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Final Report on
Bidding 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 – Pilani
02/05/2014
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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.
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Table of contents
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
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Introduction
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
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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
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.
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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
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
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would have otherwise not been available to you. Green energy products are an
example of innovative programs made possible by retailers.
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
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Timeline of energy deregulation:
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.
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[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.
[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
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firm learns and dynamically regulates its conjecture upon the reactions of its rivals
to its bidding according to available information published in the electricity
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.
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[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
[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.
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[33] In this paper, a game theoretic model for examining non-cooperative bidding
strategies for acquiring FTRs in a deregulated power market is presented.
[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.
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[41] A brief literature survey of strategic bidding in electricity markets is made in
this paper based on more than 30 research publications
[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.
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[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.
[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.
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[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.
[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.
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[71] Spatial gaming model with coalition is discussed. The stability in a coalition is
measured as profit gain for all coalition members.
[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.
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[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.
[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
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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
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.
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2.1) Bidding classifications (Single sided and double sided):
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.
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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
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) = aj𝑃𝑗
2
+ 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 =∑ 𝑃𝑡𝑗𝑗∈𝐺𝑖 , 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 𝐺𝑗 (𝑃𝑗 ) = 𝑎𝑗 + 𝑏𝑗 𝑃𝑗 for j
= 1, 2,. . .,m. where 𝑃𝑗 is the active power output, 𝑎𝑗and 𝑏𝑗 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)
∑ Pj
𝑚
𝑗=1 = 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=
𝑄 𝑜+∑ (
𝑎 𝑗
𝑏 𝑗
)𝑚
𝑗=1
𝐾+∑ (
1
𝑏 𝑗
)𝑚
𝑗=1
(5)
𝑃𝑗 =
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 follows
Maximize : F(𝑎𝑗, 𝑏𝑗) = R𝑃𝑗 − 𝐶𝑗(𝑃𝑗)
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.
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
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(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
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Where
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;
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π(s)=1 means that state s is totally possible (= plausible or unsurprising).
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 ignorance
π(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.
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Human knowledge is often expressed in a declarative way using statements to
which some belief qualification is attached. Certainty-qualified pieces of uncertain
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.
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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
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
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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
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.
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Case 2: fitness function is considered for case 2-problem.
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 17
Parent2 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 1
Parent2_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 0
Mask2 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 16
Offspring 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
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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:
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(𝛼1, 𝛽1)^……. ^𝜇𝑖−1(𝛼𝑖−1, 𝛽𝑖−1) ^……^𝜇 𝑛(𝛼 𝑛, 𝛽 𝑛)
The expected value is defined as follows
E(f(𝛼𝑖 𝛽𝑖, ᶓ)) = ∫ 𝐶𝑟{f(𝛼𝑖, 𝛽𝑖, ᶓ)
𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦
0
≥r}dr - ∫ 𝐶𝑟{f(𝛼𝑖, 𝛽𝑖, ᶓ)
0
−𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑦
≤r}dr
A fuzzy simulation algorithm can be used to estimate E(f(𝛼𝑖 𝛽𝑖, ᶓ)) . Randomly
generate 𝛼1𝑙, 𝛽1𝑙, 𝛼2𝑙, 𝛽2𝑙, … … 𝛼 𝑛𝑙, 𝛽 𝑛𝑙 and set ᶓ𝑙 = (𝛼1𝑙, 𝛽1𝑙, 𝛼2𝑙, 𝛽2𝑙, … … 𝛼 𝑛𝑙, 𝛽 𝑛𝑙)
μ(ᶓ𝑙) = μ1(𝛼1𝑙, 𝛽1𝑙)^……….^ μ 𝑛(𝛼 𝑛𝑙, 𝛽 𝑛𝑙)
where l = 1,2,3,…m , respectively form the ᶓ level sets of 𝛼𝑗, 𝛽𝑗 (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(𝛼𝑖, 𝛽𝑖, ᶓ}≥r can be estimated by
𝐶𝑟{f(𝛼𝑖, 𝛽𝑖, ᶓ}≥r =
½*( {𝜇(ᶓ𝑙)|f(𝛼𝑖, 𝛽𝑖, ᶓ) ≥ r}𝑙=1,2,3,…𝑚
𝑚𝑎𝑥
+1- {𝜇(ᶓ𝑙)|f(𝛼𝑖, 𝛽𝑖, ᶓ) < r}𝑙=1,2,3,…𝑚
𝑚𝑎𝑥
)
And similarly for r ≤0
𝐶𝑟{f(𝛼𝑖, 𝛽𝑖, ᶓ}≤r =
½*( {𝜇(ᶓ𝑙)|f(𝛼𝑖, 𝛽𝑖, ᶓ) ≤ r}𝑙=1,2,3,…𝑚
𝑚𝑎𝑥
+1- {𝜇(ᶓ𝑙)|f(𝛼𝑖, 𝛽𝑖, ᶓ) > r}𝑙=1,2,3,…𝑚
𝑚𝑎𝑥
)
4.3) Fuzzy adaptive GSA:
Now, consider a system with N agents (masses), the position of the ith agent is
deﬁned by:
𝑋𝑖 = (𝑥𝑖
1
, … 𝑥𝑖
𝑑
, … , 𝑥𝑖
𝑛
) for i = 1,2,3….N
43.
43
where 𝑥 𝑑 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:
𝐹𝑖𝑗
𝑑
(t)=G(t)
𝑀 𝑝𝑖(𝑡)𝑀 𝑎𝑗(𝑡)
𝑅 𝑖𝑗(𝑡)+ᶓ
(𝑥𝑗
𝑑
(𝑡) − 𝑥𝑖
𝑑
(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.
𝐹𝑖
𝑑
(𝑡) = ∑ 𝑟𝑎𝑛𝑑(𝑤𝑗 )𝑁
𝑗=1 & 𝑗 ≠𝑖 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.
𝑎𝑖
𝑑
(𝑡)=
𝐹𝑖
𝑑
(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
44.
44
using the map of ﬁtness. The gravitational and inertial masses are updated by the
following equations:
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.
45.
45
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
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
46.
46
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).
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.
47.
47
NFV is defined as
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.
𝐺 𝑡+1
= 𝐺 𝑡
+ ⧍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 where
N: population size =50
G: gravitational constant for GSA=100
Max_iterations=1000
The generator data for IEEE 30 bus system is as follows
48.
48
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
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
49.
49
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
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
60.
60
%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;
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];
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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;
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;
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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
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
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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
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
64.
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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
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
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