(186576222) Intro Report - Copy (Recovered)

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    Birla Institute of Technology and Sciences-Pilani

    K.K.Birla Goa Campus

    A Project Report on BIDDING

    STRATEGY IN DEREGULATED

    POWER SYSTEM In

    Partial fulfillment of Requirements of

    EEE F266 Study Oriented Project

    Submitted to Submitted by

    M.K Deshmukh Kumar Avinish

    2011A3PS050G

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    CONTENTS

    1. Introduction 3

    a. Regulation

    b. Deregulation

    2. Market Models.. 4

    a. Poolco

    b. The Open Access Model

    c. Bilateral Exchange

    3. Bidding. 5

    a. Market Clearing Price

    4. Bidding Strategy6

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    ACKNOWLEDGEMENT

    I have taken efforts in this project. However, it would not have beenpossible without the kind support and help of many individuals. I

    would like to extend my sincere thanks to all of them.I am highly indebted to Prof MK Deshmukh for his guidance andconstant supervision as well as for providing necessary informationregarding the project & also for their support in completing theproject.

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    History of space research program in India

    Though ancient Indians were known to have knowledge about rocket science- it beingused in during wars- it was only after independence that the process of exploring spacereally accelerated. It was Dr. Vikram Sarabhai who founded the Physical Research

    Laboratory (PRL) in Ahmedabad on November 11, 1947. This was the first step that Indiatook towards becoming a space power. Our first biggest success was on April 19, 1975,when India launched its first satellite into space. It was launched by the Soviet Union fromKapustin Yar using a Cosmos-3M launch vehicle. The Aryabhata was named after a 5thcentury Indian mathematician, who founded concepts of the numerical value zero andmany astronomical calculations in around 500 AD.

    After that India has sent a number of satellites into space, notably the Apple (1981),BhaskaraI (1979) and BhaskaraII (1981), INSAT-1 series (1A, -1B, -1C and -1D),INSAT-2 series (2A, -2B, -2C and -2D), IRS-Series (1A, -IB, -1E, -P2, -1C, -P3, -1D),Rohini (1A, 1B, 2 and 3) and Sross.

    Also, India has developed various Launch vehicles that make a space programmeindependent and are the most important technological measure of its advancement.Prominent among them are Satellite Launch Vehicle (SLV), Augmented Satellite LaunchVehicle (ASLV), Polar Satellite Launch Vehicle (PSLV), Geosynchronous Satellite LaunchVehicle (GSLV).

    Indian Space Research OrganizationThe Indian Space Research Organisation is the primary space agency of India. ISROis amongst the largest government space agencies in the world. Its primary objective is toadvance space technology and use its applications for national benefit.

    Established in 1969, ISRO superseded the erstwhile Indian National Committee forSpace Research (INCOSPAR). Headquartered in Bangalore, ISRO is under theadministrative control of the Department of Space, Government of India.

    Since its establishment, ISRO has achieved numerous milestones. It built India'sfirst satellite, Aryabhata, which was launched by the Soviet Union in 1975. In1980, Rohini became the first satellite to be placed in orbit by an Indian-made launchvehicle, SLV-3. ISRO subsequently developed two other rockets: the Polar Satellite

    Launch Vehicle (PSLV) for launching satellites into polar orbits and the GeosynchronousSatellite Launch Vehicle (GSLV) for placing satellites into geostationary orbits. Theserockets have launched numerous communications satellites and earth observationsatellite. In 2008, Chandrayaan-1, India sent its first mission to the Moon.

    Over the years, ISRO has conducted a variety of operations for both Indian and foreignclients. ISRO's satellite launch capability is mostly provided by indigenous launchvehicles and launch sites. In 2008, ISRO successfully launched its first lunarprobe, Chandrayaan-1, while future plans include indigenous development ofGSLV, manned space missions, further lunar exploration, mars exploration andinterplanetary probes. ISRO has several field installations as assets, and cooperates with

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    the international community as a part of several bilateral and multilateral agreements. In2013, ISRO launched its Mars Orbiter Mission, which is currently on-route to Mars.

