A Multiple Criteria Utility-based Approa h for the Unit Commitment with Wind Power and Pumped Storage Hydro

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University of Bremen, Germany

A Multiple Criteria Utility-based Approa h for the

Unit Commitment with Wind Power and Pumped

Storage Hydro

Author:

BrunoVieira

Supervisors in Portugal:

Prof. Dra. Ana Viana

Prof. Dr. Manuel Matos

Prof. Dr. João Pedro Pedroso

Supervisors in Germany:

Prof. Dr. Herbert Kopfer

Dr. Jörn S hönberger

Do ument submitted in partial fulllment of the requirements

for the degree of Master's in Ele tri al Engineering - Systems and Industrial

Planningin the

Institute of Engineering, Polyte hni of Porto, Portugal

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Finan ialsupportfor this work was provided by the Portuguese Foundationfor

S ien e and Te hnology (under Proje t PTDC/EGE-GES/099120/2008)through

the ProgramaOpera ionalTemáti oFa tores de Competitividade

(COMPETE) of the QuadroComunitáriode ApoioIII, partiallyfunded by

FEDER.

Otrabalho de investiga ão apresentado neste do umentofoi par ialmente

suportado pelaFundação para aCiên ia eTe nologia(proje to

PTDC/EGE-GES/099120/2008)através do ProgramaOpera ional Temáti o

Fa toresde Competitividade(COMPETE) doQuadro de Comunitáriode

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Abstra t

The integration of wind power in ele tri ity generation brings new hallenges to unit

ommitment due to the random nature of wind speed. For this parti ular

optimisa-tion problem, wind un ertainty has been handled in pra ti e bymeans of onservative

sto hasti s enario-based optimisation models, or through additional operating reserve

settings. However, generation ompaniesmayhavedierentattitudestowards operating

osts, load urtailment, or waste of wind energy, when onsidering the risk aused by

wind power variability. Therefore, alternative and possibly more adequate approa hes

should be explored.

Thisworkisdividedintwomainparts. Firstlywesurveythemainformulationspresented

intheliteraturefortheintegrationofwindpowerintheunit ommitmentproblem(UCP)

andpresentanalternativemodelforthewind-thermalunit ommitment. Wemakeuseof

theutility theory on epts to develop a multi- riteria sto hasti model. The obje tives

onsidered are the minimisation of osts, load urtailment and waste of wind energy.

Those arerepresentedbyindividualutilityfun tionsandaggregated inasingleadditive

utility fun tion. This last fun tion is adequately linearised leading to a mixed-integer

linearprogram(MILP)modelthat an be ta kledbygeneral-purposesolversinorderto

nd the mostpreferred solution.

In the se ond part we dis uss the integration of pumped-storage hydro (PSH) units in

theUCPwithlarge windpenetration. Those units anprovideextraexibilitybyusing

windenergy to pumpand storewater intheformofpotential energythat an be

gener-atedafter during peak loadperiods. PSH units areadded to the rstmodel, yielding a

MILP modelwith wind-hydro-thermal oordination. Results showed that theproposed

methodology is able to ree t the risk proles of de ision makers for both models. By

in luding PSHunits, the resultsaresigni antly improved.

Keywords

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Multi-Resumo

Aintegração deenergiaeóli anossistemasdegeraçãodeenergia elétri aintroduznovos

desaos no problema do Unit Commitment. A elevada aleatoriedade asso iada à

ve-lo idade do vento tem sido tratada essen ialmente através de modelos de otimização

esto ásti os baseados em enários, normalmente onservadores, ou através da denição

deníveisdereservaadi ionaisquepermitamfazerfa eaospossíveisdesviosemrelaçãoàs

previsõesdeprodução eóli a. Noentanto, asempresas geradorasde energia apresentam

diferentes atitudes perante ris os omo o orte de arga, a não satisfação dos níveis de

reserva requeridosou odesperdí io de eóli a.

Naprimeirapartedestetrabalhoéefetuadoumestudodoestadodaartedasformulações

para o problema do Unit Commitment om integração de energia eóli a e é proposto

ummodelo esto ásti omulti-obje tivo alternativo. São onsiderados omo obje tivos a

minimizar os ustos opera ionais, do orte de arga e do desperdí io de energia eóli a.

Estes obje tivossão representados individualmente através de uma função de utilidade

não-lineargenéri aquesãoagregadasnumafunçãodeutilidadeaditiva,queélinearizada

originando um problema de programação linearinteira mista. Este problema pode ser

resolvidoporsolversde otimizaçãogenéri osdeformaaen ontrarasoluçãopreferidade

umdeterminado agentede de isão, onsiderando a suaatitude perante oris o.

Numa segunda parte são adi ionadas ao modelo termi o-eóli o asunidades

hidroelétri- as de geração de energia om sistemas de bombagem e armazenamento. Pretende-se

explorar a apa idade destasunidades em utilizar energia eóli a parabombar água que

a armazenada emforma de energia poten ial, podendo posteriormente sergerada em

períodos de maior pro urade energia. Testesrealizados demonstram que a abordagem

multi- ritério proposta reete as diferentes atitudes do de isor em ambos os modelos.

Considerando unidades hídri as ombombagem osresultados sãomelhorados

signi a-tivamente.

Palavras- have

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de-A knowledgements

This work would not be possible without the help of some people, to whom I want to

addressmymost sin erea knowledgments.

Firstly I wouldlike to thankmy parents, witha spe ial thanksto mymother, for their

eort duringthese years for the ompletion ofmystudies.

I would like to thanks a lot to my s ienti advisor, Professor Ana Viana, for all the

opportunities provided, for all the help and all the onden e that she put on me. It

wasjustamazingtoworkwithatea herandadvisoratthislevel. Sheisandwillalways

bea referen e for me at a personaland professional level, either a ademi ally or inthe

industry.

I would also like to thank Professor Manuel Matos for his availability and for the help

provided wheneverI needed hisexpertise to larify some questions. In thesame way, I

want to thankProfessor JoãoPedro Pedroso for allthe helphehasprovided me.

Spe ialthankstomysupervisorsintheUniversityofBremen,ProfessorsHerbertKopfer

andJörnS hönberger,forre eivingmetodevelopandnishmyworkinBremen. Thanks

for makingme feelwel omeand for all thehelpprovided wheneverIneeded.

Imustalso thanktheUniversityof Bremenand INESCPorto forproviding mewithall

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Agrade imentos

A realização destetrabalho não seriapossívelsem aajuda de algumaspessoas,àsquais

faz todoo sentidoenviarumasin era palavra deagrade imento.

Em primeiro lugar gostaria de agrade er aos meus pais, om um espe ial obrigado à

minhamãe, peloesforçoefetuadodurante estesanosparaa on lusãodosmeusestudos.

Quero deixar uma profunda palavra de agrade imento à minha orientadora íenti a,

a professora Ana Viana, por todas as oportunidades que me propor ionou, por toda

a ajuda que me prestou e toda a onança que depositou em mim. Foi simplesmente

fantásti otrabalhar omumaprofessoraeorientadoradeste nível. Éeserásemprepara

mimumareferên iapessoaleprossionalmente, sejanomundoa adémi oou industrial.

Gostariatambém de agrade erao professorManuel Matospor toda a disponibilidade e

auxílio queme prestou sempre que pre isei dos seus onhe imentos para tirar dúvidas.

Damesmaforma agradeço aoprofessor JoãoPedro Pedroso toda aajuda prestada.

