations that have highly negatively correlated, often conflicting, and tradeoff relationships between objectives. From possible combinations of the feasible solutions, optimal schedules must be determined by the ISO. The optimal schedule could be affected greatly by the ideal goals and their allowable values, possibly working in favor of some specific market players. Therefore, when the ISO defines its objectives and their allowable ranges, an agreeable conclusion amongst the ISO and market participants is required. Fuzzy multiobjective optimization methods can be suitably applied to the scheduling of the ISO, reflecting its objectives and their allowable ranges properly. Thirdly, I have developed models to represent and forecast the price and demand of power from 3 the market participants' point of view. Such a modeling of demand and price is vital for market participants as they compete in the power market, as it is in stock or foreign currency exchanges. When the consumers are offered a direct supply (bilateral) contract, they will probably compare the offered price to the market price. Therefore, suitable methods to represent and forecast the volatile market prices are necessary. Time-series data of electricity consumption are forecasted based on possibility theory and fuzzy autoregression [5]. Two models have been developed, and both models have been applied to the forecasting of real-life data. The proposed fuzzy autoregression models preserve the rich information of the original data, including uncertainty. Electricity price is forecasted based on a fuzzy regression model [6]. The fuzzy model can represent highly volatile demand-price relations as a range, and give the possibility distribution of prices when a single value (point) of demand forecast is given. In the conventional regression analysis, the data set is represented by a single line or curve. The decision-maker's intention to focus on a particular range could be accommodated by narrowing down the fuzzy model. Based on the forecast model developed above, a procedure to help consumers decide whether to accept a bilateral transaction contract or market-based purchase of electricity has been developed [7] [8]. The same procedure can also be used by an electricity supplier or broker to determine an attractive price to offer the target consumer, when the consumer preference is uncertain. 4 1.2 Background The conventional electric power industry has been mostly monopolized by large electric companies, called vertically integrated utilities (VIUs), which own generation, transmission, and distribution systems to supply power to captive customers at regulated rates. This situation was justified because of economies of scale, which meant that larger generation plants were more efficient. However, these services, once considered as a natural monopoly for VIUs, are rapidly experiencing massive changes [9], [10]. In the conventional environment, consumers' demands were assumed independent of price, and the vertically integrated utility owned whole systems — generation, transmission, and distribution. Therefore, the objective of the utility was to minimize operating cost and meet fixed demand subject to several technical constraints. The utility set the outputs of its generators by loading the units over which the utilities have complete control, in the order of increasing marginal cost (i.e., in merit order). The loading was subject to constraints associated with the physical limitations of the generators and transmission lines, and security constraints. To improve the security or voltage profile, the utility could change the output of the generators or install new transmission lines in order to control the system conditions. Finding the optimal operation schedule subject to constraints is called the optimal power flow (OPF) problem. The deregulation process, however, has reached the electricity industry, as well as other infrastructure industries such as gas, telecommunications, and transportation. The electricity industry is changing from tight regulation of vertically integrated monopolies to limited regulation of functionally unbundled firms. The justification for deregulating the electricity industry is mainly based on the economic concept that free competition will bring about price 5 reduction through improvement in market efficiency, and partly through new technology introduced by small, efficient-plants, such as gas turbine generators that diminish economies of scale. The easy availability of inexpensive fuel, such as natural gas, also enables smaller generating units to compete against conventional generating units. In Britain, the formerly nationally-owned and centrally operated power utility was privatized, and broken up into separate companies in the generation, transmission, and distribution sectors in 1990. Norway followed suit in 1991, and several South American countries, such as Argentina, Chile, and Brazil, introduced Britain-like privatization and deregulation policies during the early 90s. Australia started a similar initiative in Victoria State in 1996. In the United States, where many utility companies are privately owned, the movement toward industry restructuring was initiated by the federal regulator in 1995, and is now being implemented in California, Texas, New England, other states, and regions where electricity prices are relatively high. Deregulation has divided the role of the electric utilities roughly into three service sectors: generation, transmission, and distribution. The generation and transmission sectors are technically challenged by deregulation. In the distribution sectors, where the monopoly over captive customers is supposed to be natural, deregulation is considered inappropriate in protecting final customers. However, in a few areas, a retail supply competition is either planned or established to allow customers to choose their suppliers. In the US, the first deregulation was enacted by the Public Utility Regulatory Policy Act (PURPA) in 1978. This legislation required the VIUs to purchase power from independent power producers (IPPs) at their avoided cost, which is the cost of producing power by VIUs 6 themselves without IPPs. It was intended by the government to promote new alternative means of energy generation and energy saving. However, it introduced competition into the generation sector amongst IPPs, thanks to the decline of gas prices, the encouragement of new alternative energy sources, and technical innovations such as high-efficiency turbines. These were usually co-generation, small-scale hydro, wind, geothermal, or other types of renewable energy sources. Present IPPs can even sell power to individual entities by participating in the generation market. The VITJ-owned generators have been divested by order of the regulatory authority, and have become IPPs in order to maintain fair competition among generation market participants. The second deregulation was initiated by the Energy Policy Act in 1992. This legislation led to the Federal Energy Regulatory Commission (FERC) order [11], and brought so-called open access, which granted anybody nondiscriminatory access to use the transmission system for delivering power. The order was aimed at facilitating competition in the generation market because competition will become possible in a wide range of areas, where the power systems are largely interconnected to other systems. On the demand side, in such places as Britain and California, individual consumers are allowed to choose the power supplier of their preference, as with long-distance telephone services. Thus, open access to the transmission systems has enabled any entity to transmit power by a third party using the transmission system on a nondiscriminatory basis. Such a trend towards a competitive, market-based electricity industry is changing transmission service from a natural monopoly to nondiscriminatory service for any entity. To ensure nondiscriminatory access to the transmission system and avoid conflicts of interest in its use, FERC proposed an independent neutral entity to operate the power system under 7 competitive circumstances, which schedules and dispatches generation plants subject to the constraints of the transmission system. Such an entity is called an independent system operator (ISO). The ISO is responsible for controlling the transmission system, and is completely independent of generation entities. The ISO has to preserve the integrity of the power system, whilst considering many new types of market participants. The ISO has already been implemented in several countries, including the US. Generating Co Generating Co Generating Co. Generating Co. Bilateral contract Power Exchange Scheduling Market clearing Schedules of Bilateral contract Power Exchange Independent system operator System dispatch Local distributor Energy service provider Figure 1 Independent System Operator The concept of the ISO was first realized in the United Kingdom. In 1990, the UK power system was privatized and separated into generating companies, distribution companies, and one transmission utility. This utility is called the National Grid Company (NGC), which owns and operates the transmission system. The NGC also functions to operate a spot electricity market, called the Pool system. The Pool system is a merit order system that dispatches generators in order of their increasing marginal prices (bids), every half-hour. The costs associated with maintaining system security and reliability are recovered by the NGC. The Pool 8 system in North America is called the Power Exchange (PX). The PX is a financial market whose participants include generators and customers. It handles power transactions while ignoring physical constraints. The ISO is a control authority that adjusts their transactions considering physical constraints. The NGC is playing both roles. However, in the US, the functions of ISO and PX are considered separate. The reason is that in some parts of the US any party can buy or sell power directly through bilateral contracts as well as through the PX. By separating PX and ISO, bilateral contract parties and PX participants can interact with the ISO in the same way, and obtain the same nondiscriminatory treatment. The main responsibility of the ISO is to provide transmission service while maintaining network security. Therefore, the ISO's objective is to operate the power system subject to several technical constraints, and to calculate how much power market participants can transmit over the network (available transfer capability). The ISO also has to provide services, called ancillary services, such as reactive power or frequency control, which are necessary in facilitating power transmission while maintaining reliable operation of the transmission system. The ISO has to incorporate market participants' behavior into its own OPF problem [12] and define its service charge. Whenever the ISO allows any transactions, it raises concerns about violating system constraints, increasing system losses, or compromising system security. Therefore, the ISO has to guarantee sufficient transmission capability to maintain a reliable and secure supply, and has to price its services in consideration of system conditions. The prices for those services must induce economic efficiency. The cost of the service should be differentiated by the configuration of the network and its inherent capability. Marginal-cost based service pricing is said to provide the incentives required for generators and customers to locate efficiently in the power system. The users of the system, the buyer and the seller of the power, have to pay for the transmission and ancillary services provided by the ISO. 9 The introduction of competition has drastically changed the power system condition and produced the concept of the ISO. However, depending on market conditions, the authority of the ISO can vary significantly. Two types of ISO roles in the power market can be considered. One type of ISO obtains all bid information from sellers and buyers, and schedules optimal transactions subject to technical factors such as transmission constraints. This ISO can be called max-ISO, since this ISO has full control over the network. The objective of this ISO is to maximize social welfare in the network. This ISO is also functioning as PX (power exchange). Another type of ISO does not obtain bid information, but instead receives preferred or desirable schedules of generator outputs and customer demands, and reschedules their outputs and demands to satisfy the technical constraints. The preferred schedules are determined outside the ISO by power exchange markets where markets are cleared based on bid information or by bilateral contracts between individual market participants. This ISO can be called min-ISO, since this ISO can only readjust schedules if necessary. Min-ISO does not play a power exchange role as max-ISO does. However, the power exchange or bilateral contracts have to give the min-ISO information about their willingness to increase or decrease their outputs, demands, or prices if their schedules are adjusted by the ISO. While power exchange markets will ideally lead to greater market transparency and greater competition, they also provide opportunities for power suppliers to engage in strategic interactions with their rivals. For example, a power supplier can misrepresent its cost schedule to the ISO in order to influence the outcome of the ISO's optimization process, and thus the market price. Such actions may be particularly effective when the network is congested or demand is close to the capacity limit, while under normal circumstances such actions would be punished by competitive market forces. 10 A notable feature of the power supply markets is their relatively high concentration. While the vertical separation of the transmission function from the generation function provides the potential for more competition in the power supply market, the reality is totally different. For example, in the UK there were only two dominant competing power suppliers at the beginning of deregulation. The limited number of suppliers gives these companies oligopolistic market power. Evidence from the UK points to an increase in profitability of electric utilities after their privatization [13]. In addition to facilitating market transactions, the primary role of the ISO is to operate the transmission network and to ensure that it meets the standards of economic and technical efficiency. By itself, the transmission network remains a natural monopoly, which is therefore subject to government regulation, if not government ownership. In practice, ISOs are virtually always run as public non-profit entities that maximize a government-mandated social welfare function. In the near future, regulatory authorities may impose various objectives or constraints on independent system operators (ISOs), which are not restricted to social welfare maximization, but, for example, also include environmental impact reduction or network congestion limitation for reliability. These constraints or objectives have to be defined by the regulators based on their policies, giving explicit weights that govern the tradeoff between conflicting objectives and constraints. Then, the ISOs have to solve multiobjective optimization problems that have negatively correlated, often conflicting, and tradeoff relationships between objectives and constraints. From possible combinations of the feasible solutions obtained by multiobjective optimizations, the ISOs must determine the optimal schedule. 11 In a newly developed power market, many market participants, such as buyers and sellers of power, will conduct their transactions in order to maximize their profits. Electricity is typically traded either in a power exchange market or by bilateral contracts, paying certain tariffs to transmission service providers. Many transactions are affected mutually, and can be often conflicting because of the externality of the physical network constraints. The condition of the network is monitored only by the system operator, and detailed information on the transmission system is rarely available to market participants. They obtain partial information such as the price or forecast of the total demand from the ISO. The most crucial information in conducting power transactions is price and demand. A direct transaction between suppliers and consumers may become attractive in some situations because of its stability of price, while in a power exchange market, gaming and speculation of participants may push up electricity prices considerably. When the consumers are offered a bilateral contract, they will probably compare the offered price to the market price. However, the market price is significantly volatile. Therefore, suitable methods for forecasting the volatile market price are necessary. Traditional forecasting methods are based on statistical and probabilistic approaches, and it may not be entirely suitable in applying purely stochastic models to the data generated by human activities such as the power exchange market. 1.3 Objectives The objective of this research is to provide three viewpoints. In the first part of this research, desirable ISO functions for different market structures are examined from a power system operators' point of view. In the second part of this research, the effect of various policies imposed by regulators on the power transaction is examined from the regulators' point of view. 12 A graphical user interface is also introduced to help non-expert personnel interpret numerical results easily. In the third part of this research, a decision-making aid in a newly developed market is examined from the market participants' points of view. The objectives are summarized as follows: 1.3.1 ISO Functions for Different Market Structures Two dominant types of ISOs, centralized and decentralized, have been considered in the power market, depending on the authority of the ISOs. The centralized ISO controls and coordinates both energy trade and transmission services. The centralized ISO works well under perfect competition on both supply and demand sides, and optimizes the system completely to increase efficiency. However, the centralized ISO ignores effects on profit-making incentives and gaming by participants. Participants can exploit their market powers and strategically manipulate price information, and the optimization may not work without strong incentives to ensure that bids reflect actual costs. On the other hand, the decentralized ISO only controls the transmission market, and allows participants to manage their own transactions by promoting competition in energy markets. The decentralized ISO works well under good scheduling decisions by each participant. The objective of "ISO Functions for Different Market Structures" is to examine the two dominant types of ISOs carefully. The characteristics of these ISOs, especially the social welfare of the market or system and vulnerability to strategic bidding by market participants, are evaluated. 13 1.3.2 Different Policies Imposed by Regulators At present, existing ISOs are mainly considering the economic factors of the power system. However, in the near future, regulatory authorities may impose different objectives on the ISO in addition to the economic factors. Then the ISO will have to take into account other factors, such as environmental impact, which can affect the economy or technical profile of the network. The impact upon system conditions must be evaluated from multiple perspectives, such as cost, environmental impact, network congestion, and security. These attributes are not independent, but most often these constraints or objectives are conflicting. Therefore, optimization of the power system has not one solution, but many possible solutions because the ISO can reduce cost at the expense of some constraints, such as the security of the system. The actual operating points amongst those solutions have to be determined by the ISO based on a certain policy. The objective of "Different Policies Imposed by Regulators" is to evaluate the tradeoff relationships between operating cost and other attributes of the power system, such as security, to help the regulators determine an operating policy based on several objectives. By showing tradeoff relationships, the regulators are able to appreciate the cost to satisfy their policies. The tradeoff relationships also enable market participants to understand the way markets will be restricted or intervened upon by the regulators' goals. Here, a graphical user interface program is developed to help non-expert regulators in understanding many numerical values showing the conditions of power systems, and to evaluate the system condition easily. 1.3.3 Decision-Making Aid for Market Participants The first phase of "Decision-Making Aid for Market Participant" is forecasting electric power consumption. It is becoming all the more important for retail suppliers to accurately forecast 14 electric power consumption under uncertainty. For example, while residential customers or commercial users are charged not on a time of use basis, but flat rate, retail suppliers have to purchase power from generation suppliers in the deregulated power market. In order to gain profit in the open access power market, forecasting electric power consumption is quite challenging. The second phase is forecasting electricity price based on PX market demand. If participants decide to join the day-ahead forward market and determine their bids, additional information such as the total market demand forecast can be obtained from the ISO. The forecast of electricity demand is studied extensively, and recent techniques can forecast the short-term (day-ahead) demand with a small error. Here, the relation between electricity demand and market price with volatility is examined. Such a model can be used to estimate the possible range of prices when demand information is given. The third phase is assisting an electricity consumer to make decisions on a bilateral power transaction under uncertain market conditions when a bilateral price is offered. In practice, bilateral transactions are observed to stabilize the potentially volatile market, but it is quite uncertain what price level would satisfy the participant and what price would be acceptable. 1.4 Methods 1.4.1 ISO Functions for Different Market Structures The centralized and decentralized ISOs are compared from two aspects. In one aspect, the effect of allowing market participants to choose their own partners in the physical network is examined. By calculating the difference of the total social welfare between centralized and 15 decentralized ISOs, the effect of free market activity can be quantified. In another aspect, vulnerability to strategic bidding is examined. Here, price parameters of a given participant are changed to include the effect of strategic bidding. By calculating the difference between the total social welfare of different cases without strategic bidding and with strategic bidding, the effect of strategic bidding can be quantified. 1.4.2 Different Policies Imposed by Regulators The effects of the ISO's different objectives imposed by regulators' policies upon a power transaction are examined. The OPF method is used to determine the optimal schedules that would maximize the total social welfare of the system subject to several technical constraints. The total social welfare is formulated as an objective function, with transmission line capacity as constraint. However, several other constraints can be considered, for example, environmental impact or network congestion. Then, those constraints can also be seen as objectives with social welfare as one constraint. Therefore, the OPF problem can be regarded as a multiobjective optimization that has negatively correlated, often conflicting, and tradeoff relationships between objectives. Finally, the optimal operational schedules are determined based on a fuzzy multiobjective optimization method. 1.4.3 Decision-Making Aid for Market Participants Firstly, based on possibility theory and fuzzy sets, two fuzzy autoregression methods are introduced. The two models are different in complexity, and the performance of the models is evaluated by vagueness and a-cuts. The proposed fuzzy autoregression models represent the rich information of uncertainty that the original data contain. Then, the proposed fuzzy autoregression methods are applied to the analysis of time-series data of electric power consumption to forecast consumption. 16 Secondly, a fuzzy regression method is introduced for determining the relationship between demand and price. Conventional regression analysis is applied to a given data set to determine the most likely relation. The regression parameters are later extended to triangular fuzzy numbers. Through such a fuzzy model, a possible range of electricity prices can be estimated, along with volatility information. Then, a fuzzy regression method is applied to actual data in a deregulated market. It is assumed that the base price of electricity predominantly reflects the marginal cost under a competitive market environment. It is also assumed that price volatility occurs mainly when the supply-demand balance is tight. When abundant sellers are found against buyers, on the other hand, sellers would compete to obtain transactions, and the prices would eventually settle down to the marginal-cost based prices. Lastly, the method of fuzzy autoregression is applied to market participants' decision making regarding electricity transactions. As an example, retail customers of electricity are assumed to have a choice between bilateral transactions and market-based purchases. Here, fuzzy autoregression analysis is performed on time-series data of electricity market prices, and uncertain market prices are represented by a band of fuzzy numbers, containing highest/lowest price range in a time-series. The consumer's somewhat vague preference is also defined by a separate fuzzy set. Then, the differences between the offered price for a bilateral contract and the market price are compared with the preference index. The overall grade of matching the price differential with the consumer's preference indicates the value of the offered price for the bilateral transaction. The thesis is organized as follows: in Chapter 2, the roles of the ISO are examined closely. The functions of the several types of ISOs are identified, and these ISOs are compared. In Chapter 17 3, the effect of the regulators' different policies toward power transactions is evaluated. In Chapter 4, several types of market structures are explained first. Then, data from existing markets are observed. Then, fuzzy autoregression methods are introduced, and applied to the forecast of electricity consumption, electricity price, and the consumer decision-making process. In Chapter 5, this research's conclusions and achievements are summarized. In Chapter 6, possible future work is proposed. 18 Chapter 2 ISO Functions for Different Market Structures In Chapter 2, two dominant types of independent system operators (ISOs) are identified. One is the centralized ISO, where all market participants receive orders from the ISO. Another is the decentralized ISO, where all participants are allowed to trade power freely. The effects of reduced market efficiency in the decentralized ISO are examined in comparison with the effects of strategic bidding in the centralized ISO, using a numerical example. 