Network and market models: preliminary ... - SmartNet project

SmartTSO-DSOinteractionschemes,marketarchitectures,andICTsolutionsfortheintegrationofancillaryservicesfromdemand-sidemanagementanddistributedgeneration

Networkandmarketmodels:preliminaryreport D2.4

Authors:ArazAshouri(N-SIDE),PeterSels(N-SIDE),GuillaumeLeclercq(N-SIDE),OlivierDevolder(N-SIDE),FrederikGeth(KULeuven/VITO/EnergyVille),ReinhildeD’hulst(VITO/EnergyVille)

DistributionLevel Confidential

ResponsiblePartner N-SIDE

CheckedbyWPleader

Date:26/04/2017MarioDžamarija(DTU)

VerifiedbytheappointedReviewers

Date:15/02/2017RitaMordido&MiguelMarroquin&DavidSánchez(ONE)CarlosMadina(Tecnalia)

ApprovedbyProjectCoordinator Date:26/04/2017GianluigiMigliavacca(RSE)

This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programmeundergrantagreementNo691405

IssueRecord

Planneddeliverydate 31/12/2016

Actualdateofdelivery

Statusandversion

Version Date Author(s) Notes

1.0 20/01/2017 N-SIDEandVITO Finaldraftsubmittedforreview

1.1 25/04/2017 N-SIDEandVITO Reviseddraftsubmittedforreview

Copyright2017SmartNet Page1

AboutSmartNetThe project SmartNet(http://smartnet-project.eu)aims at providing architectures for optimized interaction between

TSOs and DSOs in managing the exchange of information for monitoring, acquiring and operating ancillary services

(frequencycontrol,frequencyrestoration,congestionmanagementandvoltageregulation)bothatlocalandnationallevel,

taking intoaccount theEuropeancontext.Localneeds forancillary services indistribution systemsshouldbeable to co-

existwithsystemneedsforbalancingandcongestionmanagement.Resourceslocatedindistributionsystems,likedemand

sidemanagementanddistributedgeneration,aresupposedtoparticipatetotheprovisionofancillaryservicesbothlocally

andfortheentirepowersysteminthecontextofcompetitiveancillaryservicesmarkets.

WithinSmartNet,answersaresoughtfortothefollowingquestions:

• Whichancillaryservicescouldbeprovidedfromdistributiongridleveltothewholepowersystem?

• HowshouldthecoordinationbetweenTSOsandDSOsbeorganizedtooptimizetheprocessesofprocurement

andactivationofflexibilitybysystemoperators?

• Howshould the architectures of the real-timemarkets (inparticular themarkets for frequency restoration

andcongestionmanagement)beconsequentlyrevised?

• Whatinformationhastobeexchangedbetweensystemoperatorsandhowshouldthecommunication(ICT)

be organized to guarantee observability and control of distributed generation, flexible demand and storage

systems?

Theobjectiveistodevelopanadhocsimulationplatformabletomodelphysicalnetwork,marketandICTinorderto

analyzethreenationalcases(Italy,Denmark,Spain).DifferentTSO-DSOcoordinationschemesarecomparedwithreference

tothreeselectednationalcases(Italian,Danish,Spanish).

Thesimulationplatformisthenscaleduptoafullreplicalab,wheretheperformanceofrealcontrollerdevicesistested.

In addition, three physicalpilots are developed for the same nationalcases testing specific technological solutions

regarding:

• monitoringofgeneratorsindistributionnetworkswhileenablingthemtoparticipatetofrequencyandvoltage

regulation,

• capabilityofflexibledemandtoprovideancillaryservicesforthesystem(thermalinertiaofindoorswimming

pools,distributedstorageofbasestationsfortelecommunication).

Partners


ListofAbbreviationsandAcronymsAcronym MeaningAC AlternatingCurrent

aFRR AutomaticFRR

AS AncillaryService

BFM BranchFlowModel

BIM BusInjectionModel

CHP CombinedHeatandPower

CQCP ConvexQuadratically-ConstrainedProgramming

DC DirectCurrent

DER DistributedEnergyResource

DLMP DistributedLMP

DSO DistributionSystemOperator

ENTSO-E EuropeanNetworkofTransmissionSystemOperatorsforElectricity

FACTS FlexibleAlternatingCurrentTransmissionSystems

FCR FrequencyContainmentReserve

FRR FrequencyRestorationReserve

HVDC High-VoltageDirectCurrent

IIC ImprovedImpedanceCalculation

IP Interior-Point

KCL KirchhoffCurrentLaw

KVL KirchhoffVoltageLaw

LMP LocationalMarginalPrice

LP LinearProgramming

mFRR ManualFRR

MI MixedInteger

MICP MixedIntegerConvexProgramming

MILP MixedIntegerLinearProgramming

MISOCP MixedIntegerSecond-OrderConeProgramming

MV MediumVoltage

NLP NonlinearProgramming

OLTC On-LoadTapChanging

OPF OptimalPowerFlow

PQ ActivePower-ReactivePower

PSD PositiveSemidefinite

PTDF PowerTransferDistributionFactors

PV ActivePower-Voltage

QC Quadratic-Convex


QP QuadraticProgramming

RES RenewableEnergySources

SDP SemidefiniteProgramming

SOC Second-OrderCone

SOCP Second-OrderConeProgramming

SCOPF Security-ConstrainedOptimalPowerFlow

SOS1 SpecialOrderedSettype1

TF Transformer

TSO TransmissionSystemOperator

UPFC UnifiedPowerFlowController


ExecutiveSummaryTheincreasingshareofintermittentrenewableenergysources(RES)intheEuropeanelectricitygridresults

in higher need for flexibility resources providing ancillary services (AS) to compensate for the power

fluctuations.TheimbalancesintroducedbyRESarecausedbyadifferencebetweentheactualgenerationand

the forecasted values. Luckily, the rapid development of grid-connected distributed energy resources (DER)

offersnewsourcesofflexibilityforthepowersystemoperatorsatbothdistributionandtransmissionlevel.

Thesedistributedflexibilitieshavetobeharvested inaproperway, inordertostaycompatiblewiththe

current power system while avoiding the introduction of new system-imbalance, congestions, and voltage

problems.ItisthereforeofkeyimportancetoadaptthecurrentASmarketmechanismstoallowaneffective

andefficientparticipationofDERsforprovidingdifferentancillaryservices.

Inthisreport,tothisend,theproposedIntegratedReservemarketarchitectureispresented,whichaimsat

optimally leveraging the flexibility ofDERs for providing different ancillary services to the transmission grid

(ignoringcross-borderexchanges)aswellaslocalservicestothedistributiongrid,andtoputtheseresources

in competition with more traditional sources of flexibility (e.g. big power plants directly connected to

transmission grid), in a level playing-field. A design for the Integrated Reservemarket alignedwith the use

casesdefinedanddevelopedindeliverableD1.3ofSmartNetproject[1]isprovided,andthelinkismadeto

thecurrentancillaryservicemarketssettingsinthethreepilotcountriesinvestigatedintheSmartNetproject,

namelyDenmark,Italy,andSpain[2].

As shown in the figurebelow, themarket receivesbidsas input (fromaggregatorsor larger commercial

players).Then,themarketiscleared,i.e.quantitiesandpricesaredeterminedaccordingtoanalgorithm(see

blueblock)solvinganoptimizationproblemunderconstraints,possiblytakingintoaccountthenetworkmodel

of the transmission and distribution grid (see orange blocks). In the deliverable, the market products list,

networkmodels,andclearingalgorithmaredescribedseparately.

Ablock-diagramrepresentingvariousblocksinthesystem.

Themarketproducts(orbids)listdefinesthetypesofbidswhichareacceptedbythemarket.Themarket

allowsbothsimplebids(specifyingquantitiesandprices)andcomplexbids(i.e.simplebidsonwhichfurther

constraints (e.g. ramping constraints, exclusive bids) may be applied), in order to map the dynamics of


differentflexibilityresourceswhileexpressingtheconstraintsofassets,aggregators,andoperators.Theresult

isacatalogofproductswhichcanbeoptionally integrated in themarket,according to thedesireofsystem

operatorsandregionalregulations.

TransmissionandDistributionnetworkmodelscanbeusedinthemarketclearingalgorithminorder1)to

makesurethattheirconstraintsarenotviolatedwhenclearingthemarket,butalso2)tosolvecongestionor

local problems. There is a trade-off regarding the complexity of the network model to be included in the

constraintsofmarketclearingalgorithm: itcannotbeverysimplified (otherwise itcreatesabigdemandfor

countertrading because the physical constraints of network will not be taken into account in the market

clearing algorithm) but it cannot be too complex (in order to maintain the algorithm computationally

tractable).Therefore,apropernetworkmodel ischosenbasedonthetype,topology,andsizeofthepower

network. The selected model is a second-order cone programming model (SOCP) model for distribution

networkandadirectcurrent(DC)modelforthetransmissionnetwork.

In the clearing module, methods for finding the optimal values of traded quantity (of energy) and the

resulting price (of the corresponding quantity) are developed. The clearings of quantity and price are not

separable tasks,but rather sequentialproceduresbelonging to the samemodule. Themost straightforward

andmeaningfulmethodsofclearingareexplained.

Furthermore, various coordination schemes for the acquisition of ancillary services between the

transmissionsystemoperator(TSO)andthedistributionsystemoperator(DSO)havebeendefinedin[1]and

werefurtherexploredinthisdeliverableintermsofdetailedmarketdesign.Specifically,thetwomainmarket

structures,namelythecentralizedandthedecentralizedapproachesareinvestigated,asshowninthefigure

below.Usingthegenericfeaturesdevelopedformarketproducts,networkmodelsandmarketclearing,they

areadaptedforeachoftheTSO-DSOcoordinationschemes.

Diagramsshowingcentralizedanddecentralizedmarketarchitectures.

Ontopofthis,therearedifferentrequirementsregardingtheparametersettings,input/outputdata,and

arrangement of modules for each TSO-DSO coordination schemes, as well as additional local optimization

algorithms for thedecentralizedschemes (e.g. theaggregationprocedureof theDSO,whomustcluster the

flexibilityavailableinitslocalmarkettocreateabidfortheTSOmarket).


TableofContents

1IntroductionandObjectives...................................................................................................101.1 Context....................................................................................................................................101.2 Objectiveandscope................................................................................................................121.3 DocumentStructure................................................................................................................13

2MarketDesign.........................................................................................................................142.1 BiddingAspects.......................................................................................................................14

2.1.1 FeedbackfromConsultation:Bidding.................................................................................152.2 TimingAspects........................................................................................................................15

2.2.1 FeedbackfromConsultation:Timing..................................................................................182.3 ClearingandPricingAspects...................................................................................................18

2.3.1 FeedbackfromConsultation:ClearingandPricing.............................................................192.4 NetworkAspects.....................................................................................................................20

2.4.1 FeedbackfromConsultation:Network...............................................................................202.5 ProblemStatement.................................................................................................................21

2.5.1 TheGrid...............................................................................................................................212.5.2 TheSystemSteady-StateConditions...................................................................................212.5.3 TheFlexibilityProviders.......................................................................................................222.5.4 TheMarketObjective..........................................................................................................22

2.6 PotentialsofIntegratedReserve.............................................................................................232.6.1 OptionalServiceExpansion:Co-optimizationwithVoltageRegulation..............................232.6.2 OptionalTimingExpansion:FinerTime-ResolutionforFirstTime-Step..............................24

2.7 CoverageofExistingSituationsinOtherRegions....................................................................242.7.1 ExistingMarketinCalifornia...............................................................................................242.7.2 ExistingASMarketsinPilotCountries.................................................................................252.7.3 RequirementsforExtensionofCurrentAS-Markets............................................................27

3BiddingProcessandMarketProducts....................................................................................283.1 Assumptions............................................................................................................................283.2 TypeofBids.............................................................................................................................29

3.2.1 UNIT-bids............................................................................................................................293.2.2 Q-bids..................................................................................................................................313.2.3 Qt-bids.................................................................................................................................313.2.4 QuantityandPriceConventions..........................................................................................313.2.5 PrimitiveBidDefinition.......................................................................................................323.2.6 DecomposingQ-BidsintoPrimitiveBids.............................................................................343.2.7 DefinitionofAcceptance.....................................................................................................35

3.3 CommitmentofMarketDecisions..........................................................................................353.4 Intra-BidTemporalConstraints...............................................................................................36

3.4.1 Accept-All-Time-Steps-or-NoneConstraint.........................................................................363.4.2 RampingConstraints...........................................................................................................373.4.3 MaximumNumberofActivationsConstraint......................................................................383.4.4 MinimumandMaximumDurationofActivationConstraint..............................................383.4.5 MinimalDelayBetweenTwoActivations............................................................................383.4.6 IntegralConstraint..............................................................................................................38

3.5 Inter-BidLogicalConstraints...................................................................................................393.5.1 ImplicationConstraint.........................................................................................................403.5.2 ExclusiveChoiceConstraint.................................................................................................40


3.5.3 DeferabilityConstraint........................................................................................................413.6 IncludingReactivePowerinBids.............................................................................................41

4GridModelforMarketClearing:Trade-OffsinTractability,Accuracy,andNumericalAspects........................................................................................................................................444.1 Introduction............................................................................................................................444.2 ChallengesinOptimalPowerFlow..........................................................................................44

4.2.1 OptimalPowerFlow............................................................................................................444.2.2 ConvexRelaxationApproaches...........................................................................................454.2.3 ConvexRelaxationofOPF...................................................................................................464.2.4 FurtherBackground............................................................................................................48

4.3 ConvexRelaxationFormulations.............................................................................................484.3.1 CompactFormulationofExtendedSOCPBranchFlowModel............................................49

4.4 ImpactofFormulationandRelaxation....................................................................................504.4.1 NumericalStability..............................................................................................................504.4.2 Scaling.................................................................................................................................504.4.3 Variables.............................................................................................................................514.4.4 ConvexificationSlack...........................................................................................................514.4.5 Penalties..............................................................................................................................514.4.6 Objective.............................................................................................................................52

4.5 ApproximatedOPFFormulations............................................................................................524.6 NetworkSimplifications..........................................................................................................53

4.6.1 ExternalGridEquivalent......................................................................................................534.6.2 NetworkReduction.............................................................................................................54

4.7 Summary.................................................................................................................................56

5ClearingAlgorithmandNodalPricing....................................................................................585.1 BalancingObjective.................................................................................................................595.2 TheMarketObjectiveFunction...............................................................................................59

5.2.1 MinimizationofActivationCost..........................................................................................605.2.2 MaximizationofSocialWelfare..........................................................................................615.2.3 MultiTime-StepObjectiveFunction....................................................................................625.2.4 AvoidingUnnecessaryAcceptance......................................................................................63

5.3 NodalPhysicalConstraints......................................................................................................645.4 PricingApproach.....................................................................................................................645.5 NodalMarginalPricing............................................................................................................65

5.5.1 PricingoverSpaceandTime...............................................................................................665.5.2 Granularity..........................................................................................................................66

5.6 CalculatingPrices....................................................................................................................665.6.1 NodalClearedPricesasDualVariables...............................................................................67

5.7 Alternating-CurrentPowerFlow.............................................................................................685.7.1 DistributionLocationalMarginalPricing............................................................................68

5.8 MarketClearingAlgorithm......................................................................................................69

6TSO-DSOCoordinationSchemes............................................................................................706.1 CentralizedvsDecentralizedMarketArchitectures................................................................706.2 MarketSpecificitiesforEachTSO-DSOCoordinationScheme................................................71

6.2.1 CentralizedASMarket.........................................................................................................726.2.2 LocalASMarket..................................................................................................................736.2.3 SharedBalancingResponsibilityModel..............................................................................746.2.4 CommonTSO-DSOASMarket(Centralized)........................................................................75


6.2.5 CommonTSO-DSOASMarket(Decentralized)....................................................................766.2.6 IntegratedFlexibilityMarket...............................................................................................76

6.3 BiddingfromLocalMarketstoCentralMarket.......................................................................776.4 ParametricLocalMarketClearing...........................................................................................776.5 FinancialSettlementandSharingofCostsBetweenTSOAndDSO........................................78

7Conclusions..............................................................................................................................807.1 Bidding....................................................................................................................................807.2 Network...................................................................................................................................807.3 ClearingandPricing.................................................................................................................807.4 CoordinationSchemes............................................................................................................81

8References...............................................................................................................................82

9Appendices..............................................................................................................................889.1 FormulationofBidsandConstraints.......................................................................................88

9.1.1 Nomenclature:ConstantsandVariables.............................................................................889.1.2 StructureofDataforDefinitionofBids...............................................................................909.1.3 DefinitionofAcceptance.....................................................................................................919.1.4 FormulationofConstraints.................................................................................................92

9.2 NetworkModeling..................................................................................................................979.2.1 Nomenclature.....................................................................................................................979.2.2 Notation..............................................................................................................................989.2.3 PhysicsofPowerFlow.........................................................................................................989.2.4 GridElements......................................................................................................................999.2.5 Kirchhoff’sCircuitLaws.......................................................................................................999.2.6 PowerFlow........................................................................................................................1009.2.7 Nodes................................................................................................................................1009.2.8 ClassicPowerFlowFormulations......................................................................................101

9.3 ExtendedOPFFormulation...................................................................................................1059.3.1 Transformer......................................................................................................................1069.3.2 Switch................................................................................................................................1069.3.3 BranchFlowModelFormulation:DistFlow.......................................................................1079.3.4 BusInjectionModelFormulation......................................................................................1119.3.5 QCrelaxation....................................................................................................................1139.3.6 OPFApproximation:LinearDCFormulation.....................................................................1179.3.7 OPFwithAdditionalModels..............................................................................................1209.3.8 Networkmodelingformulationextra’s.............................................................................122

9.4 Illustrationofpowerflowapproximationsandrelaxations:calculationresults...................1239.4.1 Objective...........................................................................................................................1239.4.2 CaseStudies......................................................................................................................1249.4.3 ToolChain.........................................................................................................................1279.4.4 NumericalComparisonofOPFFormulations....................................................................1289.4.5 VerificationStudyoftheInactiveConstraintsMethod.....................................................1389.4.6 VerificationStudyoftheDCApproximationforTransmissionGridModeling..................141

9.5 MathematicalFormulationofMarketClearing.....................................................................1509.5.1 CouplingofPhysicsandEconomics...................................................................................1509.5.2 AdaptationofPricestoAvoidUnnecessaryAcceptance...................................................1509.5.3 NetworkConstraintsforMarketClearing.........................................................................1529.5.4 AnExampleofNodalMarginalPricing.............................................................................1539.5.5 AnExampleofZonalPricing..............................................................................................154


9.5.6 PricinginthePresenceofBinaryVariables.......................................................................1559.5.7 AlgorithmforCalculatingPriceswithPossibleIndeterminacies.......................................1559.5.8 DCversusACon2-nodeExample......................................................................................1569.5.9 ThreeApproachestoPricingwithACPowerFlow............................................................157


1 IntroductionandObjectives

1.1 Context

TheincreasingshareofDistributedEnergyResources(DERs)inthedistributiongridoffersnewsourcesof

flexibilitywhichcanbe leveragedbybothcommercialmarketparties (e.g.abalanceresponsibleparty (BRP)

tryingtobalanceitsownportfolio)andbysystemoperators(SOs).Inparticular,flexibleDERhavethepotential

to provide local services to the Distribution System Operators (DSOs) and/or ancillary services (AS) to the

Transmission SystemOperators (TSOs). The provision of AS by resources connected to the distribution grid

requires thecoordinationbetweenTSOandDSOs. In [1], severalTSO-DSOcoordination schemeshavebeen

describedandanalyzed.Theyrelyoncentralizedordecentralizedapproacheswheretheflexibilitiesfromthe

differentDERsareleveragedvia localand/orglobal(common)marketsattheDSOandTSOlevels.Asabrief

summary(see[1]formoredetails),therearefiveofthem(seeFigure1.1):

• Centralized AS market model: the TSO operates an AS market for both resources located at

transmissionanddistributionlevel,withnoorlittleinvolvementoftheDSO.

• LocalASmarketmodel: theDSOoperatesa localmarkettosolvedistributiongridproblemsand

thenaggregatesandofferstheremainingflexibilitybidstotheTSOmarket.

• Sharedbalancingresponsibilitymodel:thebalancingresponsibilitiesaredividedbetweenTSOand

DSOsaccordingtoapredefinedschedule,andeachSOorganizeshisownmarket.DERflexibilityis

notaccessibletotheTSO

• CommonTSO-DSOASmarketmodel: TSOandDSOhave thecommonobjective tominimize the

totalcostsneededtosatisfytheirrespectiveservices(ASforTSOandlocalservicesforDSO).This

objectivecanbereachedwithtwovariantsofcoordination

o acommon(centralized)marketforbothTSOandDSOneeds

o adecentralizedarchitecture,butwithadynamic integrationofa localmarketoperated

bytheDSO.

• Integratedflexibilitymarketmodel:thereisonecentralizedmarketopentoTSO/DSOsbutalsoto

commercialmarketparties(e.g.BRPs).

AsshowninFigure1.1,threeancillaryserviceshavebeenfurther investigatedin[1],amongthemultiple

current and future AS described in [2]: 1) frequency control (FCR), 2) balancing and congestion

management(bothattransmissionanddistributionlevels),and3)alsovoltagecontrol(atthetransmission

grid).


Figure1.1Mappingofancillaryservicesandcoordinationschemes(Figuretakenfrom[1])

ItisofkeyimportancetospecifyandadapttheASmarketmechanismstoallowanefficientparticipationof

DERs,aimingatleveraginginanoptimalwaytheflexibilityofDER,puttingitincompetitionwithothersources

offlexibility(e.g.bigpowerplantsconnectedtothetransmissiongrid,traditionallyprovidingancillaryservices

forsystembalancingandotherservices),inalevelplayingfield.

Importantly,eachofthefiveTSO-DSOcoordinationschemesischaracterizedbyaspecificmarketdesign.

Somehigh-levelfeaturesforthemarketdesignforeachcoordinationschemehasbeendescribedin[1]:

• marketscope:specificationwhetherresourcesfromdistributionandtransmissiongridparticipate

ornottothesamemarketsession.

• market organization: set of discrete auctions vs continuousmarket? For different reasons (see

[1]), the choice has beenmade for a set of discrete auctions and continuousmarkets are not

consideredintheremainderofthisdeliverable.

• marketoperation:descriptionofwhorunsthemarket

However,thedetailedmarketdesignneedstobedevelopedforeachTSO-DSOcoordinationscheme.


1.2 Objectiveandscope

The main objective of this deliverable is to further define the AS market design for each TSO-DSO

coordinationscheme,andestablishamathematicalmodelofthemarketclearingalgorithmneededtodefined

thedispatchedquantitiesandpricesoftheASmarket.

Whendesigningaplatformforancillaryservices,twodifferentaspectscanbeconsidered:theprocurement

oftheseancillaryservicesintermsofcapacityandtheactualactivationoftheseservicesintermsofenergy.

WhileactivationofASisbynatureareal-timeprocess,theprocurementofflexibilitycapacitycanbeayearly,

monthly, weekly, daily or even close to real-time. Furthermore, different co-optimization mechanisms

betweencapacityprocurementandactivationofAScanbeconsidered.InthecontextoftheSmartNetproject,

onlytheactivationcomponentisconsidered1:howtodesignareal-timeenergymarketwhereflexibilitiesfrom

DERsandfromassetsconnectedatthetransmissionnetworklevelareactivatedinanoptimalway.

Basedonthischoice,theFCRancillaryservice(seeFigure1.1), isnotfurtherconsideredinthefollowing,

sinceitmakesnosensetohaveareal-timemarketforFCR,sincetheactivationofthisfastfrequencycontrolis

based on local controllers measuring the network frequency, and correcting for any deviation. The same

commentappliestoaFRR,forwhichtheactivationisautomaticandtheupdateofthesetpointstoofrequent

(5-10sec)toallowamarkettodefinetheactivationoftheseproducts.

Also,itwaschosennottoinvestigatethevoltage-controlservice(seeFigure1.1)atthetransmissiongrid,

becauseofthelocalcharacteristicofsuchservice,thatcannotguaranteesufficientmarketliquidity.

Instead, inthisdeliverable,thereal-timeenergymarketdesignisdevelopedtoconsidertheactivationof

balancingandcongestionmanagementservices(forthelatter,bothintransmissionanddistributionancillary

services).Aninnovativemarketdesignisproposed,inorderto:

• leverage flexibilities coming from both transmission grid and distribution grid resources

(geographicalscopeexpansion,comparedtomostcurrentASmarkets,see[1],[2]).

• combinemultipleservices inasinglemarket:providingbalancingservicestothewholesystem,

but also providing congestion management both at the level of distribution and transmission

networks (services scope extension). In particular, this requires to take the transmission and

distributiongridconstraints(ifany)intoaccountinthemarketdesign.

• develop a generic timing extension (timing scope extension), consisting of a higher clearing

frequency(ifneeded)andconsidering(thepossibilityof)arollingtimehorizon(similartowhatis

alreadydoneinCaliforniaforinstance[3]).

1Itisassumedthatthereservehasalreadybeenprocuredand/orthatregulationenforcessomeresourcestoprovideAS.Also,freebidscanalsobeconsideredinthisenergyreal-timemarket.


In this deliverable, this generic market design is called The Integrated Reserve market design. It is

importanttospecifythatwithspecificchoicesofsettingsofthismarket,wecanreproducetheactivationof

existingFRRand/orRRreserves.However,withothersettings, itcanbeamorebenchmarkmodel,perhaps

moreambitious,butstillrealisticinthelongterm.

Thekeychallengeofthisnewmarketdesignistocreateaneconomicallyefficientbutalsocomputationally

tractablemarketclearingprocessfortheIntegratedReservemarket,forthesedifferentTSO-DSOcoordination

schemes,suchthatallpossibleconstraintscanbetakenintoaccount,includingforinstancetransmissionand

distributionnetworkconstraints,aswellasmarketproductsandtheirassociatedconstraints.

1.3 DocumentStructure

This deliverable aims at proposing a generic design for Integrated Reserve market, to be aligned and

adaptedtoeachTSO-DSOcoordinationschemesdevelopedin[1].ThetargetedmarketwillallowTSO/DSOto

leverageflexibilitiesfromDERsinanefficientwayandprovideASbothatthelocalandgloballevel.

InChapter2,wegiveageneraloverviewoftheaspectstobetackledinthemarketdesign.Timing,bidding

andpricingaspectsareaddressed togetherwith thequestionsof thesystemsteady-stateconditionsandof

thenetworkmodeling.Also,optionalextensions (in termsofASandtiming)arediscussed,andwedescribe

how the Integrated market reserve can fit to the current AS market settings in place in the three pilot

countries(Denmark,ItalyandSpain).

InChapter3,wedeepdiveintothebiddingaspects,proposingacatalogueofdifferentmarketproductsto

bidtheflexibilityofDERsinthenewASmarketanddescribingthecorrespondingmathematicalrepresentation

ofthesemarketproducts.

InChapter4,severalmodels(withvaryinglevelofcomplexity)ofthedistributionandtransmissionnetwork

modelareconsidered,andoptimalpowerflowequationsaredescribed.Networkmodelswillbeusedinthe

marketclearingalgorithminordertotakethenetworkconstraintsintoaccount.

InChapter5,wederivethemathematicaloptimizationproblemusedforclearingthemarket.Inparticular,

theacceptanceprocedure,theobjectivefunction,necessarynetworkconstraints,andotherrelatedmaterial

aredescribedindetail,togetherwiththeirmathematicalformulation.Furthermore,weexplainthederivation

of Distributed Locational Marginal Prices (DLMP) using the dual variables of the optimization problem

describedinchapter5andprovideaninterpretationforthesenodalprices.

Finally, Chapter 6 explains how the developed market modules can be used to implement the market

designofthevariousTSO-DSOcoordinationschemesdescribedin[1].


2 MarketDesign

Inthepreviouschapter,weintroducedtheideabehindtheIntegratedReservemarketandexplainedhow

it differentiates from the current solutions for balancing and other ancillary services. In this chapter, we

describe themain aspectsof themarketdesign for the IntegratedReservemarket, namelybidding, timing,

clearing,andnetworkaspects.

2.1 BiddingAspects

TheIntegratedReservemarketisorganizedasaclose-to-real-timemarketforexchangingflexibilities

provided by conventional and renewable resources at transmission and distribution level. Similar to

other energy markets, the traded quantities correspond to electricity injection or off-take. The

flexibilitiesareobtainedbymodifyingtheplannedschedule/commitmentonpreviousmarkets (suchas

day-aheadandintra-daymarkets),orbymodifyingthereal-timebaselineoperation.

Note that, as mentioned in Chapter 1, capacity procurement for the Integrated Reserve is not

consideredintheSmartNetprojectperimeterandthatwefocushereontheactivationdecisionprocess.

The Integrated Reserve market architecture aims at allowing flexible resources coming from both

transmission and distribution networks to compete in the same ancillary services market. In such a

market, what is traded is the service: the sellers are aggregators or sufficiently large flexible assets

connected at transmission or distribution levels; while the buyer is usually a grid operator that in a

centralized architecture usually coincides with the TSO that is active in the control zone, whereas it

couldalsobetheDSOinadecentralizedarchitecture,orevencommercialmarketparties(CMP)inother

proposedcoordinationschemes.

Within a givenmarket session, a flexibility providerwill place a bid in the formof a price-quantity

curve specifying the price asked for different levels of extra supply or consumption of energy (via the

extra injection or off-take of active power). Both positive and negative quantities are, therefore,

consideredtoaccountforupwardanddownwardbalancingenergy.

On top of the economical parameters, a bid on this market includes information on the physical

location of assets: for assets in the distribution grid, to which node of the medium-voltage (MV)

distribution network it is connected, and for assets in the transmission grid, to which node of the

transmissiongrid it isconnected.This locational informationhastobeavailableforthemarketclearing

algorithmtocomputeandlimitthepowerflowsinthenetwork.

If needed by the bidder, a bid on the market can also include some sort of temporal or logical

constraints,forexample:

• Rampingconstraints;


• Maximumnumberofactivationsoveratimehorizon;

• Minimumandmaximumdurationofanactivationrequest(intermsofnumberoftimesteps);

• Minimumdurationbetweentwoactivations;

• andcomplexgenerationbids(e.g.theintegralofenergyinjectionoveragivenperiod).

Adescriptionofbiddingprocessandthedetailedmathematicaldefinitionofthemarketproductsiscarried

outinChapter3.

2.1.1 FeedbackfromConsultation:Bidding

Apublic consultation survey regarding various aspectsofmarketdesignwas carriedoutonline [4] for a

durationof2months.Severalquestionswereaskedand12participatinginstitutes2providedtheirfeedbacks

forthesurvey.Furtherinformationaboutthedistributionofparticipantsisgivenintheappendix.Ingeneral,

the majority of participants believed in a future ancillary service market which is more dynamic, more

extensive,andeconomicallyefficient.

There were 4 questions about bidding aspects of market design, covering topics of allowing complex

products,theresponsiblegridparty,andthetypeofcomplexproducts.

Regardingtheintroductionofcomplexproducts,onlytwoparticipantsopposedthefundamentalidea.Half

oftheanswerssuggestedthatsuchadvancedproducts/bidsmaybeallowed,withtheconcernthat itwould

create a tradeoff between reflecting the market needs, market liquidity, and complexity of computations.

Finally, fourparticipantsapproved the ideaby reasoning that suchproductswill enhance the integrationof

distributedenergyresources(DERs).

Themajorityofparticipantswereleaningtowardstheideathattheaggregator,andnotthemarket,should

handle the dynamic and economic aspects of DER constraints. Only two respondents suggested that the

marketshouldtakecareofsuchconstraints,withoutprovidingastrongargument.

Regarding the list of suggested advanced market products, 5 participants selected products which are

implementedasapartofSmartNetproject.Furthermore,alltheadditionallyrecommendedmarketproducts

(astheanswertoaseparatequestion)werealreadyapartofSmartNetmarketdesign.

2.2 TimingAspects

Somekeyelementsofthemarketdesignliewithintimingaspects.Beforedetailingtheproposal,wefirst

definesometimingnotionsthatwillbeusedextensivelyinthisdocument(asshowninFigure2.1):

2 ACER, Clevergy, Comillas, cyberGRID, EDF, EIMV, ENEL, Epex Spot, European Commission, IBERDROLA, RTE, S&Tconsulting.


Figure2.1Therollingoptimizationlooksaheaduntiltheendofthehorizon.Dispatchandpricesforthecurrenttimesteparemandatory(greensegments),whiletheremainingdispatchdecisionsandinstructionsareadvisory(orangesegments).

• Time step: Considered time granularity for themarket clearing. Activation decisions aremade for

eachtimestepandthebehaviorofthesystemandtheflexibilityassetsinsideeachtimestepareconsidered

constantattheiraveragevalue.Thenotationδtisusedinthisdocumenttorepresenttheindividualtimestep.

• Timehorizon:Timeperiodconsideredforthemarketclearing.Thetimehorizoncanbeequal toor

greater thanthetimestep.However, itwill typicallybeamultipleof thetimestep inorder tomodel inter-

temporalconstraintsandtoclearthemarketwithsomeanticipationonthefuturetimesteps.WedenotebyH

thenumberoftimestepsconsideredinthetimehorizonandbyT,thecorrespondingtimehorizonduration

(i.e.T = H · δt).

• Frequencyofclearing: It isamultipleof timestepsanddefineshowoftenthemarket iscleared. It

canbeequaltothetimestep,equaltothetimehorizonorinbetween.Fromanetworkbalancingperspective,

weneedtoclearthemarketsufficientlyofteninordertotakeintoaccountthelatestupdateddatafromthe

systemstate.Fromanalgorithmicperspective,thefrequencyofclearingneedstobesufficientlylow,sothat

the optimization algorithm used to clear the market can generate (almost) optimal solutions within the

allowedtime.Ifahigherfrequencyisrequired,e.g.,forsecurityreasons,aneconomicallysub-optimalsolution

canbeacceptable.


• Rollingoptimization:Whenthefrequencyofclearing isshorterthanthetimehorizon,asametime

stepwill typicallybe cleared/optimizedmultiple timeswithupdatedand refined informationon the system

stateandDERavailability.Thisrollingoptimizationallowsthemarketoperatortocombinefrequentclearing,

alwaysusing the latest information and forecasts of system imbalance, and anticipationon the future time

steps.

• TemporalData:Therollingoptimizationuseshistorical,real-time,andforecasteddata.Historicaldata

is used to understand the outcome of previous market (day-ahead, intra-day, etc.) as well as previous

activations of the Integrated Reserve market (up to the current time step). Real-time data includes those

providedtothemarketbyexternalsources(suchasmeasurementsofinjectionsandoff-takesateachnodeof

thenetwork,providedbyTSO/DSO).Thesedataareusedtocalculate/estimatethesteady-stateconditionsof

thesystem,aswellastheinitialconditionsforthebeginningofthenexttimestep.Forecasteddata(suchas

forecastedsystemimbalanceateachnode)arecrucialfortherollingoptimizationtofunction.Forthisproject,

weassumethesedatatobeavailablebeforetheclearingprocessateachtimestepandtheircalculationisnot

discussed.

Using a rolling optimization approach implies that the samemarket time-step can be cleared in several

consecutive optimizations. As the time window moves forward and the state of the system evolves, the

activation decisions can be different for the same time step between two different market clearings. This

raisestheissueofthefirmnessfortheactivationdecisions:

• For the first time-step of the considered time horizon, the activation decisions generated by the

marketclearingarebynaturefirmasitisthelasttimethatthistimestepisconsideredinthemarketclearing.

• For theother times stepsof the considered timehorizon, someof thedecisions couldbe firmand

thereforenotavariableanymoreforthefuturemarketclearingsconsideringthistimestep(e.g.theactivation

of an asset with a constraint of minimum duration of activation) while other market decisions could be

indicativebutnotbinding(e.g.theexactquantityofflexibilityaskedtothisassetfortheconsideredtimestep).

Itisalsoimportanttonotethatifweonlyproposesomevaluesforthelengthoftimestep,timehorizon,

andfrequencyofclearing,buttheyaremodelparametersthatcanbechangedattheimplementationstage.

The parameters clearing frequency and time horizon can be adjusted in the course of the project

activities based on the size/complexity of the considered network, the number and complexity of the

bidsonthemarketandontheperformanceoftheclearingcomputer:

• Theclearing frequency ismainlyacompromisebetweenthedesire todealeffectivelywith the

minute-by-minute fluctuation of renewable resources and the computational burden imposed by the

highcomplexityoftheproblem.

• Asforthedurationofthehorizon,considerthatitcouldbebeneficialtoactivateanassetnow,

butitcouldbealsobeneficialtoactivateitlater.Iftherearenotemporalconstraints,wecouldevendo


both.However, ifwe imaginethatanassetcouldonlybeactivatedonceperhour, themarketneedsto

decidewhethertoactivateitnow,inthenexttimestep,orlateron.Forthis,weneedawindowonthe

futurebeyondthecurrentclearingtimestepandarollingoptimization.Ideally,thehorizonhastobeof

the order of the time duration associatedwith these constraints. Note that in our system, assets can

haveramp-up/downlimitationsormanyothertemporalconstraints.Furthermore,thehorizonshouldbe

selected considering the accuracy limitation of forecasts. This means that if, for example, the latest

“acceptable”forecast isavailableforonehourahead,thehorizonshouldnotbeselectedastwohours.

Vice versa, if the horizon is shorter that the latest acceptable forecast, the potentials of rolling

optimizationwillnotbeharvested.

The Integrated Reserve market relies on the fact that we will be able to simultaneously identify,

discriminate, leverage, commit and dispatch DERs based on their cost but also based on their

operationalconstraints.Inparticular,howfastorslowacertainDERcanreacttoactivationwillhavean

impactonwhetheritisdispatchedornot.

2.2.1 FeedbackfromConsultation:Timing

In this sectionof thesurvey,4questionswereasked.Thequestioncovered the topicsofmore frequent

marketclearing(closertoreal-time),rolling-horizonoptimization,andimbalanceforecasting.

Therewas no opinion against amore frequentmarket clearing,which supported this fundamental idea

implemented in SmartNet as a 5-minute auction market. Furthermore, more than half of the participants

confirmedthat it is feasibletomanagethe inputsandoutputswithin5minutes.Somewereskepticalabout

suchafastclearing-scheme,arguingthatthesystemisnotreadyforthischangeyet,orclaimingthatitwillnot

createanybenefitsforthemarket.Inaverage,theanswersweremoresupportivethanopposing.

Regardingtherolling-horizonoptimization,wereceiveda100%ofpositivefeedback,whichwasveryheart-

warming. However, some participants had concerns about challenges with imbalance forecasting and

transparencyofdecisions.

Thepossibilityofimbalanceforecastingwasanotherquestionedtopic.Theanswersinthistopicweresplit.

Whileafirstgroupbelievedthat it ispossibleandevenalreadyexisting,thesecondgroupdoubtedthat it is

technologicallyfeasibleand/orthatitwillbesoonavailable.Somebelievedthatimbalanceforecastingwillbe

availableforthetransmissiongrid,butnotfordistributionnodes. Ingeneral, itseemslogicaltoassumethat

suchforecastswillbeavailableinnearfutureandthatthemarketwouldbeabletotakeadvantageofthem.

2.3 ClearingandPricingAspects

Theclearingoftheconsideredmarketisdonebytakingintoaccountthenetworkconstraintsbothat

thetransmissionandthedistributionlevels.


Moreprecisely, themarket-clearing algorithmembeds aDC-power flowmodel for the transmission

network and an approximated AC-power flowmodel (convexification of the AC power flow equations)

forthedistributiongridthatincludescomplexvoltagesandpowers.ThesearedetailedinChapter4.

Theinputstotheclearingandpricingalgorithmare:

• Theforecastedinjectionatthedifferentnodesofthenetworkpriortheactivationdecisions(for

thedifferenttimestepsofthetimehorizon),

• Thedifferentavailableflexibilitybidsprovidedonthemarket(forthedifferenttimestepsofthe

timehorizon)andtheircorrespondingconstraints,

• Andthetransmissionanddistributionnetworkmodels.

Then the market optimization decides which bids (and therefore, which flexible assets) to be

activated.

Themarketassociatesaspecificpriceofupwardanddownwardflexibility(e.g.Europerkilowatt)for

each node at the level of transmission or distribution networks, using the approach of distribution

locationalmarginalpricing(DLMP).

2.3.1 FeedbackfromConsultation:ClearingandPricing

Inthissectionofthesurvey,4questionswereasked.Thetopicswerethetypeanddistributionofprices,as

wellas the typeofobjective function tobeused (amaximizationof socialwelfareversusaminimizationof

activationcosts.)