Quero também agrade er aos professores Herbert Kopfer e Jörn S hönberger, da

Uni-versidade de Bremen, por me terem a olhido e dado a oportunidade de desenvolver e

nalizar o meu trabalho em mobilidade na Alemanha. Obrigado pela hospitalidade e

pelaajudaprestada semprequepre isei.

Não posso também deixar de agrade er à Universidade de Bremen e ao INESC Porto

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Abstra t iii

Resumo v

A knowledgements vii

Agrade imentos ix

List of Figures xiii

List of Tables xv

Glossary xvii

1 Introdu tion 1

1.1 S opeand Resear h Question . . . 1

1.2 Obje tives . . . 4

1.3 Outline . . . 5

2 Wind-Thermal Unit Commitment Problem 7 2.1 UnitCommitment - Problem Des ription. . . 7

2.1.1 Obje tive andConstraints . . . 9

2.2 UnitCommitment Problem withWindIntegration . . . 10

3 FormulationsfortheShort-termWind-ThermalUnitCommitment Prob-lem 13 3.1 Sto hasti Formulations for the Wind-Thermal UnitCommitment Problem 16 3.1.1 CommonSto hasti Formulation . . . 17

3.1.2 Sto hasti Formulation Updating Data ina RollingManner . . . . 21

3.1.3 Other Formulations for theUCP withWindIntegration . . . 25

3.2 Wind-ThermalUnitCommitmentProblem-ANewlyProposedFormulation 28 3.2.1 Obje tive fun tion . . . 28

3.2.2 System onstraints . . . 31

3.2.3 Te hni al onstraints . . . 32

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4 Multi- riteria De ision Making 37

4.1 Multi-attributeDe ision Making and Multi-Obje tive Optimisation . . . . 37

4.1.1 Multi-attributeDe ision Making . . . 38

4.1.2 Multi-obje tive Optimisation . . . 39

4.2 UtilityTheory. . . 42

4.2.1 TheUnidimensional Utility Theory . . . 43

4.2.2 TheMulti-attribute UtilityTheory . . . 45

5 AMultipleCriteriaUtility-basedApproa hfortheWind-ThermalUnit Commitment 47 5.1 TheUtility-based Approa h for theWTUCP . . . 48

5.2 AMILP formulation forthe non-linear additive utility fun tion . . . 52

5.2.1 General Linearisation Formulation . . . 52

5.2.2 Setting the breakpoints(

a

ik

,

b

ik

) . . . 53

5.2.3 Methodology to xthenumberof segments . . . 56

5.3 Case study . . . 57

5.3.1 Simulated ases . . . 57

5.3.2 Results . . . 61

6 Wind-thermal UCP with Pumped Storage Hydro 67 6.1 Therole ofhydro unitswithpumped storageinpower systemsoperations 67 6.2 Wind-hydro-thermal UCP . . . 71

6.3 Pumped StorageHydro Modeling . . . 74

7 A Multiple Criteria Utility-based Approa h for the WHTUCP 77 7.1 Assumptions . . . 78

7.2 Mathemati alformulation . . . 79

7.3 Simulated ases and results . . . 81

8 Con lusions and Future Work 87

A Intera tion for Estimating the Parameter

α

for the Utility Fun tion 91 B Intera tion for Estimating the S aling Constants 95

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2.1 Single bus powersystem[1 ℄ . . . 8

2.2 Ageneralised stru turefor thethermal UCP. . . 9

3.1 Rollingplanning withs enario trees[2 ℄ . . . 23

3.2 Commonshapeof thefuel ostfun tion when onsideringthevalve-point loading ee t [3 ℄ . . . 30

3.3 Lowerapproximation ofthequadrati ostfun tion by4linearfun tions . 31 4.1 Graphi al representation of riskattitudes for a riterion to minimise . . . 44

5.1 Utilityfun tionshapefor a negative

α

. . . 50

5.2 Utilityfun tionshapefor a positive

α

. . . 51

5.3 Pie e-wise linearisationof a onvex fun tionwith lowerapproximation . . 55

5.4 Wind power fore ast(deterministi point fore ast and 10 sto hasti s e-narios)and realized windgeneration for day87 . . . 59

5.5 Dailyoperating ostsfor 30-day simulation period . . . 62

5.6 Dailyenergy notserved for 30-daysimulation period . . . 62

5.7 Total hoursof ommitment . . . 64

5.8 Dailyhours of ommitment for 30-daysimulationperiod . . . 64

5.9 Dailyutilityvalues for 30-daysimulation period . . . 65

5.10 Computational timesat the ommitment stage . . . 65

6.1 Hydro unitwithpumped storage apa ity[4℄ . . . 68

6.2 Diagram ofa pumpedstorage system[5℄ . . . 69

6.3 Constraintsrelated tothePSH units intheUCP . . . 72

6.4 Graphi al representation of thesimplied hydro power produ tion fun -tioninthree dimension spa e[6℄ . . . 73

7.1 Dailyhydro generationfor 30-daysimulation period . . . 83

7.2 Dailyoperating ostswith PSH units . . . 83

7.3 DailyENS withPSH units. . . 83

7.4 Dailyutilityvalues withPSH units . . . 83

7.5 Dailyhours of ommitment withPSH units . . . 83

7.6 DM1 - Daily ostswithand withoutPSH unit. . . 84

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7.8 DM1 - Dailyutilitywithand withoutPSH unit . . . 84

7.9 Computational timesat the ommitment stage . . . 85

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5.1 Thermal generators data-set . . . 58

5.2 Ranges onsidered for themeasurement s ales . . . 60

5.3 Parameters for3 simulated proles . . . 61

7.1 PSH unit's hara teristi s . . . 82

A.1 Valuesof parameter

α

ens

for thededu edutilitypoints . . . 94

B.1 Ranges onsidered for themeasurement s ales . . . 96

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DA Day Ahead

DM De ision Maker

ED E onomi Dispat h

EENS Expe tedEnergy Not Served

ENS Energy Not Served

EXS Energy eX essServed

GENCO Generation Company

ISO Independent SystemOperator

MADM Multi-Attribute De ision Making

MAUT Multi-Attribute Utility Theory

MCDM Multi-Criteria De ision Making

MILP Mixed-Integer Linear Programming

MIP Mixed-Integer Programming

MOCO Multi-Obje tive Combinatorial Optimisation

MOO Multi-Obje tive Optimisation

PSH Pumped StorageHydro

RAC ReliabilityAssessmentCommitment

RNS ReserveNot Served

RT RealTime

UCP Unit Commitment Problem

WPF W ind PowerFore asts

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Introdu tion

This do ument aims at partially fullling the requirements for the degree of Master's

in Ele tri al and Computers Engineering (prole of Systems and Industrial Planning).

A preliminary work and report, a ademi ally valued in 12 European Credit Transfer

and A umulation (ECTS), were developed and evaluated in Mar h in the Institute

of Engineering, Polyte hni of Porto, Portugal. That rst part was developed under

a resear h proje t COORDINATOR: algoritmos híbridos para uma gestão efe tiva

da produção de energia, em sistemas hidro-térmi os om re ursos eóli os funded by

Fundação para a Ciên ia e a Te nologia, in partnership withINESC Porto. This nal

do ument in ludes inthepreviousdo ument aresear h work a ademi allyvalued in30

ECTS,developedintheUniversityofBremen,Germany,undertheERASMUSinternship

ex hange program.

