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Final Report onBidding strategies in deregulated power market

Prepared by

K.GAUTHAM REDDY - 2011A8PS364G

A Report prepared in partial fulfilment of the requirements of the course

INSTR F266: STUDY ORIENTED PROJECT (SOP)INSTRUCTOR: K.CHANDRAM

Birla Institute of Technology and Science – Pilani02/05/2014

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Acknowledgement:

I take this opportunity to express my profound gratitude and deep regards to my guide K.Chandram sir for his exemplary guidance, monitoring and constant encouragement throughout this project. The blessing, help and guidance given by him time to time shall carry me a long way in the journey of life on which I am about to embark.

Table of contents

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1) Introduction

1.1 Introduction…………………………………………………………………………… 4

1.2 Market structure……………………………………………………………………. 5

1.3 Operation of power system under deregulation……………………. 6

1.4 Literature survey…………………………………………………………………… 8

2) Bidding

i) Single side and Double side Bidding……………………………………….. 19

ii) Various bidding strategies…………………………………………………….. 21

3)Recent algorithms for solution of bidding strategies

3.1 Conventional methods…………………………………………………………. 25

i)Game theory

ii)Nash equilibrium

3.2 Recent algorithms……………………………………………………………….. 28

i) Genetic algorithm

ii) Two level optimization

iii) Possibility theory

iv) Fuzzy adaptive gravitational search algorithm

4) Implementation of above algorithms………………………………………. 36

5) Case studies…………………………………………………………………………….. 47

6) References……………………………………………………………………………… 49

Appendix

Introduction

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In economic terms, electricity (both power and energy) is a commodity capable of being bought, sold and traded. An electricity market is a system for effecting purchases, through bids to buy; sales, through offers to sell; and short-term trades, generally in the form of financial or obligation swaps. Bids and offers use supply and demand principles to set the price. Long-term trades are contracts similar to power purchase agreements and generally considered private bi-lateral transactions between counterparties.

Wholesale transactions (bids and offers) in electricity are typically cleared and settled by the market operator or a special-purpose independent entity charged exclusively with that function. Market operators do not clear trades but often require knowledge of the trade in order to maintain generation and load balance. The commodities within an electric market generally consist of two types: power and energy. Power is the metered net electrical transfer rate at any given moment and is measured in megawatts (MW). Energy is electricity that flows through a metered point for a given period and is measured in megawatt hours (MWh).

Throughout the 20th century, control of the energy industry rested with a large group of regional monopolies-companies that were the sole providers of the supply and delivery of electricity for the areas they served. Because of the importance of these services to the public, these utilities were heavily regulated by the government.

Since the mid-1990s, a number of states and provinces have deregulated their electricity markets, allowing competition in the industry. This means that customers in those territories can choose an alternative electricity provider (different from their utility) to seek competitive rates and choose electricity products that make sense for their business.

Most people that deal with energy have heard of a deregulated electricity market. Still, many find it difficult to differentiate it from a regulated market. It can be a challenging concept to understand

The difference between the two markets is actually fairly simple. In a regulated electricity market, there is only one main company, which is commonly referred to as the utility. This utility claims ownership of the entire infrastructure including wires, transformers, poles, etc. It has two major responsibilities. The first is to

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purchase electricity from companies that generate it, and the second is to sell and distribute it to its customers.

In a deregulated market, an additional party is involved. The utility still owns the infrastructure, but now, its only responsibility is to distribute the electricity. Deregulated markets permit electricity providers to compete and sell electricity directly to the consumers.

Market structure:

Deregulation is the process of removing or reducing state regulations. It is therefore opposite of regulation, which refers to the process of the government regulating certain activities.Hence it is reduction or elimination of government power in a particular industry, usually enacted to create more competition within the industry.

The stated rationale for deregulation is often that fewer and simpler regulations will lead to a raised level of competitiveness, therefore higher productivity, more efficiency and lower prices overall. Opposition to deregulation may usually involve apprehension regarding environmental pollution and environmental quality standards (such as the removal of regulations on hazardous materials), financial uncertainty, and constraining monopolies.

Regulatory reform is a parallel development alongside deregulation. Regulatory reform refers to organized and ongoing programs to review regulations with a view to minimizing, simplifying, and making them more cost effective. Such efforts, given impetus by the Regulatory Flexibility Act of 1980, are embodied in the United States Office of Management and Budget's Office of Information and Regulatory Affairs, and the United Kingdom's Better Regulation Commission. Cost–benefit analysis is frequently used in such reviews. In addition, there have been regulatory innovations, usually suggested by economists, such as emissions trading.

Deregulation can be distinguished from privatization, where privatization can be seen as taking state-owned service providers into the private sector. Deregulation gained momentum in the 1970s, influenced by research at the

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University of Chicago and the theories of Ludwig von Mises, Friedrich von Hayek, and Milton Friedman, among others.[citation needed] Two leading 'think tanks' in Washington, the Brookings Institution and the American Enterprise Institute, were active in holding seminars and publishing studies advocating deregulatory initiatives throughout the 1970s and 1980s.[citation needed] Alfred E. Kahn played an unusual role in both publishing as an academic and participating in the Carter Administration's efforts to deregulate transportation.

Traditional areas that have been deregulated are the telephone and airline industries. In the late 1990s and early 2000s the utility industry (power companies) in North America started to deregulate.

Operation of power system under deregulation:

Deregulation allows competitive energy suppliers to enter the markets and offer their energy supply products to consumers. Energy prices are not regulated in these areas and consumers are not forced to receive supply from their utility. In deregulated markets, consumers can choose their supplier, similar to other common household service providers. The marketing of these services is still regulated.

Energy deregulation is very similar to the deregulated telephone industry, in which you may choose your long distance service provider, but your local phone company still maintains the telephone lines. The transmission and distribution portion of your electric bill (the cost to get the power to you) is still provided by the utility, but you can shop for the best prices and services available in the market for electricity supply

Deregulation gives consumers choice - the power of the buyer. A deregulated market allows you to choose your commodity supplier. It also motivates retailers to differentiate their products from the utility and those of competitors by developing innovative features, pricing plans and options that would have otherwise not been available to you. Green energy products are an example of innovative programs made possible by retailers.

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The benefits of a deregulated market is that it boosts competition among suppliers, which leads to lower prices and the chance for customers to find the best deal. Determining if your state participates is where it gets tricky. It is being executed on a state-by-state basis, but some have deregulated natural gas and others strictly have deregulated electricity. Use this interactive map to determine if your state participates

Fig 1

Timeline of energy deregulation:

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Fig 2

Literature survey:

[1] This paper describes how supplier should determine prices of offered blocks offered to grid system taking different parameters into consideration with an illustrative example.

[2] Monte-Carlo approach is used to solve the stochastic optimization model with risks for load serving entities (LSE)

[3] In this paper a framework is developed for a comprehensive evaluation of possible scenarios for the implementation of DSB into the electricity market and the assessment of the influence of DSB on total production costs, SMP profile, capacity element payments and benefit allocation between producers and consumers.

[4] Two bidding schemes i.e. max hourly benefit bidding strategy and min stable output bidding strategies are proposed and genetic based algorithm is presented for overall bidding strategy.

[5] This paper discussed the development of a competitive power pool (CPP) framework to formulate and determine the optimal bidding strategies of a bidder in the CPP under conditions of perfect competition.

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[6] A bidding strategy based on FCM (fuzzy-c-mean) and ANN (Artificial neural network) has been proposed for a bidder in a competitive power market

[7] Bidding behaviors’ under a simple auction market are modeled in this paper considering clearing price rule and making necessary assumptions

[8] Two different bidding schemes, namely ‘maximum hourly benefit coordinated bidding strategies’ and ‘minimum stable output bidding strategies’, are suggested for each hour and an optimization model is developed to describe these two schemes in this paper.

[9] Interior- point optimal power flow (IPOPF) model is proposed in this paper and this model is used to generate its optimal bids in the electricity generation auction market.

[10] A method to build bidding strategies, with which power suppliers can optimally co-ordinate their activities in the energy and spinning reserve markets, is presented. A stochastic optimization model is established and a refined genetic algorithm (RGA) based method developed for describing and solving this problem.

[11] This paper describes a new approach for optimal supply curve bidding (OSCB) using Benders decomposition in competitive electricity markets.