Firstly,participantswereaskedwhethertheythinkthatmarginalpricingorpay-as-bidshouldbethe

methodologyforpricing.Therewasnotasinglevoteinfavorofpay-as-bid,whichwasinfullagreement

with the current implementation in the SmartNet project. Therewere participants which hesitated to

choose,arguingthattheselectionshouldbemadebasedonthenumberofbids,frequencyofclearing,

etc.

Secondly, the feasibility and benefits of nodal pricing was put into 3 questions. The majority of

respondentsconfirmedthatnodalpricescanbe implementedfromatechnologicalpointofview,while

not many of them think that it is economically advantageous. The reasoning behind is that further

analyses shouldbe carriedoutbeforeopting toonemethodoranother.Also,mostof theparticipants

believethatnodalpricingisacceptablefromaregulatorypointofview.

Finally, regarding the objective function, the majority of participants believe that maximization of

socialwelfare has to be used. This is justifiable from several points of view, including system security,

operationalandinvestmentscosts,etc.However,oneparticipantarguedthatitwill increasethecostof

balancing. In general, we believe that the best objective function to be used is a combination of two


approaches.

2.4 NetworkAspects

Theclearingoftheconsideredmarketwillbebasedonanappropriatemathematicalrepresentationofthe

transmissionanddistributiongrids.Thechallengeistofindanetworkmodelwhichisatthesametime:

• Sufficiently accurate, to identify and solve the balancing and congestion issues and avoid creating

voltageproblems;

• Sufficientlyefficient,tobetractableintheoptimizationproblemwhichissolvedateverytimestep;

and

• Sufficientlysimple,toincludeonlyfunctionsthatareimportanttobuildupthemarketalgorithmand

thetestcases.

The considered approach is to represent the transmission grid with a DC power flow model and the

distributiongridwiththesecond-ordercone(SOC)relaxationoftheACpowerflowmodel.

Asaconsequence,linearconstraintsandSOCconstraintswillbeincludedintheoptimizationproblem.The

detailedequationsofboththetransmissionandthedistributionnetworkwillbepresentedinChapter4ofthis

document.

2.4.1 FeedbackfromConsultation:Network

Inthissectionofthesurvey,5questionswereasked.

First, thegeneralopinionabout the integrationof transmissionanddistributiongrids in themarketwas

questioned.Wereceivedequalnumbersofpositiveandnegativeopinions,withsomeneutralfeedbacks.The

optimistic participants believed that thesemodels can be added gradually, while the pessimistic reviewers

arguedthat itwould introducecomplexitiesandthatweshouldtakethenetworkmodelsoutofthemarket

clearingproblem.

The next 4 questions concerned the availability of data for building the model and for providing

informationonnodalinjectionattransmissionanddistributionlevels.Wereceivedmixedanswers:Ontheone

hand,someparticipantsbelievedthattherequireddataisavailable(atleastfortheTSO)inordertomakethe

modelsandthatinjectionforecastsarepossible.Ontheotherhand,wehaveparticipantswhichbelievedthat

such model data or forecast are difficult to collect and that it is not possible to ask for such data. One

participantclaimedthatimbalanceforecastsarenotevenneeded.Ingeneral,ourtakefromthesequestionsis

thatthepossibilityofhavingaccesstosuchdataishigh,butweshouldalsoconsiderscenarioswhentheyare

notavailable.


2.5 ProblemStatement

As already discussed before, the market we aim to design is a real-time market for activation of

flexibilityresources,connectedbothattransmissionanddistribution levels.Wenowdescribetheexact

optimizationproblemtobesolvedmoreindetail.

2.5.1 TheGrid

Akeyelementof theproblemstatement is the consideredgrid,which consistsofone transmission

grid anddifferent distribution grids connected to the transmission grid. Thenetwork topology is a key

inputfortheproblemstatement.

Figure2.2Exampleofagridwiththreedistributionnetworks(blue)connectedtothetransmissionnetwork(orange).

2.5.2 TheSystemSteady-StateConditions

Foragiventime-step,anotherimportantelementfortheproblemstatementisthesteady-statesituation

of the system i.e. what would be its state during and at the end of the time step in the absence of any

flexibilityactivationby themarket.The informationregarding thesteady-stateconditions isprovided to the

market by the network operators. It is natural that the complexity of calculation and the accuracy of such

information will be impacted by increasing the clearing frequency of the market. As a result, the clearing

frequencyasa“designparameter”hastobechosenconsideringsuchlimitations3.

Thissteady-statesituationisdefinedbyalltheactiveandreactivepowerinjectionsatthedifferentnodes

ofthenetwork(bothtransmissionanddistribution)resulting:

• frompreviousenergymarkets(e.g.day-aheadorintraday),

3Bysteady-statesituation,wedonotmeanthecurrentsituationofthesystembutwhatwouldbethesituationofthe

systemifnoactivationsweredoneinthemarket.

4 1

2

3

11

12

13

21

22

233

1

32

DSO1

DSO2

DSO3


• from theactivationbyTSOand/orDSOofout-of-marketancillary services (e.g. voltage regulation),

and

• fromdeviations(forecasterrors,unpredictableevents)fromtheseprevioustimeframes.

Itisimportanttonotethatthesesteady-stateconditionsareexpressedatthelevelofeachnetworknode

butnotattheresource level.The injectionsateachnodearethemselvesthecombinationsofthe individual

behaviorofmanydifferentconsumersorproducersofelectricity,butitisnotneededfortheflexibilitymarket

toknowtheseindividualbehaviorsandtomodeltheindividualsourcesofimbalances.

Itisexpectedthatthesesteady-stateconditionsaresuchthat:

• Thereisapowerimbalancebetweensupplyanddemand,

• There are potentially some instances of congestion problems at the lines of the considered

(distributionandtransmission)grids,andthat

• Therearenopre-existingvoltageproblems,asthecorrespondingancillaryserviceshavealreadybeen

activatedpriortothemarket.

2.5.3 TheFlexibilityProviders

ThedifferentDERsandtransmission-networkconnectedassetscanproposetheirflexibilitiestotheTSOat

thenodelevel,eitherasanindividualassetorinanaggregatedway.Byflexibilitywemeanthepossibilityto

changetheactivepowerinjectionatagivennodeofthenetwork.Evenifreactivepowerisnottradeditself,

theimpactofactivationintermsofmodificationofinjectedreactivepowerhasalsotobeincludedinflexibility

bids(forexampleviaapowerfactororarelationbetweenactiveandreactivepower).

Byactivatingdifferentflexibilitybids,theTSOwillmodifythesituationofthenetworkwiththeobjectiveto

solveefficientlytheimbalanceandcongestionproblemsfacedbythenetworkinitssteady-stateconditions.

2.5.4 TheMarketObjective

Inthecomingchapters,wewilldemonstrateamathematicalformulationofthemarketclearingalgorithm,

includingtheobjectivesandconstraints.Atthispoint,itispossibletodiscussthehigh-levelobjectivesofthe

IntegrateReservemarketaslistedbelow.

Given:

• theconsideredgridmodel,

• thesteady-statesituationofthegrid,and

• thedifferentbidsfromflexibilityprovidersateachnodeofthegrid,

theobjectiveofthemarketoperatoristooptimizethechoiceofbidsactivationsinorder


• tosolvetheimbalanceproblematlocalorgloballevel,

• tosolvethepossiblecongestionproblemsintransmissionanddistributionnetworks(orboth),

• nottocreatenewvoltageproblems,

• andtomaximizethesocialwelfareorminimizethecostofactivations(thesewillbeimplementedintheformofanobjectivefunction).

2.6 PotentialsofIntegratedReserve

In thissection,wedescribeacoupleofoptionalbroadenings in themarketdesignwhichcanextendthe

functionalitiesoftheIntegratedReservemarket,allowingittofitintovariousapplications.

2.6.1 OptionalServiceExpansion:Co-optimizationwithVoltageRegulation

TheIntegratedReservemarketisfocusingonbalancingandcongestionmanagementbutincludesvoltage

constraintsandreactivepowerinformationofdistributionnetworks.Thisallowsthemarkettoensurethatthe

activationintermsofactivepower,(madeinordertobalancethesystemandmanagethecongestion)does

notcreatenewproblemsintermsofreactivepowerandvoltage.

However,itisnotexpectedfromthemarkettostartfromasituationwithimportantvoltageproblemsand

tosolvethem,togetherwithbalancingandcongestionmanagement.Inparticular,ifforeachbid,thereactive

power impact of an active power activation is taken into account (via a simple power factor), the reactive

powerisseenasaside-productandnotsomethingwhichistradedonthemarket.Themarketisnotexpected

to “play” with reactive power and to leverage by itself the flexibility in the P/Q ratio of some assets. The

marketproductsarenotdesignedintermsofreactivepower.Thislimitationisjustifiedbythefactthat,while

the resources providing balancing and congestion management are typically the same, voltage regulation

impliestointegrateinthemarketothertypeofassets(e.g.assetsofferingpurereactivepowerflexibility)and

todefineothertypeofproducts(e.g.productsmodelingtheflexibilityintheP/Qratioofsomeassets).

However,asthenetworkmodelusedintheclearingmechanismtakesintoaccountthevoltageconstraints

anyway,itcouldbetemptingtoextendthescopeoftheIntegratedReservetoincludealsovoltageregulation

intheancillaryservicesitcanprovide.Buttodoso,asexplainedearlier,marketproductsshouldberedefined,

suchthatboththeirflexibilities inactiveandreactivepower, ifany,are leveraged.Somebidscouldevenbe

justpurereactivepowerbids.

Fromaneconomicalperspective,providingalltheancillaryservicestogetherinthesamemarketcouldof

course increase theeconomicefficiencyof theancillaryservicesdecisionsbutat thepriceofmorecomplex

market products and clearing mechanism and of stronger deviation from current organization of ancillary

services.


2.6.2 OptionalTimingExpansion:FinerTime-ResolutionforFirstTime-Step

A flexibility provider will bid a price-quantity curve over the decision period of and potentially, for all

decisionperiodsoverthegiventimehorizon.

Arefinementofthisapproach istoconsider, forthefirststepoftherollingoptimization, inputdataata

shortertimeperiodthanthedecisiontimestep(e.g.a1-minuteinputdatatimestepanda5-minutedecision

timestep).Hence,wecan:

• differentiateproductswith the sameaveragepowerover thedecision time stepbutwithdifferent

andirregularpowerprofilesattheinputdatastep(typicallydifferenttypesofreactivity),and

• ensure that the imbalance is correctlymatched on the input data time step, which is shorter

thanthedecisiontimestep.

Thisallowstoreducetheneedformorereactivereserve,likeaFRR,astheactivationdecisionmadebythe

IntegratedReserveallowsnotonlytobalancethesystemforeachtimestepbutalsoinsideeachtimestep.

However,pleasenotethatinthisrefinedapproach,wedonotchangethemarketclearingfrequency(e.g.

still5minutes).Therefore,evenwith30seconds inputdata timestep, IntegratedReservecannotprovidea

response timeof 30 seconds at any time (once themarket is cleared, thenext clearinghappens5minutes

later).Forthisreason,theIntegratedReservecannotreplaceaFRR,whichisstillneededtoreacttounplanned

eventshappeningbetweentwoclearingeventsoftheIntegratedReserve.

2.7 CoverageofExistingSituationsinOtherRegions

Existing ASmarkets general characteristics have been described for 8 European countries in deliverable

D1.1 of the SmartNet project [2]. Since the scope of the Integrated Reserve market is mainly about the

mFRR/RRreserve,thefocusofthissectionisonthesetypesofmarketsinthethreepilotcountries:Denmark

(area DK1), Italy and Spain. In addition, the already existing advanced real-time market in California is

discussedforcomparison.

2.7.1 ExistingMarketinCalifornia

Forthesakeoftheexampleandcomparison,webrieflydescribeoneofthemostadvancedmarketsthat

arealreadyinoperation,featuringhighclearingfrequencyandarollingwindowoperation.Forthisreason,we

alsolistCalifornia[3].ERCOTinTexasrunsasimilaradvancedmarket.

FocusingontheReal-TimeEconomicDispatch(RTED)process inCalifornia,wenoticethat ithasarolling

timehorizonofupto13time-steps(of5-minuteduration),startingexecutionevery5minutesatthemiddleof

the5-minute timesteps (1.5 timestepsbefore thedispatch).TheRTEDstartsatapproximately7.5minutes

(1.5 time steps) prior to the start of the next dispatch interval and produces an instruction for energy


injection/off-take for thenext dispatch interval and advisory dispatch instructions for asmany as 12 future

dispatchintervalsovertheRTEDoptimizationtimehorizon.

Thismeansthatthankstothispipeliningofinputdataanddecisionmaking,1)thedecisiontimestepcan

bechosenshorterthantheneededworstcaselatencyoftheclearingalgorithm,and2)theoldestinputdata

usedwill thenbeone timestepolder thanwithoutusing this trick.Basedon the feedbackof theSmartNet

simulation platform, this pipelining trick could be considered in the design of SmartNet clearing algorithm,

depending on the decision time steps chosen for the simulations and depending on the market clearing

algorithmperformance.Usingthispipelinefeature,atrade-offmustbemadebetweenthebenefitsoftaking

morefeaturesintoaccount(largernetworks,additionalconstraintsorcostfunctionterms)versusthebenefits

ofusingmorerecentinputdata.

AfurtheraspectoftheCaliforniaRTEDmarketisthatbidsarelimitedtomaximum10segmentsalongthe

quantityaxis.Interestingly,bidsspecifyingbothpositiveandnegativequantitiesareallowed.Non-curtailable

blocksarealsoallowed.Forabidassociatedtoagenerator,minimumandmaximumenergyspecifiedinabid

areonlylimitedtowhatthegeneratorcanprovide.

2.7.2 ExistingASMarketsinPilotCountries

Table2.1describesdetailedmarketsettingsparameters, thatare representative in thecurrentmFRRAS

marketsofthesecountries,andalsosomecomparisonwithaCaliforniareal-timedispatchmarket.

Thisisnecessaryfortworeasons.First,toensurethat,eventhoughwearetryingtoextendthepossibilities

ofexistingenergymarketoperations,wedonotlosesomecapabilitiesandflexibilitiesofcurrentmarkets(last

column of Table 2.1 explains that these settingswill be considered in the description of themore general

Integrated reserve market). Second, to state the ground and then determine the minimum extensions

requiredforthesemarketstobeabletoleveragetheflexibilityoftheDERinthedistributionnetworks(e.g.for

instance,inItaly,DERarenotallowedyettobidontheASmarkets),andrepresentassuchaminimumchange

ASmarketvariant.Decisiontimestepsare60minutesforeachpilotcountry,andnolargerpredictionwindow

is considered.Most countries specifyminimum (max) bid sizes to participate to themarket, aswell as bid

resolution.

InItalyandSpain,bidsareformulatedaseitherbeingupward(only)ordownward(only).So,enforcingthis

as the only allowed bid formulations in the applicable regions has to be catered for in SmartNet. In other

regions,bidswithbothpositiveandnegativeselectablequantitiesshouldbeallowed.

In the three pilot countries, economicmerit order (on price) is the first criterion for bid acceptance. In

Denmark, Spain and Italy, for twobidswith equalmerit order, the renewable energy bids have preference

overtheotherone(althoughforItaly,itappliestoenergymarkets:DA,ID.RESandDERcannotyetbidonAS

markets). Also, when there is no preference between two bids even after the renewability criteria, a bid

representing high efficiency cogeneration is preferred over the other bids. This means that SmartNet bids


should contain information on whether they are representing renewable energy or high-efficiency

cogenerationornot.

Non-curtailablebidsarenotallowed inSpain. InSection3.2, theprovidedcatalogueofmarketproducts

considersthepossibilityofbothcurtailableandnon-curtailableblocks,ForSpain,thecurtailablebidswillbe

excludedintheminimumextensionASmarket,whileareallowed.

4 For Italy, thedecision time step is actually a set-point resolution, and thepredictionwindow-size is the lengthof thewindowforwhichtheset-pointsarecalculated,whichisalsoamarketsessionfrequency.

5Differentvaluesarelistedin[114]forRRandFRRservicesinItalyandSpain.

--------------------------------CurrentPractice-------------------------------- Target

Setting/Parameter Denmark Italy Spain California SmartNet

CurrentTSO-DSOcoordinationscheme

CentralizedASmarket

CentralizedASmarket

CentralizedASmarket

/ AllTSO-DSOcoordinationschemesareconsidered.

DecisionTimeStepSize(minutes) 60 154 60 5 parameter,definedinminutes

PredictionWindowSize(minutes) 60 2404 60 55-65 parameter,definedinminutes

Clearingalgorithmlatencyislargerthanthedecisiontimestep?

N N N Y,1.5time-step [pipelining,parallelism]parameter

BidMinTimebeforethealgorithmstarts(minutes)

45 n.a. 60 2.5 parameter

MinBidsize(MW) 10 1-105 0-105 𝑃()* ofgenerator parameter

MaxBidsize(MW) None n.a. None 𝑃(+, ofgenerator parameter

Productresolution(MW) 0.1 n.a. 0.1 10segments parameter

Bidsmusthaveeitherpositiveornegativequantity(notboth)

Y Y N N Allowingbothsignsofquantityinonebidwhenallowed.

Bidswithnegativequantitiescanonlyhavenegativeprices

N Y N n.a. Allowallcombinationsofquantityandprice(andrestrictwhennecessary)

1.EconomicMeritOrder Y Y Y Y Y

2.Forequallyinterestingbids,renewablesfirst

Y n.a. Y n.a. parameter+needtoknowinbid

3.Forequallyinterestingbids,high-efficiencycogenerationfirst

Y n.a. Y n.a. parameter+needtoknowinbid

Non-curtailableblocksallowed? N n.a. N Y(self-dispatch) parameter

Bid-SelectionObjective MinimumCost MinimumCost

MinimumCost

Minimumcost(demandfixedtoCa-ISOforecast)

MinimumcostsorMaximumSocialWelfare(withlimitedunnecessaryactivations)

RemunerationScheme UnifiedZonalPrice(Payasclear)

PayasBid UnifiedZonalPrice

UnifiedNodalPrice

UniformNodalPrice(PayasClear)

NetworkBalancing Y Y Y Y Y

Congestionmanagementfortransmissionnetwork

Y Y Y Y Y

Congestionmanagementfordistributionnetwork

N N N N Y,preferably

Voltagecontrol N N N N N(Notaggravatingvoltageproblems.Parameter.)

Y:YesN:Non.a.:Datanotavailable


Table2.1Comparisonofmarketsinthethreepilotcountries,CaliforniamarketandwhatisplannedintheframeworkofSmartNetintegratedreserve.

Inallthreepilotcountries,theobjectivefunctionofthemarketclearingalgorithmistominimizeactivation

costs. The remuneration scheme in Italy is of the pay-as-bid-type. Other regions (Denmark and Spain) are

remuneratingaccordingtoauniformmarginalpricescheme.

CurrentancillaryservicesinEuropeancountriesaremainlybalancingandcongestionmanagementonthe

transmissionnetwork.SmartNetwantstotakethisonestepfurthertoalso includecongestionmanagement

fordistributionnetworks.Ontopof that,makingsure thatvoltageproblemsarenotcreatedby themarket

clearingoutputwillbeconsideredaswell.

2.7.3 RequirementsforExtensionofCurrentAS-Markets

GiventhecurrentsituationofmFRR/RRmarketsinthethreepilotcountries,welistthenecessaryrequired

adaptations inorder tominimallyextend themarketsand facilitate the integrationofDER.This is shown in

Table2.2.

Parameter Currentvalue® Valueafteradaptation Comment

Denmark DecisionTimeStepSize(minutes)

60 5-15 MorefrequentclearingallowscapturingthedynamicsofRESandDER.

MinBidsize(MW) 10 1 AllowRESandDERtoparticipateinASmarket.

Italy DecisionTimeStepSize(minutes)


UseUniformmarginalpriceforremunerationofbidders

Pay-as-bid Uniformnodalprice Pay-as-bidcouldbeconsideredasanextstep,butwoulddeeplyimpactthebiddingstrategydefinedinD2.2.

MinBidsize(MW) <10 1 AllowRESandDERtoparticipateinASmarket.

Spain DecisionTimeStepSize(minutes)


MinBidsize(MW) <10 1 AllowRESandDERtoparticipateinASmarket.

Table2.2ListofnecessaryadaptationsofparametervaluesforASmarketsinpilotcountries.


3 BiddingProcessandMarketProducts

The Integrated Reserve market is a market for ancillary services aiming to leverage the flexibility from

assetslocatedbothatthedistributionandthetransmissiongridlevels.Themaingoalofthismarketistosolve

imbalances between energy supply and demand in near real time. A side goal is congestionmanagement,

although it ispossiblethatbalancingandcongestionaredealtwith indifferentmarkets.Flexibilityproviders

bidonthemarketusingthemarketproductsthatareproposed.Themarketoperatorselectsbidsbyrunning

analgorithm,calledthe(market)clearingalgorithm.

While this deliverable takes care of defining the market products and the functionality to clear the

IntegratedReservemarketandtoprice theactivated flexibility in themostefficientway,deliverable2.3 [5]

takescareofaggregatingtheflexibilityofalargeportfolioofDERsandofofferingthisflexibilityonthemarket

usingtheproposedmarketproducts.

Thedefinitionofthemarketproductsisthereforeakeyelementforanefficientcoordinationbetweenthe

aggregatorsandthemarketoperator:

Figure3.1Coordinationbetweentheenduser,theaggregator,andthemarket.

3.1 Assumptions

Thefollowingassumptionsaretakenintoaccountintherestofthischapter:

1. The Integrated Reserve is a discrete/closed-gate market, cleared very frequently (e.g. every 5

minutes)andusingarollingoptimization(e.g.1hour)totakeadvantageofforecasts.

2. A bid can be associated to a specific physical asset (e.g. a large power plant connected to the

transmissiongrid)ortoanaggregatedportfolioofDER.

3. Abid is associated to a specific nodeof the distributionor transmission grid.Multiple bids canbe

presentedatasamenodebymultiplemarketparticipantsbut thesamebidcannotcover resourcescoming

fromdifferentnodesofthenetwork.

4. Amarketparticipantcansubmitbidsfordifferentnodes.

5. A market participant can submit multiple bids for the same node but by default these bids are

considered completely independent. The market can accept all of them, some of them or none of them


without linking constraints. If twobidsof the same issuer are intended tobeexclusively accepted, thenan

exclusive-choiceconstraintwillhavetobeconstructed.

6. The bids are expressed in terms of active power injection/offtake but, associatedwith it, a power

factorwillalsobeprovidedbytheflexibilityproviders.Thisallowsthemarketoperatortomodeltheimpacton

the reactive power side of the different activation decisions, which means that the reactive power is

accountedforinthemodel.Iftheresourcehastheflexibilityforthereactivepower,itcansubmitseveralbids

with the sameactivepowerbutdifferent reactivepowers.Then, theclearingalgorithmwill select themost

suitable bidof this asset. It is important to remember that the goal of IntegratedReservemarket is not to

optimizethereactivepowerflows,buttoavoidcreatingadditionalproblemsrelatedtothat(suchasvoltage

problems).

3.2 TypeofBids

Threemaintypesofbidswithrespecttotheirdimensionaredefined,namelytheUNIT-bid,theQ-bid,andthe

Qt-bid.Eachofthesetypescanbecurtailableornon-curtailable.Acurtailablebidallowsthemarkettoselect

andactivateanyportionofitsquantity(thatisthepowervolume),whileincontraryanon-curtailablebidcan

either be accepted at the given quantity or be rejected. In the rest of this section, each type of the bid is

described.

3.2.1 UNIT-bids

Figure3.2showsasummaryofbidtypes.


Figure3.2Differenttypesofstandardbids(shownonlyfor𝒑, 𝒒 > 𝟎).Standardbidsonlymentionpriceandquantityand

formthe(mandatory)basisofanybid.Threefamiliesofbidsareconsidered:Unit-bids,Q-bids,andQt-bids.

AUNIT-bidisdefinedforaspecifictimestepandissimplydefinedbyapriceandaquantity,asshownin

thefirstrowofFigure3.2.Threesub-typesofUNIT-bidsareconsidered:

1. Non-curtailableUNIT-bid: Thevery simplestbid (seecolumn1ofFigure3.2)hasoneprice forone

quantity,meaningthatthemarketoperatorcaneitheracceptorrejectthetotalenergyquantity𝑄3ataprice

of𝑃3orhigher.Thismarketproductiscallednon-curtailablebecauseeitherthetotalblockhastobeaccepted

or theblock isnotacceptedatall. Themarketcannotaccepta fractionalpartother than0%or100%of it.

Priceandquantitycanbenegativeorpositive.Positivequantitiesstandforenergyinjectionsintothenetwork

andnegativequantities forenergyofftake fromthenetwork.Amarketparticipantcouldsend twoseparate

bids,oneofferinganenergyinjectiontothenetworknodeandanotheroneofferinganenergyofftakefrom

thenetworknode.

2. STEP curtailable UNIT-bid: This UNIT-bid has a single price 𝑃3 = 𝑃4 for a range between two

quantities𝑄3and𝑄4 (seethecolumn2ofFigure3.2).Here,themarketoperatorcanacceptanyquantity in

between𝑄3and𝑄4foratleastthemarginalcostof𝑃3 = 𝑃4(expressedinEUR/MW6).

PWL (piecewise linear) curtailableUNIT-bid: ThisUNIT-bidhas a changingmarginal price𝑃3 to𝑃4 for a

rangebetweentwoMWquantities𝑄3and𝑄4(seethecolumn3ofFigure3.2).Here,themarketoperatorcan

accept any quantity in between 𝑄3 and 𝑄4 for at least the marginal cost on the 𝑃3-𝑃4-line. Detailed

specificationofUNIT-bidsisprovidedintheappendixinSection9.1.2.1.6 Thequantities inMWrepresent theaverageactivepower injection,definedas the ratioof the integralof thepowerprofile/trajectorytotheactivationtimestep(e.g.5minutes).AllthequantityandpriceareexpressedinMWandEUR/MWbutcanbeconvertedinMWhandEUR/MWhusingthefixedtimestepofthemarket.


3.2.2 Q-bids

Q-bidsaresimilartoUNIT-bids,butcanprovidelongervectorsofQandPalongtheQ-axis,asshowninthe

secondrowofFigure3.2.TheobjectiveofaQ-bidistoallowtheCMPtodirectlyexpressanaggregatedcurve

offlexibilitywithvaryingmarginalcost.

SimilartoUNIT-bids,theQ-bidscanbeeithernon-curtailable,stepcurtailableorpiece-wisecurtailableand

theyareassociatedtoaspecifictimestepofthemarketclearing.Also,thepowerfactorisconstantpertime

step,i.e.perQ-bid.DetailedspecificationofQ-bidsisprovidedintheappendixinSection9.1.2.2.

3.2.3 Qt-bids

TheUNIT-bids andQ-bids are associated to a specific time-step and do not allow offering flexibility for

consecutivetimesteps,whichmeansthatthetime-dimensionismissing.

Qt-bids essentially offer aQ-bid for a series of time stepswithin thewindowof optimization and allow

expressinginadvancetheavailabilityofflexibilityforthefuturetimesteps,asdemonstratedinthethirdrow

ofFigure3.2.

Theywill alsobe servingas the standardbidonwhichadditional temporal constraints canbeadded. In

addition, Qt-bids can have a different power factor at each time step. Detailed specification of Q-bids is

providedintheappendixinSection9.1.2.2.

Thenumberofdecision time steps inaQt-bid is called thebid-horizon7. If aQt-bidhasa largerhorizon

thanthetimewindowhorizonconsideredbytheclearingalgorithm,therearetwooptionsforthemarket.The

firstoptionisthatthebidwillnotbeconsideredandthebidderwillbenotifiedofit.Inthiscase,thebidders

arediscouragedtosubmitbidswhichgobeyondthemarkethorizonandtheclearingalgorithmissparedfrom

unnecessarylongbids.Thesecondoptionistocutsuchbidsaccordingtothemarkethorizon.Forexample,if

thehorizonconsistsof12timestepsandaQt-bidcontains13timesteps,themarketconsiderstheQ-bids1to

12forthecurrentmarketclearingandtheQ-bids2to13forthenextmarketclearing. Inthiscase, longbut

feasiblebidsarenotrejected.

3.2.4 QuantityandPriceConventions

In thisdocument,weavoidusing the termsbuyersandsellers.Thereason is that inamarketwithboth

positiveandnegativevaluesforpricesandquantities,thisterminologytendstobeconfusing,sincetheycanbe

definedbasedon thepriceoron thequantity. In this context, a helpful convention is todefine the flowof

things(powerquantitiesinMWorMWh/time-step,andrespectivepricesinEURO/MW)asfollowing:

7Therefore,onecanconsidertheQ-bidasaQt-bidwithonlyonetime-interval(orabidhorizonofone),andsimilarlytheUNIT-bidasaQ-bidwithonlyonequantity-interval.


𝑝 > 0:Thebidderwantstoreceivemoney.

𝑝 < 0:Thebidderwantstopaymoney.

𝑞 > 0:Thebidderinjectsmoreoroff-takeslessenergy(upwardsflexibility).

𝑞 < 0:Thebidderoff-takesmoreorinjectslessenergy(downwardsflexibility).

Ifastandardsupply-demandmarketwouldusethisconvention,ageneratorwouldusuallysubmitbidswith

𝑝, 𝑞 > 0andaconsumerwouldusuallysubmitabidwith𝑝, 𝑞 < 0.However,evenintheEuropeanday-ahead

electricitymarket,bothpositiveandnegativevaluesof𝑝areacceptable8.

Insomelesscommonmarkets,allfourpossiblecombinationsof𝑝and𝑞canappearinbids.Asanexample,

an hour of high wind may cause the network to contain more energy than predicted. In such a scenario,

removingtheextraenergyisconsideredtohaveapositivecontributiontowelfare.Therefore,generatorswill

beintheunusualpositionofbiddingapositive𝑞(injectingenergy)withanegative𝑝(payingmoney).Similarly,

energyconsumerscansubmitbidswithanegative𝑞(off-takingenergy)withapositive𝑝(receivingmoney).

In the IntegratedReservemarket,weare interested in theassetsproviding flexibility inbothdirections.

Therefore, it is not the absolute supply and demand, but the relative increase/reduction in generation or

consumptionthatisimportant.Asanexample,ageneratorreducingitsgenerationlevelisprovidingthesame

typeofflexibilityasaconsumerincreasingitsconsumptiondoes.Theseassetswillbothsubmitbidswithq < 0

and are considered as downwards flexibility. Vice versa, an increase in generation or a decrease in

consumptionisregardedasupwardsflexibilityandisrepresentedwithq > 0bids.Therefore,oneshouldnot

usethephrases‘supply’and‘demand’inthismarket.

3.2.5 PrimitiveBidDefinitionInthissection,wedefinetwoprimitivebidstypesandexplainhowQ-bidsaredecomposedintoprimitive

bids.Thedescriptionismostlyadaptedforcurtailablebids.Fornon-curtailablebids,theonlydifferenceisthat

thequantityisacceptedatvalues0orfull.Therefore,weusethesameshapefordescribingbothcurtailable

andnon-curtailableprimitivebids.

Wedefinetwoprimitivebidtypes:arectangularoneandatrapezoidaloneascanbeseen inFigure3.3.

Both bid types offer a quantity of energy injection or energy off-take𝑞. Of this quantity q, any fraction𝑥

between0and1canbeaccepted(forcurtailablebids).Fornon-curtailablebids,𝑥willbeaBooleanvariable.

8BothpositiveandnegativevaluesofqareallowedwithinasingleQt-bid.


Figure3.3Twoprimitivebidtypes.

Usingtheconvention,therewouldbeonlyoneformatfortheprimarybid: q, p ,wherepisdefinedbythe

twoparametersp3andp4whichalwayshavethesamesign:

p > 0p3, p4 > 0

p < 0p3, p4 < 0

In principle, bothp3 < p4 andp3 > p4 are accepted. Also,q can take positive or negative values. As a

result,fourtypesofprimarybidscanexist:

𝑞 > 0, 𝑝 > 0 Commonbid

𝑞 < 0, 𝑝 < 0 Commonbid

𝑞 < 0, 𝑝 > 0 Uncommonbid

𝑞 > 0, 𝑝 < 0 Uncommonbid

Note that at any moment, we initially consider the scenario that the four types of bids can be

simultaneously offered and outstanding in the market. At every time step, the market clearing algorithm

decidesonwhichbidstobeaccepted.Thecategoriesoftheacceptedbidswillbemainlybasedonthesignof

the global network imbalance. Local exceptions can occur, certainly if the market imbalance situation is

shiftingfromupwardtodownwardorvice-versa.

Forarectangularbid,p3 = p4andwhenafractionxofqisaccepted,itholdsthatthecostf(i.e.moneyto

payorreceive)forthebidderisafunctionoftheparametersp3, q,andxequals

𝑓 𝑝3, 𝑞, 𝑥 = 𝑝3 ⋅ 𝑞 ⋅ 𝑥 (1)

ForaUNIT-bid,thecostisalsodefinedasin(1),withthedifferencethatthevariable𝑥becomesaBoolean,

onlyacceptingvalues0or1.


Foratrapezoidalbid,thecost𝑓iscalculatedas

𝑓 𝑝3, 𝑝4, 𝑞, 𝑥 = 𝑝3 ⋅ 𝑞 ⋅ 𝑥 + 𝑝4 − 𝑝3 ⋅ 𝑥 ⋅ 𝑞 ⋅ 𝑥 2 (2)

whichisthesumofareasoftherectangleandthetrianglecontainedinthetrapezoid.Infact, 𝑝4 − 𝑝3 ⋅ 𝑥

istheheightofthetriangleand𝑞 ⋅ 𝑥isthewidthofit.Thecostin(2)canbeconsideredasthemoregeneral

caseof(1).

3.2.6 DecomposingQ-BidsintoPrimitiveBidsPreviously,aQ-bidwasdefinedasaseriesofrectangularortrapezoidalbidsalongtheq-axis.Theideais

that bidders can submit a Q-bid by specifying a price curve as a piecewise linear function of incremental

quantities.TwoexamplesofthesebidsareshownatthetopofFigure3.4.

Thisexampleisshownforbidswith 𝑞 > 0, 𝑝 > 0 .However,theextensiontootherbidtypesshouldbe

trivial. The leftmulti-blockmarginal cost curve is non-decreasing andnon-continuous. The rightmulti-block

curveisincreasingperblock,butnotasatotalcurve.Inprinciple,themarketonlyrequiresthattheseparate

blockseachhavenon-negative slope.This isadvantageousbecause themarket clearingalgorithm,will then

selecttheblockinorderofincreasingx,sincethisisthemeritorder.Thismeansthat,perblock,nosplitupnor

Booleansconstraintswillhavetobeadded,exceptfornon-curtailableblocks.

Figure3.4Q-biddecompositionintoprimitivebids.Whenablockontheright-sideofanotherblockcontainsapricelowerthanthemaximumpriceoftheblockleftofit,anexplicitlogical‘outofmeritorder’constraint(bluearrows)is

automaticallygenerated.

Contrary to that, for the totality of the Q-bid, decomposition may be necessary. Prior to entering the

marketclearingalgorithm,suchacomplexQ-bidisbrokendowninprimitivebidsasshowninFigure3.4.This

holdsforbothstepwiseandpiecewiselinearQ-bids.


IntheleftcaseofFigure3.4,anyblockhasequalorhigherpricesthananyofthepricespresentinitsleft

neighborblock.Thismeansthattheyarealreadyorderedineconomicmeritorder,whichistheorderthatthe

marketwill select theseblocks.Therefore, there isnoneedtoaddconstraints thatablockwillbeaccepted

beforeitsrightneighborblock.Thisessentiallymeansthatthebidderwantsblockstobeacceptedoutofmerit

order.Thiswillhavetobeindicatedbytheadditionofalogicalconstraintthatforcestheleftneighboringblock

ofablocktobeacceptedbeforetheblockitself.

In the right case in Figure 3.4, this leads to a constraint forcing acceptance of the entire second block

beforethethirdcanbeevenpartiallyacceptedandsimilarlyaconstraintforcingthefirstblockbeingentirely

acceptedbeforeanyfractionofthesecondblockcanbeaccepted.Whenx3,x4,xE,andxFaretheacceptance

fractionsoftheblocksdrawnfromlefttorightinFigure3.4,italwaysholdsthatx3 ≥ x4 ≥ xE ≥ xF,sothese

inequalitiescanbeappliedasconstraintsinthemodel.Theseconstraintsareofthelogicalimplicationtypeas

discussedlateron.

Also,weimpose xE ≥ xEx4 = 1 and x4 ≥ x4

x3 = 1 fortherightcaseofFigure3.4,wherexH

is theminimal resolution of the acceptance fraction 𝑥 of the bid 𝑖, with an exemplary value of 0.01. This

minimal resolution brings advantages both in physical activation-limit of assets and in computational

efficiency.Anexampleofthefirstcaseiswhenabatteryassetisactivatedtoinjectwitha0.0001percentof

thebidquantity,whichmaynotbepossibleduetolimitedprecisionofpowerelectronics.Forthesecondcase,

theminimalresolutionhelpsavoidingstorageofvaluestoahighdecimalprecision,allowingroundedvalues

forpowerflows.Thisisindeedincompliancewiththecurrentmarketregulations.

3.2.7 DefinitionofAcceptance

Tobeabletoformulatetemporalconstraintsregardingthenumberordurationofactivationsoverthetime

horizonofaQt-bid,onehas to formulateaconceptof ‘acceptance’of thatbid foreach timestep.Another

reasontodefinethisistobeabletoconstructlogicalrelationsbetweenacceptanceofQ-bidsandbetweenQt-

bids.Note that, for bidswhich are not subject to these constraints and also only composed ofQ blocks in

economicmeritorder,nosingleBooleanvariableisdefined.

ForUNIT-bids,acceptanceofthebidsimplymeansthatanon-zeroquantityisactivated.Ontheotherside,

thelevelofacceptance(fullyorpartially)referstotheactualquantitywhichisactivated.ForQ-bidsandQt-

bids,theformulationofacceptanceisprovidedintheappendixinSection9.1.3.

3.3 CommitmentofMarketDecisions

The rolling optimization of the Integrated Reserve market, as demonstrated in Figure 2.1, raises the

questionofthecommitmentforthemarketdecisions.

To keep amarket as reactive to system state change as possible,we assume that, by default, only the

decisions for the first time-step are firm. The activation information for the other time steps are not a


commitment requestedby themarketoperator to the flexibilityproviders, but informationabout themost

likely future activation decisions if everything in the system stays as planned. By committing the activation

onlyforthefirsttime-step,themarketoperatorkeepsopenenoughdoorstoadapttheactivationdecisionsin

caseofchangeinthesystemstate.

However,someflexibilityassetshavetemporalconstraintsorneedtobeinformedoftheactivationsome

timeinadvance.Namely,biddingcapacityinmarketattimeNisafunctionoftheclearingprocessinmarketat

N-1. For this reason, optional temporal constraints can be added to the market bids and will lead to

anticipative commitment from the market operator. We describe these optional constraints in the next

sections.

3.4 Intra-BidTemporalConstraints

Wedefineamarketwherebidderscanoptionallysupplysometemporalconstraintswitheachoftheirbids.

Therearemultipletypesoftemporalconstraints,butonlyoneconstraintisallowedperbidandperconstraint

type.Notethat inthissection,allconstraintsaredefinedonbids intheformthebiddersupplies(b ∈ BNO),

andnotonthedecomposedprimitivebids(β ∈ ΠNOST).

Thesimplestbalancingmarketwouldnotkeeptrackofanytemporalconstraintsof itsbidders. Insucha

marketsetup,theaggregatorswouldhavetoensurethemselvesthatallpowerquantitiesandallcombinations

ofselectablepowerquantitiesoverthewholebidtimehorizonarefeasibleforthem.

However, doing somay severely limit the offered feasible region of the quantities bidders can present,

sincesometimes their feasible regionsareconditionalon temporal constraints, like, say rampconstraints. If

biddingaggregatorskeeptrackofthesetemporalconstraintsthemselves,theycannotofferquantitiestothe

marketbeyondthefirsttime-step,sincetheydonotknowwhatdecisionthemarketwilltakeinthefirsttime-

step.Therefore,thiswouldmeanonlybidsforthefirsttime-stepwillbeoffered.

This inturndoesnot leverageatall theadvantagesoftwoaspectsoftheproposedoptimization:(i) the

rolling horizon aspect and (ii) the global optimization across all market participants, each with their own

temporalconstraints.Forthesetworeasons,webelievethatofferingthemorecommontemporalconstraints

as a service in the market clearing algorithm is reasonable. Therefore, the approach taken is to include

complexity on the market clearing algorithm, rather than on the bidding algorithm (see discussion about

consultationinSectionFeedbackfromConsultation:Bidding).

In thenextsections,we introduce the6 intra-bid temporalconstraintswhichareallowed in themarket.

