Delft University of Technology
Robust unit commitment with dispatchable wind power
Morales-España, Germán; Lorca, Álvaro; de Weerdt, Mathijs M.
DOI
10.1016/j.epsr.2017.10.002
Publication date
2018
Document Version
Final published version
Published in
Electric Power Systems Research
Citation (APA)
Morales-España, G., Lorca, Á., & de Weerdt, M. M. (2018). Robust unit commitment with dispatchable wind
power. Electric Power Systems Research, 155, 58-66. https://doi.org/10.1016/j.epsr.2017.10.002
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ContentslistsavailableatScienceDirect
Electric
Power
Systems
Research
jou rn a l h om ep a g e :w w w . e l s e v i e r . c o m / l oc a t e / e p s rRobust
unit
commitment
with
dispatchable
wind
power
Germán
Morales-Espa ˜
na
a,∗,
Álvaro
Lorca
b,
Mathijs
M.
de
Weerdt
aaAlgorithmicsGroup,DepartmentofSoftwareTechnology,DelftUniversityofTechnology,2628CDDelft,TheNetherlands
bDepartmentofElectricalEngineering,DepartmentofIndustrialandSystemsEngineering,UCEnergyResearchCenter,PontificiaUniversidadCatólicade
Chile,Vicu˜naMackenna4860,Macul,Santiago,Chile
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received25January2017
Receivedinrevisedform4September2017 Accepted1October2017 Keywords: Stochasticoptimization Robustoptimization Dispatchablewind Windcurtailment Unitcommitment
a
b
s
t
r
a
c
t
Theincreasingpenetrationofuncertaingenerationsuchaswindandsolarinpowersystemsimposes newchallengestotheunitcommitment(UC)problem,oneofthemostcriticaltasksinpowersystems operations.Thetwomostcommonapproachestoaddressthesechallenges—stochasticandrobust opti-mization—havedrawbacksthatrestricttheirapplicationtoreal-worldsystems.Thispaperdemonstrates that,byconsideringdispatchablewindandaboxuncertaintysetforwindavailability,afullyadaptive two-stagerobustUCformulation,whichistypicallyabi-levelproblemwithoutermixed-integer pro-gram(MIP)andinnerbilinearprogram,canbetranslatedintoanequivalentsingle-levelMIP.Experiments ontheIEEE118-bustestsystemshowthatcomputationtime,windcurtailment,andoperationalcosts canbesignificantlyreducedintheproposedunifiedstochastic–robustapproachcomparedtobothpure stochasticapproachandpurerobustapproach,includingbudgetofuncertainty.
©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense
(http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Inrecentyears, higherpenetrationof variableanduncertain generation(e.g.,windandsolarpower)haschallengedindependent systemoperators(ISOs)tomaintainareliableandstilleconomical operationofpowersystems.Toachievethisandtobepreparedfor futuredemand,ISOsdecideaboutstartupandshutdownschedules ofgeneratingunitssometime(typicallyaday)upfrontbysolving theso-calledunitcommitment(UC)problem,whosemain objec-tiveistominimizeoperationalcostswhilemeetingpowersystem constraints.Theseoperationalcostsarecommitment-relatedcosts, and dispatchcosts ofboth thermal and wind generatingunits, wherethelattercanpresentnegativecost(orbids)[1].High lev-elsofvariableanduncertaingenerationsignificantlyincreasethe uncertaintyinthenetforecastedfuture demand,increasingthe differencebetweena fail-safesolutionandaneconomical solu-tion,andthisdilemmatherebyincreasesthecomplexityoftheUC optimizationproblem[2].
ThetwomainapproachesfordealingwiththeuncertaintyinUC problemsarestochasticandrobustoptimization.Stochastic opti-mization(SO)[2–5]typicallyconsistsofminimizingexpectedcosts overa setof possiblescenariosfor uncertainparameters. How-ever,SOcanbecomeimpracticalunderhigh-dimensionalproblems
∗ Correspondingauthor.
E-mailaddress:g.a.moralesespana@tudelft.nl(G.Morales-Espa ˜na).
mainlybecauseofaheavycomputationalburden[2].Additionally, themaingoalforISOsistoensureasafeoperationofthesystem,and SOdoesnotgivesufficientguaranteesonmeetingtheconstraints inrealizationsoftheuncertainty.Moreover,SOrequiresa large numberofscenariostobereliableandtheirassociatedprobability distributionishardtoobtain.
In robust optimization(RO) [6–10]the costsare minimized maintainingfeasibilityunderallpossiblerealizationsofuncertain parameterswithinsomespecifieduncertaintyset.Consequently, theresulting schedules couldturn out tobe over-conservative under a large uncertainty set: although the probability of the worst-caseevent is virtually nil, thechosen scheduleis robust forthisevent,andhencemuchmorecostlythanwhatisactually required.Onewaytoreduceover-conservatismistousea bud-getofuncertainty, whichmodelsa smalleruncertaintysetin a flexibleway[6].ArobustUC typicallyrequiressolvingabilevel optimizationproblem,where theouterlevelis amixed-integer linearprogram(MIP),andtheinnerlevelisusuallyabilinear pro-gram,whichisnon-deterministicpolynomial-timehard(NP-hard) [6].Findingoptimalrobustsolutionsintime forlarge-scale sys-temsisstillamajorchallenge;inparticular,solvingtherespective bilinearproblemsusuallyrequiressub-optimalheuristicmethods orcomputationallyexpensiveexact methods[6,11–14].Further, thisbilinearproblemistypicallysolvedmultipletimesaspartof aniterativealgorithmsuchascolumn-and-constraintgeneration [11],whichalsorequiressolvingadifficultmasterproblem
multi-https://doi.org/10.1016/j.epsr.2017.10.002
pletimes.Thiscomputationaldifficultyhighlightsthechallengeof creatingefficientmethodstofindeffectiverobustUCsolutions.