[12] This paper presents the concept of conjectural variation and its applications to strategic bidding of generation firms in the electricity spot market. It is shown that the conjectural variation based bidding strategy can help Gencos to maximize their profits based on available imperfect information

[13] In this paper, bilevel programming formulation of a deregulated electricity market is proposed. Bilevel program also indicates the magnitude of the error that can be made if the electricity market model studied does not take into account the physical constraints of the electric grid

[14] A conjectural variation based learning method is proposed in this paper for generation firm to improve its strategic bidding performance. In the method, each firm learns and dynamically regulates its conjecture upon the reactions of its rivals to its bidding according to available information published in the electricity

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market, and then makes its optimal generation decision based on the updated conjectural variation of its rivals.

[15] In this paper, a new approach is proposed for presenting the best GENCO bidding strategy. The proposed method is based on the behavior of participants and the demand model, with regard to the ISO’s objective function.

[16] This paper presents an evolutionary algorithm to generate cooperative strategies for individual buyers in a competitive power market. The paper explores how buyers can lower their costs by using an evolutionary algorithm that evolves their group sizes and memberships.

[17] This paper presents a bilevel programming formulation to the problem of strategic bidding under uncertainty in electricity markets. A nongame approach is adopted and it is considered that the agent being optimized can obtain bidding scenarios (price-quantity) for its competitors.

[18] In this paper oligopolistic electricity markets were taken as non-linear dynamical systems and used –discrete-time Nash bidding strategies.Cournot model is proposed to solve bidding problem where LSEs decide on demand quantities and MCP is the marginal cost of producing electricity.

[19] This paper has proposed related theories focusing on the uncooperative BLMF decision model. An extended Kuhn–Tucker approach for solving the specific kind of BLMF decision problems is then developed.

[20] In this paper we compare Nash equilibria analysis and agent-based modelling for assessing the market dynamics of network-constrained pool markets.

[21] This paper presents an evolutionary algorithm to generate cooperative strategies for individual buyers in a competitive power market. The developed agent-based model uses Power World simulator to incorporate the traditional physical system characteristics and constraints while evaluating individual agent’s behavior, actions and reactions on market dynamics.

[22] This paper presents bidding strategy in the highly uncertain market with the help of linear programming model. Its main aim is to include uncertainty into optimization

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[23] In this paper a methodology is proposed that enables a strategically behaving bidder to estimate the profit maximizing bid underprice uncertainty considering a multi-unit pay-as-bid procurement auction for power systems reserve.

[24] This paper employs supply function equilibrium (SFE) for modeling bidding strategy to obtain equilibrium points for reliability.

[25] The model-based approach and the QL algorithm are used to find the optimal bidding strategy for a supplier in electricity PAB auction.

[26] Based on the stepwise bidding rules in electricity power markets, the impact of different numbers of bidding segments on the bidding strategies of generation companies is studied in this paper.

[27] Normal form game theory approach and adopted cost based unit commitment program is discussed in this paper to formulate bidding strategies.

[28] This paper presents a dynamic bidding model of the power market based on the Nash equilibrium and the supply function.

[29] In this paper, two PSO techniques are used to determine bid prices and quantities of power market. One is conventional PSO and other is decomposition technique in conjunction with PSO approach

[30] This paper compares Particle swarm optimization method with marginal cost method on determining price-quantity pairs that will be submitted to the day-ahead markets.

[31] This paper has presented a literature review on conjectural equilibrium models, very often used for electricity market modeling.

[32] In this paper, linear function model is applied to find supply function equilibrium and also proposed a new approach to find SFE for network constrained electricity markets.

[33] In this paper, a game theoretic model for examining non-cooperative bidding strategies for acquiring FTRs in a deregulated power market is presented.

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[34] In this paper, PSO is combined with simulated annealing (SA) to formulate the bidding strategies of Gencos where pay as bid payment is followed and other information is insufficient for analysis.

[35] This paper discusses how profit gets effected by choosing a pricing method (Pay as bid pricing or marginal pricing) by utilizing bilevel optimization technique and game theory concepts

[36] In this paper, a new GA-approach was presented for bidding strategy in a day-ahead market from the viewpoint of a GENCO for maximizing its own profit as a participant in the market. Two approaches were considered based on two different GENCO’s point of view: as a supplier wishing to maximize the profit without considering rival’s profit function, and as a supplier wishing to maximize the profit considering rival’s bidding and profit functions.

[37] This paper presents comparative analysis between market clearing price and pay as bid mechanisms and also proposes a complex model of a multi-agent game in an electricity market based on CAS (complex adaptive system) theory.

[38] Possibility theory based method is proposed in this paper, which could accommodate uncertainties and incomplete and insufficient information. Given any estimated rivals bidding behaviors represented by fuzzy sets , the method could be used to develop a bidding strategy for the subject generation company.

[39] The bidding decision making problem is studied from a supplier’s viewpoint in a spot market environment. The decision-making problem is formulated as a Markov Decision Process - a discrete stochastic optimization method

[40] This paper introduced the framework of a BE solution approach to the EPEC problem of NE in strategic bidding in short-term electricity markets. The BE scheme is used to transform the nonlinear, nonconvex, NE problem into a mixed integer linear problem, which can be solved by commercially available computational systems.

[41] A brief literature survey of strategic bidding in electricity markets is made in this paper based on more than 30 research publications

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[42] This focus of this paper is to present a framework in which strategies may developed for the individual participants in an energy brokerage.

[43] This paper analyzes the effect of minimum output on the result of competition in a deregulated environment. A criterion is presented to evaluate the result when there is competition for commitment among suppliers with different minimum outputs.

[44] In this paper a strategic bidding procedure based in stochastic programming is decomposed using the Benders technique.

[45] In this paper it is modeled the bidding strategies of spot users as a normal form game. We have then shown that for a band of bid values the game is equivalent to the prisoner dilemma game.

[46] The method proposed in this paper was game based procedure to estimate opponents' behavior in market considering a risk aversion degree for GENCOs' bidding strategy.

[47] In this paper, the dynamics resulting from line capacity constraints on a two-agent strategic bidding is focused on in light of linear supply function model in centralized electricity markets. Global attractors for this non-smooth bidding model in different cases are analyzed in order to help the two generators make bidding regulation.

[48] This paper introduces a stochastic programming model that integrates strategic bids or offers for electricity (in quantities and prices) in a deregulated electricity market.

[49] In this paper, the impact of line capacity constraints on the strategic bidding is focused on in light of linear supply function model in centralized electricity markets. Dynamic analysis for this non-smooth bidding model with respect to the change of the market equilibrium is given in order to help the generators to make bidding regulation.

[50] In this paper, application of the PSO method for strategic bidding of an electricity supplier in an oligopolistic power market is proposed. The power market model has been postulated for both block bidding and linear bidding.

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[51] A novel procedure introduced in this paper for strategic bidding of Gencos in power market. The method is fully compatible to pay as bid markets . The market model assumed as Bertrand model

[52] This paper has applied bilevel programming and swarm algorithms to model the competitive strategic bidding decision making in the electricity markets in order to obtain solutions.

[53] A method to build optimal bidding strategies for competitive power suppliers in an electricity market is presented in this paper.

[54] In this paper, the problem of developing bidding strategies in oligopolistic dynamic electricity double-sided auctions has been studied. Attention was given to strategic bidding of the GF and LSEs in these markets.

[55] Model supporting the construction of a bidding curve that fits the rules of the Nord Pool day-ahead market was suggested. The intended user is a retailer having end users with price-sensitive demand.

[56] In this paper, A combined centralized economic dispatch model and a decentralized bidding strategy model are used to solve the energy trading problem in competitive electricity markets

[57] This paper presents a comprehensive approach to evaluate the performances of the electricity markets with network representation in presence of bidding behavior of the producers in a pool system.

[58] This paper presented a BE solution approach to the problem of strategic bidding under uncertainty in short-term electricity markets.

[59] The strategies encoded in the GP-Automata are tested in an auction simulator in this paper

[60] This paper models bidding behaviors of suppliers in electricity auction markets under clearing pricing rule and with some simplified bidding assumptions.

[61] Built on an existing hydrothermal scheduling approach, an innovative model and an efficient Lagrangian relaxation-based method are presented to solve the bidding and self-scheduling problem.