DetailedmathematicalformulationofeachconstraintisshownintheappendixinSection9.1.4.

3.4.1 Accept-All-Time-Steps-or-NoneConstraint


This constraint associates to aQt-bid and ensures that either the bid is accepted for all the time steps

consideredinthebidorthatitisnotacceptedatall.Notethatthedefinitionofacceptanceisdifferentfora

non-curtailable,stepcurtailableorpiecewiselinearcurtailablebid.

Intermsofcommitment,thistypeofproductimpliesananticipativecommitmentbythemarketoperator:

ifduringthemarketclearing,aQt-bidwith“Accept-All-Time-Steps-or-None”constraint isaccepted,thenthe

market operator already commits to activate it for all the time steps considered in the Qt-bid and to stay

consistentwiththatfortheconsecutivesoptimizations(atmostuntiltheendofhorizon).

This constraint allows, for example, to submit a specific load-profile that has to be followed when the

correspondingasset isactivated.Furthermore,thisconstraintcanbeusedwhenaccountingfortherebound

effect.MathematicalformulationofthisconstraintisprovidedintheappendixinSection9.1.4.2.

3.4.2 RampingConstraints

A physical energy generation or consumption asset typically has a ramp-up or/and ramp-down energy

production/consumption profile. After aggregation of many assets this may sometimes disappear, but can

sometimes still bepresent.We formulate the ramp-upand ramp-downconstraintsat the resolutionofone

singlemarket clearing decision step, since that is themost accurate level possible. The constraints can be

formulatedas,forexample:

MAX_RAMP_UP_PER_STEP=3

MAX_RAMP_DOWN_PER_STEP=4.

Thismeansthatthemarketclearingalgorithmwilltakecarethat,fromanydecisionsteptothenextone,

injected/extractedactivepower,inabsolutevalue,doesnotincreasebymorethan3MWandthatpowerdoes

notdecreasebymore than4MW.Note that thebid issuers are awareof the fixeddecision time step and

shouldformulatetheirrampingconstraintsaccordingly.Notethatonecanalsospecifyjustoneoftheramping

constraintsseparatelywithouttheother.

Regardingcommitmentaspects,theactivationdecisionsforQt-bidsareonlyfirmforthefirsttime-stepof

therollingoptimizationbuttherampingconstraintlimitswhatitwillbepossibletodointhenextclearing.

Inaninterestingapplication,rampingconstraintscanbesetto

MAX_RAMP_UP_PER_STEP=0,

MAX_RAMP_DOWN_PER_STEP=0,

whichmeansthatthesameamountofpowershouldbeinjected/off-takenforeverytimestepwithinthe

bid-horizon of the (Qt-) bid. In this case, the ramping constraint intrinsically includes an “Accept-All-Time-

Steps-or-None”criterion,asdescribedbefore.

MathematicalformulationoftherampingconstraintisprovidedintheappendixinSection9.1.4.3.


3.4.3 MaximumNumberofActivationsConstraint

Abidissuermaywanttorestrictthenumberofactivationswithinagivennumberofdecisiontimesteps.A

typicalexamplewouldbetoavoidwearandteareffectonmechanicalswitchesoronincandescentlights.This

canbe specifiedbyMAX_N_ACTIVATIONS_PER_STEPS= (X,Y)whereX represents themaximumnumberof

activationsandYtheconsideredtimehorizonforthebid(thatcanbeshorterthanthemarkettimehorizonif

needed).MathematicalformulationofthisconstraintisprovidedintheappendixinSection9.1.4.4.

3.4.4 MinimumandMaximumDurationofActivationConstraint

Abiddermaywanttorestricttheactivationdurationtoagivenminimumnumberofdecisiontimesteps.

Thiscanbespecifiedby

MIN_N_STEPS_PER_ACTIVATION=5.

SupposethataQt-Bidcontains10decisiontimesteps.Thenifthebidisactivated,ithastobeactivatedfor

at least5subsequenttimestepswithinthese10timesteps.Obviously,MIN_N_STEPS_PER_ACTIVATIONcannot

be larger than the number of time steps contained in the Qt-bid. The default value for

MIN_N_STEPS_PER_ACTIVATIONis0.

Similarly, abiddermaywant to restrict theactivationduration toa givenmaximumnumberofdecision

timesteps.AtypicalexampleofthisconstraintisthecaseofanHVACsystemwherethereisamaximumtime

period that theHVACcanbestoppedbefore thediscomfort level isunacceptable for theusers.Thiscanbe

specifiedby

MAX_N_STEPS_PER_ACTIVATION=10.

Thedefault value forMAX_N_STEPS_PER_ACTIVATION is equal to thenumberofdecision time steps in the

bid. The time a bid is active is defined between the beginning of activation and the end of that same

activation.Notethatasinglebidcanhaveseveralactivationsoveritsbidtimehorizon.

MathematicalformulationoftheseconstraintsisprovidedintheappendixinSection9.1.4.5.

3.4.5 MinimalDelayBetweenTwoActivations

Similar to theprevious constraints, abiddermayneed to specify theminimumtimedelaybetween two

consecutiveactivations,whichcanbeusefulforexamplewhentheassethasacool-downorstart-upduration.

MathematicalformulationofthisconstraintisprovidedintheappendixinSection9.1.4.6.

3.4.6 IntegralConstraint


For some types of flexible assets, we want to express temporal constraints in terms of the integral of

injection/off-takeofactivepowerovertheconsideredtimehorizon.

Thisconstraintcanforcethepower-integraltobeatafixedvalue(e.g.zero)ortobebetweenalowerand

anupperbound,specifiedby:

INTEGRAL_Target=E

or

INTEGRAL_Bound=(L,U)

where𝐸representstheenergytargetintermsofintegralofactivepowerinjection/offtakeoverthetime

horizonandwhere𝐿and𝑈representsthepotentiallowerandupperboundonthisintegral,respectively.

Intermsofcommitment,theactivationdecisionisonlyfirmforthefirsttime-stepbutthisfirstactivation

decisionhasimplicationonwhatcanbedoneforthenexttimestepsandnextclearings,aswekeeptrackof

theintegralconstraint.

For a consumer bid, the integral constraint is intended to formulate that the total energy consumption

shouldremainunchangedoverthebidhorizon.Forsheddableloads(suchasthermostaticloads),itismeantto

assurethattheset-point(andhencethecomfortorschedule)ismaintained.Forshiftableloads(alsoknownas

atomic loads, suchasa factoryproduction lineoranaggregationofwetappliances), it guarantees that the

profileofdemandisfollowingthescheduledpattern.

Formulatingsuchaconstraintonabidreflectsthelimitsonbothupwardanddownwardflexibilityofthe

bidder’sassetswithinthebidhorizon.Additionally,thisconstraintcanbeappliedtogeneratorassets,incase

theownerwouldliketohaveaguaranteedamountofproductionperhorizon(andhenceachieveaminimum

amountofrevenue).


3.5 Inter-BidLogicalConstraints

Intheprevioussection,wehavedescribeddifferentoptionaltemporalconstraintsthatcanbeassociated

withoneQt-bid.Wecanalsoallowlogicalconstraintsthatlinktheacceptanceornon-acceptanceofmultiple

bidssubmittedbythesamebidder.Thismeansthatthelogicalconstraintswouldbeinter-bidconstraints.

These constraints are applied to two or more separate bids and define a logical relation between the

Booleanswhichruletheiracceptance.NotethatacceptanceforaQt-bidisdefinedasacceptanceforanytime

step.Abiddercanchangethisdefinitiontoacceptance-at-all-time-stepsbyspecifying“acceptalltimestepsor

none”perbid,asdiscussedbefore.

Foreverybidthatoccursinalogicalconstraint,aBooleanvariablewillthenautomaticallybemodelledin

themarketclearingmodel.Thisvariablerepresentsthatabid isaccepted,partiallyortotally.Sinceinter-bid


constraintsneedtorefertomultipledifferentbids,weneedasystemtouniquelyidentifythem.Weusethe

(BID_ISSUER_ID,BID_ID)-pairsalreadydiscussedforthatpurpose.

3.5.1 ImplicationConstraint

Itisacommonpracticetoallowthemarkettoselectabidonlyifanotherbidhasbeenacceptedaswell.

Thiscouldforexamplerepresenttwo interrelatedflexibleprocesseswherethesecondonecanbeactivated

onlyifthefirstoneisalsoactivated.

This constraint, called the implication, is modelled by introduction of Boolean variables bool1, bool2

associatedwiththebidsb1andb2(1ifthebidisaccepted,0otherwise)andaddingtheconstraintbool1

bool2. This then means that if b1 is accepted, b2 will also be accepted. Implication constraints are only

possiblebetweenbidssubmittedforthesamenetworknodeandbelongingtothesamebidder.

Sucharelationcouldbeformulatedbyabidissueras,forexample:

(BID_ISSUER_ID=100.101.102.1,IMPL_ACCEPT,BID_ID=22,BID_ID=21)

Thismaybe the case for a primitive bid (within aQ-bid string) that has a lowermarginal cost than the

primitivebidinalowerquantity,ashasbeendiscussedintherighthalfofFigure3.4.Inthiscase,thebidders

do not generate the acceptance implication constraint. Instead, at the bid decomposition stage, these

constraintsareautomaticallygenerated.

Anothersituationwhereaccesstoactivationofasecondbidisconditionalontheactivationofafirstbidis

givenhere.Inaproductionprocess,asecondprocessmaybesimultaneous,butoptionaltoafirstone.Inthat

case, theactivationof thebid representing thesecondprocess implies that thebid representedby the first

processisactivated.

If the implication constraint holds in both directions between bids, one should consider the question if

then, these bids cannot simply be aggregated into one bid instead. Looped implication constraints (i.e. the

acceptanceofbid1subjecttobid2,acceptanceofbid2subjecttobid3,andacceptanceofbid3subjectto

bid1)areallowed,assuchloopsexistintoday’sEuropeanday-aheadelectricitymarket.


3.5.2 ExclusiveChoiceConstraint

Theexclusivechoiceconstraint(alsoknownasEXORconstraint)isusedtoindicateanexclusiveacceptance

betweenasetofbids.Forexample, ifanaggregatorhasanassetthatcouldmovetoanumberofexclusive


statesduringthenexttimestep(suchasanatomicload),thecorrespondingbidsshouldbeaccompaniedbyan

exclusivechoiceconstraint9.

Sucharelationcouldbeformulatedbyabidissueras,forexample:

(BID_ISSUER_ID=100.101.102.1,EXOR_ACCEPT,BID_ID=22,BID_ID=7,BID_ID=8,BID_ID=30)


3.5.3 DeferabilityConstraint

Ifthebidderhastheflexibilityofdeferringitsbid,itcandosobyspecifyingthemaximumdeferability(i.e.

themaximumdelayintheacceptance)intermsofthenumberoftimesteps.Themarketcanalsoleveragethis

additionalflexibilitytodeterminethetime-shiftthatbestresolvesitsimbalanceproblem.

Anexampleisaprocesswithagivenfixedloadprofile(definedusingaQt-bidwithonegivenquantityper

time and an “Accept-All-Time-Steps-or-None” constraint) which has a flexible starting time, allowing the

markettoactivateitforthecurrenttimesteporshiftitifneeded.


3.6 IncludingReactivePowerinBids

Untilnow,weonlydiscussedtheformulationofbidsandconstraintswhicharerelatedtotheflexibilityof

activepower.However, if it is intended that themarketprevents violationof voltage constraints, theasset

havetoprovideinformationabouttheirrangeofflexibilityoninjection/off-takeofreactivepower.Inaddition,

anoptimalmanagementofthereactivepowerwouldallowahigherexploitationoftheactivepowerpotential

ofdistributionresources.

The relationship between the active and reactive power for a certain asset (or a group of aggregated

assets)isdefinedusingthepowerfactor(PF).Thereareseveraloptionstodefinethelimitationofthepower

factor:

1. Fixedpowerfactor

2. TriangularP-Qlimits

3. RectangularP-Qlimits

4. CircularP-Qlimits

Figure3.5providesagraphicalrepresentationoftheseoptions.

9TheExclusiveChoiceconstraintisdefinedpertimestepandisnotautomaticallycarriedontothenexttimestep.


Figure3.5GraphicalrepresentationofvariousP-Qlimitdefinition:a)fixedpowerfactor,b)triangularlimit,c)rectangularlimit,d)circularlimit.

Somepointsofattentionabouttheprovisionofinformationonreactivepowerandthemarketclearingatthe

presenceofreactivepowerbidsarelistedhere.

• Thereactivepowershouldhaveazeroprice.Giventhis, themarketwillhavethefreedomofactivating

reactivepowerresourcestoreachfeasibilitywithoutsacrificingthesocialwelfare.Itisalsocriticalthatthe

reactivepowerisactivatedonlywhenitisnecessarytoavoidviolatingaconstraint:

o Theprovisionofreactivepowerflexibilityatzerocostisconsideredrealisticanditisalready

amandatoryserviceinsomeoftheEuropeancountries.

o The“activationonlyinnecessity”ismeanttoavoidreactive-powerexchangewhenitisnot

necessary.Thiscanbeimplementedbyassigningasmallpricetoreactive-powerexchange.

Thishastobefurtherinvestigated.

• Themarketclearingalgorithmhasthepossibilityofincludingreactivepowerinthebids:

o Rectangularandtriangularcapabilitiescanbeeasilyimplemented.

o Implementing fixed power-factor capabilities are more difficult, as they may still lead to

infeasibilities.However,thisproblemcanbetackledbymodelingthegrid-ownedassetssuch


as capacitor banks, which can be used to compensate for the lack of reactive-power

flexibility.


4 GridModel forMarketClearing:Trade-Offs inTractability,Accuracy,andNumericalAspects

4.1 Introduction

In the SmartNet project, it is the aim to develop amarket clearingmethodology that takes the power

system’sphysicallimitsintoaccount.

In current best practices, the part of the power system participating in the market is considered as a

copperplate,whereastheinteractionwithotherzonesisconsideredthroughanetworkflowapproximation.

To avoid problems due to the limited accuracy of such models, a post-clearing AC power flow check is

performed and re-dispatch canbe performed in case of expectedoperational difficulties. Such actionsmay

lead to inefficiency. Including a more accurate network model in the market clearing will help to avoid

countertradingandtheassociatedcosts,especiallywhenconsideringdistributiongrids.

However, taking thephysicsofpower flow intoaccount,while guaranteeing solution times in real time,

demands pragmatic approaches. Convexification and linear approximation of the equations describing the

physical system are two such approaches to improve computational tractability, which will be detailed

throughoutthischapter.

Generically, including power system’s physics into a market clearing methodology implies solving an

OptimalPowerFlow(OPF)problem.Therefore,thischapterstartswithanoverviewonchallengesinoptimal

powerflowcalculations,andrecentadvancesinOPFrelaxationsandapproximations.

4.2 ChallengesinOptimalPowerFlow

4.2.1 OptimalPowerFlow

Optimal power flow (OPF) has been a topic of interest for 50 years,with Carpentierwriting his seminal

paper in 1962 [6]. He, amongst other things, concluded that the convexity of the problem was to be

researchedfurther.DommelandTinney[7]in1968developedpracticalcalculationmethods,andsolvedcase

studiesof500nodes.AlsacandStottwidened the scopeof thesemethods in1973 through the conceptof

security-constrainedOPF(SCOPF)[8],[9].AreviewofrecentSCOPFliteratureispublishedbyCapitanescu[10].

Ingeneral,OPFencompassesanypowersystemoptimizationproblemwithphysicalpowerflowaspartof

themodel.Thisdoesnotexcludemuch,thereforeOPFingeneralcanbe:multiperiod,security-constrained,DC

orAC,orbasedonapproximations.


Power flow is generally considered as a complex-valued problem. Two formulations of the power flow

equationshavebeenusedwidely:aquadraticone,basedontherectangular-complexnotation,andapolar-

complexone,usingsineandcosinefunctions.Finally,BaranandWudevelopedaninterestingformulationof

thepowerflowequationsin1989[11],whichsimplifiedpowerflowmodelinginradialgrids.Thisformulation

ofthepowerflowequationsiscalledDistFlow.

4.2.2 ConvexRelaxationApproaches

Sincequite awhile it hasbeenunderstood that the truedistinctionbetweeneasy-to-solve andhard-to-

solveproblemsdoesnotalignwithlinearprogramming(LP)versusnonlinearprogramming(NLP)problemsbut

withconvexversusnonconvexoptimization. In theoryand inpractice, largeconvexproblemscanbesolved

reliably (convergence guarantees), quickly (polynomial-time) and to global optimality. With nonconvex

(smooth) optimization, one largely has to choose between solving problems quickly but locally optimal or

globallyoptimalbutslowly.Withthedevelopmentofpracticalsemidefiniteprogramming(SDP)methods[12],

effortshavebeenmadeto leveragetheirexpressivepowertofindstrongSDPapproximationstononconvex

problems.Ahierarchyofcontinuousoptimizationcomplexityclassescanbedevelopedasfollows:

LP ⊂ QP ⊂ CQCP ⊂ SOCP ⊂ SDPabNcde

⊂ NCQCP ⊂ NLPNbNabNcde

Anynonconvexquadratically constrainedoptimizationproblem (NCQCP) canbe reformulated as an SDP

problem with the addition of a nonconvex rank constraint. After removal of the rank constraint (Shor’s

relaxation [13]), a SDP problem remains, which can be solved with SDP solvers. This step of removing

nonconvexconstraintsisreferredtoasthe‘convexrelaxation’.Thepowerflowequationsareeasilywrittenas

asetofnonconvexquadraticequations;therefore,thisstrategycanalsobefollowed.Onetherebyobtainsthe

SDPrelaxationofOPF.

Relaxations,byaprocessofonlyremovingequationsfromthefeasiblesetofanoriginalproblem,provide

strongqualityguaranteesonthesolutionofbothproblems:

• iftheoriginalproblemisfeasible,therelaxedproblemisfeasible;

• iftherelaxedproblemisinfeasible,theoriginalproblemisinfeasible;

• theoptimumof the relaxedproblemwill be a lowerbound (minimization) for theoptimumof the

originalproblem.

Approximations(whichcannotbeshowntoberelaxations),donotprovidesuchguarantees.Nevertheless,

the idea is that they are sufficiently accurate, but only under certain conditions. More generally, convex

relaxationscanbeobtainedforpolynomiallyconstrainedpolynomialprograms,forwhichmomentrelaxation

strategies have been developed [14]. Moment relaxations are tighter than the previously-discussed SDP

relaxations,asthisSDPrelaxationisjustthefirst-ordermomentrelaxation.


AnySDPformulationcanberelaxedfurthertoobtainasecond-orderconeprogramming(SOCP)problem

[15]. For SOCP, state-of-the-art solutionmethods are faster andmore scalable. Furthermore,mixed-integer

(MI) SOCP solvers aredeveloped commerciallywhereas commercialMISDP solvers donot yet exist.Mixed-

integer Linear Programming (MILP) is very commonly used for optimization of energy systems [16].Mixed-

integerconvexprogramming(MICP)referstotheoptimizationofproblemswith integervariables, forwhich

thecontinuousrelaxationisconvex.ThisgeneralizesMILP,MISOCPandMISDP.Generalsolutionmethodsfor

MICP are a topic in current research. For example, the solver Pajarito [17], released recently, uses lifted

polyhedral relaxations to handle the convexnonlinear constraints. It is noted that any convexquadratically

constrainedprogramming(CQCP)orconvexquadraticprogramming(QP)problemcanreformulatedasaSOCP

problem[18].

Finally, Ben-Tal and Nemirovski developed a general (noniterative) approach to reformulate SOCP

problemsasLP,toanarbitraryaccuracy[19].Thispolyhedralrelaxationtechniquecanbeusedtoobtaintrade-

offsbetweenaccuracyandspeedforanySOCPproblem.

4.2.3 ConvexRelaxationofOPF

It can be shown that for radial grids, under balanced power flow, the SDP relaxation and the SOCP

relaxation are equivalent. Related to radial grids, two formulations have been of primary interest.One has

beendevelopedin2006byRabihJabr[20],theotheroneisderiveddirectlyfromtheDistFlowformulationof

WuandBaran [11].The Jabr formulationbelongs to thebus injectionmodel (BIM)class, theWuandBaran

formulationbelongstothebranchflowmodel(BFM)class.Ithasbeenshownthatbothapproachesobtainthe

equivalentsolutionset[18],albeitindifferentvariables.

Ultimately, itwas shownthatSOCPBIMandBFMformulationsareexactundermildconditions in radial

grids [21]. The notion of exactness refers to equivalence between the optimized decision variables of the

originalproblemandthoseoftherelaxedproblem.Therefore,fromeveryoptimalsolutionoftherelaxation,

onemustbeabletorecoveranoptimalsolutiontotheoriginalproblem[21].

Inmeshed grids, SDP relaxations are tighter than SOCP.Nevertheless, they arenot exact, so efforts are

madetoimproveaccuracyfurther.Eventhoughcomputationaltractabilityislimited,anumberofpublications

haveappliedmomentrelaxationmethodologiestoOPF[22]–[24].Inthelimited-sizecasestudiesthatcanbe

solved,typically,solvingthesecondordermomentrelaxationisalreadysufficienttofindtheglobaloptimum.

Jabr furthermore developed approaches to iteratively integrate angle variables [25], transformers and

UPFCs [26], and grid reconfiguration [27].Hijazi et al. extended the reconfiguration formulations to include

radialandmeshedgrids[28],whileprovingboundsonoptimality.

CoffrinpublishedanarchiveofOPFtestcases,calledNESTA[29],whichiscommonlyusedtocomparethe

effectiveness of convex relaxation methods. Authors developing stronger and more tractable relaxation

methodsoftenusethisarchiveforbenchmarkpurposes,e.g.[30].


CoffrinandHijazialsodevelopedaconvexrelaxationmethodbasedonthepolarformulation,convexifying

thesineandcosine functionsdirectly,calledtheQuadraticConvexorQCrelaxation[31], [32].Furthermore,

theauthorsdevelopedtheoreticalinsightsincopperplateandnetworkflowmodelsasrelaxations[33]ofAC

OPF.

Figure4.1:VenndiagramofthefeasiblesetsforvariousrelaxationstoACOPFdiscussedinthisdeliverable(nottoscale).

AcomparisonofthetightnessofdifferentoptimalpowerflowrelaxationsdiscussedisshowninFigure4.1

[31]. The SOCP relaxation – exact undermild conditions in radial grids – dominates the Ben-Tal polyhedral

relaxation(oftheSOCPmodel),whichinturndominatesthenetworkflowrelaxation,whichinturndominates

thecopperplaterelaxation.Ifthefocusistodealwithmeshedgrids,tighterrelaxationscanbeconsidered.For

instance, by adding the QC constraint set to the SOCP relaxation. The SDP relaxation dominates the SOCP

relaxation,butisneitherdominatedbyordominatestheSOCP+QCrelaxation.TheSDP+QCrelaxationisfinally

the tightest relaxation discussed within the scope of this report. Table 4.1 summarizes the discussed

formulations.

Complexity Meshed/Transmission Exact? Radial/Distribution Exact?

AC Nonconvex Balancedpolar Y Balancedpolar Y

Balancedrectangular Y Balancedrectangular Y

Unbalancedrectangular[34]–[36] Y

BalancedDistFlow[37] Y

UnbalancedDistFlow[38] Y

Rankrelaxation[13] SDP Moment[22],[24] Y*

BalancedBIM&BFM[39]–[42] N UnbalancedBIM&BFM[38],[43] Y

PSDrelaxation[15] SOCP BalancedBIM&BFM N BalancedBIM[21]&BFM[20],[44] Y

Ben-Talrelaxation[19] LP BalancedBIM&BFM N BalancedBIM&BFM[45] Y*

Convexhull SOCP QC[31] N

Linearapproximation LP DC N SimplifiedbalancedDistFlow[37] N

LP LPAC[46] SimplifiedunbalancedDistFlow[38] N

Networkflow LP Relaxation[33] N Relaxation[33] N

Copperplate LP Relaxation[33] N Relaxation[33] N


Table4.1Categorizationofformulations.

4.2.4 FurtherBackground

Theupcomingparagraphs discuss a number ofworkswhich summarize and generalize a number of the

previouslydiscussedarticles,aswellasdevelopcasestudiesbasedontheproposedmethods.

An in-depthreviewofOPFmethodologieswasperformedbyFERCandaseriesof reportswaspublished

[47],includingarticlessuchas[48].Morerecently,Capitanescupublishedareviewofdevelopmentsrelatedto

ACOPFcalculation,includingadiscussionontheeffectivenessofconvexificationstrategies[49].

Early2015,JoshuaTaylorpublishedabookonconvexoptimizationmethodsforpowersystems[18].This

wellwrittenworksummarizesthestate-of-the-artofconvexOPFmethodsinasingleformalism,whichmakes

thisbookaneasyentrypointtothefield.

Steven Low developed a convex OPF literature study [50] and published a tutorial-style article to get

startedwith the convex OPF formulations [51]. Lingwen Gan, a former doctoral student of Low, wrote his

doctoraldissertationontheapplicationoftheconvexificationmethodsinradialgrids[52].

GanandLowregularlydiscussedthe‘exactness’oftheconvexformulations[53]–[55].JavadLavaeifocused

hiseffortsontheSDPformulationandpublishedanimplementationofthisproblemforMatlab[56],inwhich

healsodevelopsapenalizationapproach.Healsopublishedsomeinsightsintotheeffectivenessoftheconvex

relaxationsofOPFbasedoninsightinthephysicsunderlyingtheproblem[57].

Anya Castillo, in her doctoral dissertation, developed a number of ACOPF formulations (nonconvex,

convex),methods (global, local) and case studies (storage) [58]. In his dissertation, titled ‘Optimization and

control of power flow in distribution networks’, Masoud Farivar develops convex models for OPF in

distributiongrids[59].Heincludesvolt-varcontrolanddevelopsanumberofcasestudies.

4.3 ConvexRelaxationFormulations

Twomain formulations of convex relaxations of the power flow equations have been of interest in the

scientific literature: the bus injection model (BIM) and the branch flow model (BFM) formulation. These

relaxationshavebeenshowntobeequivalent[50].

TheQuadraticConvexorQCformulation,beingbasedonadifferentrelaxationprocess(combiningconvex

hulls) [30], is not equivalent to those formulations. Therefore, adding any non-dominatedQC constraint to

either the BFM or BIM formulation can result in a tighter overall formulation, as the formulations can be

combined.


A complete derivation of these three relaxations is given in Appendix 9.2, for an extended grid model

includingtransformers,andswitches.Table4.2givesanoverviewofall theparametersrequiredtomodela

networkusingtheextendedformulationsproposed.

Line/cable shuntadmittance,seriesimpedance,ratedvoltagelevel,ratedcurrent,ratedpower

Transformer shuntadmittance,seriesimpedance,voltageratio,ratedvoltagelevels,ratedcurrents,ratedpower

Tap-changingtransformer

shuntadmittance,seriesimpedance,voltageratiorange,ratedvoltagelevels,ratedcurrents,ratedpower

Phaseshifter shuntadmittance,seriesimpedance,voltageratio;shiftratiorange,ratedvoltagelevels,ratedcurrents,ratedpower

Nodes ratedvoltage,operationalvoltagebounds.

Table4.2Parametersforgridelements

4.3.1 CompactFormulationofExtendedSOCPBranchFlowModel

Asareference,thecomplete(compact)formulationoftheSOCPbranchflowmodelrelaxationisgivenbelow.Thisformulationwillbeusedintheprojecttoincludedistributiongridconstraintsinthemarketclearing.Allvaluesandparametersarereal-valued.

Foreachnodei:

0 ≤ (𝑈)()*)² ≤ 𝑢) ≤ (𝑈)(+,)² ≤ 𝑀²(𝑈)k+lmn)² (3)

Foreachlinel=ij:

0 ≤ 𝑎)p()* ≤ 𝑎)p ≤ 𝑎)p(+, (4)

0 ≤ 𝑎p)()* ≤ 𝑎p) ≤ 𝑎p)(+, (5)

0 ≤ 𝛼r()* ≤ 𝛼r ≤ 𝛼r(+, ≤ 1, 𝛼r ∈ 0,1 (6)

𝑃)pE + 𝑄)pE ≤ (𝑆)pk+lmn)², 𝑃p)E + 𝑄p)E ≤ (𝑆p)k+lmn)² (7)

−𝑆)pk+lmn ≤ 𝑃)p ≤ 𝑆)pk+lmn, −𝑆p)k+lmn ≤ 𝑃p) ≤ 𝑆p)k+lmn (8)

−𝑆)pk+lmn ≤ 𝑄)p ≤ 𝑆)pk+lmn, −𝑆p)k+lmn ≤ 𝑄p) ≤ 𝑆p)k+lmn (9)

0 ≤ 𝑖)p,t ≤ 𝑀² ∙ (max 𝐼)pk+lmn, 𝐼p)k+lmn )² (10)

0 ≤ 𝑢)∗ ≤ 𝑀² ∙ (𝑈)pk+lmn)²𝛼r (11)

0 ≤ 𝑢p∗ ≤ 𝑀² ∙ (𝑈p)k+lmn)²𝛼r (12)

0 ≤ (𝑎)p()*)²𝑢)∗ ≤ 𝑢)z ≤ (𝑎)p(+,)²𝑢)∗ ≤ 𝑀² ∙ (𝑈)pk+lmn)² (13)

0 ≤ (𝑎p)()*)²𝑢p∗ ≤ 𝑢pz ≤ (𝑎p)(+,)²𝑢p∗ ≤ 𝑀² ∙ (𝑈p)k+lmn)² (14)


−𝑀E 𝑈)pk+lmnE1 − 𝛼r ≤ 𝑢) − 𝑢)∗ ≤ 𝑀E 𝑈)pk+lmn

E1 − 𝛼r (15)

−𝑀E 𝑈p)k+lmnE1 − 𝛼r ≤ 𝑢p − 𝑢p∗ ≤ 𝑀E 𝑈p)k+lmn

E1 − 𝛼r (16)

𝑃)pE + 𝑄)pE ≤ (𝐼)pk+lmn)²𝑢)𝛼r (17)

𝑃p)E + 𝑄p)E ≤ (𝐼p)k+lmn)²𝑢p𝛼r (18)

𝑃)p + 𝑃p) = 𝑔)p,t|𝑢)z + 𝑟r,t𝑖)p,t + 𝑔p),t|𝑢pz (19)

𝑄)p + 𝑄p) = −𝑏)p,t|𝑢)z + 𝑥r,t𝑖)p,t − 𝑏p),t|𝑢pz (20)

𝑃)p − 𝑔)p,t|𝑢)zE+ 𝑄)p + 𝑏)p,t|𝑢)z

E≤ 𝑖)p,t𝑢)z (21)

𝑢pz − 𝑢)z = −2 𝑟r,t 𝑃)p − 𝑔)p,t|𝑢)z + 𝑥r,t 𝑄)p + 𝑏)p,t|𝑢)z + (𝑟r,tE + 𝑥r,tE )𝑖)p,t (22)

Note that thenodebalanceequations still need tobeadded inorder toobtain the completeOPFproblemformulation.

4.4 ImpactofFormulationandRelaxation

TheSOCPconvexrelaxationsofthepowerflowequations(givenintheappendixinSection9.3.3.2),havea

number on consequences on the possible result of the initial optimal power flow problem. Also, the

formulationsgivenmayperformdifferentlyfromanumericalpointofview.Adiscussionofanumberofthese

consequencesisfoundbelow.

4.4.1 NumericalStability

LingwenGannotesthattheBFM(DistFlow)formulationhasnumericaladvantagesabovetheBIM[38]for

unbalanced power flow formulations (SDP). This suggests that this may also be the case for balanced

formulations,eventhoughtheeffectmayonlybeobservedforlarge-scaleor‘difficult’problemcases.

In theBFM formulation, thecomplexcurrentandvoltagevariablesare replacedby squaredcurrentand

voltage magnitude variables. Note that the accuracy of these squared variables is impacted by the

reformulation process. For instance, for a numerical accuracy of 10-6 of the reformulated problem in the

squaredcurrentandvoltagemagnitude,inpost-processing,anaccuracyofonly 10�� = 10�Fisobtainedfor

thecurrentandvoltagemagnitude.

4.4.2 Scaling


Thescalingofthesquaredvariablesisimportant.Therefore,isbesttomakesurethatthesquaredvariables

are scaled so that they have a numerical value around 1. This is commonly performed as part of the

transformationofSIunitstotheper-unitbase,butmoreaccuratebasescanbeenvisionedifnecessary10.

4.4.3 Variables

Ascurrent finds thepathof least resistance,KCLcanalsobe thoughtofasminimizing thegrid losses in

electriccircuits[60]–[62].InmeshedACgrids,however,thisisnotgenerallyassimpleduetothevoltagephase

angleconstraints(addinguptozero)thatapplyinloops.IntheDistFlowderivation,thevoltageanglevariables

are substituted out in the SOCP formulations. The SOCP formulations introduce phase shifters in any grid

element,furthermoremakingthevoltageangleconstrainttriviallyfeasible.Voltageanglevariablescanonlybe

partiallyincorporatedagainthroughtheQCfeasibleset.

4.4.4 ConvexificationSlack

The SOCP relaxation by definition introduces nonphysical solutions: those forwhich (𝑃)p,t)E + (𝑄)p,t)E is

strictlylessthan(𝑖)p,t ∙ 𝑢)z)inthefeasibleset.Non-negligibleconvexificationslackinvalidatestheoverallresult

fromthephysicalperspective.Slackmaybeobservedwhenitisbeneficialtohaveincreasedlossessomewhere

inthegrid.Slackrepresents‘free’nonphysicallosses,ontopofthenormalphysicalpowertransferlosses.Such

controllable‘losses’,likeadispatchableload,canbeusedtoeasethegridoperationindifficultcircumstances.

For example, in case of overvoltage, such losses actually help you to bring down the voltage, allowing the

solvertofind(mathematical)solutionsthat‘comply’witha‘difficult’overvoltagelimit.Similareffectscanbe

observedwithcurrentandapparentpowerbounds. It isnotedthat thisslackcannotbeobservedwhenthe

operationalenvelopesareabsent.

If the convexified OPF is infeasible, this guarantees that the original problem is infeasible. However,

infeasibility of the original problem does not guarantee that the relaxed problem returns a certificate of

infeasibility. Incertaincases,anumericalbutnonphysical solution is returned,due to the interactionof the

powerflowandtheoperationalenvelope[21]. Inthiscase, infeasibilityoftheunderlyingproblemhastobe

confirmed.

4.4.5 Penalties

TheBIMSDPandSOCPrelaxationareequivalentforradialgrids.Furthermore,aSOCPconvexrelaxationin

theBFMexpressionisobtainedbydroppingarankconstraintinthenon-convexquadraticconstraint.10 For instance, due to static transformer taps (e.g. 1.1pu), theper unit base canbeoff-set (e.g. nominal 20 kV)withrespecttothebestnumericalbase(e.g.closerto22kV).


Toobtain rank-1 solutions in SDPproblems, rankminimizationheuristics [63]–[65]weredeveloped. The

rankconstraintistherebyreplacedwithapenalty.Insomecases,ahiddenrank-1solutionisrecoveredinthe

original(global)optimum,inothers,someactualincreaseinthecostsisobservedduetothepenalization.The

applicationofsuchheuristicisillustratedby[41],[66],[67],forthebalancedBIMSDPformulation.

Therefore, in case the relaxation is not exact, one can choose to penalize as such, to obtain a solution

withoutconvexificationslack.Itisnotknownwhichpenaltymakesiteasiesttorecoveraglobalornear-global

optimal solution.Twoaspects canbe considered: the formulationand scalingof thepenalty, aswell as the

applicationofthepenaltytoaselectionortoallgridelements.Intheimplementation,onehastochooseto

penalizejustthelinesforwhichtheconvexrelaxationisnotexact,versusbydefaultpenalizingalllinesorno

linesatall.

4.4.6 Objective

Thegoalistofindasolutiontocostminimizationproblem,wheretheobjectiveisaconvexcostfunctionin

thedecisionvariables.Forexample,theobjectiveistominimizethecostofdispatchsubjecttothepowerflow

physicsofadistributiongrid. IntheconvexifiedSOCPOPF, theoriginalproblemhastobecombinedwitha

penalization objective. This penalty is there tomake sure that a solution is found on the boundary of the

solution space (and is thus a feasible solution). The convexification penalties defined are convex and the

feasiblesetistheintersectionofconvexcones(polyhedra,spectahedra).Forradialgrids,itisknownthatthe

convexrelaxationisexactandaphysicalsolutioncanberecoveredifitislocatedonthesurfaceoftheSOCs.

Therefore,theeffectiveobjective isaconvexcombinationoftheoriginalobjectiveYcostandthepenalty

YPFconvex.Thereisachallengeinfine-tuningtheweightofthepenaltyintheproblemobjective.Forapenalty,

whichistoolow,theresultisnotphysical.Forapenalty,whichistoohigh,resultsarephysical,butthecosts

aresuboptimal.Forthelowestpossiblepenaltyforwhichtheslackontheconvexificationis(closeto)zero,the

optimumisobtained.Notethat thecostscanneverthelessbeconstant forawiderangeofpenalties,which

means that fine-tuning of the penaltyweightmay not be required inmost cases.Nevertheless, in cases of

overcurrentandovervoltage,itisexpectedthattheobjectivemaybeverysensitivetothepenaltyweight.

Asearchalgorithmcanbeusedto findthispenaltyweight.Oneapproachto findsuchapenaltyweight,

without use of derivatives, is the bisection method, as proposed by [67]. Note that due to the nature of

interiorpoint algorithms, it is hard toobtain solutions very close to (oron) theedgeof the solution space.

Conversely, simplex-based methods naturally find solutions on the edge of the solution space (but only

supportpolyhedralfeasiblesets,thereforetheBen-Talreformulationtechniquemustbeappliedfirst).

4.5 ApproximatedOPFFormulations

Approximations, conversely to relaxations, do not offer strong general quality guarantees, nevertheless,

theyarewidelyusedbecauseoftheirsimplicity.Twocommonapproximationsexist,thelinearized‘DC’OPF,


commonlyusedintransmissionsystemmodellingandthesimplifiedDistFlow,commonlyusedindistribution

systemmodelling. The feasible setsobtained inboth cases are linear, allowing thedirect applicationof the

morewidelyunderstoodLPmodelingtoolsandsolvers.

InSmartNet, the linearized ‘DC’approximationwillbeusedtomodel thephysicalgridconstraintsof the

transmissiongrid. In linearized ‘DC’OPF, voltagemagnitudes are assumed tobe close to1pu.Conduction

lossesareneglected.Furthermore,resistanceisassumedtobesmallcomparedtoreactance.Finally,voltage

angledifferencesovergridelementsareassumedtobesmall.

Note thatpower flow indistributiongridsviolates theseassumptions:voltagemarginsare larger than in

transmissiongrids,andresistanceandreactancearesimilarinmagnitude.

Coffrin and Van Hentenryck discuss the validity of the linearized OPF in [46]. They note that the ‘DC’

approximationworks reasonablywell activepower flow through largely inductive lines.However, the same

quality of approximation does not hold for the reactive power balance, where stronger nonlinearities are

observed.Therefore,itishardtoincludevoltagemagnitudeconstraintsin‘DC’OPF.

In the simplified DistFlow formulation, it is assumed that the series conduction losses are negligible.

Simplified DistFlow can be considered as the outcome of a low-accuracy Ben-Tal relaxation applied to the

convex DistFlow formulation. Losses are neglected, which results in an error on the power flow of a few

percent.As thepowerconsumedbygrid lossesalsoneedtobetransferredto theplacewhere it isactually

lost,thevoltagedropsarealsounderestimated.

Network flow neglects all electrical physics and reduces the gridmodel to lossless power transfer. This

approximatemodel is still switchableand theswitching isnowdirectlyenforcedon thepowerbalance.The

apparent power flow limit is typically linearized by taking the bounds. Ben-Tal or naive polyhedral

approximationscanbeusedtoimproveaccuracyinthemodeloftheapparentpowerflowconstraint.

Finally,powerflowcanbefurthersimplifiedasacopperplate(slackbus,nonetwork),whichcanbealso

consideredasarelaxation[33].Thisisneverthelessoutofscopeduetotheabsenceofanyphysicalvariables

relatedtotheunderlyinggridmodel.

4.6 NetworkSimplifications

Inordertoreducethedimensionsofthemarketclearingproblem(oranyOPF),itcanbeusefultosimplify

thenetwork. Anumberofmethodstoreducethenumberofvariables inthephysicalsystemequationsare

describedbelow.ThesewillmainlybeusedtomodelthetransmissiongridintheSmartNetapproach.