1.1 S ope and Resear h Question

In power systemsoperations, the short-term s heduling of powergeneration units (also

known as Unit Commitment Problem) is done in two stages, where de isions taken in

a rst stage inuen e a se ond one. The rst stage is the unit ommitment, that is

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de iding whi h power generating units must be ommitted/de ommitted in the

day-ahead to meetthe load. Dueto te hni al limitationsofmost thermal units,thatdonot

provide enough operational exibility,the pre-planned states annot be hanged inthe

day-ahead operation. The se ondstage is thee onomi dispat h,that is taken minutes

beforetheimplementation and onsistson de idingthemost e onomi produ tionlevel

for the ommittedunits, whi his inuen ed bythe ommitment de isions. Thisse ond

stage is also usedto deal withun ertainties by allowing to adjustthe produ tion levels

of thermal units a ording to the real wind values veried in ea h time period in the

intra-dayperspe tive.

The UnitCommitment Problem (UCP)isa mixed-integer non-linear ombinatorial

op-timisation problem: it deals with integer variables, su h as the units status, and with

non-linear fun tions(thermal generation ost fun tions), fallingin the lassof NP-hard

problems. In its standard format the problem handles only thermal power generators,

but several extensions have been proposed to in orporate hydro units. More re ently,

notonlyduetothe ontinuousin reaseoffuel ostsbutalsoforenvironmental on erns,

there is a trend to take advantage and in lude as mu h renewable energy as possible

inele tri ity generation. Those sour es of energy are generally heaper and have lower

environmental impa t. Among the renewable sour es of energy, wind energy is theone

thatisgrowingthe mostthroughout theworld. However, dueto thesto hasti behavior

ofthewindspeed,thewindpowerprodu tionishighlyun ertain. Theneedfora urate

fore asting tools for the wind speed arises, but even thebesttools available are unable

toavoidtheun ertaintyasso iatedwiththewind powerprodu tion. TheUCPbe omes

more ompli atedwiththelargepenetrationofwindenergysour es,sin ethewindunits

arenon-dispat hableand their produ tion levels depend ontherandom wind speed.

Theun ertaintyinherenttothewindenergymayhavedierentimpa tsinthe ontextof

theUCPwithwind integration. An unexpe ted downward deviation inthewind power

produ tion may erupt into load demand or reserve urtailments, due to the

ramping-up limitations of the thermal units. On the other hand, a big upward deviation inthe

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ramping apabilities. These wind urtailments may happen mostly at night, when the

wind isusually strongerwhile theloaddemandis lower.

Ingeneral, powersystemsoperationsaresubje ttoother sour esofun ertaintybesides

those brought up by the renewable sour esof energy. Thesein lude loaddemand

devi-ations or for edoutages ofthe equipments. Thewind un ertaintyhave been ta kled by

means of providing additional operating reserve, or by onsidering sto hasti programs

where thereserve is ommittedimpli itly. Theseapproa hesappearmostlyin

onserva-tive environments. They try to avoid loador reserve urtailments bynding s hedules

that over the possible deviations from the wind power fore asting over a set of

pre-determined s enarios, assuming an aversion attitude from the de ision makers towards

risk of urtailments and usually leading to higher operating osts. Some approa hes

aim at penalising the amount of energy/reserve not served in the obje tive fun tion.

However, generation ompanies (GENCOs) and/or system operators (ISOs) may have

dierent attitudestowards risk ausedbythewindpowerintegration intheUCP.Some

mayprefer to risk loaddemand urtailments ifthatmeans a relevant enough redu tion

in operating osts, and others may prefer to payto avoidthe possibility of not serving

demanded energy. These risk proles may also hange over time due to e onomi al or

politi alissues.

In this way, the large penetration of wind farms in power systems with thermal units

introdu es the question of how should the thermal UCP be adapted to in orporate the

high variability of wind speed. There is the need for innovate approa hes that an

balan e the unexpe ted surpluses or de itsof wind powera ording tothe preferen es

of de isionmakersduring their operations inpowersystems.

Generation te hnologies have appeared to help to a ommodate as many wind energy

aspossible intopowersystems. Fromthose te hnologies, hydro unitswithpumping and

energystorage apa itieshaveprovedtobeaninterestingsolutionduetoitsexibilityof

operation,byusingsurplusesofenergy generatedbywind topumpandstore waterthat

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then a topi of interest, parti ularly if the GENCO wants to analyze thepossibility of

investments inpumped storagehydro units.

1.2 Obje tives

This work fo uses on the day-ahead short-term UCP with wind integration that an

be either undertaken by ISOs in de entralised markets or by GENCOs in entralised

non- ompetitive environments. Theambitofour workfallsintwomainareas: 1) power

systemsand 2) multi- riteria de ision making.

A few ontributions are expe ted in the power systems area. We intend to des ribe

the thermal UCP and the main steps involved in the short-term operation of power

systems. We also briey des ribe the impa t of the integration of wind power in the

thermal UCP and surveythe main sto hasti formulations thathave been presentedin

the literature sofar. Making use of the survey, we aim at proposing and des ribing an

alternative sto hasti WTUCP modelbased on themodel presented in[7℄, thatproved

to be ee tive to a hieve the optimal solution for small and large-s ale thermal UCP.

Firstly,weintegratethewindpowerinas enario-basedapproa hthataimsatminimising

expe ted ostsrepresentedbythree omponents: operating osts,energy not served (or

load urtailment) andwasteof wind energy (orenergy ex essserved).

We will develop a multi-obje tive optimisation model for the WTUCP to handle the

impa t of wind un ertainty in power systems operations. Operating osts, load

ur-tailment and waste of wind energy are assumed to be targets to minimise by ISOs or

GENCOs. Theseobje tives arerepresented by an individualnon-linear utility fun tion

proposedin [8℄that should appropriatelyrepresent thesatisfa tion level of thede ision

makertowardsthe feasiblelevelsforoperating osts,load urtailmentand wasteofwind

energy. The individual utility fun tions are linearised by a xed number of segments.

The linearised utility fun tions are aggregated in one single additive utility fun tion,

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The nal sto hasti multi-obje tive model for solving the WTUCP with this new

ap-proa h allows to integrate the ISOs or GENCOs preferen es and prole hara teristi s

forndingthemostpreferredsolutionfollowingthemaximumexpe tedutilityparadigm.

In ase ond stage, hydro units withpumping andstorage apa itiesare in luded inthe

model, yielding a model with wind-hydro-thermal oordination. We aim at nding the

impa t of in luding those units when omparing to the results found by solving the

wind-thermalmodel.

1.3 Outline

This work is organised in eight hapters. This hapter presents the s ope, relevan e

and maingoals ofthis work. Chapter2 ontains a briefintrodu toryexplanation ofthe

thermal UCPand statesthe impa tof wind integration. Chapter3 reviews the urrent

resear h inintegrating wind intotheUCP andproposesan alternative sto hasti model

to solvethe wind-thermalunit ommitment problem. Chapter4reviews multi-attribute

de ision making theory, inparti ular utility theory. Chapter 5 des ribes a new

utility-based approa h developed to solve the wind-thermal UCP problem and presents the

simulations and results obtained for three dierent de ision maker proles. Chapter 6

providesanintrodu tionaboutpumpedstoragehydrounitsandreviewstheliteratureon

theintegration ofthoseunits into theUCP.Chapter7 des ribestheproposedapproa h

and mathemati al modeling for the integration of pumped storage hydro units in the

previous wind-thermal UCP model, and presents the simulations and results obtained

withthewind-hydro-thermal model. Finally, hapter8 providesnal on lusionsof this

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Wind-Thermal Unit Commitment

Problem

2.1 Unit Commitment - Problem Des ription

Theunit ommitment problem isahard ombinatorial optimisation problemwhose

ob-je tiveistodetermine as heduleto asetofpowergeneratingunits,whi hmustbe

om-mited/de ommitedoveraplanninghorizon,insu hawaythattotal ostsareminimised.