Recently,theso-calledunifiedstochastic–robust(SR) optimiza-tionapproach hasbeenproposedtostrike avery goodbalance betweenrobustnessandefficiency[7].Thisapproachincreasessafe operationofthesystemwhilemitigating theover-conservatism ofrobustoptimization.However,sincethisformulationcombines thestochasticformulation(whichrequiresoptimizationovermany scenarios)withthebilinearrobustformulation,thecomputational challengeisevenstronger.
Thispaperconsiderswinddispatchflexibilitybyallowing cur-tailmentintheUCformulation.Invariouspreviousworks,wind powerhasalsobeenmodeledasdispatchablebyallowingwind curtailmentinUCorexpansionplanningmodels[1,12,15–19].In thispaperwewillexploitsomeconsequencesthatareobtainedin therobustUCproblemwhenwindpowerisdispatchable,which willleadtoverycomputationallyefficientformulations.
ThispaperrisesuptovariouscurrentchallengesintheUC prob-lembyprovidingthefollowingcontributions:
1.Under thecase inwhich wind poweravailabilityis modeled throughaboxuncertaintyset,andassumingthatwindpower canbecurtailed,weprovethattheusuallynon-linearand NP-hard [6] secondstageof a fullyadaptive robust UC problem hasan equivalentlinear program (LP) representation, which solvesinpolynomialtime.Consequently,inthiscasethefully adaptive two-stage robust UC formulation can be translated intoanequivalentsingle-levelMIPproblem.Thisallowssolving realistically-sizedprobleminstancesveryclosetoglobal opti-malitysignificantlyfasterthanthetraditionalbi-levelrobustUC. Whencomparedwithatypicalrobustformulationbasedon bud-getofuncertainty,theproposedformulationprovidessimilar resultsbutsolvessignificantlyfaster.
2.Wealsoshowthatconsideringdispatchablewindcancontribute toefficientlysolvinga unifiedstochastic–robustoptimization problem[7]bylinkingthewinddispatchbetweenthestochastic andtherobustparts.Thus,theproposedSRformulationprovides acheaperoperation,higherrobustness,lesswindcurtailment, whilesimultaneouslyhavinglowercomputationalburdenthan atypicalROformulationbasedonbudgetofuncertainty.Also, theSRneedsveryfewscenariostoprovideverysimilarresults toascenario-richSO,hencenaturallyavoidingthehigh com-putationalburdenassociatedtoconsideringalargenumberof scenarios.
Theremainderofthispaperisorganizedasfollows.Section2 detailstheproposedrobustUCreformulationwithdispatchable wind,andshowshowtocomplementstochasticUCby incorporat-ingtherobustpart.Section3providesanddiscussesresultsfrom severalexperiments,whereacomparisonbetweenrobust, stochas-ticandunifiedUCformulationsismade.Finally,mainconclusions aredrawninSection4.
2. Mathematicalmodelsandstructuralresults
Thissectionformulatesthemathematicalmodelsandpresents asetofresultsthatexploitthestructureoftherobustUCwith dis-patchablewind.Section2.1definesthisproblem,andSection2.2 presentsageneralstructuralresultthatcharacterizesthesubsetof elementsoftheuncertaintysetthatcanachievetheworstcase. Sec-tion2.3studiesthisstructuralresultunderthewidelyusedbudget andboxuncertainty sets.Finally,Section2.4 studiesthe conse-quencesofthesestructuralresultsintheunifiedstochastic–robust UC.
2.1. RobustUCwithdispatchablewind
Weextendthe3-binarysettingUCformulation[20]toarobust UCwithdispatchablewind.ThecompactformofthisrobustUCis expressedas min x∈X
bx+max ∈ min (p,w)∈˝(x,) cp+dw (1) where X=x∈{0,1}3|G||T|: Ax≤a (2) and ˝(x,)={(p,w): Ep+Fw≤g+Gx (3a) w≤}. (3b)Here,xisavectoroffirst-stagedecisionsincludingthebinary on/off,start-upandshut-downdecisionsofconventional gener-ators.These decisionsare constrained throughset Xdefined in (2),whichincludesthelogicalrelationsbetweenon/off,start-up andshut-downvariables,aswellasminimumupanddowntimes. In(2),GisthesetofconventionalgeneratorsandTisthesetof time periods.Vector contains alluncertainparameters in the problem, correspondingtotheavailabilityof windpowerat all windfarmsandtimeperiods,i.e.,=
it:i∈W,t ∈T ,where itistheavailablewindpoweratbusiandtimet,andWisthesetofbusescontainingwindproduction.Theclosedsetisan uncertaintyset thatdescribestherealizationsof. Vectorsp,w aresecond-stagepowerdispatchdecisionsforconventional gen-eratorsandwindfarms,respectively,i.e.,p=
pgt:g ∈G,t ∈T andw= (wit:i∈W,t ∈T),wherepgtandwitarethepowerout-putofconventionalgeneratorgandofwindfarmsatbusi,attime t, respectively. These power dispatchdecisions are constrained through set˝(x,) definedin (3).In ˝(x, ),Eq. (3a)involves dispatch-relatedconstraintssuchaspoweroutputboundsfor con-ventionalgenerators,nonnegativityofpoweroutputatwindfarms, rampingconstraints, transmissionlinecapacity constraintsand energybalanceconstraints.Eq.(3b)representstheupperboundfor poweroutputatwindfarms,dependingonavailablewindpower, thatis,wit≤itforalli,t.Finally,theobjectivefunctionofproblem
(1)consistsofminimizingthesumofno-load,start-upand shut-downcosts,givenbybx,andworst-casedispatchcosts,givenby theinnermax-minproblem,whereccontainstheproductioncosts ofconventionalgeneratorsandwheredistheproductioncostsof windfarms,whichisusuallyzeroornegativerepresentingnegative bids(resultinginwindcurtailmentpenalization)[1].