4.6.1 ExternalGridEquivalent

Thenetworkareasexternaltothenetworkareainfocuscanbemodelledindifferentways,accordingto

theavailabledata. Iftheexternalnetworkdataareavailable,detailedrepresentationcouldbeapplied[68]–


[70].However, theexternalnetworkdataareusuallynotavailable. Intheparticularcaseofthisproject, the

transmissionnetworksof theneighboringcountriesarenotavailable, since the related transmission system

operatorsarenotpartnersoftheproject.

AsimplerpossibilityisconsideringtheboundarynodesasPQ,PVorslacknodes.ThePQnodesrepresent

the lines in which the exchange of power is controlled. In some cases, this is true, if the power rate is

controlledbythemeansofFACTSdevicesorHVDClinks.ThePVnodesaresimilartothePQones,butinthese

cases,thereactivepowercanchange.However,PVnodesarenotpresentintheDCmodelofthenetwork.The

slack node usually represents the external networkwith the highest power rating. If some interconnection

linesaretopologicallyveryclose,thentheycanbeconsideredconnectedtothesameslacknode(considering

forexampletheexternalnetworkasacopperplate).InthecaseoftheDCpowerflow,iftheintervention(that

isthevariationofpowerexchangeineachline)oftheexternalnetworkisknown,multipleslackbusseswith

differentweightscouldbeconsidered.

Consideringthesesimplifications,thenecessarydataare:

• Theexchangeofpowerwiththeexternalnetworkineachinterconnection

• Theslacknode,orifnecessarythedistributionofslacknodes

• Iftheexchangesofpoweroftheinterconnectionchange,howthevariationcanbeevaluated

4.6.2 NetworkReduction

The size of the network in focus affects the computational effort needed to solve the power flow

equations.Ifalargeareaofthetransmissionnetworkisaddressed,itcanbenecessarytoconsiderthousands

of transmission lines. All these variables slow down the computational process, possibly leading to

unacceptablecalculationtimes.

However, it is usual that not all lines represent a constraint for themarket clearing, since only a small

percentageofthemreacheshighloadingconditions.Itisthereforepossibletoexcludethesenon-constrained

lines to reduce the computational complexity of the problem. This can be done by directly simplifying the

initialnetworktopologyordiscardingtheinactiveconstraints.

IncaseoftheDCpowerflow,thelatterisaparticularlyeffectiveprocedure,sinceitispossibletodirectly

express the power of each linewith the PTDFmatrix. It is therefore possible to consider only the relevant

constraints,excludingalltheinactiveconstrainsandreducingalsothenumberofvariables.

Finally, thesimplificationof thenetworkor thechoiceof the lineswith inactiveconstraintscanbedone

consideringthehistoricalbehaviorandthenormaloperationalconditionsofthenetwork.Inthissection,four

methodsfornetworkreductionareexamined.

4.6.2.1 ClusteringMethod


The clustering method, which initially reduces the complexity of the network, performs aggregation of

nodeswithsimilarcharacteristics.Thismethodreducesdirectlythenumberofnodesandlinesatthecostof

highererrorsinevaluatingthevoltagesandthepowerflows.Thereareseveralmethodstoclassifyandreduce

thenodes[68],[71]–[75],whichtakeintoaccountdifferentmetricsandboundaryconditions.

Thesemethodsassumetheknowledgeoftheinitialstateofthenetwork,whichisnotnecessarilyavailable.

Anotherproblemwith the applicationof thesemethods is that the informationof the loading rateof each

singlelineislost.Infact,onlytheexchangeofpowerbetweentheclustersofnodesispreserved,butnotthe

powerflowthrougheachline.

Theclusteringmethodscanbeusedtorealizeanautomaticproceduretosimplifythenetwork,performed

withagivenfrequency(e.g.onceperhour).Otherwise,thebasicprincipleexpressedwithinthemethodscan

beusedtosimplifytheinitialnetwork,whichwouldremainequaluntilsomereconfigurationoccurs.

4.6.2.2 InactiveConstraintsMethod

Theinactiveconstraintsmethodsallowtoidentifyandthentodiscardtheinactiveconstraintsbeforethe

powerflowcalculation.Theseinactiveconstraintsarethelimitationsofthetransmissionlinesthatinnoneof

thepossibleoperationalconditionscanexceedthepowerrating.

Thesemethodsarewell fitted for theDCpower flow, inwhich there isadirectconnectionbetweenthe

power of the nodes and the power flow of the lines. Simplified techniques can in some cases achieve a

reductionrateabove85%[76],[77].Othermoreexacttechniquesallowachievinghighreductionratesatthe

costofamorecomplexandheavycomputationalefforttofindthelineswithinactiveconstraints[78],[79].

Themethod “Fast identificationof inactive security constraints in security constrainedunit commitment

problems” [89], has shown good performance and it is applied to the network (see the appendix) in some

example. The method is quite simple and it uses the PTDF matrix. For each transmission line, the largest

elementofthePTDFisidentifiedandthecorrespondinggenerators(inaspecificloadcondition)areincreased

tothemaximumrateinordertoidentifyifthemaximumpowerratecanbereachedforthespecificline.

4.6.2.3 HistoricalData

Athirdwaytoidentifytheinactivelinesistostudythehistoricalprofiles.Iftheloadingrateofonelineis

alwayswell below the limit in all theoperational conditions, it is then very improbable that the linewould

experimentanyprobleminthefuture,unlessaveryimportantreconfigurationoccurs.

4.6.2.4 OperationalExperience


Anothermethod,quitesimilartothepreviousone, istousetheexperienceofthenetworkoperators. In

fact, they know at first hand the network characteristics and can contribute to determine the lines with

inactiveconstraints.

4.6.2.5 LimitingtheAreaoftheNetwork

If thenumberof variables is still toohighafternetwork reductionhasbeenapplied,analternative is to

controlonlythebranchesonacertainzone, forexamplethezoneofthepilot. If themethod isappliedand

onlythelinesinthezoneofthepilotareselected,thenumberoflinesinfocuscanbesignificantlyreduced.

4.7 Summary

Chapter4servesastheoutputoftask2.1oftheSmartNetproject. Thegoalofthistaskwastoexamine

the trade-off of computational tractability versus model accuracy and complexity of integrating a physical

modelofthenetworkinthemarketclearingprocess.

Table4.3summarizesthepropertiesofthepowerflowformulationsbasedonthetheoreticalgroundwork

andsupportedbythenumericalexperiments.

Complexity nature penalty losses under-

voltage

over-

voltage

over-

current

dual

var.

quality tractability optimality algorithm

ClassicAC Nonconvex exact No exact medium medium medium hard high* low local IP

DistFlow SOCP exact

relax

Yes exact easy hard hard hard high high global IP

DistFlow

Ben-Tal

LP IP,

simplex

Simplified

DisfFlow

LP approx.. No neglected hard easy easy easy medium high IP,

simplex

Linearized

‘DC’

LP approx.. No neglected hard hard hard easy low high IP,

simplex

Table4.3Propertiesofproposedformulationsforradialgrids(*Nonlinearsolversusuallyhavegoodtractabilityforpower

systemoptimizationwithonlycontinuousvariables,buttractabilityislowforproblemsincludingintegervariables).

Thefollowingrecommendationsarederivedforincludingnetworkmodelsinmarketclearing:

1. Duetothefactthatintegervariablesarerequiredinthemarketclearingconstraints,theapproach

developed tomodel the power flow physics needs to be tractable in combinationwith integer

variables. Mixed-integer non-convex optimization is not tractable within the scope of the

SmartNetproject.Theapproachesproposedinthischapterdependonconvexapproximationsof

thepower flowphysics.Thisallows todevelopanoverallmodelwhich ismixed-integerconvex,

whichhassuperiortractability.

2. TheSOCPBFM (DistFlow)offers thebestaccuracyandveryhigh computational tractability,but

requires the tuning of a penalty. In almost all realistic cases, optimality and feasibility of this

formulation isequal to theoriginalnon-convexproblem.Furthermore, in somecases, theSOCP


solution (global optimum) is actually superior to those obtained by local non-convex solvers.

Nevertheless, the tuning of the penalty can be cumbersome in difficult situations (overvoltage,

overcurrent).

3. ThesimplifiedDistFlowformulationoffersahigh-qualityapproximationintermsoffeasibilityand

optimality.Themajoradvantagesarethatnopenalty finetuning is required in thismethodand

thattheproblemformulationisimmediatelylinear.Themajordisadvantageisthatnoguarantees

can be provided on feasibility. It is noted that simplified DistFlow is the natural equivalent of

linearized‘DC’OPFfordistributiongrids.

4. ApplyingtheBen-TalreformulationtechniqueselectivelytotheSOCPconstraintsisaninteresting

pathfor fine-tuningaccuracyselectively.Furthermore,Ben-Tal throughout isan interestingpath

toleveragewarm-startcapabilityofthesimplexalgorithm.

5. There is no opportunity for slack in the simplified DistFlow formulation. However, due to the

nature of the approximation, this results in an underestimation of the undervoltages (which

results in uncleared undervoltage problems) and overcompensation of what potentially could

havebeenovervoltages.

6. Conversely,theSOCPcanbeshowntobealwaysexactwithrespecttoundervoltage,buttheslack

introduced by the convexification can be misused to avoid overvoltage in nonphysical ways

(creating fictitious losses brings down the voltage).However, if there are sufficient controllable

resources,theycanbedispatchedtoavoidthis.

7. The‘QC’feasiblesetcanbeaddedtotheBFMformulationinanextstageifimprovedaccuracyis

soughtformeshedgrids.Nevertheless,theBFMgenerallyoffersahigh-qualityapproximationof

OPFinmeshedgrids,evenwithout‘QC’.

Based on these observations and recommendations, the choice was made to include the BFM SOCP

formulationinthemarketclearingprocess,forthedistributiongrids.

Transmission grids will be modeled with the traditional DC approximation, however, where suited for the

foreseensimulationplatform,networksimplificationwillbeappliedtoreducethedimensionsoftheproblem.


5 ClearingAlgorithmandNodalPricing

Once the bids are pre-processed and the networkmodel is updated with themost current values, the

marketisreadytoruntheoptimizationprocessknownasthe“marketclearing”.Thisisthecorealgorithmin

theheartofthemarketwhichisresponsibletocalculatetheoptimalvolumeofpowerexchange(knownasthe

clearedquantity)andtheassociatedvalueofpowerinjectionoroff-takeateachnode(knownasthecleared

price).

Before discussing themarket clearing algorithm, we start by listing several assumptions (either new or

repeatedasareminder)thatareemployedbythemarket.

1. The transmission and distribution networks are defined as connected graphs11. The parameters of

thesegraphsareassumedtoremainconstantduringonetime-stepoftheoptimization.Thisimpliesthatthe

powerswitches(interrupters)ofthenetworkremainintheirinitialstateandthelineparameters(impedance

andreactance)donotchange.Sincedifferent levelofcomplexity isutilizedfortransmissionanddistribution

networkmodels,itshouldbeindicatedtowhichnetworkalinebelongs.

2. Anactor(suchasanaggregator)whichbidsinthemarket,isallowedtoplaceabidatthemarketat

eachtimestep.Theminimuminformationinabidarethequantityofflexibility(intermsof injectionoroff-

take)andthecorrespondingmarginalprice.Abidisassignedtoaspecificnodeofthenetwork.

3. Themarketclearingalgorithmwillonlyprocessthebidsuptotheoptimizationhorizon.Thisimplies

thatthetemporalconstraintshouldbedefinedwithinthishorizon.Furthermore,thesystemimbalance(thatis

the expected different between net injection and off-take over the whole network) is forecasted for the

durationoftheoptimizationhorizon.Thevaluesforresolution(orthedurationofatimestep)andhorizonare

chosenaccordingtomultiplefactors, includingthemaximumtimeallowedforthemarketclearingalgorithm

totake.

4. Themarketclearing isarolling-horizonoptimization,whichclearsovermultipletime-stepsatonce.

Theobjective functionateach timestep isweightedbyadiscount factor,whichdecreasesbymoving from

neartofarfuture.Thisensuresthatthemore-certainnear-futuredecisionsareprivilegedoverthefar-future

decisionswhicharebasedonuncertainforecasts.

5. It canhappen that thedecision for the current time-step is related todecisionsor states from the

past,duetoatemporalconstraint.Inthiscase,itisassumedthatsuchhistoricalinformationisavailable.This

canbeachievedbyeithermakingthattheoptimizationalgorithminternallystoreitspreviousdecisions,orby

requiring the bidder to supply such information explicitly in the bid. For the scope of this report, it is not

necessarytodecideonthemethod,hencebothoptionsareacceptableforthemarket.

11Agraphiscalledconnectedifapathexistsbetweenanyarbitrarypairofnodes.


5.1 BalancingObjective

The primary goal of the Integrated Reserve market is balancing, that is to remove the global power-

imbalanceprobleminthenetwork.Thisgoaltranslatestoapower-balanceequation,whichguaranteesthat

thetotalamountofpowerinjectionandpoweroff-take(consideringtheinitialsystemstateandtheaccepted

bidsofthemarket)sumuptozero.Thisrequiresacouplingofmarketeconomicswiththenetworkphysics:

Thequantity in themarket is the same as thepower (or nodal injection and off-take) in the network. The

mathematicaldescriptionofsuchcouplingisprovidedinSection9.5.1.

It is important todiscuss thetypeofsubmittedbids,accordingtothep − q signconvention. In theday-

aheadenergymarket,aconsumer-bidrepresentsanactorwillingtoreceiveapositivequantityofenergywith

awillingnesstopaymoney,hencea q < 0, p < 0 bid.Astandardproducer-bidrepresentsasupplyofenergy

(quantity)andadesiretobepaid,hencea q > 0, p > 0 .

However, in other markets, such as the Integrated Reserve market, an opposite behavior can

simultaneouslyexist.Someactorsmaybewillingtoprovidedown-regulatingservice(byconsumingmoreor

generatingless)andaskformoneyinreturn,hencea q < 0, p > 0 bid.Anexampleisthecaseofelectricity

storage(suchasthoseinafleetofelectricvehicles)whichwouldgetdegradedbyprovidingsuchanancillary

service,thereforeaskingtobecompensated.Correspondingly,whenanactorhasanexcessofenergy,itmay

bewillingtoinjectit(byreducingconsumptionorincreasinggeneration)whileagreeingtopaymoney,hence

a q > 0, p < 0 bid.Anexamplecanbethesameelectricitystorage(inthepreviousexample),whichwantsto

returntoitsinitialstateofcharge.

5.2 TheMarketObjectiveFunction

The market objective function can be defined in several ways, targeting various economic and

environmental measures. The most common economic objectives (which are utilized in other energy and

ancillarymarkets)areminimizationofactivationcostandmaximizationofsocialwelfare.Theformeraimsat

reducingtheoperatingcostofthegridoperatorwhichisresponsibletosolvegridproblemsusingthemarket

product,whilethelatterisdesignedtoincreasethewelfareforbothsellersandbuyersofflexibility.

Figure5.1Theprice-quantitydiagramshowingtheavailablebidsforup-anddown-regulationoffers.

p[€/kW]

q[kW]

Upbid

Upbid

Upbid

Downbid

Downbid

Downbid


In order to better understand the two variations ofmarket objective, let us stepback and consider the

situationwhereallbidshavearrivedatthemarket,aresorted,andaregroupedbasedonthesignofq,asseen

inFigure5.1.Asareminder,bidswithq > 0correspondtoanup-regulation(hence,calledanupbid),andbids

withq < 0representadown-regulation(therefore,adownbid).

5.2.1 MinimizationofActivationCost

When the market objective is to minimize the activation cost, one group of the bids in Figure 5.1 are

useless.AsshowninFigure5.2,inascenariowithaneedforupregulation,thedown-bidsareneglectedand

themarket is cleared according to the required up-regulation quantity (shown as up ask). In the opposite

scenario,theneedfordownregulationisappliedtothedownbids,whiletheupbidsareignored.Theblack

circles inFigure5.2 represent theclearingpoint,corresponding to theclearedquantity (q∗)andthecleared

price(p∗).

Figure5.2Minimizationofactivationcostswhenoperatorasksforupregulation(right)andfordownregulation(left).

Whenthisobjectiveisused,nounnecessaryexchangeofpowercanhappenbeyondthebalancingneedof

theoperator.Thishelpstoreducetherisksassociatedwiththeunnecessaryactivations,asthenewlyactivated

devicesmayfailtodispatchasexpected,introducingadditionalimbalancewhichcouldevenexceedtheinitial

imbalanceproblems.

However, according to the guidelines of ENTSO-E (the European TSO), it is recommended that the

newlydesignedmarketsincorporatethemaximizationofsocialwelfareastheobjective.Inaddition,by

definingtheobjectiveasaminimizationofactivationcosts,atechnicaldifficultyappears.Theactivation

costisdefinedastheproductofclearedquantityandclearedprice12:

𝑐𝑜𝑠𝑡+�l = 𝑝∗ ∙ 𝑞∗ (23)

12Theactivation-costobjectivehastobeMinimized.

p

q

Downbid

Downbid

Downbid

Downaskp

q

Upbid

Upbid

Upbid

Upask


Since both𝑞∗ and𝑝∗ are decision variables to be optimized, the resulting optimization becomes a

linear problem. This is not desired and should be avoided if possible. As a result, we investigate the

secondoption,thatisthemaximizationofsocialwelfare.

5.2.2 MaximizationofSocialWelfare

If the market objective is defined as maximization of social welfare, such technical difficulty is

avoided. Inorder toelaborate,we startbyanexplaining the ideaofcurve-crossing formaximizing the

welfare.We want to accept bids offering the up- and down-regulation such that the total quantities

match.However,thesystemimbalance,thatistheneedforup-ordown-regulationhastobetakeninto

account.Forthis,weintroducepseudo-bidsrepresentingsuchaneedforflexibility:Anupask isaneed

forup-regulation,whileadownaskisaneedfordown-regulation.Then,thepseudo-bidsareattachedto

thenormalbids.ThisisshowninFigure5.3.Itislogicallyassumedthatthesystemoperatorsarewilling

toofferamoreinterestingpricecomparedtothebidsinthesamegroup,sincetheyneedtheserviceto

resolvethegridproblems.

Figure5.3Representationofflexibilityneedsintheformsofpseudo-bids(or“asks”).

WeobservethatinFigure5.3itisnotpossibletoapplythecurve-crossingtechnique,sinceaccording

totheconvention,thebidsarenotinthesamehalf-planesofthe𝑝 − 𝑞diagram.Therefore,thebidson

one side (in this case, on the left half-plane) are rotated to the other side. This is demonstrated in

Figure5.4.Afterwards,thejunctionoftwocurvesdefinestheclearingpointofthemarket,revealingthe

optimal values for theclearedquantityand theclearedprice.The socialwelfare is theareamarked in

green.

p[€/kW]

q[kW]

Upbid

Upbid

Downask

Upask

Downbid

Downbid


Figure5.4Therotationofbids,resultinginacurve-crossing.Theclearingpointtomaximizethesocialwelfareis

locatedatthejunction.

Itisshownthatinthiscase,theoptimizationproblemisnotnon-linearanymore,astheclearedprice

isnotanexplicitvariableintheobjective.Thesocialwelfareasafunctionoftimeisdefinedas

𝑆𝑊 𝑡 = 𝑓 𝑝�,3, 𝑝�,3, 𝑞�, 𝑥��∈��,��

− 𝑓 𝑝�,3, 𝑝�,3, 𝑞�, 𝑥��∈��,��

+ 𝑓 𝑝�,3, 𝑝�,3, 𝑞�, 𝑥��∈��,��

− 𝑓 𝑝�,3, 𝑝�,3, 𝑞�, 𝑥��∈��,��

(24)

whereSWisthesocialwelfareandthevariablesare𝑥� ,indicatingtheportionofbids(between0and

1)thatisaccepted13.Thesignofan𝑓-functiondependspurelyonthesignofthe𝑞element(accordingto

(1)and (2)). Therefore, the two terms in (24) relating to𝑝 < 0haveapositive signand the two terms

relatingto𝑝 > 0haveanegativesign.Thisimpliesthattheeffectofatransactiononthesocialwelfare

(i.e. increasing or decreasing it) does not depend on the direction of power flow, but rather on the

directionofmoneyflow.

5.2.3 MultiTime-StepObjectiveFunction

Tosolveimbalancesinthereal-timeIntegratedReservemarket,theclearingalgorithmhastorunat

eachtimestep.However,thelogicalandtemporalconstraintswhichcanbedefinedovermultipletime

stepsrequirethatthemarketclearinglooksbeyondthecurrenttimestepandintothehorizon.Infact,it

isnotintendedtomakeamyopicdecisionandsolvetheimbalanceproblemsolelyforthecurrenttime

step,butalsotoensurethatthemarketisabletoresolveimbalanceproblemsforfuturetimesteps.This

requires that the market has access to imbalance forecasts for the future time steps, which can be

providedbythesystemoperatororathird-partyprovider.

13The socialwelfareobjectivehas tobemaximized. This objective is never non-linear. It is either linear (if thereexistonlyrectangularbids)orquadratic(iftherearesometrapezoidalbids).

p[€/kW]

q[kW]

Upbid

Upbid

Downask

Upask

Downbid

Downbid

p[€/kW]

q[kW]

Socialwelfare

Clearingpoint

q*

p*


Inaddition,weintroducediscountfactors𝛾l inthesocialwelfareobjective,foralltimesteps𝑡over

theoptimizationhorizon.This isdoneduetothefactthatthenear-futureimbalancescanbepredicted

withmore certainty than the far-future imbalances. Such 𝛾l factorsmay gradually decrease, starting

from1(fort=1)andtendingto0byreachingtheendofhorizon.Asaresult,thetime-discountweighted

social-welfareoverthewholetime-horizonisdefinedas

𝑆𝑊 = 𝛾ll∈�

𝑆𝑊 𝑡 (25)

whichisnotanymore,afunctionoftime.

5.2.4 AvoidingUnnecessaryAcceptance

Themain goal of the IntegratedReservemarket is to resolve the power-imbalance problem in the

system. Once the accepted bids are enough to eliminate the imbalance, it is not logical to allow

unrestricted acceptation of additional bids [80], even though it may contribute to an increase of the

social welfare. The reason behind this avoidance is that accepting extra bids and allowing additional

trading between actors may introduce imbalances even higher than those previously existing in the

network.Itcanalsointroducethepossibilityofarbitrage.

In order to elaborate the issue, let us have a better look at the problem using a curve-crossing

approach. In the example shown in Figure 5.5, the up-regulation need of the system operator is

presentedasupask. If nounnecessary acceptance is allowed, themarket is clearedat𝑞4∗ (left sideof

Figure5.5). Inthiscase,noneofthedownbidsareaccepted. Incontrast, ifsocialwelfare ismaximized

(accordingto(24)),themarketwillbeclearedat𝑞E∗ instead(rightsideofFigure5.5).Itcanbeseenthan

inthelattercase,thesocialwelfareisincreased,butsomeunnecessaryacceptancehasoccurred.

Figure5.5Necessary(left)andunnecessary(right)activationofbidsandthecorrespondingsocialwelfare.

As mentioned before, it is not logical to switch tominimization of activation cost, because such

modificationwouldmaketheprice𝑝avariableintheobjectivefunction,whichwouldcreateanon-linear

p[€/kW]

q[kW]q*2

Unnecessaryacceptance

Unnecessaryacceptance

MaximumSocialwelfare

Downbid

p[€/kW]

q[kW]q*

Upask

UpbidUpbid

Upbid

Downbid

1

Socialwelfare


optimizationproblemandwould introduceadditionalcomplexities.Therefore,keepingtheobjectiveas

maximizationofsocialwelfare,weintroduceatechniquetoavoidunnecessarybid-acceptance.

Thetechniqueistoadaptthepricesofthosebidswhichhaveawrongsignofquantity(i.e.oppositeto

whatisneededbythesystem).Wecallthem“wrongbids”fromnowon.Theideaistomakesuchbids

noteconomicallyinterestingtobeaccepted,bynotmatchingthemwiththebidswhichhaveausefulsign

of quantity. However, if there is a non-economic reason to accept the bids (for example to solve a

congestion problem), it will be allowed. The consequence would be a reduced social welfare. When

adapting the prices ofwrong bids, it is required that themerit order of the bids remains unchanged.

Figure5.6showssuchadaptationappliedtotheexampleofFigure5.5.

Figure5.6Adaptationofpricesforavoidingunnecessaryacceptance.Themeritorderispreserved.

After the prices of wrong bids are adapted, it is possible to use the standard formulation for

maximizationof socialwelfare,without the riskofunnecessaryacceptance. Thedetailedalgorithm for

priceadaptationisdiscussedinSection9.5.2.

5.3 NodalPhysicalConstraints

The network models for the transmission and distribution grid were obtained in Chapter 4. The

relevantequations(suchasthepowerbalanceandKirchhoff’slaws)areinsertedintothemarketclearing

optimization intheformofconstraints.Asummaryoftheconstraintswhichareofmostrelevanceand

importanceforthemarketclearingisshowninSection9.5.3.

5.4 PricingApproach

Thedesignedmarketaimsnotonlyatdeterminingtheactivationofflexibilitiesbutalsotoassociateaprice

totheseactivations.Differentpricingapproachescanbeconsidered:

• The “Pay as bid” approachwhere the activated bids simply receive the price corresponding to the

activated quantity in the bidding curve. This approach is simple and intuitive for the different market

stakeholders. However, it does not give incentive for themarket participants to bid using the real cost of

flexibility,creatinganeconomicdistortionintheactivationdecision.

p[€/kW]

q[kW]q*

Socialwelfare

p1

p2p'2p'1


• The“Payascleared”or“locationalmarginalprice(LMP)approach”wheretheactivatedbidsreceive

the same price per MWh (or MW over a time step), corresponding to the most expensive (or the least

economicallyinteresting)activatedflexibility.Thisapproachremovestheriskofmarketparticipantbiddingin

termsofwhattheywanttoreceiveinsteadoftheirrealcostofflexibility.

Asthemotivationofthedesignedmarketistoallowafairandcost-efficientcompetitionbetweendifferent

sourcesofflexibilities, inparticularthoselocatedatthedistributionlevel, itappearsnaturaltousethemost

economicallyefficientapproachofthemarginalpricing[81].

However, as the considered system is not a perfect copper plate, network constraints, both at the

transmissionanddistributionlevels,havetobetakenintoaccount.

Marginalpricingcanbeadaptedtoasystemwithnetworkconstraintsindifferentways:

• ANodalapproachwhereapriceforflexibilityisassociatedtothemostgranularlevelinournetwork

representationi.e.toeachnodeofthedistributiongrid

• AZonalapproachwhereapricefor flexibility isassociatedtoazonecoveringdifferentnodes.Each

zonecanhaveadifferentpricebutthenodesinthesamezonehavethesameprice.

Asthenetworkconstraints,bothatthetransmissionanddistributionlevels,areofkeyimportanceforan

ancillary service market ensuring the satisfaction of congestion and voltage constraints, it is natural to

encompassinthepricemechanismthesenetworkfactorsinthemostaccurateandeconomicallyefficientway.

Moreover,ancillaryservicemarketsareclose-to-real-timemarketandtheirdecisionscannotbecorrectedbya

market afterwards. For these reasons, a nodal approach is proposed for the considered Integrated Reserve

markettobedesigned.Further justificationsandthederivationofthedistributionlocationalmarginalprices

fortheconsiderednetworkmodelanditsinterpretationisdetailedintherestofthechapter.

5.5 NodalMarginalPricing

Thetheoryofnodalmarginalpricingispredicatedonthemoregeneraleconomicconceptofmarginal

costpricing,whichisanapproachthatachievesanefficientallocationofresourcesundertheassumption

ofperfect competition.The ideaofmarginal costpricing is simple,anddictates thatconsumerswitha

well-definedwillingness to pay for a commodity should consume as long as their incremental benefit

exceeds the incremental cost of the additional production. In an economy where consumers can be

stackedinorderofdecreasingvaluationandsupplierscanbestackedinorderofincreasingmarginalcost

itiseasytoseewhythissystemallocatesresourcesefficiently:consumerswiththehighestwillingnessto

payarematchedwithsupplierswiththe lowestmarginalcostofproduction,uptothepointwhereno

additionalmutualbenefitsfromtradecanbeachieved.Inordertorespectcompatibilitywithourmarket,

oneshouldinterpreta“consumerbid”asabidwith𝑝 < 0anda“supplierbid”asonewith𝑝 > 0.


5.5.1 PricingoverSpaceandTime

The complicating factor of electric power systems is the fact thatmarginal cost cannot be defined

uniformly for all locations, or for a single time period, but instead depends on the configuration of

demand over the network and the availability and physical characteristics of power generation,

transmissionanddistributionequipment,andchangescontinuously(overminutes,hours,days,…).This

ledtothetheoryofspotpricingofelectricityoverspaceandtime,championedbytheseminalworkof

Schweppe,Caramanis,andTabors[82],[83].

Amajorreasonwhymarginalcostcannotbedefineduniformlyforalllocationsofapowersystemis

the physical laws that govern power flow, losses, and limits on transmission and distribution lines. A

numericalexampleisshowninSection9.5.4.

5.5.2 Granularity

There has been a longstanding debate about the granularity of pricing in electricitymarkets.More

specifically,nodalpricinghasbeencriticizedforthinningupmarketsandcreatingopportunities forthe

exerciseofmarketpower,aswellas forcreatingapricingsystemthat is toocomplexorunnecessarily

granular.Successfulexperienceofnodalpricinginsomeofthelargestmarketsworldwide,includingthe

PennsylvaniaJerseyMaryland(PJM)marketwhichprices8700locationswithafrequencyof15minutes,

have discarded concerns of complexity. Market power mitigation by market monitors has further

softenedconcernsregardingtheexerciseofmarketpower.

Analternativetonodalpricingthathasbeenadvocatedonthegroundsofsimplicityiszonalpricing,

whereby nodes are aggregated into zoneswith a commonprice (which implicitly assumes away intra-

zonal congestion) and inter-zonal lines are aggregated into a single link. This aggregation creates a

challengeofdefiningcapacityandcreatingaproxyfortherepresentationofpowerflowlaws,asshown

in a numerical example in Section 9.5.5. The conclusion is that a zonal model will either sacrifice

efficiencyorwelfare,sinceitignoresthephysicallawsthatgovernpowerflow.Remediestothisissueby

meansof re-dispatchinghaveexposedthemarket toseveregamingopportunities (including theEnron

“Decgame”)[84],resultinday-aheadschedulinginefficiencies[85],andcreatefree-ridingopportunities

forcertainzonesonthetransmissiongridsofotherzones[86].

5.6 CalculatingPrices

Thestandarddefinitionofmarketequilibriumineconomicsisapairofpricesandquantitiessuchthat,

given the prices, agents maximize profits, and prices are such that supply and demand are equal. It

follows from standard results in convex optimization that this definition ofmarket equilibrium implies

thatthepriceofelectricalpowerintheoptimalpowerflowproblemcanbeobtainedasthedualoptimal


multiplier of thepowerbalance constraints14. Theseprices canbeobtainedby solving thedual of the

optimalpowerflowproblem,whichisformulatedasasetofconstraintsinthemarketclearingalgorithm.

Therefore,theclearingofquantitiesiscalledthe“primaryproblem”,andtheclearingofnodalmarginal

pricesisreferredtoasthe“dualproblem”.

Awide rangeofoptimizationalgorithms that areused in commercial applications for solving linear

programs, second-order coneprograms, andmoregenerallynon-linearprograms, solve thedual of an

optimizationproblemasaby-productofsolvingtheprimalproblem.Thisimpliesthat,inordertoobtain

prices,itisnotnecessarytosolvethedualproblemexplicitly,sincethisisanywaysaside-productofthe

optimization process. Commercial solvers (such as CPLEX) will then typically furnish primal optimal

variablesthatareusedfordecidinghowtodispatchanasset,howmuchpowershouldflowoveraline,

whatthevoltageofanodeis,aswellaspricesthatshouldbeassignedtoanexchangeofquantity.

5.6.1 NodalClearedPricesasDualVariables

The nodal dual variable associatedwith the nodal power-balance constraint (see (216)) equals the

marginal price of injecting/off-taking one unit of power in that node. In principle, for a linear

programming(LP)problem,theclearedpricep∗canbefetchedbyrequestingtheLP-solvertoprovidethe

valueofthedualvariable.

In reality, themarket clearing problem contains binary variables. In such a case, the clearedprices

need tobederivedusing adedicatedalgorithm. Theprice is not explicitlypresent as a variable in the

model of the primal problem; if it was, the primal problem would belong to a category of problems

knownasMathematicalProblemswithEquilibriumConstraints (MPEC),whichareknowntobehardto

solve since their feasible region is not necessarily convex or connected [87]. Further explanation is

providedinSection9.5.6.

Byavoidingthesituationofhavingpricesasexplicitvariablesintheprimalproblem,suchdifficultyis

avoided.Italsomeansthatthedualproblemdoesnothavetobesolvedexplicitly,exceptinthecaseof

indeterminaciesorparadoxicalacceptance.

In general, indeterminacies can happen for both quantity and price. Figure 5.7 compares the

situationswithnoindeterminaciesandthosewithindeterminacies.Incaseofaquantityindeterminacy,

the rule of volumemaximization applies, resulting in the selection of clearing point at the end of the

horizontalcrosssection. Incaseofaprice indeterminacy, therule is tochoosethemiddlepointof the

verticalcrosssection.Section9.5.7explainsthealgorithminmoredetailsusinganumericalexample.

14 This statement is true for convex problems. Problems with binary decision variables pose deeper theoreticalquestionsofdefiningwhatamarketclearedpriceis,asdiscussedinthenextsection.


Figure5.7Priceandquantityindeterminaciesandthecorrespondingclearingpoint.

Afterhandlingthe indeterminacies,there isonlyonemorereasontosolvethedualproblem(forall

nodestogether),andthatisthecaseofaparadoxicallyacceptedblockinanynode.Theseareblocksthat

are accepted even though they are out of themoney (i.e. the bidder is losingmoney because of the

clearedprice).ThissituationdoesnotoccurinanLPmodel,butcanbecausedbythepresenceofbinary

variables(non-curtailablebids,outofmerit-orderacceptance,rampconstraints,etc.).

Forexample,therelaxedproblemwouldaccept90%ofablockwhichisinthemoney,butbecauseof

thebinaryvariablerepresentingthenon-curtailabilityofthatblock,weshouldaccept0%or100%ofit.

Theproblem tobe solved is almost thedual problem; thedifference is that, per block, a constraint is

addedtoaccountforthefactthattheblockhastobeintheformofmoney.However,thiscouldmake

thesolvedprobleminfeasible.Ifthathappens,somebranchandboundalgorithmisnecessary.

5.7 Alternating-CurrentPowerFlow

Nodalpricingtheoryiswelldevelopedinlinearizedmodelsoftransmissionpowerflowwherereactive

power flows are ignored, voltage is normalized across all buses, and real power losses are ignored or

simplified. By contrast, the increasingly important role of distributed resources in power system

operationsnecessitatesanexplicit considerationof thenon-linearpower flowequations inoperations,

sincetheassumptionsemployedforthelinearizationofKirchhoff’spowerflowlawswhicharereasonably

accurateforthehigh-voltagegridarenotadequatefor low-voltagedistributiongrids.This issuewillbe

discussed using an example in Section 9.5.8, but the important thing to note here is that the general

theoryofmarginalpricingremainsthesame:onesolvestheoptimalpowerflowandobtainsthepriceat

acertainnodeasthedualoptimalmultiplierofthepowerbalanceconstraintatacertainnode.

5.7.1 DistributionLocationalMarginalPricing

Thepricingapproachesusedinthemarketclearingresultinobtainingdistributionlocationalmarginal

prices (DLMPs). In the early days of electricity market deregulation, locational marginal pricing in DC

networks presented interpretation difficulties, due to the fact that it deviated from simpler pricing

modelsasaresultofthelinearizationofKirchhoffpowerflows.TheinterpretationofDLMPsaddsalayer

Noindeterminacy

p[€/kW]

q[kW]q*

Clearingpoint

p*

Priceindeterminacy

p[€/kW]

q[kW]q*

Clearingpointp*

Quantityindeterminacy

p[€/kW]

q[kW]q*

Clearingpoint

p*


of complexity, since voltage, real power losses, and reactive power flows are accounted for, whereas

theseaspectsoftheproblemareeithersimplifiedorignoredinDCpowerflow.Variousmethodscanbe

usedforunderstandingandinterpretingthevalueofDLMPs,withtheobjectiveof justifyingtheirvalue

onphysicalgrounds.InSection9.5.9weshowthreedifferentapproachestoanswerthequestionofwhy

DLMPsobtainthespecificvaluefurnishedbythemarketclearingalgorithm.

5.8 MarketClearingAlgorithm

The clearing algorithm is an optimization problem which is solved at every time step. Here we

summarize the comprising routines. The following multi time-step clearing algorithm over the time

horizonissetup:

1. Initialization:Settheparametersandinitialvalues(basedontheoutputoftheoldmarketclearings);

2. Bidacquisition:Receivebidsandprocessthemtocreateprimitivebids;

3. Networkmodeling:Receivethenetworkstate(forexamplethepower injectionateachnode)and

lineparameters,thencalculatethetotalsystemimbalanceandidentifycongestionproblems;

4. Problemformulation:Createtheobjectivefunctionandeconomicandphysicalconstraints;

5. Clearing:Solvetheoptimizationproblemandfindtheclearedquantityandclearednodalprices;

6. Finalization:Transmittheinformationregardingclearedbids;

7. Waituntilthenexttimestepandreturntostep1.

Themarketresolution(i.e. lengthofatimestep), thetimehorizon,andthefrequencyatwhichthe

optimizationisrunning,aredesignparametersthatcanbeadjustedattheimplementationstage.


6 TSO-DSOCoordinationSchemes

Previous sections have focused on describing the mathematical formulations needed to clear a

market: market products specification, simplified network model, clearing algorithm, and pricing

mechanism.

Inthecentralizedapproach,wheretheDSOhasnolocalmarket, it istheTSO(orthemarketoperator in

general) who takes the activation decisions for flexibilities proposed, not only at the transmission network

level,butalsoatthelevelofthedistributionnetworks.

Twodifferentvariantsofthiscentralizedmodelcanbeconsidered:

• Full-observabilityvariant:Inthisapproach,theTSOclearstheglobalbalancingmarketbytakinginto

account congestions using (simplified) models of the transmission network, but also of the different

distributionnetworks.

• No-observabilityfromTSO:TheTSOclearstheglobalancillaryservices(IntegratedReserve)market

withoutexplicitlytaking intoaccountthemodelsofthedistributionnetworks. It ispossibletoconsidersuch

network constraints, butmore computation timewill be required.Ablocking/vetoprocess from theDSO is

thereforeneededafterthemarketclearingduringthecounter-tradingphase.Themarketclearingwillnotrun

again,butratherthecounter-tradingishandledoutsidethemarket,withalimitedtimeallowedforvetoes.In

caseofaveto, theTSO (or theDSO)canuse itsownresources tocompensate for theassetswhicharenot

activated.

Inthischapter,wedescribethespecificitiesandrequirementsofthemarketarchitecturesforthedifferent

TSO-DSOcoordinationschemesinvestigatedintheSmartNetproject[1].

6.1 CentralizedvsDecentralizedMarketArchitectures

Firstofall,animportantdifferencebetweenTSO-DSOcoordinationschemesiswhetheracentralized

ordecentralizedarchitecture(seeFigure6.1)isconsidered.Table6.1showswhicharchitectureisusedin

eachcoordinationscheme.

Centralizedarchitecture DecentralizedmarketarchitectureCentralizedASmarket LocalASmarketCommonTSO-DSOASmarket(centralized) CommonTSO-DSOASmarket(decentralized)Integratedflexibilitymarket Sharedbalancingresponsibilitymodel

Table6.1Listofcentralizedanddecentralizedarchitectures.

Inthecentralizedarchitecture,there isone(big)market,whilefordecentralizedarchitecture,many

localmarkets(managedbyDSOs)co-existwithaglobalASmarket.Forthedecentralizedarchitecture,we

have:


• Either no interaction between the markets (Shared balancing responsibility model TSO-DSO

coordinationscheme):inthiscase,theTSOandDSOsagreeinadvanceonascheduledpowerexchange

profileattheirinterconnection(basedonthemethodexplainedin[1]).

• Either (as indicated by the blue arrows in Figure 6.1) the DSOs are responsible to aggregate

(eitherdirectlyorthroughanaggregator)thebidssubmittedonthelocalmarketsandtransferbidstothe

global AS market. This is the case for the Local AS market and Common TSO-DSO AS market

(decentralized) coordination schemes. In this case, the DSO may cluster the bids having the same

locational tag (i.e. connected to the same distribution network node), but also across all the network

nodes. Inbothcases,theDSOneedstocheckthatnetworksconstraintsareobeyedwhenconsolidated

acrossthenodes,sincethereisnodistributionnetworkmodelintheglobalASmarket.