    ISRO

    Established August 15,1969

    Headquarters Bangalore

    PrimarySpaceport

    Satish Dhawan Space Centre

    Administrator K.Radhakrishnan

    Budget 6,600 crore

    Website www.Isro.org

    GOALS AND OBJECTIVES

    The prime objective of ISRO is to develop space technology and its application tovarious national tasks. The Indian space programme was driven by the vision of DrVikram Sarabhai, considered the father of Indian Space Programme.

    Indias economic progress has made its space programme more visible and activeas the country aims for greater self-reliance in space technology. Hennock etc.hold that India also connects space exploration to national prestige, further stating:"This year India has launched 11 satellites, including nine from other countriesand it became the first nation to launch 10 satellites on one rocket." Indian spaceResearch Organisation (ISRO) has successfully put into operation two majorsatellite systems namely Indian National Satellites (INSAT) for communicationservices and Indian Remote Sensing (IRS) satellites for management of naturalresources; also, Polar Satellite Launch Vehicle (PSLV) for launching IRS type ofsatellites and Geostationary Satellite Launch Vehicle (GSLV) for launching INSATtype of satellites. On July 2012, former president, A. P. J. Abdul Kalam said thatresearch by ISRO and DRDO is under way for developing cost reduction

    technologies for access to space.

    LAUNCH VEHICLE FLEETDuring the 1960s and 1970s, India initiated its own launch vehicle programme owing togeopolitical and economic considerations. In the 1960s1970s, the country successfullydeveloped a sounding rockets programme, and by the 1980s, research had yielded theSatellite Launch Vehicle-3 and the more advanced Augmented Satellite Launch Vehicle(ASLV), complete with operational supporting infrastructure. ISRO further applied itsenergies to the advancement of launch vehicle technology resulting in the creation of

    PSLV and GSLV technologies.

    http://www.isro.org/http://www.isro.org/http://www.isro.org/
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    SATELLITE LAUNCH VEHICLE(SLV)

    The Satellite Launch Vehicle, usually known by its abbreviation SLV or SLV-3 was a 4-stage solid-fuel light launcher. It was intended to reach a height of 500 km and carry apayload of 40 kg.Its first launch took place in 1979 with 2 more in each subsequent year,and the final launch in 1983. Only two of its four test flights were successful.

    LEFT TO RIGHT:SLV,ASLV,PSLV,GSLV,GSLVIII

    Augmented Satellite Launch Vehicle (ASLV)

    The Augmented Satellite Launch Vehicle, usually known by its abbreviation ASLV was a5-stage solid propellant rocket with the capability of placing a 150 kg satellite into LEO.This project was started by the ISRO during the early 1980s to develop technologiesneeded for a payload to be placed into a geostationary orbit. Its design was based onSatellite Launch Vehicle. The first launch test was held in 1987, and after that 3 others

    followed in 1988, 1992 and 1994, out of which only 2 were successful, before it wasdecommissioned.

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    Polar Satellite Launch Vehicle (PSLV)

    The Polar Satellite Launch Vehicle, usually known by its abbreviation PSLV, is

    an expendable launch system developed to allow India to launch its Indian RemoteSensing (IRS) satellites into Sun synchronous orbits, a service that was, until the adventof the PSLV, commercially viable only from Russia. PSLV can also launch small satellitesinto geostationary transfer orbit (GTO). The reliability and versatility of the PSLV isproven by the fact that it has launched 65 satellites / spacecraft ( 30 Indian and 35Foreign Satellites) into a variety of orbits so far. In April 2008, it successfully launched 10satellites at once, breaking a world record held by Russia.

    On 5 November 2013 the PSLV flew its 24th consecutive successful launch mission.]Itsonly failure in 25 flights was its maiden voyage in September 1993, providing the rocketwith a 96 percent success rate.