TherobustUCproblem(1)isanadaptive robustoptimization problem[21].Inthisproblem,pandwareadaptivedecision vari-ableswhosevaluescandependontherealizationofthevectorof uncertainparameters,whilexisa“here-and-now”decisionthat istakenbeforeisrealized.Thisadaptiverobustframeworkforthe UCproblemwasfirstproposedin[6,22,23].Thissectionfocuseson studyingtheconsequencesofconsideringwindpowertobe dis-patchable,thatis,thatwindpoweroutputwitcantakeanyvalue
between0MWanditsavailabilityit.Inwhatfollows,weprovide
ageneralresultthatcharacterizesasubsetoftheuncertaintyset thatnecessarilycontainstheworst-caserealizationof.
2.2. Worst-caseisachievedinthesetofminimalelements Theinnermax–minproblemintherobustUC(1) max
∈ (p,w)min∈˝(x,)
cp+dw
(4) hasa specialstructure:theonly dependenceof ˝(x,) on is throughconstraint(3b),namely,w≤.Wecanusethisstructuretocharacterizeasubsetofthatincludestheworst-case.Weneed thefollowingdefinition.Thesetofminimalelementsofisgiven by
ME()={ ∈: ϕ ∈ s.t. ϕ≤andϕ=/} (5) thatis,ME()correspondstotheelementsofforwhichthere isnootherelementinthatislessthanorequaltoinall com-ponentsofthevector,andstrictlylessinatleastone.
AnimportantpropertyofME()thatweneedisgiveninthe followinglemma:
Lemma 1. Suppose that *∈\ME(), then there exists ˆ∈
ME()suchthat ˆ≤∗.
Proof. Define the subset S of all elements ϕ∈ suchthatϕ≤∗andϕ=/ ∗. This set is nonempty by defini-tionof*.Sinceisclosed,Sisalsoclosed,thusME()isalso
nonempty. Next, consider an element ˆ ∈ME(S). It thus holds that ˆ≤∗.Toshowthat ˆ ∈ME(),supposethereisanelement ϕ∈suchthatϕ≤ ˆandϕ= ˆ./ Thenϕ∈S,becauseϕ≤ ˆ≤∗. However,this contradictsthat ˆ ∈ME(S). Therefore,thereisno suchϕandthus ˆ ∈ME().䊐
Oneimportantresult thatleadstosignificantcomputational savingsisthattheworst-caseintherobustUCwithdispatchable wind(1)hastobewithinthesetofminimalelementsofthe uncer-taintyset.Belowweprovethismainstructuralresult:
Proposition2. Thefollowingequalityholds: max ∈ min (p,w)∈˝(x,)
cp+dw = max ∈ME() min (p,w)∈˝(x,) cp+dw. (6) Proof. Defineheref()= min
(p,w)∈˝(x,)
cp+dw
. (7a) SinceME()⊂itfollowsthatmax
∈f()≥∈maxME()f(). (7b)
Thus,weonlyneedtoshowthereverseinequality.Let*∈be
theoptimalsolutionof f(∗)=max
∈
f(). (7c)
If*∈ME()thenwehave
max
∈f()=f(
∗)≤ max
∈ME()f(). (7d)
Otherwise,if*∈/ME(),byLemma1thereexists ˆ∈ME()such
that ˆ≤∗.Thisdirectlyimpliesthat˝(x, ˆ)⊂˝(x,∗)duetothe definitionof˝(x,·).Consequentlygiventhedefinitionoff(·)we havethatf(ˆ)≥f(∗).Thus
max ∈
f()=f(∗)≤f(ˆ)≤ max ∈ME()
f() (7e)
whichcompletestheproof.䊐
2.3. Resultsforthebudgetandboxuncertaintysets
ThestructuralresultinProposition2isnotveryusefulifwe cannotcharacterizeME()inexplicitform.Fortunately,forthe
mostwidelyuseduncertaintyset,thebudgetuncertaintyset,this ispossible.Thebudgetuncertaintysetisdefinedas
bud
= : ˜ it− ˆit≤it≤ ˜it+ ˆit∀
i∈W,t ∈Ti∈W |it− ˜it| ˆit ≤t
∀
t ∈T (8)where ˜itisthenominalavailablewindpoweratbusi,timet, ˆit
isthealloweddeviationofavailablewindpowerfromitsnominal value,and t ∈[0,|W|]isthebudgetofuncertaintyattimet ∈T,
determiningthesizeoftheuncertaintysetandthusthe conserva-tivenessoftherobustUC(1).
Now,letusconsiderthefollowingset ˆbud,whichweshowis
exactlythesetofminimalelementsofbud:
ˆ bud
= : ˜ it− ˆit≤it≤ ˜it∀
i∈W, t ∈Ti∈W it− ˜it ˆit =−t
∀
t ∈T . (9)Thisisformallyshownasfollows.
Proposition3. Thefollowingholds:ME(bud)= ˆbud.
Proof. Let us first show that ˆbud⊆ME(bud). Consider ∈
ˆ
bud.Supposethatthereexistsϕ∈budsuchthatϕ≤andϕ=/ .
Thatis,theremustbesomei ,t suchthatϕi t <i t .Then,wemust
have
i∈W ϕit − ˜it ˆit <
i∈W it − ˜it ˆit =−t (10a)
whichimpliesthat
i∈W
|ϕit − ˜it |
ˆit
>t (10b)
whichisacontradiction,sinceϕ∈bud.Therefore,thereisnoϕ∈
budsuchthatϕ≤andϕ=/ ,whichmeansthat∈ME(bud).