Also, regarding decentralized architecture,market settings can be different for eachmarket. As an

example,thepricingschemecanbepay-as-bidinthelocalmarketandpay-as-clearinthecentralmarket

[88],orthemarketcanclearatdifferentfrequencies.Thisdegreeoffreedomcanbeanadvantagebut

alsoaninconvenience.Inanycase,thepossibilityofhavingdifferentmarketsettingsdoesnotchangethe

mathematical description of how themarkets are cleared (see previous chapters), but instead itmay

introducenewbiddingstrategiesandcauseachangeofparadigm[89],consideringtheproblemfroma

“gametheory”pointofview.

Figure6.1Diagramsshowingcentralizedanddecentralizedmarketarchitectures.

6.2 MarketSpecificitiesforEachTSO-DSOCoordinationScheme

Table6.2showsthespecificities regardingthemarketclearing foreachcoordinationscheme.Some

similaritiescanbeseenbetweenthecoordinationschemesAandD1,withthedifferentbeingthatinD1

themarket considers the problem of both transmission and distribution networks. It implies that the

schemeD1probablyhasthemostcomputationallycomplexsingle-clearingproblemtobesolved.

SimilaritiesarealsonoticeablebetweencoordinationschemesBandD2.Infact,themaindifferenceis

thatthemarketclearingsinBcanbedonesimultaneously,whileinD2theyhavetobecarriedoutinan

iterativemanner.


Coordinationscheme SpecificitiesA.CentralizedASmarketmodel

• TSOistheonlybuyer:NoneedtorepresenttheimbalanceofTSO’scontrolareabyanexplicitbid.

• Marketobjective:minimizeactivationcostsofTSOtosolvebalancingandcongestion.

Onlytransmissiongridconstraintsareconsidered.

B.LocalASmarketmodel

• Differentmarketproductsinlocalmarket(tunedtodistributiongridsettings)ordifferentmarketparameters(e.g.minsizeofabid).

• Needtodevelopaggregationmethodtoaggregatelocalmarketbidsnon-acceptedtosolvelocalneeds,whiletakingnetworkconstraintsintoaccount.

Local(central)marketobjectiveistominimizeactivationcostsofDSO(TSO).C.Sharedbalancingresponsibilitymodel

• Localandglobalmarketsfullydecoupled:easierBUTneedonadditionalinput:scheduledexchangedprofilebetweenTSOandDSO.

Local(central)marketobjectiveistominimizeactivationcostsofDSO(TSO).D1.CommonTSO-DSOASmarketmodel(centralized)

• TSOandDSOarebuyerstosolveglobalandlocalproblemstogetherObjectiveistominimizeactivationcostsforTSO+DSOàneedmethodtosharecosts

betweenTSOandDSO(financialsettlement).

D2.CommonTSO-DSOASmarketmodel(decentralized)

• TSOandDSOarebuyerstosolveglobalandlocalproblemstogether.• ObjectiveistominimizeactivationcostsforTSO+DSO.Fromamarketclearingalgorithmpointofview,samechallengesaslocalASmarket

E.Integratedflexibilitymarketmodel

• TSO,DSOandBRPsareflexibilitybuyersonthemarket+TSO/DSOcansellpreviouslyprocuredflexibility.

• Marketobjectiveistomaximizewelfare.TSOandDSOneedtorepresenttheirimbalanceintheformofexplicitbidsonthe

market.

Table6.2ListofTSO-DSOcoordinationschemesandthecorrespondingspecifications.

Inthenextsections,weexplainhoweachcoordinationschemeisusingthemodulesdefinedearlierin

this document and also in other parts of the project. More detailed description of the coordination

schemesisfoundin[1].

6.2.1 CentralizedASMarket

TheblockdiagramrepresentingthiscoordinationschemeisshowninFigure6.2.

Figure6.2Ablock-diagramrepresentingthecentralizedASmarket.

BiddingblockBiddingblock

Centralmarket

PriceclearingQuantityclearing

aggregatedbids

DispatchingDevicesAggregation

unaggregatedbids

model schedule

clearedquantities

clearedbids

Networkblock

Imbalances&Congestions Txmodel


Thediagramshowsasetofblocksrepresentingmodules,andarrowswhichcorrespondtodataflows.

Theblocksareconnectedsequentiallyinaclosedloop,implyingthattheprocedureisrepeatedateach

timestep.HereweprovideadescriptionofeachblockinFigure6.2:

• Devices:Thisblockincludesthemodelsofavailableflexibleassets inbothdistributionnetwork(Dx)

and transmission network (Tx). Input to this block is the dispatching schedule for the following time step.

Outputs from this block are themodels and parameters (including internal states) of flexible assets.More

informationaboutthecontentsofthisblockcanbefoundinD1.2[90].

• Bidding: This block contains the algorithms required for aggregation and dispatching (or

disaggregationplusconsideringthephysicallayer)offlexibledevices.Theaggregationmodulereceivesdevice

models and produces bids. The dispatching module receives the cleared bids and generates dispatching

schedulesforselectedflexibleassets.Moreinformationaboutthecontentsoftheseblockscanbefoundin[5].

• Network:Thisblockcontains thenetworkmodel, includingnetwork topology, lineconnectionsand

parameters, power-flows and voltage constraints, forecasts of system imbalance, etc. As described in

Chapter 4, different models are considered for distribution (Dx) and transmission (Tx) grids. In this

coordinationscheme,only theTxmodel is considered, since theDxproblemsarenot solved in themarket.

This model is used in the quantity and price clearing blocks, in order to locate the submitted bids and to

calculatethenodalprices.

• Market: This block represents the clearing processes regarding quantity and price, which are

thoroughlydiscussed inChapter5.Several typesofmarketsexist indifferentcoordinationschemes,suchas

local(DSO)market,central(TSO)market,andcommon(TSO-DSO)market.

6.2.2 LocalASMarket

TheblockdiagramrepresentingthiscoordinationschemeisshowninFigure6.3.Blocksareorganized

into four categories as in centralizedASmarket. However, alternative block connections and different

sequenceofrunningisusedinthiscase:

• Devices:Inthiscoordinationscheme,DxdevicesparticipateinthelocalmarketrunbyDSO(whileTx

devicesparticipateinthecentralmarketrunbyTSO).

• Bidding: Since devices connected directly to Tx are large, no aggregation is needed for them in

general.Also,separatedispatchingstepsareusedforDxandTxdevices.

• Network:Here,DxandTxmodelsareusedinthelocalandcentralmarket,respectively.Congestion

problems are treated by the corresponding network operators in both local and centralmarket. However,

imbalancesareonlyconsideredinthecentralmarket.


Figure6.3Ablock-diagramrepresentingtheLocalASmarket.

• Market:ThecentralmarketblockisorganizedsimilartotheblocksincentralizedASmarket.Onthe

contrary, the local market block includes two additional components; DSO aggregation, and DSO

disaggregation.FirstDSOclearsafractionofbidstosolvethecongestionproblemsinDx.Then,theuncleared

bidsareprocessedbytheDSOaggregationblockwhichusesaparametricoptimization(oranotheralgorithm)

toconditionallyclearthebidsandtoaggregatethem.Thisstep is furtherexplainedinSection6.3.Afterthe

clearance of central market, the cleared DSO-bids return to the DSO disaggregation blocks, when the

previouslycalculatedparametersareusedtodeterminewhichnon-aggregatedbidshavetobelocallycleared.

6.2.3 SharedBalancingResponsibilityModel

TheblockdiagramrepresentingthiscoordinationschemeisshowninFigure6.4.Inthisscheme,DSO

andTSOsolvetheirowncongestionand imbalanceproblemsusingthedevicesconnectedtotheirown

networks.However,additionalconstraintsonDxandTxareusedtoscheduleprofilesofpowerexchange

betweenDx and Tx at connecting nodes. These profiles are generated prior to any optimization using

forecastsofpowerinjectionateachnodeofthenetwork.

Networkblock

Biddingblock

Biddingblock

Biddingblock

Biddingblock

Localmarket

PriceclearingQuantityclearing

Centralmarket

Txdevices

PriceClearing

aggregatedbids

QuantityClearing

ImbalancesCongestions Txmodel Dxmodel

AggregationDxdevices

unaggregatedbids

Dispatching

DSODisaggregation

Dispatchingmodel schedule

clearedquantities

clearedDSO-bids

unclearedquantities

aggregated bids(conditionallycleared)

Parameters

clearedTSO-bids

schedule

clearedaggregatedDSO-bids

unaggregatedTSO- bids

DSOAggregation(Parametricoptimization)


Figure6.4Ablock-diagramrepresentingthesharedbalancingresponsibilitymodel.

6.2.4 CommonTSO-DSOASMarket(Centralized)

The block diagram representing this coordination scheme is shown in Figure 6.5. Similar to the

centralizedASmarket scheme, in this setupacommonmarketcleared thebids fromalldevices in the

network,inordertosolveimbalanceandcongestionproblemsinbothTxandDx.

Figure6.5Ablock-diagramrepresentingthecommonTSO-DSOASmarket(centralized).

Biddingblock

Biddingblock

Biddingblock

Networkblock

DSOmarket

Aggregation

Priceclearing

Dxdevices

Quantityclearing

TSOmarketTx

devices PriceClearingQuantityClearing

ProfileScheduling

ImbalancesCongestions DxconstraintsTxconstraints

unaggregatedbidsaggregatedbids

Txmodel Dxmodel

Dispatching

Dispatching

Biddingblock

model schedule

schedule

clearedquantities

clearedbids


Commonmarket

Priceclearing

aggregatedbids


unaggregatedbids

model schedule

clearedquantities

clearedbids

Networkblock

Imbalances&Congestions TxandDxmodels

Quantityclearing


6.2.5 CommonTSO-DSOASMarket(Decentralized)

The block diagram representing this coordination scheme is shown in Figure 6.6. The blocks are

organizedinasettingsimilartolocalASmarketscheme.However,thereexistsonlyonemarketforDSO

andTSO:thecommonmarket. Inthisscheme,theDSOblockperformstheconditionalclearingusinga

methodsuchasparametricoptimization,withoutclearinganybidsinordertosolvethelocalcongestion

problems.Therefore,thecommonmarketisresponsibletosolvetheimbalanceandcongestionproblems

inoneclearing.Similar to localASmarketscheme,aDSOdisaggregation isused forpost-processingof

clearedaggregatedDSO-bids.

Figure6.6Ablock-diagramrepresentingthecommonTSO-DSOASmarket(decentralized).

6.2.6 IntegratedFlexibilityMarket

TheblockdiagramrepresentingthiscoordinationschemeisshowninFigure6.7.Inthisscheme,TSO,

DSO, and BRPs are allowed to buy flexibility on a common market. The market is operated by an

independentoperator,whichhasaccesstoinformationregardingDxandTXparameters.

DSO

Biddingblock


Commonmarket

PriceClearing

aggregatedbids

Remainingbids

QuantityClearing

Networkblock

ImbalancesCongestions Txmodel Dxmodel

AggregationDispatchingDxdevices

unaggregatedbids

Dispatching

DSODisaggregation

aggregated bids(conditionallycleared)

clearedDSO-bids

clearedTSO-bids

clearedaggregatedDSO-bids

Biddingblock

Txdevices

unaggregatedTSO- bids

schedule

model schedule

ParametersDSOAggregation

(Parametricoptimization)


Figure6.7Ablock-diagramrepresentingtheintegratedflexibilitymarket.

6.3 BiddingfromLocalMarketstoCentralMarket

Ifadecentralizedmarketarchitectureisusedandifthelocalmarketoperators(DSOs)havetotransferthe

(additional) flexibility,offeredbyDERsorbyaggregators, fromthe localmarket to thecentralASmarket,a

methodology is needed on how the DSO would perform this task. In the framework of the coordination

schemes,suchDSOaggregationisneededintwomodels:LocalASmarketandCommonTSO-DSOASmarket

(decentralized).

TheDSOreceivesbidsfromDERs(aggregatedornot)onthelocalmarketandisresponsibletoclusterthem

andsubmitoneaggregatedbid(orpossiblyseveralbids)totheTSOmarket.However,amaindifferencewith

respecttocommercialaggregatorsisthatDSOneedstoperformthisaggregationundertheconditionthatthe

distributionnetworkconstraints(suchascongestionsandvoltagelimits)arenotviolated.Onewaytosolvethis

aggregation problem is through parametric local market clearing. In the next section, we propose an

algorithmfortherealizationofsuchclusteringandaggregation.

6.4 ParametricLocalMarketClearing

Givenallthebidssenttothelocalmarket,theparametriclocalmarketclearingmethodamountstosolve

the localmarketclearing fordifferent flexibilitydemands∆PHat theHV-MV(highvoltage -mediumvoltage)

connectionpointbetweenthedistributiongridandthetransmissiongrid: for instance,adiscrete intervalof

[−5MW,… ,−1,0, 1, … , 5MW]canbeusedfor∆PH.Themethodologyisquitesimilartotheprocedureapplied

in[91].Notethatthedemand∆PH istheexchangeofactivepowerwithrespecttothebaselineactivepower

exchange at that HV-MV node. The parametric algorithm is explained in Table 6.3 for the two relevant

coordination schemes. For eachmarket clearing, the goal is tominimize activation costwhile satisfying the

distributiongridconstraints.

Biddingblock(TSO)Biddingblock

Commonmarket(operatedbyIMO)

Priceclearing


Networkblock

Imbalances&Congestions TxandDxmodels

QuantityclearingBiddingblock(DSO)

Aggregation

Biddingblock(BRPs)

Aggregation


LocalASmarket CommonTSO-DSOASmarket(decentralized)• Foreachdemand∆𝑷𝒊,theresultofthemarketclearingisobtainedforanabsolutecost𝑪𝒊(=thevalueofthe

objective).Atsomepoint,westopiteratingwhenthelocalmarketclearinghasnosolution(i.e.notpossibletoprovidethecorrespondingdemandattheHV-MVnode,usingthebidsinthenetworkANDsatisfyingdistributiongridconstraints)

• Priortotheaggregationstepdescribedhere,theDSOrunsthelocalmarkettosolvelocal(e.g.currentcongestions)problems.Therefore,inthisstepofparametriclocalmarketclearing,therearenolocalproblemstosolve

• Potentiallocalproblemsarenotsolvedasapriorstep.Instead,theDSOaggregatesallbids(asdescribedabove),notonlyobeyingdistributiongridconstraintsbutalsosolvinglocalproblems(i.e.localproblemsaresolvedatanyvalueofacceptedquantity∆𝑃)).

• Forademandof0MWattheHV-MVnode,thecostC_0is0sincethereis,bydefinition,noneedtoactivateanyflexibleassets(nolocalproblemstosolve);

• Inthiscase,thecostforaflexibilitydemandof0MWattheHV-MVnodeisnotzeroanymore,itisequaltoC3,whichrepresentsthecostto:

o Solvelocalproblems(e.g.congestion)o andcorrectthechangeinthebalance

usinglocalresourcessuchthatthereisnovariationofpowerattheHV-MVnode

• Thus,DSOgeneratesbidsatquantity∆𝑷𝒊andatprice𝝀𝒊 =

𝑪𝒊∆𝑷𝒊€/MWh

• However,theTSO(buyeronthecentralASmarket)wouldnotwanttopayacostC3 foractivatingnothing:thatiswhyforagivenpowerexchange∆PH,thetotal(absolute)priceofbiddingofDSOshouldbeequaltoCH − C3 andthusthebidpricecorrespondingtoeachquantity∆PHwillbedecreasedaccordingly(comparedtoatotalcostofCH).Ofcourse,thiseffectinbidpricewillbestrongestforsmallofferedquantitieswhileitwouldbequitedilutedforlargequantities:λH =(CH − C3)

∆PH€/MWh

• DSOsubmitsoneaggregatedbid • Usingthisstrategy,theDSOpaysatmostthelocalresourcetocompensatetheimbalancelocally,butmostlikelyabidwillbeacceptedbytheTSOmarketandiftheTSOmarketclearingpriceishigherthanthebidpriceoftheDSO,itmeansthattheDSOwillhavebalanceditscongestionmanagementactionusingcheaperresourcesfromtheTSOgrid.

• Inthenextstep,TSOclearsthecentralASmarketandprovidestheresulttotheDSOontheacceptedquantity∆𝐏𝐜𝐥𝐞𝐚𝐫𝐞𝐝andthepriceoftheclearing,𝛌𝐜𝐥𝐞𝐚𝐫𝐞𝐝€/MWh.

• TheDSOdisaggregationstepissimple15,sinceinthebid-constructionstep,theDSOhassolvedthelocalmarketclearingforeachquantity∆𝑷𝒊,including∆𝑷𝒄𝒍𝒆𝒂𝒓𝒆𝒅.

Table6.3Thealgorithmforparametriclocalmarketclearing.

6.5 FinancialSettlementandSharingofCostsBetweenTSOAndDSO

TSOsandDSOscouldbeseenasbeinginvolvedinagame,sinceitisclearthattheactionsofeachofthem

hasrepercussionsontheutilityfunctionoftheothers,andreciprocally.Basically,thismeansthattheyshare

conflictingobjectives.Wecanalsoconsiderthattheycontributetoglobalpublicgoods,suchasequalservice

provisiontotheendusers,constantimprovementofthequalityofservice,etc.,thatwouldrequiretoperform

subsequent investments at thenetwork side (both transmissionanddistribution).Onmanyof these issues,

evenexistingfederationsofagentsoftenchoosetoactinadecentralizedway.Themembersofthefederation

15 In practice, however, there is the question of the granularity of∆PH and how we tackle the problem for values inbetween.Thesetopicswillbeinvestigatedinthenextphaseoftheproject.


can choose to provide the good in a centralized way. This implies that they will delegate the voluntary

provisiontoacentralized level,whichmaximizes their total surplus. If theychoosetoprovidethegood ina

decentralizedway,eachmembermaximizesitsownsurplusindependently.

Thereexistmanywaysofsharingthesocialwelfare(orthesumoftheactivationcosts)betweenDSOsand

theTSO.Themechanismsthatareproposed inthecooperativegametheory literaturerelyonbothfairness

andstabilitycriteria(Shapleyvalue,Banzhaf index,nucleolus,etc.).Thistopicwillbefurtherexplored inthe

nextphaseoftheproject.


7 Conclusions

In this document, we provide the mathematical description of necessary modules which comprise the

IntegratedReservemarket,namelybidding,network,clearing,andpricing.Thesecomponentscanbearranged

andconnected invarioussetups, inorder toadapt themarket fora specific coordinationschemebetween

TSOandDSO.Thekeymessagesineachpartaresummarizedbelow.

7.1 Bidding

InChapter3wedefineacatalogueofmarketproducts(bids)andtheaccompanyingconstraintswhichare

allowedintheIntegratedReservemarket.Thebidsrepresentthedesiretoexchangeaquantityofpowerata

givenprice.Theaccompanyingconstraintsareusedtoimpressthosephysicallimitationsofflexibleresources

whichcannotbedescribedbythebiditself.Welistandformulatelogicalandtemporalconstraintswhichare

mostrelevanttoresourcesavailableatthemomentorinforeseeablefuture.

7.2 Network

Chapters4providethemodelsofpowernetwork,withanoptimization-orientedpointofview.Inaddition

to themathematicalmodeling, several topics such as the trade-off betweenmodel complexity and level of

detail, identificationandestimationofnetworkvariables (power flows,voltages,etc.)andcostofmodelling

errors are discussed. Since integer variables exist within the market clearing constraints, the approaches

proposed depend on convex approximations of the power flow physics, which allow to develop a mixed-

integerconvexmodelhavingsuperiortractability.

It is shown that in somecases, theSOCPBFMDistFlowsolution (globaloptimum) is actually superior to

those obtained by local non-convex solvers. Also, the simplified DistFlow formulation offers a high-quality

approximationintermsoffeasibilityandoptimality.

7.3 ClearingandPricing

The mathematical formulation of market clearing and pricing algorithms, which is the core of the

optimization problem, is described in Chapter 5. A first keymessage is that the objective of themarket is

highly affected by several pre-defined conventions, such as types of bids allowed (simple and complex

products),quantity/priceconvention(whetherbothpositiveandnegativequantitiesandpricesexist),network

model complexity (linearity and convexity), and the type of objective (cost minimization versus welfare

maximization).Furthermore,manyparametervaluescanaffectthebehaviorandalsotheperformanceofthe

clearingalgorithm.Someimportantones includethedurationoftimestep,thelengthofpredictionhorizon,

andthefrequencyofclearing.


We show that it is possible to avoid unnecessary activation of resources, hence reducing the costs of

system operators, by adapting the prices of some bids. In this method, the objective will be kept as

maximizationofsocialwelfare,whichisrecommendedbyENTSOE.

Ingeneral, thepricingprocess,which isdefinedasadualproblemofthequantityclearing,canhavethe

same complexities as theprimal problem.However, there are severalmethods to avoid these complexities

andfindthelocationalmarginalpricesinamorestraight-forwardandintuitiveway.Insightintothistopicare

providedinChapter4andamorethoroughdiscussionwillbeprovidedinthefinalversionofthisdocument,

i.e.deliverableD2.2.

7.4 CoordinationSchemes

Chapter6summarizesthestructureofmarketundereachTSO-DSOcoordinationscheme.Inthisschematic

presentation, it is shown how different market modules can be used to create a setup suitable for each

coordinationscheme.Thesedescriptionswillbeadaptedasthemarket-implementationstageoftheSmartNet

projecthasbegun,throughacontinuousfeedbackloop.FinalizeddetailswillbeavailableindeliverableD2.2.


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9 Appendices

9.1 FormulationofBidsandConstraints

9.1.1 Nomenclature:ConstantsandVariables

In this section, constants and variableswhich are used to define the constraints and the objectives are

listed.

Constants(Scalars,SetsandMappings)

1. 𝛿𝑡:timesteporperiodatwhichtheenergymarketclearingisperformed.

2. 𝑇:numberoftimestepsconsideredinonerollingwindowoptimization.

3. 𝑁: the set of all nodes in the network (graph). For the considered tree-shaped distribution sub-

networks,𝑁denotesthesetofnodesofthetree-shapednetwork,exceptfortherootnode.

4. 𝛾l:timediscountfactorthat, intheobjectivefunction,weighsnear-futuretimeslotobjectiveterms

higherthanfar-futuretime slots.

5. 𝐴*:thesetofactors(aggregatorsoragents)atnode𝑛 ∈ 𝑁.

6. ∀𝑛 ∈ 𝑁, ∀𝑎 ∈ 𝐴*:

(a) 𝐵*+:setofbids.

(b) 𝐼*+:setofpairsofbidswithan‘implication’constraint.

(c) 𝑋*+:setofsetsofbidswith‘exclusivechoice’constraints.

7. ∀𝑛 ∈ 𝑁, ∀𝑎 ∈ 𝐴*, ∀𝑏 ∈ 𝐵*+:

(a) 𝑃𝐹*+´:powerfactor.

(b) 𝑥*+´: the minimal acceptance fraction for the variable 𝑥*+´. It holds for all primitive bids it

generates.

(c) 𝑎*+´,�4:theassumedcurrentstateofactivationoftheofferedbid.

(d) 𝛼*+´:maximalnumberofactivations(start-ups)intheoptimizationwindow.

(e) 𝛿*+´µ :minimalduration(timesteps)oftimeinwhichtheunitmustbeactivated.

(f) 𝛿*+´µ

:maximalduration(timesteps)oftimeinwhichtheunitcanbeactivated.


(g) 𝛿*+´¶ :minimalduration(timesteps)oftime inwhichtheunitmustbedeactivated.Theparameter

𝛿*+´¶ has0asitsdefaultvalue.

(h) 𝛼*+´,·:Booleansindicatingthestartofactivationpriortocurrenttimestep.Thesemustbedefinedif

constraint (e) or (f) exists. For constraint (f), 𝜏 ∈ −𝛿*+´µ

+ 1, … , −2, −1 , and for constraint (e), ∀𝜏 ∈

−𝛿*+´µ + 1, … , −2, −1 .

(i) 𝜔*+´,·:Booleansindicatingstopofactivation(i.e.deactivation)priortocurrenttimestep.Thesemust

beavailableifconstraint(g)exists,inwhichcase𝜏 ∈ −𝛿*+´¶ + 1, … , −2, −1 .

(j) 𝛥𝑞*+´:maximaldecreaseofenergyquantityfromanytimesteptothenextone.

(k) 𝛥𝑞*+´:maximalincreaseofenergyquantityfromanytimesteptothenextone.

(l) 𝑇*+´:setof(consecutive)timestepsinaQtbid.

8. ∀𝑛 ∈ 𝑁, ∀𝑎 ∈ 𝐴*, ∀𝑏 ∈ 𝐵*+, ∀𝑡 ∈ 𝑇*+´:

(a) 𝑞*+´l:maximalavailablequantityofactivepowerforaprimitivebid(step/trapezoidal).

(b) 𝑝*+´l:biddingpriceatchosenquantity(correspondingtothemarginalcostsofthebidder).

(c) 𝑚*+´l: ℝ → ℝ:mappingofpowerquantity𝑞tomarginalcost(price)𝑝.Itimpliesthat𝑝 = 𝑚*+´l(𝑞)

(note that reactivepowerhasnodirectlyspecifiablemarginalcosthere.Abidder takes this indirectly into

accountviathepowerfactor𝑃𝐹*+´though.)

(d) 𝛱 = 𝛱*+´l:setofprimitivebids,resultingfromdecompositionontheQ-axis

(e) 𝛱k)À|l:subsetof𝛱,includingcommonbidswiththerightsignof𝑞(usefulforeliminationofsystem

imbalance).

(f) 𝛱ÁkÂ*À:subsetof𝛱, includingcommonbidswiththewrongsignof𝑞 (notusefulforeliminationof

systemimbalance).

Variables

Variablescanberealorbinary(orinteger,tobemoregeneral).Binaryvariablestakevaluesin 0,1 .

1. ∀𝑛 ∈ 𝑁, ∀𝑎 ∈ 𝐴*, ∀𝑏 ∈ 𝐵*+:

(a) 𝑎𝑎*+´ binary whichequals1 if thebid isactivatedforanytimestepwithinthehorizon.Thenwe

callthebid‘accepted’.

2. ∀𝑛 ∈ 𝑁, ∀𝑎 ∈ 𝐴*, ∀𝑏 ∈ 𝐵*+, ∀𝑡 ∈ 𝑇*+´:

(a) 𝑎*+´l(binary)whichequals1ifthebidisactivatedforthetimestept.


(b) 𝛼*+´l(binary)whichequals1 for time stepatwhich thebidgoes fromnon-activated toactivated

(activationbegins).

(c) 𝜔*+´l(binary)whichequals1 for timestepatwhich thebidgoes fromactivated tonon-activated

(activationends).

3. ∀𝑛 ∈ 𝑁, ∀𝑎 ∈ 𝐴*, ∀𝑏 ∈ 𝐵*+, ∀𝑡 ∈ 𝑇*+´, ∀𝛽 ∈ 𝛱*+´l:

(a) 𝑥*+´l�(real): 0 ≤ 𝑥*+´l� ≤ 1:fractionofquantity𝑞*+´l�ofprimitivebid𝛽thatisactivated.

9.1.2 StructureofDataforDefinitionofBids

9.1.2.1 DefinitionofUNIT-bids

PracticallyUNIT-bidsincolumns1,2and3canrespectivelybespecifiedas:

1. (BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,STEP,NON_CURTAILABLE,POWER_FACTOR,Q0,P0),

2. (BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,STEP,CURTAILABLE,POWER_FACTOR,Q0,P0,Q1),

3. (BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,PWL,CURTAILABLE,POWER_FACTOR,Q0,P0,Q1,P1)

Someclarificationsontheindividualfieldsareprovidedhere:

1. ThementionedfieldBID_ISSUER_IDistheuniqueIDofeachmarketparticipant

2. The mentioned field BID_ID represents an identifier that uniquely identifies the bid amongst the

(current) bids of that particular bid issuer. Bid issuers are responsible for the uniqueness of BID_ID. If an

answertothisbidissentbacktothebidissuer,itwillcontainthisBID_IDsothatthebidissuerknowswhatbid

the answer is a response to. Note that the couple (BID_ISSUER_ID, BID_ID) uniquely identifies a bid in the

marketbutBID_IDalonedoesnot.

3. TheargumentUNIT_BIDdefinesthetypeofthebidregardingitsdimension,meaningthatitisnota

Q-bidnoraQt-bid.

4. Theidentifiers{STEP,CURTAILABLE},{STEP,NON_CURTAILABLE},and{PWL,CURTAILABLE}arethere

todistinguishamongthetypesofbidsregardingthequantity.Acurtailablestepbidcanonlyprovideafixed

quantity(orvolume)ifaccepted.Acurtailablestepbidmaybeacceptedatanyquantitybetweenzeroanda

maximumquantity(foraUNIT-bid).Apiece-wiselinearcurtailablebidissimilartoacurtailablestepbid,with

thedifferencethatitspricechangeslinearlywithchangingquantity.

5. ThementionedfieldNETWORK_NODE_IDcontainsavaluethatuniquelyidentifiesanodewithinthe

electricitynetworkmodelledinthemarketclearing.

6. The mentioned POWER_FACTOR gives the ratio of reactive power to the active power and is

supposedtobeconstantpertimestep.


7. ThementionedfieldsQiandPicontainnumericalvaluesexpressedinenergyunitandcurrencyunit.

Forsimplicity,itisassumedthattheyarefixed(forallbids)toMWandEUR/MW,respectively.

9.1.2.2 DefinitionofQ-bids

Q-bidswithseveralpairsof 𝑞, 𝑝 incolumns1,2and3ofFigure3.2canrespectivelybespecifiedas:

1. (BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,STEP,NON_CURTAILABLE,POWER_FACTOR,Q0,P0,

Q1,P1,...,Qn-1,Pn-1),

2. (BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,STEP,CURTAILABLE,POWER_FACTOR,Q0,P0,Q1,P1,

...,Qn-1,Pn-1),

3. (BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,PWL,CURTAILABLE,POWER_FACTOR,Q0,P0,Q1,P1,

...,Qn-1,Pn-1)

Becausewewanttoallowanaggregator topotentiallyofferbothpower injection (positiveQ)aswellas

power offtake (negative Q) from the network, we allow a general Q-bid to specify prices for a range of

quantitiesthatcangenerallygofromanegativetoapositivevalue.

It follows from the principle of themerit-order acceptance of bids, thatwe require, that the Pi on the

positiveenergysideareincreasingwithincreasinglypositiveQ,andthePionthenegativesideareincreasing

withincreasinglynegativeQ.(TheactualSTEPcurvecanstillhaveflatparts).Notethatthereisnorestriction

onpositiveornegativepricesandnoapriorirestrictiononconvexityoftheassociatedcurve.

9.1.2.3 DefinitionofQt-bids

Qt-bidsincolumns1,2and3ofFigure3.2canbespecifiedas:

(BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,STEP,NON_CURTAILABLE,

t=0,Q0,P0,Q1,P1,...,Qn-1,Pn-1,POWER_FACTOR,

t=1,Q0,P0,Q1,P1,...,Qn-1,Pn-1,POWER_FACTOR,

…

t=11,Q0,P0,Q1,P1,...,Qn-1,Pn-1,POWER_FACTOR)

andsimilarly,forthe(BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,STEP,CURTAILABLE,…)and

(BID_ISSUER_ID,BID_ID,NETWORK_NODE_ID,UNIT_BID,PWL,CURTAILABLE,…)forms.Here,t=0referstothe

firstdecision timestepand t=11 to the twelfthone (ifweassume5-minute time stepwitha1hour rolling

optimizationhorizon).Notethatthedecisiontimestepsizewillbefixedinthemarketsystemandknownto

marketparticipants.

9.1.3 DefinitionofAcceptance


9.1.3.1 AcceptanceofaQ-bid

WecallaQ-bid‘accepted’,ifanyofthesequenceofitsprimitivebidsitiscomposedofisactivatedfor

a fraction 𝑥 larger than 0. The Boolean variable 𝑎*+´l indicates whether the bid is activated or not.

Whetherthisisthecaseornotcanbederivedfromtheacceptanceofitscomposingprimitivebidsbythe

constraints

𝑎*+´l ⋅ 𝑥*+´ ≤ 𝑥*+´l�

�∈�ÉÊËÌ

≤ 𝑎*+´l ⋅ 𝛱*+´l (26)

where 𝛱*+´l isthenumberofelementsintheset𝛱*+´land𝑥*+´istheminimumacceptancefraction

atwhichacceptanceisconsideredrelevant,forexample0.01. If intheleft-handsideof(26),𝑎*+´l = 1,

the summation in themiddle part is bigger than 0, so at least one𝑥*+´l� is forced to be non-zero. If

𝑎*+´l = 0,theright-handsideforcesthesummationinthemiddleparttobe0,soall𝑥*+´l�areforcedto

bezero.

Theconstraints

∀𝛽 ∈ 𝛱*+´l,𝑥*+´l� ≤ 𝑎*+´l (27)

can be added on top, but are not strictly necessary. If they turn out to be speeding up the

optimization,theycanbeadded.

9.1.3.2 AcceptanceofaQt-bid

WecallaQt-bid‘accepted’iftheQ-bidforany(atleastone)ofitstimestepshasbeenaccepted.This

isimposedbythefollowingconstraints:

𝑎𝑎*+´ ⋅ 𝑥*+´ ≤ 𝑎*+´ll∈�ÉÊË ≤ 𝑎𝑎*+´ ⋅ 𝑇*+´ . (28)

When𝑎𝑎*+´ = 0, theright-handsideforces𝑎*+´l tobe0forall𝑡 ∈ 𝑇*+´.When𝑎𝑎*+´ = 1, the left-

handsideisbiggerthanzeroandthesummationinthemiddleisforcedtobebiggerthanzero,soatleast

one𝑎*+´lhastobe1(accepted).

Theconstraint

∀𝑡 ∈ 𝑇*+´,𝑎*+´l ≤ 𝑎𝑎*+´ (29)

can be added to themodel, but it is not strictly required. If it speeds up solving the problem, it is

useful. Thevalues𝑎𝑎*+´will allowus todefineBoolean relationsbetween theacceptanceofdifferent

bids. Examples are given in the next sections, especially in the definition of “Accept All Time Steps or

None”constraint.

9.1.4 FormulationofConstraints


9.1.4.1 HelperActivationandDeactivationPulseVariables

Tobeabletodefineconstraintsinthenextsections,wedefinesomehelperBooleans.αNOSTforthestart

of activationandωNOST for the stopof activation (i.e. deactivation). These variables relatewith the already

definedactivationlevelaNOSTas

∀𝑡 ∈ 𝑇*+´\{0},𝑎*+´l = 𝑎*+´,l�4 + 𝛼*+´l − 𝜔*+´l (30)

To avoid that, for a given bidnab, bothαNOST andωNOST are true for the same time step, we add the

constraints

𝛼*+´l + 𝜔*+´l ≤ 1 (31)

Equations(30)and(31)areaddedtotheoptimizationmodelandtogetherimplythatwhenabidisalready

activated (whenaNOS,T�4 = 1), then it cannotbeactivatedagain (αNOST = 1 cannotoccur). Similarly,whena

bid is de-activated (aNOS,T�4 = 0), it cannot be deactivated again (αNOST = 0 cannot occur). Therefore, the

signalsαNOST,ωNOST, andaNOST are compatible andwell defined. This relation is graphically represented for

someexamplewaveformsinFigure9.1.ItshowsafirstpulseonαNOSTmarkedas1,whichleadstoaNOSTgoing

high.ThefirstpulseonωNOSTmarkedas1,makesaNOSTgoingdownagain.Thefirstactivationperiodismarked

asa4.SimilarstartandendpulsesoccurforthesubsequentactivationsmarkedasaEandaF.

As mentioned before, we assume the activation state of the bid available as the value of the variable

aNOS,�4.Theinitialconditioncorrespondingtothegeneralcasein(30)is

𝑎*+´,3 = 𝑎*+´,�4 + 𝛼*+´,3 − 𝜔*+´,3 (32)

Ifabidisofferedforthefirsttimetothemarket,inthattimestep,thevariableaNOS,�4willbe0.Forbids

thatarerepeated ineverytimestep,thevariableaNOS,�4couldbeeither0orcouldbe1already.Eitherthe

market tracks the bid repetitions and then calculates aNOS,�4 as aNOS,3 from the previous time step or the

bidderhimselfsuppliesthisstateofaNOS,�4tothemarket.Therepetitionofbidsovertimestepsisrecognized

bythemulti-time-stepclearingalgorithmbynotingthattheyhavethesamebididentificationnumber.

Figure9.1Relationofpulses𝜶𝒏𝒂𝒃𝒕and𝝎𝒏𝒂𝒃𝒕toactivationlevel𝒂𝒏𝒂𝒃𝒕.


9.1.4.2 Accept-All-Time-Steps-or-NoneConstraint

Whentheactorspecifies that thebid is ‘Accept-All-Time-Steps-or-None’, the followingconstraintwill

beset:

𝛿*+´µ = #𝑇*+´ (33)

where#𝑇*+´ isthebidhorizon(i.e.numberoftime-stepsintheQt-bid).Notethatall𝛿′𝑠areconstant

parametersandnotvariables.

9.1.4.3 RampingConstraints

Wesupposetherampingconstraintsareexactlythesameforeverytwoconsecutivetimestepsofthe

bid.Itholdsthat:

𝑞*+´,l ≤ 𝑞*+´,l�4 + 𝛥𝑞*+´ (ramp-uplimit) (34)

and

𝑞*+´,l ≥ 𝑞*+´,l�4 + 𝛥𝑞*+´ (ramp-downlimit) (35)

Note that 𝛥𝑞*+´ ≥ 0 and 𝛥𝑞*+´ ≤ 0. This can also be used to allow specification of the ramp

constraintsatthefirsttime-stepinthebid:

𝑞*+´,3 ≤ 𝑞*+´,�4 + 𝛥𝑞*+´ (36)

𝑞*+´,3 ≥ 𝑞*+´,�4 + 𝛥𝑞*+´ (37)

9.1.4.4 MaximumNumberofActivationsConstraint

Tobeabletocountthenumberofstartsandstopsofactivationsduringthebidtimespan,thevariables

αNOSTandωNOSTdefinedpreviouslyareused.Theconstraintfortheupperboundonthenumberofactivations

isthen

𝛼*+´l

l∈�ÉÊË

≤ 𝛼*+´ (38)

NotethatthelimitsαNOSandωNOSareonaperbidbasis,andarenotgloballyconstrainingallbidsofthe

bidder.

9.1.4.5 MinimumandMaximumDurationofActivationConstraint

As for the minimum duration, the following equation expresses, that whenever, for a certain τ = t −

δNOSÛ + 1,αNOSTis1(beginningofactivation),aNOSTwillbe1(active)foratleasttheδNOSÛ timeslotststartingat

τ = t − δNOSÛ + 1andendingatτ = t.

∀𝜏 ∈ 𝑡 − 𝛿*+´µ + 1 ,𝛼*+´,· ≤ 𝑎*+´,l (39)


Notethattheconstraints(39) implythatforthefirsttime-step,t = 0,valuesofαNOS,Ü for δNOSÛ − 1time

stepsbeforet = 0stillhavetobeknown.Theycaneitherbebiddersuppliedorhavetoberememberedfrom

thesamebidintheprevioustimestep.Thelatterassumesthatthemarketclearingalgorithmwouldbeableto

trackonebid in subsequentdecision time-steps.This couldbedonebymaking thebid ID remain the same

between subsequent time steps. For the scopeof this document,we assume this extra informationwill be

availablesomehow,eitherrememberedbythemulti-stepclearingalgorithmorsuppliedbythebidderdirectly.

Therefore, thevaluesαNOS,Ü, ∀τ ∈ δNOSÛ + 1, … , −2, −1 areassumedtobeavailable.Theirdefaultvaluesat

start-upshouldbezero.

SettingδNOSÛ = 0leadstononeoftheconstraintsoftype(39)beinggenerated,whichisfine,butitisnota

usefulcase.SettingδNOSÛ = 1 leads tooneconstraintper timestep tbeinggeneratedαNOS,Ü ≤ aNOS,T.This is

alsofineandthiswillbethedefaultvalueforδNOSÛ whentheuserdoesnotexplicitlyspecifyavalue.

Asforthemaximumduration,theconditionthataNOS,Tshouldnotbe1anylongerthanδNOSconsecutive

timesteps,cansimplybeenforcedby:

𝑎*+´,·l·Ýl�ÞÉÊË

ß ≤ 𝛿*+´µ

(40)

Indeed,the left-handsideof (40) isasumofδNOSÛ

+ 1 termsandshouldsumuptoatmost δNOSÛ

socan

neverbeequaltoδNOSÛ

+ 1.

ThevaluesaNOS,Ü, ∀τ ∈ −δNOSÛ

, … , −2, −1 areassumedtobeavailable.