    TYPICAL PARAMETERS OF PSLV

    Lift-off weight 295 tonne

    Pay Load 1600 kg in to 620 km Polar Orbit,1060 kg in to Geosynchronous

    Transfer Orbit (GTO)

    Height 44 metre

    Geosynchronous Satellite Launch Vehicle (GSLV)

    Geosynchronous Satellite Launch Vehicle(GSLV)-Mark I&II ,is capable of

    placing INSATII class of satellites (2000 2,500 kg) intoGeosynchronous Transfer Orbit (GTO). GSLV is a three stage vehicle

    GSLV is 49 m tall, with 414 t lift off weight. It has a maximum diameterof 3.4 m at the payload fairing. First stage comprises S125 solid booster

    with four liquid (L40) strap-ons. Second stage (GS2) is liquid engine andthe third stage (GS3) is a cryo stage. The vehicle develops a lift off

    thrust of 6573 kn.The first flight of GSLV took place from SHAR on April 18, 2001 by

    launching 1540 kg GSAT-1. It was followed by six more launches , GSLV-D2 on May 8, 2003 (GSAT-2 1825 kg), GSLV-F01 on September 20,

    2004 (EDUSAT 1950 kg), GSLV-F02 on July 10, 2006, GSLV-F04 on

    September 2, 2007 (INSAT-4CR 2130 kg), GSLV-D3 on April 15, 2010,GSLV-F06 on December 25, 2010 and GSLV-D5 on January 05, 2014

    (GSAT-14 1982 kg).

    TYPICAL PARAMETERS OF GSLV

    http://en.wikipedia.org/wiki/Indian_Space_Research_Organisation#cite_note-19http://en.wikipedia.org/wiki/Indian_Space_Research_Organisation#cite_note-19http://en.wikipedia.org/wiki/Indian_Space_Research_Organisation#cite_note-19http://en.wikipedia.org/wiki/Indian_Space_Research_Organisation#cite_note-19
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    Lift-off weight 414 tonnePay Load 2 to 2.5 Tonne in to Geosynchronous

    Transfer Orbit (GTO)Height 49 metre

    Geosynchronous Satellite Launch Vehicle Mark-III (GSLV III)

    The Geosynchronous Satellite Launch Vehicle Mark-III is a launch vehicle currently underdevelopment by the Indian Space Research Organisation. It is intended to launch heavysatellites into geostationary orbit, and will allow India to become less dependent onforeign rockets for heavy lifting. The rocket, though the technological successor tothe GSLV, however is not derived from its predecessor.

    TYPICAL PARAMETERS OF GSLV-III

    Lift-off weight 630 Tonne

    Pay Load 4 Tonne in to Geosynchronous Transfer Orbit(GTO)

    Height 42.4 metre

    LANDMARK ACHIEVEMENTS IN ISROS LAUNCH VEHICLEDEVELOPMENTS

    1 PSLVhas 24 consecutively successful flights out of 25 launches

    2 PSLVused for launching a total of 27 satellites for foreign customers under commercialagreements, demonstrating its multi-satellite launch capability

    3 PSLVused to launch Space capsule Recovery Experiment(SRE-1),Chandrayaan-1and ISRO's exclusivemeteorological satellite, KALPANA-1, proving its versatility

    4 GSLVwith five successful flights of eight launches can launch 2 to 2.5 tonne satellite into Geo-synchronousTransfer Orbit (GTO)

    5 Successful testing of indigenously developed cryogenic upper stage on November 15, 2007.

    ISROS LAUNCH FLEET

    1 ISRO developed two experimental satellite launch vehicles, SLV-3and ASLV

    2 Polar Satellite Launch Vehicle(PSLV)commissioned in 1997

    3 Geosynchronous Satellite Launch Vehicle (GSLV-Mk I) commissioned after second successful flight in May2003

    http://www.isro.org/satellites/chandrayaan-1.aspxhttp://www.isro.org/satellites/chandrayaan-1.aspxhttp://www.isro.org/satellites/chandrayaan-1.aspxhttp://www.isro.org/satellites/chandrayaan-1.aspx
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    4 GSLV - MK II will use indigenously developed cryogenic Upper Stage

    5 GSLV - MK IIIis under development

    TECHNOLOGICAL DEVELOPMENT

    For the past four decades, ISRO has launched 70 satellites for various scientific andtechnological applications like mobile communications, Direct-to-Home services,meteorological observations, telemedicine, tele-education, disaster warning, radionetworking, search and rescue operations, remote sensing and scientific studies of thespace.

    ISRO has established two major space systems, the Indian National Satellite System

    (INSAT) series for communication, television broadcasting and meteorological serviceswhich is Geo-Stationary Satellites, and Indian Remote Sensing Satellites (IRS) system forresources monitoring and management which is Earth Observation Satellites. ISRO haslaunched many Experimental Satellites which are generally small comparing to INSAT orIRS, Space Missions to explore the space and Navigation Satellite to provide accurateposition information service to users.