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[62] This paper describes an environment in which distribution companies (discos) and generation companies (gencos), buy and sell power via double auctions implemented in a regional commodity exchange

[63] This paper reports upon the mathematical models and implementation of the Scheduling, Pricing, and Dispatch (SPD) application for the New Zealand Electricity Market (NZEM).

[64] This paper shows how Information Gap Decision Theory (IGDT) can serve as a decision support tool that assists in quantifying severe uncertainty when information is scarce and expensive.

[65] A model of an electricity generation bidding system has been analyzed. In this article the bidding system is formulated as a control problem by introducing the idea of multiple bidding rounds.

[66] Game theoretical approach to the problem of pricing electricity in deregulated energy marketplaces is presented in this paper

[67] This paper presents a methodology to design an optimal bidding strategy for a generator according to his or her degree of risk aversion.

[68] This paper describes a method for analyzing the competition among transmission-constrained Generating Companies (GENCOs) with incomplete information.

[69] This paper compares the behavior of Generating Companies (Gencos) in the two competing pricing mechanisms of uniform and pay-as-bid pricing in an electricity market. Game Theory is used to simulate bidding behavior of Gencos and develop Nash equilibrium bidding strategies for Gencos in electricity markets.

[70] In this paper, application of the PSO method for strategic bidding of an electricity supplier in an oligopolistic power market is proposed. The power market model has been involved for linear bidding.

[71] Spatial gaming model with coalition is discussed. The stability in a coalition is measured as profit gain for all coalition members.

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[72] A practical and efficient MPEC-based procedure for calculating oligopolistic price equilibria for an electric power market has been developed and illustrated

[73] An algorithm that allows a market participant to maximize its individual welfare in electricity spot markets is presented.

[74] In this paper,we have presented a mathematical programming approach to derive optimal offers for a generation company operating in an electricity spot market consisting of a sequence of market mechanisms.

[75] This paper has described SGO, a management information system for bidding in deregulated electricity markets. It has been developed for the Spanish Market with the Electrical Utility ENDESA.

[76] This paper describes an agent based computational economics approach for studying the effect of alternative structures and mechanisms on behavior in electricity markets.

[77] A fuzzy-controlled crossover and mutation probabilities in GA for optimization of PECs has been proposed. They are determined adaptively for each solution of the population. It is in the manner that the probabilities are adapted to the population distribution of the solutions.

[78] In the problem we model , Gencos and Discos interact through an electronic bulletin board, posting bids and offers until agreement has been reached.

[79] This paper models the interaction of long-term contracting and spot market transactions between one Genco and one or more Discos. The basic model proposed allows the Genco and Discos to negotiate bilateral electric power contracts and then, on the day, to sell or buy in an associated spot market.

[80] In this paper, we resort to a set of comparison indexes that allows to measure market power comparing the oligopoly outcome with the ideal benchmark represented by perfect competition.

[81] This paper focuses on a procedure that uses particle swarm optimization with Time varying acceleration coefficients (PSOTVAC) to analyze the bidding strategy of Generating Companies (Gencos) in an electricity market about their opponents.

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[82] This paper introduces the methodology and techniques of the new bidding strategy, illustrates the formation of the optimal price-production pairs, constructs the optimal bidding curve for the particular participant

[83] In this paper, we propose a novel forecasting approach, which can handle both nonlinear and heteroscedastic time series and thus is suitable for interval forecasting of the electricity price.

[84] The paper models bidding behaviors of power suppliers under the assumptions of costs uncertainty. The market clearing price is the result of the bidding strategies and it is determined by using a merit order dispatch procedure.

[85] This paper presents a game theory application for analyzing power transaction in a deregulated energy market place such as poolco, where participants, especially, generating entities, maximize their net profit through optimal bidding strategies

[86] This paper presents a bidding strategy based on the theory of ordinal optimization that the ordinal comparisons of performance measures are robust with respect to noise and modeling error

[87] This paper presents a methodology for the development of bidding strategies for electricity producers in a competitive electricity marketplace.

[88] In this paper, the deregulation, power supply and bidding in Turkish market are examined.Two models for each bidding methodology are proposed for a price taker unit that aims to maximizes its profit under uncertain market prices.

2) Bidding

Bidding is an offer (often competitive) of setting a price one is willing to pay for something or a demand that something be done. A price offer is called a bid. The term may be used in context of auctions, stock exchange, card games, or real estate. Bidding is used by various economic niche for determining the demand and hence the value of the article or property, in today's world of advance technology, Internet is one of the most favorite platforms for providing bidding facilities, it is

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the most natural way of determining the price of a commodity in a free market economy.

Biddings are arranged by first disclosing the time and space location of the place where the bid is to be performed, so that more interested bidders may participate and the most "true" price of the commodity may come out, in terms of bidding on Internet the time frame for posting the bids may be a topic of interest.

For some auction houses, bidding is meant to be fun and enjoyable, but remember that each bid you place enters you into a binding contract. All bids are active until the auction ends.

Many similar terms that may use or may not use the similar concept have been evolved in the recent past in connection to bidding, such as reverse auction, social bidding, or many other game class ideas that promote them self as bidding. Bidding is also sometimes used as ethical gambling in which the prize money is not determined solely by luck but also by the total demand that the prize has attracted towards itself.

Restructuring of the power industry mainly aims at abolishing the monopoly in the generation and trading sectors, thereby, introducing competition at various levels wherever it is possible. But the sudden changes in the electricity markets have a variety of new issues such as oligopolistic nature of the market, supplier’s strategic bidding, market power misuse, price-demand elasticity and so on.

Theoretically, in a perfectly competitive market, supplier should bid at their marginal production cost to maximize payoff. However, practically the electricity markets are oligopolistic nature, and power suppliers may seek to increase their profit by bidding a price higher than marginal production cost. Knowing their own costs, technical constraints and their expectation of rival and market behavior, suppliers face the problem of constructing the best optimal bid. This is known as a strategic bidding problem.

2.1) Bidding classifications (Single sided and double sided):

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The underlying assumption we make when modeling auctions is that each bidder has an intrinsic value for the item being auctioned; she is willing to purchase the item for a price up to this value, but not for any higher price. We will also refer to this intrinsic value as the bidder’s true value for the item.

Auctions with just one seller and multiple buyers (or vice versa) are called single sided auctions. Double sided auctions have multiple buyers and sellers. Klemperer names four standard single sided auction types: (i) ascending (ii) descending (iii) first price sealed bid and (iv) second price sealed-bid. Wurman proposed a classification of five classic auctions by differentiating the attributes (i) single vs double sided (ii) Open vs sealed (iii) Ascending vs descending.This classification is showed in figure below

Fig.3-Classification of auctions

1) Single sided auctions:

i) Ascending-bid auctions, also called English auctions. These auctions are carried out interactively in real time, with bidders present either physically or electronically. The seller gradually raises the price, bidders drop out until finally only one bidder remains, and that bidder wins the object at this final price. Oral auctions in which bidders shout out prices, or submit them electronically, are forms of ascending-bid auctions.

ii) Descending-bid auctions, also called Dutch auctions. This is also an interactive auction format, in which the seller gradually lowers the price from some high

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initial value until the first moment when some bidder accepts and pays the current price. These auctions are called Dutch auctions because flowers have long been sold in the Netherlands using this procedure.

iii) First-price sealed-bid auctions. In this kind of auction, bidders submit simultaneous “sealed bids” to the seller. The terminology comes from the original format for such auctions, in which bids were written down and provided in sealed envelopes to the seller, who would then open them all together. The highest bidder wins the object and pays the value of her bid.

iv) Second-price sealed-bid auctions, also called Vickrey auctions. Bidders submit simultaneous sealed bids to the sellers; the highest bidder wins the object and pays the value of the second-highest bid. These auctions are called Vickrey auctions in honor of William Vickrey, who wrote the first game-theoretic analysis of auctions (including the second-price auction ). Vickery won the Nobel Memorial Prize in Economics in 1996 for this body of work.