SettingδNOSÛ

= 0willnotallowanyaNOS,Ttobe1foranyt.Thiswilldisallowanyactivationofthebid,which

canbeusefulfortestingpurposes,butnototherwise.SettingδNOSÛ

= TNOS allowsactivationtolastaslongas

thebidhorizon,whichistakentobethedefaultiftheuserdoesnotspecifyδNOSÛ

explicitly.

9.1.4.6 MinimalDelayBetweenTwoActivations

Theequationsreflectingminimumdelaybetweentheendofanactivationandthebeginningofthenext

activationaresimilartotheonesforthelimitsonthewindowbetweenbeginningandendofactivationinthe

lastsection.Essentially,onereplacesαNOSTbyωNOST,δNOSÛ

byδNOSà

,δNOSÛ byδNOSà ,andaNOSTby1 − aNOST.

Thisgivesfortheminimumdelaybetweenendandbeginningofnextactivation

∀𝜏 ∈ 𝑡 − 𝛿*+´¶ + 1,… , 𝑡 : 𝜔*+´,· ≤ 1 − 𝑎*+´,l (41)

Itisinterestingtonotethatforamaximumdelaybetweenactivations,theconstraintwouldbe

1 − 𝑎*+´,·l·Ýl�ÞÉÊË

á ≤ 𝛿*+´¶

. (42)


However,wecannotallowabiddertoimposeamaximumonthetimeofbeingnon-activesincethenthe

biddercouldexploit this trick to force themarket toactivatehisbid.Therefore, theaboveconstraint isnot

addedtothemodel.

Asexplainedbefore,thevaluesaNOS,Ü, ∀τ ∈ −δNOSà + 1, … , −2, −1 arealsoassumedtobeavailableand

alsodefaulttozero.ThevaluesaNOS,Ü, ∀τ ∈ −δNOSà

, … , −δNOSà arenotusedandmaynotbedefinedatall.

9.1.4.7 IntegralConstraint

Asmentionedbefore,themarketclearingalgorithmwillonlyacceptbidswithabidhorizonTNOSsmalleror

equaltotheoptimizationtimewindowT.Theintegralconstraintisformulatedas:

𝑒*+´) ≤ 𝑞*+´l ⋅ 𝛿𝑡l∈�ÉÊË

≤ 𝑒*+´)

(43)

Note that,q (power quantity) is inMWande (energy quantity) is inMWh.When the lower andupper

boundsareequal,asingleequalityconstraintresults.Asaspecialcase,whentheboundsaresettozero,the

integralconstraintguaranteesthattheasset(usuallyathermalorelectricalstorage)returnstoitsinitialvalue

attheendofthebid-horizon,oritsenergycontentremainswithinthepre-definedbounds.

9.1.4.8 ImplicationConstraint

TheimplicationconstraintisdefinedinamatrixINO ⊂ BNO×BNO.Itisformulatedas:

∀´å´æ 𝑏, 𝑏z ∈ 𝐼*+, 𝑎𝑎*+,´ ≤ 𝑎𝑎*+,´æ (44)

IfaaNO,S = 1thenaaNO,Sæ isforcedtobe1.IfaaNO,S = 0,thennothingisenforcedonaaNO,Sæ.Therefore,the

implicationb → bzholds.

9.1.4.9 ExclusiveChoiceConstraint

This limitation is setupby firstcreatingasublist lwithina listXNOofbid-identifiers,andthenwithin the

sublist,requiringthatonebidcanbeacceptedatmost.Thisisenforcedbythefollowingconstraint:

∀𝑙 ∈ 𝑋*+, 𝑎𝑎*+´´∈r

≤ 1 (45)

Also,thisconstraintcanonlybeenforced,bythebidder,betweenbidsatthesamenodeandbelongingto

thisbidder.

9.1.4.10 DeferabilityConstraintWedonotformulateanindependentconstraintfordeferability,sinceitcanbedonebysubmittingtheQt-

bidandthedifferentshiftedversionsofitandthenaddinganexclusive-choiceconstrainttoit,ensuringthatat

mostoneoftheshiftedversionsofthebidisaccepted.


Toeasetheprocess formarketparticipant,weproposeaproductthatallowsexpressingthedeferability

explicitlyusing:

DEFERABILITY=Max.

Thisfielddescribesthenumberofdecisiontimestepsthattheofferedbidcouldbemaximallydelayed.The

value isboundedby theoptimizationhorizon𝑇.Theconstraint is then internallyconverted toanexclusive-

choiceconstraint.

9.2 NetworkModeling

9.2.1 Nomenclature

GridElementParameters NodeVariables𝑧r,t Seriesimpedance(Ω) 𝑈) Nodevoltagemagnitude(V)𝑟r,t Seriesresistance(Ω) 𝑈)z Nodevoltagemagnitude(V)𝑥r,t Seriesreactance(Ω) 𝑈)∗ Nodevoltagemagnitude(V)𝑦r,t Seriesadmittance(S) 𝜃) NodeVoltageangle(rad)𝑔r,t Seriesconductance(S) 𝜃)z NodeVoltageangle(rad)𝑏r,t Seriessusceptance(S) 𝜃)∗ NodeVoltageangle(rad)𝑧)p,t| Shuntimpedance(Ω) 𝑆) Aggregatedapparentpower(VA)𝑟)p,t| Shuntresistance(Ω) 𝑃) Aggregatedactivepower(W)𝑥)p,t| Shuntreactance(Ω) 𝑄) Aggregatedreactivepower(var)𝑦)p,t| Shuntadmittance(Ω) 𝐼) Aggregatedcurrent(A)𝑔)p,t| Shuntconductance(S) Nodesquaredvariables𝑏)p,t| Shuntsusceptance(S) 𝑢) Nodesquaredvoltage(V²)

GridElementBounds 𝑢)z Nodesquaredvoltage(V²)𝑎)p()* Mintapratio(-) 𝑢)∗ Nodesquaredvoltage(V²)𝑎)p(+, Maxtapratio(-) Sets𝜑)p()* Minphaseshift(rad) 𝒰 Setsofunits𝜑)p(+, Maxphaseshift(rad) ℐ Setsofnodes𝛼r()* Stateboundofsection(0,1) ℐïð Setsofslacknodes𝛼r(+, Stateboundofsection(0,1) ℐ*Â*ïð Setsofnon-slacknodes

GridElementRatings 𝒥 Setsofgridelements𝑈)pk+lmn VoltageratingatsideI(V) Othersymbols𝐼)pk+lmn CurrentratingatsideI(A) 𝑀 bigM𝑆)pk+lmn Apparentpowerratingatside𝑖(VA) Gridelementsquaredvariables

GridElementVariables 𝐼r,tE Squaredseriescurrentmagnitude(A²)𝑃)p Activepowerflow(W) 𝑖)p,t Variablesquaredseriescurrentmagnitude𝑄)p Reactivepowerflow(var) NodeParameters𝑃rrÂtt Activepowerloss(W) 𝑈)

kmò Voltagereference(V)𝑄rrÂtt Reactivepowerloss(var) 𝜃)

kmò Voltageanglereference(rad)𝐼)p GridElementcurrent(A) NodeBounds𝐼)pkm Realcurrent(A) 𝑈)()* Minboundonvoltagemagnitude(V)𝐼)p)( Imaginarycurrent(A) 𝑈)(+, Maxboundonvoltagemagnitude(V)𝐼)p,t seriescurrent(A) Noderatings 𝐼)p,t| shuntcurrent(A) 𝑈)k+lmn Nodevoltagerating(V)𝜃)p Voltageangledifference(rad) 𝛼)p stateindicatorforgridelement(0,1)


𝑎)p instantaneousvoltagetransformationratio(-)

𝜑)p phaseshift(rad)

9.2.2 Notation

With the exception of Yalmip [92] and CVX, common algebraic modelling languages only support real-

valued variables. Intermediately, complex variables will be used in derivations, but ultimately, real-valued

constraintsetsareformulated.NotethatquadraticrepresentationofSOCPconstraintsisused,butthesecan

beconvertedtothe2-normvectorrepresentation.

𝑥4E + 𝑥EE ≤ 𝑥FE, 𝑥F ≥ 0 𝑥4E + 𝑥EE ≤ 𝑥F

𝑥4𝑥E E

≤ 𝑥F (46)

Finally, a typographicdistinction ismadebetween scalar variables, e.g.Wij andmatrix variables, e.g.W.

Valued indexes are subscripted and stylized in a specific font, e.g. ‘i’, whereas descriptive indexes are

superscriptedandstylizedasnormaltext,e.g.‘s’.

9.2.3 PhysicsofPowerFlow

In power system modeling, the system is typically represented as a single-wire diagram. This

representationmatchesnicelywiththeconceptofmathematicalgraphs.Mathematicalgraphsarecomposed

of nodes/vertices/buses and edges/arcs/lines. To convert circuits tomathematical graphs, edgemodels for

conductingelementsaredeveloped.

Gridnodesandelementsareassignedindexesasfollows:

• Gridnodesi;j∈ ℐ = ℐïð ∪ ℐ*Â*ïð

• Slackbusnodesi∈ ℐïð

• Non-slackbusnodesi∈ ℐ*Â*ïð

• Gridelementsl,ij∈ 𝒥

Forsymmetricvariablesandparameters,theindexlispreferredtoijorji.

Nodesareconnected toeachother throughconductiveelements.Conductinggridelement technologies

include:overheadlines,undergroundcables,classictransformers,on-loadtap-changingtransformers,phase-

shiftingtransformers,switches,andbreakers.


Figure9.2Circuitrepresentationoftheclassicπ-element

9.2.4 GridElements

In this section, the focus ison cablesand lines; inupcoming sections, switchesand transformerswill be

added.Naturalnotation isseries impedance,zl,s,andshuntadmittance,yl,sh.Nevertheless,admittanceforms

can be freely converted to impedance forms and vice versa, except for degenerate cases (negligible

admittance/impedance).Itisnotedthatinthescientificliterature,papersswitchthesignofbl,sh(e.g.y=g+

jb[26]vs.y=g-jb[20]).

Figure9.2displaysthecircuitrepresentationoftheclassicπ-element,whichiscommonlyusedtorepresent

lines&cables.Theseriesimpedance(symmetricij=ji=l)isdefinedas:

zl,s=rl,s+j.xl,s (47)

Theshuntadmittanceisdefinedas:

yl,sh=gl,sh+j.bl,sh (48)

Generally,theshuntsareassumedtobesymmetric,dividedequallytotheiandjsideoftheπ-element.

9.2.5 Kirchhoff’sCircuitLaws

The Kirchhoff voltage law (KVL) states that the voltage drops in any (circuit) loop add up to zero. The

Kirchhoffcurrentlaw(KCL)statesthatthecurrentsatanynodeadduptozero.Thisarticleusespolarnotation

forvoltagevariables,withUi thevoltagemagnitudeandθi thevoltageangle.Forcurrents, thevariablesare

.

𝑈)∠𝜃) − 𝑈p∠𝜃p = 𝐼)p,t𝑧r,t (49)


𝐼)p,t| = 𝑦)p,t| 𝑈)∠𝜃) (50)

𝐼p),t| = 𝑦p),t| 𝑈p∠𝜃p (51)

𝐼)p = 𝐼)p,t + 𝐼)p,t| (52)

𝐼p) = 𝐼p),t + 𝐼p),t| (53)

𝐼)p,t + 𝐼p),t = 0 (54)

HerewehaveOhm’slaw(49),thecurrenttotheshuntelements(50)-(51),andKirchhoff’scurrentlaw(52)-

(54) As is easily verified, Kirchhoff’s circuit laws are linear in voltage and current for constant impedances.

Notethatthisdescribesonlyonegridelement.Inmeshedgrids,KVLalsohastobeappliedinthemeshloops.

9.2.6 PowerFlow

As the purpose of the grid is to transport energy, stakeholders are interested in quantities of energy,

power,andcost.Power inACgridsconsideredascomplexrectangularvariableSij ,withPij theactivepower

andQijthereactivepower.Thecomplexconjugateoperatorisasuperscripted‘*’.EventhoughKLsappliedto

π-element results in a linear set of equations in current and voltage, adding equations tomodel (complex)

powerflowmakesthesetofequationsnonconvex:

𝑆)p = 𝑃)p + 𝑗𝑄)p = 𝑈)∠𝜃) 𝐼)p∗ (55)

Historically,OPFhasbeenbasedonthestaticapproximationtothepowerflowequations.Thismeansthat

inthepowerflowequations,therearenotimederivatives.

Voltage angles and reactive power do not have any useful meaning in the DC power flow domain.

Nevertheless,theequationsremainvalid(withθij=0;Qij=0;xl,s=0;bij,sh=0).

With interacting DC and AC grids, AC-DC converter models (for reactive power control of the AC-DC

converter)aretobeincluded.

9.2.7 Nodes

9.2.7.1 AnyNodeNodalaggregatepoweristhesumofallthepowersflowsawayfromnodeitoanyconnectednodej:

Voltageboundscanbeappliedatanynode.Theseusersuppliedboundscanbeusedtoreflectqualityof

supplystandardssuchasEN50160[93].

𝑈)()* ≤ 𝑈) ≤ 𝑈)(+, (56)


9.2.7.2 NormalNodesComplex power flows Pi and Qi are determined by the units connected to the node. Voltage Ui∠𝜃)

consequentlyvaries.Someunitsmayhavecontrollerswhichregulatethevoltage.Commonly,thebusesthat

havegeneratorswithsuchcontrollers,arecalledPVbuses.Typically,then,reactivepowerdispatchischanged

tokeepthevoltagesteady.Moregenerally,activepowerdispatchcanalsobeadaptedtoregulatethevoltage.

9.2.7.3 VoltageanglereferencenodeOnenodeisrequiredtohaveavoltageanglereference:

𝜃)kmò ≤ 𝜃) ≤ 𝜃)

kmò (57)

Often,aslacknodeisusedinsteadofjustavoltageanglereferencebus.

9.2.7.4 SlacknodeAslacknodeisnotrequiredingeneral.However, inaradialgrid,theslacknodeisusedtorepresentthe

interaction with the external nodes, e.g. the transmission system. As the interface with the transmission

system is generally stronger than thedistribution grid connected to it, it is assumed that the slack bus can

support a large complex power flow at a fixed voltage. Conversely, in amicrogrid, one can choose not to

includeaslacknode,butmerelyavoltageanglereferencenode.

Itgenerallymakessensenottoconnectanyloadsorgeneratorsdirectlytothisnode,astheseunitsdonot

influencethepowerflowphysics.Ataslacknode,voltagemagnitudeandanglearefixed,buttherearefree

variables(slack)forthecomplexpowerflow:

𝜃)kmò ≤ 𝜃) ≤ 𝜃)

kmò (58)

𝑈)kmò ≤ 𝑈) ≤ 𝑈)

kmò (59)

−∞ ≤ 𝑃) ≤ ∞ (60)

−∞ ≤ 𝑄) ≤ ∞ (61)

9.2.8 ClassicPowerFlowFormulationsThepowerflowderivation inthissection isbasedontheclassicπ-elementrepresentationasdepicted in

Figure9.2.Threeformulationsarederived:rectangular,polarandDistFlow.Anyofthoseformulationscanbe

usedstraightawayinNLPsolvers.Commonly,thepolarformulationisimplemented.

9.2.8.1 PowerFlowRectangularThepowerbalanceandseriesandshuntcurrentsaredefinedastheproductofcomplexnumbers.

𝑃)p + 𝑗𝑄)p = 𝑈)∠𝜃) (𝐼)p,t| + 𝐼)p,t (62)

𝐼)p,t = ( 𝑈)∠𝜃) − 𝑈p∠𝜃p )𝑦r,t (63)


𝐼)p,t| = 𝑈)∠𝜃) 𝑦)p,t| (64)

Combiningtheequationsaboveresultsintherectangularformulationofpowerflow:

𝑃)p + 𝑗𝑄)p = 𝑈)∠𝜃) 𝑈)∠𝜃) ∗ − 𝑈p∠𝜃p∗𝑦r,t∗ + 𝑈)E𝑦)p,t|∗ (65)

9.2.8.2 PowerFlowPolarThepowerflowequationsareformulatedinpolarformwithtrigonometricfunctions

𝜃)p = 𝜃) − 𝜃p (66)

𝑃)p = 𝑔r,t + 𝑔)p,t| 𝑈)E − 𝑔r,t𝑈)𝑈p cos 𝜃)p − 𝑏r,t𝑈)𝑈psin(𝜃)p) (67)

𝑄)p = − 𝑏r,t + 𝑏)p,t| 𝑈)E + 𝑏r,t𝑈)𝑈p cos 𝜃)p − 𝑔r,t𝑈)𝑈psin(𝜃)p) (68)

Thephaseangledifferenceoveranysectionisgenerallylimitedto90°:

−𝜋/2 ≤ 𝜃)()* ≤ 𝜃)p ≤ 𝜃)(+, ≤ 𝜋/2 (69)

9.2.8.3 PowerFlowDistFlowTheDistFlowreformulationprocesstakesthesquareofOhm’slaw,andderivesinanonconvexquadratic

equationinthevariablesofUi;Pij;QijandIij,s.

𝑈pE=𝑈)E − 2 𝑟r,t𝑃)p,t + 𝑥r,t𝑄)p,t + (𝑟r,tE + 𝑥r,tE )𝐼r,tE (70)

IntheDistFlowreformulation,thevoltageanglevariablesgetsubstitutedout.Thevoltagemagnitudesand

either end of a grid element can be related purely based on complex power flow and current magnitude

observations.Suchapproximationdoesnotchangetheresultsofthepowerflowinaradialgrid,asthevoltage

anglevariablescanstillbe recovered throughaprocesswhichgivesauniquesolution.However, inmeshed

gridsthisisnotthecaseduetovoltageangleconstraintapplyingtoanyloop.

Thepowerbalanceequationsarederivedasfollows:

Rectangularpowerflow

FeasibleSetI:(65)

Polarpowerflow

FeasibleSetII:(66)-(68)

DistFlowpowerflow

FeasibleSetIII:(70)


0 ≤ 𝑃)p + 𝑃p) = 𝑃rrÂtt = 𝑃r,t|rÂtt + 𝑃r,trÂtt (71)

𝑄)p + 𝑄p) = 𝑄rrÂtt = 𝑄r,t|rÂtt + 𝑄r,trÂtt (72)

𝑃r,trÂtt = 𝑟r,t𝐼r,tE (73)

𝑄r,trÂtt = 𝑥r,t𝐼r,tE (74)

𝑃r,t|rÂtt = 𝑔r,t|𝑈)E (75)

𝑄r,t|rÂtt = −𝑏r,t|𝑈)E (76)

𝐼r,tE = (𝑃)pE + 𝑄)pE ) 𝑈)E (77)

NotethatUiand 𝐼)p,t onlyappearassquaredvariables.Thevoltageanglesshouldsatisfy:

𝜃)p = 𝜃) − 𝜃p = ∠(𝑈)∠θ))( 𝑈)∠θ) − 𝑧r,t𝐼)p,t)∗ (78)

𝜃)p = 0𝑚𝑜𝑑2𝜋rÂÂþ

(79)

Thevoltageangles inany loophavetoaddupto0mod2π,whichistriviallysatisfiedinaradialgridbut

whichrequiresadditionaltreatmentinmeshedgrids[44],[59].

9.2.8.4 OperationalEnvelopes

9.2.8.4.1 GridelementratingsBoundsonapparentpower,currentmagnitudeandvoltagemagnituderelatedtolines:

0 ≤ 𝑆)p ≤ 𝑆)pk+lmn (80)

0 ≤ 𝐼)p ≤ 𝐼)pk+lmn (81)

0 ≤ 𝑈) ≤ 𝑀 ∙ 𝑈)pk+lmn (82)

It isnotedthattheratingsdonotneedtobesymmetric(ij≠ ji),e.g.voltageratingsbeingdifferentwith

transformers. It is furthermorenoted that such voltage ratings, (which are generally always known, i.e. the

rating depends on the cablematerial, insulation level, etc.), can be used to derive generic bounds on the

voltages,butonlybyapplicationofabigM.

9.2.8.4.2 NoderatingsBoundsvoltagerelatedtonodes:

Gridelementratings

FeasibleSetIV:(80)-(82)


0 ≤ 𝑈)()* ≤ 𝑈) ≤ 𝑈)(+, ≤ 𝑀 ∙ 𝑈)k+lmn (83)

Again, it is assumed that the voltage rating (voltage level) towhich nodes belong, is always known, i.e.

𝑈)k+lmn.Thiscanbeusedtodefineanupperboundonthevoltages,eveninabsenceofuser-suppliedbounds

UH"HN, 𝑈)(+,.

Noderatingandbound

FeasibleSetV:(83)


9.3 ExtendedOPFFormulation

A commonly used model for the conducting elements is the π-element, as discussed in the previous

section,whichincludesaseriesimpedanceandshuntadmittancesateachendoftheline.Thisπ-elementcan

againbegeneralizedtoincludevoltagetransformation,phaseshiftingandswitching[94].Suchaformulation

will bedeveloped in thisworkand is referred to as the ‘extended symmetricπ-element’ representation, as

depictedinFigure9.3.Figure9.4furthermoredisplaysthecomplex-valuedgraphrepresentationcorresponding

totheextendedsymmetricπ-element.Finally,Figure9.5illustratesthereal-valuedgraphrepresentation.

Figure9.3Balancedπ-element-representationofaconductingelement.

Figure9.4Complex-valuedgraphrepresentationoftheextendedbalancedπ-element-representation.

Figure9.5Real-valuedgraphrepresentationsoftheextendedbalancedπ-element-representation.


9.3.1 Transformer

Theeffectivetransformationratioisdefinedas:

𝑎)p = 𝑈)z

𝑈)∗ (84)

𝑎p) = 𝑈pz

𝑈p∗ (85)

Thisvoltagetransformationratioisboundedbyaminimumandmaximumvalue:

0 ≤ 𝑎)p()* ≤ 𝑎)p ≤ 𝑎)p(+, (86)

0 ≤ 𝑎p)()* ≤ 𝑎p) ≤ 𝑎p)(+, (87)

Tooperateataknownratio,thelower𝑎)p()*;𝑎p)()*andupperbounds𝑎)p(+,; 𝑎p)(+,aregivenequalvalues.

Minimumandmaximumphaseshiftaredefinedas:

0 ≤ 𝜑)p()* ≤ 𝜑)p ≤ 𝜑)p(+, (88)

0 ≤ 𝜑p)()* ≤ 𝜑p) ≤ 𝜑p)(+, (89)

𝜃)z = 𝜃)∗ + 𝜑)p (90)

𝜃pz = 𝜃p∗ + 𝜑p) (91)

Tooperateataknownphaseshift,thelower𝜑)p()*; 𝜑p)()*andupperbounds𝜑)p(+,; 𝜑p)(+,aregivenequal

values.Negativephaseshiftscanbehandledbyassigningtheshifttothecorrespondingsideoftheelement.A

negativephaseshiftfromitojcorrespondstoapositivephaseshiftfromjtoi.

Having a transformer at eachendallows you to: define the impedancemodel in any referencebase, to

separate voltage transformation and tap changing, and also to model tap changing at either side of the

transformer.

9.3.2 Switch

Foreachgridelementtohaveaninterpretablestatevariable,switchesarerequiredatbothends. Ifonly

oneswitchisused,powermaystill flowinthegridelementduetothecontributionsoftheshuntelements.

Onlyifswitchingoccursatbothsides,powerflowisguaranteedtobeinterrupted.

Stateαl=0meansbothswitchesareinopenpositionandthegridelementdoesnotconduct.Stateαl=1

means the switches are in closed position and the grid element does conduct. A disjunctive formulation is

developed,whichrequiresanupperboundonthevoltage(‘bigM’notationused).Instateαl=0,thevoltage

𝑈)∗isforcedto0,and𝑈)zisindependentfrom𝑈)∗.Instateαl=1,thevoltage𝑈)∗isnotforcedto0,butequality

of𝑈)zwith𝑈)∗isenforced.Forcing𝑈)∗to0issufficienttostoppowerflowthroughtheelement.


𝛼r ∈ 0,1 (92)

0 ≤ 𝑈)∗ ≤ 𝑀 ∙ 𝛼r ∙ 𝑈)pk+lmn (93)

0 ≤ 𝑈p∗ ≤ 𝑀 ∙ 𝛼r ∙ 𝑈p)k+lmn (94)

0 ≤ 𝑈) − 𝑈)∗ ≤ 𝑀(1 − 𝛼r) ∙ 𝑈)pk+lmn (95)

0 ≤ 𝑈p −𝑈p∗ ≤ 𝑀(1 − 𝛼r) ∙ 𝑈p)k+lmn (96)

Ifaswitchisintheclosedposition,thevoltageanglesateitherendoftheswitchareequal.Iftheswitchis

open,thevoltageanglesarenotrelated.

−𝜋/2 ∙ 𝑀 ∙ (1 − 𝛼r) ≤ 𝜃) − 𝜃)∗ ≤ 𝜋/2 ∙ 𝑀(1 − 𝛼r) (97)

−𝜋/2 ∙ 𝑀 ∙ (1 − 𝛼r) ≤ 𝜃p − 𝜃p∗ ≤ 𝜋/2 ∙ 𝑀(1 − 𝛼r) (98)

−𝜋/2 ∙ 𝑀 ∙ 𝛼r ≤ 𝜃)z ≤ 𝜋/2 ∙ 𝑀 ∙ 𝛼r (99)

−𝜋/2 ∙ 𝑀 ∙ 𝛼r ≤ 𝜃pz ≤ 𝜋/2 ∙ 𝑀 ∙ 𝛼r (100)

Switchesaredynamicelementsandswitchingactionscontributetooperationalcosts.Thefollowingtypes

ofconstraintsareoftenconsideredinthiscontext:

• switchramprates

• switchcosts

• switchingrestrictedtocertaintimeslots

Tofixthestateoftheswitch,bounds𝛼r()*and𝛼r(+,canbeused:

0 ≤ 𝛼r()* ≤ 𝛼r ≤ 𝛼r(+, ≤ 1 (101)

Itisnotedthattheswitchpositiondoesnotneedtobeconsideredasavariable(throughfixingthebounds

andeliminationandsubstitution),eventhoughitcanbeusedasavariable.Nevertheless,thevariableswitch

formulationisalsousedinthemodellingofdiscretetransformertaps,seefurtherfordetailsonthis.

9.3.3 BranchFlowModelFormulation:DistFlow

9.3.3.1 Substitution

The complex current and voltage variables are replaced by squared current and voltage magnitude

variables.Therearenovariablesforcurrentorvoltageangles.

0 ≤ 𝑖)p,t = 𝐼)p,tE (102)

0 ≤ 𝑖)p = 𝐼)pE (103)


0 ≤ 𝑢) = 𝑈)E (104)

0 ≤ 𝑢)z = 𝑈)zE (105)

0 ≤ 𝑢)∗ = 𝑈)∗E (106)

9.3.3.2 SOCPConvexFormulation

TheDistFlowequationsarereformulatedinthesquaredvariables:

0 ≤ 𝑃)p + 𝑃p) = 𝑃rrÂtt = 𝑃r,t|rÂtt + 𝑃r,trÂtt (107)

𝑄)p + 𝑄p) = 𝑄rrÂtt = 𝑄r,t|rÂtt + 𝑄r,trÂtt (108)

𝑃r,trÂtt = 𝑟r,t𝑖r,t (109)

𝑄r,trÂtt = 𝑥r,t𝑖r,t (110)

𝑃r,t|rÂtt = 𝑔r,t|𝑢)z (111)

𝑄r,t|rÂtt = −𝑏r,t|𝑢)z (112)

(𝑃)p,tE + 𝑄)p,tE ) ≤ 𝑢)z ∙ 𝑖r,t (113)

(𝑢)z − 𝑢pz) = −2(𝑟r,t𝑃)p,t + 𝑥r,t𝑄)p,t) + 𝑧r,tE∙ 𝑖r,t (114)

Withrespecttothenonconvexformulation,theconvexificationisthereplacementoftheequalitysymbol

in (77) by the inequality in (113), obtaining a second-order cone. Basically, (114) reflectsOhm’s law / KVL.

Furthermore,KCLisreflectedinthecomplexpowerbalanceequations(107)-(112)and(113)linksthepower,

voltageandcurrentvariablestogetherthroughtheconvexrelaxation.

Switchconstraintsarereformulatedinthesquaredvariables:

𝛼r ∈ 0,1 (115)

0 ≤ 𝑢)∗ ≤ 𝑀² ∙ 𝛼r ∙ (𝑈)pk+lmn)² (116)

0 ≤ 𝑢p∗ ≤ 𝑀² ∙ 𝛼r ∙ (𝑈)pk+lmn)² (117)

0 ≤ 𝑢) − 𝑢)∗ ≤ 𝑀²(1 − 𝛼r) ∙ (𝑈)pk+lmn)² (118)

0 ≤ 𝑢p − 𝑢p∗ ≤ 𝑀²(1 − 𝛼r) ∙ (𝑈)pk+lmn)² (119)

Thepowerflowthroughtheswitchthereforesatisfies:

SOCPDistFlow

FeasibleSetVI:(107)-(114)


𝑃)pE + 𝑄)pE ≤ (𝐼)pk+lmn)² ∙ 𝑈)E ∙ 𝛼r (120)

This formulation effectively limits the total current magnitude 𝐼𝑖𝑗 to a maximum of 𝐼)pk+lmn , but doesn’t

require an explicit variable for the total current or total current magnitude. As the current rating isn’t

necessarily symmetrical, the equation is also considered for ji. This equation, after substitution, can be

formulatedinthenewvariablesas:

𝑃)pE + 𝑄)pE ≤ (𝐼)pk+lmn)² ∙ 𝑢) ∙ 𝛼r (121)

Thisconvexexpressionisarotatedsecond-ordercone.

Finally,thevoltagemagnitudetransformationisreformulatedinanexactmanneras:

(𝑎)p()*)²𝑢)∗ ≤ 𝑢)z ≤ (𝑎)p(+,)²𝑢)∗ (122)

(𝑎p)()*)²𝑢p∗ ≤ 𝑢pz ≤ (𝑎p)(+,)²𝑢p∗ (123)

It is noted that all the constraints only contain real-valued variables and parameters, therefore no

additional reformulation is required. In post-processing, complex-valued variables 𝐼)p, 𝑈)∠𝜃); 𝑆)p are again

obtained.

Thisrelaxationcanbepenalized,tomakesure𝑖)p,tand𝑢)zlayontheconesurface:

min𝑌%&�Â*'m, =1𝑛𝒥

𝑖)p,t(𝐼)pk+lmn)²r∈𝒥

(124)

Notethat𝑃)p,t or𝑄)p,t,also in(113),cannotbeeasilypenalizedduetotheirunknownsign.Theobjective

𝑌%&�Â*'m, is considered as a penalty in OPF problems with a wider scope, e.g. a multiperiod OPF with

operationalcostminimization.FormoreinsightsonthepenalizationoftheSOCPrelaxation,seesection4.4.5.

9.3.3.3 OperationalEnvelopes

Thevoltage, currentandcomplexpowerenvelopesare representable in the squaredvariablesby taking

thesquareoftheoriginalbounds.

(𝑈)()*)² ≤ 𝑢) ≤ (𝑈)(+,)² (125)

0 ≤ 𝑢)z ≤ 𝑀² ∙ (𝑈)pk+lmn)² (126)

0 ≤ 𝑖)p,t ≤ 𝑀² ∙ (𝐼)pk+lmn)² (127)

𝑃)pE + 𝑄)pE ≤ (𝑆)pk+lmn)² (128)

Switchinginsquaredvoltagemagnitudevariables(MILP)

FeasibleSetVII:(115)-(121)


Notethattheapparentpowerflowboundoncomplexpowerflow(83)isacircularareaconstraint(𝑥4E +

𝑥EE ≤ 𝑟E), which is convex and representable as a SOCP constraint. Note also that by including expression

(127)anexplicitformulationforthecurrentboundisaddedtotheconstraintset.

Ultimately,theoverallfeasiblesetisderived:

9.3.3.4 Post-Processing

Theslackontheconvexificationisdefinedasfollows:

𝜖)p%&�Â*'m, = 𝑖)p,t ∙ 𝑢)z − 𝑃)p,tE − 𝑄)p,tE (129)

If𝜖)p%&�Â*'m, ≈ 0thenaphysicallymeaningfuloptimumisobtained.Duetonumericaleffects,theslackwill

notbe0exactly,andcanevenbeslightlynegative.If𝜖)p%&�Â*'m, ≉ 0thentheresultsarenotphysical.Slackon

theconvexificationimpliesartificiallyincreasedgridlosses.

Thevoltageangledifferencemustadduptozeroinanycycleinameshedgrid[59].

𝜃)pz = 𝜃)z − 𝜃pz = ∠ 𝑢)z − 𝑧r,t𝑆)p (130)

𝜃)p = 0𝑚𝑜𝑑2𝜋)p∈rÂÂþ

(131)

This condition is trivially satisfied in a radial grid, but the error has to be checkedwhen this formulation is

appliedtoameshedgrid.

Nopost-processing is required for𝑃)p, 𝑄)p, 𝛼r.Theoptimizedvariables𝑢), 𝑢)∗, 𝑢)z, 𝑖)p,t need tobemapped

back to the original variables 𝑈), 𝑈)∗, 𝑈)z, 𝜃), 𝜃)∗, 𝜃)z, 𝐼)p . A possible procedure for post-processing is the

following.Start fromslackbus,andsolvegridelementbygridelement,whilewalkingdownthe treeof the

network.

Itisnotedthatagenericpowerflowsolvercanalsobeusedtorecovertheresults.Powerflowsolverstake

as input the optimized dispatch of generators and loads (𝑃+;𝑄+) and the grid model (topology and

parameters),tocalculatevoltagesandcurrentsthroughoutthesystem.

ExtendedSOCPDistFlowenvelopes

FeasibleSetVIII:(122)-(124),(125)-(128)

ExtendedSOCPDistFlow

FeasibleSetIX:VI,VII,VIII


9.3.4 BusInjectionModelFormulation

9.3.4.1 Substitution

Thecomplexvoltagevariablesarereplacedbycross-productsofnodevoltages.Therearenovariablesfor

voltageanglesandcurrent.Complex-valuedvariablesWijareelementsinthematrixW(notethatthisimplies

𝑖, 𝑗 ∈ ℕ).Theelementsareassignedtheproductsofvoltagevariables(𝑈)z∠𝜃)z)and(𝑈pz∠𝜃pz):

𝑊)p = 0 + 0𝑗∀𝑖𝑗 ∉ 𝒥 (132)

𝑊)p = 𝑈)z∠𝜃)z 𝑈pz∠𝜃pz∗∀𝑖𝑗 ∈ 𝒥 (133)

𝑊)p = (𝑈)z)² (134)

Nonexistent connections ij have Wij set to zero (132). Therefore, the matrixW is sparse if the degree of

meshednessofthenetworkislow.Notethatthis𝑛×𝑛matrixisHermitian𝑾 ∈ ℍ*,asitsatisfies𝑊)p = 𝑊p)∗.

ThisequalitybetweenthenaturalvoltagevariablesandWij(133)-(134)canalsobeformulatedasconstraints

onthismatrixvariable:

𝑾 ≽ 0 (135)

𝑟𝑎𝑛𝑘 𝑾 = 1 (136)

Thepositivesemi-definitenessisencodedas(135)andtherank-1requirementas(136).Thepowerbalance

canbeformulatedasfollows:

𝑆)p = (𝑦r,t + 𝑦)p,t|)∗𝑊)) − (𝑦r,t)𝑊)p (137)

Overall, this formulation then is a rank-constrained SDP problem. Only due to the rank constraint, the

problem isnonconvex.ToobtaintheSDPformulation, therankconstraint issimplydropped.Similar to Jabr

formulation[26],thevariables𝑅)p and𝑇)p areusedtomodeltherealandimaginarypartsofthevoltages,as

follows:

𝑊)p = 𝑅)p + 𝑗𝑇)p (138)

𝑅)p = 𝑅𝑒 𝑊)p = 𝑈)z𝑈pzcos(𝜃)pz ) (139)

𝑇)p = 𝐼𝑚 𝑊)p = 𝑈)z𝑈pzsin(𝜃)pz ) (140)

Itisnotedthatduetothesubstitution,thephase-shiftvariablesarenotexplicitintheconvexformulation.

Furthermore,voltageanglesrelatebythefollowingnonlinearequation:

𝜃)z − 𝜃pz = 𝑎𝑡𝑎𝑛2𝑇)p𝑅)p (141)

Aphaseangledifference(PAD)constraintcanthereforebeformulatedinthesevariablesasfollows:


tan(𝜃)pz()*)𝑅)p ≤ 𝑇)p ≤ tan(𝜃)pz(+,)𝑅)p (142)

To obtain a tight convex hull, bounds implied on𝑅)p and 𝑇)p , through bounds on𝑊)),where𝑊)) are

derivedfrom(138)-(140),assuming𝜃)p ∈ [−𝜋/2, 𝜋/2]:

0 ≤ 𝑅)p ≤ 𝑀² ∙ 𝑈)pk+lmn𝑈p)k+lmn (143)

−𝑀² ∙ 𝑈)pk+lmn𝑈p)k+lmn ≤ 𝑇)p ≤ 𝑀² ∙ 𝑈)pk+lmn𝑈p)k+lmn (144)

Thevoltageboundsarereformulatedas

0 ≤ (𝑈)()*)² ≤ 𝑊)) ≤ (𝑈)(+,)² ≤ 𝑀E 𝑈)k+lmn ² (145)

9.3.4.2 SDPConvexRelaxation

Droprankconstraint(136)fromthefeasibleset.Ifthisrelaxedformulationsatisfies𝑟𝑎𝑛𝑘(𝑊) = 1inpost-

processing, the global optimum is found. The dropped rank constraint can be replaced with a rank

minimizationheuristicpenalty,basedonthetraceofthematrix[95]:

min 𝑡𝑟 𝑾 = 𝑊)))∈ℐ

(146)

min𝑌%&�Â*'m, =𝑊))

(𝑈)k+lmn)²)∈ℐ

(147)

9.3.4.3 SOCPConvexRelaxation

Thepositivesemi-definiteness(PSD)matrixconstraintcangenerallynotberepresentedinSOCPinanexact

manner.Anotherrepresentationisdevelopedforthesquaredvoltagemagnitude:

𝑊)p𝑊)p∗ = 𝑈)z∠𝜃)z 𝑈pz∠𝜃pz

∗ 𝑈)z∠𝜃pz

∗𝑈pz∠𝜃)z (148)

𝑊)p ² = 𝑊))𝑊pp (149)

Thisnonconvexequationisconsequentlyrelaxed:

𝑊)p ² ≤ 𝑊))𝑊pp (150)

The above constraint is a rotated second-order cone. Note that this formulation is equivalent to a PSD

constrainton2x2principalminors[15],[63](foralleffectivelinesij)intheoriginalmatrixW:

𝑊)) 𝑊)p

𝑊)p∗ 𝑊pp

≽ 0 𝑊)p ² ≤ 𝑊))𝑊pp(151)

SDPBIM

FeasibleSetX:(135),(137)


Theresultingreal-valuedpowerflowequationsare:

𝑃)p = 𝑔r,t + 𝑔)p,t| 𝑊)) − 𝑔r,t𝑅)p − 𝑏r,t𝑇)p (152)

𝑄)p = −𝑏r,t + 𝑔)p,t| 𝑊)) + 𝑏r,t𝑅)p − 𝑔r,t𝑇)p (153)

𝑊))𝑊pp ≥ 𝑅)pE + 𝑇)pE (154)

Notethat𝑅)p = 𝑅p)but𝑇)p + 𝑇p) = 0.Therefore,thisrelaxationcanbepenalizedasfollows:

min𝑌%&�Â*'m, =1𝑛𝒥

𝑅)p)∈ℐ

(155)

9.3.4.4 Post-processing

Theslackontheconvexificationisdefinedasfollows:

𝜖)p%&�Â*'m, = 𝑊))𝑊pp − 𝑅)pE − (𝑇)pE) (156)

Nopost-processingisrequiredforPij,Qij,𝛼r.TheoptimizedvariablesRij,Tij,Wiineedtobemappedbackto

theoriginalvariablesUi,𝜃),Iij.Startfromslackbus,andcalculatelinebyline,whilewalkingdownthetreeof

thenetwork.

9.3.5 QCrelaxation

The Quadratic Convex or QC formulation is based on the use of convex hulls of the sine and cosine

function. Notation f(. ) is used to indicate the convex hull of the function f(.). Instead of variables being

bound by functions in equalities, variables can also be formulated as lying in the convex hull of nonlinear

functions.Astheoriginalfunctioniscontainedwithinthatconvexhull,throughthismethodology,arelaxation

oftheoriginalproblemisperformed.