2.1 Deregulated ISO Functions and Market Structures In deregulated power industries, market activity is separated from the control of the power system, and power exchange markets (PXs) are established. In PX, each supplier submits a bid to sell electricity in the form of a price curve, the supplier's marginal cost curve, which must be non-decreasing in accordance with the law of diminishing returns [14]. If these bids are aggregated, the supply curve can be drawn as in Figure 2. The supply curve shows the quantity (power) that would be sold at various market prices. This is similar to the cost curves in conventional economic dispatch. The upward slope of the supply curve leads to the concept of producer surplus. At point B (where supply curve and price line intersect), the marginal cost of the power is equal to the market price. However, for all power sold before point B, the cost of the power is smaller than the price. This difference between cost and price is the producer surplus (area BCD in Figure 2). In conventional electric utility industries, power consumption has been relatively insensitive to the power price (rate) because the price has been determined by the utilities irrespective of the consumer response. In a deregulated environment, however, power consumption is elastic, and 19 can be expressed as functions of price. Therefore, as each supplier does, each buyer submits a bid to purchase electricity in the form of a price curve, which is non-increasing in accordance with the law of diminishing marginal utility [14]. If these bids are aggregated, the demand curve can be drawn as in Figure 2. The demand curve shows the quantity (power) that would be purchased at various market prices. The downward slope of the demand curve leads to the concept of consumer surplus. At point B (where the demand curve and price line intersect), the marginal value of power is equal to the market price. However, for all power purchased before point B, the value of the power is greater than the price. This difference between value and price is consumer surplus (area ABC in Figure 2). Price L D Market clearing price N ^  Power p Figure 2 Clearing process of power exchange market Each market participant seeks the best price so as to maximize its surplus. For example, each generator compares the market price to its own marginal cost, and determines its output. Similarly, each customer compares the market price with its own marginal benefit and determines its demand. An iterative procedure of changing the market price will eventually balance total demand and supply; hence the market will be settled in equilibrium ("cleared") 20 where the demand and supply curves intersect (point B in Figure 2). By determining a uniform market-clearing price, the market settles upon transactions that maximize social welfare, the total of producer and consumer surplus (area ABD in Figure 2). In deregulated power systems, competition among market participants such as generators and customers is performed on the physical network. The system operator is required to give non-discriminatory access to all participants. However, there may be some occasions when their preferred transactions are infeasible. When the demand for a transmission link exceeds its safe transfer capacity, it is called 'congested'. Congestion is inevitable since the network is constrained in several ways including those of transmission line capacities. The ISO must coordinate their preferred transactions and assign transmission resources to them ahead of time so as to manage congestion within the power system. For this reason, major PX market power transactions are conducted at least one day-ahead to allow the ISO to intervene. Therefore, the PX market is actually a (day-ahead) "forward" market, not a real-time market. Thus, the energy market is organized by the PX, and the transmission resource assignment or transmission market is managed by the ISO. Congestion management in the transmission market is the main responsibility of the ISO. Details of market functions are discussed in section 4.1. Two distinct models exist for actual power markets. One is called the pool model. In the pool model, such as in UK, all suppliers and customers transact with the pool, where the most efficient source is dispatched subject to network constraints based on price and quantity information from suppliers and customers. Therefore, the responsibilities of a power exchange (PX) (determining generators' outputs, or economic dispatch functions) is performed by the ISO. This idea is based on the concept that a natural monopoly would ensure the least cost dispatch of all generators in the system. Another dominant model is called the bilateral model. 21 In the bilateral model, such as in California or NordPool (Norway) [15], all suppliers and customers transact with each other independently. Then, their schedules are coordinated by the ISO, a separate authority from the PX. This idea is based on the concept that regulation of the commercial market should be minimized, and that market efficiency is achieved by customers choosing their own suppliers. In the pool model, all generators are dispatched so as to avoid network congestion, based on price bids, which may be quite different from the actual costs as determined by bidders' strategies. The pool design requires an incentive for market participants to reveal their true costs for market efficiency. Consequently, a nodal pricing scheme was introduced, where energy prices are adjusted reflecting locational values caused by loss and congestion. The locational price can be calculated if all the price data and network data are given to the ISO. The charges for transmission between buses are based on the locational prices. The original concept was proposed in [16], and extended to the pool model in [17]. The pool model is also supported in [18], based on the reason that property rights cannot be well-defined in the electricity market. However, in [19], the locational pricing method was tested on the western US system, and was concluded to be questionable since the locational price is too sensitive to network conditions. In the bilateral model, if all preferred schedules determined by the bilateral model are feasible, or no congestion exists in the physical power system, those schedules can be accommodated by the ISO easily. However, if congestion occurs, schedules must be adjusted so as to alleviate congestion of the system. Therefore, some coordination among transactions is necessary. Study of [ 19] supported the bilateral model, and suggested that the strategic behavior can drive up the locational price in the pool model and that bilateral contracts can eliminate locational price 22 differences. The coordination of bilateral and multilateral transactions was examined in [20], and the underlying conflicts were shown. In [21], the adjustment of multilateral schedules due to network constraints was determined by a "willingness to pay price." The multilateral transactions can be considered the preferred schedules of a power exchange, since total supply is equal to total demand in each multilateral transaction. Several curtailment strategies were compared with variations of the willingness factors. The resultant adjusted generations and loads were examined closely. However, the effect of market size was not examined. In [22], the coexistence of bilateral and pool models was examined, where a two-stage transmission dispatch model was explored with priority given to either model. 2.2 Mathematical Models for System Operators Finding the optimal operation schedule subject to constraints is called the optimal power flow (OPF) problem [23]. The ISO has to apply the OPF problem to the target system. The formulation of the original OPF is expressed as follows: 2.2.1 Optimal Power Flow (OPF) 2.2.1.1 Objective Function The objective of the conventional OPF problem was to minimize the total supply cost (TSC) of generators with a given inelastic demand. where PGi is the active power output of the i -th generator, and generators' costs are expressed as quadratic functions of its output. These cost coefficients of the units are a/, bi, and c/. Nc is (1) 23 the total number of generators assumed on-line in the system. 2 . 2 . 1 . 2 Constraints Active and reactive power at each bus must observe the following load flow constraints [23]: AC load flow constraints are expressed such that = rJX(G„cos(e, - ey)+*, since, -e,.)) (2) Qf = vifjvj(Gijsin(Qi -e,)-^cos(e,. -e.)) (3) where Vf and 9/ are the bus voltage magnitude and phase angle at the i -th bus (slack bus acts as reference; thus, at slack 6/ =0). Af is the total number of buses. Gtj and By are the conductance and susceptance elements of the bus admittance matrix Y . Pf and Qf are the net active and reactive power bus injections and are calculated at each bus as: Ps = P - P r i r G i r D , 4^") Q! = QG-QC (5) At generator buses, active and reactive power outputs are bounded, and at compensator buses only reactive power output is bounded by upper and lower limits as below: P™ <PGj< P™ ( 6 ) amin * Qo, * QT (7) The voltage magnitudes on all but slack buses are bounded by operating upper and lower limits such as: 24 MTI ax vr < v, < v; (8) The balance of the generation and load must be always satisfied. The generation-load power balance is given as below: 1=1 /=! (9) where PL is the total active power loss, and is a function of Kand 0. Power flow on the transmission line is constrained by its maximum flow limit. The limit is defined by thermal or stability factors. The active power flow on the line between buses / and j can be expressed as follows: v2 vv Pij = ~Gij -T + — \Gu c o s ( 9 / ~Qj) + Bu sin(9- " 9,)} a Q. / J (10) where a is the off-nominal tap ratio of a transformer when buses / and j are connected by the transformer. In the case of transmission lines, a is equal to 1. Active power flow (MW) is often approximated as the MVA flow on a line, neglecting reactive power flow. Then, security constraints on branches are expressed as follows: p < pmax v- v (11) 2.2.2 Optimization Methods The optimal power flow problems introduced in 2.2.1 include many nonlinear constraints such as AC load flows or power balance. The optimization problem with nonlinear constraints can be described as follows: 25 Minimize objective function f(x) Subject to Ci(x)=0, i e E : set of equality constraints Ci(x)>0, / e l : set of inequality constraints x = vector of real numbers (12) The local minimizer of the above problem can be found using the Kuhn-Tucker conditions. A Lagrange function is introduced using Lagrange multipliers1 X as follows: L(x,X) = f(x) - 2^,c,0) (13) The Kuhn-Tucker optimum conditions for the point x°, X° can be described as follows: • A set of partial derivatives of the Lagrange function that must be equal to zero at the optimum: — (x°,V) = 0 dx (14) • Equality constraint condition on the problem: cj(x°) = 0 for all ieE • Inequality constraint condition on the problem: c, . (x°)>0 for alii el (16) • Complementary slackness condition: A.°c, (x°) = 0, X° > 0 for all i el (17) A Lagrange multiplier of any constraint measures the rate of change in the objective function if its constraint function changes. It therefore indicates the sensitivity (marginal value) of the objective function to a change in the constraint. 26 Methods of optimization depend on characteristics of objective functions and constraints. The methods, which use second order derivative information such as a Hessian matrix, are called Newton methods. Because calculating the Hessian matrix directly is computationally expensive, many efficient methods approximate the Hessian matrix, and are called Quasi-Newton methods. One of the most efficient quasi-Newton methods is called Sequential Quadratic Programming (SQP), where second order derivative information from the Kuhn-Tucker conditions are used to reach the optimal solution. In the SQP method, the Hessian of the Lagrange function is approximated to a quadratic function using the quasi-Newton method, and the nonlinear constraints are linearized. The optimization of the quadratic objective function and linear constraints are called a quadratic programming (QP) problem, and can be solved efficiently in a finite number of steps. Therefore, in the SQP method, the original problem is approximated and transformed into a QP problem iteratively so as to reach the optimal solution. Appendix I explains the SQP method [24] thoroughly. The solutions of an original SQP program were compared with those of the MATLAB Toolbox [25]. The original program gave almost the same accuracy within a tolerable error, though it required more time to converge to a solution. This seems to be caused by its less-efficient way of finding active constraint sets among all constraints. 2.3 ISO Functions The independent system operator (ISO) provides transmission service to market participants. The ISO is also required to provide several supporting services to facilitate competitive power transactions, called "ancillary services." Ancillary services are explained later in 3.5.2. Among 27 these services, scheduling of transmission resources to market participants is very important in maintaining reliability within the power system. The ISO must manage congestion ahead of time by coordinating power transactions. The ISO has to schedule optimal transactions for maximizing the benefits associated with them. The maximum benefit is achieved by minimizing the total operation cost and maximizing the total consumption benefit to the power system. Therefore, the responsibility of the ISO is to maximize the system's total social welfare subject to several technical constraints, such as transmission line capacity or voltage limits. Unlike the objective function of conventional OPF, the ISO's objective includes demand elasticity. The effect of demand elasticity upon the power system constraints was first addressed in [26] without numerical examples. In [27], instead of total welfare maximization, individual welfare maximization was examined. 2.3.1 Constraints for ISO The following constraints must be observed in order to maintain the physical integrity of the power system. • Bus power balance: at all nodes (buses) total inflow and total outflow are equal; • Output bounds: all generators have feasible bounds for their output; • Total balance: Total demand plus transmission loss is equal to total supply; and • Line capacity constraint: All line flows must be within capable ranges. Among these constraints, line capacity constraints will affect power markets significantly. Even if each preferred schedule can be acceptable individually to the system, their combinations may cause violation of the transmission line constraints. The transmission line capabilities or available transfer capabilities (ATC) of the interconnected transmission systems for a 28 commercially viable electricity market are defined thoroughly by the North American Electric Reliability Council (NERC) in [28], along with definitions of total transmission capability (TTC) and transmission reliability margin (TRM). TTC is the amount of power that can be transferred over the interconnected transmission system in a reliable manner with the conditions of normal thermal, voltage, and transient stability limitations. TRM is the amount of transfer capability to ensure that the transmission system is secure under a reasonable range of uncertainties in system conditions. ATC is the difference between TTC minus TRM and the current loading on a specific path. When the demand for a transmission link exceeds its safe transfer capacity, it is called 'congested'. An uncongested transmission system is the pool, which unites all suppliers and consumers in a single market place. Congestion management is the main function of any ISO and is required to ensure that all the transactions are feasible without violating its operating limit. Violations will have a direct impact on the reliability or security of a power system. However, if improperly implemented, congestion management can work in favor of specific market players or thwart free electricity trades. Congestion management is thoroughly explained in [29]. 2.3.2 Max-ISO (Centralized system) Overall or comprehensive optimization is necessary in order to minimize the total cost or to maximize the total social welfare of a power system by coordinating generation and transmission. For the purpose of overall optimization, strong management of power markets by a system operator is justified, and such a system operator is called "max-ISO" here. Figure 3 shows the max-ISO model and its functions. 29 Seller Buyer Seller Buyer Power Exchange Generation Scheduling System operation Transmission coordination Max-ISO Figure 3 Max-ISO model Participants reveal their supply costs, demand values, and various technical constraints, such as generator output limits. The ISO optimizes both generation and transmission markets simultaneously subject to constraints, including inter-temporal factors such as startup commitments and ramping rates. The ISO serves an economic dispatch function, deciding which generators will be committed. In the max-ISO, from an engineering point of view, the ISO reduces flows or produces counterflows by directing various generators to decrease or increase their outputs. From an economical point of view, the ISO induces the counterflows by differentiating energy prices according to location. The energy price at a node is derived from optimization as the shadow prices of constraints, and reflects the marginal value precisely. The objective of conventional OPF was to minimize the total supply cost (TSC) of generators with inelastic demand known, and was expressed as follows: T S C ^ a ^ + b ^ + c ] (1) 3 0 In the case of the new deregulated market, the customers' demand is not assumed to be fixed. Since customers can express their willingness to consume power via benefit functions, as generators can do via cost functions, the objective of the max-ISO is to maximize social welfare in the system, subject to technical constraints such as transmission congestion. To calculate the total social welfare, each participant's supply or demand function must be given to the max-ISO. The ISO maximizes the total social welfare (TSW) of generators and customers by bringing the following economic objective function to maximum: The second term is the same as the total supply cost of generators as before. PDI is the active power demand of the i -th customer, and customers' benefits are also expressed as a quadratic functions of their demand. Those benefit coefficients of the customers are di, e/, and fi. No is the total number of customers assumed on-line in the system. This formula is quite similar to the conventional OPF objective function. It is different in that customers' benefits are included. Here, let fg(Pc) be the derivative of the sellers' supply function, namely the marginal cost function, and PG is the output of the generator. Then, the generator's cost can be expressed as follows: " r TSW = Y[diP^ + eiPDi+fi]-Y[aiP>i+biPGi+cl (18) i=\ cost = (19) In the same way, let fidfPo) be the derivative of the buyers' demand function, namely the marginal benefit function, and PD is the demand of the buyer. Then, the consumer's benefit can 31 be expressed as follows: benefit = ^  fd(PD)dP Pn fd(pD)dP o J d K D ^ (20) Then, the total social welfare (TSW) is given by: T S W =E \c f* (pD, )dp\ - z \ r { p°. ) d p (21) If no constraints exist, the optimal supply and demand are determined when the system (aggregated) marginal cost is equal to the system (aggregated) marginal benefit. However, mainly due to the transmission line capacity constraints, the supply and demand need to be occasionally adjusted by the ISO to alleviate transmission congestion. Since the ISO has all the price information of market participants, the optimization process is performed solely by the ISO, and all particpants receive distpatch orders from the ISO. If the customers' demand is not elastic enough, some loads must be shed to alleviate constraints in the power system [30]. 2.3.3 Min-ISO (Decentralized system) In decentralized systems, the ISO does not control the energy markets, and is called "min-ISO" here. The min-ISO will try to accommodate preferred schedules settled outside the ISO, and will only manage the transmission resource assignment or transmission market. The ISO alleviates congestion by intervening in the energy markets only when congestion occurs. However, unlike simultaneous optimization by the max-ISO, the min-ISO will not coordinate markets for energy and transmission explicitly, therefore, markets for energy and transmission operate sequentially. In the min-ISO, inter-temporal or start-up costs and ramp-up/down constraints are not considered explicitly. Therefore, market participants must internalize those costs or constraints. Figure 4 shows the min-ISO model and its function. 32 Seller tt Buyer Seller tt Buyer Generation Scheduling Preferred schedules Adjustment bids System operation Transmission coordination Min-ISO Figure 4 Min-ISO model In the min-ISO, preferred schedules for generators and customers are determined outside the ISO. Free market activity allows each generator and customer to negotiate with each other and determine their preferred schedules (bilateral contracts). Therefore the min-ISO does not have any (price) information regarding the contracts, and only receives requests to use the transmission lines from bilateral or multilateral (if the number of parties involved in the contract is more than two) contracts. However, if congestion occurs, some preferred schedules must be adjusted in order to satisfy the physical constraints. To alleviate congestion, the min-ISO uses adjustment bids made by market participants. The bid of the generator expresses the amount of output that the generator is willing to increase above the preferred schedule, and specifies the cost incurred by increasing its output. It is called an incremental bid for the generator. Conversely, the bid of the generator also expresses the amount of output that the generator is willing to decrease below the preferred schedule, and specifies the savings obtained by decreasing its output. It is called a decremental bid for the generator. The combination of incremental and decremental bids is called an adjustment bid. 33 The objective function of the min-ISO is to minimize the total adjustment cost (TAC), and can be defined as follows: TAC = Y ]cg(PG)dP -I 1=1 jcd(PD)dP (22) where the preferred schedules for generators and customers are expressed as Pcio and Poio, respectively. The adjustment bid for the generator is cg(Pa) (an incremental bid if the generator's output is to be increased, and a decremental bid if its output is to be decreased). The adjustment bid for a customer is cd(Poi) (an incremental bid if its demand is to be decreased, and a decremental bid if its demand is to be increased). Therefore, the adjustment bids can be considered as marginal cost and marginal benefit functions. The objective function tries to minimize the deviation from the preferred schedules. Here, if the preferred schedules are fully observed, the objective function is minimized to zero. If the generator gives consistent cost information, ("consistent" implies that the adjustment bid is the derivative of supply cost function), and the generator output is readjusted from the preferred schedule Pao to Pa, the cost incurred or saved by the generator is equal to its marginal cost, and can be expressed as follows: Pa, pc, jcg(PGi)dP= \fg(PGi)dP % %o (23) Similarly, if the customer gives consistent cost information and the customer demand is readjusted from the preferred schedule Pom to PDI, the cost incurred or saved by the customer is equal to its marginal benefit, and can be expressed as follows: 34 P,„ PD, \cd{PDi)dP= \fd(PDi)dP Pi>H) P<M) (24) These adjustment bids can be different from actual cost or benefit information, due to the obligation of contracts or other considerations. In bilateral contracts, free transactions create a gain, leading to demand for transmission use. Using adjustment bids to alleviate congestion is not necessarily efficient when the curtailed trades are not the ones with the smallest gains from the trade. Here, the adjustment bids imply the value of the transmission capacity use, because generation or demand price differential at two locations gives the value of the transmission capacity. Then, any entity using the congested lines has to pay congestion fees based on the difference between the locational marginal costs. In [31], a method using explicit appreciation of the transmission capacity use was compared with the implicit adjustment bids method, and the implicit approach was preferred since the number of variables in the optimization can be smaller. If all generators and customers provide consistent cost and benefit information regarding adjustment bids, the objective function of the min-ISO is expressed as follows: TAC = Y, i=l ]fAPG,)dP 1=1 jfd(Po,)dP (25) or TAC = Y \ fg(Po,)dP \fd(PD,)dP + const. (26) The first two terms in the right hand side of (26) have exactly the opposite sign of the objective 35 function of the max-ISO. Therefore, if all generators and customers give consistent information to the min-ISO and the max-ISO, both ISOs yield exactly the same generator outputs and customer demands under congestion conditions. However, bilateral contracts will not be observed anymore in the min-ISO. Even when the contract specifies the transfer of a certain amount of power from one seller to one buyer, the output of the generator is not equal to its partner's (customer's) demand, since both of them are redispatched by the ISO. This is equivalent to central dispatching, since less efficient generators are replaced by more efficient ones based on information from the adjustment bids. Therefore, if congestion occurs, the min-ISO intervenes in the energy market, and forces participants to trade power unwillingly. In other words, adjustment bids create a locational marginal price at each node. Raising the price in the importing node and reducing the price in the exporting node adjusts the demand and supply at these nodes. The difference between these two nodal energy prices is the usage fee charged for flows from the exporting node to the importing node. 2.3.4 Mix-ISO In the min-ISO, preferred schedules are determined by market participants themselves based on their negotiations, and will be adjusted by the min-ISO if congestion occurs. In the max-ISO, generators and customers submit their price information to the ISO without any preferred schedules, and only follow schedules determined by the max-ISO. At least the min-ISO process has a voluntary negotiation process among market players when no congestion occurs in the physical network. However, as mentioned in 2.3.3, bilateral contracts are not observed, and all participants in the min-ISO are treated as if in the max-ISO system if congestion occurs. If market activity is separated from the control of the power system, as in the case of the min-36 ISO without congestion, a power exchange (PX) market can be formulated in addition to bilateral contracts. The preferred schedules from the