Typi ally,thepre-dispat h problem, ne essaryto evaluate thes heduling de isions,is a

subproblem of the UCP. The pre-dispat h problem determines theprodu tion levels at

whi hthe ommitedunits mustoperate inorderto meetthe fore astedsystemdemand

andreserverequirements,whilesatisfyingasetofoperational onstraintsandminimising

the overall operating osts. Later, in a shorter period basis (5 to 15 minutes ahead of

real dispat h) optimal produ tion levels are determined for the units that were set to

ONbythe UCP. Thisoptimisation problemisknownasE onomi Dispat h(ED).In a

real power system,both GENCOsand onsumers arelo ated indierent pla es, sothe

network is omposedbyseveralbuses and nodes. However,in theUCPa simpli ation

takes pla e: GENCOs and onsumers are onsidered to be onne ted through a single

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Figure2.1: Singlebuspowersystem[1℄

TheUCPtypi allyrefersto ashort-terms heduling,usuallyfrom1 dayto2weekssplit

inperiodsofone hourand takespla eonaday-ahead (DA)stage,ratherthan real-time

(RT)stages. Moreextendedplanning horizons anbe onsidered, yieldingthemid-term

and long-termUCP,upto one year. Those problems areoutsidethes opeof thiswork.

In entralised non- ompetitive environments, the UCPis ofmajor pra ti al importan e

forGENCOs,whoarelookingfore onomi s hedulesthat anmeettheloadandreserve

requirements, satisfyingthe onstraintsat minimumoperating osts. Thefa tofhaving

themonopoly ofthe energyprodu tion anddistribution allows these ompanies to seta

pri e thatprovidesthem the required prot.

Some de entralised ( ompetitive) markets have a similar stru ture to the entralised

ones, andtheUCPis entrally arriedout bytheISOonadailybasis. However,rather

than minimisingthe operating osts, thegoal isabout maximisingprot.

Inele tri itymarketoperations threestagesareusually arriedout bytheISOs: 1) the

DA stage, where the market pri es are dened by solving the UCP a ording to the

supplyand demandbids. 2)the reliabilityassessment ommitment (RAC)stage,where

the ommitment status of some units is revised loser to real-time, in order to address

theupdatedinformation about someun ertainvariablessu h astheloaddemand, units

outages,availabilityofrenewable energyorothernan ial issuesaboutthemarket. The

ommitment offast-start units may hange inthis stage; 3) thereal-time market. Here

thestatusofthepowergeneratingunitsisxedandtheEDmodeldeterminestheoptimal

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2.1.1 Obje tive and Constraints

There aredierent mathemati al models to solve the thermal UCP problem. Dierent

assumptions and onstraints may be onsidered, depending on the environment of

op-eration ( entralised or de entralised), hara teristi s of the power generating units or

features of the power system. A general stru ture for the thermal UCP is shown in

Figure2.2.

Obje tive fun tion:

Minimise(Produ tion osts+start-up osts+shut-down osts)

Subje t to:











System onstraints

(

Load requirements Reserverequirements Te hni al onstraints

(

Minimumup anddowntimes

Generation limitsand ramps

Network onstraints

Figure 2.2: Ageneralisedstru tureforthethermalUCP

Theobje tivefun tion orrespondstotheminimisationoffuel ostsforprodu ingele tri

energy, plus start-up and shut-down osts of thermal units over the planning horizon.

The set of onstraints in lude system onstraints, te hni al onstraints and network

onstraints.

System operation onstraintsare relatedto loadand spinningreserve requirements

sat-isfa tion. Thetotalthermalgenerationmustmeettheloaddemand,andenough reserve

levels provided byupward apabilities ofthermal units must meet some pre-dened

re-quirements.

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the thermal units must be kept ON/OFF; 2) generation limitations: in lude the

fea-sible maximum and minimum produ tion level of ea h thermal unit as well as ramp

onstrains. Ramp onstraints limitthe maximumin rease or de reaserate ofgenerated

power between onse utive periods, due to te hni al restri tionsof thermalunits.

Network onstraints are responsible for the reliability and stability of the system. A

more detailed explanation of the UCP obje tive and onstraints and a mathemati al

formulation areprovided inse tion3.2.

2.2 Unit Commitment Problem with Wind Integration

The expansion of wind power plants all over the world has experien ed a big apa ity

growth rate in the last de ade, more a entuated in the ountries lo ated in Europe

and North Ameri a. Despite the predi table de rease of the growth rate, the installed

apa itywill ontinuegrowing up a hieving almost500 GWof installed apa itybythe

end of 2016 [9℄. This fa t reates new hallenges for the UCP, for both GENCOs and

ISOs.

As known, the wind power produ tion depends on the wind speed, whi h depends on

some omplexfa torssu hasthe limateor geodesy. Thus,itisveryhard topredi tthe

speedofthewindandgivea urate windpowerfore asts(WPF), ne essaryto al ulate

the available wind power at ea h hour of the day-after. The wind may verify rapid

and unpredi table hanges on its speed insmall time periods bringing un ertainty, and

onsequently risk, to the de ision.

For thethermal UCP,the main sour es of un ertainty onsidered arethe load demand

andfor edoutagesofunits. However,errors onsideringthepredi tedandrealisedvalues

inherenttothese sour esareusuallylow. Inthis way,itislegitimatethatthese

parame-tersareusually onsideredasknownwith ertainty(inputs)whensolvingtheUCP.When

thewindpowerprodu tion is onsidered su hsimpli ationisnot reasonable. Thehigh

(29)

ur-additional supporting methods thattake into a ount thewind un ertainty, have to be

usedwhen windis in luded intheUCP.

Previousstudiesshowthatthea ura yoftheWPFhasasigni antimpa tintheUCP

andEDde isions,sin emorea urate fore astswouldprovidebetterandmoree onomi

s hedules[2,1014℄. Thismayberelatedeithertothereservelevelsdenedthatdepend

on the WPF errors [15℄ or to the less onservatism of the solutions obtained , sin e

more a urate WPF means less risk deviations (s enarios) from the inputs onsidered

[1012,16℄.

Nevertheless,theimportan eofa urateWPFmightdependonthepowersystem

onsid-ered. For instan e, Ummels etal. performedin[17℄asimulation for theWind-Thermal

Unit Commitment Problem (WTUCP) in the Dut h system, using an auto-regressive

moving average pro ess for the WPF onsidered. They surprisingly on luded thatfor

theDut h systemthe wind powerlimited predi tability does not require additional

re-servelevels. Theyalso on luded thatthewind powervariabilitydoesnothave a

signif-i ant ee ton thesystem osts, urtailments, waste ofenergy oremissions. Howeverwe

shouldkeep inmind thatthese resultsmight be relatedto thespe i hara teristi s of

thepowersystem.

The interests related to the a ura y of the fore asts may vary among groups, inside

the power system. For example, ISOs are more on erned with the a ura y related

to possible rapid hanges between time periods, the so- alled ramp events, in order to

maintainthe reliability andstabilityofthesystem. Ontheotherhand,GENCOsmight

be more on erned in a urate WPF for the overall planning period, in order to nd

good s hedules and minimise theoperating ostsusing allthe availableresour es inthe

most e onomi manner.

The problem has been ta kled in two ways: deterministi and sto hasti approa hes.