2) Double sided auctions:

The classic double sided auction formats are the Continuous Double Auction (CDA) and the Call Market. In both auction types multiple buyers and multiple sellers participate. The bids either comprise offers to buy or offers to sell. Bids in double sided auctions are also called “order”.Orders are collected in an order book. The order book has two sides, one for the buy orders and one for the sell orders.

i) In the case of a CDA, each incoming order is either matched with the best possible order on the opposite side of the order book or it is put into the order book. The order book can be open , which means that all (or at least a specified number of )orders currently outstanting are displayed.

ii) The classic case of a Call market works with a closed order book. All incoming orders are put into the order book until the matching process starts. The price determination in both cases depends on the defined institutional rules. Double sided auctions are commonly used in stock exchange.Besides described auction formats there are various other subtypes, extensions or additional auctions formats.

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2.2) Various bidding strategies

a) Equilibrium oriented bidding strategy

While an equilibrium point (also known as Nash Equilibrium ) is reached, no GENCO could increase its profit by unilaterally changing its behavior, e.g. its output. In many equilibrium-oriented models, conjectural variation (CV) methods are used to model the interactions among market players . The essentials of conjectural variation value are to capture the behavioral response of competitors to the action change by the observed GENCO in the market. It has been well accepted that CV enables a more powerful representation of GENCO bidding behaviors and is capable of modeling various degrees of market competitions, ranging from perfect competition (CV = 1), Cournot game (CV = 0), to collusion (CV = 1) and other variants .

CV methods provide a quantified measure to analyze the bidding behavior of GENCOs. A duopoly market has been analyzed for a pool spot market , where CV is used to model/estimate forward market behavior.

Two main categories of approaches to estimate CV values according to publicly available historical information, namely explicit fitting and implicit fitting. In an implicit fitting procedure a closed-form which employs historical available market data has been developed for energy and transmission price response . However, those CV values only reflect system historical status and can only be used to analyze the static market behaviors of GENCOs within a predefined market setting. Moreover, in day-ahead markets or repeated markets based on regular time intervals, as each GENCO aims to maximize

profits, normally they have incentives to learn from bidding history and public market data and hence, they gradually evolve their bidding behavior. Those static approaches fail to answer what CV value set will be reached in a dynamic market with multiple GENCOs.

In order to research the dynamic interaction among strategic GENCOs, a CV-based learning method is proposed, based on which GENCOs evolve their bidding behavior in a spot market. It has been proved that the equilibrium reached during

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the learn learning process is a Nash Equilibrium. The approaches typically assume a commonly agreed market model, e.g. a common market price demand function. However, for a practical electricity market, no such function exists. Each GENCO has to analyze publicly available information and its own private information to build its own market model. As the market model is typically influenced by many stochastic factors, e.g. changes of demand curves and behaviors of generators, each generator uses its own way to interpret the market data to construct its own estimated market model, based on which market behaviors are predicted. The models held by individual GENCO may be inconsistent with the real market model by minor variations.

b) Competitive strategic bidding

The competitive mechanism of day-ahead markets is a very important research issue in electricity market studies, which can be described as follows: Each generating company submits a set of hourly (half-hourly) generation prices and the available capacities for the following day. According to this data and an hourly (halfhourly) load forecast, a market operator allocates generation output for each unit.

b) Strategic Pricing Model for Generating Companies

Each generating company is concerned with how to choose a bidding strategy, which includes the generation price and the available capacity. Many bidding functions have been proposed. For a power system, the generation cost function generally adopts a quadratic function of the generation output, i.e., the generation cost function can be represented as

Cj(Pj) = ajP j2 + bjPj + cj

where Pj is the generation output of generator j and aj , bj , and cj are the coefficients of the generation cost function of generator j. The marginal cost of generator j is calculated by

λ = 2ajPj + bj .

It is a linear function of its generation output Pj . The rule in a goods market may expect each generating company to bid according to its own generation cost.

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Therefore, we adopt this linear bid function. Suppose that the bidding for the jth unit at time t is

Rtj = αtj + βtj*Ptj

where t ∈ T is the time interval, T is the time interval number, j represents the unit number, Ptj is the generation output of unit j at time t, and αtj and βtj are the bidding coefficients of unit j at time t. According to the justice principle of “the same quality, the same network, and the same price,” we adopt a uniform marginal price (UMP) as the market clearing price. Once the energy market is cleared, each unit will be paid according to its generation output and UMP. The payoff of the ith generating company is

where Gi is the suffix set of the units belonging to the ith generating company. Each generating company wishes to maximize its own profit Fi. In fact, Fi is the function of Ptj and UMPt, and UMPt is the function of all units’ bidding αtj , βtj , and output power Ptj , which will impact on each other. Therefore, we establish a strategic pricing model for the generating companies as follows:

Where L is the number of generating companies, Pti =∑j∈Gi

Ptj , t = 1, 2, . . . , T.

The profit calculated for each generating company will consider both Ptj and UMPt, which can be computed by a market operator according to the market clearing model.

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Mathematical formulation:

Consider total of ‘m’ suppliers participating in bidding where MCP is employed.Assume that each supplier is required to bid a linear supply function to the pool. The jth supplier bid with linear supply curve denoted by G j (P j ) = a j + b j P j for j = 1, 2,. . .,m. where P j is the active power output, a jand b j are non-negative bidding coefficients of the jth supplier.After receiving bids from suppliers, the pool determines a set of generation outputs that meets the load demand and minimizes the total purchasing cost. It is clear that generation dispatching should satisfy the following Equations.aj + bj Pj = R, j = 1, 2, . . . , m ……………………………..(1)

∑j=1

m

P j = Q (R) ……………………………………..(2)

Where R is the market clearing price (MCP) of electricity to be determined,Q(R) is the aggregate pool load forecast as follows:Q(R) = Qo −KR ……………………………………………….(3) Where Qo is a constant number and K is a non-negative constant used to represent the load price elasticity. When we solve the above equation we get the solutions as

R=Qo+∑

j=1

m

(a j

b j

)

K+∑j=1

m

( 1b j

) (5)

P j = R−a j

b j (6)

The jth supplier has the cost function denoted by Cj (Pj ) = ej Pj + fj P2 , where

ej and fj are the cost coefficients of the jth supplier.

Hence our main objective is to maximize profits which is the difference between the selling price and the production price which is as followsMaximize : F(a j , b j) = RP j−C j(P¿¿ j)¿Subject to : Eqs. (5) and (6)

The objective is to determine bidding coefficients aj and bj so as to maximize F(aj,bj) subject to equations 5 and 6.

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The bidding coefficients (aj, bj) are interdependent; therefore one of the coefficient make as a constant and other is randomly varied using probability density function (pdf). Let, from the ith supplier’s point of view, rival’s jth (j / = i) bidding coefficients (aj, bj) obey a joint normal distribution with pdf given by:

Based on historical bidding data these distributions can be determined. The probability density function Eq. (8) represents the joint distributions between aj and bj, the task of optimally coordinating the bidding strategies for a supplier with objective function Eq. (7), and constraints (5) and (6), becomes stochastic optimization problem. The proposed Fuzzy Adaptive gravitational search algorithm (FAGSA) is applied to solve the above stochastic optimization problem.

3) Recent algorithms for solution of bidding strategies

3.1) Convectional algorithms:

3.1.1) Game theory

A game-theoretic auction model is a mathematical game represented by a set of players, a set of actions (strategies) available to each player, and a payoff vector corresponding to each combination of strategies. Generally, the players are the buyer(s) and the seller(s). The action set of each player is a set of bid functions or reservation prices (reserves). Each bid function maps the player's value (in the case of a buyer) or cost (in the case of a seller) to a bid price. The payoff of each player under a combination of strategies is the expected utility (or expected profit) of that player under that combination of strategies.

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Game-theoretic models of auctions and strategic bidding generally fall into either of the following two categories. In a private value model, each participant (bidder) assumes that each of the competing bidders obtains a random private value from a probability distribution. In a common value model, each participant assumes that any other participant obtains a random signal from a probability distribution common to all bidders. Usually, but not always, a private values model assumes that the values are independent across bidders, whereas a common value model usually assumes that the values are independent up to the common parameters of the probability distribution.

When it is necessary to make explicit assumptions about bidders' value distributions, most of the published research assumes symmetric bidders. This means that the probability distribution from which the bidders obtain their values (or signals) is identical across bidders. In a private values model which assumes independence, symmetry implies that the bidders' values are independently and identically distributed (i.i.d.).