    SPACE SYSTEM ESTABLISHED BY ISRO GEO STATIONARY SATELLITES

    EARTH OBSERVING SATELLITES

    NAVIGATION SATELLITES

    SPACE MISSIONS

    EXPERIMENTAL SATELLITES

    GEOSTATIONARY SATELLITES

    SATELLITE LAUNCH DATE LAUNCH VEHICLE

    GSAT-14 05.01.2014 GSLV-D5

    GSAT-7 30.08.2013 Ariane-5VA-215

    INSAT-3D 26.07.2013 Ariane-5VA-214

    GSAT-10 23.09.2012 Ariane-5VA-209

    GSAT-12 15.07.2011 PSLV-C17

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    GSAT-8 21.05.2011 Ariane-5VA-202

    EARTH OBSERVING SATELLITES

    Indian Remote Sensing (IRS) satellite system was commissioned with the launch of IRS-1A, in 1988. With eleven satellites in operation, IRS is the largest civilian remote sensingsatellite constellation in the world providing imageries in a variety of spatial resolutions,spectral bands and swaths. The data is used for several applications covering agriculture,water resources, urban development, mineral prospecting, environment, forestry, droughtand flood forecasting, ocean resources and disaster management.

    SATELLITE LAUNCH DATE LAUNCH VEHICLE

    SARAL 25.02.2013 PSLV-C20

    RISAT-1 26.04.2012 PSLV-C19

    Megha-Tropiques 12.10.2011 PSLV-C18

    RESOURCESAT-2 20.04.2011 PSLV-C16

    CARTOSAT-2B 12.07.2010 PSLV-C15

    NAVIGATION SATELLITES

    Indian Regional Navigation Satellite System (IRNSS)IRNSS is an independent regional navigation satellite system being developed by India. Itis designed to provide accurate position information service to users in India as well asthe region extending up to 1500 km from its boundary, which is its primary service area.IRNSS will provide two types of services, namely, Standard Positioning Service (SPS)and Restricted Service (RS) and is expected to provide a position accuracy of better than20 m in the primary service area.

    SATELLITE LAUNCH DATE LAUNCH VEHICLE

    IRNSS-1A 01.07.2013 PSLV-C22

    SPACE MISSIONS

    Indian space programme encompasses research in areas like astronomy,

    astrophysics, planetary and earth sciences, atmospheric sciences and

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    theoretical physics. Balloons, sounding rockets, space platforms and ground-based facilities support these research efforts. A series of sounding rocketsare available for atmospheric experiments. Several scientific instrumentshave been flown on satellites especially LAUNCH VEHICLE to direct celestialX-ray and gamma-ray bursts.

    Major space missions are Chandrayaan-l and Mars Orbiter Mission.

    SATELLITE LAUNCH DATE LAUNCH VEHICLEMars Orbiter Mission

    Spacecraft05.11.2013 PSLV-C25

    Chandrayaan-1 22.10.2008 PSLV-C11

    EXPERIMENTAL SATELLITES

    ISRO has launched many small satellites mainly for the experimentalpurposes. This experiment include Remote Sensing, Atmospheric Studies,Payload Development, Orbit Controls, recovery technology etc.

    SATELLITE LAUNCH DATE LAUNCH VEHICLE

    Jugnu 12.10.2011 PSLV-C18

    SRMSat 12.10.2011 PSLV-C18

    STUDSAT 12.07.2010 PSLV-C15

    INSAT-1B SARAL SATELLITE

    Technology demonstrationThe Space Capsule Recovery Experiment (SCRE or more commonly SRE or SRE-1) isan experimental Indian spacecraft which was launched using the PSLV C7 rocket, alongwith three other satellites. It remained in orbit for 12 days before re-entering the Earth'satmosphere and splashing down into the Bay of Bengal. The SRE-1 was designed to

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    demonstrate the capability to recover an orbiting space capsule, and the technology forperforming experiments in the microgravity conditions of an orbiting platform. It was alsointended to test thermal protection, navigation, guidance, control, deceleration andflotation systems, as well as study hypersonic aero-thermodynamics, management ofcommunication blackouts, and recovery operations. ISRO also plans to launch SRE-2

    and SRE-3 in the near future to test advanced re-entry technology for future mannedmissions.