i∈W
it − ˜it
ˆit
>−t (10c)
• Case1:≤ ˜.Selecti suchthati t > ˜i t − ˆi t anddefine
=max
⎧
⎨
⎩
˜i t − ˆi t , ˜i t − ˆi t⎛
⎝
t +i=/i it − ˜it ˆit
⎞
⎠
⎫
⎬
⎭
. (10d) • Case2:= ˜./ Selecti suchthati t > ˜i t anddefine= ˜i t .Lemma4. Thefollowingequalityholds: max ∈bud (p,w)min∈˝(x,)
cp+dw = max ∈ ˆbud min (p,w)∈˝(x,) cp+dw. (11) Atthispoint,itisimportanttounderstandthecomputational consequencesoftheaboveLemma.Howmuch“simpler”is ˆbudthanbud?Onewaytolookatthisisthroughthenumberof
vari-ablesandconstraintsrequiredtorepresentthesepolyhedralsets. Wecanfirstobservethattoformulate ˆbudwerequire|W|T
vari-ables,2|W|T inequalityconstraints,andTequalityconstraints.By removingequalityconstraints,wethenrequire|W|T−Tvariables, and 2|W|T inequality constraints.Now,torepresentthebudget
uncertaintysetbudwecanuseanextendedformulationto
han-dletheabsolutevalues.Infact,budistheprojectionofasetwith
2|W|Tvariablesand4|W|T+Tinequalityconstraints.Inshort,bud
requiresatotalof6|W|T+T variablesandinequalityconstraints, while ˆbudonlyrequiresatotalof3|W|T−T .
Finally,animportantspecialcaseariseswhenthebudget uncer-taintysetcorrespondstoa“box”set,i.e.,t=|W|forallt.Inthis
case,thesetofminimalelementsofthebudgetuncertaintyset containsasingleelementandtheoverallmax-minproblem con-sequentlycollapsestoa singleminimization problem.The next theoremsummarizesthisresult.
Theorem5. Ift=|W|forallt,then ˆbud=
˜ − ˆ
andtherobust UCwithdispatchablewind(1)becomesmin x∈X
bx+ max ∈bud(|W|)(p,w)min∈˝(x,) cp+dw = min x∈X,(p,w)∈˝(x,˜−ˆ) bx+cp+dw. (12)Proof. Theonly ∈ ˆbudsatisfying(9)when
t=|W|foralltis
givenby= ˜− ˆ.TherestfollowsdirectlyfromLemma4.䊐 Theimportanceofthisresultliesinthefactthatavery diffi-cultmin-max-minprobleminthegeneralcasebecomesamuch simplersingle-levelMIPproblemundertheassumptionsthat(i) windpowerisdispatchable,andthat(ii)theuncertaintysetisa boxset.Thismeansthatundertheseassumptions,itisnot neces-sarytoimplementcomputationallydifficultalgorithmstosolvethe robustUC,sinceitcanbedirectlysolvedbyMIPsolvers.
2.4. Consequencesfortheunifiedstochastic–robustUC
Theunifiedstochastic–robustUCapproachwasfirstproposedin [7].Inthispaper,weexploretheconsequencesofthespecial struc-tureofwinddispatchabilityintheproblem,toshowthatunderthis settingaverypowerfulandefficientstochastic–robustUCapproach isobtainedwithjustafewscenarios,exploitingthestructureof themax-minproblemrepresentingworst-casedispatchcosts.We thenproposeaconstraint thatrelatesthestochasticandrobust partscreatingaformulationwheretherobustpartprovides robust-nesstothedispatchscenariosofthestochasticpart.Althoughthis constraintcanincreasetherobustnessofaunifiedUCitcanalso increaseoperationalcosts.Section3.3.2discussestheeffectsofthis extraconstraintintheunifiedUCformulations.
Theunifiedstochastic–robustUCisdefinedhereas min x∈X {b x+(1−˛)max ∈ min (p,w)∈˝(x,)
cp+dw +˛ Ss=1 s min (ps,ws)∈˝(xs,s) cps+dws } (13)
wheretherearetwocharacterizationsofuncertaintyforavailable windpower:theuncertaintyset,asinproblem(1),andS sce-narioss,withrespectiveprobabilities
s.Now,forthestochastic
part,psandwsarethedispatchdecisionsofthermalandwindunits
forscenarios,respectively.Theobjectiveistominimizethesumof commitmentcostsbx,andaweightedcombinationofworst-case dispatchcost(withweight1−˛)andexpectedcost(withweight ˛),whereparameter˛∈ [0,1].
Motivatedbythefactthattheworst-casewinddispatchisin thesetofminimalelements,inthispaperweproposetorelatethe dispatchofthestochasticandworst-casescenariosas
ws≥w
∀
s∈S (14)which guarantees that all the wind dispatch scenarios for the stochasticpartaregreaterthanorequaltotheworst-casewind
dispatchoftherobustpart.Thepurposeofcombiningtherobust andstochasticcomponentsofthemodelinthiswayisensuringthat thestochasticsolutionisindeed“protected”bytherobustsolution. Withthisapproach,anyuncertainwindrealizationabovewis pro-tectedsince,intheworst-case,itcanbecurtailedtow.Thisrelation betweenthestochasticandrobustpartsthrough(14)can signifi-cantlyimprovetherobustnessofthemodel,asshowninSection 3.3.2.
3. Computationalexperiments
Tovalidatetheproposedformulations,wecomparethe perfor-manceofthefollowingnetwork-constrainedUCmodels:
RO:TherobustUCformulationunderboxuncertaintyset(see (12)inTheorem5)whereitslevelofconservatismisadjustedby shrinkingtheuncertaintyset.Forthis,weintroducetheparameter tocontrolthelevelofconservatism:[˜−(ˆ), ˜+(ˆ)].Sincefor RO,theworst-casescenarioliesonthelowerboundofthe uncer-taintyset,thenbychangingfrom0to1theworst-casescenario changesfrom ˜ to(˜− ˆ).
ROB:TherobustUCformulation(1)includingbudgetof uncer-tainty(8).SimilarlytoRO,thelevelofconservatismiscontrolledby ∈ [0,1],thusexpressingthebudgetofuncertaintyast=|W|.
Thetwo-leveloptimizationproblemissolvedusinga column-and-constraintgenerationalgorithm[11]andthealternatingdirection method[13,14].See[13,11]forimplementationdetailsof these algorithms.