In the deterministi models only one wind s enario is onsidered and the un ertainty

of the wind power is not in luded. Con erning the sto hasti models, some authors

(30)

developedrigid modelsthat overallpossibledeviationsbetweens enarios,whileothers

onsiderprobabilitydistributionsforthewindprodu tioninput. There areothermodels

that are adjusted in an intra-day perspe tive a ording to the more a urate fore asts

that are provided. A few multi- riteria approa hes an also be found in the literature

[1821℄.

In this hapter, we survey the main formulations presented in the literature fo using

on sto hasti models. We will start by introdu ing models previously proposed in the

literature, followed by a proposal of a omplete model that is adapted from the work

(31)

Formulations for the Short-term

Wind-Thermal Unit Commitment

Problem

Notation

Constants

• T

lengthof the planning horizon.

• U

numberof thermalunits.

• W

numberofwind units.

• S

numberof s enarios.

• T = {1, . . . , T }

setof planningperiods.

• U = {1, . . . , U }

set ofthermal units.

• W = {1, . . . , W }

u

.

• C

ens

ost ofenergy not served.

• C

rns

ost ofreserve not served.

• C

exs

ost of wasteof energy.

• D

t

systemloadrequirements inperiod

t

.

_t

expe tedwind powerprodu tion fore astinperiod

t

.

• pw

p

Day

ut

day-ahead s heduled powergeneration for thermalunit

u

inperiod

t

.

p

up

uts

up regulation of the produ tion level of thermal unit

u

, in period

t

, s enario

s

.

p

down

uts

down regulation of theprodu tion level of thermal unit

u

,in period

t

,s enario

s

.

•

Auxiliary variables:

x

on

ut

, x

off

s

.

S(x

off

ut

, y

ut

)

start-up ostof unit

u

inperiod

t

.

(34)

3.1 Sto hasti FormulationsfortheWind-Thermal Unit

Com-mitment Problem

Theaim ofthis se tionis to explainhow wind powerprodu tion an bein luded inthe

formulation ofthe thermal UCP,and ompare formulationspreviously proposed.

Several workspresentedinthe literature showthatbynegle ting un ertaintyand using

deterministi valuesforthewindpowergenerationmoreexpensives hedulesareobtained

[2, 10, 14, 2225℄. Therefore, several sto hasti approa hes were developed where the

un ertaintyofthe windenergy isin ludedinthemodelusingtheavailableWPF.These

fore asts are usually integrated in the models in two dierent ways: using umulative

distributionfun tionsofthefore astedwindpower,orusingasetofgenerateds enarios.

Theformer maybeusedto obtain umulativeprobabilities to beinserted asparameters

inthe model[15℄. The lattermay be integrated using multiplepre-dened s enarios for

thewholesetofperiods [10℄,orusinga s enariotreetoolwhere thenumberofs enarios

in reases with the length of the planning period [2, 1214℄. These topi s are dis ussed

later.

Some approa hes assume that the WPF errors follow a normal distribution. However,

those errors do not really follow that kind of distribution [26℄. Thus the best way for

integration ofthe errors ofthe WPF intheWTUCP isstill an ongoingdis ussion.

Inthis hapter we willdes ribe the obje tive fun tionand thesystem onstraintsofthe

WTUCP.Theremainingte hni al onstraintsareintrodu edinthese tion3.2sin ethey

arerelatedto the unitstatesvariables,independent of thes enarios andnot relevant in

this se tion. The loadis onsidered asa sto hasti parameter, sin e itisnot possibleto

know with ertaintythe valueof demand inea h period of thefollowing day. However,

as the relative error observed in the real-time ompared with the fore asted values is

not relevant, itis ommonly onsideredasa knownparameter. Reserve isusedto over

(35)

reserve. Thesimplest approa h is to dene axed per entageof theload. Thepurpose

of thereserve isto overgeneratoroutages and/or loaddeviations.

3.1.1 Common Sto hasti Formulation

A ommon pra ti e in the ontext of the WTUCP is to introdu e probabilities in a

s enario-basedapproa h. As enariorepresentsapossiblewindrealisationinea hofthe

planningperiodsanditisassumedthatonlyoneofthegenerated s enarioswillo urin

thefollowing day. Thisis a strong assumption sin e itis knownthat it isvery unlikely

thatthewindrealisationsforea hhouroftheplanninghorizonwillfollowonlyoneofthe

S

s enarios onsidered. Nevertheless, this approa h allows us to integrate the possible deviationsaround thefore astedwind speed values. The generationof s enarios should

ensure atemporalrelation between onse utive periods. Thismeans thatthegenerated

values of wind powerprodu tion for ea h period should be relatedto theprevious and

the following ones, andnot beindependently generated.

Ideally, a dierent s hedule should be obtained for ea h s enario so that the de ision

maker (DM) gets an overview of the possible events and hooses one of the possible

s enario-indexed s hedules, knowing thenegative impa ton hisobje tive ifea h of the

other possible s enarios o urs [19℄. He/she an, following this pro edure, implement

a solution a ording to his/her preferen es and attitude towards the risk. However,

this methodology seems to be not pra ti al for real and large-s ale appli ations, due to

the high omputational time needed to solve ea h single (s enario indexed) problem.

Therefore, the most ommon pra ti e is to develop a model that leads to a s hedule

that is feasible for all possible s enarios (but not ne essarilyoptimal ifea h s enario is

analysed separately).

Wang et al. followed this approa h rstly in [23℄ and later expanded it in [10, 22, 27℄.

They integrated the wind energy hara teristi s in the thermal UCP model proposed

(36)

Sin ethemodelrepresentsasto hasti s enario-basedapproa h,theobje tive isto

,areindexedbys enario, and onstraints for e thatthe ommitment to be found is the same for all the possible

wind realisations. An e onomi pre-dispat h is intrinsi ally run for ea h s enario. A

generalversionofthe obje tiveofthisformulation ispresentedin(3.1). Thevalueofthe

obje tive fun tion obtained is only an indi ator of the expe ted osts for theobtained

ommitment, sin e the real oststo be observed would be dierent from this expe ted

value. Note thatonly the ontinuousvariables

p

uts

areindexedby s enario.

min

P

s∈S

prob

s

P

t∈T

P

u∈U

(F (p

uts

) + S(x

off

ut

, y

ut

) + H

ut

).

(3.1)

Thetraditionaldeterministi UCPformulation oin ides withthesto hasti oneby

on-sidering only one s enario with probability equal to one. In deterministi approa hes,

the only s enario onsidered for the wind produ tion input omes from the fore asted

wind poweror the expe tedvalueof anumberof generated s enarios.

To redu e the onservatism of the deterministi model, whi h assumes that load and

reserve have to be served, it is ommon to introdu e some exibility in the

formula-tion by in luding other indi ators su h as the possibility of load or reserve urtailment

[10, 12℄. The urtailment events are introdu ed to give more exibility to the model,

a ommodating bigdeviations that an be veried between dierent s enarios. Energy

not served (ENS) or load urtailment events an o ur in s enarios where the sum of

available windpowerisnot enough to meettheload. Theseeventsmayo urwhenthe

available produ tion annot absorb su h variations. It is also possible to have reserve

urtailment, yielding to the so- alled reserve not served (RNS)values. The RNS values

maybe dividedinspinningand repla ement reservesla ks,oronly asa singleoperating

reserve, as dis ussed later. Both omponents ENS and RNS ommonly have an

asso- iated ost perMW, represented by

C

ens

and

C

rns

, respe tively. These osts may, for

(37)

After integrating these omponents, the obje tive fun tion isnow to minimise the sum

oftheexpe tedprodu tionand start-upandshut-down osts,plustheexpe ted ostsof

energy and reserve urtailments, asshownin (3.2).

min

P

s∈S

prob

s

P

t∈T

P

u∈U

(F (p

uts

) + C

ens

ts

+ C

rns

ts

) +

P

t∈T

P

u∈U

(S(x

off

_ut

, y

ut

) + H

ut

)

(3.2)

Note that for reserve to be urtailed ahead of load, the

C

ens

penalty value should be

bigger than the

C

rns

penalty.