    Development of crew vehicleThe Indian Space Research Organisation (ISRO) is working towards a maiden mannedIndian space mission vehicle that can carry three astronauts for seven days in a nearearth orbit. The Indian manned spacecraft temporarily named as Orbital Vehicle intendsto be the basis of indigenous Indian human spaceflight programme. The capsule will bedesigned to carry three people, and a planned upgraded version will be equipped with arendezvous and docking capability. In its maiden manned mission, ISRO's largelyautonomous 3-ton capsule will orbit the Earth at 248 miles (400 km) in altitude for up toseven days with a two-person crew on board. The crew vehicle would launch atop ofISRO's GSLV Mk II, currently under development. The GSLV Mk II features anindigenously developed cryogenic upper-stage engine. The first test of the cryogenicengine, held on 15 April 2010, failed as the cryogenic phase did not perform as expectedand rocket deviated from the planned trajectory. However the second test of theindigenous cryogenic engine was successful on 5 January 2014.

    FUTURE PLANS

    ISRO plans to launch a number of new-generation Earth Observation Satellites in thenear future. It will also undertake the development of new launch vehicles and spacecraft.ISRO has stated that it will send unmanned missions to Mars and Near-Earth Objects.ISRO has planned 58 missions during 201217; 33 satellites missions in next two yearsand 25 launch vehicles missions thereafter, costing 20000crore(US$3 billion)

    FORTHCOMING SATELLITES

    Satellite Name Details

    ASTROSAT ASTROSAT is a first dedicated Indian Astronomy satellite mission, which will enablemulti-wavelength observations of the celestial bodies and cosmic sources in X-ray

    and UV spectral bands simultaneously. The scientific payloads cover the Visible

    (35006000 ), UV (13003000 ), soft and hard X-ray regimes (0.58 keV; 3

    80 keV). The uniqueness of ASTROSAT lies in its wide spectral coverage extending

    over visible, UV, soft and hard X-ray regions.

    GSAT-6 / INSAT-

    4EThe primary goal of GSAT-6/INSAT-4E, which is a Multimedia broadcast satellite, is

    to cater to the consumer requirements of providing entertainment and information

    services to vehicles through Digital Multimedia consoles and to the Multimedia mobile

    Phones. The satellite carries a 5 spot beam BSS and 5 spot beam MSS. It will be

    http://en.wikipedia.org/wiki/Crorehttp://en.wikipedia.org/wiki/Crorehttp://en.wikipedia.org/wiki/Crorehttp://en.wikipedia.org/wiki/Indian_rupeehttp://en.wikipedia.org/wiki/Crore
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    positioned at 83 East longitude with a mission life of 12 years.

    GSAT-7/INSAT-4F It is a multi-band satellite carrying payloads in UHF, S-band, C-band andKuband.The satellite weighs 2330 kg with a payload power of 2000W and mission life of 9

    years

    GSAT-9 GSAT-9 will carry 6 C band and 24Kubandtransponders with India coverage beam.The satellite is planned to be launched during 201112 with a mission life of 12 years

    and positioned at 48 East longitude. ThisI-2Ksatellite has a liftoff mass of 2330 kg

    and payload power of 2300 W

    GSAT-11 GSAT-11 is based on I-4K bus which is under advanced stage of development. Thespacecraft can generate 1012 KW of power and can support payload power of 8KW.

    The payload configuration is on-going. It consists of 16 spot beams covering entire

    country including Andaman & Nicobar islands. The communication link to the user-

    end terminals operate in Ku-band while the communication link to the hubs operate in

    Ka-band. The payload is configured to be operated as a high data throughputsatellite, to be realised in orbit in 2013 time frame

    GSAT-15 GSAT-15 is an Indian communication satellite similar to GSAT-10 to augment thecapacity of transponders to provided more bandwidth for Direct-to-Home television

    and VSAT services. The satellite will be the 10th one in the series of GSAT satellites.