PX are quite similar to the max-ISO schedules except for the consideration of network constraints. However, free market activity also allows each generator and customer to negotiate with each other and determine their preferred schedules (bilateral contracts) outside the PX. There can even be several PXs. Consequently, the ISO receives requests to use the transmission lines from several PXs and from bilateral contracts. Such a system operator is called "mix-ISO" here. Figure 5 shows the mix-ISO model and its functions. Seller Buyer Seller tt Buyer Portfolio SG=SD Power Exchange Generation Scheduling Preferred schedules Adjustment bids Portfolio G=D System operation' Transmission coordination Mix-ISO Figure 5 Mix-ISO model To alleviate congestion, the mix-ISO uses adjustment bids given by market participants as in the min-ISO model. Under congested conditions, the min-ISO could give exactly the same generator outputs and customer demands as the max-ISO could. However, the bilateral contracts were not observed in the min-ISO, since less efficient generators were replaced by more efficient ones based on information from the adjustment bids. In the mix-ISO model, even though the amount of transferred power may be altered, the output of the generator is kept equal to the demand of the customer in each bilateral contract. For each PX the total output of 37 the generators is kept equal to the total demand of the customers. Therefore, if congestion occurs, the mix-ISO intervenes in the energy market, but does not force any participants to trade power amongst themselves. As in the min-ISO, adjustment bids create a locational marginal price at each node. The difference between these two nodal energy prices is the usage fee charged for flows from the exporting node to the importing node. In the mix-ISO, energy and transmission markets operate sequentially as in the min-ISO. 2.4 Existing ISOs A highly centralized ISO or the max-ISO type can be found in the Pennsylvania-New Jersey-Maryland (PJM) market and in the UK. In the UK, early stages of deregulation favoured centralized design because the organization was close to a vertically integrated utility or regional power pool in that all controls were centralized. In a large system like the PJM market, the fully locational marginal pricing method requires setting prices at many nodes. Practically, setting nodal prices only at major nodes is sufficient for the PJM market. Figure 6 shows the market structure of the PJM ISO. 38 Seller (Generator) Buyer (Customer) price information Power Marketer (Aggregator) Power Exchange Scheduling ISO System Dispatch (Supply and demand adjustment) Figure 6 PJM market structure The decentralized ISO or mix-ISO type can be found in Californian and Scandinavian systems. In California, two types of entities request transmission use from the ISO. One is the power exchange (PX), and the others are called scheduling coordinators (SC), who aggregate requests for transmission services from bilateral contracts. When these requests or preferred schedules from the P X and SCs cause congestion in the transmission lines, the ISO reschedules participants' schedules based upon the adjustment bids, which participants voluntarily provide to the ISO when asking for the use of the transmission lines. The ISO, based on the adjustment bid information, gives the transmission use to whoever values it most. Then any entity using the congested lines has to pay congestion fees based on the difference between locational marginal costs. Figure 7 shows the market structure of the California ISO. 39 Seller Buyer (Generator) (Customer) Power Marketer (Aggregator) Scheduling Coordinator Bilateral contracts Preferred schedules Adjustment bids (Demand = Supply) ISO System Dispatch (Supply and demand adjustment) Figure 7 California market structure The congestion management process in the California market is thoroughly explained in [32]. When adjusting schedules to alleviate congestion, the California ISO keeps each participant's generation and demand transactions in balance. In other words, the ISO keeps each scheduling coordinator's portfolio (supply and demand) separate. Therefore, the California ISO is of the mix-ISO type. By keeping all participants' transactions in balance, the ISO clears the transmission market without trading power with any participants or forcing any participants to trade energy amongst themselves. Even if the ISO can reduce total cost or increase social welfare by creating trades among participants, the ISO tries not to get involved in the energy trade. The transmission loss of each scheduling coordinator is estimated and included in its portfolio. The total balance of power is maintained in real time by the ISO as an ancillary service. The details of power balance in the California ISO is explained in section 3.5.2. NordPool in Norway is the min-ISO type, where more than 80 percent of electricity is traded 40 through bilateral contracts [33]. In NordPool, bids in the supply and demand functions are submitted as offers and are used to increment or decrement energy outputs. However, in the California market, bidders are allowed to submit adjustment bids to the ISO that differ from their bids in the PX. 2.5 Challenges of ISO The max-ISO controls and coordinates both the energy and transmission markets. The max-ISO works well under fair competition, and optimizes the system completely so as to attain higher efficiency. However, the max-ISO ignores speculation and gaming by participants. Participants with market power can manipulate price information, and the optimization may not work without strong incentives to ensure that bids reflect actual costs. Punitive sanctions and penalties are required to suppress the incentive to deviate prices from actual costs, but cannot be a perfect and sufficient means. The min-ISO only controls the transmission market. The energy markets promote efficiency through competition amongst participants so as to manage their own transactions. The min-ISO works well under good scheduling decisions by each participant. However, the min-ISO intervenes in the energy market in order to alleviate congestion, which breaks the bilateral transactions and forces participants to trade power based on adjustment bid information regardless of their incentives. The mix-ISO is a compromise in that it only controls the transmission market, and allows participants in the energy market to negotiate their transactions freely, but does not force participants to trade power regardless of their incentives. To keep each contract's portfolio in balance, the mix-ISO does not force bilateral contract holders to trade power amongst them, 41 resulting in less optimal social welfare. That is the price the mix-ISO market has to pay in order to allow participants to manage their own transactions. 2.6 Comparison of ISO Models The max-ISO and the min-ISO give exactly the same results if each contract's portfolio (supply and demand) is not kept separate. The mix-ISO keeps each portfolio separate and in balance. Here, the max-ISO and the mix-ISO are compared in terms of two points. Firstly, the effect of keeping each portfolio separate in one physical network system is examined. Secondly, the vulnerability to market strategic bidding is observed. 2.6.1 Methods To examine the effect of keeping each portfolio separate (plural PXs), firstly, some generators and customers are combined to create several PXs. Then their (power) transactions are determined in each market. If only one generator and one customer are combined, they can be considered a bilateral contract. In the next step, the ISO checks whether all the preferred transactions are feasible simultaneously in the target system. If congestion (line overload) happens, some generators and customers are redispatched under the condition that the demand and supply is balanced in each PX by using adjustment bids, which are consistent with the price bids given to the PX. By calculating the difference between the total social welfare of one PX (max-ISO) and plural-PX market conditions, the effect of the plural PXs is quantified. To observe the effect of strategic bidding under plural PXs, exactly the same procedure is followed as was done for the effect of plural PXs. Namely, the creation of several PXs and determination of the preferred schedules are followed by a feasibility check and redispatching by the ISO. However, price parameters of one generator are changed to include the effect of 42 strategic bidding. By calculating the difference between the total social welfare of the cases without strategic bidding and with strategic bidding, the effect of strategic bidding under plural PXs is quantified. 2.6.2 Assumptions 2.6.2.1 Network and Model Since the electrical parameters of the target network should be well known, an IEEE-30 bus test system was used with practical modifications. Additional generators were connected at bus 8 and 24. No fixed amount of base load was included in the test system. All the data is shown in Appendix II. Figure 8 shows the configuration of the network. 30 29 Figure 8 Network configuration (30 bus system) 43 For simplification, the DC model2 is assumed to simulate the power system. In the actual case of the California market, where several nodes are aggregated as zones3, interzonal congestion is relieved by the ISO using the DC model. The objective function of the optimization is to minimize the total adjustment costs (min-ISO) as explained in 2.3.3, or maximize the total social welfare (max-ISO) as explained in 2.3.2, which should give the same results if congestion exists in the network. The maximization of the total social welfare can be expressed as in (18). In the DC model, the following conditions are assumed [23]: • Line resistance is negligible (no transmission loss); • Voltage angle difference is small; QrQj=0; Then, sin(Q,-Qj)=QrQj, cos(QrQj)=J • Voltage is the same at all buses and one per unit; i.e. VrLO for all i • Reactive power is ignored. Then, the active power constraints explained in 2.2.1.2 are simplified as follows: 2The DC model is good for calculating MW flows in transmission lines and transformers. Since the model ignores reactive power flows, practically it is an 'approximated real power flow' model. It is, however, called the 'DC model' here following [23]. 3 Zones are defined as areas where congestion is infrequent, and therefore, can be priced on an average-cost basis. Gij=0 N (27) 44 P.s is net active power bus injection and calculated at each bus as: Nm Nm *=i *=' (28) Nm is the total number of markets (PXs and bilateral contracts). Pcki is the output of the generator at bus / belonging to the k-th PX. Similarly, PDti is the demand of the customer at bus / belonging to the £-th PX. All generators have upper and lower limits to their outputs. P™n < PGi < P c m a x (29) Similarly, all customers have upper and lower limits to their demands. pmin < p < pmax D: - rD, — rD, (30) If several PXs exist in the target network, the balance of demand and supply must be satisfied at each PX. The generation-load power balance is given as below: Z^=iX, fork = \...Nm-\ '=' (31) Since no transmission loss exists in the DC model, the total generation always balances the total demand if all active power bus injection constraints are observed. Therefore, if the balance of generation and.load is maintained at JVm-1 PXs, the remaining PX inevitably balances the generation and load. Each power flow on the transmission lines is constrained by their flow limits. The limit is defined by thermal or stability limits. In the DC model, the active power flows on the line 45 between buses / and j are simplified as follows: (32) Then, security constraints on branches are expressed as follows: p pmax IJ ~ u (33) 2.6.2.2 Generators and Customers In actual power markets, a few large companies often own substantial market shares with HHI4 values above 2500 [34]. Here, five generators and five customers are assumed in the target system, resulting in the HHI value of 2000. Each generator and customer has its own price and benefit function. Price functions of generators are assumed as in Table 3. Table 3 Price function coefficients of generators Bus# a($/MWh2) 6($/MWh) c($/h) r> min PG r> max *G 1 0.00533 11.669 2 0 200 2 0.00889 10.333 2 0 200 5 0.00741 10.833 2 0 200 8 0.00850 10.833 2 0 200 24 0.00850 10.833 2 0 200 Benefit functions of customers are assumed as in Table 4. 4 HHI stands for Herfindahl-Hirschman Index. The HHI is calculated by summing the squares of the individual market shares of all the participants based on the total sales or capacity currently devoted to the relevant market. 46 Table 4 Benefit function coefficients of customers Bus# 4$/MWh2) e($/MWh) ./($/h) PDmin r> max 10 -0.00533 14.000 2 0 200 12 -0.00889 13.000 2 0 200 15 -0.00741 13.000 2 0 200 21 -0.00741 13.000 2 0 200 24 -0.00741 13.000 2 0 200 2.6.2.3 Power Exchange Market Cases Twelve, cases are considered and are shown in Table 5, where G is a set of numbers of buses at which generators are located, and D is a set of numbers of buses at which customer are located. Each power exchange is assumed to not mutually reveal its participants' information. Case 0 has only one PX and, therefore, can be considered to follow the max-ISO model. Case 5 is a combination of all one-to-one pairs and, therefore, can be considered to follow the bilateral market model. 47 Table 5 Power exchange market combination Case #of PXs Power exchange market combination 0 1 G=[ 1,2,5,8,24] D=fl 0,12,15.21.241 1 2 G=[l,2,5] D=ri0,12,151 G=[8,24] D=r21,241 2 2 G=[ 1,2,8,24] D=H 0,12,21,241 G=[5] D=ri51 3 3 G=[l,2] D=ri0,121 G=[5] D=ri5i G=[8,24] D=f21,241 4 4 G=[l,2] D=|"10,12] G=[5] D=ri51 G=[8] D=[21] G=[24] D=[24] 5 5 G=[l] G=[2] D=[10] D=[12] G=[5] D=[15] G=[8] D=r211 G=[24] D=f241 6 2 G=[l,2] D=[10,12] G=[5,8,24] D=[ 15,21,241 7 3 G=[l,5] D=ri0,151 G=[2,8] D=[ 12,21] G=[24] D=[241 8 3 G=[ 1,5,8] D=[10,15,211 G=[2] D=[121 G=[24] D=[24] 9 4 G=[l,5] D=ri0,15] G=[2] D=[12] G=[8] D=[21] G=[24] D=r24] 10 2 G=[ 1,5,8] D=[10,15,2 1] G=[2,24] D=r 12,241 11 4 G=[l,8] D=r 10,21] G=[2] D=[121 G=[5] D=[151 G=[24] D=[241 12 4 G=[l,24] D=[ 10,24] G=[2] D=ri2] G=[5] D=[151 G=[8] D=f211 2.6.2.4 Parameters for Strategic Bidding One generator at bus 2 is assumed to change the price information strategically. Its parameters are adjusted as follows: Table 6 Parameters of generator at bus 2 after adjustment for strategic bidding Bus# A b c r> nun p^rnax 2 0.00889 12.000 0 0 200 Here, generator 2 turns from the most efficient into the least efficient one in the system without reducing its market share substantially. 48 2.6.3 Results 2.6.3.1 Effect of Plural PX By comparing the total social welfare with that of case 0, the reduced total social welfare caused by plural PXs has been quantified as shown in Table 7. Table 7 Total social welfare of plural PX Case # o f PXs Total social welfare ($) Reduced social welfare ($) 0 1 495.71 -1 2 495.71 0.00 2 2 492.92 2.79 3 3 470.90 24.81 4 4 466.48 29.24 5 5 431.86 63.85 6 2 473.04 22.67 7 3 465.79 29.93 8 3 488.16 7.55 9 4 465.56 30.16 10 2 490.33 5.38 11 4 465.77 29.95 12 4 436.89 58.83 As can be seen in Table 7, the plural-PX condition reduces the total social welfare as the number of PXs increases. This is caused simply because each transaction is optimized within its PX locally, and the ISO does not force any trade among PXs. Therefore, combinations of local optimizations are always suboptimal compared with global optimization, as is done in case 0. 2.6.3.2 Effect of Strategic Bidding with Plural PX Similarly, by comparing the total social welfare with and without strategic bidding (Table 7), the reduced total social welfare caused by strategic bidding under plural PXs has been quantified as shown in Table 8. 49 Table 8 Total social welfare after strategic bidding Case #of PX Total social welfare ($) Total social welfare (from Table 7) Reduced social welfare ($) 0 1 393.71 495.71 102.00 1 2 r 393.71 495.71 102.00 2 2 389.29 492.92 103.64 3 3 342.35 470.90 128.55 4 4 339.55 466.48 126.93 5 5 339.55 431.86 92.31 6 2 346.34 473.04 126.71 7 3 404.90 465.79 60.89 8 3 405.40 488.16 82.76 9 4 379.42 465.56 86.14 10 2 406.61 490.33 83.72 11 4 377.06 465.77 88.71 12 4 343.80 436.89 93.09 As can be seen in Table 8, case 5 (bilateral transactions) is not the case with the least reduced social welfare. This seems to be because congestion suppressed the effect of strategic bidding. Congestion is closely examined later. 2.6.3.3 Relationships of Strategic Bidding and of Plural-PX Conditions Relationships between the effects of strategic bidding and of plural-PX condition are examined simultaneously. Combining the results of Table 7 and Table 8, Table 9 was created. 50 Table 9 Relationship of plural power exchange market and strategic bidding Reduced social Reduced social Case #ofPX welfare by- welfare by plural PX strategic bidding 0 1 - 102.00 1 2 0.00 102.00 2 2 2.79 103.64 3 3 24.81 128.55 4 4 29.24 126.93 5 5 63.85 92.31 6 2 22.67 126.71 7 3 29.93 60.89 8 3 7.55 82.76 9 4 30.16 86.14 10 2 5.38 83.72 11 4 29.95 88.71 12 4 58.83 93.09 Figure 9 was drawn based on the result of Table 9. 3 • 4 12 • i i 5 _ , , _ _ f_2 , , , 1 10 20 30 40 50 60 70 Reduced social welfare by separated power exchanges ($) Figure 9 Relationship between strategic bidding and plural-PX condition Case 5 is the mix-ISO with bilateral contracts, and case 0 is the max-ISO with a single power exchange market. As can be seen, many mix-ISO models have less social welfare. However, many of them are resilient to strategic bidding, compared with max-ISO. The ranges of both 3 •8 140 130 120 -110 100 8 90 -80 -j 70 60 51 axes are approximately 70 ($). The rest of the cases are also plotted on the graph. Although the total social welfare increases monotonically as the size of the PX increases, or the number of PXs decreases, a larger market does not necessarily mean that it is vulnerable to strategic bidding by one generator. This is caused by the conditions of congestion. 2.6.3.4 Effect of Network Congestion To see the effect of congestion, a simulation ignoring network constraints was done, and Figure 10 shows the result. « 3 4 6 • 11 • 8 • 9 • • 5 • 1 0 • 7 l z 10 20 30 40 50 60 70 Reduced social welfare by separated power exchanges ($) Figure 10 Relationship between strategic bidding and plural-PX condition without network constraints In Figure 10, similar cluster patterns as in Figure 9 can be observed, and each cluster has its own characteristic. Cases 5, 8, 9, 11, and 12 have the bilateral contract comprised of G=[2] and D=[12]. Therefore, in these cases, exactly the same amount of total social welfare is reduced by strategic bidding as in Figure 9. The difference among them is caused by the combinations of the remaining G=[ 1,5,8,24] and D=[l0,15,21,24]. Case 5 is all bilateral contracts, and its social welfare is reduced the most by separated PXs. Cases 3, 4, and 6 have a PX comprised of 52 5^ X> ID M 140 130 120 -110 100 90 -80 70 60 • 2 t"0" G=[l,2] and D=[10,12]. The strategic bidding effect was comparatively large because the output of G=[l, 2] was changed approximately from [70, 120] to [100, 40] by the strategic bidding (G[2] changed from the most efficient to least efficient source). Cases 7 and 10 have a PX comprised of G=[2,8/24] and D=[12, 21/24]. Here, G[8] and G[24], D[21] and D[24] have the same price functions respectively. The difference between case 7 and 10 is caused by the combinations of the remaining G=[ 1,5,8/24] and D=[l0,15,21/24]. In case 10, the remaining G=[l,5,8] and D=[l0,15,21] are grouped together, and the reduced social welfare caused by plural PXs is smaller than with case 7. In case 1 and 2, G[2] is a member of a large PX, and the reduced social welfare is small. Table 10 summarizes the congested lines in all cases. Lines expressed as italic numbers were alleviated by the strategic bidding effect, and lines expressed as bold numbers were congested by the strategic bidding effect. Table 10 Lists of congested lines Case Congested lines Without strategic bidding (original b parameter) With strategic bidding (adjusted b parameter) 0 4-12 12-15 21-22 6-9 12-15 21-22 22-24 1 12-15 21-22 28-27 12-15 21-22 28-27 2 4-12 12-15 21-22 22-24 12-15 22-24 3 12-15 21-22 24-25 12-15 21-22 4 12-15 21-22 12-15 21-22 5 4-12 12-15 21-22 12-15 21-22 6 12-15 21-22 28-27 10-21 12-15 10-21 7 4-12 21-22 12-15 21-22 8 4-12 21-22 12-15 21-22 9 4-12 21-22 12-15 21-22 10 4-12 21-22 25-27 12-15 21-22 22-24 11 4-12 12-15 12-15 12 4-12 12-15 21-22 12-15 21-22 Congestion affects the reduced social welfare by plural PXs more than by strategic bidding. In 53 cases 8, 10, and 11 congestion seems to have improved the reduced social benefit caused by plural PXs. This is because the social welfare of the base case (case 0) itself was reduced by congestion, and the relative reduction of social welfare compared with other cases seemed smaller than the reduction without congestion. On the contrary, in cases 3, 4, and 6 the congestion deteriorated the social welfare by plural PXs, or forced the G[l] outputs to decrease. In cases 11 and 12, although the results were the same without congestion, case 12 reduced the social welfare more as shown in Figure 10 because the line 21 -22 was congested as in Table 10. In case 7 only, congestion greatly affected the reduced social welfare caused by strategic bidding. Through strategic bidding, the output of G[2] changed from 64 to 7 MW, and was accompanied by the compensation of the G[8] output from 39 to 76 MW. The adjustments were necessary in order to alleviate the congestion, and resulted in reduced output by G[2], and, therefore, the effect of the strategic bidding was smaller. Similarly, in cases 1 and 2, the reduced social welfare caused by the strategic bidding was mitigated by congestion. 2.7 Summary - ISO Functions for Different Market Structures Several types of ISOs have been established and operated in different power systems. In Chapter 2, two dominant types of ISOs, namely centralized and decentralized, have been compared. One is the max-ISO, where all market participants receive orders from the ISO based on the price information participants bid to the ISO. Another is the mix-ISO, where all participants are allowed to trade power freely, while the ISO keeps each PX's portfolio (demand and supply) in balance and does not force participants to trade power amongst themselves. The mix-ISO inevitably reduces the market efficiency because of lack of trade among PXs or bilateral contracts. However, freedom of trade can protect market participants from strategic bidding caused by powerful players, who may have greater effects upon the 54 max-ISO. The effect of reduced market efficiency in the mix-ISO was examined in comparison with the effect of strategic bidding in the max-ISO. If no powerful players exist, the max-ISO would achieve the desired economic efficiency. However, the advocacy of free trade and the fear of strategic bidding justify the existence of the mix-ISO. Reduced market efficiency is the price markets have to pay to prevent strategic bidding. 55 Chapter 3 Different Policies Imposed by Regulators In Chapter 2, the optimal power flow (OPF) method was used to determine the optimal schedule that maximizes the total social welfare of the system subject to technical constraints. In the near future, regulatory authorities may impose various objectives or constraints upon independent system operators (ISOs) that are not restricted to social welfare maximization but may be, for example, environmental impact reduction or network congestion limitation (for reliability). These constraints or objectives have to be defined by the regulators based on their policies, giving explicit weights that govern the tradeoff between conflicting objectives and constraints. Therefore, the OPF problem can be regarded as a multiobjective optimization problem that has negatively correlated, often conflicting, and tradeoff relationships between objectives and constraints. From possible combinations of the feasible solutions obtained by multiobjective optimizations, the ISOs must determine the optimal schedules. Even in conventional environments, emission control was an important objective of electric utilities [35] [36] [37]. Environmental and economic dispatch algorithms under conventional environments are well summarized in [38]. The conflicting goals of minimizing costs and emissions are handled through several methods. In [39], an interactive method was applied to determine the generation dispatch schedules that satisfy emission constraints. In [40], a fuzzy multiobjective approach was applied to include post-contingency corrective scheduling. In [41], two objectives of minimum fuel cost and minimum environmental impact were solved through a fuzzy method. Concerning emission trade, in [42], a market value of emission allowance was compared with the additional cost of switching fuels using heuristics-guided evolutionary algorithms. 56 In this chapter, firstly, the effects of different objectives on power transactions are observed. To examine the effects of different objectives, the pool (max-ISO) model is used in the multiobjective optimization because the overall optimization is controlled by one authority, max-ISO, and social costs can be included in the optimization process easily. In the case of the min-ISO model, special markets have to be established not only for power, but also for other products such as environmental emissions. The ISO will have to interact and coordinate these conflicting power exchanges' goals. A numerical example is given using the fuzzy multiobjective optimization method. Secondly, DC and AC models are compared. The cause of differences such as transmission loss and reactive power are examined in detail. The effects of the differences on the power transactions are observed through a numerical example. Lastly, a graphical user interface for power transaction simulation for policy makers is introduced. 3.1 Possible Objectives of Regulators At present, existing ISOs are considering economic and security factors of the power system. The transmission line capacity is usually determined by thermal or stability limits. However, these limits directly affect market activity, since a too conservative attitude toward security can impair free trade. Furthermore, in the near future, the ISO may have to take into account other factors such as environmental impact as well. This will also affect market activity. In this chapter, the size of the market is examined when environmental emission and aggregated network security are binding market activities. 57 The method presented in this chapter is summarized as follows: the ISO optimizes the total social welfare in the target system, ignoring other constraints such as environmental emission; then the proposed constraints are included in the optimization process. At the optimal schedule, the emission or network congestion and the total social welfare are calculated. The relationship between them can, therefore, be drawn. Finally, the optimal operational schedules are determined based on a fuzzy multiobjective optimization method. 