Con erning wind un ertainty, opposite events may also o ur. An unforeseen upward

windrealisationmayo urinseverals enarios,yieldingawasteofwindenergyorenergy

ex ess served (EXS), when the ommitted thermal units are operating intheir feasible

minimum. This wind power surplus happens mostly at night, when wind is usually

stronger and the system load is low. The wind energy may be then spilled in order to

maintain the normal operation of the slow-start units, su h as oal and nu lear, due

to the physi al onstraints ofthose units, and simultaneously ensure thereliability and

stabilityof the systemdue to rampand/orinertia te hni al and network onstraints.

In this formulation the integer variables and the te hni al onstraints related to the

thermal units remain independent of the s enarios. This means that the start-up and

shut-down onstraints, as well as the minimum on and minimum o time onstraints,

des ribedfurtherinthese tion3.2,arethesameforalls enarios. Thesystem onstraints

(see(3.3)-(3.5))mustbesatisedforea hs enario. Constraints(3.3)statethatthetotal

energy produ tionprovided bythethermalandwind unitsmeetstheloaddemandwith

possibility of load urtailment. Constraints (3.4) ensure the reserve satisfa tion if not

urtailed. Note that

p

max

ut

is onsidered instead of

P

max

u

, due to the te hni al ramp limits of the thermal units, further detailed in the se tion 3.2. Constraints (3.5) state

(38)

X

u∈U

p

uts

+

X

w∈W

pw

wts

= D

t

− ens

ts

, ∀t ∈ T , ∀s ∈ S

(3.3)

X

u∈U

(p

max

_uts

− p

uts

) ≥ D

t

.R

t

− rns

ts

,

∀t ∈ T , ∀s ∈ S

(3.4)

pw

wts

+ cw

wts

= F W

wt

,

∀w ∈ W, ∀t ∈ T , ∀s ∈ S

(3.5)

X

w∈W

cw

wts

= exs

ts

,

∀t ∈ T , ∀s ∈ S

(3.6)

Note that the EXS value for ea h period/s enario is given bythe sum of the urtailed

windenergyonea hunit

w

(3.6). Constraints(3.3)-(3.4) ouldbedevelopedwithoutany ofENSandRNSvaluesin onjun tion withobje tive(3.1),asdonebyWang,

Shahideh-pourandZ.Liin[29℄. Theauthorsdevelopedadeterministi formulation onsideringthe

expe tedwindgenerationasaknownparameterintheobje tivefun tionshownin(3.1)

todeterminethe unit ommitment. Theyalsoadded onstraintstoensurethatea h

s e-nariodispat h remainswithin afeasiblerange fromthe dispat hpreviously determined.

They usedBender's de omposition to solve the problem adding uts iteratively until a

feasiblesolution isfound. However, thismodeldemonstratesto be too onservative and

itmaybe omedi ult to ndfeasible solutions,sin e thede isionspa e wouldbe ome

very restri ted.

Followingthes enario-basedapproa hes,Zhang,Ze hunandLiangzhongpresentedin[30℄

arobuststo hasti WTUCPtodealwiththespinningreserverequirementsfromonehour

tothe next,inorderto overwindpowervariations between s enarios. Theauthors

pre-ferred to onsider only three s enarios whose dis rete probabilities are dedu ted from

the ontinuousCDF ofthe WPF.Toree t their on erns inthepresented formulation

two additional onstraints would be needed, (3.7) for upward dispat hes and (3.8) for

downward dispat h of thermalunits.

P

u∈U

(p

max

u,t+1,f

− p

u,t+1,f

) ≥

P

w∈W

(pw

wte

− pw

w,t+1,f

),

for

t = 1 . . . T − 1, ∀e, f ∈ S, e 6= f

(3.7)

(39)

P

u∈U

(min(p

u,t+1,f

− P

u

min

, r

u

down

) ≥

P

w∈W

(pw

w,t+1,f

− pw

wte

),

for

t = 1 . . . T − 1, ∀e, f ∈ S, e 6= f

(3.8)

The onsideration ofthese new onstraintswould over therisk introdu ed bythewind

speed variations between periods. The number of s enarios should be small to ensure

the omputational e ien y of the model. The authors did not onsider ENS or RNS

valuesin onstraints(3.3)-(3.5). Sotheproblem,withthisadditional onstraints,besides

being non-exible and very exigent for the solver to nd a feasible solution is also too

onservative andresults inhighoperational osts.

3.1.2 Sto hasti FormulationUpdating Data in a Rolling Manner

The ommitment de ision is usually made in a day-ahead perspe tive, in a short term

horizon, typi ally for 24 hours. However, more updated information be omesavailable

during the day, whi h should be taken into a ount, espe ially in systems with large

wind penetration. Inthis way, ommitment anddispat h de isionsshould be allowedto

be hanged in an intra-day perspe tive, in order to in orporate the updated fore asts,

hanging the day-ahead de isions ina rolling planmanner. For systemoperations with

large-s ale wind power, more a urate near real-time wind power measurements and

ontinuousre- al ulationareessential inthe ontextoftheUCPandED[17℄. Thislogi

isusedintheWindPowerIntegrationintheLiberalisedEle tri ityMarkets(WILMAR)

proje t, presentedby Meibomet al. in[2,1214℄.

WILMARis aproje tinitiallydevelopedto study the hangesin Nordi systemenergy

marketsduetothelargeamountofwindpower. Therstapproa hwasinitiallypresented

by Barth et al. in[14℄. The authors presented a modelthat doesnot orrespond to an

unit ommitment model, but rather to a planning tool that aims to optimise a given

input s hedule for 5 dierent markets. In their previous work an e onomi dispat h is

(40)

markets, ndingoptimalprodu tion levels forgiven ommitments, evaluatingvariations

inpri esand system osts.

Further, WTUCP algorithms were developed in the ontext of the WILMAR proje t.

Tuohyet al. extendedthe previous workin[2,12,13,31℄ onsidering unit ommitment

variables and integrating system, te hni al and network onstraints. The aim was to

analyse theimpa t of sto hasti wind and load on the unit ommitment and dispat h

of power systems with high levels of wind power. The model al ulates the UC and

ED de isions ina day-ahead rolling planapproa h, usingmultiple s enarios ina

multi-stage s enario tree. The ommitment de isions aredivided instages, typi ally about 1,

3 or 6 hours long ea h. In the rst stage there is only one root node where the wind

power produ tion and loadareassumedto beknownwith ertainty,yielding the

"here-and-now" de isions. In the following stages dierent paths with a given probability of

o urren earegeneratedbyas enariotreetool,ndinga ommitment forea hs enario

path. Ea hUCPrunndsas hedulebasedonthefore astedinformationforthela king

planning periods, starting at noon and nishing at the end of thefollowing day (36h).

An illustration of the rolling planning and de ision stru ture onsidering 3 hours long

stages an be seeninFigure 3.1.