3.1.2) Nash equilibrium:In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.[1] If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium

Game theorists use the Nash equilibrium concept to analyze the outcome of the strategic interaction of several decision makers. In other words, it provides a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome depends on the decisions of the others. The simple insight underlying John Nash's idea is that one cannot predict the result of the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do, taking into account the decision-making of the others.

NE bidding strategy in a bilateral trading environment is studied below

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A bilateral trading environment in which multiple sellers (generators) and multiple buyers (loads) are involved. A first price sealed bid auction mechanism is adopted to achieve a transaction. Generators submit bids to a load. The lowest bid is the winning bid if this bid price is lower than the load’s willingness to pay. The following assumptions and rules apply.

Complete and perfect information is assumed, i.e., each bidder (generator) knows its own cost and all the other bidders’ cost. All the generators’ cost and the loads’ willingness to pay are common knowledge.

2) A generator is responsible to pay system losses and transmission charge. Therefore, a generator’s costs of supplying different loads could be different even if the loads have the same size.

3) Each generator can supply only one load; i.e., each generator can only win one bid. This assumption can be justified by the generator’s capacity constraints. If a generator wins more than one load, this generator chooses to supply the load that gives it the highest profit. If the generator is indifferent in terms of profit, this generator chooses the load that achieves system-wide cost minimization.

4) If two or more generators place the same cheapest bid for a load, the load randomly chooses one of them.

5) For any load, if all the bid prices are higher than the load’s willingness to pay, the load will withdraw from the bilateral market. No re-bid takes place. It is assumed that the load could rely on its own resources or buy electricity from another market, e.g., spot market

3.2) Other algorithms:

3.2.1) Genetic algorithm:

In the computer science field of artificial intelligence, genetic algorithm (GA) is a search heuristic that mimics the process of natural selection. This heuristic (also sometimes called a metaheuristics) is routinely used to generate useful solutions to optimization and search problems.

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Genetic algorithms belong to the larger class of evolutionary algorithms (EA), which generate solutions to optimization problems using techniques inspired by natural evolution, such as inheritance, mutation, selection, and crossover.

In a genetic algorithm, a population of candidate solutions (called individuals, creatures, or phenotypes) to an optimization problem is evolved toward better solutions. Each candidate solution has a set of properties (its chromosomes or genotype) which can be mutated and altered; traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possible.

The evolution usually starts from a population of randomly generated individuals, and is an iterative process, with the population in each iteration called a generation. In each generation, the fitness of every individual in the population is evaluated; the fitness is usually the value of the objective function in the optimization problem being solved. The more fit individuals are stochastically selected from the current population, and each individual's genome is modified (recombined and possibly randomly mutated) to form a new generation. The new generation of candidate solutions is then used in the next iteration of the algorithm. Commonly, the algorithm terminates when either a maximum number of generations has been produced, or a satisfactory fitness level has been reached for the population.

A typical genetic algorithm requires:

1) A genetic representation of the solution domain,

2) A fitness function to evaluate the solution domain.

A standard representation of each candidate solution is as an array of bits. Arrays of other types and structures can be used in essentially the

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same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size, which facilitates simple crossover operations. Variable length representations may also be used, but crossover implementation is more complex in this case. Tree-like representations are explored in genetic programming and graph-form representations are explored in evolutionary programming; a mix of both linear chromosomes and trees is explored in gene expression programming.

Once the genetic representation and the fitness function are defined, a GA proceeds to initialize a population of solutions and then to improve it through repetitive application of the mutation, crossover, inversion and selection operators.

Initialization of genetic algorithm

Initially many individual solutions are (usually) randomly generated to form an initial population. The population size depends on the nature of the problem, but typically contains several hundreds or thousands of possible solutions. Traditionally, the population is generated randomly, allowing the entire range of possible solutions (the search space). Occasionally, the solutions may be "seeded" in areas where optimal solutions are likely to be found.

Selection

During each successive generation, a proportion of the existing population is selected to breed a new generation. Individual solutions are selected through a fitness-based process, where fitter solutions (as measured by a fitness function) are typically more likely to be selected. Certain selection methods rate the fitness of each solution and preferentially select the best solutions. Other methods rate only a

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random sample of the population, as the former process may be very time-consuming.

The fitness function is defined over the genetic representation and measures the quality of the represented solution. The fitness function is always problem dependent. For instance, in the knapsack problem one wants to maximize the total value of objects that can be put in a knapsack of some fixed capacity. A representation of a solution might be an array of bits, where each bit represents a different object, and the value of the bit (0 or 1) represents whether or not the object is in the knapsack. Not every such representation is valid, as the size of objects may exceed the capacity of the knapsack. The fitness of the solution is the sum of values of all objects in the knapsack if the representation is valid, or 0 otherwise.

In some problems, it is hard or even impossible to define the fitness expression; in these cases, a simulation may be used to determine the fitness function value of a phenotype (e.g. computational fluid dynamics is used to determine the air resistance of a vehicle whose shape is encoded as the phenotype), or even interactive genetic algorithms are used.

Genetic operators

The next step is to generate a second generation population of solutions from those selected through a combination of genetic operators: crossover (also called recombination), and mutation.

For each new solution to be produced, a pair of "parent" solutions is selected for breeding from the pool selected previously. By producing a "child" solution using the above methods of crossover and mutation, a new solution is created which typically shares many of the characteristics of its "parents". New parents are selected for each new

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child, and the process continues until a new population of solutions of appropriate size is generated. Although reproduction methods that are based on the use of two parents are more "biology inspired", some research suggests that more than two "parents" generate higher quality chromosomes.

These processes ultimately result in the next generation population of chromosomes that is different from the initial generation. Generally the average fitness will have increased by this procedure for the population, since only the best organisms from the first generation are selected for breeding, along with a small proportion of less fit solutions. These less fit solutions ensure genetic diversity within the genetic pool of the parents and therefore ensure the genetic diversity of the subsequent generation of children.

Opinion is divided over the importance of crossover versus mutation. There are many references in Fogel (2006) that support the importance of mutation-based search.

Although crossover and mutation are known as the main genetic operators, it is possible to use other operators such as regrouping, colonization-extinction, or migration in genetic algorithms.

It is worth tuning parameters such as the mutation probability, crossover probability and population size to find reasonable settings for the problem class being worked on. A very small mutation rate may lead to genetic drift (which is non-ergodic in nature). A recombination rate that is too high may lead to premature convergence of the genetic algorithm. A mutation rate that is too high may lead to loss of good solutions unless there is elitist selection.

Termination

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This generational process is repeated until a termination condition has been reached. Common terminating conditions are:

A solution is found that satisfies minimum criteria Fixed number of generations reached Allocated budget (computation time/money) reached The highest ranking solution's fitness is reaching or has reached a

plateau such that successive iterations no longer produce better results

Manual inspection Combinations of the above

3.2.2) 2 level optimization:

In two-level optimization problem participants try to maximize their profit under the constraint that their dispatch and price are determined by the OPF. Hence an efficient numerical technique, using price and dispatch sensitivity information available from the OPF solution, to determine how a market participant should vary its bid portfolio in order to maximize its overall profit.

i.e

Where

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3.2.3) Possibility theory:

Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic.

Basic Notions

A possibility distribution is a mapping π from a set of states of affairs S to a totally ordered scale such as the unit interval [0,1] . The function π represents the knowledge of an agent (about the actual state of affairs) distinguishing what is plausible from what is less plausible, what is the normal course of things from what is not, what is surprising from what is expected. It represents a flexible restriction on what the actual state of affairs is, with the following conventions: π(s)=0 means that state s is rejected as impossible; π(s)=1 means that state s is totally possible (= plausible or unsurprising).

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If the state space is exhaustive, at least one of its elements should be the actual world, so that at least one state is totally possible (normalization). Distinct values may simultaneously have a degree of possibility equal to 1.

Possibility theory is driven by the principle of minimal specificity. It states that any hypothesis not known to be impossible cannot be ruled out. A possibility distribution is said to be at least as specific as another one if and only if each state is at least as possible according to the latter as to the former (Yager 1983). Then, the most specific one is the most restrictive and informative.