𝑊))𝜖 (𝑈)z)² (157)

𝑅)p𝜖 𝑈)z𝑈pz cos(𝜃)pz ) (158)

𝑇)p𝜖 𝑈)z𝑈pz sin(𝜃)pz ) (159)

9.3.5.1 ConvexHullforCosineFunction

Tight convex hulls of sine, cosine and quadratic functions are derived in [30], [32]. The phase angle

differenceoveranysectionisconsideredlimitedto90°:

SOCPBIM

FeasibleSetXI:(152)-(154)


−𝜋/2 ≤ 𝜃)pz()* ≤ 𝜃)pz ≤ 𝜃)z(+, ≤ 𝜋/2 (160)

𝜃)pz+´t(+, = max( 𝜃)pz()* , 𝜃)pz(+, ) (161)

A visualization of the convex hull of the cosine function is shown in Figure 9.6. Variable 𝑐)pmodels the

resultofthecosinefunction.

cos 𝜃)pz ≡

𝑐)p ≤ 1 −1 − cos(𝜃)pz+´t(+,)

(𝜃)pz+´t(+,)²𝜃)pzE

𝑐)p ≥ cos 𝜃)pz(+, + cos 𝜃)pz()* − cos 𝜃)pz(+,

𝜃)pz()* − 𝜃)pz(+,(𝜃)pz − 𝜃)pz()*)

(162)

Figure9.6Polyhedralconvexhullofcosinefunctionintherange− 4E≤ 𝜃)pz ≥

4E[32].

ItisnotedthatthisconstraintsetisSOC-representable.Notethatthetangenttothecosineinthisdomain

canberepresentedas

𝑡 ∈ 1,2, … , ℎ (163)

𝑡𝑑 =

2𝜃)pz(+,

ℎ + 1

(164)

𝑡𝑑 ∈ −4E:

4E

(165)

𝑐)p = − sin 𝑡𝑑 𝜃)pz − 𝑡𝑑 + cos(𝑡𝑑) (166)

Therefore,apolyhedralrelaxationofthecosinefunctioncanbedefinedas:

cos 𝜃)pz ≡ (167)


𝑐)p ≤ − sin 𝑡𝑑 − 𝜃)pz(+, 𝜃)pz − 𝑡𝑑 +𝜃)pz(+, + cos(𝑡𝑑 −𝜃)pz(+,)

𝑐)p ≥ cos 𝜃)pz(+, + ab6 789

æ:8É �ab6 789æ:Ê;

789æ:8É�789

æ:Ê; (𝜃)pz − 𝜃)pz()*)

It is noted that another polyhedral relaxation can be obtained through the application of the Ben-Tal

polyhedralrelaxationtechniquetotheSOCPrepresentationofthequadraticconvexterms.

9.3.5.2 ConvexHullforSineFunction

Avisualizationoftheconvexhullofthesinefunction isshowninFigure9.7.Variable𝑠)p modelsthesine

function:

sin 𝜃)pz ≡

𝑠)p ≤ cos𝜃)pz+´t(+,

2𝜃)pz −

𝜃)pz(+,

2+ sin(

𝜃)pz+´t(+,

2)

𝑠)p ≥ cos𝜃)pz+´t(+,

2𝜃)pz +

𝜃)pz(+,

2− sin(

𝜃)pz+´t(+,

2)

𝑠)p ≥ sin 𝜃)pz()* + 6HN 789

æ:8É �6HN 789æ:Ê;

789æ:8É�789

æ:Ê; 𝜃)pz − 𝜃)pz()* 𝑖𝑓𝜃)pz()* ≥ 0

𝑠)p ≤ sin 𝜃)pz()* + 6HN 789

æ:8É �6HN 789æ:Ê;

789æ:8É�789

æ:Ê; 𝜃)pz − 𝜃)pz()* 𝑖𝑓𝜃)pz(+, ≥ 0

(168)

Itisnotedthatthisconstraintsetispolyhedral.

Figure9.7Polyhedralconvexhullofsinefunctionintherange− 4

E≤ 𝜃)pz ≥

4E[32].Notethat𝑠𝑖𝑛 𝜃)pz = 𝜃)pz onlylieswithin

theconvexhullforsmallangles.

9.3.5.3 ConvexHullforQuadraticVariables

AvisualizationoftheconvexhullofthequadraticfunctionisshowninFigure9.8.


𝑥< ≤ 𝑥 ≤ 𝑥=

x² ≡ 𝑧 ≤ 𝑥²𝑧 ≥ 𝑥< +𝑥= 𝑥 − 𝑥<𝑥=

(169)

ItisnotedthatthisconstraintsetisSOCrepresentable.

Figure9.8Convexhullofquadraticfunction[32]illustratedinthedomain𝑥 ∈ [0.9; 1.1].

9.3.5.4 McCormick’sEnvelopesforBilinearTerms

𝑥4< ≤ 𝑥4 ≤ 𝑥4=

𝑥E< ≤ 𝑥E ≤ 𝑥E=

x4𝑥E ≡

𝑦 ≥ 𝑥E=x4 + 𝑥4=xE − 𝑥4=𝑥E=

𝑦 ≤ 𝑥E<x4 + 𝑥4=xE − 𝑥4<𝑥E=

𝑦 ≤ 𝑥E=x4 + 𝑥4<xE − 𝑥4=𝑥E<

𝑦 ≥ 𝑥E<x4 + 𝑥4<xE − 𝑥4<𝑥E<

(170)

Variableymodelstheresultofthebilinearmultiplication.Itisnotedthatthisconstraintsetispolyhedral.

9.3.5.5 OverallQCFormulation

𝑃)p = 𝑔r,t + 𝑔)p,t| 𝑢)) − 𝑔r,t𝑟)p − 𝑏r,t𝑡)p (171)

𝑄)p = − 𝑏r,t + 𝑏)p,t| 𝑢)) + 𝑏r,t𝑟)p − 𝑔r,t𝑡)p (172)

𝑢)) ∈ 𝑈)zE (173)

𝑢)p ∈ 𝑈)z𝑈pz (174)

𝑐)p ∈ cos(𝜃)pz ) (175)


𝑠)p ∈ sin(𝜃)pz ) (176)

𝑟)p ∈ 𝑢)p𝑐)p (177)

𝑡)p ∈ 𝑢)p𝑠)p (178)

Thetightertheanglebounds𝜃)pz()*; 𝜃)pz(+,areset,thetightertheformulation.

It is noted that the LPAC formulation [46] uses a convex-hull formulation of the cosine terms, in

combinationwithalinearizationofthesineterms.Apiecewiselinearizationofthisconvexhullisused,which

canbeconsideredasapolyhedralrelaxationofthequadratichullproposedhere.ThisLPACapproximationhas

astrongcorrelationwiththeNLPACformulationintransmissionsystems,andissuperiortothelinearized‘DC’

OPFformulation.

9.3.6 OPFApproximation:LinearDCFormulation

Thelinear‘DC’approximationisusedinSmartNettodescribethetransmissiongridphysicalconstraintsin

themarket-clearingalgorithm.

9.3.6.1 ModelFormulationTheDCformulationhasthefollowingassumptions.

• Branches can be considered lossless. In particular, if branch parameters 𝑟r,t, 𝑏r,t| and 𝑔r,t|are

negligible:

𝑦r,t = 1

𝑟r,t + 𝑗𝑥r,t≈

1𝑗𝑥r,t

,𝑏r,t| ≈ 0,𝑔r,t| ≈ 0 (179)

• Allbusvoltagemagnitudesareclosetoonep.u.

𝑈) ≈ 1 (180)

• Voltageangledifferencesacrossbranchesaresmall:

sin(𝜃) − 𝜃p + 𝜑)p) ≈ 𝜃) − 𝜃p + 𝜑)p (181)

• All theshuntelementscanbeconsideredasPQloadatconstantvoltage(e.g.nominalvoltage)and

addedtotheloadsvector.

Theapproximaterealpowerflowisthen:

QC

FeasibleSetXII:(158)-(178)


𝑃)p ≈𝑎)p𝑥r,t

(𝜃) − 𝜃p + 𝜑)p) (182)

Asexpected,giventhelosslessassumption,asimilarderivationforthepowerinjectionatthetwoendsof

thelineleadsto𝑃)p = −𝑃p).

9.3.6.1.1 HVDCLinksTheHVDClinkscanbemodelledbytheuseofgeneratorswithoppositeinjection.ThelosseswithintheDC

cable,infirstapproximation,canbeneglected.

9.3.6.1.2 FACTSApproximationsIntheDCapproximation,all reactivepowercontroldevicesaredisregarded.Otherdevicesthatareused

onlyforspecificapplications(e.g.oscillationdampingorthe increaseofthetransmissioncapability)arealso

neglectedintheDCapproximation.Theonlydevicestobemodelledarethosewhodirectlyaffecttheactive

powerflow.Therearedifferentmodelswhichcanbeusedforeachsingledevice[96],[97].However,fromthe

pointofviewoftheDCmodel,theeffectoftheactivephaseshiftersistochangetheactivepowerflowofa

line,independentlyfromtheparticularmodelorcontrolscheme.Then,fortheDCmodelitisonlynecessaryto

knowtheactivepowerortheequivalentphaseshiftingasshownin[68],[71],[98].

9.3.6.2 IncludingLossesIncludingtheactivepowerlossesgivesabetteralignmentoftheACandDCmodels. Itwasshownthata

significantpartoftheerroroftheDCapproximationcomesfromnotconsideringthecontributionofnetwork

losses.

It ispossibletoimprovetheDCmodelincludingafirstapproximationofthelosses.Themainproblemto

include them is that it is necessary to know them before calculating the power flow. Either the losses are

alreadyknownfromconventionalvalues(basedontheoperationalexperience),oritisnecessarytocalculatea

completeACpowerflow,whichconsidersboththereactivepowerflowandtheresistanceofthelines.Once

thelossesareknown,itispossibletotakethemintoaccount,eitherbyincreasingtheloadsorbydecreasing

thegeneratorpowerinjections.

Ifonlythetotalamountoflossesisknown,itispossibletoscalealltheloadsorgenerators(usuallywitha

proportionalmultiplicationfactor).Ifthespecificlossesofeverysinglelineareknown,itispossibletoassign

the lossesof a linedirectly to the loadsor generatorsof the twobussesbetweenwhich the specific line is

connected[68],[99].

Thefinalproceduretoincludethelossesintheplatformcalculationswilldependalsoontheavailabledata

and on the structure of the platform itself. To show the effectiveness of the procedure five methods of

consideringthenetworklossesareaddressedhereandbetterdescribedintheAppendix.


9.3.6.2.1 Method1-FlatLoss-ProfileMethod1usesaflatloss-profile,whichisaddedproportionallytotheload.Thevalueoftheflatloss-profile

parameterisselectedtoafixedvalue,e.g.2.5%(aboutthepercentageoflossesintheSpanishnetworkunder

normalconditions).

9.3.6.2.2 Method2-FlatLoss-ProfileRecomputedThepreviousmethod failswhen the lossesaredifferentwith respect the fixedvalue, in these cases the

errorisconcentratedparticularlyintheslackbus.Thenthelossesarerecomputedforeachparticularscenario.

9.3.6.2.3 Method3-LossesProportionaltoeachBusInmethod 3, the losses of each line in theDC power flow are assigned to the loads at the two busses

connectedbythelinewitha50%distribution.ThismethodshouldreducetheerroroftheDCapproximation,

sinceitconservesthegeographiclocationofthelosses.

9.3.6.2.4 Method4-ErrorProportionaltoeachGenItislikethemethod2,butinthiscasethelossesareassignedproportionallytothegeneratorproductions.

9.3.6.2.5 Method5-ErrorProportionaltoEachGenandGeneratorinHalfIt is like the method 2, but in this case the losses are assigned 50% proportionally to the generator

productionsand50%totheload.

9.3.6.3 ImprovedImpedanceCalculationItispossibletousethefollowingapproximationintheDCpowerflow:

𝑦r,t = −𝑗𝑥r,t

𝑟r,tE + 𝑥r,tE≈

1𝑗𝑥r,t

,𝑏r,t| ≈ 0,𝑔r,t| ≈ 0 (183)

This equation represents the Improved Impedance Calculation (IIC) and it should give a more accurate

behavior since it also takes the resistance of the lines into account.However, the achievable improvement

dependsonthestructureandtheparametersofthenetwork.

9.3.6.4 DCModelAccuracyItisimpossibletoevaluateaprioritheaccuracyoftheDCmodel:itdependsfromtheparticularnetwork

andscenarioconsidered.However,someconsiderationscanbemadeinordertoknowwheretheerrorscome

from[99].

• Neglectingthelossesbringsanerrorofafewpercent.However,ifthenetworkisbigandthereisonly

one slackbus, theerror in theproximityof the slackbus can increase considerably. Then it is necessary to

distributethelossestoallthenetworkbusses.

• Themaximumangledifferencebetweentwobussesisusually40°,thentheerrorsbetween and

isusuallylessthan8.6%.

• Ifthevoltagelimitsaretakenintherangeof0.75to1.4pu,themaximumerrorisabout44%.

( )qsin

q


• Theerrorofneglectingtheresistanceofthelinesdependsonther/xratioofthelines.Forr/xratios

ofabout1/3theerrorisabout11%,whichincreasesforhigherratio.HoweverusuallytheerrorofaDCmodel

ismuchlowerandtheaccuracyoftheDCpowerflowisaround5%oftheACpowerflow,butinparticularlines

theerrorcangreatlyincrease[71].Inparticularifther/xratioissmallerthan0.25,thentheerrorinthepower

flowsacrossthelines,whenconsideringACandDCsolution,doesnotexceed5%forthevastmajorityoflines

[100]. This because usually the r/x ratio is very small and similar for all the lines, then the flows in theDC

modeldivide themselves in the samewayof theACmodel.Besides that, thevoltageandangledifferences

between two consecutive busses are usually low. These approximations are not true with particular non-

linearitiesoftheACmodel(e.g.voltagedependenceoftheload,effectsofshuntelements,differencesinhigh

meshednetwork…).Theseerrorscanbeusuallyneglectedforthemarketapplications[68].

9.3.7 OPFwithAdditionalModels

Additionalmodelsandtheirfeasiblesetsaredevelopedinthissection.

9.3.7.1 PartialSlackNodetoLinkDistFlowwithLinearizedOPF

In linearized ‘DC’ OPF, active power flow and voltage angles are represented, but reactive power and

voltagemagnitudesarenotmodelled.Therefore,tolinkthelinearized‘DC’formulation,usedfortransmission

gridmodeling, with the DistFlow formulation, used in distribution gridmodeling, a new slack node type is

definedattheircommoninterface:

𝑈)kmò ≤ 𝑈) ≤ 𝑈)

kmò (184)

−∞ ≤ 𝑄) ≤ ∞ (185)

Avoltageanglereferencenodecanstillbedefinedatanynodeinthemodelofthetransmissionsystem.

9.3.7.2 OLTCTransformerwithDiscreteSteps

An on-load tap changing (OLTC) transformer is already modelled in a continuous way using the

formulationsdevelopedinSection9.3.1.Often,tapchangershavealimitednumberofdiscretesettings,e.g.

𝑎)p ∈ {0.90, 0.92, . . . , 1.08, 1.10}.Suchintegralityconstraintisconsideredasa‘specialorderedsetoftype1’

(SOS1)constraint,aMILPformulationtechniquewhichmodelsstepwisefunctions.Oneadds,foreachpossible

tapsettingl,theswitchstatevariableαl:

𝛼r ∈ 0,1 (186)

𝛼rr

= 1 (187)


For each tap,onealsoaddsa transformer to theoptimizationmodelwith a known tap𝑎)p()* = 𝑎)p(+, ∈

{0.90, 0.92, . . . , 1.08, 1.10}.All these transformersarevirtuallyput inparallel.TheSOS1constraintenforces

thatonlyoneofthosetransformerswillactuallybeused.Thefactthatthevaluesinthetapsethaveanorder

canbeexploitedbycertainsolvers.

9.3.7.3 ZIPLoadModels

Genericloadbehavior,definedbyareference𝑃+kmòisoftendescribedintermsofconstantimpedance(Z),

current(I)andpower(P)parts,respectivelythroughparameters𝑎+@%; 𝑎+A%; 𝑎+%%.

SuchZIPloadmodelis(foractivepower)formulatedasfollows:

0 ≤ 𝑎+@% ≤ 1; 0 ≤ 𝑎+A% ≤ 1; 0 ≤ 𝑎+%% ≤ 1 (188)

𝑎+@% +𝑎+A% +𝑎+%% = 1 (189)

𝑃+ = 𝑎+@%

=B=BCDE

E

+ 𝑎+A%=B=BCDE + 𝑎+

%% 𝑃+kmò

(190)

Here,Uuisthevoltagemagnitudeseenbytheunit,namelythevoltageofthenodeUitheunitisconnected

to (there exists a mapping of u to i). From this formulation, it can be immediately seen that the Z and P

componentsarecompatiblewiththeconvexifiedACOPFformulationspreviouslydeveloped.Therefore,with

𝑎+A% = 0, themodel is exact in those variables. Nevertheless, there is no variable representing the voltage

magnitudeUu in that set of variables, only a variable for (Uu)². An approximation𝑈+r)* ≈ 𝑈+ is developed

basedontheTaylorseriesintheneighborhoodof𝑈+kmò.

𝑈+r)* = 𝑈+kmò + 4

E=BCDE(𝑢+ − 𝑈+

kmò E) (191)

TheapproximatedZIPformulationthenbecomes:

𝑃+ = 𝑎+@%

+B=BCDE ²

+ 𝑎+A%=BF8É

=BCDE + 𝑎+

%% 𝑃+kmò

(192)

This formulation is linear in the squared voltagemagnitude variable𝑢+. It is noted that ZIPmodels are

inherently approximate in any case, as it abstracts behavior of individual consumers and distributed

generationunits.AnequivalentZIPformulationissometimesdevelopedforreactivepower.

9.3.7.4 RadialityEnforcementinReconfiguration

Ifthegridelementstatevariablesarefree,onecandevelopoptimizationformulationsforreconfiguration.

Acommonobjectiveforgridreconfigurationisminimizationofgrid(energy)losses.

min 𝑃rÂtt (193)


It is noted that equivalent formulations exist to enforce the radiality [101], which could be more

computationallyefficient,dependingonthecase.Suchconstraintsetsinclude‘singlecommodityflow’,‘multi-

commodityflow’,and‘spanningtree’[101].Below,thespanningtreeformulationisillustrated.Gridelements

areeitheractive,inactive,ortheswitchstatescanbeoptimized(194).Onlylimitednumberofgridelements

are active at the same time, to satisfy a necessary condition for radiality: the number of lines equals the

number of nodesminus one (195). For each line, there is a parent node indicator variablewhich is binary

(196).Alinehasatmostoneparentnode(197)andifithasaparentnode,thenthelineisinitsonstate.Every

non-slacknodeisaparentnodeonce(198)-(199).

0 ≤ 𝛼r()* ≤ 𝛼r ≤ 𝛼r(+, ≤ 1 (194)

𝛼r = 𝑛ℐ − 1r∈𝒥

(195)

𝛽)p ∈ 0,1 , 𝛽p) ∈ 0,1 (196)

𝛽)p + 𝛽p) = 𝛼r (197)

∀𝑖 ∈ ℐn,ïð: 𝛽)p = 0 (198)

∀𝑖 ∈ ℐn,*Â*ïð: 𝛽)p = 1 (199)

As the binary nature of αl is already guaranteed due to the integrality constraint (194), the integrality

constraint(96)canberemoved.

9.3.7.5 UnitModelingApproaches

FrameworksforMILPandMISOCPunitmodels,forconvexmultiperiodOPF,aredevelopedin[102],[103].

Furthermore,volt-varcontrolcanbeincorporatedinthemodelsofinvertersorsynchronousmachines[102],

[104].

9.3.8 Networkmodelingformulationextra’s

9.3.8.1 Definitionatan2

atan2 is a variation on the arctangent function with two arguments instead of one. This allows to

distinguishthesignofbotharguments.

𝑎𝑡𝑎𝑛2 𝑦, 𝑥 = (200)


arctan𝑦𝑥𝑖𝑓𝑥 > 0

arctan𝑦𝑥+ 𝜋𝑖𝑓𝑥 < 0𝑎𝑛𝑑𝑦 ≥ 0

arctan 𝑦, 𝑥 − 𝜋𝑖𝑓𝑥 > 0𝑎𝑛𝑑𝑦 > 0

+𝜋2𝑖𝑓𝑥 = 0𝑎𝑛𝑑𝑦 > 0

−𝜋2𝑖𝑓𝑥 = 0𝑎𝑛𝑑𝑦 < 0

𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑𝑖𝑓𝑥 = 0𝑎𝑛𝑑𝑦 = 0

9.3.8.2 RotatedSOCP

The2-normrepresentationofarotatedSOCisthefollowing

𝑥4E + 𝑥EE ≤ 𝑥F𝑥G, 𝑥F ≥ 0, 𝑥G ≥ 0

2𝑥42𝑥E

𝑥F − 𝑥G≤ 𝑥F + 𝑥G

2𝑥4 E + 2𝑥E E + 𝑥F − 𝑥G E ≤ (𝑥F + 𝑥G)²

2𝑥4 E + 2𝑥E E + 𝑥FE − 2𝑥F𝑥G + 𝑥GE ≤ 𝑥FE + 2𝑥F𝑥G + 𝑥GE

(201)

Aquadraticconstraint 𝑥4 E + 𝑥E E ≤ 𝑥Fwith𝑥F ≥ 0canbewrittenassucharotatedSOCwith𝑥Gkmò ≤

𝑥G ≤ 𝑥Gkmò.

9.4 Illustration of power flow approximations and relaxations:calculationresults

This section describes the results obtained by the optimal power flow calculations using the different

optionsforapproximationorrelaxationdescribedabove.

9.4.1 ObjectiveTheobjective for the optimal power flowproblemsdealtwith in the case studies discussedbelow is to

determinetheminimalcurtailingcostwhilestillmaintainingthebranchpowerflowsandbusvoltageswithin

their respectiveratings.Theassumptionsof thecurtailment flexibility inthecasestudiesareassumedtobe

thefollowing:

• Theactivepowerofthegeneratorsinthenetworkcanbecurtailedupto50%ofitsinitialvalue.The

boundsonthereactivepoweroutputofthegeneratorsaresettobethereactiveoutputatapowerfactorof

0.8 at the initial, i.e. maximal, active power generation. The constraints on the generator power are

implementedasboxconstraintswiththeupperandlowerboundsofactiveandreactivepowerspecifyingthe

boxedges.


• The curtailable loads can be curtailed up to 20% of their initial value. The curtailable loads are

assumedtohaveaconstantpowerfactor;hencetheirreactivepowerconsumptionisdependentontheactive

powerconsumption.

Theminimalcurtailmentcost(€)isexpressedas:

𝑲𝒄𝒐𝒔𝒕 = (𝒄𝒖𝒄𝒖𝒓𝒕,𝑷 𝑷𝒖 − 𝑷𝒖

𝒓𝒆𝒇 + 𝒄𝒖𝒄𝒖𝒓𝒕,𝑸 𝑸𝒖 − 𝑸𝒖

𝒓𝒆𝒇 )𝒖∈𝓤

(202)

TheobjectiveusedintheoptimizationproblemYcostisnormalizedasfollows:

𝑌�Âtl = 𝐾�Âtl

(𝑐+�+kl,%𝑃+

kmò + 𝑐+�+kl,P𝑄+

kmò)+∈𝒰 (203)

with𝑃+kmò and𝑄+

kmò the initial power (active and reactive) consumption of the𝑢 ∈ 𝒰 flexible units (here:

dispatchablegeneratorsandcurtailable loads).Curtailingcostsareassumedtobeequal foreachcurtailable

load and generator (𝑐+�+kl,% = 10€/MW curtailed). A curtailing cost is also applied for a change in reactive

poweroutputofthegenerators(𝑐+�+kl,P=10€/Mvaradapted).Itisnotedthatthiscostfunctionisconvex.

9.4.2 CaseStudiesTheOPFcalculationsareappliedondifferentcasestudies.Allcasestudiesbuildonanadaptationof the

Atlantidetestgrid,i.e.aMVdistributionnetworkprovidedbyENELDistribuzionethroughtheAtlantideproject

[105]. It is a typical radial distribution networkwhere commercial, residential and industrial customers are

connected.Theradialnetworkhas101AClinesconnectingthe100ACbuseswith15kVasratedvoltage.The

total lengthofthe lines is120.42km,divided inoverhead lines(53.32km)andcables(67.10km).Atotalof

128loadsareconnectedtothenetwork,dividedinresidential,commercialandindustrialloads.Inadditionto

this,28generatorsareconnectedtothenetwork,theyareeitherrotating(3windgeneratorsand3CHPs)or

static(22photovoltaicsolarpanelinstallations)generators.AschematicofthenetworkisshowninFigure9.9,

the generators are indicated as blue circles, the loads as greenhouse-shaped nodes. Someof the loads are

curtailable,theseareindicatedaslight-greenloads.

Threecase studiesweredefined: ‘undervoltage’, ‘overvoltage’and ‘overcurrent’.Dependingon thecase

study,thepowerdemandandgenerationofeachload/generatorisadaptedsothatthegridsuffersfrombus

voltagesbelowthevoltagelimit,abovethevoltagelimitand/orbranchpowersexceedingbranchratings,ifno

curtailingwouldbeapplied.Figure9.10showsthecalculatedbusvoltagesineachcasestudy,Figure9.11shows

the calculated branch powers, and Figure 9.12 shows the number of each type of constraint violation

(undervoltage,overvoltageorpowerlimitexcess).


Figure9.9SchematicoftheadaptedAtlantideMVdistributiongrid.


Figure9.10Busvoltagesforeachcasestudy(withoutcurtailing).

Figure9.11Branchpowerforeachcasestudy(withoutcurtailing).

Figure9.12Thenumberofconstraintviolationsforeachscenario.


9.4.3 ToolChainAll power flow and optimal power flow calculations are implemented in Python. All variables and

parameters are expressed in pu-values, with a power base equal to the rated power of the substation

transformer(25MVA),andavoltagebaseequaltotheratedvoltageoftheMVnetwork(15kV).

Theoptimalpower flowcalculationsusinga convex relaxationor linearapproximationare implemented

using CVXPY [106],with ECOS [107] used as solver. PYPOWER [108] is used to calculate the nonconvex AC

optimalpowerflowresult.ThePYPOWERpackageincludestheMIPSsolver,whichisanimplementationofan

interior-point method for nonconvex problems. MIPS only guarantees that the solution is locally optimal.

Following the optimization, the obtained result, i.e. the obtained active and reactive power output from

curtailable loadsandgenerators, is fed intoapowerflowcalculation,alsoprovidedbyPYPOWER[108].This

load flow result is used to check the feasibility of the optimization result. The tool-chain and calculation

methodologyareillustratedinFigure9.13.

Figure9.13Illustrationoftool-chain&methodology

Thefollowingformulationsarecomparedintheupcomingsectionwithnumericalresults:

• AC–NLP

• DistFlow-SOCP

• DistFlowBen-Tal-LP

• SimplifiedDistFlow-LP

• Linearized‘DC’–LP


9.4.4 NumericalComparisonofOPFFormulations

9.4.4.1 ScalingofSOCPPenaltyFactor

As indicated in Section 4.3, a penalization factor𝑤 ∙ 𝑌%&�Â*'m, needs to be added to the optimization

objectiveoftheSOCrelaxationoftheoptimalpowerflow,sothattheobtainedresultliesontheconesurface.

The SOCP result is then physicallymeaningful and hence a feasible result (when applied to radial grids). In

ordertotestifanobtainedresultisfeasible,thetotalrelaxation‘slack’𝜖%&�Â*'m,iscalculated.Ifthecalculated

slack𝜖%&�Â*'m, ≈ 0,theoptimizationresultliesontheconesurface.Computationaltestswereperformedto

see how sensitive the obtained results are for the weight w that is given to the penalty factor in the

optimizationobjectivefunction.Thepenalizationfactorusedinthetestsis:

𝒘 ∙ 𝒀𝑷𝑭𝒄𝒐𝒏𝒗𝒆𝒙 = 𝒘𝒏𝓙

𝒊𝒊𝒋,𝒔𝒍∈𝓙

(204)

with𝑖)p,texpressedasapu-value. Thegivenweight to thepenalty factor shouldbehighenough so that

slackisremovedfromtherelaxation,butatthesametimeshouldbenottoohighsothatisoptimizedtowards

minimalcurtailingandnottowardsminimalcurrent.

Figure10.9showstheoptimizationresultswhentheweightwisvariedforeachcasestudy.Thetopplotof

these figuresshowstheobtainedobjectivevalue,consistingof thecurtailedcostvalueandthevalueof the

penalization term(indicated in red).Themiddleplotshowsthe total slack𝜖%&�Â*'m,of theSOCPrelaxation.

ThebottomplotshowstheobtainedcurtailingcostoftheoptimizationresultKcost,obtainedfrom(224).

Figure9.14Sensitivityofpenalizationfactorinthe‘undervoltage’casestudy.

Itcanbeseenthatforthe‘undervoltage’casestudy,nopenaltyfactorisneededinordertofindafeasible,

optimal solution. This complies with the observation that relaxation slack is found when additional ’non-

physical’ losses would be beneficial for a solution, however, undervoltages can never be solved by these

additionallosses.Thiscanbeseenbyobserving(99).


Figure9.15Sensitivityofpenalizationfactorinthe‘overvoltage’casestudy.

In the ‘overvoltage’ case study, the optimization result is quite robust in terms of penalty weight: the

optimizationresultliesontheconesurfaceanddoesnotchangewithaweightwrangingfrom±164to±250.

Whenusingapenaltyweightwithinthisrange,theobtainedresultistheoptimalone.

Figure9.16Sensitivityofpenalizationfactorinthe‘overcurrent’casestudy.

Inthe‘overcurrent’casestudy,however,theoptimizationismuchlessrobustagainstthepenaltyweight:

theresultisonlyoptimal(intermsofcurtailingcost)withapenaltyweightwithinaverysmallrangearound

39.Thisindicatesthatforeveryscenarioandcasestudy,thepenaltyweightshouldbecarefullychosensothat

theobtainedresultisfeasible,andisoptimalatthesametime,especiallywhenovervoltagesandcurrentlimit

constraintviolationsarefound.

9.4.4.2 RelaxationandApproximationApproachesCompared

A comparison ismade of the results found by the SOCP relaxation, the DCOPF approximation and the

SimplifiedDistFlowapproximationoftheoptimalpowerflowproblemforthethreecasestudies.Anonconvex

ACoptimalpowerflowisalsoaddedtothecomparison.


9.4.4.2.1 ‘Undervoltage’CaseStudy

Figure9.17andFigure9.18showacomparisonoftheresultsfoundofthedifferentOPFapproachesofthe

‘undervoltage’ case study. Figure 9.17 shows the active and reactive power consumed or generated by the

flexibleresourcesinthesolutionsobtainedthroughthedifferentoptimizationapproaches.Figure9.18givesan

overview of the resulting costs of each solution, the corresponding grid losses for each solution and the

constraintviolations foundof thepost-processedresultsofeachoptimizationapproach.Fromtheseresults,

wecanconcludethefollowing:

• The solution found by the DCOPF approximation is, although in cost the cheapest solution, not a

feasibleone.Theoverall numberof constraint violations found in theoriginal situationdoesnot changeby

applyingtheDCOPFapproximation.

• ThesolutionfoundthroughtheSimplifiedDistFlowapproachisalsonotafeasiblesolution,butisstill

able to solve some of the constraints found in the original situation. Generally speaking, voltage drops are

underestimatedintheSimplifiedDistFlowapproximation,hence

• thenon-feasiblesolution.Inthiscasestudy,thecostoftheSimplifiedDistFlowsolutionislowerthan

thecostoftheSOCPsolution,butgridlossesarehigher.

• TheSOCPapproachyieldsafeasiblesolutionwithleastgridlossesandthelowestcurtailmentcost.

• The solution of the nonconvex AC OPF is not the optimal solution (cost is higher than the SOCP

solution),indicatingthatinthiscasestudythenonlinearsolverreachesalocaloptimum.

Figure9.17Comparisonofsolutionsfoundbydifferentformulationsforthe‘undervoltage’casestudy.


Figure9.18Theresultingcostsandgridlossesofthedifferentsolutionsinthe‘undervoltage’casestudy.

9.4.4.2.2 ‘Overvoltage’CaseStudy


‘overvoltage’casestudy.Fromtheseresults,wecanconcludethefollowing:

• The solution found by the DCOPF approximation is, although in cost a very cheap solution, not a

feasible one. This solution solves almost none of the constraint violations that were found in the original

situation.

• ThesolutionfoundthroughtheSimplifiedDistFlowapproachisafeasiblesolution.However,thecost

of this solution is higher than the cost of the SOCP solution. This is mainly because the overvoltages are

overcompensatedinthissolution,leadingtoahighercost.

• TheSOCPapproachyieldsafeasiblesolutionwithleastgridlosses.However,theresultfoundbythe

(nonconvex)ACOPFhasaslightlylowercostthantheSOCPsolution.Inthiscase,theinclusionofthepenalty

termdoesnot immediately lead toa feasibleaswellasglobaloptimal solution, indicatingnumerical issues.

Theseareprobablycausedbyabadlyscaledpenaltyfactor.Thisindicatesthat,asmentionedinSection4.4.4,

scaling of the SOCP penalty is non-trivial. In terms of combined grid losses and curtailment cost, the SOCP

solutionstillperformsbest.


Figure9.19Comparisonofsolutionsfoundbydifferentformulationsforthe‘overvoltage’casestudy.

Figure9.20Theresultingcostsandgridlossesofthedifferentsolutionsinthe‘overvoltage’casestudy.

9.4.4.2.3 ‘Overcurrent’CaseStudy


‘overcurrent’casestudy.Fromtheseresults,thefollowingcanbeconcluded:

• The solution found by the DCOPF approximation is, although in cost the cheapest solution, not a

feasibleone,asitfailstosatisfythebounds.


• ThesolutionfoundthroughtheSimplifiedDistFlowapproachisafeasiblesolution inthiscasestudy.

ThecostofthissolutionishoweverhigherthanthecostoftheSOCPsolution.

• TheSOCPapproachyieldsafeasiblesolutionwithleastgridlosses.

• ThecostofthenonconvexACOPFquasiequalsthecostoftheSOCPsolution,thegridlossesofthis

solutionarehoweverhigherthanthesolutionfoundbytheSOCPapproach.Oneagain,inthiscase,numerical

issues related to the scalingof thepenalty factor lead to a slightly higher cost of the SOCP solution found.

However,intermsofcombinedcostandgridlosses,theSOCPsolutionstillperformsbetter.

Figure9.21Comparisonofsolutionsfoundbydifferentformulationsforthe‘overcurrent’casestudy.

Figure9.22Theresultingcostsandgridlossesofthedifferentsolutionsinthe‘overcurrent’casestudy.


9.4.4.2.4 Discussion

Overall seen, it can be concluded that the DC OPF approximation is not a suitable approach to solve

optimal power flow problems in a distribution grid, since inmany (ormost) cases it will not yield feasible

solutions.Thereasonforthisisthatreactivepowerisnotproperlyaccountedforinthisapproximationwhen

usedindistributiongridmodeling.

ComparedwiththeDCOPF,theSimplifiedDistFlowapproximationyieldsmuchbetterresults intermsof

feasibility. Feasibility issues aremostly related to undervoltage issues, since these are under-compensated

usingtheSimplifiedDistFlowapproximation.Intermsofsolutioncost,thesolutionsobtainedbytheSimplified

DistFlowapproximationinmanycasesleadtohighercoststhantheoptimal.Thisismainlyduetothefactthat

overvoltagesareovercompensatedintheobtainedsolutions.

TheSOCPrelaxationyields,inallcases,afeasiblesolution.Theobtainedsolutionistheguaranteedglobal

optimalsolutionwithrespecttothecombinedminimumofcostandgridlosses.Gridlossesareaccountedfor

intheobjectivebecauseoftheinclusionofthepenaltytermforcingtheobtainedsolutiontolieonthecone

surface and thus be physically feasible. Finding a penalty factor that effectively leads to the global optimal

solutionintermsofcurtailmentcostprovestobenon-trivialin2cases.Inthecaseofundervoltageissues,the

penaltyfactorislesscritical,anditisthuseasiertofindtheglobaloptimum.

ThenonconvexACOPFdoesnotgiveaguaranteedoptimum,thesolvermayreachalocaloptimum,witha

highercostandhigherlossesthanthesolutionfoundbytheSOCPrelaxation.

9.4.4.3 Ben-TalPolyhedralRelaxationofSOCP

The Ben-Tal polyhedral relaxation technique [19] can be used to reformulate any SOCP problem as an

equivalent LP problem to an arbitrary, chosen accuracy. The SOCP DistFlow formulation includes 3 SOC

constraints(seesection4.3):

𝑃)p,tE+ 𝑄)p,t

E≤ 𝑖)p,t ∙ 𝑢)z (205)

𝑃)pE+ 𝑄)p

E≤ (𝑆)pk+lmn)² (206)

𝑃p)E+ 𝑄p)

E≤ (𝑆p)k+lmn)² (207)

Thepolyhedralrelaxationaccuracyϵisafunctionofauserchosensettingν:

𝜈 ≥ 1, 𝜈 ∈ ℕ

𝜖 𝜈 = 1

cos( 4E\]^)

− 1

(208)

For all three case-studies, the three SOC constraints were relaxed using the Ben-Tal technique with a

varying accuracy. Accuracy setting ν4 is used to define the accuracy of the polyhedral relaxation of SOC


constraint (205), while νE is the accuracy setting of the polyhedral relaxation of SOC constraints (206) and

(207). A different relaxation accuracy for the different SOC constraints makes sense, since power rating

constraints,i.e.(206)and(207),canbetreatedaslessstrictconstraints,whiletheaccuracyofSOCconstraint

(205)definesthefeasibilityoftheobtainedresult.Itmustbenotedthatthemoreaccuratetheresulthasto

be,i.e.thehighertheaccuracysettingν,morevariablesareintroducedintheoptimizationproblem,possibly

leadingtomemoryorothercomputationaltractabilityissues.

Figure9.23andFigure9.24showtheobtainedresultsforthe‘undervoltage’casestudy.Itcanbeseenthat

for an accuracy setting of ν4 = 4 and νE = 3 relatively accurate results are obtained. Using these accuracy

settings only one branch power constraint violation remains, but the power rating is only exceededwith a

relativelylowvalue.Theundervoltageissuesarereducedto3issues,buttheremainingundervoltageisquite

closetothevoltagelimit.Also,thecostoftheobtainedresultdiffersfromtheSOCPobtainedresultwithonly

4%.

Figure9.23Theresultingcosts,slackvalueandgridlosswithvaryingBen-Talaccuracysettingforthe‘undervoltage’case

study.Theobtainedcost,slackvalueandgridlossoftheSOCPresultisalsoshown.


Figure9.24FeasibilityofthesolutionsobtainedwithvaryingBen-Talaccuracysettinginthe‘undervoltage’casestudy.Theleftfiguresshowthenumberofconstraintviolationsfoundandtherightfiguresshowthebiggestconstraintviolation.

Figure 9.25 and Figure 9.26 show the obtained results for the ‘overvoltage’ case study. In this case, an

accuracy setting of𝜈4 = 2 and𝜈E = 1 leads to relatively accurate results. Using these accuracy settings, no

branch power constraint violation nor any overvoltage issue remains. Also, the cost of the obtained result

differs from the SOCPobtained resultwith only 5%. Itmust be noted that the achieved accuracy does not

necessarily increase monotonously with the accuracy settings given: note that the accuracy setting 𝜈

determinesanupperboundoftheachievedaccuracy,aloweraccuracycanbereachedusingaloweraccuracy

setting𝜈.

Figure9.25Theresultingcosts, slackvalueandgrid losswithvaryingBen-Talaccuracysetting for the ‘overvoltage’casestudy.Theobtainedcost,slackvalueandgridlossoftheSOCPresultisalsoshown.


Figure9.26FeasibilityofthesolutionsobtainedwithvaryingBen-Talaccuracysettinginthe‘overvoltage’casestudy.Theleft figures show the number of constraint violations found, the right figures show how large the biggest constraintviolationis.

Figure 9.27 and Figure 9.28 show the obtained results for the ‘overcurrent’ case study. In this case, an

accuracy settingof𝜈4 = 2 and𝜈E = 1 leads to relatively accurate results.Using these accuracy settings, no

branch power constraint violation remains. Also, the cost of the obtained result differs from the SOCP

obtainedresultwith7%.

Figure9.27Theresultingcosts,slackvalueandgridlosswithvaryingBen-Talaccuracysettingforthe‘overcurrent’case

study.Theobtainedcost,slackvalueandgridlossoftheSOCPresultisalsoshown.