SO:ThestochasticUCformulation(equivalenttothesecondpart of(13)).
SR:Theproposedunifiedstochastic–robustUCformulation(13) and(14),whereROisusedfortherobustpart.
SRB:Theunifiedstochastic–robustUCformulation(13)and(14), whereROBisusedfortherobustpart.
Inthissection,SRandSRBinclude(14)sinceitimprovestheir robustness,asfurtherdiscussedinSection3.3.2.
Afterdetailingtheexperimentalsetup,thissectionstudiesthe impactofdifferentscenariosonthestochasticandunified formu-lations.ForSRandSRB,wethenestablishanoptimalweight˛in theobjectivefunction.Alsoweanalyzetheeffectofchangingthe penaltyforwindcurtailment.Finally,wecomparetheperformance oftherobustandunifiedformulationswhenchangingtheirlevel ofconservatism.
3.1. Experimentalsetup
Asacasestudy,weusetheIEEE118-bustestsystem,which wasadaptedtoconsiderstartupandshutdownpowertrajectories [24].Thissystemhas186transmissionlines,54thermalunits,91 loads,and threebuseswithwindproduction.The penaltycosts fordemand-balanceandtransmission-limitsviolationsaresetto 10000 $/MWhand 5000$/MWh [25], respectively. In addition, wind bidsaresettod=−300$/MWh,abouttentimes the aver-age wind bid in somemarkets[26,1], tosimulatecases where curtailmentisundesired.Althoughtheoptimizationproblemsare solvedusingdw,thefinalvaluesshownhereareobtainedusing d(w−)toonlyreflectcurtailment(penalization)costs[1].
All experiments are solved using CPLEX 12.6.3with default parametersandarerunonanIntel-Xeon3.7-GHzpersonal com-puterwith16GBofRAMmemory.Allinstancesaresolveduntil theyreachanoptimalitytoleranceof5×10−4.
TocomparetheperformanceofthedifferentUCmodels,we make a clear distinction between the scheduling and (out-of-sample)evaluationstages.Theschedulingstagesolvesthedifferent UC modelsand obtains theircommitment policyusinga small representativenumberofwindscenarios(upto50)forthe
stochas-Fig.1.Windproductiononbus69.Exampleof30generatedscenariosforthe stochasticformulations.Uncertaintysetdefinedby98.8%confidencelevel.The1000 out-of-samplescenariosusedfortheevaluationstageareshownonthebackground.
ticformulations,anduncertaintysetsfortherobustformulations. Theevaluationstage,foreachfixedcommitmentpolicy,solvesa network-constrainedeconomicdispatchproblemrepetitivelyfora setof1000out-of-samplewindscenarios(seeFig.1),thus obtain-inganaccurateestimateoftheexpectedperformanceofeachUC policy.
Togeneratethescenarios,weassumethatwindproduction fol-lowsa multivariatenormal distribution, which istruncated for nonnegativevalues.FortherobustformulationsRO,ROB,SRand SRB, theuncertainty setis defined bya nominalvalue ˜ and a deviation ˆ correspondingto98.8%confidencelevel(2.5standard deviations).For thestochasticformulationsSO, SR andSRB, the scenariosaregeneratedfollowingapredictednominalvalueand volatilitymatrix.Tooptimallydistributethesamplesandexplore thewholeexperimentalregion,weemployLatinHypercube Sam-pling(LHS)[18].Thatis,agivennumberofscenariosisgenerated withminimalcorrelationandmaximalmutualdistance[27].Fig.1 showsanexampleofsuchasetofthescenariosgeneratedforone ofthethreebusescontainingwindproduction.
3.2. Robust,stochastic,andunified
Table1showsanoverviewoftheproblemsizeforRO,andfor SOandSRfor1,5and30scenarios.ThesizesofROBandSRBarenot listedinTable1sincetheyarenotconstant.Theystartwithaninitial sizesimilartoROandSR,respectively,buttheirmasterproblems increasethroughtheiterationsofthecolumn-and-constraint gen-erationalgorithm[11,13].Notethatforagivennumberofscenarios, SRisslightlylargerthanSObecauseSRaddsanextrascenario rep-resentingtheworst-casescenario(sincetheworst-casescenarioof SRisunique,asshowninTheorem5).Despitethenumberof scenar-ios,alltheUCshavethesamenumberofbinaryvariables,because theyallobtainthesamenumberofbinary(first-stage)decisions.
Inthissection,wesetthelevelofconservatism()ofRO,ROB, SRandSRBat0.6,0.5,0.6and0.4,respectively.Becausethe formu-lationsperformthebestatthesevaluesof,asdiscussedinSection 3.3.Similarly,wesettheweight˛ofSRandSRBat0.9,asdiscussed inSection3.2.3.
3.2.1. Stochastic(SO)vs.robust(ROandROB)formulations
InTable2,weassesstheperformanceofthedifferentUCpolicies onsizeaspects.Fortheschedulingstage:(1)thefixedproduction costs(FxdCost[k$]),which includesnon-load,startupand shut-downcosts,and(2)thetimerequiredtosolvetheUCproblem(CPU Time[s]).Fortheevaluationstagewerecord(3)theaverageofthe
totalproductioncostsincludingthewindcurtailmentpenalization (AvgTC[k$]);(4)themaximumtotalcostofthe1000out-of-sample scenarios,representingtheworst-casescenario(WorstTC);(5)the totalaccumulatednumberofviolationsinbothdemand-balance andtransmission-limitsconstraints(#Viol);and(6)theaverage percentageofwindthatwascurtailed(%WCurt).