The more distant from the de ision stage are the planning periods, more un ertainty

exists,and onsequentlymores enariosareneeded. Asmorea urateWPFareavailable,

more s hedules areable to be found at morerealisti levels. The ommitment de isions

frompaststagesareinputstothe modelinordertondthesolutions forthesubsequent

periods. In this way, the length of the fore ast horizon whi h the system is optimised

overis redu ed for subsequent planning periods. In Figure 3.1 we an seethat at ea h

3hours (startingat 12 AMandnishingat midnight ofthefollowing day) theplanning

period onsideredinthemodelisredu ed. Thewindpower produ tionisassumedtobe

knownfortherst3hours,ves enariosaregeneratedforthefollowing3hours,andfor

(41)

Figure 3.1: Rollingplanningwiths enariotrees[2℄

In termsof obje tive fun tion,it aimsto minimise theexpe tedoperatingand start-up

and shut-down ostsaswell astheload andreserve urtailments, asshownin(3.9).

min

P

s∈S

prob

s

P

t∈T

P

u∈U

(F (p

Day

_ut

+ p

up

_uts

− p

down

_uts

) + C

ens

_ens

int

ts

+ C

rns

spin

rns

spin

_ts

+C

rns

rep

rns

rep

_ts

) +

P

t∈T

C

ens

_ens

day

t

+

P

t∈T

P

u∈U

(S(x

off

_ut

, y

ut

) + H

ut

).

(3.9)

The s enarios of the s enario tree tool and the respe tive probabilities are then used.

Penalties are applied to avoid load and reserve urtailments. In terms of reserve, it is

divided into spinning and repla ement reserve, with dierent penalties,

C

rns

spin

(42)

spinningreserve and

C

rns

rep

for the repla ement, a ording to theIrish ode. Both are

).

Ea h day at 12 AM a day-ahead onstraint is added into the model inorder to set the

day-ahead pri es, sin e they typi ally must be dened and provided to the ISO from

12h to 36h before the operatingday. Theexpe tedENS valueis minimised inthis step

by adding the respe tive penalty ost to the obje tive fun tion. Constraint (3.10) is

addedto modelthe ENSat theday-ahead stage. Deterministi values forwindandload

(averagevalueofthefore asteds enarios)areusedtondthe ommitment thatsatises

the onstraintsat minimum ost.

X

u∈U

p

Day

_ut

+

X

w∈W

pw

_wt

Exp

= D

t

− ens

day

t

, ∀t ∈ T

(3.10)

The UC model onsiders a xed produ tion level per period for ea h thermal unit for

the rolling plan horizon at the day-ahead stage. However, up and down regulations in

relation to the predened level are onsidered in the intra-day operations, in order to

integratetheupdateddataof theWPF.

In the intra-day perspe tive, and onsidering the deviations related to ea h s enario,

onstraints (3.11) areadded to the model. Here,

pw

Exp

wt

is theexpe ted wind powerfor ea h timeperiod,introdu ed asa parameter.

P

u∈U

(p

up

_uts

− p

down

_uts

) −

P

w∈W

cw

wts

=

P

w∈W

(43)

Aswe an see, the deviations between thewind generationinea h s enario are overed

bytheupordownregulations(auxiliaryvariables),dedu ingthenthewind urtailment.

Load urtailment providesthe ne essaryexibilityinto themodel.

Additional onstraints to dene thevarious reserves onsidered arealso provided inthe

referredpaper.

The main on lusions of the WILMAR proje t developments, on erning theWTUCP

formulations, arethatthesto hasti optimisationisableto redu ethe ostandprodu e

better performing s hedules than the traditional deterministi approa h. Res heduling

more often means that more reliable and e onomi solutions are a hieved. The

un er-taintyisminimisedbe ausemorewindandloadfore astsarebeingupdated,parti ularly

whenfast-startunitsareavailable. Theirexibilityallowsto oversomeofthevariability

ofwindpoweroutput. Additionalstorageofele tri itydidnotappeartobringanyextra

benets in their study. The a ura y of the WPF has an important role on planning

de isionswhenintegrating windenergy,sin emoree onomi s hedules maybeobtained

iftheWPF aremore a urate.

A limitation of themodelis thatitis ne essaryto assume perfe t fore asts for therst

stage, whose asso iated errors may have a big inuen e in the following ommitment

stages. Furthermore, the model doesnot onsider network onstraintsthat are

parti u-larly important for some markets. The model isstill mainly a planning tool and isnot

beingusedby real-timemarket operators.

3.1.3 Other Formulations for the UCP with Wind Integration

In [11℄, Jiang, Wang and Guan presented a robust optimisation model for the thermal

UCP in the day-ahead market. The obje tive is to minimise the total ost under the

worst wind power s enario, applying a Bender's de omposition algorithm to obtain a

solution. Pumped storage hydro units are in luded in the model. The wind power

(44)

the model is ontrolled by a variable managed bythe DM, however it an be hard for

him/herto dene this valueand thesolutionmayeasily be onservative.

A model developed for the day-ahead wind-thermal UCP for the system operators in

deregulated powersystemsispresentedbyXieet al. in[15℄. The modeldoesnotfollow

a s enario-based approa h, and onsiders the Expe ted Energy Not Served (EENS) as

a fun tion of WPF un ertainty and thermal generators outages. The EEES, relatedto

a possible waste of wind energy, isa fun tion of wind un ertainties, that areexpressed

intermsof theunit ommitment variables and onsequently dene thespinningreserve

levels to set. Spe ialised formulations for these two indi ators arepresented, whi h are

initially non-linear and depend on the umulative probability based on the fore asted

wind power. In order to integrate the EENS and EEES indi ators, two steps are

re-quired. Firstly the sto hasti variables EENS and EEES are set to be under a dened

thresholdvalueatall time periods. However, besidesthelossofexibility,there issome

di ulty inherent to the denition of the eilings, that an turn themodelless exible

and introdu e extra onservatism. The ost may in rease exponentially if the EENS

thresholdisset too lowor,on theother hand,huge amountsofENS maybe introdu ed

ifthe threshold isset too high. A ost-benet variable is reated and added to the

ob-je tive fun tion to balan e EENSand EEES valueswiththereserve amount. With this

ost-benetbalan e,thespinningreservedetermination anbalan etheleastEENSand

EEES in ea h time period. However, the di ulty of setting the thresholds as well as

thepenaltyvalues isstill adrawba k.

Botterudetal. [16℄improvedthestudydes ribedinse tion3.1.1andintegrateddemand

dispat h to the modelpresentedin [10℄. They onsidered a exible loaddemand inthe

intra-day marketthat responds to the pri espra ti ed in ea h of theplanning periods.

Flexibleloaddemand anhelpwiththeintegrationofwindpowerwhenthereisasurplus

intheprodu tion,sin ethepri epra ti edinthemarketde reases. Insteadofa

s enario-based approa h, the authors developed a deterministi model that onsiders the wind

generationasthe50%quantileoftheWPF.Thewindun ertaintyisintegratedbysetting

(45)

aresolved basedon the supplyand demandbids thatarepreviously knownand remain

always the same. The updated WPF areavailable at ea h stage. Within a ase study

for the ele tri ity market of Illinois the authors on lude that the exibility from the

demand dispat h improves the ability to handle wind power un ertainty. A dynami

spinning reserve adjusted depending on the level of un ertainty of the WPF leads to

more e ient s hedules ofresour es ompared with thetraditional xedreserves.