Figure9.28FeasibilityofthesolutionsobtainedwithvaryingBen-Talaccuracysettinginthe‘overcurrent’casestudy.Theleft figures show the number of constraint violations found, the right figures show how large the biggest constraintviolationis.

To conclude, in all tested case studies, relatively accurate Ben-Tal relaxation results are obtained with

relatively low accuracy settings. Accuracy settings of 𝜈4 and 𝜈E = 4 in all tested cases lead to acceptable

results.Thisimpliesthatwithouttheinclusionofmuchmoreextravariables,LPsolverscanbeusedtosolve

theSOCPproblemwithoutlosingtoomuchaccuracy.Aloweraccuracysettingmightbebetter,dependingon

thespecificcasestudy,sincetheaccuracysettingonlydeterminesanupperboundoftheachievedaccuracy.

9.4.5 VerificationStudyoftheInactiveConstraintsMethod

The inactive constraints method as network simplification method for transmission grid modeling is

illustratedinthischapter.Themethodisappliedtotheavailablenetworks,inordertofindtheinactivelines.

Tomaketheverificationmoregeneral,themethodisappliedtolargerangeofloadvalues.Theloadistreated

asavariablebetween10%and300%(dependingonthenetworkconsidered)ofthenominalvalue.Inthisway,

themethodisvalidatedindifferentloadingconditions.

InthecaseofSmartNet,alsothepossiblebalancingcomingfromtheflexibilitiesondistributionnetwork

shouldbeconsidered.Theproblemisthattherangeofvariationoftheseresourcesisnotknownatthisstage

oftheproject.

Theuseofmediumvoltageresourcescanreducetheuseofthehighvoltageresources.Theimpactofthe

medium voltage resources also depends on their position. However, the increase of distributed generation

shouldlimitthepowerflowinthetransmissionsystem,thenneglectingtheireffectshouldnotintroducegreat

errors.

Finally,we have to take into account thatwe are considering the balancingmarket. In thismarket, the

changesof theproduction/consumption,whicharecausedby relievingnetworkcongestionsandshort-term


fluctuationsofsustainablepowersources,areonlyafewpercentofthetotalload,sothebalancingshouldnot

introducegreatvariationofthepowerflow.Forthispreliminaryanalysis,onlythehighvoltagegeneratorsare

takenintoaccount.Thissimplifiedapproachisconsideredtobesufficienttohighlighttheweakestareasofthe

transmissionnetwork.

9.4.5.1 DanishNetwork

The results for the twoasynchronousareasof theDanishnetworkare reported inFigure9.29. It canbe

seenthatinalltheloadingconditions,thelineswithactiveconstraintsareusuallynomorethan10%(which

areabout40lines)ofthetotal.Thenominalloadinthiscaseisabout5GW.

Figure9.29:InactiveconstraintsinthetwoasynchronousDanishnetworkareas

9.4.5.2 SpanishNetwork

9.4.5.2.1 BaseCase

The results for the Spanish network are reported in Figure 9.30. In this network, there aremore active

constraintsthanintheDanishnetwork:Thelineswithactiveconstraintsareusuallynolessthan20%(about

600lines).Thenominalloadinthiscaseisabout40GW.

Figure9.30:InactiveconstraintsintheSpanishnetwork


9.4.5.2.2 LimitedMaximumPower

Not all the active power of the generator is usually available in the balancingmarket, but only a small

fractionofit.Therefore,thecalculationsarerepeatedconsideringthatthegeneratorscanincreasetheactive

powerofonlythe20%(conventionalvalue)ofthemaximumpower.

Itcanbeseen(Figure9.31)thatinthisconditionthepercentageoflineswithinactiveconstraintsincreases

toabout90%.Onlyforhighvaluesoftheload(almost80GW),thepercentageofactiveconstraintsincreases

above10%(about300lines).Withthenewmaximumcapability,thenumberoflineswithinactiveconstraints

arelowerforveryhighloadingsituations.Infact,thecurrentlimitviolationsareduetothehighloads,where

the lower values of the generators capabilities increase the probability of the current flow at high load

situationshitsthelimits.

Figure9.31:InactiveconstraintsintheSpanishnetworkwithadjustedmaximumgeneratorcapability

Also,consideringthatasmallfractionofthelinecapacityhastobepreservedforthereactivepower,itis

clearthatthereductionofthenumberoflinesisconsiderable.

9.4.5.2.3 StatisticalAnalysis

To furtheranalyses, the inactiveconstraintsmethod,astatisticalanalysis iscarriedout.Thetotal load is

increasedfrom10%to200%andforeachloadscenario50casesareanalyzedmultiplyingeachsingleloadbya

randomnumberbetweenthe75%and125%.Consideringthesestochasticvariations,thenumberoflineswith

inactiveconstraintsdecreasesabout5%


Figure9.32:InactiveconstraintsintheSpanishnetworkwithstochasticloadvariation

9.4.5.2.4 LimitingtheAreaoftheNetwork

Ifthereducednumberofvariablesisseenasstillbeingtoohighafternetworkreductionhasbeenapplied,

an alternative is to control only the branches on a certain zone, for example the zone of the pilot. In the

examplecaseoftheSpanishpilot,onlyabout40lineswithactiveconstrainshavetobeconsidered.

9.4.6 VerificationStudyoftheDCApproximationforTransmissionGrid

Modeling

It isnecessarytoverifythattheDCapproximationdoesnot introducetoolargeerrors inthepowerflow

calculation.Forthispurpose,sometestsarecarriedoutindifferentconditionsofoperation.

Inthisverificationstudy,thecalculatederrorconcerntheactivepowerflow.Allerrorsaregivenasmean

values.ThecalculatedACpowerflowistakenasthereferencevalue,andthedeviationofthecalculatedDC

powerflowapproximationtowardstheACpowerflowisconsideredaserror.Allrelativeerrorsareexpressed

withrespecttothemaximumpowerrating.

The total load (and generation) is varied with a load-scaling factor, ranging from 50% to 150% of the

nominalpower,with11stepsof10%.foreachstep.10randomconfigurationsaremade,whereeachloadand

generator aremade variablewith a Gaussian distributionwith a standard deviation of 20%. This gives 110

differentcasestobecalculated.

The resulting error data are three-dimensional depending on the applied load-scaling factor, the line

numberand the randomdistribution.Forvisualizationpurposes, thedataareplottedas setsof curves,and

twopossibilitiesareused:

• Acurveforeachcombinationof linenumberandrandomdistribution,withthe load-scalingfactors

onthehorizontalaxis


• Acurveforeachcombinationof load-scalingfactorandrandomdistribution,withthe linenumbers

onthehorizontalaxis

9.4.6.1 SpanishNetwork

StudyingtheSpanishnetworkhasshown,thattheerrorbetweentheDCandACpowerflow, intermsof

powerflowonthelines,islowonaverage.However,intheareaofthenetworkneartheslackbus,theerror

increases and become too relevant to be discarded (almost 500MW). In the following section reports the

resultofthemethodsintroduced.

9.4.6.1.1 Method1

Thefollowingprocedureisperformedforeachrandomsample:

1. Theloadsandgeneratorsareobtainedbytherandomprocedurepreviouslydescribed.

2. OneACPFisperformedtoobtainthelosses.

3. The losses are added proportionally to the generators. In this way, the balancing of the losses is

distributedbetweenallthegeneratorsandnotonlyintheslack.

4. TheACPFisrepeatedtakingintoaccountthenewgenerationprofilewiththelosses.

5. Flatlossesareaddedproportionallytotheload.

6. TheDCpowerflowiscomputed.

7. TheACandDCpowerflowarecompared.

Thevalueoftheflatloss-profileparameterisselectedto2.5%,whichisaboutthepercentageoflossesin

theSpanishnetworkundernormalconditions.

Themeanerrorinthiscaseis4.1MW.Themeanpercentageerror,computedwithrespectthepowerlimit

ofthelinesis1.3%.

Themeanerrorisquitelow,butitisnotacceptableinsomelines,inparticularwhenthereisahugeerror

neartheslackbus(Figure9.33;Figure9.34),sincetheslackhastobalancethelosses.

Theerrorisminimumwhentheloadandthelossesarenearthenominalcondition.Theyincreaseforthe

otherloadingconditions.Theerrorbecomesveryhighwhentheloadincreases(Figure9.34),sincethelosses

are quadraticwith respect the currents. It is clear that it is necessary to take into account, at least in first

approximation,thevalueofthelossestocomputetheexactvalueoftheload.


Figure9.33:Relativeerrorvs.linenumber(Method1,Spain)

Figure9.34:Relativeerrorvs.loadscalingfactor(Method1,Spain)

9.4.6.1.2 Method2

Inordertoreducetheerror,thelossesarenotkeptfixed,butarecomputedineachconditionofoperation

considered.Theprocedureisdescribedbythefollowingsteps:






5. Thelossescomputedatthepreviousstepareaddedproportionallytotheload.



500 1000 1500 2000 25000

200

400

600

800

1000Mean error

line

Pow

er [%

]

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

200

400

600

800

1000Mean error

load

Pow

er [%

]


Using this method, the error strongly decreases: the mean error in this case is 3.3MW and themean

percentageerror,computedwithrespectthepowerlimitofthelinesis1.1%.Besides,thepowererroronthe

linesisalwayslowerthan150MWandthegreaterrorneartheslacknodedisappears.The95-percentileerror

isbelow12.7MWand4.1%(Figure9.35;Figure9.36),thenumberof lineswithhigherroris limited.Theline

with the maximum absolute errors correspond to the line with the highest power flow (Figure 9.37), but

considering the percentage error, the errors in these lines is below the 5%. This is good, since the highest

absoluteerrorcorrespondswiththelineswiththehighestcapacity.



500 1000 1500 2000 25000

5

10

15

20

25

30

35Mean error

line

Pow

er [%

]

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

5

10

15

20

25

30

35Mean error

load

Pow

er [%

]


Figure9.37:Meanactivepowerflowvs.linenumber(Method2,Spain)

9.4.6.1.3 Method3

In this case, the losses in theDCpower flowarenotaddedproportionally to the load,but the lossesof

each line are assigned to the two ending busses. This should increase the accordance between the two

models.Theprocedurefollowthesteps:






5. Thelosses,computedatthepreviousstep,ofeachlineareaddedtotheendingbussesoftheline.



Theerrordecreaseswithrespectthepreviouscases:themeanerrorinthiscaseis2.3MWandthemean

percentageerror,computedwithrespectthepowerlimitofthelinesis0.84%.Besides,thepowererroronthe

linesisalwayslowerthan140MWandthegreaterrorneartheslacknodedisappears.The95-percentileerror

isbelow8.4MWand3.4%(Figure9.38;Figure9.39),thenumberoflineswithhigherrorislimited.Thismethod

demonstratesverygoodagreementsbetweenthetwocalculationmethods.

500 1000 1500 2000 25000

500

1000

1500

2000

2500

3000Mean power

line

Pow

er [M

W]




9.4.6.1.4 Method3withIIC

Themeanerrorinthiscaseis2.3MWandthemeanpercentageerror,computedwithrespectthepower

limitofthe lines is0.84%.Besides,thepowererroronthelines isalways lowerthan100MWandthegreat

error near the slack node disappears. The 95-percentile error is below 8.7MW and 3.3% (Figure 9.40;

Figure9.41),thenumberoflineswithhigherrorislimited.

Thetwomethodspresentverysimilarresults,soitisdifficulttosaywhichmethodisbetter.Inthisspecific

case,thepreviousmethodperformsalittlebetter.

500 1000 1500 2000 25000

5

10

15

20

25

30Mean error

line

Pow

er [%

]

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

5

10

15

20

25

30Mean error

load

Pow

er [%

]


Figure9.40:Relativeerrorvs.linenumber(Method3withIIC,Spain)

Figure9.41:Relativeerrorvs.loadscalingfactor(Method3withIIC,Spain)

9.4.6.1.5 Method4

Thismethod is like themethod2,but the lossesareaddedproportionally to thegeneratorproductions.

Themeanerrorinthiscaseis3.4MWandthemeanpercentageerror,computedwithrespectthepowerlimit

ofthelinesis1.15%.Besides,thepowererroronthelinesisalwayslowerthan150MWandthegreaterror

neartheslacknodedisappears.The95-percentileerrorisbelow12.8MWand4.2%,thenumberoflineswith

higherrorislimited.

9.4.6.1.6 Method5

This method is like the method 2, but the losses are added 50% proportionally to the generator

productions and 50% to the load. Themean error in this case is 3.0MW and themean percentage error,

500 1000 1500 2000 25000

5

10

15

20

25Mean error

line

Pow

er [%

]

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

5

10

15

20

25Mean error

load

Pow

er [%

]


computedwithrespectthepower limitofthe lines is1.05%.Besides,thepowererroronthe lines isalways

lowerthan140MWandthegreaterrorneartheslacknodedisappears.The95-percentileerrorisbelow11.1

MWand3.7%,thenumberoflineswithhigherrorislimited.

9.4.6.2 DanishNetwork

ThetestsarerepeatedforthetwopartsoftheDanishnetwork.TheerrorbetweentheACandDCpower

flowissmallerforthisnetwork,sotheanalysisissimpler.Onlymethod3isusedhere.Theerrorsforthewest

part of the network are reported (Figure 9.42). The errors for the east part of the network are reported

(Figure9.43).Thelowererrorisprobablyduetothesmallerdimensionofthenetwork,thelowerlossesandto

thegenerallyhigherx/rratio.

Figure9.42:Relativeerrorvs.linenumber(Method3,westernDenmark)

Figure9.43:Relativeerrorvs.linenumber(Method3,easternDenmark)

9.4.6.3 Summary

TheDCapproximationisagoodapproximationforthisnetwork.Theerrorsarelimitedalsoforthesimpler

methods. The principal open point is the contribution of the losses. In fact, the AC and DC approximation

20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5Mean error

line

Pow

er [%

]

20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12Mean error

line

Pow

er [%

]


return thesameresultsonly if the lossesarecorrectly taken intoaccount.Thereareprincipally twoway to

takeintoaccountthelossesintheprojectframework:

• Thebalancingmarketsimulatesthenetworkcondition inthereal-timemarket.Thenthe inputdata

(profiles of load and generators) should already take into account the losses. Then it is possible to add the

losses in the DC approximation with different methods (proportionally to the load or to the generation).

However, since the AC power flow is not performed it ismore difficult to associate the losses to the lines

(futureinvestigation).

• The input data does not take into account the losses (for example if the data are fromday ahead

marketanditdoesnottakeintoaccountthelosses).Inthiscase,itisnotnecessarytoaddthelossesintheDC

formulation.


9.5 MathematicalFormulationofMarketClearing

9.5.1 CouplingofPhysicsandEconomics

We startwith thephysical apparent-powerequilibrium. The following constraint expresses that𝑆*l, the

totalabsoluteapparent-powerinjectioninnode𝑛attime𝑡,isthesumof𝑆*l,thepowerinjectioninprevious

markets (therefore representing the imbalance), plus 𝑆*l( , the compensating power injections that the

consideredbalancingmarketdoes.

𝑆*l = 𝑆*l + 𝑆*l( (209)

Then, the coupling of (network) physics and (market) economics is formulated using (210). This

constraint links the apparent-power injections of the considered market in node 𝑛 at time 𝑡, to the

acceptedactiveandreactivepowerquantitiesforallbids.

𝑆*l( = 𝑞*+´l� ⋅ 𝑥*+´l� ⋅ 1 + 𝑗 ⋅ 𝑃𝐹*+´�´+

(210)

In(210),bothleftandrighthandsidesarecomplexquantities(defining𝑗 = −1),soitequatesboth

the real and imaginary part, representing active and reactive power respectively. Note also that the

physical limits on total injection will be formulated in terms of the total absolute apparent-power

injection𝑆*l.

9.5.2 AdaptationofPricestoAvoidUnnecessaryAcceptance

Thetechniqueisperformedbyfirstfindingtheminimumbidpriceamongthecommonbidswiththe

rightquantitydirectionandthennegatingitsvalue:

𝜓 = −minimum�∈�C8abÌ

{𝑝�} (211)

Theresult(called𝜓)isthenusedforcreatingthenew‘manipulatedprice’forthecommonbidswith

thewrongquantitydirection,inthefollowingway:

∀𝛽 ∈ 𝛱ÁkÂ*À,𝑝�*mÁ = 𝜓 + 𝜖 + 𝜖E ⋅ 𝑝�Ârn (212)

where𝜖isaverysmallpositivenumber(0 < 𝜖 ≪ 1).

Asareminder,thedefinitionofrightandwrongcommon-bidsispresentedhere:

𝛱k)À|l:𝑠𝑔𝑛 𝑞 = 𝑠𝑔𝑛 𝑝 commonbid 𝑠𝑔𝑛 𝑞 ≠ 𝑠𝑔𝑛 𝑟𝑒𝑎𝑙 𝑆 (useful) 𝛱

ÁkÂ*À:𝑠𝑔𝑛 𝑞 = 𝑠𝑔𝑛 𝑝 commonbid 𝑠𝑔𝑛 𝑞 = 𝑠𝑔𝑛 𝑟𝑒𝑎𝑙 𝑆 (notuseful)


Usingthistrick,thepricesofbidsinthewrongquantitydirectionbecomeuninterestingtobematched

withanyotherbids.Asaresult,thesebidsareONLYacceptediftheycansolveanotherphysicalproblem

inthenetwork(suchascongestionandvoltageproblems).

The application of price manipulation is demonstrated using several examples in the rest of this

section.At first,we explore the casewhenonly commonbids are submitted to themarket. Then,we

examinethecasewithbothcommonanduncommonbids.Forallexamples,𝜖 = 0.01.Also,thebidsare

consideredtobecurtailable.

Example 1. Consider a market with an imbalance of−1 MW (requiring upwards regulation). The

followingtableshowstheoriginalandmanipulatedprices.

Oldbids 𝒒, 𝒑 𝝍 Newbids 𝒒, 𝒑

Right𝒒

bid1: 1,10 𝜓= −min 10,11,12= −10

bid1: 1,10 bid2: 2,11 bid2: 2,11 bid3: 2,12 bid3: 2,12

Wrong𝒒

bid4: −2,−13 bid4: −2,−9.9913 bid5: −2,−15 bid5: −2,−9.9915

Naturally,all5bidscouldbeaccepted,butwewouldliketoavoidunnecessaryacceptance.Therefore,

weapplythemanipulationtricksothatonly‘bid1’isaccepted.

It is observed that after the manipulation of prices, neither of the bids 4 or 5 are economically

interesting tobeaccepted.However, ifoneof themcansolveaphysicalproblem in thenetwork, they

willbeactivatedaccordingtothenewprices.

Furthermore, (212)doesnotchange themeritorderof thebidswithwrongquantitydirection.This

means that if, for example, ‘bid 2’ has to be accepted to remove a congestion problem, it would be

matchedwith‘bid5’,whichinitiallyandfinallyoffersamoreinterestingbidcomparedto‘bid4’.

Example 2. Assume now that the imbalance changes to+2MW (so that downwards regulation is

needed),usingthesamebidsand𝜖value,theresultofprice-manipulationisasshownintablebelow.


Wrong𝒒

bid1: 1,10 𝜓= −min −13, −15= 15

bid1: 1,15.0110 bid2: 2,11 bid2: 2,15.0111 bid3: 2,12 bid3: 2,15.0112

Right𝒒

bid4: −2,−13 bid4: −2,−13 bid5: −2,−15 bid5: −2,−15

Similar to the previous example, the acceptance of ‘bid 5’ is enough to remove the imbalance

problem,hencethepricesofbids1,2,and3aremanipulatedtoavoidunnecessaryacceptance.Again,if

ithappensthat‘bid4’shouldbeacceptedforaphysicalreason,itwillbematchedtothebidswiththe

mostinterestingmanipulatedprices.Inthiscase,itwouldbe‘bid1’plusahalfof‘bid2’.


Finally,weshowtwoexamplesforthecasewhenall4typesofbidsaresubmittedatatimestep.

Example3.Amarketwithanimbalanceof−1MW.Thisisamodificationofexample1.


Right𝒒

bid1: 1,10

𝜓= −min 10,11= −10

bid1: 1,10 bid2: 2,11 bid2: 2,11 bid3: 2,−5 bid3: 2,−5

Wrong𝒒

bid4: −2,−13 bid4: −2,−9.9913 bid5: −2,−15 bid5: −2,−9.9915 bid6: −1,7 bid6: −1,7

Inthisexample,twouncommonbids,‘bid3’and‘bid6’,areadded.Wenoticethatthemeritorderof

bids4,5,and6hasnotchanged.Thisisbecause‘bid3’isnotusedinfindingthevalueof𝜓.Also,‘bid6’

notmanipulated. Themost economically interesting action is a full acceptanceof ‘bid 3’ and a partial

acceptanceof‘bid4’.Thereasonisthat‘bid3’isindeedveryinteresting.However,suchbidsarerarein

practice.

Example4.Amarketwithanimbalanceof+2MW.Thisisanextensionofexample2.


Wrong𝒒

bid1: 1,10

𝜓= −min −13, −15= 15

bid1: 1,15.0110 bid2: 2,11 bid2: 2,15.0111 bid3: 2,−5 bid3: 2,−5

Right𝒒

bid4: −2,−13 bid4: −2,−13 bid5: −2,−15 bid5: −2,−15 bid6: −1,7 bid6: −1,7

Again, the merit order of bids 1,2, and 3 remains the same, as desired. Also, the price of the

uncommon‘bid6’doesnotcontributetothevalueof𝜓.

Theonly importantdifferenceinthiscaseisthat‘bid3’hasaveryinterestingpricebutnotauseful

quantity. The algorithmwill fully accept the bids 3, 4, and 5. Although such acceptance could create

additional risk of imbalance, the financial gain of it can compensate for the disturbance. If such

unnecessaryacceptanceisnotdesiredatall,itcanbeavoidedbydisallowingbidswithawrong𝑞.

9.5.3 NetworkConstraintsforMarketClearing

WedefineSN"Tasthepowersentfromnodenthroughtheedgenm,andRN"Tasthepowerreceived

innodenthroughedgemn.Then,sNTz isthelocalpoweroff-takefromnoden,andrNTz isthelocalpower

injection into noden.We can state the power balance at a node by equaling outgoing and incoming

powerateverytimestepnas


𝑆*(l(:*→( + 𝑠*lz = 𝑅*(l(:(→* + 𝑟*lz (213)

Thepowerreceivedthroughedges,R"NT,islessthanthepowerdepartingontheedgemnatnodem,

sincepowerlossesoccuronthelines(oredges).Infact,morespecifically

𝑅*(l = 𝑆*(l − 𝑧(* ⋅ 𝐼(*lE (214)

wherez"NisthelineimpedanceandI"NTrepresentstheelectriccurrentintheline.

WhenwedefineSN"Tasthelocalnetpowerinjection,thensNT = rNTz − sNTz ,andweget

𝑆*(l(:*→(

= 𝑆*(l − 𝑧(* ⋅ 𝐼(*lE

(:(→*

+ 𝑠*l (215)

Now,sNT" canbeexpressedintermsofmarketbidsaswesawin(210).Substitutingthisinto(215)gives

𝑆*(l(:*→(

=

𝑆(*l − 𝑧(* ⋅ 𝐼(*lE

(:(→*

+ 𝑆*l + 𝑞*+´l� ⋅ 𝑥*+´l� ⋅ (1 + 𝑗 ⋅ 𝑃𝐹*+´)�´+

(216)

9.5.4 AnExampleofNodalMarginalPricing

Consider theexampleof Figure9.44 [109]. The systemconsistsof two loads and threegenerators.

Each line of the network has identical characteristics and line 1-3 has a flow limit of 120 MW. The

demandandmarginalcostofthegeneratorsareshownintheleftpartofthefigure.Theresultingprices

areshownintherightpart.AlinearizationofKirchhoff’spowerflowlawsthroughastandardDCpower

flowmodelimpliesthateachMWofpowershippedfromanynodetoanyothernodewilldistributeitself

so that one third takes the ‘long’ path (i.e. the path involving two lines) and two thirdswill take the

‘short’path (i.e. thepath involvingone line). It canbeshownthat theprice isequal to40€/MWh,80

€/MWh,and120€/MWhforlocations1,2,and3,respectively.

Figure9.44Athree-nodenetwork.Thenetworkconsistsoflineswithidenticalelectricalcharacteristics.Line1-3hasatransmissionlimitof120MW.Off-takesareinelasticandthemarginalcostsofgeneratorsareindicatedonthefigure.


The first thing to note is that the price is different for all three locations, and that even locations

connectedbyanuncongestedline(locations1and2)haveadifferentprice.Whatisalsonoteworthyis

thatnogeneratorinthenetworkhasamarginalcostof120€/MWh,neverthelessthepriceinlocation3

is120$/MWh.

Toseehowthismarginalcostemerges,notethattheminimumcostdispatchthatsatisfiesoff-takesis

equalto160MWinlocation1and140MWinlocation2. Inordertosatisfyamarginal increaseinoff-

take in location3by1MWatminimumcost,while satisfying transmission constraints,we reduce the

injectionofgenerator1by1MW,andincreasetheinjectionofgenerator2by2MW,soastosatisfythe

additionaloff-takewhileensuringthattheflowonline1-3remainsatthe120MWlimit.Thisresultsina

marginalcostincreaseof120$/MWh,whichisthenodalpriceatlocation3.Thispricesendsthecorrect

signaltoloadsatlocation3aboutthemarginalcostimpactoftheirconsumption.

Theaboveexampleisusedforthesakeofillustration,butitisbasedonamathematicallyformalized

approach towards defining nodal prices. In particular, nodal prices are obtained as the dual optimal

multipliersofpowerbalanceconstraintsinoptimalpowerflowproblems.Optimalpowerflowproblems

are solved routinely in short-term (day-aheadand real-time)operations, therefore the computationof

nodalprices isfullycompatiblewithexistingoperatingpracticesandiswellwithinthereachofexisting

computationalsoftwareforindustrialscalemodels.Nodalpricescanthenbeusedasthebasisforsettling

longer-termforwardcontracts,andprovidingappropriatesignalsfortheinvestmentintransmissionand

generationinfrastructure.Intheaboveexample,thepriceatnode1isequalto40$/MWh,indicatinga

profitable investment opportunity for an increase in the capacity of line 1-3 which coincidentally

increasesthevalueoftradebetweenproducers in location1andconsumers in location3.Owingto its

compatibilitywithexistingoperationsanditssoundtheoreticalfoundations,nodalpricinghasbecomea

standardcomponentofelectricitymarketdesigninanumberofmarketsworldwide,includingtheUnited

States (e.g. PJM, Texas, MISO, NYISO, California ISO) and certain European markets (e.g. Poland and

Greece).

9.5.5 AnExampleofZonalPricing

ConsidertheexampleofFigure9.45[110].Anaggregationofnodes1and2intoasinglegenerating

zonecreatesthechallengeofdefininghowmuchcapacitycanbecarriedfromtheinjectionzonetothe

off-takezone(whichiscomprisedoflocation3).Withaviewtowardsrespectingthethermallimitofline

1-3regardlessoftheconfigurationofproduction,wewouldimposealimitof900MWbetweenzones,so

as to ensure that even if all injections are sourced from location 1, the limit of line 1-3 is respected.

However,thiseliminatesopportunitiesfortrade,incasenode2hascheaperinjectionsthanlocation1.If,

ontheotherhand,wewishtomaximizeopportunitiesfortradethenwewoulddefinethetransmission


limitas1800MW(ifalloff-takesweretobesourcedfromlocation2),howeverthiswouldthreatenthe

thermallimitofline1-3ifitturnsoutthatthepowerissourcedfromlocation1.

Figure9.45Athree-nodenetworkwithidenticallinecharacteristics,injectionsinnodes1and2,andanoff-takeinnode3.Line1-3hasalimitof600MW.

9.5.6 PricinginthePresenceofBinaryVariables

Introducing binary variables that indicate on/off decisions into an optimal scheduling or dispatch

problem (and, more generally, introducing features to the problem that render it non-convex)

complicates the definition of market clearing prices, because market equilibrium, as described in the

previous paragraph (a pair of decisions and prices such that agents maximize profits and such that

markets clear) may not exist due to ‘jumps’ in agent decisions as a function of price. An alternative

definition of market clearing price under such conditions has been debated extensively in the power

systemeconomicscommunity.Welisthereanumberofpossibilities:(i) It ispossibletoproceedwitha

resolutionoftheconvexrelaxationoftheschedulingproblemandusethedualmultipliersofthepower

balanceconstraintsaslocationalprices;(ii)itispossibletosolvetheschedulingproblem,fixdecisionsfor

thepresent interval, solve thedispatchproblemwithbinarydecisions fixedto theiroptimalvalue,and

obtainthepricefromthepowerbalanceconstraints;(iii)convexhullpricing[111]couldbeemployed,(iv)

theproposalofO’Neill[112]couldbeused,or(iv)aschemelikeEuphemia[113]couldbeimplemented,

whereby the payments to paradoxically accepted bids are minimized. The relative advantages and

disadvantagesof these approaches arewell documented in literature. These approaches remain tobe

testedinournumericalexperiments.Theapproachespresentedinthesecondsectionofthisdocument

pertaintoconvexoptimizationmodelsand,hence,donotaccommodatebinaryvariables.

9.5.7 AlgorithmforCalculatingPriceswithPossibleIndeterminacies

First, theprimalproblem issolvedandquantitiesare found.Then, thealgorithmfor findingcleared

price𝑝∗atnode𝑛goesasfollows:

Every totally-acceptedandpartially-acceptedbid issortedaccordingto themeritorder.Thisshould

result inallpartially-acceptedbidsbeingattheendofthesequence,andforthebidswiththesamep,

thepartially-acceptedoneshouldcomeafterthetotally-acceptedones.Foreachnoden,weconcentrate

onthe lastacceptedbids,𝛽jO6T (notethatweusetheplural form“bids”,becausethe lastacceptedbid


couldexistforbith𝑞 > 0and𝑞 < 0).IfthereexistapartiallyacceptedbidinβjO6T(suchthat0 < 𝑥 < 1),

theclearedpriceisfoundbylookingatthevalueofthepriceinthatbid:

𝑝∗ = 𝑝∗ 𝑥�∈�klmnoOpTHOj (217)

If𝛽jO6T contains no partially-accepted but only totally-accepted bids (𝑥 = 1), then there is a price

indeterminacy.Insuchacase,theclearedpriceislocatedinthemiddleofthe𝑝4valuesforthe𝛽jO6Twith

𝑞 > 0andthe𝛽jO6Twith𝑞 < 0:

𝑝∗ = 12 ⋅ 𝑝4

qr3 + 𝑝4qs3 (218)

Payattentiontothefactthatp4valuescorrespondtothenon-zeroquantityofaprimitivebid(please

refertoFigure3.3).

AnexampleisshowninFigure9.46.Inthiscase,thebidsa4andb4arethelasttotally-acceptedbids.

Figure9.46Priceindeterminacyattheverticaledgeofthecrossing.

Therefore,theclearedpriceiscalculatedas

𝑝∗ = 12 ⋅ 38 + 30 = 34

9.5.8 DCversusACon2-nodeExample

In order to illustrate the influence of the exact representation of AC power flow equations on

locationalmarginalprices,considerthetwo-nodesystemexampleofFigure9.47.

Figure9.47Atwo-nodenetworkwithaninjectionatconstantvoltageatnode1andaninelasticoff-takeatnode2.Thebasisis100MW.Theper-unitcharacteristicsofthelineareR=0.0065andX=0.062.Themarginalcostatnode1

is50€/MWh.


AnoptimalDCpowerflowformulationofthemodel,wherereactivepowerandrealpowerlossesare

ignored andwhere voltage is normalized to 1 per unit16,would result in amarginal price equal to 50

€/MWhatlocation2,aslongasthecapacityofthetransmissionlineisgreaterthan0.95perunit.AnAC

formulationcanbedescribedas:

𝑣E − 2 0.0065𝑓𝑙𝑜𝑤𝑃 + 0.062𝑓𝑙𝑜𝑤𝑄 + 𝑙 0.0065E + 0.062E = 𝑣4

𝑓𝑙𝑜𝑤𝑃E + 𝑓𝑙𝑜𝑤𝑄E

𝑙≤ 𝑣E

𝑞 = 0.0065𝑙 − 𝑓𝑙𝑜𝑤𝑃

𝑓𝑙𝑜𝑤𝑃 + 0.95 = 0

𝑞km+�l)'m = 0.062𝑙 − 𝑓𝑙𝑜𝑤𝑄

𝑓𝑙𝑜�𝑄 + 0.5 = 0

where𝑣) isthevoltagemagnitudesquaredinlocation𝑖,𝑓𝑙𝑜𝑤𝑃 istherealpowerflowovertheline,

𝑓𝑙𝑜𝑤𝑄isthereactivepowerflowovertheline,𝑞km+�l)'m isthereactivepowerinjectioninnode1,and𝑙is

the currentmagnitude squared. The locationalmarginal price at node2becomes50.679€/MWh. The

differencecomparedtotheDCmodelisduetotherealpowerlossesthatoccuroverthelineconnecting

nodes1and2.

9.5.9 ThreeApproachestoPricingwithACPowerFlow

The above example illustrates the complexity introduced by the AC power flow equations. This is

crucialfortheintuitivenessofderivednodalprices,whichwillhelptojustifythedifferencesinthenodal

prices.Inageneralnetwork,itisimportanttobeabletoexplainhowdifferentfactorscontributetothe

formationofprice.Wehavedevelopedthreedifferentapproachestowardsansweringthisquestion,with

each approach providing a different point of view for the same problem. During the implementation

phase, it will be revealed which approach has the advantage in terms of intuitiveness, computation

efficiency,andaccuracy.

9.5.9.1 Method1:ContributionofConstraintstoPriceFormation

ThefirstmethodgeneralizestheapproachofSchweppe,CaramanisandBohn[82],[83].Thepriceata

certainlocationisobtainedasthecontributionofaglobalpowerbalanceterm,atermrelatedtobinding

voltageconstraints,andatermrelatedtobindingtransmissionconstraints:

𝜆 = GlobalBalanceConstraintTerm + VoltageConstraintTerm +

TransmissionConstraintTerm + CongestionTerm(219)

16 The basis for voltage is irrelevant since all relevant quantities can be scaled accordingly,whatmatters is thatvoltageisnotexplicitlyoptimized.


Eachofthefourtermsabovedependonnetworkparameters(forexample,impedanceandreactance

oflines)andthepartialderivativeofthedecisionvariableswithrespecttoanetinjectionofrealpowerin

thenodewhosepricewearecomputing.Thispartialderivativebecomesanalyticallycomplextoderive

evenforthesimpletwo-nodenetworkofFigure9.47,andispracticallyimpossibletoderiveanalytically

for a network with arbitrary topology. Therefore, each of these terms needs to be approximated

numerically by re-solving theoptimalpower flowproblemwith a slightperturbation in theamountof

powerinjectioninthenodewhosepricewearecomputing.

Thekeytotheaboveresultistouseawell-knownresultinpowersystemanalysis:allpowerflowscan

bedescribedasafunctionoftherootnodevoltagemagnitude,andrealandreactivepowerinjectionsat

thenodes.Inintuitiveterms,thismeansthatifweweretoldthevoltagemagnitudeattherootandthe

real and reactivepower injections at all locationsof thenetwork thenwe coulddetermine thepower

flows over the entire network.We can thenwrite out the optimization problem in terms of the root

voltagemagnitude,realinjectionsandreactiveinjections.Relaxingthevoltageconstraints,transmission

constraints,andglobalpowerbalanceconstraintoftheproblemyieldsaLagrangianfunction,whosefirst-

orderconditionswithrespecttorealpowerinjectioncanbeusedtoderivetherelationshipabove.

9.5.9.2 Method2:ContributionofLossestoPriceFormation

Thestartingpointofthesecondmethodisthesensitivityinterpretationofprices:therealpowerprice

ofacertainlocationisthesensitivityofcosttoamarginalchangeinrealpowerinjectionsinthatsame

location.Sincecostcanbeexpressedasafunctionoftotalrealpower,whichinturncanbeexpressedas

thesumofrealpowerconsumptionpluslossesintheentirenetwork,weobtainthefollowing:

𝜆 ≃ 𝜆((1 +n<Âttmt8n|É)∈} ) (220)

whereλ" is themarginal cost of themarginal generator,E is the set of lines of the network, and��b66d6��

is the total derivativeof the real power losseswith respect to real power consumption in the

locationwhosepricewearecomputing.Themarginalgeneratoristhegeneratorthatisproducingstrictly

above its technical minimum and strictly below its technical maximum, and whose output would be

expected tochange inorder tosatisfyachange in the realpowerconsumption for the locationwhose

pricewearecomputing.Asinthecaseofthepreviousmethod,thefullderivativescannotbecomputed

analyticallyandneedtobeapproximatednumerically.

Itisworthnotingthattherelationaboveisnotexact,sinceitdoesnotaccountforthesecond-order

effectthatnon-marginalgeneratorrealpoweradjustmentshaveontheobjectivefunction.Nevertheless,

experimentalexperienceonaCIGRE15-busnetworkshowthatitpredictspriceswithanaccuracywhich

isnoworsethan0.1€/MWh.Additionaltestsonnetworksofdifferentsizesandtopologieswouldhaveto

beperformedinordertoverifytheprecisionoftheaboveformula.


Theabovedecompositionprovides themost intuitivewayofunderstanding the formationofprice.

Onceweunderstandhowchangesinrealpowerinjectionatacertainlocationimpactrealpowerlosses

onlines,wecanexplainhoweachoftheselosstermscontributestowardstheformationofprice.Note

thatthesecontributionsmaybepositiveornegative.Anegativecontributiontolossesoccurswhenreal

power injection at a certain locationbrings the voltagemagnitudeof this location closer to thatof its

neighbors (i.e. if the neighbors have a higher voltage magnitude than the node whose price we are

computing),and thecontributionbecomesnegativeotherwise (i.e. theneighborshavea lowervoltage

magnitude than the nodewhose pricewe are computing).Note that the price formation involves the

entirenetwork,andcannotbeexplainedonlyintermsofthelinesthatareadjacenttothenodewhose

priceisbeingcomputed.

9.5.9.3 Method3:RecursiveRelation

Thethirdmethodbreaksdownthepriceasfollows:

𝜆 = 𝛼 ∙ 𝜆� + 𝛽 ∙ 𝜇 + 𝛾 ∙ 𝜇� (221)

whereλ�isthepriceofrealpoweroftheancestornode,µisthepriceofreactivepowerofthenode

inquestion,andµ�isthepriceofreactivepoweroftheancestornode.Theconstantsα,βandγdepend

onnetworkparametersandthedecisionvariablesoftheoptimalpowerflowsolution,andcantherefore

beeasilycomputedoncethemathematicalprogramhasbeensolved.Statedotherwise,thepriceatthe

currentnodecanbeexpressedasalinearcombinationofthereactivepowerpriceatthecurrentnode,

andthepriceofrealandreactivepoweroftheancestornode.Thisprovidesuswitharecursiverelation

forcomputingpricesfromthetoptothebottomofaradialnetwork.Experimentalexperiencesuggests

thatthetermα ∙ λ�hasthegreatestinfluenceintheformationoftheprice.

9.5.9.4 Cross-comparisonofThreeMethods

Thethreemethodsdescribedabovecanbecomparedintermsofhowintuitivetheyareinexplaining

prices,howeasilythepricebreakdowncanbecomputed(howmanytimestheoptimalpowerflowneeds

to be solved and whether the conic relaxation is adequate or a non-linear power flow needs to be

solved), and exactness (whether the derived identities are exact and whether these identities can be

computed inclosed formorneed tobeapproximatednumerically).Table9.1 summarizes the relevant

information. Ithighlights the tradeoffs in the threemethods, since there isnodominantmethod inall

criteria.

Physicalintuition

Scalabilityofcomputation Exactness

Method1:Constraintcontribution

+ -(twoNLPs,ortwoSOCPsifconicrelaxationistight)

+(partialderivativesneedtobeapproximatednumerically)

Method2:Losses ++ +(twoSOCPsevenifconicrelaxationisnottight)

-(originalidentityisapproximate,andtotalderivativesneedtobeapproximatednumerically)

Method3:Recursive - ++(oneSOCP) ++(validevenifconicconstraintisnottight)


Table9.1Comparisonofthreemethods.

Ourrecommendationforthepreferableapproachismethod2,basedonlosses,sinceitisthemethod

bestgroundedonphysical intuition,whilebeingacceptablyaccurateandscalable(itrequirestwicethe

amountofcomputationcomparedtomethod3,andreliesontheresolutionofanSOCPrelaxationwithin

an operationally acceptable timeframe, which is a working assumption of the SmartNet project). The

accuracy of the method has been verified through experiments: for the CIGRE 15-node model the

methodpredictspriceswithanaccuracyof0.1€/MWh.

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Network and market models: preliminary ... - SmartNet project