In the possibilistic framework, extreme forms of partial knowledge can be captured, namely:

Complete knowledge: for some state s0 ,π(s0)=1 and π(s)=0 for other states s (only s0 is possible)

Complete ignoranceπ(s)=1,∀s∈S ,

(all states are totally possible).Given a simple query of the form does an event A occur?, where A is a subset of states, or equivalently does the actual state lie in A, a response to the query can be obtained by computing degrees of possibility and necessity, respectively (if the possibility scale is [0,1] ):

The possibility degree Π(A) evaluates to what extent event A is consistent with the knowledge π , while N(A) evaluates to what extent A is certainly implied by the knowledge. The possibility-necessity duality is expressed by N(A)=1−Π(Ac), where Ac is the complement of A. Generally, Π(S)=N(S)=1 and Π(∅)=N(∅)=0 . Possibility measures satisfy the basic maxitivity property:Π(A∪B)=max(Π(A),Π(B)).

Necessity measures satisfy an axiom dual to that of possibility measures, namely N(A∩B)=min(N(A),N(B)). On infinite spaces, these axioms must hold for infinite families of sets.Human knowledge is often expressed in a declarative way using statements to which some belief qualification is attached. Certainty-qualified pieces of uncertain

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information of the form A is certain to degree α can then be modelled by the constraint N(A)≥α. The least specific possibility distribution reflecting this information assign possibility 1 to states where A is true and 1−α to states where A is false.Apart from Π , which represents the idea of potential possibility, another measure of guaranteed possibility can be defined

It estimates to what extent all states in A are actually possible according to evidence.Notions of conditioning and independence were studied for possibility measures. Conditional possibility is defined similarly to probability theory using a Bayesian like equation of the form :Π(B∩A)=Π(B∣A)⋆Π(A)

However, in the ordinal setting the operation ⋆ cannot be a product and is changed into the minimum. In the numerical setting, there are several ways to define conditioning, not all of which have this form. There are several variants of possibilistic independence. Generally, independence in ordinal possibility theory is neither symmetric, nor insensitive to negation. For non-Boolean variables, independence between events is not equivalent to independence between variables. Joint possibility distributions on Cartesian products of domains can be represented by means of graphical structures similar to Bayesian networks for joint probabilities. Such graphical structures can be taken advantage of for evidence propagation or learning.

3.2.4) Gravitational search algorithm:

Gravitational search algorithm (GSA) is an optimization algorithm based on the law of gravity and mass interactions. It follows two basic laws

i) Law of gravity. Each particle attracts every other particle and the gravitational force between two particles is directly proportional to the product of their masses and inversely proportional to the distance ‘R’ between them.

ii) Law of motion. The current velocity of any mass is equal to the sum of the fraction of its previous velocity of mass and the variation in the velocity. Variation

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in the velocity or acceleration of any mass is equal to the force acted on the system divided by mass of inertia.

In this algorithm, agents are considered as objects and their performance is measured by their masses. The gravitational forces influence the motion of these masses, where lighter masses gravitate towards the heavier masses (which signify good solutions) during these interactions. The gravitational force hence acts as the communication mechanism for the masses (analogous to ‘pheromone deposition’ for ant agents in ACO and the ‘social component’ for the particle agents in PSO ). The position of the masses correlates to the solution space in the search domain while the masses characterize the fitness space. As the iterations increase, and gravitational interactions occur, it is expected that the masses would conglomerate at its fittest position and provide an optimal solution to the problem.

4) Implementation of the algorithms:

4.1) Genetic algorithm:

Suppose that there are two independent GENCOs participating in electricity market in which the sealed auction with a pay-as bid MCP is employed. It is assumed that GENCOS have information about forecasted load, forecasted price and expectations of rival bids. Suppose first GENCO wish to participate at market and to maximize its own profit. Since, this problem is going to be solved from two points of view; therefore, it is needed to define different objective for each problem.

Case 1: In this case, the problem is going to be solved from GENCO’s point of view that doesn’t consider his rival’s bid. This GENCO only wish to maximize his profit as objective function. Suppose player 1 wish to maximize his profit without considering rival’s bid. So, following single objective is used:

Case 2: In this case, the problem is going to be solved from GENCO’s point of view considering his rival’s bid. Therefore, there is a problem with multi objective function to solve. In this case, GENCO wish to maximize his profit while consider his rival wishing to maximize his profit, too. So, GENCO is going to solve two

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maximization problems.One approach for solving such problems is transferring two objectives to a single objective by giving some coefficient to each objective based on its important.

The evolutionary computation is appropriate to solve complex and non-convex bidding strategy problem. GA is used as an important evolutionary computation to solve this optimization problem evolutionary computation. The proposed methodology consists of following components.

Population size: Here, the population represents a sample that is chosen to be representative of the whole solution set. Typically the population size of a GA is kept at a fraction of the whole solution set. The number of chromosomes in a generation will direct the time for result an optimal solution to a given problem. If there are too few chromosomes, there are few possibilities to carry out crossover and only a small part of the search space is explored. This may result in GA finish with a suboptimal solution.

Representation: The solution process begins with a set of identified chromosomes as the parents from a population. For this problem, the proper offered quantities for both GENCOs are selected as control variables in the problem. Each chromosome in this proposed GA-approach consists of these 6 variables and can be expressed as follows:

Pji Show offered quantity of jth GENCO in ith segment. qji Show cumulative quantities for jth GENCO in ith segment.

Fitness function: In this study, the value of the objective function (profit) is used to designate the fitness of each chromosome.

Case 1: fitness function is considered as for case 1-problem.

Case 2: fitness function is considered for case 2-problem.

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Initialization: The population of chromosome is randomly initialized within the operating range of the control variables.

Ex: We considered an electricity market with two players for bidding. Suppose player 1 decides to submit his bidding by three segments. Suppose that forecasted demand and forecasted market clearing price by player 1 are 30 MVh and 7.5$, respectively. Also, price cap is equal to 8$. Suppose this player considers bidding caps for each segment’s quantity equal to 5, 10, and 15 (MVh). Also, he determines his offered price for each segment based on historical data, price cap and forecasted price. For example, he select prices equal to 2, 5, 7 ($) for each segment, respectively. Also, suppose player 1 knows that his rival wishes to bid on three segments. Player 1 expect three bidding caps for quantity bidding of his rival equal to 6, 10, 20 (MVh) and three offered price equal to 2, 5, 7 ($). Suppose minimum and maximum generation for player 1 is equal to 11 and 15 (MVh), respectively. Also minimum and maximum generation for player 2 is equal to 16 and 20 (MVh), respectively. Total cost of player 1 and player 2 are considered fixed and are equal to 10 and 15, respectively. Therefore, profit functions of two players can be defined as follows:

where cji is offered price by jth GENCO in ith segment. Also, Pji show offered quantity of jth GENCO in ith segment. Based on this approach, profit functions for given players are obtained as follow:

Now, player 1 try to choose parameters based on constraints for maximizing own profits. The some unreality data were used to solve this example, because of simplifying problem to show efficiency of proposed GA solving such problems. In follow, a GA approach is proposed for solving this problem.

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The crossover operator is carried out according to a rate of crossover. In this study crossover rate is defined as 0.7. Two parents are selected from Matingpool as parents, randomly. A number is selected from interval (0, 1), randomly and uniformly. If random number is less than crossover rate then crossover operator create two new chromosomes as offspring from parents, else parents will be copied in offspring chromosomes cell by cell. Proposed crossover in this paper is described by an example as follows. First, two parents are selected.

Parent1 1 5 13 0 4 17Parent2 3 3 14 1 9 16

Then, parents are encoded as on the base of scale of four. For example, 5 is a decimal number. Based on definition of numbers in scale of four, 11 show 5. Four scale-based coding is selected because selected quantity values are small in this paper. Although, binary code is not proper because the size of chromosomes would be large.

Parent1_1 0 0 1 0 1 1 0 3 1 0 0 0 0 1 0 1 0 1Parent2_2 0 0 3 0 0 3 0 3 2 0 0 1 0 2 1 1 0 0

After that, two random integer array between [0,1] called mask are produced as follows:

Mask1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 1 0Mask2 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1

Then, if jth cell content of mask i is equal to 1, copy jth cell content of parent1_1 into jth cell of offspring i. Else, copy jth cell content of parent1_2 into jth cell of offspring i.