3.1.1 Environmental Emission Assume a regulator of the power industry will have to consider the environmental impacts of power generation in the near future. In the US, the Environmental Protection Agency is controlling sulfur dioxide emissions by passing the Clean Air Act (1990) [43]. About two thirds of sulfur dioxide emissions are said to be caused by the coal-fired electric power generating industry. The energy sources of generators can be various, ranging from natural gas, small hydro to waste heat. New generators introduced into the market are dominantly natural-gas fired generators, due to their easy availability of fuel and short construction period. If generators are assumed to be thermal units, the environmental impact can be calculated in terms of CO2, SOx, or NOx emissions through nonlinear equations. The CO2 or SOx emission from thermal generators can be calculated through the following equation: E(PGj) = aEiP2Ci+bEiPG+cEi ( 3 4 ) This is similar to the generator cost functions, but the coefficients are, naturally, different. The NOx emission is a little more nonlinear in nature. 58 3.1.2 Aggregated Network Security Transmission line capacity has been considered for each line under normal operating conditions up to this point. However, transmission systems can fall under emergency operating conditions at any moment after a major fault happens. Under these conditions, line capacities must be reserved conservatively in order to prepare for several line outages. If large enough transmission capacities are maintained in all transmission lines, the system can be considered a secure network because the system can easily adapt to different power flows depending on the load demand [44]. Therefore, a network congestion index is introduced here, which is measured by active power capacity margins based on branch flow constraints. The branch-loading rate is defined by: P, R„ IJ ij pmax ^ (35) If the loading rate Ry for branch i-j is 1.0, the branch is loaded up to the capacity, and is thus congested; and if the value of Ry is smaller than 1.0, some capacity margin remains. These indices are calculated for each line, and have to be aggregated to define the index for the total system. The definition for congestion, or the overload performance index for the system is defined by the following equation [23]: v ( P, ^ congestion = > all branches \ pmax (36) where n can be any integer larger than 0. In the following examples, for simplicity, n=l is used. 3.2 Methods The ISO has to solve multiobjective optimization problems that have not only one solution, but a set of many possible solutions because, at the expense of cost, other objectives such as line 59 capacity margin can be improved. These solutions, where one objective cannot be improved without the deterioration of the other objectives, are called Pareto optimal solutions. Multiobjective optimization is explained in detail in Appendix III. 3.2.1 Interactive Multiobjective Programming If the Pareto optimal solutions are found by some means, then decision-makers have to decide which solution should be practically applied. In the process, where the satisfactory level of one objective can be improved only when another objective is sacrificed, trying to make all objectives satisfactory is quite difficult. If the decision-makers do not want to sacrifice any one of the objectives, then they may not be able to reach a final solution. However, iterations between the decision-maker and the solution algorithm can settle the problem by a compromise. 3.2.2 Fuzzy Multiobjective Programming In power system operation, the goal of the decision-makers is not deterministic, but fuzzy, because no specific rules of compromise for security exist. They will probably want to indicate the level of satisfaction for each objective rather than to specify the tolerance amount in such cases as transmission line capacity. Their goals are expressed such that 'the power flow should be substantially less than some value,' or 'the voltage should be in the vicinity of some value.' If the level of satisfaction of optimizing some goals can be expressed as a numerical function, then finding the optimal point becomes an optimization of these functions [45] [46]. In this research, fuzzy membership functions are introduced. The upper and lower limits of membership functions are the allowable minimum and maximum values for the decision-maker, which means that at the extremes of the functions, are either completely satisfactory (1), 60 or completely unsatisfactory (0). As an example, a linear membership function u / fi(x)) [47] can be defined as the following equation: M/,(*)) = - f.° o M*)*/? (37) where ff or denotes the value of the objective function fi(x), such that the degree of membership function is 0 or 1 respectively. Figure 11 illustrates the graph of a possible shape of the linear membership function. Figure 11 Linear membership function 3.2.3 Interactive Fuzzy Multiobjective Programming If the decision-maker specifies the membership functions and follows the fuzzy decision process of Bellman and Zadeh [47], the original multiobjective programming problem can be interpreted as: maximize min {co ,• u.,- (/) (*))} subject to Ax < b, x > 0 ^g) where co, is a weighting coefficient assigned to the /-th objective function. 61 This formulation assumes that the fuzzy decision process of Bellman and Zadeh reflects the decision-makers' fuzzy preference. However, their intention may change when combining the fuzzy goals and constraints. In the case of power systems, the decision-maker may focus on the economic cost at one time and the security margin at another time, depending on the system conditions. Therefore, in addition to specifying membership functions for fuzzy goals, the decision-makers have to specify the appropriate aggregate function and weighting coefficients that well reflect his preferences. A combination of interactive and fuzzy multiobjective programming is suitable. This method determines membership functions based on the decision-makers' preference, and finds the optimal solution for them. 3.3 Assumptions 3.3.1 Network and Model In this example, the target network is the modified IEEE-30 bus test system which was used in Chapter 2. Figure 8 shows the configuration of the network. 3.3.2 Generators and Customers Generators are assumed to use several types of fuel. Each generator's cost is represented as a quadratic function of MW generation. C O 2 emissions (kg/MWh) are also represented as a similar function of MW generation. Five generators are operated in the network. The environmental scenario is also considered. All generators are considered as natural-gas fired, coal-thermal, or hydro. [48]. Table 11 shows each generator's economic and environmental characteristic coefficients. Because the AC model is simulated, reactive power balance of supply and demand must also be defined for (5). The customers' reactive power demand is assumed to be proportional to its 62 active power demand and is fixed at a power factor of 0.95 (lagging). In addition to five generators in the target system, two synchronous compensators are assumed at buses 11 and 13, and their reactive power capabilities for (7) are shown in Table 12. Because no incentive is given to the reactive power generation in the simulation, declaring reactive power capabilities are voluntary. Therefore, reactive power capabilities differ depending on the generator. Their reactive power outputs are assumed to follow the ISO's order within ranges specified by the units. Voltage settings are also assumed to be given by the ISO. Table 11 Generator characteristics (active power) Bus# a ($/MWh2) b ($/MWh) c ($) a-Ei (kg/MWh2) bet (kg/MWh) CEi (kg) p min (MW) pmax (MW) 1 0.00533 11.669 2 0 875 0 0 200 2 0.00889 10.333 2 0 7.0 0 0 200 5 0.00741 10.833 2 0 7.5 0 0 200 8 0.00850 10.833 2 0 395 0 0 200 24 0.00850 10.833 2 0 395 0 0 200 Table 12 Generator characteristics (reactive power) Bus# Voltage (p.u.) min (MVA) max Qc (MVA) 1 1.060 -35 100 2 1.045 -35 100 5 1.010 -20 50 8 1.010 -30 100 11 1.082 -6 24 13 1.071 -6 24 24 1.071 -6 24 As was explained in (8), voltages at load buses (where no generators are connected) are bounded at 0.9~1.1 p.u. Similarly, the benefit of the customer is represented as a quadratic function of MW consumption. Five customers are assumed, and Table 13 shows the customers' economic characteristic coefficients. 63 Table 13 Customer characteristic J3us# d ($/MWh2) e ($/MWh) / ($/h) rt mm (MW) r> max (MW) 1 -0.00533 14.000 2 0 200 2 -0.00889 13.000 2 0 200 5 -0.00741 13.000 2 0 200 8 -0.00741 13.000 2 0 200 24 -0.00741 13.000 2 0 200 3.4 Results 3.4.1 Two Dimensional Relationships The relationship between social welfare and emission was examined. Figure 12 shows the results, which were obtained for AC and DC models. The DC model is an approximation of the AC model. The diamond marks show the optimal operating points as determined by the interactive fuzzy multiobjective programming explained in section 3.2.3. 100000 80000 |? 60000 o 1 40000 e w 20000 0 600 500 400 300 200 100 Social welfare($) DC - - - -AC Figure 12 Tradeoff relationships in AC and DC systems Table 14 shows each objective value at extreme points and at an "optimal" operating point. Point A was obtained by maximizing the total social welfare of the target system. If emission is minimized without conditions, the optimal solutions are no output by any generator and no demand by any customer. Point B was obtained by minimizing emission with customer demand 64 being fixed to that obtained at point A. For the interactive fuzzy multiobjective optimization method, parameters ft andf1 (upper and lower limits, or the allowable minimum and maximum values) of the linear fuzzy membership function are also shown in Table 14. Here, the same values of weighting coefficients were given to social welfare maximization and emission minimization. Table 14 Extreme and operating points in Figure 12 (a) AC system A B fi fi Optimal social welfare($) 233.2 147.3 147.3 233.2 200.6 emission(kg) 51,157 10,352 51,157 10,352 25,845 (b) DC system A B fi f! Optimal social welfare($) 511.5 486.9 486.9 511.5 502.8 emission(kg) 88,607 43,301 88,607 43,301 59,295 Figure 13 and Figure 14 show generator outputs and customer demands at two points on the tradeoff curves of both AC and DC systems. All generators' outputs and customers' demands are reduced. 300 200 100 0 A B Optimal • G-l BG-2 DG-5 DG-8 rjG-24 (a) Generator outputs 65 300 A B Optimal ID-11 HD-12 OD-15 DD-21 • D-24 (b) Customer demands Figure 13 Generator outputs and customer demands in AC model A B Optimal I G-1 H G-2 • G-5 • G-8 • G-24 (a) Generator outputs B Optimal HD-11 HD-12 DD-15 DD-21 DD-24 (b) Customer demands Figure 14 Generator outputs and customer demands in DC model 66 In both AC and DC models, optimal outputs and demands were defined using the results of solution A and solution B. Their outputs and demands are quite different; especially in the AC model where the transmission losses must be supplied by generators. In this section, the max-ISO is assumed, therefore the ISO dispatches generators to compensate for the transmission loss based on the merit order of the generators. Table 15 shows the transmission losses in the AC system. The effects of transmission loss are closely examined in 3.5. Table 15 Transmission losses in A C system A B Optimal Transmission loss (MW) 14.81 16.87 15.59 Loss/Total supply (%) 6.5 9.1 7.7 For a typical transmission system, the average transmission loss is two to three percent of the total system load [49]. In [49] the loss reported by the Electric Reliability Council of Texas was shown to range from below 1% to more than 11%. The assumed system had relatively large transmission loss due to the distance and direction of generators and customers. Losses can vary greatly as the conditions of the system change. 3.4.2 Three-Dimensional Tradeoff Relationships The relationship among social welfare, emission, and transmission congestion was examined. Figure 15 shows the result, which was obtained only for the DC model. A diamond mark shows the optimal operating point determined by the interactive fuzzy multiobjective programming explained in 3.2.3. 67 trade-off C o n g e s t i o n 0 0 E m i s s i o n ( k g ) Figure 15 Three-dimensional relationship Table 16 shows each objective value at extreme points and at the operating point. Extreme points were obtained by changing the emission and congestion constraints value to 10% each of their maxima (B and D), and both simultaneously to 10% of their respective maxima (C). If emission or congestion was minimized the optimal solutions would be no output by any generator and no demand by any customer. For example, point D was obtained by changing the emission constraint from 87,942 to 8,794.2 (kg). The parameters ff and f/ (upper and lower limits, or the allowable minimum and maximum values of the linear fuzzy membership function as Figure 11) for the interactive fuzzy multiobjective optimization method are also shown in Table 16. Here, the same values of weighting coefficients were given to social welfare maximization, emission minimization, and congestion minimization. 68 Table 16 Extreme and operating points in Figure 15 A B C D fi fi Optimal Social welfare($) 511.5 318.9 263.9 395.0 263.9 511.5 432.6 Emission(kg) 88,607 88,607 8,861 8,861 88,607 8,861 34,264 Congestion 12.104 1.210 1.210 12.10 12.104 1.210 4.681 At the above operating points, generator outputs and customer demands were determined as in Figure 16 and Figure 17. 400 i 1 A B C D Optimal • G-1 • G-2 • G-5 • G-8 • G-24 Figure 16 Generator output at operating points 400 i A B C D Optimal • D-11 HD-12 rjD-15 DD-21 • D-24 Figure 17 Customer demand at operating points At present, all existing ISOs are considering only economic factors. Therefore, point A is the most likely operating point in the target system. However, once environmental constraints are 69 introduced, the point moves from A toward D, reducing the social welfare gradually. As the congestion constraint is introduced, the point also moves toward B, reducing the social welfare even more. Furthermore, the decrement of generator output is not similar if generator outputs at point A and B are compared, mainly due to the locational differences of the generators. Therefore, the conservative use of transmission capacity can greatly hinder free power trades, and possibly work in favor of some specific generators. In this study, the optimal operating point was determined using the interactive fuzzy multiobjective method. If new constraints, such as environmental constraints, are included in the ISO's objectives in the future, the ISO will have to develop an agreeable method to use with market participants. The participants are unable to check whether the ISO decision is reasonable, since only the ISO can obtain all the market information. The interactive fuzzy multiobjective method proposed here can function if weighting coefficients are determined by the ISO and market participants without distorting the power trade. However, no dominant method exists to determine proper weights at present. 3.5 Comparison of AC and DC Models In the California market, several nodes are aggregated as zones (zones are defined as areas where congestion is infrequent, and can be priced on an average cost basis). The DC model is actually used for interzonal congestion management where the main transmission system constraints are thermal and stability limits, expressed as MW limits. The AC model is used only for infrequent intrazonal congestion management, where the main constraints are expressed as MVA limits. However, the necessity of considering reactive power is stressed in [50]. In this section, the difference between the AC and DC models is examined in detail. 70 3.5.1 Difference between AC and DC Models In Figure 12, points A on the AC and DC curves (the maximum social welfare and maximum emission) are compared in order to find the cause of the differences, because these two points are the most probable operating points in terms of the existing ISO's goals. Table 17 Social welfare and emission Case Social welfare ($) Emission (kg) DC model 511.5 88,607 AC model 233.2 51,157 71 As Figure 18 and Figure 19 show, the total generator output and total customer demand decreased significantly in the AC model when compared to the DC model. This can be interpreted as a reduction in volume of power transactions. The active constraints for both cases are summarized in Table 18. Table 18 Active constraints for DC and AC models Case Bus voltage Reactive power (MVA) Line DC model - - 15-18, 10-22, 15-23, 28-27, 13-12, 15-14 AC model -G2 (-35), G5 (-20) G8(-20), C511 (24) C13 (24) none A large difference in the results can be found between the AC and DC models. The limited capability of reactive power generation seems to have caused the difference. Therefore, marginal social welfare of reactive power (Lagrange multipliers of reactive power equality constraints) was examined and shown in Table 19. 5 compensators 72 Table 19 Marginal social welfare of reactive power ($) Case bus 2 bus 5 bus 8 bus 11 bus 13 AC model 0.0039 0.0557 0.1400 0.0867 0.3168 The Lagrange multiplier expresses the marginal social welfare by per unit reactive power increment in the bus. Because the difference in total social welfare between the AC and DC models are approximately $280 in Table 17, the reactive power is not the primary reason to reduce social welfare or to cause a large difference between AC and DC models. In the target network, the insufficient reactive power supply capability was not severe enough to increase the marginal social welfare of reactive power. However, in [51] or [52], reactive power is also considered as a product, and its market structure is proposed. In this research, reactive power is assumed to be provided voluntarily by market participants. Assumptions are made in reactive power and transmission loss in order to find the reason for the difference between AC and DC models. Firstly, the AC system is assumed lossless. Secondly, the reactive power capability of all the generators is assumed unlimited. The results are shown as follows: Condition 1: Lossless lines • Resistance in all transmission-line data are made equal to zero; and • Off nominal turns ratios in all transmission lines are made equal to 1 (Susceptance matrix is made symmetric). Table 20 shows the results of both models under condition 1. 73 Table 20 Social welfare and emission under Condition 1 Case Social welfare ($) Emission (kg) DC model 511.5 88,607 AC model 487.4 63,880 Differences between AC and DC cases still exist. Condition 2: Lossless lines + infinite reactive power generation capability • Condition 1; and • Reactive power supply capabilities at all generators are unlimited. Table 21 shows the results of both models under assumption 2. Table 21 Social welfare and emission under Condition 2 Case Social welfare ($) Emission (kg) DC model 511.5 88,607 AC model 510.5 86,917 When no transmission loss exists, and reactive power generation capability is unlimited (condition 2), the results of both the AC and DC models coinside well. The primary difference in social welfare was caused by transmission loss. A minor difference was caused by the reactive power capability of the AC system. Due to the loss, the bus marginal cost at the generator bus decreases, and leads to the reduction of generator output in order to balance the generator's marginal cost and its bus marginal cost. Similarly, due to the loss, the bus marginal benefit at the customer bus decreases, and leads to the reduction of customer demand in order to balance the customer's marginal benefit and its bus marginal benefit. These effects lead to reducing the trade volume of transactions, or reduced social welfare. 3.5.2 Ancillary Services Under either the max-ISO or the mix-ISO model, the ISO is required to provide several 74 coordinated services, called "ancillary services," in order to facilitate competitive power transactions. The Federal Energy Regulatory Commission (FERC) defined these services as falling into six categories [53]: scheduling and dispatch, reactive supply and voltage control, regulation and frequency response, real power loss (transmission loss compensation), and operating reserves. As Table 15 shows, transmission loss compensation can be a large part of ancillary services. In [54], the total cost of ancillary services was calculated using the actual data of twelve US investor-owned utilities, and transmission loss cost comprised approximately 30% of the total ancillary service cost. Although almost all ancillary services are considered to be provided by the ISO, FERC identified that transmission loss can be provided by any entity, including the participant itself, not uniquely by the ISO. Therefore, the ISO must instruct market participants to provide the required transmission loss compensation by themselves. In the max-ISO model, the ISO organizes the PX (energy market) and assigns transmission resources (transmission market) simultaneously. If transmission loss must be compensated for, the max-ISO simply dispatches several generators from an engineering point of view. Therefore, the total sum of generation output becomes equal to the total demand plus the transmission loss. From an economic point of view, the price of active power at each bus is differentiated as being the Lagrange multiplier of the active power constraint, and the locational marginal price reflects the value of active power, which are different according to location due to the effects of transmission loss and congestion. In the mix-ISO model, each portfolio of PXs (or bilateral contracts) must be kept separate. If all the portfolios are kept separate in the AC model, or the total generator output is always equal to the total customer demand, the system cannot find any feasible operation schedules, since no entity will compensate for transmission loss. 75 In practice, the PX is required to provide transmission loss compensation as calculated by the ISO. For example, in the case of the California ISO, its protocol [55] says, The ISO will... specify GMMs (Generator Meter Multipliers) for each energy supply source to account for the energy lost in transmitting power from generating units to load. GMMs are calculated based on the sensitivity of injecting energy at generator bus to serve an increment of demand distributed proportionately throughout the area. GMMs are calculated several times as follows: • Estimated GMMs are calculated based on the forecast of the total area demand by the ISO before PX or SC submit preferred schedules; • Revised GMMs are calculated to allocate re-estimated transmission loss based on the final schedules; and • Any difference between scheduled and re-estimated transmission loss is considered as an imbalance energy deviation and will be purchased or sold in the real-time market. In the case of the mix-ISO, the loss is estimated in advance in order to allow market participants to provide it themselves or purchase it from a third party. However, the estimated loss can be different from the actual loss, consequently requiring not a "forward," but near "real-time" market to solve the imbalance at the final stage. In the case of the max-ISO where the energy and transmission markets are controlled simultaneously, comprehensive optimization can be conducted as frequently as possible, solving the imbalance without resorting to a real-time market. 76 3.6 Graphical User Interface for Power Transaction Simulation The ISO defines the optimal operating point based on the information from power systems. Numerical data based on optimal power flow (OPF) must be handled efficiently, and the data must be meaningful even to regulators of the power systems without an engineering background. In this research, a simple graphical user interface was developed to facilitate non-expert personnel to understand the impact of power transactions upon the power system, and is named PSOS (Power System Operation simulation Support environment). PSOS can also be used as an interface between the system and the operator or for electrical power system studies. This section gives an overview of the function and necessary input/output data file structure of PSOS. 3.6.1 PSOS Structure The primary function of PSOS is to present a user environment which efficiently shows the information from numerical simulations obtained by separate programs such as OPF programs. The information includes voltages (magnitude/angle), power (active/reactive) of each node in the target system, and power flow (active power) on the transmission lines, which are displayed as both bar charts and tables (numerical values). Values are compared with built-in lower/upper limits (upper limit only for power flows), and if a value exceeds limits, PSOS alerts operators by changing the color of the bar charts. PSOS also allows users to see detailed text output for information regarding the numerical analysis. PSOS helps users to edit selected input data files for OPF programs in special windows so that the numerical simulation programs can run consecutively with different case settings. PSOS also launches different numerical programs in order to run the simulation. 77 3.6.2 PSOS Interface PSOS shares its input/output (PO) files with the OPF program. Input/output file names may be different from one simulation case to another. File names are defined in the problem definition file. PSOS also displays the configuration of the target power system in a user-prepared graphic file. The graphic files are either in Windows metafile or bitmap file format. All the other files are ASCII text format. Figure 20 shows the input/output file structure for the PSOS/simulation environment. Figure 21 shows the graphical user interface of the PSOS program. O P F [Input] Infile i m (numerical) (*.dat or *.cdf) EDfile PBfile CNTfile (*) ,(*) SSfile (*optional) Defile.dat Infile EDfile PBfile CNTfile SSfile OUTfile SLTfile BFLfile P S O S (GUI) Figure 20 File structure [Output] OUTfile SLTfile BFLfile Config (*.bmp or *.wmf) 78 Bu T t e a PL . • 1 i.oGooo ; 0.00000 0.79397 -0.12269 O.OU * 1.06120 -0.01302 1.30361 0.25679 0.21 « 1.03765 -0.077m 0.00000 0.00000 oo: ! 