Wecanclearlyobservethat,first,thehigherthenumberof sce-narios,thebettertheSOperformance(lowerAvgTCandViol),as expected.Second,ontheonehand,thestochasticformulationSO using30scenariosguaranteesrobustness(Viol=0),andaSOwith ahighernumberofscenariospresentsasmallimprovementatthe expenseofhighercomputationalcost:from30to50scenarios,SO improvesAvgTCinlessthan0.01%andtakesmorethan2×longer tosolve.Ontheotherhand,therobustformulationsROandROB guaranteerobustnessbyonlyoptimizingfortheworst-case sce-nario,buttheyscheduletoofewreserves(lowerFxdCost)hencenot ensuringthathigherwindproductionlevelscouldbedispatched (WCurtwasaround10×largerthanSO).Therobustformulations ignorethepossibilityof optimistic(highwind)scenarios hence nottakingadvantageofthem.However,comparedwithSOwith 30scenariosandROB,theformulationROproposedinthispaper solvesmorethananorderofmagnitudefaster(above71×and13×, respectively).Furthermore,sinceROonlyconsidersonescenario, itsolvesinsimilartimeasSOwithonescenario,butROreduces theviolationstozero,lowerstheAvgTCandWorstTCby20.5%and 83.4%,respectively,althoughincreaseWCurtbymorethantwice. 3.2.2. Stochastic(SO)vs.unifiedrobust–stochastic(SR)
formulations
Table2showshowtherobustpartofSRdrasticallyimproves theperformanceofthestochasticformulationsSOregardlessof thenumberofscenariosused.Evenwhenveryfewscenariosare considered,SRpresentsasignificantlybetterperformancethanSO: forthecaseofoneandfivescenarios,SRpresentsnoviolations, insteadof698and191,acostreductionofmorethan24%and4%, andaworst-case(WorstTC)reductionofmorethan84%and71%, respectively.
Themostimportantpart,however,isthat,comparedwithSO, addingtheworst-casescenariointoSRcomesatalmostnoextra costsintermsofruntime(evenlowerinsomecases).
3.2.3. Unifiedrobust–stochasticformulations,SRvs.SRB
Forfiveormorescenarios,SRandSRBperformsimilarlyinthe out-of-sampleevaluation.SRBpresentsviolationswhenonlyone (thenominal)scenarioisconsidered,becausethelevelof=0.4is toolowtoproviderobustnessforthisone-scenariocase;however, fora=0.5,SRBwithonescenarioachieves0violations,anAvgTCof 769.39k$,aWorstTCof890.41k$,andaWCurtof0.34%.Although SRandSRBperformsimilarlyintheout-of-sampleevaluation,SR solvesmorethan6.8×fasterbecauseitcanbesolveddirectlyas single-levelMIPproblem,insteadofrequiringad-hocalgorithmsto solvethetwo-levelMIPproblemincludingabilinearinnerproblem. NumberofscenariosneededbySRandSRB.Unlikethestochastic SOformulation,whichneedsalargenumberofscenariosto guar-anteerobustness,thestochastic–robustformulationsSRandSRB arerobustandfewscenarioscanbeusedtoobtainagood perfor-mance.AlthoughSRandSRBusingonescenariopresentalready abetterperformancethanSOusing25scenarios,henceforth,we use5scenariosforthestochasticpartofSRandSRBsinceit fur-therdecreasesAvgTC,WorstTCandWCurt,andtheystillsolvein lessthanoneandfourminutes,respectively.Moreover,considering morethan5scenariosaddsaverylittleperformanceimprovement atahighcomputationalcost,e.g.,using10,tookmorethan3×and 2.7×longer,respectively.
ObjectiveweightsforSRandSRB.Weaimtoestablishthe opti-malbalancebetweenthecostsoftheworst-casecomponentand
Table1
ProblemsizecomparisonofUCformulations.
UC SC# Constraints Continuousvariables Binaryvariables Nonzeroelements
SO 1 15855 7128 6390 305938 5 56367 35640 6390 1413190 30 309567 213833 6390 8332689 RO 1 15855 7128 6390 305938 SR 1 26055 14256 6390 582985 5 66855 42761 6390 1689430 30 321848 220954 6390 8614565 Table2
ComparisonbetweendifferentUCformulations.
UC SC# Scheduling Out-of-sampleevaluation
FxdCost[k$] CPUTime[s] AvgTC[k$] WorstTC[k$] Viol# WCurt%
SO 1 54.02 14.9 1023.88 5793.01 698 0.40 5 65.09 38.0 805.63 3075.12 171 0.22 10 63.20 184.8 782.10 1971.05 86 0.23 15 62.89 277.0 782.16 1972.61 86 0.25 20 66.33 266.8 780.13 1973.04 86 0.16 25 65.04 460.8 772.68 1577.55 50 0.20 30 64.52 980.6 763.82 875.10 0 0.16 50 64.56 1971.2 763.79 875.05 0 0.16 RO 1 59.22 13.8 813.16 961.93 0 1.05 ROB 1 63.22 184.4 816.78 959.83 0 1.11 SR 1 60.13 20.2 769.76 891.64 0 0.34 5 66.48 50.9 765.86 883.49 0 0.21 10 62.08 161.9 765.41 884.51 0 0.21 15 62.91 139.0 764.88 882.15 0 0.22 20 65.65 289.6 764.47 876.23 0 0.14 25 64.73 554.8 764.43 880.02 0 0.15 30 68.82 570.9 764.42 879.2 0 0.14 50 66.98 1932.5 764.17 879.71 0 0.17 SRB 1 62.95 239.4 771.95 1190.16 18 0.36 5 65.94 233.9 765.34 883.30 0 0.17 10 63.74 654.8 765.23 882.41 0 0.17 15 65.01 940.9 765.04 880.61 0 0.14 20 65.61 1388.4 764.76 880.08 0 0.15 25 66.64 2557.4 764.56 877.53 0 0.14 30 66.34 3359.3 764.17 880.55 0 0.17 50 66.47 8705.4 763.92 880.33 0 0.17
Fig.2. Differentlevelsof˛forSRandSRBusing5scenarios.Uppergraph: Aver-agetotalcostsoverthe1000out-of-samplescenarios.Lowergraph:Averagewind curtailmentofthe1000scenarios.
theotherscenariosintheobjectivefunctionoftheSRandSRB for-mulations.Fig.2presentstheresultsofSRandSRBusing5scenarios
forobjectiveweight˛rangingfrom0(allweighttorobust)to1(all weighttostochastic).