Offspring 1

0 0 1 0 1 3 0 3 2 0 0 0 0 2 0 1 0 0

Offspring 2

0 0 3 0 1 1 0 2 1 0 0 1 0 2 1 1 0 1

As mentioned, the mutation operator is carried out according to the rate of mutation. In this study, mutation rate is considered as 0.02. Off springs will be

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copied in an array as future parents, so it must return to first step for fitness calculating. Mutation operator changes two cell contents of off springs if random number be lower than mutation-rate. In this step, off springs are transferred to origin decimal numbers.

Offspring 1 1 7 14 0 8 16Offspring 2 3 5 13 1 9 17

Here, algorithm will be completed by the determined number of repetitions equal to 20 runs.

4.2) Possibility theory:

The credibility of a fuzzy event is defined as the average of its possibility and necessity, as detailed in the following definition.

It is the average of possibility and necessity of A

A fuzzy event may not happen even though its possibility is 1, and may occur even though its necessity is 0. However, the fuzzy event will be sure to happen if its credibility is 1 , and will surely not happen if its credibility is 0. There are many ways for defining the expected value of a fuzzy variable. In the following work, the definition given detailed below will be employed.

Let ᶓ be a fuzzy variable on the possibility space (U, F, π) , then the expected, value of ᶓ is defined by

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The ith supplier’s profit, f(αi,βi,ᶓ), depends on his estimations of rivals’ bidding behaviors ᶓ and his own bidding coefficients αi,βi. f(αi,βi,ᶓ) is a fuzzy variable since 5 is a fuzzy one. E() represents the expected value.

The ith supplier’s estimations of the rivals’ bidding coefficients αj,βj, (j = 1,2; *, n; j ≠ i) represent his fuzzy and qualitative (or roughly quantitative) knowledge of the rivals’ behaviors. The membership functions of αj,βj can be obtained through structure and parameter identifications. However, due to the insufficiency of historical data and limited knowledge; it is difficult to obtain the joint membership functions directly. In this work, the membership functions of αj and βj are represented by two one-dimensional Gaussian functions respectively, however, other forms of membership functions can be accommodated in the proposed method as well.

The correlation between αj and βj is as follows

is the possibility that Pj is y given that a is x . In fact, the fuzzy correlation is defined as a kind of conditional possibility distributions. Based on this, the possibility that the jth supplier will choose (αj,βj)= (xj,yj) is modified as:

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From the ith supplier’s point of view, the possibility that the rivals bid the parameters included in ᶓ=(α1,β1,…., αi-1βi-1,….. αnβn ) is u(ᶓ) = μ1 ¿¿)^…….¿ μi−1¿¿) ^……¿ μn ¿¿)

The expected value is defined as follows

E(f(α iβ i , ᶓ)) ¿∫0

∞

Cr {f (αi , βi , ᶓ )≥r}dr - ∫−∞

0

Cr {f (α i , β i , ᶓ )≤r}dr

A fuzzy simulation algorithm can be used to estimate E(f(α iβ i , ᶓ)) . Randomly generate α 1 l , β1 l , α 2 l , β2 l ,……α nl , βnl and set ᶓ l = (α 1 l , β1 l , α 2 l , β2 l ,……α nl , βnl)

μ(ᶓ l) = μ1(α 1 l , β1 l)^……….¿ μn(α nl , βnl)

where l = 1,2,3,…m , respectively form the ᶓ level sets of α j , β j (j= 1,2,3,…n). Here ᶓ is a sufficiently small number, and m is a sufficiently large positive integer representing sampling times. The integration terms from the above eqn can be obtained by discrete integration for H times, and H is a sufficiently large number.

Hence for given r≥0 , the credibility Cr {f ¿}≥r can be estimated by

Cr {f ¿}≥r = ½*( {μ(ᶓl)∨f (α i , β i , ᶓ )≥r }l=1,2,3 ,…mmax +1-

{μ(ᶓl)∨f (α i , β i , ᶓ )<r }l=1,2,3 ,…mmax )

And similarly for r ≤0

Cr {f ¿}≤r = ½*( {μ(ᶓl)∨f (α i , β i , ᶓ )≤r }l=1,2,3 ,…mmax +1-

{μ(ᶓl)∨f (α i , β i , ᶓ )>r }l=1,2,3 ,…mmax )

4.3) Fuzzy adaptive GSA:Now, consider a system with N agents (masses), the position of the ith agent is defined by:

X i=(x i1 ,… x id ,…, x i

n )for i = 1,2,3….N

where xd presents the position with N agents (masses), the position of the ith agent in the dth dimension and n is the space dimension.At a specific time ‘t’ we define the force acting on mass ‘i’ from mass ‘j’ as following:

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F ijd(t)=G(t)

M pi(t )M aj(t)Rij (t )+ᶓ (x j

d ( t )−x id ( t ))

where M aj is the active gravitational mass related to agent j, M pi is the passive gravitational mass related to agent i, G(t) is gravitational constant at time t, ε is a small constant and Rij(t) is the Euclidian distance between two agents i and j.The total force acting on each mass i is given in a stochastic form as the following.

F id (t )= ∑

j=1∧ j ≠ i

N

rand (w j❑)Fij

d(t)

where rand(wj) ∈ [0, 1] is a randomly assigned weight. Consequently, the acceleration of each of the masses,   is then as follows.

a id (t )=

F id ( t )

Mii❑ ( t )

where Mii is the inertial mass of ith agent. The next velocity of an agent is considered as a fraction of its current velocity added to its acceleration. Therefore, its position and its velocity could be calculated as follows:

vi (t + 1) = randi × vi (t) + ai (t)

xd (t + 1) = xd(t) + vd(t + 1)

where randi is a uniform random variable in the interval [0,1]. This random number to gives randomized characteristic to the search.

Until all candidate solutions are at their highest fitness positions and the termination criterion is satisfied, these iterations are then sustained.

The gravitational constant, G, is initialized at the beginning and will be reduced with time to control the search accuracy. Hence, G is a function of the initial value (G0) and time (t):

G = Go¿eℷ∗iter /itermax

Gravitational and inertia masses are simply calculated by the fitness evaluation. Here G0 is set to 100. A heavier mass means a more efficient agent. This means that better agents have higher attractions and walk more slowly. Assuming the equality of the gravitational and inertia mass, the value of masses is calculated using the map of fitness. The gravitational and inertial masses are updated by the following equations:

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It is obvious that for maximizing the profits of a supplier, bidding coefficients aj, and bj cannot be selected independently in other words, a supplier can fix one of these two coefficients and then determine the other by using an optimization procedure. In this regard, GSA is applied to find the optimal bidding coefficients and profit of each supplier.

Fuzzification and defuzzification:

Metaheuristic which include the GSA method, are approximate algorithms designed to be applied to engineering problems. It is clearly desirable that these algorithms be applicable to real optimization problems without the need for highly skilled labor. However, till date, their application has required significant time and labor for tuning the parameters, and hence, from engineering perspective, it is desirable to add robustness and adaptability to these algorithms. The latter adaptability property is especially important from the viewpoint of practical applications.

Two significant relationships must be understood in order to add adaptability to an optimization algorithm. One is the analysis of the qualitative and quantitative relationship between parameters and the behavior of the algorithm. The other is the analysis of the qualitative and quantitative relation between the behavior of the algorithm and success, or failure, of the search. The modification of the algorithm due to the results of these analyses should be carefully weighed so that an ideal algorithm behavior may be determined relative to the success of the search, so that an adaptive algorithm which feeds back the conditions of the search in order to maintain this behavior may be understood.

In GSA, force acting on the masses is related to the value of gravitational constant (G). Hence, the acceleration of the agent varies by varying the value of gravitational constant . Therefore, the gravitational constant determines the

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influence of agent’s previous velocity in the next iteration and also the search ability of GSA is reduced when the scale of the problem becomes large, because the search finishes before the phase of searching shifts from diversification to intensification. Suitable selection of the gravitational constant (G) provides a balance between global exploration, local exploration and exploitation, which results in less number of iterations on average to find a sufficiently optimal solution. Although the GSA algorithms can converge very quickly toward the nearest optimal solution for many optimization problems, it has been observed that GSA experiences difficulties in reaching the global optimal solution.

The gravitational constant (G) characterizes the behavior of agents, and experience shows that the success or failure of the search is heavily dependent on the value of the gravitational constant.