1.03236 -0.09232 0.00000 0.00000 "obi Figure 2 1 Graphical user interface of PSOS 3.7 Summary - Different Policies Imposed by Regulators In this chapter, firstly, the effect of different objectives imposed by regulators on social welfare in the market has been observed. The various objectives of the regulators are not restricted to social welfare maximization, but can include environmental impact or network reliability. Therefore, the ISOs have to solve multiobjective optimizations that have negatively correlated, often conflicting, and tradeoff relationships between objectives and constraints, which can be expressed by as a tradeoff curve or plane if the number of objectives is two or three. From possible combinations of solutions, the optimal schedule is determined by the fuzzy 79 multiobjective optimization method. When the method was applied to the target system, the maximum and minimum allowable values had to be defined by the regulator. The optimal schedule could be affected greatly by the objectives and their allowable values, possibly working in favor of some specific market players. Therefore, when the regulator defines its objectives and their allowable ranges, an agreement among the regulator and market participants is required. If all the information concerning the transmission systems is open to the participants, each participant can find a combination of objectives that always gives preference to the participant itself. Regarding environmental regulations, mainly three types of regulatory interventions can be considered. The first means of regulation is explicitly constraining each generator's output. This regulation can be simulated easily in the developed multiobjective method as additional constraints of output variables. The second means is by issuing a limited number of tradable permits or pollution rights [43] to the ISO service area. Instead of constraining the output of each generator, the total amount of emission is constrained. The developed multiobjective method assumed this second regulation, dealing with objectives of different units ($ and ton). The third means is by introducing "green" taxes. Here, pollution has a monetary value (cost) synonymous with electric energy. Instead of constraining the total output explicitly, the market will adapt, considering both environmental and economic conditions. This regulation can be simulated by including the total environmental cost (pollution amount multiplied by the green tax: $/ton) into the objective function. The max-ISO model was used in the simulation of the multiobjective optimization, because the overall optimization by one authority can include additional factors, such as emission constraints, easily. The min-ISO model would reach the same solution as the max-ISO if 80 similar adjustment bids for emission is submitted to the min-ISO as well as the adjustment bids for active power by participants. The mix-ISO model also requires adjustment bids for power and emission. If no monetary value is attached to emission, the mix-ISO finds the optimal schedules by adjusting the outputs and demands in order to reduce total emission within its constraints. Here, the portfolios of each PX are kept separate, but emission is traded among PXs or the dispatch of the mix-ISO, which in turn increases or decreases the total emission in each PX. In 1995, the Environmental Protection Agency, following the Clean Air Act in the US, set up a market for the pollution rights to emitting sulfur dioxide (SO2). Electric utilities were assigned many pollution allowances at the beginning, and are presently allowed to trade through an auction process run by the Chicago Board of Trade (CBOT). Now, the mix-ISO cannot force each PX to trade pollution rights among PXs from the point of free trade. Each PX has to obtain the right in the pollution right market before asking the ISO to include their preferred schedules. Consequently, the mix-ISO minimizes the total adjustment costs, keeping each PX's portfolio separate, and keeping each PX's emission below its pollution right. The two market systems in the mix-ISO will guarantee that emission is below the total amount constraints. However, the solution of the mix-ISO is always suboptimal compared with that of the max-ISO, because the mix-ISO cannot coordinate the trade of pollution rights among PXs, as it cannot for power. Therefore, as the number of objectives in the ISO increases, the mix-ISO will have a smaller chance of achieving optimal solutions, compared with that of the max-ISO. Secondly, the differences between the AC and DC models have been considered. The primary difference is caused by transmission loss. A minor difference is caused by the reactive power capability of the AC system. These effects lead to the reduction of transactions, or reduced 81 social welfare. In the case of the max-ISO, the loss could be compensated for easily by the ISO, using a central distpatch order. However, in the case of the mix-ISO, the loss must be estimated before the actual delivery of the transaction, and the resultant difference between the estimated and actual loss must be solved in the near "real-time" market. Lastly, a graphical interface program, PSOS (Power System Operation simulator Support environment) was developed. Based on the results of optimal power flow (OPF), PSOS aims to facilitate non-expert personnel, possibly regulators of the ISO, to understand the impact of power transactions upon the power system. 82 C h a p t e r 4 D e c i s i o n - M a k i n g A i d f o r M a r k e t P a r t i c i p a n t s In Chapter 2, the market models were examined closely. In this chapter, the markets are examined from the point of market participants. The available information for them is mainly the price at their locations. Therefore, market participants make power transactions based on the forecasted price data. In the max-ISO (independent system operator) model, they use the locational marginal price information given by the ISO. In the mix-ISO model, they use the clearing price information given by power exchanges (PXs). Although both energy and transmission markets can be considered with regard to market participants, the energy market is mainly focused on in this chapter. Research on this aspect of power industry deregulation is mainly reported on the supplier side. The bidding behavior of suppliers (generators) considering bilateral contracts was analyzed in [56]. The bidding behavior of suppliers considering spot markets was examined in [57]. The bidding profiles were constructed for a group of generators with similar characteristics in [58]. Financial risk management in electricity markets has begun to see focus in [59]. For distributors, a fuzzy regression analysis was applied in [60] to estimate peak load in distribution systems. This chapter is structured as follows: Firstly, time-series data for two actual power exchange markets is observed. Secondly, the mathematical formulation of fuzzy autoregression models is introduced. Two fuzzy autoregression (AR) models, which are different in complexity, are presented. Thirdly, the fuzzy AR method is applied in order to forecast time-series electricity consumption data in the California area, which has recently opened a commodity-like market called Power Exchange [61]. Fourthly, the fuzzy regression method is applied to the 83 relationship between demand and price. Finally, the fuzzy AR method is applied to time-series electricity price data in order to help in the decision-making process of consumers who have a choice between bilateral transaction of power and market-based purchases. 4.1 Electricity Markets Electricity markets are incomplete because electricity is a flow (not a stock), and storing potential energy is expensive [62] [63]. The market is also imperfectly competitive because flows are constrained continuously by operational limits such as transmission capacity. If no transmission constraints existed, the power system could be operated based on only the schedules determined by the electricity spot market, as other commodities are traded in various spot markets. However, the externality of transmission constraints makes it impossible to operate the power system in a short-time frame, as in the spot market. Therefore, the system operator must intervene in the energy market activity as real-time operations approach. In most power systems, energy markets can be divided into spot, forward, and possibly, futures markets based on the timing to real-time operations. In both max-ISO and min-ISO models, the forward market is the primary one where energy transactions are determined in advance to satisfy the physical constraints of the transmission system. 4.1.1 Forward Market In 2.3, three ISO models were defined, based on the coordination of the energy and transmission markets. Similarly, two forward markets can be considered, namely for energy and transmission. Since the flow of electricity in a transmission system must obey Kirchhoff s Laws, the ISO must intervene in the energy market and operate the transmission system in order to alleviate congestion created by the energy market. In the energy markets of most power systems, a day-ahead notice is required in order to provide enough time to alleviate 84 anticipated congestion on transmission systems. Forward markets can be considered as the intersection of economic and engineering judgement. In the max-ISO model, both the energy and transmission markets are treated simultaneously based on the principle that a centralized control can achieve market efficiency. There, all bid information is submitted to the ISO, and congestion is alleviated by controlling participants' output or demand, resulting in energy prices differentiated according to location. In the min- or mix-ISO model, energy and transmission markets are treated sequentially, based on the principle that a free energy market will achieve economic efficiency through competition for electricity. In the energy market, a (day-ahead) power exchange market creates aggregated supply and demand curves from individual supplier curves and individual demand curves, and the market is cleared. Then, in the transmission market, the adjustment bids mentioned in 2.3.3 are used to alleviate congestion, resulting in a transmission usage fee or difference between two locations. The power transactions in the forward market are termed physical, because delivery is expected, but in reality all forward transactions can be financial if purchases or sales in the spot market can reverse commitments. Market participants usually contract forward based on expectations, and adjust their schedules based on conditions arising the next day. Both the California and PJM markets use day-ahead markets to balance supply and demand in the system so the ISO must deal with a small amount of deviation in real time. Real-time energy demand can be typically predicted day-ahead within 3% for each hour, so day-ahead scheduling is largely sufficient. 85 4.1.2 Spot Market In the max-ISO model, no options from market participants are voluntary, and the ISO can control resources in real time based on the information obtained from the market. There, participants must submit all capabilities and prices to the ISO, and must accept its orders. Therefore, the deviation from schedules determined in the forward markets is compensated for by the central dispatch of the max-ISO, and is charged penalties and sanctions. In the max-ISO mode, no spot or "real-time" market is necessary. In the min- or mix-ISO model, the system operator does not control their resources directly, so participants can deviate from the schedules determined in the forward market, leaving the system operator responsible for balancing the system. The spot market is the final method in real time to balance supply and demand throughout the system in order to maintain reliability. The spot market is operated continuously, using options such as regulations and spinning reserves, or offers submitted to the spot market. The deviation from schedules determined in the forward market is priced in the spot market. The deviation can be large, caused by intentional behavior of market participants. As an example of the mix-ISO model, a real-time operation in the California ISO is explained in [64] and [65]. The California ISO obtains ancillary services such as regulation and operating reserves in the forward market, and also receives "supplemental energy bids" that are submitted to the ISO 30 minutes prior to real time. Then, the ISO uses them in real time based on the merit order of energy bid price. By the externalities of transmission line constraints and requirement of balancing demand and supply, a perfect spot market controlled by the min- or mix-ISO is infeasible. From the point of view of real-time operations or spot markets, the max-ISO is advantageous, since the system 86 operator can re-optimize the entire system as many times as necessary. 4.1.3 Futures Market Several electrical futures products are transacted in the US and Norway [15]. All of the futures products are defined for specific delivery locations (nodes). However, due to the characteristics of transmission lines, futures products are affected greatly by the availability of transmission services. Unlike other futures products that can be delivered via several possible routes and stored ahead of schedule, electricity must be delivered at the time specified in its contract. An inherent problem with electricity is retarding the development of futures markets. Longer-term contracts extend beyond the operational time horizon. Large projects such as the construction of new transmission lines can change nodal prices significantly, and affect many market participants. The uncertainty of transmission expansion planning is another reason for retarding the development of a futures market. Under the deregulated environment, game-theoretic approaches are examined in order to allocate transmission expansion cost in many studies [66] [67] [68]. However, at present, no definitive method has been found to solve the network expansion problem. 4.1.4 Financial Right The energy market can be organized as spot, (day-ahead) forward, and futures markets as above. However, the transmission market can be organized only as a (day-ahead) forward market. For longer-term transmission markets, various financial rights are created by contracts such as 'transmission rights.' They can be considered as the right to inject or withdraw power from the transmission grid at specific locations. A financial right gives the right holder reimbursement of the usage fee for transmission whether or not it transmits energy. Creating 87 these rights can make the price of the transmission market certain. Since the right is defined as a transmission from point-to-point, resale in secondary markets is difficult. In actual markets, the longer-term rights are handled via periodic auctions. In the PJM market, a financial right specifies injection and withdrawal points. In the California market, each right pertains to the interface between two zones, and includes the priority for scheduling. Financial rights are strongly supported due to risk-aversion and long-term bilateral contracts. 4 .2 Market Observation From 4.1, it can be said that the forward market dominates a large part of the transactions, and it is more difficult to predict the market behavior of participants. The market structure can create excessive fluctuation in electricity prices, as seen widely at the stock-exchange and currency-exchange markets. Under the conventional industry structure, utility companies or public authorities have determined electricity prices largely based on cost. On the other hand, under the deregulated environment, generators are operated by independent power suppliers, and they are free to set the price of electricity. Such an unregulated market structure inevitably yields volatility in electricity prices in the market due to speculation and gaming. Here, two typical markets, the California power exchange market (mix-ISO model) and the PJM market (max-ISO model) are observed. 4.2.1 History of PJM and California Markets In the PJM market, the max-ISO controls both energy and transmission. The PJM energy market opened a bid-based market on April 1, 1997 [69]. Marginal cost pricing was calculated based on marginal cost in order to meet demand from resources dispatched by the PJM ISO. When the transmission lines were constrained, the market-clearing price was calculated based 88 on hypothetical unconstrained dispatch. Since congestion charges were averaged over all users, market participants lost their incentive to mitigate congestion. This problem led the PJM system to introduce locational marginal prices on April 1, 1998. Locational marginal prices are calculated based on actual system operating conditions. When the transmission lines are congested, locational marginal prices are different, reflecting generation marginal cost and transmission congestion cost at each node. Transmission congestion charges are based on marginal price differences between source and sink locations. In the California market, the mix-ISO does not operate a day-ahead market, but a power exchange or scheduling coordinators do instead [70]. The California market opened a bid-based market on April 1, 1998. The California power exchange is the largest scheduling coordinator (approximately 80% of the total volume in the market), and operates both 'one day-ahead' and 'one hour-ahead' markets. The day-ahead market of the power exchange uses simple price/quantity bids (offers), and ignores all operational constraints. Then, the transactions preferred by each scheduling coordinator are sent to the ISO. The ISO keeps each scheduling coordinator's generation and demand in balance when adjusting their preferred transactions. Therefore, the ISO does not force each scheduling coordinator or power exchange to trade amongst themselves for buying or selling power. When some transmission lines are congested, market participants scheduling flow on congested paths are charged the marginal-cost based price for using the path, as is done in the PJM market. The sizes of the two markets [71] are compared in Table 22. 89 Table 22 Market size of PJM and California ISO Peak MW Number of units Number of participants PJM 51,550 535 145 California 45,000 600 30 4.2.2 Forward Market Figure 22 shows the main part of the PJM market. Figure 23 shows some of the locational prices in the PJM market [72]. Figure 22 Geographical area of PJM market 90 1/1 1/31 m. 4/1 5/1 531 • 630 7/30 809 928 1028 11/27 12/27 —a—PJM -->--PSEG - 4 - P B 0 0 - -^ - -PPL - « - B G E PFNFI FC ^ - P E P C O - — - D P L Figure 23 Locational prices of P J M market in 1999 Figure 24 shows the main part of the California market. Figure 25 shows some of the locational (zonal) prices in the California market [61] [73]. 91 NW HUMB Figure 24 Geographical area of California market Figure 25 Locational prices of California market in 1999 92 In the California market, more than half of the volume of energy is traded through the Power Exchange (PX) market, where the unconstrained market-clearing price (UMCP)6 and locational prices are determined. While the rest is negotiated through "bilateral contracts" between a supplier and a retailer/consumer. In the California market, for the first five years of operation, generators that are owned by the existing three investor-owned utilities (Pacific Gas & Electric, Southern California Edison, and San Diego Gas & Electric) are required to bid only through the PX [74]. This fact of large day-ahead and hour-ahead market-based trade makes the California electricity prices particularly volatile and unpredictable, compared with other matured markets such as Norway where more than 80% of energy is traded through bilateral contracts [33]. Both markets show seasonal variations of high- and low-demand seasons, namely the large volatility. The difference between the two markets can be caused by market characteristics or speculation of powerful market participants.. 4.2.3 Futures Market Several futures products are traded in NYMEX. In these products, its delivery point is specified, since the price of the power depends on its location. Here, two kinds of data, namely COB (California Oregon Border) futures in the California market, and PJM (Pennsylvania-New Jersey-Maryland Interconnection) Western Hub futures in the PJM market were examined. The futures contracts for both markets are issued 18 months ahead. Futures Contract Specifications are attached in Appendix IV. 1 [75] [76]. For example, in March 2000, futures from April 2000 to September 2001 were traded. However, the prices of these futures are available in [77] for 6 The unconstrained market-clearing price and its schedules are determined ignoring all network constraints. Then, the schedules are adjusted to satisfy the constraints resulting in locationally different (zonal) prices. 93 only three months ahead. For example, in August, the prices of the futures for the following September, October, and November are available. Figure 26 and Figure 27 show prices of NYMEX Electricity futures contracts from June 1999 through May 2000. Each line represents its price change during the three months before its delivery, over 18 months of trading. 200 180 160 140 120 100 80 60 40 20 0 4 ] \1 --•V os Os os os os Os Os os Os Os Os Os Os Os O P O O O O O O O O C N C N — — July - - Aug. - • - A - • • Sept. - • - • - O c t . - ' •* - • Nov. —•—Dec. -•"—-Jan. Feb. - • • - - -Mar. - - A - - Apr. —"— May --"--June - - -° - - July - • - - - A u g . Figure 26 NYMEX Futures at PJM Western Hub 94 100 80 60 20 ON ON V:' i: .'t U ON Os ON ON ON ON ON ON ON ON ON ON o o o o p p o o o o CN m July Dec. May Aug. • Jan. • June - • - A - - • Sept. Feb. -••<»••• July •"---Oct. -«--- Mar. - - - -Aug. - - * - - Nov. Apr. Figure 27 NYMEX Futures at COB Markets for futures contracts influence shorter-term markets because they are used mainly as hedges against the exchange price. Exchange markets for longer-term contracts, such as futures markets, must ensure a transparent and liquid market whose price can be benchmarks less volatile than short-term market prices. Daily volatility of the futures was examined for one year. The method of volatility calculation [78] is briefly explained in Appendix IV.2. Table 23 and Table 24 show the results. 95 Table 23 Daily volatility of N Y M E X Futures at PJM Western Hub Month -> Futures^ June 99 July 99 Aug. 99 .Sep. 99 Oct. 99 Nov. 99 Dec. 99 Jan. 00 Feb. 00 Mar. 00 Apr. 00 May 00 July 99 6.3% Aug. 99 3.5% 5.3% Sept. 99 1.7% 1.4% 3.7% Oct. 99 1.2% 0.5% 1.1% Nov. 99 0.3% 0.6% 1.5% Dec. 99 0.5% 0.7% 1.2% Jan. 00 0.0% 2.2% 3.1% Feb. 00 2.9% 1.9% 2.5% Mar. 00 1.7% 0.3% 1.7% Apr. 00 7.6% 1.6% 1.2% May 00 0.6% 0.9% 4.2% June 00 1.0% 2.3% 8.5% July 00 1.5% 5.3% Aug. 00 5.2% Table 24 Daily volatility of N Y M E X Futures at COB Month -» Futures>l June 99 July 99 Aug. 99 Sep. 99 Oct. 99 Nov. 99 Dec. 99 Jan. 00 Feb. 00 Mar. 00 Apr. 00 May 00 July 99 4.3% Aug. 99 2.5% 2.6% Sept. 99 2.4% 2.3% 2.5% Oct. 99 1.4% 2.3% 2.5% Nov. 99 1.6% 2.4% 1.7% Dec. 99 1.7% 1.8% 3.1% Jan. 00 1.5% 4.7% 2.0% Feb. 00 1.6% 1.8% 2.7% Mar. 00 2.1% 1.4% 2.0% Apr. 00 1.6% 2.4% 1.5% May 00 2.2% 1.4% 1.5% June 00 1.3% 6.1% 5.6% July 00 1.4% 5.7% Aug. 00 4.0% The volatility of two futures products cannot be compared easily, since (besides the difference in the ISO structure) the number of generators and their types or the availability of their fuel 96 sources are completely different. However, i f only the volatility is compared, the PJM futures market has more volatility in summer, and less in winter than the COB futures market. 4.3 Forecasting Method for Market Participants In 4.2, the time-series data of electricity price have been observed. Because a day-ahead forward market for energy is the most dominant market, forecasting the price is very important to market participants. In this section, a new approach, based on possibility theory [79] and fuzzy autoregression (AR) [80], is introduced and applied to the analysis and estimation of time-series data of electric power consumption and price. Various time-series data analysis methods have been proposed and applied to the analysis of economic and other human activities. Traditional methods are based on statistical and probabilistic approaches, but it may not be entirely suitable to apply purely stochastic models to data generated by human activities such as bids in power exchange markets. For example, market-based transactions of electricity tend to make demand (price) more volatile on the high volume-side than on the low-volume side, devaluating the traditional assumption of Gaussian distribution of random data [81]. The proposed fuzzy model assumes that the time-series data of power consumption or price reflects human decision-making activities rather than a stochastic process as assumed in conventional models. This approach can represent the rich information contained in the original data set, including certain levels of possibility, while conventional models are aimed at converging to a single point representing the most probable point of occurrence at any time. As in the conventional approach, the autocorrelation function is calculated in order to identify the model's structure. The model can reflect cyclic patterns such as weekly or seasonal variations. The regression parameters are given by fuzzy numbers [82], which are represented by triangular fuzzy sets 97 having crisp parameters. Such a fuzzy model can encompass a wide range of possible data with varying degrees of possibility for each time stage, from the most likely value to a less possible value. The fuzzy parameters for autoregression are determined by linear programming. The objective of the optimization is to minimize the vagueness of the model while covering the time-series data as broadly as possible. 4.3.1 Fuzzy AutoRegression (FAR) Model 4.3.1.1 Symmetrical Fuzzy AR Model (FAR-S) A fuzzy autoregression (FAR) model represents a possibility of occurrence of a certain set of data in the future, when the present data are dependent to some degree on the past data. Possibility theory is explained in Appendix V.2 [47] [83]. Let x(t) be a time-series datum observed. Then, the fuzzy A R model is represented by the following equation: X[t) = A0 + AX- x(t -1) + A2 • x(t - 2)+- • -+Am • x(t - m) (39) where t is discrete time of an equal interval, m is the order of model description (m < t), At is a fuzzy coefficient, and x(t) is an observed variable. The estimated value x(t) is a fuzzy number and is capitalized to distinguish it from x(t). There can be various ways to define the fuzzy parameters [82]. One of the simplest ways is to use symmetric triangular fuzzy sets represented by two crisp parameters such that: Ai = <ai,c>,(ci>0) (40) Figure 28 shows the triangular fuzzy set. For convenience, this fuzzy autoregression model is called the FAR-S model. 98 Figure 28 Symmetrical triangular fuzzy set The crisp parameters a, and ct will be determined so that the model will include all or most of the observed data such that: X^x(t) (41) To minimize the vagueness of the model, linear programming is applied to determine the parameters such that: n Minimize: ^ |c 0 + c, • \x{t -1)|+- • -+cm • \x{t-m)\} (42) t=m+\ subject to: a0 +a, -x(m) + a2 -x(m-l)H—hzm-x(l) +c0 + c, • \x(m)\ + c2 • |x(m -1)|+- • •+cm • |x(l)| > x(m +1) aQ+a^-x(m + {) + a2-x(rri)-\—\-am-x{2) +c0 + c, • \x(m +1)| + c2 • |x(m)|H vcm • |JC(2)| > x(m + 2) a0 + a, • x(n -1)+a2 • x(n - 2)H—vam • x(n - ni) +c0 + c, • |x(/? -1)| + c2 • \x(n - 2)|H Ycm • \x(n - m)| > x(n) a0+a{- x{rri) + a2 • x{m - 1)H—Vam • x(l) - j c 0 + c, • |x(m)| + c2 • |x(m- 1)|H—hcm • |*(l)|} ^ x(m +1) a 0 + a, • x{m + \) + a2- x(m)-\—Yam • x(2) -jc0+c,-|x(/w + l)|+ c2-|x(/n)|H Hcm-|x(2)|j < x(m + 2) 99 a0 + ax • x(n — l) + a2 • x{n - 2)H—vam • x(n — ni) —[c0 + c, • \x(n -1)| + c2 • \x(n - 2)|H hcm • \x{n - m)|J > x(«) and c0,c,,c2,---,cm >0 (43) where n is the total number of observed data, and m is the order of model description. 