Weobservethatas˛increases,theaveragetotalcostsdecreases. Thisisbecausetheproblembecomeslessconservativewhenthe robustpartofSRhasasmallerweight.Thesamebehaviorwas pre-viouslyobservedin[7].Similarly,windcurtailmentdecreasesas ˛increases,andbothSRandSRBachievetheirlowestvaluewhen ˛=0.9.Wealsoobservethatresultsarenotverysensitiveto˛:in therange[0.2,1]weseeadifferenceintotalcostslowerthan1.5%. ItisimportanttohighlightthatnoneoftheSRandSRBcases presentanyviolations,evenwhentherobustpartisignoredinthe objectivefunction(˛=1).Thisisbecausetheworst-casescenario isstillinthesetofconstraints,henceguaranteeingrobustness.
3.2.4. Windpenalization
Theapproachweputforwardinthispaperisenabledbywind curtailment. In somepowersystems, wind curtailmentmay be undesirable;however, violationsof thedemandbalanceand of transmission-capacitylimitsareevenworse.Theobjectiveofthis experimentistostudytheeffectofdifferentpenaltiesonwind cur-tailment,whichisequivalenttodifferentnegativevaluesofwind bids[1].
Fig.3showstheperformanceofthefollowingformulationsfor differentpenaltiesonwindcurtailment:therobustformulationsRO andROB,thestochastic–robustformulationsRSandRSBwith5 sce-narios,andthestochasticformulationsSO.IngeneralforagivenUC formulation,asthewindcurtailmentpenalizationincreases,wind
Fig.3.Effectofpenalizationsofwindcurtailment.
curtailmentdecreases.Consequently,weneedtoschedulemore resources(upanddownreserves)tobetteraccommodate differ-entwindrealizations.This,inturn,alsoincreasestheaveragetotal costs.
Whenwindcurtailmentisnotpenalized(zeropenalization)the UCapproachesaccommodatetheleastquantityofwind,hencethey alsocarrytheleastquantityofdownwardsreserves.Consequently, SOwith30scenariosreportedviolationsforthecaseinwhichthe penalizationofwindcurtailmentwassettozero.Therefore,the leastquantityofreserveswasinsufficienttoavoidviolations in theout-of-sampleevaluationstage.Surprisingly,whenwind cur-tailmentpenaltyisequaltozero,ROandROBobtaincurtailment andaveragecostssimilartothestochasticandstochastic–robust formulations.
The robust formulations RO and ROB present very simi-lar performance, but RO solves 17× faster. The stochastic and stochastic–robustformulationsalsopresentsimilarperformance toeachother,whereSOpresentslightlylowercostsforpenalties differentthanzero;however,SRsolvestheproblemsmorethan7× fasterthanSRB,andmorethananorderofmagnitudefaster(20×) thanSO.
Fornon-zerowind-curtailmentpenalties,ROandROBpresent thehighestcurtailment.Thisisaresultoftheirconservativepolicy, whichavoidsinfeasiblesolutionsbutcannotguaranteethatahigh windproductionwillbedispatched.ThisalsoleadsROandROBto havehigheraveragetotalcostscomparedtoSR,SRBandSO. 3.3. Levelofconservatism:budgetandboxofuncertainty 3.3.1. Robustformulations,ROvs.ROB
Asdetailedatthebeginningofthissection,thelevelof conser-vatismoftherobustformulationsROandROBcanbecontrolled throughtheparameter ∈ [0,1].Fig.4showstheaveragetotal costsandtheworst-casescenarioofthe1000out-of-sample sce-nariosforROandROBwhenchangingthelevelofconservatism from0to1.Bearinmindthatbothformulationsareexactly equiv-alentwhenthebudgetofuncertaintyis0(nominalscenario)or 1(complete box).RO presentsviolations (non-servedenergy & transmissionlimits)forlevelsofconservatism(box-size)lowerand equalto0.5,whileROBpresentsviolations forlevelsof conser-vatism(budgetofuncertainty)below0.4.
Forlowvaluesof,below0.4,therobustnessofROBdominates presentingfewerviolations,henceloweraveragecostsandalso
Fig.4. Comparisonofrobustformulations.
lowerworst-casescenario(seelowergraphinFig.4).Thisisthe mainadvantageofROBwhichisprotectingagainstaworst-case scenarioevenwhenisverylow,consideringthefulllimitsofthe boxofuncertainty,insteadofreducingtheboxcompletelyasRO does.
Ontheotherhand,forlevelsofconservatismabove0.4,both formulations present similar average and worst-case costs. It is important to highlight that the ideal value of conservatism shouldbelowtoavoidexpensiveoperationcostsduetoan over-conservativepolicybuthighenoughtoavoidviolationscompletely. Thatiswhyweset=0.6forROand=0.5forROB,whicharethe valueswheretheaveragecostsarethelowestwhileavoidingany possibleviolation.
3.3.2. Unifiedstochastic-robustformulations
Apartfromtheprevioustwostochastic–robustformulationsSR andSRB,thefollowingtwoformulationsarealsoimplemented:
SRI:The sameas SR but disregarding constraint (14), hence makingtheset ofconstraints ofthe stochasticpartcompletely independentfromthoseoftherobustpart.