The main causes of the search failures are given by the following:

• The velocity of the agents (masses) increase rapidly, and agents go out of the search space.

• The velocity of the agents (masses) decrease rapidly, and agents become immobile.

• Agents (masses) cannot escape local optimal solutions.

In order to avoid these undesirable situations, it is important to analyze the relationship between the parameters and the behavior of agents (masses), with special regard to divergence and convergence of agents (masses). Therefore, the fuzzy adaptive GSA is proposed, to design a fuzzy adaptive dynamic gravitational constant using fuzzy “IF/THEN” rules for solving the optimal bidding problem. In FAGSA concept, the velocity and position update equations are same as in the case of GSA. But the gravitational constant is dynamically adjusted, as iteration grows, using fuzzy “IF/THEN” rules. The fuzzy inference system maps crisp set of input variables into a fuzzy set using membership functions. According to the predefined logic, the output is assigned based on these fuzzy input sets. The variables selected as input to the fuzzy inference system are the current best performance evaluation (normalized fitness value) and current gravitational constant; whereas output variable is change in the gravitational constant

To obtain a better gravitational constant value under the fuzzy environment, two inputs are considered: (i) normalized fitness value (NFV); (ii) current gravitational constant (G) and output is the correction of the gravitational constant (dG).

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The fuzzy rules are designed to determine the change in gravitational constant (dG).

From the characteristics it can be observed that if the NFV is smaller than the G, then NFV is to be increased to meet G which can be achieved by increasing the gravitational constant. If NFV is greater than G, then NFV is to be decreased to meet G which can be achieved by decreasing the gravitational constant. To incorporate these, three linguistic variables ‘Negative’, ‘Zero’ and ‘Positive’ (NE, ZE, PE) are considered. Therefore, nine (3 × 3 = 9) fuzzy rules can be designed from Table 1.

NFV is defined as

47

The fitness value (FV) calculated from Eq. (7) at the first iteration may be used as FVmin for the next iterations, whereas FVmax is a very large value and is greater than any acceptable feasible solution

The value of the parameter ‘G’ is large at the beginning of the search process and gradually it becomes small as the iterations are increasing. The change in gravitational constant (dG) is small and requires both positive and negative corrections.

Gt+1 = Gt + ⧍G

After we get a new value of G, GSA is repeated until iteration reaches their maximum limit. Return the best fitness (optimal bid value bj) computed at final iteration as a global fitness. Using bj values, calculate MCP from Eq. (5).

5) Case studies:In order to evaluate the performance of proposed FAGSA for solving optimal bidding problem, IEEE 30-bus system are considered .In this work, the parameters used for GSA are as follows whereN: population size =50G: gravitational constant for GSA=100Max_iterations=1000

The generator data for IEEE 30 bus system is as follows

The IEEE 30-bus system consists of six suppliers, who supply electricity to aggregate load. The generator data is shown in Table 4.Qo is 500 with inelastic

48

load (K = 0), considered for aggregated demand. The bidding parameters obtained by FAGSA are optimum compared to GSA, PSO, GA and GSS method. the time taken for the convergence of the proposed method is drastically reduced because of the fuzzification of gravitational constant (G). The gravitational constant adjusts the accuracy of the search, so it decreases with the time, which leads to a fast convergence rate compared to reported methods. In GSA the optimum selection of gravitational constant (G) is tedious and improper selection of gravitational constant results the velocity of the agents (masses) decreases rapidly, and agents become immobile. The performance of the PSO greatly dependent on the inertia weight, therefore, improper selection of the inertia weight may lead to premature convergence of the particles. GA has limitation of sensitivity of the choice of the parameters such as crossover and mutation probabilities.

MCP and profits of IEEE 30 bus system for FAGSA:

Generator Power (MW) Profit

1 42.0907 0.6313 e+07

2 182.94 2.2498 e+07

3 103.44 1.5516 e+07

4 158.3987 1.4999 e+07

5 6.6215 0.1500 e+07

6 6.5014 0.1500 e+07

MCP = 1.499e+05

In this paper, a new optimization algorithm called fuzzy adaptive gravitational search algorithm (FAGSA) has been proposed to achieve a better balance between global and local searching abilities of the agents (masses). The result of gravitational search algorithm (GSA) greatly depends on gravitational constant (G) and the method often suffers from the problem of being trapped in local optima. To overcome this drawback, gravitational constant has been adjusted dynamically and nonlinearly by using fuzzy “IF/THEN” rules in order to reach the global solution.

The performance of the proposed FAGSA is tested on IEEE 30- bus system. The test results of proposed method are compared with the well-known heuristic search methods reported in literature. From the test results, it is observed that, the proposed FAGSA converge to global best solution due to fuzzification of gravitational constant. Proper selection of gravitational constant makes a great intensity of attraction as a result the agents tend to move toward the best agent

49

compared to gravitational search algorithm (GSA), particle swarm optimization (PSO) and genetic algorithm (GA). The proposed FAGSA takes minimum execution time due to the gravitational constant has been dynamically adjusted using simple “IF/THEN” rules and also FAGSA outperformed the reported algorithms in a statistically meaningful way. Therefore, in conclusion, the proposed FAGSA outperform the GSA, PSO and GA reported in literature in terms of global best solution, standard deviation and computation time. Thus, the proposed FAGSA is more effective for the optimal bidding strategy in giving the best optimal solution in comparison to the GSA, PSO and GA with respect to total profit and computation time.

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Appendix:

Matlab code

%v:velocity

%x:position

%xmin and xmax:limits for the position

%N:No of agents

%dim:dimensions

%G:Gravitational constant

%G0:Initial value of G

%max_it:maximum no of iterations

%t:iterations

P_limits = [20 160;

15 150;

10 120;

10 100;

10 130;

10 120];

max_it=100;

60

N=6;

dim=1;

V=zeros(N,dim);

force=zeros(N,dim);

a=zeros(N,dim);

t=0;

G = zeros(1,max_it);

DG=0;

x1 = 0;

x2 = 0;

X = mvnrnd(2600.17,304.18,6)

e = [2;1.75;1;3.25;3;3];

f = [0.00375;0.0175;0.0625;0.00834;0.025;0.025];

Qo = 500;

a1 =0.05;

for t = 1:max_it

b1 = X;

x1 = sum(sum(a1./b1,1),2);

x2 = sum(sum(1./b1,1),2);

R = ones(N,dim).*((Qo+x1)./x2)

for i = 1:N

P = (R-a1)./b1;

61

if (P(i,1)>P_limits(i,2))

P(i,1)=P_limits(i,2);

end

if(P(i,1)<P_limits(i,1))

P(i,1)=P_limits(i,1);

end

end

fuel_cost = e.*P+f.*(P.*P);

revenue = P.*R;

fitness = revenue-fuel_cost;

best = max(fitness);

worst = min(fitness);

z(t)=sum(fitness);

FVmax = max(z);

FVmin = min(z);

NFV = (z-FVmin)/(FVmax-FVmin);

%Gravitational constant

alfa=20;G0=100;

G(1,t)=DG+G0*exp(-(alfa*t)/max_it);

%Mass

62

for i = 1:N

m(i) = (fitness(i) - worst)./(best-worst);

w = sum(m);

end

for i = 1:N

M(i)= m(i)./w;

end

%Force

for i=1:N

for j=1:dim

for k=1:N

if (i~=k)

force(i,j)=force(i,j)+(rand()*G(1,t)*M(k)*M(i)*(X(k,j)-X(i,j))./abs(X(k,j)-X(i,j)));

end

end

end

end

%acceleration

for i=1:N

for j=1:dim

63

if(M(i)~=0)

a(i,j)=force(i,j)/M(i);

%Velocity and position

V(i,j)=rand().*V(i,j)+a(i,j);

X(i,j)=X(i,j)+V(i,j);

end

end

end

DG = 0.1;

if(NFV <0 )

if G<0.4

G=G+0;

elseif 0.4<G<1

G=G-DG;

else

G=G-DG;

end

end

if(0<NFV<1)

if G<0.4

G=G+DG;

elseif 0.4<G<1

64

G=G+0;

else

G=G-DG;

end

end

if(NFV>1)

if G<0.4

G=G+DG;

elseif 0.4<G<1

G=G+0;

else

G=G-DG;

end

end

end

65