4.3.1.2 Fuzzy AR Model Based on Crisp AR (FAR-A) The fuzzy AR model described in 4.3.1.1 is obviously simple and powerful as will be shown later, but one of the problems is that the center value at of the fuzzy parameters does not necessarily represent the most probable data in the time-series. On the other hand, conventional autoregression methods are quite established [84], and some popular programs are publicly accessible via the World Wide Web (e.g., Web Decomp [85]). Therefore, it is reasonable to apply time-series data obtained through conventional AR as part of the fuzzy time-series model. Let such a conventional AR-based fuzzy model be referred to as FAR-A here. In the FAR-A model, the results of conventional AR is used to obtain a median value at of the triangular fuzzy number. By determining the median value, the triangular fuzzy set representing the fuzzy parameters need not be symmetrical, and has the advantage of reflecting the possibility distribution of the time-series data better than FAR-S, particularly for data set which have non-Gaussian distribution. Although the crisp AR method applied to determine the median value still assumes Gaussian distribution inherently, such assumption would cause little problem because data points far from the median are infrequent, and have little influence on determining the median values. Figure 29 shows the unsymmetrical fuzzy set used in the FAR-A model. 100 Figure 29 Unsymmetrical fuzzy set With the two different parameters c ; + , and c- (c ( + for upper deviation and cf for lower deviation), the F A R - A parameters are determined by solving an extended linear programming problem such that: n Minimize: ^ jcg + c,+ • \x(t -1)|+- • -he* • \x(t - m)\ + c~ + c\ • \x[t -1)|+- • -+c~ • \x(t - m)\ j (44) /=m+l subject to: a0 +«, •x(m) + a2 -X(/W-1)H—l-am-x(l) + C Q + C , + • |x(m)| + C 2 + -|jt(m-l)|-l-••+<;* - |jc(l)j > x(m +1) a 0 +<3, - x ^ - l - l) + a 2 -X(/W)H—i-am-x(2) + C Q + c[ • \x(m +1)| + c2 • \x(m)\-\ he* • |JC(2)| > x[m + 2) a0 +a, • x(n-\) + a2 -x(n-2)-\—Yam -x(tt-m) +CQ + c,+ • |x(n -1)| + c\ • \x{n - 2)|H he* • \x{n - w)| > x(n) a0 +a, -x(m)-i-a2 -x(m-l)-(---l-flm -x(l) - JCQ + cj" • + c2~ • |x(m - 1)|H he" • |*(l)|} ^ x(m +1) aQ+ax-x(m + \) + a2-x(m)-{—ham-x(2) - | C Q + cj" • |x(/n +1)| + c2 • \x(mj^—hc~ • |x(2)|j > x(m + 2) a0 +a[ •x(n-\) + a2 •x(n-2)+---+am-x(n-m) -jcg + cj" • \x(n -1)| + c2 • |x(n — 2)|+- • -+c~ • |x(« - rnj.j > x(n) 101 and C 0 ' C 0 ' C \ > C \ >• • • > C,» ' Cm - 0 (45) where n is the total number of observed data, and m is the order of model description. 4.3.2 Fuzzy Regression Model As fuzzy sets are applied to autoregression models, fuzzy sets can be applied to model a relation between two values [86]. One of the most popular methods for representing the relationship between two sets of data is regression analysis. In regression analysis, the relationship between parameters x and y are represented by the following polynomial equation [84]: y = a0 +a,x + a2x2-\ \-amxm (46) By choosing the order m we can represent nonlinear relationships. Parameters are determined so that the distance (or error) between a data point and the corresponding point on the polynomial will be minimal. In the fuzzy regression model proposed here the parameters of the polynomial are replaced with fuzzy numbers as shown below, in order to cover a wide range of data. Y = A0 + A,x + A2x2+---+Amxm ' (47) The parameters A 0 , A l 5 ... are so determined that the observed datay are encompassed by the fuzzy model Y, and the resultant left-hand variable (in this case, price) is also a fuzzy number and has an interval of rlata covered in a varying degree of likelihood. Figure 29 shows the triangular fuzzy set representing the fuzzy number A/ with three parameters, namely a/, c/ +, cf ( cf~, cf > 0). The expression Af = < a/, cf~, cf > is used again to represent such a triangular 102 fuzzy number hereafter. (48) To determine the parameters for the fuzzy numbers, linear programming is applied in order to fit the model to the given data [87]. The dimension m is empirically determined by a human observation on the data set or on the results of (conventional) regression analysis. Minimize: n £ {cl + clx(t)+- • •+c„>(0m + c~ 4crx(0+- • •+c~x(t)m} (=1 subject to: a0 + a,x(l) +a2x(Y)2+---+amx(l)m +c+0 + c,+x(l) + c 2 +x(l) 2 +• • •+c>(ir > y(l) a0+alx(n)+a2x(n)2-\ hamx(n)m + C Q + c*x(n) + c2x(n)2 +• • •+c+mx(n)m > y(n) aQ + a,x(l) + a 2 x(l ) 2 +• • •+amx(\)m -c+Q -c;x(l)-c+2x(l)2 — -c+mx(ir<y(\) a0 +aix(n) + a2x(n)2 H \-amx(n)"' -c0 + -c+xx(n) -c+2x(n)2-• - c > ( « ) m < y(n) and where « is the total number of observed data, and m is the order of model description. (49) The optimization is aimed at minimizing the vagueness of the fuzzy regression model, in order to fit the model within as small an area as possible, while covering all the data considered within the range. This phase of modeling does not need experience and data-specific knowledge. 103 The fuzzy model shows the possibility distribution or a possible interval of data at a point where each crisp input is chosen. 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 away from the most probable point. The range in the model represents the volatility information contained in the given set of data. The volatility can be measured by the area of the range encompassed by the fuzzy model. If the range is divided by the conventional regression line/curve into two, say upper and lower, the difference in volatility indicates either an upward trend or a downward inclination of the observed data. Conventional (crisp) regression results are contained in the fuzzy model, and the fuzzy regression model is an extension of the conventional model, preserving the uncertainty of the original data. 4.3.3 a-cuts for Evaluation of Models In the fuzzy A R or fuzzy regression models, a-cuts of fuzzy parameters can be used to evaluate the performance of encompassing the possibility distribution of the obtained models. The a-cut of fuzzy sets is explained in Appendix V. 1. To some decision-makers, the resultant region of data covered by the fuzzy model may look too wide because the parameters are determined to include all the data observed in the region. In order to narrow down the band of possible intervals, a-cuts of the fuzzy parameters A[ are applied such that the higher the a value becomes, the more focused on high possibility the fuzzy region tends, sacrificing some points 104 of data with low possibility. Also, the suitability of the shape of a given triangular fuzzy number may be evaluated with varying a-levels, for example, by counting the missed points (i.e., data outside of the region) at specified a-levels. For the FAR-A model, a-cuts are applied as follows: AQ = < a0, c0+x(l-a), c0"x(l-a) > Ai - < a\, ci+x(l-a), cfx(l-a) > Am = <am, cm+x(l-a), cm~x(l-a) > (50) where 0 < a < 1. From one perspective, the a-level of the fuzzy models reflects the willingness of the user of the fuzzy AR model to take risk. At a = 1, the possibility of occurrence of a particular value is the highest, but the chance of missing the point is also the highest because the interval of the possible data strips down to a point. On the other hand, at a = 0, the possibility of occurrence of a large deviation is the lowest (but not nil for the forecast), and the region encompasses all the observed data. Therefore, as the fuzzy region is narrowed by a-cuts, a = 0 can be interpreted as the most risk-avert, and the a-level increasing toward 1 as more risk-taking. Where to settle down for a particular risk level is entirely up to the decision-maker's heuristics — willingness to take risk versus tolerance against vagueness. The fuzzy based model provides such flexibility in decision making. The proposed fuzzy-AR models are aimed at representing the meaning of data as a whole, including volatility, while conventional (crisp) methods are designed to represent only the most probable data points. 105 4.4 Electricity Consumption Forecasting In the advent of electric power industry deregulation and restructuring [88], it is becoming all the more important for local distributors to accurately forecast electric power consumption under uncertainty. While residential or commercial users are charged flat rates, local distributors must purchase power or close contracts with generation suppliers in the deregulated power market. To gain profit in the open access power market, forecasting electric power consumption is quite challenging. 4.4.1 Electricity Consumption Model The most direct method of constructing a fuzzy AR model of electricity consumption would be to apply the intuition and experience of an expert observer on a given set of data. However, to obtain an effective model, the modeler needs to have experience and data-specific knowledge, and the model structure cannot be implied quantitatively. Therefore, the conventional Box-Jenkins approach is partly followed in order to determine the structure of the fuzzy AR model. In a typical conventional approach, the structure of the autoregression model for time-series data can be determined by removing trends and other seasonal fluctuations from the original data in order to extract steady-state data, and by observing the autocorrelation function values. Parameters are optimized subsequently, and the model is evaluated in the final stage. The model is successively modified, for example, by changing the orders of differences [89] in order to improve accuracy. Figure 30 shows an example of raw data of electricity consumption. In this section, the data recorded at the California Power Exchange is used to evaluate the performance of the model and forecasting. In the state of California, electricity trade is deregulated, and non-utility entities (e.g., "power brokers") can freely set the electricity price depending, primarily, on the 106 conditions of supply and demand. Therefore, the volume of trade at the exchange market tends to be quite volatile, and the forecast of electricity demand is vitally important in the decision making of such trades. This particular set of data is the maximum consumption of electrical energy (MWh) recorded every day for a three-month period (January - March 1999) and available from the California Power Exchange web site [61 ]. 30000 27000 1/1/99 1/15/99 1/29/99 2/12/99 2/26/99 3/12/99 3/26/99 Figure 30 Recorded electricity consumption in California Figure 31 shows the autocorrelation function values of the same California data. The data in the first two months are analyzed. It is obvious that the weekly cyclic pattern should be reflected in the model structure. Most of the industrial plants (volume consumers of electricity) shut down on weekends, start up on Monday, and their activity peaks mid-week. Day-ahead data is also important because electricity consumption is much influenced by temperature. The autocorrelation function values justify such heuristic knowledge of electricity consumption. Thus, the series of t-\ and t-1 time lags are used such that: X(t) = A0 + Arx(t-l) + A2-x(t-l) (51) 107 where AQ,A\, and A2 are fuzzy numbers and the time-series x(t) are real numbers. Therefore, the resultant estimated value x(t) would also be a fuzzy number. 1 0.8 -0.6 -0.8 ---1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Time lag Figure 31 Autocorrelation function values of the California data 4.4.2 Parameter Identification Parameters of fuzzy sets that construct the target fuzzy AR model can be determined through linear programming. A method similar to possibility regression [86] has been proposed, and many applications have been developed. In an analogous fashion, in this section, LP is applied to determine the parameters of the fuzzy autoregression model. The target of the linear programming problem is to minimize the region of the fuzzy time-series while covering the given time-series as much as possible. The variables to be determined are the crisp values that define the fuzzy numbers as shown in Figure 28 and Figure 29. For the California PX case, the LP problem using the data for the first two months (52 points) of those shown in Figure 30 is stated as follows (only FAR-S model is shown to save space): 59 Minimize: ^{c 0 +c, -|x(r-l)| + c2 -|x(r-7)|| (52) (=8 subject to: a0 + ax • x(l) + a2 • x(l) + c0 + c, • |x(7)| + c2 • |x(l)| > x(8) a0 + ax • x(8) + a2 • x{2) + c0 + c, • |JC(8)| + c2 • |JC(2)| > x(9) 108 a0 + ax • x(58) + a2 • x(52) + c0 + c, • |JC(58)| + c2 • |JC(52)| > x(59) a0 + a, • x(7) + a2 • x(l) - |c 0 + c, • |x(7)| + c2 • |x(l)| j < x(8) a0 + a, • x(8) + a2 • x(2) - |c 0 + c, • |x(8) |+c2-|x(2)|}<x(9) x(58) + a2 • x(52) - {c0 + c, • |x(58)| + c2 • |x(52)|} < x(59) and c0,c,,c2>0 (53) The results of the optimization are shown in Table 25. Underlined parameters in the FAR-A column have been obtained through conventional AR. Table 25 Parameters of fuzzy numbers FAR-S FAR-A a0 9182 2078 ai 0.295 0.312 a2 0.309 0.597 + 0 0 cd 0 + Cl 0.074 0.104 c{ 0.071 + C2 0 0 C2 0 Note that the CQ and C2 parameters for Ao and A2 are both zeroed. This fact indicates that the fuzziness is only represented by A \ for this particular set of data. 4.4.3 Forecasting through Fuzzy A R Model In this section, the fuzzy AR models obtained above are applied in forecasting electricity consumption data in the near future. Figure 32 shows the results of the forecast. Model parameters are determined as above using the first 52 data. (I.e., from t = 8 to t =59; Note that 109 the data for the t = 1...7 period are not represented by this model.) The estimated data are for the period from t = 60 to t =90. Points plotted on thick real lines are the recorded original data, including the forecasted period, and faint lines show the encompassed possibility region of the data. As evident in the figures, most of the data in the forecasted period are covered by possible regions in both FAR-S and F A R - A models. The time-series of the conventional (crisp) A R model is plotted as the median value of the F A R - A model. In the conventional approach, the difference between the estimation and the real data are interpreted as errors. Various ways are established to make the crisp A R model more accurate but we are not concerned as far as the proposed FAR models (mostly) cover the possible regions of the target data. The criteria that decide whether the F A R models are acceptable or useful depend on what the decision-maker expects from the information represented by the time-series data. For example, in the F A R - A model, possibility distributions that are wider on the high demand side (c,+ > cf) may indicate a larger volatility to an increase than a decrease in demand. Some performance analyses of the fuzzy A R models'are discussed in the next section. 110 30000 15000 1/8/99 1/22/99 2/5/99 2/19/99 3/5/99 3/19/99 Demand FAR (a) Time-series data of FAR-S 30000 15000 1/8/99 1/22/99 2/5/99 2/19/99 3/5/99 3/19/99 Demand AR FAR (b) Time-series data of F A R - A Figure 32 Time-series data of electricity consumption by fuzzy autoregression models 4.4.4 Comparison of Fuzzy A R Models One quantitative indication of the performance comparison of the two fuzzy A R models is the vagueness represented by each model. In this section, the vagueness of the fuzzy time-series models is measured by computing the area of the regions covered by the time-series models. I l l Table 26 below compares the vagueness for the first two-month period used to tune the parameters, and Table 27 compares the vagueness of the forecasted month. It is interesting to note that the FAR-A model, which is based on crisp AR is more vague than the FAR-S model, at least for this set of data. However, the FAR-A model tells us more information on the difference between the vagueness of the area covered by the upper possibility region (the demand data that are higher than the crisp AR value) and the area in the lower possibility region. It may be interpreted that the volatility of the possible data is different between larger demand and smaller demand. Table 26 Vagueness of models objective function value (MWh) FAR-S 2E(co+c/x) 179863 FAR-A E(c0++c/ +x) (upper) 126637 Z(co"+c/"x) (lower) 87069 Table 27 Vagueness of forecasted data Area of forecasted data (MWh) FAR-S 105305 FAR-A 125119 Another indication of performance is given by the a-cuts of the fuzzy models. By changing the a-level from 0 to 1, the observer of the data can narrow down the region of possibility distribution, a = 0 means all or most of the data are encompassed by the fuzzy model, while at a = 1 the model sees only one point (in FAR-A model, the crisp AR data). Figure 33 compares the change of the error rate for both FAR-S and FAR-A models. The error rate is calculated by the number of points outside of the fuzzy region divided by the total number of data. As the a-level increases, the error rate increases. The nonlinear rate of increase of the error rates indicates a non-monotonous distribution of possibility. In comparison, generally, the FAR-A 112 model is better in following the data deviation. However, the difference in this particular set of data is slight. Considering the simpler model structure, this can be seen as an advantage of the FAR-S model. According to Figure 33, for example, i f the decision-maker accepts a 10% error rate in the observed data, the fuzzy region of F A R - A can be narrowed down to about half of the a-level from the original (i.e., a = 0) fuzzy region. Alpha level - • - F A R - A —a— FAR-S (a) Encompassed data Alpha level FAR-A -a— FAR-S (b) Forecasted data 113 30000 - i 27000 -18000 15000 -I 1 1 1 1 1 ' 1/8/99 1/22/99 2/5/99 2/19/99 3/5/99 3/19/99 - * - Demand A R F A R (c) Time-series data of FAR-A (a=0.5) Figure 33 Forecast by fuzzy AR models 4.5 Electricity Price Forecasting Based on Demand Data This section presents an approach to representing the relationship between electricity demand and market prices with a measure of volatility on the latter data. Such a model can be used to estimate the possible range of prices when demand information is given. The forecast of electricity demand is an extensively studied area under the conventional monopoly utility structure (because the suppliers' primary concern is the security of the power supply). By the application of recent techniques such as neural networks, the short-term (e.g., one day-ahead) demand forecast can achieve small error [90]. Therefore, the demand data is assumed to contain minimum uncertainty unless special occasions exist, such as storms or other irregular events that have impacts on socio-economic activities. It is also assumed that price volatility occurs mainly when the supply-demand balance is tight [91]. When there is an abundance of sellers compared to buyers on the other hand, sellers would compete to obtain transactions, and the prices would eventually settle down to the marginal-cost based prices. 114 The procedure in this section is summarized as follows: Conventional regression analysis is applied to a given data set in order to determine the most likely demand-price relationship. The regression parameters are later extended to triangular fuzzy numbers with the crisp (conventional) regression parameters used as a median. The shapes of the fuzzy parameters are determined through linear programming to encompass most of the actual demand-price data. Through such a fuzzy model, a possible range of electricity prices can be estimated with information of volatility. Example data analysis is carried out using actual data from the California Power Exchange. 4.5.1 Electricity Demand-Price Model Market data from the California Power Exchange (CalPX) was used as a typical example of demand-price relation representation and analysis. Their daily market activity data, such as trade volume and prices, are made publicly available through the Web [61]. The electric energy is traded on the market on an hour-by-hour basis. The independent power suppliers and/or brokers bid to supply power, meeting the estimated demand for power each hour. Figure 34 shows an example of time-series data of electricity demand and price in California for the period from April, 1998 to March, 1999. 115 40000 c £ Q 35000 4-30000 25000 20000 15000 10000 4-5000 0 Demand 200 O O O O O O O O O O O O O O O O O O O O O O O V O N C T S ' - i - H ( ^ r « - ) o o ^ f O \ - ' 3 - O v O - - i t ^ < N t ^ < / - > r < " > >/-> VO t~- <3\ O — ' CN Date ^ _H Figure 34 Market data of CalPX in 1998 This set of data shows the daily maximum power demand in MW and the unconstrained market-clearing price in US dollars per MWh. "Unconstrained" means that it shows a winning bid, but is subject to practical constraints such as the maximum capacity of transmission. Therefore, the actual prices settled may occasionally be different from the market-clearing price, but it is assumed that the unconstrained price reflects the original intention of the power marketers to obtain the highest possible profit. 4.5.2 Parameter Identification 4.5.2.1 Conventional Regression Model Identification Firstly, the conventional regression method was applied to the relationship between demand and price. Figure 35 shows the results of conventional regression analysis using (a) linear and (b) second-order polynomials. 116 (a) Linear 200 o I m i in , , 1 14000 22000 30000 38000 Demand [MWh] (b) Nonlinear (quadratic) Figure 35 Regression applied to California demand-price data The demand-price data from the California power market shows that a large cluster of data exists in the low demand area where prices that shoot up out of the cluster do not heavily influence the determination of the regression line/curve. It may be interpreted that the linear regression line is representing the system marginal cost. However, as the demand increases, the trend to push up the price becomes apparent, and there are many points where prices reach several times that of the nominal marginal cost line. Two reasons can be considered: 1) As the demand goes higher, the chance that high-cost generators (such as gas turbines, which are 117 several times more costly than the base steam turbine generators) will win the bid is increasing; and 2) The last successful bidder determines the final price because of the "market-clearing price" rule. Therefore, as the demand goes higher and options to fill the demand get scarcer, more chance for speculative high-price bidding exists. The quadratic curve shown in Figure 35 (b) follows such price trend to a degree, but it is far from complete in representing volatile market prices under heavy demand. Here, the weakness of the conventional approach based on regression is apparent. Fitting the polynomial to the given data set aims at converging to a single line or curve. The polynomial might be interpreted as the most likely relationship between the given data. However, the rich information contained in the original data, such as the extent of the data range and the degree of volatility, is lost. From the conventional viewpoint, the data that are far away from the regression line/curve are simply considered as errors, and they do not contain any meaning. In addition, when the relationship of the given data sets is indistinct, the line/curve is not considered to be a "fit" to the data. 4.5.2.2 Fuzzy Regression Model Identification 4.5.2.2.1 Non-preconditioned Data The demand-price relation was analyzed based on the result of 4.5.2.1. The fuzzy parameters are determined through linear programming, as was explained in 4.3.2, in order to encompass all the price-demand data. Table 28 and Table 29 list the determined parameters, and Figure 36 shows the range of the model. In Figure 36 (a) a linear regression line, and in Figure 36 (b) a quadratic curve was used as median. 118 Table 28 Parameters for first-order model 1 st order constant a 3.2198xlO"J -45.039 c+ 4.9247 xlO"3 0 c~ 1.9047 xlO"4 17.665 Table 29 Parameters for second-order model 2nd order 1 st order constant a 2.2490 xlO" 7 -7.2375 xlO" J 7.2595 xlO" J c+ 0 5.0228 x!©"4 2.8194 xlO" 3 c~ 7.6428 xl0"y 8.9356x10"* 2.1437 x10"' 0 V- •• -2^-— I 14000 22000 30000 38000 Demand [MWh] (a) Linear 14000 22000 30000 38000 Demand [MWh] (b) Nonlinear (quadratic) ure 36 Fuzzy relationship between demand and price at CalPX 119 As shown in the graphs, the fuzzy model covers quite wide areas because the parameters are determined so that all the data are included in the range of the fuzzy model. The lower boundary curve may be considered reasonable, but the upper boundary seems too far from the median. Particularly, it is inconvenient that only one data point (unusually a high price) is making the range too vague. 4.5.2.2.2 Preconditioned Data Data pre-conditioning is quite important in obtaining a reasonably fit model. It is effective to apply a data-specific feature in order 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 preconditioning demand-price data, features representing the California climate should be focused upon. Air temperatures greatly influence the fluctuation of electrical power demand. In a warm climate region, the peak demand appears at mid-day during summer days, mainly for air-conditioning load, while in a cool climate region, cold winter nights result in high demand for power for heating. The state of California belongs to a warm climate, and as the time-series graph of demand shows in Figure 34, California has roughly two climate'periods: summer and non-summer. Therefore, demand-price points were divided into two clusters, namely, high-demand period and low-demand period, empirically setting up a threshold at 25,000 MWh-peak consumption. If a week's peak demand exceeds this threshold, all the data for the week and the following weeks are classified as "high-demand" period data, until the weekly peak demand goes below the threshold. The other demand-price data are classified as "low-demand." In the high-demand period, speculative high-price bidding is particularly active because of a possible power 120 shortage, thus increasing the price volatility. High-demand days appeared from the 11th to 28th week of the yearly data. In addition to the empirical clustering scheme, some peculiar days' data associated with social activities was cropped. In the time-series graph of price and demand in Figure 34, the price shows a high spike in late December, while the demand data does not indicate any irregularity. It can be considered that there could have been some kind of speculative activities that took place for the Christmas holiday season. Therefore, this peculiar high-price data was removed from the low demand cluster. Figure 37 shows such clusters defined by empirical analysis. 200 14000 18000 22000 26000 30000 34000 38000 Demand [MWh] |^  • Low demand data High demand data * Irregular data Figure 37 Data clusters The proposed fuzzy model was applied to the data preconditioned as described above. Table 30 and Table 31 list the parameters determined by the optimization on each cluster, and Figure 38 shows the range of the model using (a) linear regression and (b) quadratic model, respectively. 121 Table 30 Parameters for first-order model order Low Demand High Demand a c+ c- a c+ c-1 0 0.0024 0.0000 1.20xl0"4 -27.037 31.936 2.0093 0.0040 0.0039 8.89X10"4 -65.852 0.0000 0.0000 Table 31 Parameters for second-order model order Low Demand High Demand a c+ c- a c+ cr 2 1 0 0.0000 0.0000 0.0000 3.15X10"4 0.0000 8.03X10"4 -34.940 31.816 2.4466 4.24x10"' 1.06x10"' 0.0000 -0.0173 0.0000 9.18X10"4 193.935 0.0000 10.0533 14000 22000 30000 38000 Demand [MWh] ^^•Crisp (low) — C r i s p (high) —•— Fuzzy (low-upper) —*— Fuzzy O i^gh-upper) —•— Fuzzy (low-lower) —•*— Fuzzy fliigh-lower) (a) Linear 122 200 14000 22000 30000 38000 Demand [MWh] (b) Nonlinear (quadratic) Figure 38 Fuzzy models on preconditioned data Compared with Figure 36, Figure 38 shows a narrower range of possible relationships between demand and price. The preconditioning of data was effective enough to capture the characteristic relationship between demand and price. 