SRBI:ThesameasSRBbutdisregardingconstraint(14). Fig.5comparesthefourunifiedrobust–stochasticformulations. Thehighestaveragecostof thefourstochastic–robust formula-tions(782.29k$forSRIandSRBIat=0.3,seeupperpartofFig.5) islowerthanthelowestaveragecostofROandROB(792.12k$ forROat=0.1,seeupperpartofFig.4).Thisismainlybecause thestochasticpartoftheunifiedformulationshelpstominimize expectedcostbyalsoaccommodatingdifferentscenariosofwind (andnotjustminimalelementsoftheuncertaintyset),whichatthe endresultinlowerwindcurtailmentreducingoperatingcosts,as showninFig.6.Thatis,purerobustplanningleadstoconsiderably morecostlyoperation,evenwhenincludingbudgetofuncertainty, comparedwithstochastic–robust.Alsotherobustnessofthe solu-tionisconsiderablyimprovedwhenaddingtherobustparttothe stochasticpart:noticethatforanygivenlevelofconservatism,the highestworst-costscenariooftheunifiedformulations(lowerpart Fig.5)isalwayslowerthanthoseofthepurerobustformulations (lowerpartFig.4).
Similarlytothepurerobustformulations,allunified formula-tionsSR,SRB,SRIandSRBIpresentsimilaraveragecostsforlevelsof conservatismabove0.5;however,between0.4and0.5,thebudget ofuncertaintymakeSRBandSRBImorerobust.Ontheotherhand,
Fig.5.Comparisonofunifiedrobust–stochasticformulations.
Fig.6.Curtailmentcomparisonofdifferentformulations.
forvaluesbelow0.4,SRandSRBpresentfewerviolationsandlower worst-casecost(lowerpartFig.5)thanSRIandSRBI.
Whenadding(14)totheunifiedformulations,therobustness improvesconsiderablyforlowvalues of.Forhighervalues of (above0.5),SRandSRBareverysimilartoSRIandSRBI, respec-tively,because(14)becomesinactive.However,forvaluesbetween 0.4and 0.5,(14)startstodominateforcingthat theworst-case dispatchwindscenarioremainsbelowallthestochastic scenar-ioseventhoughthisworst-casescenariocouldbeverysimilarto themeanwindvalue.Forvaluesbelow0.4,thelowerelementsof thestochasticscenariosdominateincreasingtherobustnessofthe solutiondespitethelowvaluesof.Thatis,theunifiedformulation protectsitselfusingthelowerenvelopeofthewindscenariosofits ownstochasticpart.
Wesetthelevelsofconservatismto=0.6forSRandto=0.4 forSRB.Thesearethevalueswheretheaveragecostsarethelowest whileavoidinganypossibleviolations.
3.3.3. Computationalperformance
AlthoughthetwoformulationsROandROBsolvearobustUC, ROisasinglelevelMIPformulationcomparedwiththetwo-level MIPofROB,whichalsoincludeabilinearterm,henceROsolves morethan6×fasterthanROBonaverage.Moreover,theslowest caseofRO(=0.5)solvesin39.8s,isfasterthanthefastestcaseof ROB(=0.3),43.3s.Thesecomputationalcomparisonsaremade fordifferentthan0,sinceROandRSarecomputationally equiv-alenttoROBandRSB,respectively,when=0.Similarly,although SRandSRBpresentsimilarperformanceinaveragecosts,wind cur-tailmentandrobustness,SRsolves3.9×fasterwhereitsslowest
case(=0.5),85.9s,isfasterthanthefastestcaseofSRB(=0.2), 181.2s.
Asmentionedabove,therobustnessof thestochastic–robust formulationsimprovesbylinkingthestochasticandrobustparts using (14). However,this extraconstraint also brings an extra computationalburden,resulting,onaverage,in60%extra compu-tationalburdenforSRandSRB.
ItisinterestingtonotethatSRoutperformsROBinallaspects, averagecosts,robustnessandwindcurtailment,despitethevalue chosenfor,whilesimultaneouslySRsolvestheproblemsalways faster(1.6×onaverage).
4. Conclusionsandfuturework
This paper presentsa single-level mixed-integer linear pro-gramming formulation (MIP) for fully adaptive robust unit commitment (UC) with dispatchable wind. We show that this ispossiblebyallowingwindcurtailment andconsideringabox uncertaintysetforwindavailability.Consequently,theproposed formulationsolvesconsiderablyfasterthantraditionalrobust for-mulations,whichusuallyrequiretosolveaMIPbi-levelandbilinear problem.
Moreover,thelevelofconservatismcanbecontrolledby shrink-ing the box of uncertainty, providing similar robustness to a formulationconsideringabudgetofuncertainty,asshowninthe numericalexperiments.
We also show how to link the wind dispatch constraints between thestochastic and the robustparts further increasing therobustness oftheunified stochastic–robustUC formulation. Thisunifiedformulationovercomesthedisadvantagesofboththe stochasticandtherobustUCs.Itreducestheover-conservatismof purerobustUCbecauseanexpectedvalueisnowoptimizedonaset ofscenarios,anditdoesnotrequirethelargenumberofscenariosto guaranteefeasibilityaspurestochasticUCdoes.Moreimportantly, thecomputationalburdenoftheunifiedapproach remainslow, sincetheproposedrobustpartjustaddsasingleextrascenarioto thestochasticUC.Moreover,theproposedunifiedUCoutperforms atraditionalpurerobustUC,includingbudgetofuncertainty,inall aspects,averagecosts,robustnessandwindcurtailment,despite thelevelofconservatismchosen,whilealsosolvingsignificantly faster.
In short, by considering dispatchable wind, solving a stochastic–robust UC becomes computationally feasible and hasagoodperformancewithjustafewscenarios.
Astraightforwardapplicationoftheresultsinthispaperwould betoincorporatetheworst-casesolutiontoanydeterministicUC formulation,e.g.,basedonreserves,therebygreatlyimprovingits robustnesswithoutsignificantlyaffectingitscomputational bur-den.Anotherinterestingdirectionforfurtherresearchistoobtain computationally efficientrobust formulationsconsidering other typesofuncertaintysets,inparticularthosemodelingspacialand temporalcorrelations.
Acknowledgments
TheworkpresentedinthispaperisfundedbytheNetherlands OrganisationforScientificResearch(NWO),aspartofthe Uncer-taintyReductioninSmartEnergySystemsprogram.NWOhadno directinvolvementintheprocessleadingtothispaper.
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