    GSAT-16 GSAT-16 will be the 11th Indian communication satellite similar to GSAT-15 meant toincrease the number of transponders that in turn enhance the satellite based

    telecommunication, television, VSAT services in India

    FUTURE LAUNCH VEHICLES1.GSLV-Mk III

    GSLV-Mk III is envisaged to launch four tonne satellite into geosynchronous transferorbit. GSLV-Mk III is a three-stage vehicle with a 110 tonne core liquid propellant stage(L-110) and a strap-on stage with two solid propellant motors, each with 200 tonnepropellant (S-200). The upper stage will be cryogenic with a propellant loading of 25tonne (C-25). GSLV Mk-III will have a lift-off weight of about 626 tonne and will be 43.43m tall. The payload fairing will have a diameter of 5-metre and a payload volume of 100cubic metre. GSLV Mk III is planned to be launched in April, 2014. It will weigh 640tonnes at the time of lift-off, which will make it the heaviest rocket ever to be built in India.

    2.Reusable Launch Vehicle-Technology Demonstrator (RLV-TD)As a first step towards realising a Two Stage To Orbit (TSTO) fully re-usable launchvehicle, a series of technology demonstration missions have been conceived. For thispurpose a Winged Reusable Launch Vehicle technology Demonstrator (RLV-TD) hasbeen configured. The RLV-TD will act as a flying test bed to evaluate varioustechnologies viz., hypersonic flight, autonomous landing, powered cruise flight andhypersonic flight using air-breathing propulsion. First in the series of demonstration trials

    is the hypersonic flight experiment (HEX).

    http://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/I-2Khttp://en.wikipedia.org/wiki/I-2Khttp://en.wikipedia.org/wiki/I-2Khttp://en.wikipedia.org/wiki/I-2Khttp://en.wikipedia.org/wiki/Ku_bandhttp://en.wikipedia.org/wiki/Ku_band
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    Extraterrestrial exploration

    India's first mission beyond Earth's orbit was Chandrayaan-1. ISRO plans to followup Chandrayaan-2 with unmanned missions to Mars, Venus and Near-Earth objects suchas asteroids and comets.

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    Bidding

    As per the layman language bidding stands for the auction. During

    the bidding the suppliers of the power (electricity and energy) bid at

    what rate they want to sell their product and the buyers bid at what

    rate they want to buy the product. The sellers and buyers submits

    bidfor energy buy and sell. The bids are generally in the form of

    priceand quantity quotations and specify how much the seller or

    buyer is willing to buy or sell and at what price. Once the buyer and

    seller bid the amount of energy and the price, the power exchange

    form an aggregate supply bid curve for suppliers and aggregatedemand bid curve for consumers. The curves are plotted on the

    coordinates of, supply and demand energy and price as shown in

    figure. The point of intersection of the two curves determines the

    market-clearing price (MCP). The MCP is the price of electric energy

    that is paid by consumers at all the places. The sellers are also paid

    the price equal to the MCP.

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    Bidding strategies

    While a genco has no control over the bids of its competitors and the

    energy demand, it can make its own strategy to place such a bid that

    provides it highest profit at lowest risk. A bid is called highly risky if it

    can yield large profit but its probability of being selected is low, a low

    risk bid is such bid, which may have lower profit earning capability

    but high probability of being selected. The methods by which the

    optimal bidding problem is solved are as follows:

    1. Optimization based bidding strategies: A binary representation

    scheme is adopted in order to seal with the complementarily constraints. A

    commercial mixed linear-integer solver is used to solve the resulting

    problem.

    2. Game theory: Several game theories have been applied to the problem of

    deriving the optimal bidding strategies for a set of generators. Few

    algorithms have also been developed to solve the Nash-Equilibrium in a

    bilateral trading.

    3. A genetic algorithm based method: Sampling rivalsbidding strategieswith the Monte carlo method. A probability distribution is thus obtained for

    the companysprofits both in day-ahead and the spinning reserve market.

    4. Markov Decision Process :A Markov Decision Process (MDP) model

    contains:

    A set of possible world states S

    A set of possible actions A

    A real valued reward function R(s,a)

    A description T of each actions effects in each state

    5. Nash equilibrium: A universal solution in game theory. Each players

    strategy is an optimal response with respect to the other entire playersstrategy

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