4.5.2.3 a-cuts and Error Rate In the fuzzy regression model, a cuts of fuzzy numbers are used in order to narrow the range of demand-price data as was done for the FAR model in 4.4.4. At a = 0, the model is the original fuzzy model. As the a-level increases toward 1, some data becomes neglected and falls out of the range. The tradeoff between the vagueness of the range and the likelihood of missing data can be measured by the error rate. Figure 39 shows the error rate of the fuzzy model for the pre-conditioned data set. As the a-level increases from 0, the incremental error is relatively small, but as a goes beyond a certain level, the error rate drastically increases. This curve indicates, for example, that if the decision-maker takes a risk of 0.1 error rate, a can be around 0.5, thus reducing the possible range of data into half, which still retains some possibility distribution of 123 volatile price but is much more focused. This is obvious from Figure 40, which shows the possible range with an a level of 0.5, in comparison with Figure 38 (a) with a = 0. An acceptable level of a-cut naturally depends on human decision under various circumstances. Figure 3 9 Error rates versus a-cut 22000 30000 Demand [MWh] 38000 •Crisp (low) • Fuzzy (low-upper) • Fuzzy (low-lower) "Crisp (high) Fuzzy (high-upper) Fuzzy (high-lower) Figure 40 Fuzzy demand-price model with a = 0.5 4 . 5 . 3 Forecasting through Fuzzy Regression Model An application of the proposed fuzzy model for estimation of possible ranges of electricity prices is demonstrated here. One of the major interests in the fuzzy model is to predict the 124 ranges of variation in electricity prices once an estimate of electricity demand is given. Figure 41 shows an example of such data represented by the fuzzy model. A one-week period in mid-August was chosen, where the electricity prices are most volatile. For clarity, only the linear regression model is used in this example; a-cut is not used. The median value (denoted by the thin real curve) is determined through a conventional (crisp) regression from the corresponding demand data (not shown). The range of the fuzzy model (denoted by broken curves) encompasses the actual price (denoted by a thick curve) with possible intervals on the upper and lower side of the curve, while the conventional model misses the price values largely in the first three days. The wider margin on the upper side estimation means that the price goes more possibly higher than lower. 250 8/12/98 8/13/98 8/14/98 8/15/98 8/16/98 8/17/98 8/18/98 Rice Crisp - — Fuzzy (upper) Fuzzy (bwer) Figure 41 An example of price time-series estimated by the fuzzy model 4.6 Consumer Decision-Making Aid After deregulation, electricity is typically traded either in market on spot prices or through bilateral contracts between suppliers and consumers, paying certain tariffs to transmission service providers. Such a non-utility, market-driven trade of electricity was first introduced in the wholesale sector, but opportunities are rapidly expanding to the retail sector, from large industrial customers down to individual households. 125 The strong appeal of power industry deregulation is the potential to lower the price of electricity, due mainly to increased competition among suppliers. However, "free market" activities can make the spot prices excessively volatile because of gaming and speculation by market participants [92]. In such a situation, a relatively long-term (a few months to a year) bilateral transaction of power under a firm contract would be attractive to some consumers for its stability of price over a fixed period. In practice, bilateral transactions are observed to stabilize the potentially volatile market [56]. The objective of this section is to assist an electricity consumer in making decisions regarding bilateral power transactions under uncertain market conditions when a bilateral price is offered. A typical target consumer is, for example, a small- to medium-scale industrial user of electricity, where the production process must run regardless of electricity prices and where the managers of the plant are interested in lowering the electricity cost. When consumers are offered a bilateral contract, they will probably compare the offered price to the market price. Market activity information is assumed to be commonly available via the Internet, for example. If the offered price is too high compared with the market price, the bilateral offer would be unattractive. However, the market is volatile and it is difficult to focus on a particular index. In addition, the consumer's preference may not be a simple threshold. Generally, the lower the price, the better, but it is quite uncertain what price level would satisfy the consumer and what price would be acceptable even i f the offered price is slightly higher than the uncertain market index. Previous studies on bilateral transactions of electricity [93] [21] focus on the power system operator's point of view. These studies are mainly based on a traditional central control 126 framework of power systems. The operation of transmission systems is an important problem under a deregulated industry structure, but detailed information, on the transmission systems is rarely available to individual customers. Therefore, it is assumed that only price information can be obtained by market participants. The procedure is summarized as follows: The fuzzy autoregression analysis explained in 4.3.1 is performed on time-series data of electricity market prices. Uncertain market prices are represented by the band of fuzzy numbers that contain the most likely price and highest/lowest price range at each time stage. The consumer's preference is also represented by a separate fuzzy set. Then, the differences between the offered price for a bilateral contract and the market price are calculated at each time stage and compared with the preference index. The composite grade of fuzzy sets overlapping in the price differential domain will indicate the value of the bilateral contract offered. 4.6.1 Evaluation Process through Fuzzy Sets 4.6.1.1 Fuzzy Autoregression Analysis of Market Activity The first stage is to grasp the outline of the market activities. It is necessary to know whether the market price will generally be high or low over the same period because a bilateral transaction is considered a long-term firm contract. The accurate prediction of volatile market activity is a long-unresolved problem. Market records, however, can be focused on in order to obtain a suitable range of data to reflect the possible future. An outline of the market price data can be represented by the following FAR time-series as was done in 4.3.1.2: Pm(t) = A0+Alpm(t-l) + A2pm(t-2)+-+AHpIB(t-n) ( 5 4 ) where Pm (t), A{ are fuzzy numbers as is shown in Figure 29, representing the possible band of 127 ) prices and coefficients of the AR model respectively. The median of the fuzzy parameters A{ is calculated through a conventional AR model. The fuzzy parameters c ( +, and cf (c* for upper deviation and cf for lower deviation) are obtained so that the highest and lowest prices are included in the outline, with minimum deviation over the total period of consideration. The observed market prices are evaluated at each discrete time stage t = t,, t2, t„, n > 0. If a one-year period is considered, n can be set to 365 and daily high and low prices determine the range of the outline at each time stage (i.e., day). 4.6.1.2 Preference Index While the reference market outline is available from the time-series analysis described in the previous section, another key in making a decision on a bilateral transaction is the preference of the consumers for the price offered by the supplier. Naturally, it is desirable to obtain a lower price than market, but market prices can be occasionally lower than a bilateral offer because of competition. On the other hand, an attraction of bilateral contracts is stability of price. Therefore, the consumer may settle for a slightly higher price than what the market indicates. Again, the offer is unacceptable if the price for the bilateral transaction is too high compared with the uncertain market price. Hence, the preference for a (possibly low) price can be a range of acceptable prices with varying degrees of satisfaction. Figure 42 shows a fuzzy set that represents such a consumer's preference as a piecewise linear function of price differential dp: dp=p^-pm{t) (55) The market price pm changes over time, while the offered price, poff, is a fixed value throughout 128 the contract period. Assuming that the preference of the consumer does not change during the period, the trapezoidal fuzzy set G is defined by two parameters, g, and g 0, representing the consumer's preference for low price. The grade of membership to G can be interpreted as the degree of consumer satisfaction. If the price differential is smaller than g, (negative 5p means the offered price is lower than market price), the offer is 100 percent satisfactory (p G =1). The degree declines as the differential widens, and at the high price offered becomes unacceptable, thus the membership value p G = 0. Hence, go -Sp] p . G = Min 1, Maxi 0, (56) 0 Figure 42 A fuzzy set to represent the consumer's preference 4.6.1.3 Composite Evaluation Index While consumer preference is defined by the price differential between the offered price for the bilateral transaction and the market price, the market price is given by the time-series as represented by fuzzy numbers in (54). For the overall evaluation of an offered bilateral price, a procedure in the fuzzy number domain, is as follows: 1) At each time stage, calculate the price differential D(t) using fuzzy numbers as follows: D(t) = p°« - Pm(t) ( 5 ? ) The non-fuzzy offered price poff is mapped into a fuzzy number and the resultant price differential will also be a fuzzy number, represented by a triangular fuzzy set D(t). 129 2) At each time stage, compare the preference fuzzy set G and the price differential fuzzy set D(t), and measure the maximum degree pD G(0 of overlap between the two fuzzy sets. This may be interpreted as a fuzzy decision that represents the idea that, "the price differential is to be small (low price) and as far as possible into the preferable range." If no overlap exists, the degree is 0 (8p is unacceptable). On the other hand, i f D(i) is completely enclosed by G, the degree is 1.0 or 100% satisfactory. Figure 43 illustrates this operation [94]. 3) Repeat step 1) and 2) for all time stages t = t,,... tn. 4) The overall evaluation of the offered price is given by the following defuzzification process: where pD G(0 is the maximum degree of overlap of the two fuzzy sets D(t) and G at time t: P - D G W = Max[Min{\iG,\xD(tj\] (59) go [VMWh] > & Figure 43 Measurement of price differential There may be variations in the aggregation-defuzzification process in the last step of the proposed procedure. It is, however, convenient to show the performance of an offer as a single (crisp) value, scaled to 0 through 1, rather than a fuzzy set. There may be a more meaningful aggregation method, but the sum and divide approach of (58) is simple and easy to implement. 130 4.6.2 Numerical Example To demonstrate the proposed procedure some examples are presented below. Part of the California power exchange market data shown in Figure 34 is used here. Based on the market data of January and February 1999, the daily market data from March 1999 is forecasted. The optimization is applied to this time-series data in order to determine the fuzzy number parameters as follows: Pm (t) = A0+ A]Pm (t -1) + AlPm {t-1) ( 6 0 ) Figure 44 shows the original market data (the thick line with markers) and the outlines obtained through the fuzzy time-series analysis from January through February. The outlines for these months enclose the original data in order to define the fuzzy parameters. o o f N v o o ^ r o o ^ - i p . a N m r - - — ~ C C C C f>i (N CN CN Date ^ « Pj N <N CN CM CN • Price •AR Upper Lower Figure 44 Price range determination from January and February 1999 Figure 45 shows the comparison of the forecasted (thin lines) and actual (the thick line with markers) prices in March based on the fuzzy parameters determined by the data from January and February. 131 ~ ~ ~ ~ Date —*—Price — — A R Upper Lower Figure 45 Forecasted price range for March 1999 It is clear that even though conventional AR converges to a flat curve, the established possible range covers most of the actual data. The preference parameters are then set arbitrarily: g, = $2/MWh and g0 = $5/MWh. This is interpreted as "the contract is satisfactory if the price differential is within $2/MWh; quite acceptable if within $2~5/MWh; and unacceptable if more than $5/MWh". Then, the composite performance index p D G for p o f f = $32.5/MWh are calculated. From the time-series data, the overall evaluation y = 0.659 was obtained. Several other cases were also tested and the results are summarized in Table 32. The offered price was varied to three levels: $30.0, $32.5, and $35.0/MWh, while the preference parameters have been changed. Case-1 assumes the above-mentioned preference parameters g, = 2 and gg = 5 [$/MWh]; Case-2 uses g[ — 0 and go — 2 [$/MWh], reflecting an aggressive attitude of the consumer. Case-3 is based on a more conservative attitude with g, = 5 and g0 = 10 [$/MWh]. 132 Table 32 Performance evaluation (y) Offered price Case-1 (moderate) Case-2 (aggressive) Case-3 (conservative) $30.0/MWh 0.659 0.350 0.990 $32.5/MWh 0.367 0.031 0.771 $35.0/MWh 0.077 0.000 0.535 The outlines in Figure 45 lie on the conservative side, giving a wide range, and the resultant price index can be said "somewhat fuzzier." This market data is still a rough estimate of actual market activities to which the bilateral transaction performance would be compared for scrutiny. For practical applications spanning long terms, several years of data are necessary, and background information on social and economic activities particular to a target year for estimation will be important. Table 32 shows that the bilateral contract of $35/MWh is not attractive to moderate and aggressive consumers to a similar extent, because both the market price and consumer preference are treated as uncertain. If the decision-maker's attitude for preference is adjusted again, the result makes a larger difference among contracts. Therefore, determining preferable contract among many possible contracts requires defining the preference parameter and applying the above-mentioned method iteratively in order to reduce the number of preferable contracts. In such a process, the decision-maker's preference can be clarified, or expert knowledge may possibly be acquired for use in other contracts. 4.7 Summary - Decision-Making Aid for Market Participants In this chapter, firstly, a new approach for time-series data analysis, based on possibility theory and fuzzy autoregression, was introduced and applied to the analysis of time-series data of electric power consumption. Two models have been developed, and both models have been applied to the forecasting of real-life data. The proposed fuzzy autoregression models preserve 133 the rich information of the original data, including uncertainty. Such information is useful when a degree of uncertainty is important in decision making. The fuzzy autoregression models can be constructed through a relatively simple and inexpensive procedure, based on autocorrelation analysis and linear programming. Secondly, a fuzzy regression model was introduced to represent highly volatile demand-price relationships in a deregulated electricity market. Conventional regression was applied in order to determine the most likely data model, and then fuzzy parameters were determined through linear programming. The fuzzy model represents a highly volatile data set as a range and gives the possibility distribution of prices when a single value (point) of demand forecast is given. On the other hand, conventional regression analysis only represents the data set as a single line or curve. A decision-maker's intention to focus on a particular range could be accommodated by narrowing down the fuzzy model through a levels. Thirdly, a procedure to help consumers decide between a bilateral transaction and market-based purchases of electricity was developed. Forecasting of the market price of electricity was based on the method developed above. The same procedure can also be used by an electricity supplier or broker in order to determine an offering price which is attractive to the target consumer, when consumer preference is uncertain. 134 Chapter 5 Conclusion From System Operators'Point of View I have explored desirable system operation and power market structure. Two typical ISO models, centralized and decentralized, have been identified and compared, based on actual ISO models. In the centralized system, all generators and customers receive orders from the ISO based on the price information they gave to the ISO. In the decentralized system, all participants are allowed to trade power freely by keeping their portfolios in balance through the ISO. The inevitable effect of reduced market efficiency in the mix-ISO was examined vis-a-vis the effect of strategic bidding in the max-ISO. If no powerful players exist, the max-ISO will achieve high market efficiency. However, the advocacy of free trade justifies the existence of the mix-ISO, where freedom of trade protects market participants from strategic bidding by powerful players. Reduced market efficiency is the price markets have to pay in order to prevent strategic bidding. Any two existing ISO models or market structures cannot be compared simply in terms of energy price, because the composition of generators and the availability of their fuel sources, and the number of players and their market powers are different for each ISO. Therefore, identifying the most desirable market structure is challenging. After long years of market operation, empirical study may indicate the ISO model and market structure to be the most desirable. 135 From Regulator's Point of View I have examined the effect of the objectives imposed by regulators on the ISO upon social welfare. The policies of the regulators can be various, and are not restricted to social welfare maximization. The ISO needs to solve multiobjective optimizations that have negatively correlated, often conflicting, and tradeoff relationships between objectives. From possible combinations of feasible solutions, the optimal schedule was determined by the fuzzy multiobjective optimization method. When the method is applied to the target system, the maximum and minimum allowable values have to be defined by the decision-maker or regulator. The optimal schedule could be affected greatly by the ideal goals and their allowable values, possibly working in favor of some specific market players. Therefore, when the regulator defines its objectives and their allowable ranges, an agreeable conclusion among the regulator and market participants is required. Fuzzy multiobjective optimization is easy to implement once the ideal goals are defined. The regulators, then, would have to update their ideal goals periodically, reflecting their needs. However, to stabilize the power market, these updates must be foreseeable and infrequent, since no entity will enroll in power markets with unpredictable rules. From Market Participants' Point of View I have developed models to forecast the price and demand of power. Such forecasting of price and demand is vital in order for market participants to play in the power market. Firstly, time-series data of electricity consumption was forecasted based on possibility theory and fuzzy autoregression. The proposed fuzzy autoregression models preserve the rich 136 information of the original data, including uncertainty. The fuzzy autoregression models can be constructed based on autocorrelation analysis and linear programming. Secondly, electricity price was forecasted based on a fuzzy regression model. The fuzzy model can represent highly volatile demand-price relations as a range, and gives the possibility distribution of prices when a single value (point) of demand forecast is given. Finally, a procedure to help consumers decide between a bilateral transaction or market-based purchases of electricity was developed. Forecasting of the market price of electricity was based on the method developed above. The same procedure can also be used by an electricity supplier or broker in order to determine an offering price which is attractive to the target consumer when consumer preference is uncertain. These forecasting methods depend on historical data from the power market. However, once a new policy is imposed by regulators, and new operation rules are implemented by ISOs, the price data may have completely different characteristics, such as price spikes. Therefore, in the transitional period from old rules to new ones, there is a greater uncertainty in forecasting the price for all market participants. 137 Chapter 6 Future work Ancillary Services The ancillary service markets are organized as a single-sided process in many established markets at present. Namely, the ISO purchases the required amount of all ancillary services such as reactive power, voltage support, or frequency regulation, for the benefit of all market participants in order to maintain the reliability of the power system. The ISO is the only customer, unlike the energy market where many customers exist. Both max-ISO and min-(mix-) ISO apply the same single-sided process. Players with strong market power can be easily created in the ancillary market because of the difficulty of transferring reactive power with the reactive power market, and because of generators' response characteristics in the frequency regulation market. The effects of strategic bidding by players with strong market power in the ancillary service market should be examined in a similar way as in the energy market. Emission Trade If the total amount of emissions is constrained to an area controlled by the ISO, for example pollution rights, the max-ISO model can include external factors, such as emission constraints, easily in the multiobjective optimization. However, in the case of the mix-ISO model, pollution rights can be traded in the market outside the ISO. In that case, the ISO cannot force each P X to trade pollution rights among PXs from the point of free trade. Consequently, the mix-ISO minimizes the total adjustment cost, keeping each PX's portfolio separate, and keeping each PX's emission below its pollution right. The two market systems in the mix-ISO will guarantee that the emission is below the total amount constraints. However, the solution through the mix-138 ISO is always suboptimal compared with that of the max-ISO because the ISO cannot coordinate among PXs the trade of the pollution rights as well as the power. Therefore, as the number of objectives of the ISO increases, the mix-ISO will result in fewer and fewer optimal solutions compared with the max-ISO. This reduced market efficiency can be examined using several hypothetical constraints. Fuzzy Clustering When the fuzzy A R approach is applied to practical forecasting, the treatment of irregularity of the observed data (for example, holidays and special social activities) is quite important because the forecasting assumes the cyclic recurrence of observed data. However, in order to obtain a reasonably "fit" fuzzy model, some target-specific heuristics are necessary. Such heuristics are difficult to generalize. Data clustering can be required in order to preprocess the data. If price demand data is divided into two clusters, namely low- and high-demand clusters, and the two clusters are connected through fuzzy modeling, forecasting can be improved, reflecting the different behavior of speculators in each cluster range. Consumer Decision-Making Aid For consumers, prices may not be the only factor in decision making. The model should include other consumer preference indices, such as the environmental impact [95] of power generation and the evaluation of reliability of bilateral power supply under uncertainty. The probability of curtailment of a "preferable" bilateral transaction by a transmission system operator should also be an important index in decision making regarding bilateral transactions. 139 Chapter 7 Bibliography [1] T. Nakashima and T. Niimura, "Market Plurality and Manipulation: Performance Comparison of Independent System Operators Market plurality," submitted to IEEE Transactions on Power Systems, March 2000. [2] T. Niimura, T. Nakashima, F. Moreira, C. V . Cavati, and R. Yokoyama, "Multiple Attribute Performance Evaluation of Independent Power Producers in a Deregulated Power c System Based on Fuzzy Sets," Proceedings of IEE Intelligent Systems Applications to Power Systems Conference, Rio de Janeiro, April 1999, pp. 122-126. [3] T. Nakashima and T. Niimura, "Performance Evaluation of Independent Power Producers in a Deregulated Power System by Fuzzy Sets," IEEE 1997 North American Power Symposium, Laramie, Wyoming, October 1997, pp. 486-492. [4] T. Nakashima and T. Niimura, "Multiple-Impact Assessment of Independent Power Producers in a Deregulated Environment," 1997 IEEE Canadian Conference on Electrical and Computer Engineering, St. John's, Nfld., May 1997, pp. 273-276. [5] K. Ozawa, T. Niimura, and T. Nakashima, "Fuzzy Time-series Model of Electric Power Consumption," accepted for publication, International Journal of Advanced Computational Intelligence, December 1999. [6] T. Nakashima, T. Niimura, M . Dhaliwal, and K. Ozawa, "Fuzzy Data Representation for Demand-Price Relations Under Deregulated Electricity Market," submitted to IEE of Japan Transactions on Electronics and Information, February 2000. [7] T. Nakashima, T. Niimura, and K. Ozawa, "Fuzzy Estimation of Deregulated Electricity Market Prices," Proceedings of IEEE North American Power Symposium, San Luis Obispo, C A , October 1999, pp. 28-32. 140 [8] T. Niimura, T. Nakashima, and K. Ozawa, "A Consumer Decision-making Aid for Electricity Purchases by Possibility Regression and Fuzzy Decision," Proceedings of IEEE Systems, Man, and Cybernetics Conference, Tokyo, October 1999, pp. 1019-23. [9] S. Hunt, "Unlocking the Grid," IEEE Spectrum, Vol . 33, No. 7, July 1996, pp. 20-25. [10] V. V . Subramanian and J. D. Hammerly, "New Faces and Functions in a Competitive Market," IEEE Computer Applications in Power, January 1997, pp. 47-52. [11] J. W. Williams, "Open Transmission Access," IEEE Power Engineering Review, Vol . 16, No. 5, May 1996, pp. 17-19. [12]R. Mukerji, W. Neugebauer, R. P. Ludorf and A. Catelli, "Evaluation of Wheeling and Non-utility Generation (NUG) Options Using Optimal Power Flows," IEEE Transactions on Power Systems, Vol . 7, No. 1, February 1992, pp. 201-207. [13] A . E. Boardman and C. Laurin, "Factors Affecting the Stock Price Performance of Share Issued Privatisations", Applied Economics (to be published). [14] John M . Levy, "Chapter 3: Supply and Demand," Essential Microeconomics for Public Policy Analysis, pp. 23-43, Praeger Publishers, Westport CT, 1995. [15]0. B. Fosso and A . Gjelsvik etc., "Generation scheduling in a deregulated system. The Norwegian case," IEEE Transactions on Power Systems, Vol . 4, No. 1, February 1999, pp. 75-80. [16]F. C. Schweppe, M . Caramanis, R. Tabors and R. Bohn, Spot Pricing of Electricity, Kluwer Academic Publishers, Boston, 1988. [ 17] W. W. Hogan, "Contract networks for electric power transmission," Journal of Regulatory Economics, Vol . 