Ruiz,Philbri kandSauer[24℄presentedaday-aheads hedulingapproa husinga

sto has-ti modelbasedons enariosthatrepresentthreeun ertaintysour es: generationoutages,

loadand wind power. The modelis divided intwo stages. The rst before thes enario

realisationwhenthe ommitmentde isionsaretakenfortheslow-startunits. These ond

for theED and ommitment of fast-start units after verifying whi h s enario has been

realised. Thework ombinestwostrategiestoa ommodatethewindpowerun ertainty:

thes enarioanalysisand adynami reserve leveldenition. Theaimistoobtain robust

solutions. Numeri alresults obtained througha ase study onPubli Servi e Company

system, Colorado, showed that the most signi ant dieren e between sto hasti and

deterministi poli ies isinthewind power urtailment. Thus,thesto hasti approa hes

revealedtobeveryappropriateforthesystemswithlargeamountsofinstalledwind

gen-eration withhigh un ertainty and without too many exible units su h as the thermal

fast-startand pumped storagehydro units.

Abreuetal. [32℄presentedaUCPmodelfor ompetitiveenvironmentswiththeobje tive

of maximisingtheprotof GENCOsand settingthepri esfor theenergy,theso- alled

pri e-based UCP. They onsidered only wind and as aded hydro units, exploring the

oordinationbetween them (windpowersurplus an be usedto store waterin as aded

hydro units). The errorsinherent to the WPF are integrated using s enarios managed

through a Monte Carlo simulation. The model provides also an assessment of risk to

dene the estimatedpay-o on erning the market un ertainties. The riskis relatedto

thedieren esbetween the targetedandthereal pri e,andtheobje tiveis to al ulate

anexpe tedpayothatsatisestheGENCOandsimultaneouslymaintainstheexpe ted

(46)

oordina-expe ted payoofthe GENCO.The sto hasti approa h would lowertheexpe ted risk

ofthe GENCO omparing withthedeterministi one andinun oordinated ases would

resultinhigher payos.

3.2 Wind-Thermal Unit Commitment Problem - A Newly

Proposed Formulation

Thisse tionaimsatproposingamixed-integerprogramming (MIP)modelforthe

short-term WTUCP, and at presenting a brief explanation of the model obje tive and

on-straints. Themodelpresentedin[7℄provedtobee ienttoa hievetheoptimalsolution

forsmallandlarge-s aleappli ationsfortheUCPwiththermalunits. Inthisse tionitis

adaptedtodevelopamodelwithwindenergy produ tionintegrationinasto hasti way,

ina s enario-based approa h. Themain hara teristi s found inthe literature, su h as

load urtailment,reserve urtailmentandwasteofenergy arealsoin luded andmodeled

to provide exibility.

The presented model an be used either by GENCOs or by ISOs (that entrally

man-age thesystem), and provides a day-ahead s heduling. The ON/OFF states annot be

hanged on ethe ommitment de isionisdone. A pre-dispat hisimpli it inthemodel,

needed to evaluate a unit ommitment solution. However, it should also be onsidered

thatotherEDsarerunfrequentlyinaintra-dayperspe tive,inorderto overdeviations

aused byloadun ertainty andunit for edoutages.

3.2.1 Obje tive fun tion

Theobje tive isto minimisethetotalexpe tedprodu tion ostsovertheplanning

hori-zon. Those osts in lude fuel, start-up and shut-down osts, plus the osts of load and

(47)

min

P

s∈S

prob

s

P

t∈T

P

u∈U

(F (p

uts

) + C

ens

ts

+ C

rns

ts

+C

exs

_exs

ts

) +

P

t∈T

P

u∈U

(S(x

off

ut

, y

ut

) + H

ut

).

(3.12)

The fuel osts represented by

F (p

uts

)

refer to the thermal units, sin e the wind power

produ tion is onsidered at no osts. The fuel onsumption of thermal units is not

represented bya linear fun tion of the generated power.

F (p

ut

)

an be represented by

theequation (3.13).

F (p

uts

) =







c

u

p

2 uts

+ b

u

p

uts

+ a

u

+ |e

u

sin(f

u

(P

min

− p

uts

))|

if

y

ut

= 1,

0

otherwise

.

(3.13)

where

a, b, c, e, f

areparametersofthefuel ostfun tion. Thisfun tiontakesintoa ount thevalve-pointloading ee trepresentedbytheabsolute omponent ofthefun tionand

theparameters

e

and

f

. Thisee t isdened byasetofvalvepoints. Theareabetween onse utive points is on ave, asshown inthe ontinuous lineof Figure 3.2,for 5 valve

points.

However,beingnon- ontinuousandnon- onvex, thistypeoffun tionbe omesveryhard

to optimise,due to the onsiderablede reaseofe ien y of MIPsolversto handle

non- ontinuousand non- onvex fun tions.

Thus, a quadrati approximated fun tion is generally used (see equation 3.14). The

shape ofthe quadrati ost fun tionis depi ted inFigure3.2indashed line.

F (p

uts

) =







c

u

p

2 uts

+ b

u

p

uts

+ a

u

if

y

ut

= 1,

0

otherwise

.

(3.14)

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Figure3.2: Commonshapeofthefuel ostfun tionwhen onsideringthevalve-point

loadingee t [3℄

ost fun tion,usedmeta-heuristi s/evolutionary programming algorithmsor hybridised

heuristi -based methods with MIP solvers, in order to a hieve better omputational

performan es. Inallofthese asestheya epttheriskofndingasub-optimalsolution.

Morere ently, Viana andPedroso [7℄proposedan iterative linearmodelthat onverges

to global optimality.

In orderto take advantage ofthe e ien y of MILPsolvers, inthis work alinear lower

approximationofthequadrati fuel ostfun tion(3.14)isperformed. Thefun tion

F (p)

is approximated by a set of linear fun tions dened by the tangent lines to

F (p)

in a

pre-dened set of produ tion levels

p

, so- alled breakpoints. The rst linear fun tion is tangent to

F (p)

at the minimum feasible power

(P

min

_{, F (P}

min

₎₎

and the last linear

fun tion is tangent to

F (p)

at the maximum feasible power

(P

max

_{, F (P}

max

₎₎

. All the

additional produ tion levels

p

dened to set the tangent lines to

F (p)

are equidistant

between the interval

[P

min

_{, P}

max

_]

. The total number of segments approximating

F (p)

is dened bythe user. Figure 3.3 shows an example of a lower linear approximation of

F (p)

by4segments (redlines).

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Figure3.3: Lowerapproximationofthequadrati ostfun tion by4linearfun tions

OFFbeforestart-up. These osts an be modeled as:

S

x

off

_ut

, y

ut

= a

hot

_u

s

hot

_ut

+ a

cold

_u

s

cold

_ut

.

(3.15)

where

s

hot

ut

and

s

cold

ut

are binary variables and the onstants

a

hot

u

and

a

cold

u

are set as follows:







a

hot

u

if

γ

off

ut

≤ t

cold

u

,

a

cold

_u

otherwise

,

(3.16) with

γ

off

ut

as the number of onse utive periods that thermal unit

u

was OFF before period

t

.

3.2.2 System onstraints

Thesystem onstraintsremainthesameasthosedes ribedinse tion3.1.1. Onlyaxed

spinning reserve level, whi h will be used to over load deviations and for ed outages

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maximumfeasiblegenerationin onse utiveperiodsmaydierandvariables

p

max

ut

should be onsideredinsteadof onstant

P

max

u

. Thosevariablesareusedwhensettingthereserve onstraints,asshownin(3.18). Aswe ansee,theloadissatisedbythethermalplusthe

windprodu tions. Onlythermalunits an provide reserve,sin ewind powerprodu tion

levelisdispat hable.

Some exibility is introdu ed in the model by allowing wind power produ tion to be

urtailed, if ne essary, as shown in (3.19). The wind energy produ ed plus the wind