4, 1992, pp. 211-242. [18]H. Rudnick, R. Varela, and W. Hogan, "Evaluation of alternatives for power system coordination and pooling in a competitive environment," IEEE Transactions on Power 141 Systems, Vol . 12, No. 3, May 1997, pp. 605-613. [19] H. Singh, S. Hao and A . Papalexopoulos, "Transmission Congestion Management in Competitive Electricity Markets," IEEE Transactions on Power Systems, Vol . 13, No. 2, May 1998, pp. 672-680. [20] A . K. David, "Dispatch Methodologies for Open Access Transmission Systems," IEEE Transactions on Power Systems, Vol . 13, No. 1, February 1998, pp. 46-53. [21]R. S. Fang and A. K. David, "Optimal Dispatch under Transmission Contracts," IEEE Transactions on Power Systems, Vol . 14, No. 2, May 1999, pp. 732-737. [22] R. S. Fang and A. K . David, "Transmission Congestion Management in an Electricity Market," IEEE Transactions on Power Systems, Vol . 14, No. 3, August 1999, pp. 877-883. [23] A . J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control 2nd ed., John Wiley, 1996. [24] R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, 1987. [25]T. Coleman, M . A. Branch and A. Grace, User's Guide: Optimization Toolbox For use with MATLAB Version 2, The Math Works Inc., 1-999. [26] R. Rajaraman, J. V . Sarlashkar and F. L. Alvarado, "The effect of demand elasticity on security prices for the poolco and multi-lateral contract models,' IEEE Transactions on Power Systems, Vol . 12, No. 3, August 1997, pp. 1177-1183. [27] J. D. Weber and T. J. Overbye, " A Two-Level Optimization Problem for Analysis of Market Bidding Strategies," Proceedings of the 1999 IEEE Power Engineering Society Summer meeting, pp. 682-687. [28] North American Electric Reliability Council, "Available Transfer Capability Definitions and Determination," June 1996. [29] R. D. Christie, B. F. Wollenberg, and I. Wangensteen, "Transmission Management in the 142 Deregulated Environment," Proceedings of the IEEE, Vol . 88, No. 2, February 2000, pp. 170-195. [30] T. Niimura, T. Nakashima, and R. Yokoyama, "Branch Overload alleviation by flexible coordination of generation rescheduling and load shedding," International Journal of Engineering Intelligent Systems for Electrical Engineering and Communications, March 2000, pp. 67-72. [31] P. R. Gribik, G. A . Angelidis and R. R. Kovacs, "Transmission Access and Pricing with Multiple Separate Energy Forward Markets," IEEE Transactions on Power Systems, Vol . 14, No. 3, August 1999, pp. 865-872. [32] Z. Alaywan, "Facilitating the Congestion Management Market in California," Proceedings of the 1999 North American Power Symposium, pp. 1-9. [33] W. Mieiczarski, "Quality of Electricity Supply in Competitive Electricity Markets," Workshop III CONLADIS, September, 1999, pp. 1-60. [34] Office of Economic, Electricity and Natural Gas Analysis, Office of Policy, US Department of Energy, Horizontal Market Power in Restructured Electricity Markets, March 2000. [35] J.K. Delson, "Controlled Emission Dispatch," Paper T 74 156-6, pp. 1359-1366. [36] J. B. Cadogan and L. Eisenberg, "Sulfur Oxide Emission Management For Electric Power Systems," IEEE Transactions on Power Apparatus and Systems, Vol . PAS-96, 1977, pp. 393-401. [37] M . R. Gent and J.W. Lamont, "Minimum emission dispatch," IEEE Transactions on Power Apparatus and Systems, Vol . PAS-90, 1971, pp. 2650-2659. [38] J. H . Talaq, Ferial and M.E. El-Hawary, " A Summary of Environmental/Economic Dispatch Algorithms," S M 465-5 PWRS 1993. 143 [39] W. Y . Spens, and F. N . Lee, "Interactive Search Approach to Emission Constrained Dispatch," S M 507-4 PWRS 1996. [40] V . C. Ramesh and L. Xuan, " A Fuzzy Multiobjective Approach to Contingency Constrained OPF," IEEE Transactions on Power Systems, Vol . 12, No. 3, August 1997, pp. 1348-1354. [41] C. Huang, H. Yang, and C. Huang, "Bi-Objective Power Dispatch Using Fuzzy Satisfaction-Maximizing Decision Approach," IEEE Transactions on Power Systems, Vol . 12, No. 4, November 1997, pp. 1715-1721. [42] D. Srinivasan and A. G. B. Tettamanzi, "An evolutionary algorithm for Evaluation of emission compliance Options in view of the Clean Air act Amendments," IEEE Transactions on Power Systems, Vol . 12, No. 1, February 1997, pp. 336-341. [43] John M . Levy, "Chapter 12: Selling the Right to Pollute," Essential Microeconomics for Public Policy Analysis, pp. 177-187, Praeger Publishers, Westport CT, 1995. [44] H . Glavitsch and F. Alvarado, "Management of Multiple Congested Conditions in Unbundled Operation of A Power System," IEEE Transactions on Power Systems, Vol . 13, No. 3, August 1998, pp. 1013-1019. [45] M . Sakawa, Fuzzy sets and Interactive multiobjective optimization, Plenum Press, 1993. [46] T. Niimura, "Intelligent Operation and control of Electrical Power Systems Based on Operators' Heuristics," Ph.D. Dissertation, Tokyo Metropolitan University, July 1992. [47] H . J. Zimmerman, Fuzzy Set Theory and Its Applications (2nd ed.), Kluwer Academic Publishers, Dordrecht, 1991. [48] T. Ackermann, G. Andersson and Lennart Soder, "What is Distributed Generation? " Proceedings of the 1999 North American Power Symposium, pp. 232-239. [49] B. Kirby and E. Hirst, "6. Ancillary Services," Open Access Impact on Energy Scheduling, 144 Pricing, and Control, Tutorials of the 1997 IEEE PICA conference. [50JW.W. Hogan, "Chapter 7: Markets in real electric networks require reactive prices," Electricity Transmission Pricing and Technology, pp. 143-182, Kluwer Academic Publishers, Boston, 1996. [51] S. Hao and A. Papalexopoulos, "Reactive Power Pricing and Management," IEEE Transactions on Power Systems, Vol . 12, No. 1, February 1997, pp. 95-102. [52] A . A . El-Keib and X . Ma, "Calculating Short-run Marginal Costs of Active and reactive power production," IEEE Transactions on Power Systems, Vol . 12, No. 2, May 1997, pp. 559-564. [53] Federal Energy Regulatory Committee, Promoting Wholesale Competition Through Open Access Non-discriminatory Transmission Services by Public Utilities; Recovery of Stranded Costs by Public Utilities and Transmitting Utilities, Final Rule, Dockets No. RM95-8-000 and RM94-7-001, Order 888, 1996 April. [54] B. Kirby and E. Hirst, "6. Ancillary Services," Open Access Impact on Energy Scheduling, Pricing, and Control, Tutorials of the 1997 IEEE PICA conference. [55] ISO Tariff Original Sheet No. 557, "Scheduling Protocol," http://www.caiso.com/pubinfo/tariffs. [56] W. Mielczarsky, G. Michalik and M . Widjaja, "Bidding Strategies in Electricity Markets," Proceedings of the 1999 IEEE PICA conference, pp. 71-76. [57] H. Song, C. C. Liu, and J. Lawarree, "Decision making of an Electricity Supplier's Bid in a Spot Market," Proceedings of the 1999 IEEE Power Engineering Society Summer meeting, pp. 692-696. [58] W. Mielczarski, M . Widjaja, "Analysis of Bids and Re-bids of Generators in the Australian national Electricity Market," Proceedings of the 1999 IEEE Power Engineering Society 145 Winter meeting, pp. 833-838. [59] R. Bjorgan, C. C. Liu, and J. Lawarree, "Financial Risk Management in a Competitive Electricity Market," IEEE Transactions on Power Systems, Vol . 14, No. 4, November 1999, pp. 1285-1291. [60] J. Nazarko and W. Zalewski, "The Fuzzy Regression Approach to Peak Load Estimation in Power Distribution Systems," IEEE Transactions on Power Systems, Vol . 14, No. 3, August 1999, pp. 809-814. [61] Website http://www.calpx.com: Sponsored by the California Power Exchange. [62] R. Wilson, "Market Architecture," Presidential address of the 1999 Far Eastern Meeting of The Econometric Society, pp. 1-22. [63] H. Outhred and J. Kaye, "Chapter 9: Incorporating network effects in a competitive electricity industry: An Australian perspective," Electricity Transmission Pricing and Technology, pp. 207-227, Kluwer Academic Publishers, Boston, 1996. [64] ISO Tariff Original Sheet No. 417, "Dispatch Protocol," http://www.caiso.com/pubinfo/tariffs. [65] ISO Tariff Original Sheet No. 506, "Schedules and Bids Protocol," http://www.caiso.com/pubinfo/tariffs. [66] J.W. Marangon Lima and E.J. de Oliveira, "The long-term impact of transmission pricing," IEEE Transactions on Power Systems, Vol . 13, No. 4, November 1998, pp. 1514-1520. [67] J. Contreras, " A cooperative Game Theory Approach to Transmission Planning in Power Systems," Ph.D. Dissertation, University of California, Berkeley, 1997. [68] Y Tsukamoto and I Iyoda, "Allocation of Fixed Transmission Cost to Wheeling Transactions by Cooperative Game theory," IEEE Transactions on Power Systems, Vol . 11, No. 2, May 1996, pp. 620-629. 146 [69] A. Ott, "PJM: A Full Service ISO Market Evolution," Proceedings of the 1999 IEEE Power Engineering Society Summer meeting, pp. 746-752. [70] H. Singh, " The California Electricity Market: An Assessment of the first year of operations," Proceedings of the 1999 IEEE Power Engineering Society Summer meeting, pp.753-755. [71] PJM Interconnection, L. L. C , "Serving PJM Peak Load July 6, 1999." [72] Website http://www .pirn. com: Sponsored by the PJM Interconnection, L. L. C. [73] Website http://www.calpx.com/prices/index_prices_davahead trading.html: Sponsored by the California Power Exchange. [74] F. Albuyeh and Z. Alaywan, "California ISO Formation and Implementation," IEEE Computer Applications in Power, Vol . 12, No. 4, October 1999, pp. 30-34. [75]Website http://www.energvonline.com/Restructuring/energydb/conspec.html: Sponsored by the L C G Consulting. [76] Website http://www.nymex.com/markets: Sponsored by the New York Mercantile Exchange. [77] Website http://www.energvonline.conVRestmcmring/energydb/estathme.html: Sponsored by the L C G Consulting. [78] John C. Hull, Options, Futures, and other Derivatives, Prentice Hall. [79] L. A . Zadeh, "Fuzzy Sets as a Basis for a Theory of Possibility," Fuzzy Sets and Systems, Vol . l , N o . 1, 1978, pp. 3-28. [80] M . Kanke, K. Ozawa, T. Niimura, T. Watanabe, "Fuzzy Model-Based A R and Its Application," Proceedings of the 1998 Vietnam-Japan Bilateral Symposium on Fuzzy Systems and Applications, pp. 374-377. [81]G. Michalik, M . E. Khan and W. Mielczarski, "Statistical Analysis of Electricity Demand 147 and Spot Prices in Competitive markets," Proceedings of the 4th International Conference on Advances in Power System Control, Operation and Management, Hong Kong, November 1997, pp. 457-462. [82] D. Dubois and H. Prade, Fuzzy Sets and Systems: The.ory and Applications, Academic Press, 1980. [83]H. Tanaka, "Possibility Model and its Applications," Fuzzy Logic Foundation and Industrial Applications, pp. 93-110, Kluwer Academic Publishers, Boston, 1996. [84] J. D. Hamilton, Time Series Analysis, Princeton University Press, 1994. [85] Website http://ssnt.ism.ac.jp: Sponsored by the Institute of Statistical Mathematics. [86] H. Tanaka, S. Uejima, and K. Asai, "Linear Regression Analysis with Fuzzy Model," IEEE Transactions on SMC, Vol . 12, No. 6, 1982, pp. 903-907. [87] J. Watada, "Possibilistic Time-series Model and Analysis of Consumption," in D. Dubois, H. Prade, R. Yaeger (ed.), Fuzzy Information Engineering, John Wiley and Sons, New York, 1996. [88] J. Flory, "Electricity Transactions in an Open Access Market," IEEE Power Engineering Review, January 1996, pp. 15-18. [89] G.E.P. Box and G .M. Jenkins, Time Series Analysis, Forecasting and Control, Holden-Day, Revised Edition, 1982. [90] D. C. Park, M . A . El-Sharkawi, R. J. Marks, Electric Load Forecasting Using An Artificial Neural Network, IEEE Transactions on Power Systems, Vol . 6, No. 2, May 1991, pp. 442-448. [91]M. Illic, F. Galiana, and L. Fink, Power Systems Restructuring, Kluwer Academic Publishers, Boston, 1998. [92] R. Green, "The Political Economy of The Pool," in M . Ilic, F. Galiana and L. Fink (ed.), 148 Power Systems Restructuring, pp. 148-164, Kluwer Academic Publishers, Boston, 1998. [93] F. Wu, "Coordinated Multilateral Trades for Electric Power Networks," Proceedings of 1996 Power Systems Computation Conference, pp. 786-792. [94] K. Tanaka, An Introduction to Fuzzy Logic for Practical Applications, Springer, New York, 1996. [95] T. J. Hammons, et al., "Market Forces Propelling Renewable Energy Technologies," IEEE Power Engineering Review, January 1999, pp. 12-22. 149 Appendix Appendix I Sequential Quadratic Programming Nonlinear constrained optimization problems with their objective and constraint functions f(x) and c(x) respectively, can be solved efficiently through iterative solutions of certain approximations to f(x) and c(x). Then, Newton's method can be applied in order to find the stationary point of the approximations of the Lagrange function (Lagrange-Newton method). The Lagrange function is defined in terms of the variables x and X as shown in (13). Thus, a possible solution generates a sequence of approximations x(k'\ X(k) to both the solution vector x* and the vector of optimum Lagrange multipliers X*. Z ( * A ) = / ( * ) - _ _ V , ( * ) '«*•' (13) A Taylor series for VZ, about x(k), X(k) gives VL(x(k) + 5x,Vk) +6X) = VL{k) + [V2L{k)] (*) - \7r<*> 72 r(k) \8XJ +• (61) Neglecting higher order terms and setting the left-hand side to zero in (61) gives the iteration [V2Z,(*>]| = -v/J (*) (62) This is solved through Newton's method to give corrections dx and 5X. Formulae for VZ, and V 2 Z are readily obtained from (13), giving the system V§x\ (_ „ ( * ) _ L AM\ W\ _ A W T Aw 0 \o~Xj -gw+AwX( (63) where A(k) is the Jacobian matrix of constraint functions c(x) evaluated at x(k), and g(k) is the first order derivative of the objective function f(x) evaluated at x(k) respectively. W{k) =V2f(x(k))-YJ'X(i)V2ci{x{k)) i is the Hessian matrix W2XL(xw,X(k)). (64) It is more convenient to write X(k+l) - X(k) +8X and 8 < A ) =b~x, and to solve the equivalent system " Ww -Aw -A(k)T 0 in order to determine b(k) and X(k+,). (6) yXJ 150 Appendix Then, x(k+l* is given by: jc ( i + , ) = xlk)+8lk) (66) The method requires initial approximations x(,) and X(1), and uses (65) and (66) to generate the iterative sequence {x(k\ X^}. This method can be restated as a subproblem where the minimization of a quadratic function is involved. Consider the subproblem: Minimize q(k)(b) (67) subject to I(k)(5)=0 where qw(5) = - 5 TW(k)8 + g(k)T3 + f(k) 2 S (68) and / w ( 8 ) = A(k)T8 + c(k) ( 6 9 ) Thus, the following iterative method known as sequential quadratic programming is suggested, given initial estimates and Fork=l,2... (i) Solve (67) to determine b(k> and let X(k+I) be the vector of Lagrange multipliers of the linear constraints. (ii) S e t x ( A + , ) = J c w + 5 ( t ) (70) The constraints in (69) are obtained by replacing the nonlinear constraints c(x)=0 by their first order Taylor series approximation /k)(5)=0 about x<k>, given by (69). Likewise the objective function f(x) in (13) is replaced by the quadratic function q(k)(b) in (68). This is a second-order Taylor series approximation of f(x) about x ( k ), but with the addition of constraint curvature terms in the Hessian. To solve the nonlinear inequality constraint problem, this method is restated as follows: Minimize q(k)(b) (71) subject to I(k)(5)>0 (72) 151 Appendix Appendix II IEEE 30 Bus System II. 1 Bus data BUS N A M E Vs Bsh Vmin Vmax 1 Glen Lyn 1.060 0 - -2 Claytor 1.045 0 (0-9)1 (1.1) 3 Kumis - 0 0.9 1.1 4 Hancock - 0 0.9 1.1 5 Fieldale 1.010 0 (0.9) (1.1) 6 Roanoke - 0 0.9 1.1 7 Blaine - 0 0.9 1.1 8 Reusens 1.010 0 (0.9) (1.1) 9 Roanoke - 0 0.9 1.1 10 Roanoke - 0.190 0.9 1.1 11 Roanoke 1.082 0 (0.9) (1.1) 12 Hancock - 0 0.9 1.1 13 Hancock 1.071 0 (09) (1.1) 14 Bus 14 - 0 0.9 1.1 15 Bus 15 - 0 0.9 1.1 16 Bus 16 - 0 0.9 1.1 17 Bus 17 - 0 0.9 1.1 18 Bus 18 - 0 0.9 1.1 19 Bus 19 - 0 0.9 1.1 20 Bus 20 - 0 0.9 1.1 21 Bus 21 - 0 0.9 1.1 22 Bus 22 - 0 0.9 1.1 23 Bus 23 - 0 0.9 1.1 24 Bus 24 1.071 0.043 (0.9) (1.1) 25 Bus 25 - 0 0.9 26 Bus 26 - 0 0.9 1.1 27 Cloverdl - 0 0.9 1.1 28 Cloverdl - 0 0.9 1.1 29 Bus 29 - 0 0.9 1.1 30 Bus 30 - 0 0.9 1.1 7 Voltage limit when reactive power capability at the bus is exhausted, and the bus is considered to be a load bus. 152 Appendix II.2 Line data BUS# BUS# R X Csh Tap ratio Maximum flow 1 2 0.0192 0.0575 0.0528 - 100 1 3 0.0452 0.1852 0.0408 - 100 2 4 0.0570 0.1737 0.0368 - 100 3 4 0.0132 0.0379 0.0084 - 80 2 5 0.0472 0.1983 0.0418 - 80 2 6 0.0581 0.1763 0.0374 - 80 4 6 0.0119 0.0414 0.0090 • - 80 5 7 0.0460 0.1160 0.0204 - 100 6 7 0.0267 0.0820 0.0170 - 100 6 8 0.0120 0.0420 0.0090 - 100 6 9 0 0.2080 0 0.978 100 6 10 0 0.5560 0 0.969 100 9 11 0 0.2080 0 - 100 9 10 0 0.1100 0 - 100 4 12 0 0.2560 0 0.932 100 12 13 0 0.1400 0 - 100 12 14 0.1231 0.2559 0 - 30 12 15 0.0662 0.1304 0 - 30 12 16 0.0945 0.1987 0 - 30 14 15 0.2210 0.1997 0 - 30 16 17 0.0824 0.1923 0 - 30 15 18 0.1073 0.2185 0 - 30 18 19 0.0639 0.1292 0 - 30 19 20 0.0340 0.0680 0 - 30 10 20 0.0936 0.2090 0 - 30 10 17 0.0324 0.0845 0 - 30 10 21 0.0348 0.0749 0 - 30 10 22 0.0727 0.1499 0 - 30 21 22 0.0116 0.0236 0 - 30 15 23 0.1000 0.2020 0 - 30 22 24 0.1150 0.1790 0 - 30 23 24 •0.1320 0.2700 0 - 30 24 25 0.1885 0.3292 0 - 25 25 26 0.2544 0.3800 0 - 25 25 27 0.1093 0.2087 0 - 25 28 27 0 0.3960 0 0.968 25 27 29 0.2198 0.4153 0 • - 25 27 30 0.3202 0.6027 0 - 25 29 30 0.2399 0.4533 0 - 25 8 28 0.0636 0.2000 0.0428 - 25 6 28 0.0169 0.0599 0.0130 - 25 153 Appendix III Multiobjective Optimization Appendix 111.1 Multiobjective Optimization The following is the definition of the multiobjective programming problem. minimize/(x) = (fl(x),f2(x),...,fk(x))T subject to x e l = {x GR"\gj(x) < 0,j = \,...,m} ( 7 3 ) where fi(x),...fk(x) are k distinct objective functions, gi(x),...gk(x) are m inequality constraints and X is the feasible set of constrained decisions. Multiobjective optimization problems have Pareto optimal solutions, where one objective cannot be improved without the deterioration of the other objectives. The following is the definition of several optimal solutions. • Complete optimal solution x* is said to be a complete optimal solution to the problem i f and only i f there exists x e Xsuch that ft(x*) < f;(x),i = \,...,k for all x e X. However, when the objective functions conflict with each other, a complete optimal solution does not always exist, and hence, the Pareto optimality concept is defined as follows: • Pareto optimal solution x* is said to be a Pareto optimal solution to the problem i f and only i f there does not exist another x e Xsuch that f.(x) < ft(x*) for all / and f-(x) * fj(x') for at least one j-In a nonlinear programming problem, only local optimal solutions are guaranteed unless the problem is convex. • Local Pareto optimal solution x* is said to be a local Pareto optimal solution to the problem i f and only i f there exists a real number 5 > 0 such that x* is Pareto optimal in the 8 neighborhood of x* (||x-x*|| < 5 ). A number of options in optimizing multiobjective functions have been proposed, aiming at harmonization among many conflicting objectives. Two options, s-constraint and weighting methods, have often been applied in order to calculate Pareto optimal solutions. 111.2 Constraint method s-constraint technique chooses one specific objective as the objective function to be optimized from the set of objectives that have tradeoff relationships. The other objectives are considered 154 Appendix constraints. Thus, the objective function can be minimized within some constraints. If values of the constraint are changed step-by-step, Pareto optimum solutions can be found. The s-constraint problem is defined as follows: Minimize fj(x) (74) subject to fj(x)<£j, i=\,..., k; (75) The Lagrange function L(x,X) for the constraint problem is formulated as L(x, \) = f. (x) + £ X,,. (f. (x) - 8,.) (76) ;=1 where Xjni = \,2,...,k;ij denote the corresponding Lagrange multipliers. The Lagrange multiplier Xji represents the tradeoff rates between Jj(x) and fi(x) as follows: dfj(x) , . . X„= ,i = \,...,k;i j. " (77) The drawback of this method is that it requires the specification of the tolerance parameters s associated with the conflicting constraints that represent the secondary objectives. These parameters are not easy to specify in the case of power systems. For example, the system operator may be reluctant to specify a tolerable security margin. III .3 Weighting method If all objectives can be converted into one-unit values and then added, the resultant total value can be defined to form one objective function. Consequently, a conventional solution of the optimization problem can be obtained for that total objective function. This is one Pareto optimum solution, and i f the converted values are changed step-by-step, reflecting the tradeoff between conflicting objectives, Pareto optimal solutions can be found. The weighting method for obtaining Pareto optimal solutions is to solve the following weighting problem, formulated by taking the weighted sum of all the objective functions of the original multiobjective nonlinear problem. Minimize wf(x) = £ wifi (x) (78) 1=1 where w=(wi,...wk) is the vector of weighting coefficients assigned to the objective functions, and assumed to be w=(wi,...Wk) > 0. The weighting coefficients of the weighting problem give the tradeoff rate information between the objective functions as follows: .%.-Hii-2 k V, w . (79) 155 Appendix The drawback of this method is that finding suitable weighting coefficients in order to combine objectives is difficult. III.4 Fuzzy multiobjective programming Assuming that the decision-maker has fuzzy goals for each of the objective functions, it is possible to soften the rigid constraints or objective functions of the decision-maker through fuzzy goals. In such a situation, the optimization problem may be softened into the following fuzzy version: minimize f(x) = (f{ (x), f2 (x), ...,fk (x)) 7 subjecttox GX= {X eR"\gj(x)<0,j = l,...,m}\ (80) where the symbol '~ ' denotes a relaxed or fuzzy version of those of 'minimize' and ' < meaning that the objective function should be minimized under some constraints. Such fuzzy requirement can be quantified by eliciting the membership functions \ii(f(x)), \ii(gi(x)), i=\,...,m, from the decision-maker for all the functions f(x) andgt(x), i=l,...m. Reflecting the satisfaction of the membership, the decision-maker must determine the as-yet undetermined membership function \ij(f(x)), which is a monotonically decreasing function with respect to f(x) in the following way. M/,(*)) = d,{x) ;//£/,(x)£/, 0 o ; / ; . (x)>/; 0 (81) where jf or f/ denote the value of the objective function f,(x) such that the degree of membership function is 0 or 1 respectively. To elicit a membership function from the decision-makers for each of the objective functions, an individual minimum and maximum of each objective function (ideal or Utopia solutions) should be calculated under the given constraints. Then, based on those results, the decision-makers can determine the membership function, which well-reflects their preference. Moreover, i f the tradeoff information between those fuzzy goals can be provided, the decision process can be quite simplified. Such information is available from the Lagrange multipliers in the same way as was mentioned in section 111.2. d\ijWj(x)) Xj — = —,i = h-..,k;i * j. (82) SM,C/,(x)) X, This relation assumes that all the constraints are active. Therefore, i f some inactive constraints exist, it is necessary to replace jl, for inactive constraints with p,. (ft (x*)) and solve the corresponding minimax problem in order to obtain the Lagrange multipliers. 156 Appendix Appendix IV Market Observations IV.l NYMEX Electricity Futures Contract Specifications Trading Unit 736 MWh delivered over a monthly period Trading Hours 10:30 A . M . - 3:30 P .M. for the open outcry session After-hours trading will be conducted via the N Y M E X ACCESS® electronic trading system Monday through Thursday, 4:15 P .M. to 7:15 P.M. Trading Months 18 consecutive months. Price Quotation Dollars and cents per MWh. Minimum Price Fluctuations $.01 (10) per M W h ($7.36 per contract). Maximum Daily Price Fluctuation None Last Trading Day .Trading will terminate on the fourth business day prior to the first day of the delivery month. Delivery Rate 2 M W throughout every hour of the delivery period (can be amended upon mutual agreement between the buyer and seller). Delivery Period Sixteen on-peak hours: hour ending 0700 prevailing time to hour ending 2200 prevailing time (can be amended at the time of delivery by mutual consent of the buyer and seller). Scheduling COB futures: Buyer and seller must follow Western Systems Coordinating Council scheduling practices. IV.2 Volatility estimate To estimate the volatility of the electricity price index empirically, the price is observed over fixed intervals of time. Define n+1: number of observations St: electricity price at end of /-th interval (/=0,1,.. .n) x : length of time interval in years and let «f = ln(-^-) S<-1 for/=l,2,...n. (83) Since Si = S^e"1, w, is the continuously compounded return in the /-th interval. The usual estimate, s , of the standard deviation of the w, is given by .H-^z^-*)2 V * - 1 ' - (84) 157 Appendix or ( s = - 1 f t " » ( » - » ( 8 5 ) where 1 " " '=• (86) is the mean of The standard deviation of w„ s is o-Vx (87) The variable, s, is therefore an estimate of (87). It follows that a itself can be estimated as s*, where s (88) 158 Appendix V Fuzzy Set and Possibility Theory Appendix V . l Fuzzy set and a-cuts A fuzzy set A on the universe X is a set defined by a membership function u A representing a mapping: UA:X->[0,1] (89) Here the value of p A (fuzzy set A) at x e X is called the membership value or the grade of membership of x in A . The membership value represents the degree of x belonging to the fuzzy set A . Then, a-cuts of the fuzzy set A can be defined as follows: A a = {x | pA(x) > a}, a e [0,1 ] (90) The a-cut A a is an interval projected to the X domain as shown in Figure 46. Aa2 Figure 46 An example of a-cuts V.2 Possibility theory and possibility regression analysis Possibility theory uses possibility distributions as statistical theory uses probability distributions. Possibility theory is more suitable than probability theory in dealing with uncertainty arisen from expert judgement because possibility theory systematically treats uncertainty, which is not random in nature. Since membership functions of fuzzy sets can be viewed as possibility distributions, fuzzy sets, which can express expert knowledge, are equivalent to possibility distributions. Triangular fuzzy numbers have often been used as possibility distributions. Possibility distributions can be identified, based on expert knowledge and on fuzzy data analysis, such as regression analysis. 159 Appendix A simple possibility regression model can be expressed as follows: Y — A^X] 4- A2x2 4—' "\-A/Jxfl — Ax (91) where x, is an input variable, and At is a coefficient expressed through fuzzy numbers as follows: A( =< a,, ci > where {a|a,. - c(. < a < at + cj} Then, (91) can be expressed as follows: Y =< a'x,c'\x\ > where a = (at,a2,...,an)', c = (c , ,c 2 , . . . ,c„) ' , and \x\ = (|x,|,|x 2|,...,|x„|)' If the data is given as (yj, xj),j=\,...,m, where Xj = (xy.,, xj2,..., xJU)' Then the given output should be included in the estimation interval as follows: a'xj-c'\xj\<yj <alXj+c'\xj\, j = \,...,m (92) (93) (94) The spread of the models is defined by: m J=YC'\XJ\ (95) The regression analysis is to determine the coefficients Ah i=l,...,n so that J is minimized subject to (94) and c > 0. 160 

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