A mixed-integer linear programming approach

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Electric Power Systems Research

jou rn al h om e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / e p s r

A linearized formulation of AC multi-year transmission expansion planning: A mixed-integer linear programming approach

Tohid Akbari

^∗

, Mohammad Tavakoli Bina

FacultyofElectricalEngineering,K.N.ToosiUniversityofTechnology,Tehran,Iran

a r t i c l e i n f o

Articlehistory:

Received21October2013

Receivedinrevisedform28March2014 Accepted16April2014

Keywords:

AC-OPF

Linearizedpowerflow

Mixed-integerlinearprogramming Transmissionexpansionplanning

a b s t r a c t

ThispaperpresentsamethodinexpansionplanningoftransmissionsystemsusingtheACoptimalpower flow(AC-OPF).TheAC-OPFprovidesamoreaccuratepictureofpowerflowinthenetworkcomparedto theDCoptimalpowerflow(DC-OPF)thatisusuallyconsideredintheliteraturefortransmissionexpan- sionplanning(TEP).WhiletheAC-OPF-basedTEPisamixed-integernonlinearprogrammingproblem, thispapertransformsitintoamixed-integerlinearprogrammingenvironment.Thistransformationguar- anteesachievementofaglobaloptimalsolutionbytheexistingalgorithmsandsoftware.Theproposed modelhasbeensuccessfullyappliedtoasimple3-buspowersystem,Garver’s6-bustestsystem,24-bus IEEEreliabilitytestsystem(RTS)aswellasarealisticpowersystem.Detailedcasestudiesarepresented andthoroughlyanalyzed.SimulationsshowtheeffectivenessoftheproposedmethodontheTEP.

©2014ElsevierB.V.Allrightsreserved.

1. Introduction

Transmissionexpansionplanning(TEP)addressestheproblem ofaugmentingtransmissionlinesofanexistingtransmissionnet- work;theobjectiveistooptimallyserveagrowingelectricload whilesatisfyingasetofeconomical,technicalandreliabilitycon- straints[1].Ingeneral,theTEPisconsideredasmakingastochastic decisiononwhen(thetime),where(thelocation),andwhichtypes oftransmissionlinestobeinstalled.In[2,3],aclassificationscheme categorizesthesubjectspublishedinthisarea.

A mixed-integer linear programming (MILP) is used in [4], for the TEP problem that considers power losses. We suggest a linearization methodfor theAC-TEPbased onthemethodin [4]. However, both active and reactive powers are included in ourproposedformulation.In[5],amulti-yearTEPmodelispre- sentedusingadiscreteevolutionaryparticleswarmoptimization approach.Aguadoetal.[6],presentanovelTEPmodelthatcon- sidersa multi-yearplanninghorizonina competitiveelectricity market.BoththeTEPand generationexpansionplanning(GEP) problemsareanalyzedtogetherin[7,8].Also,transmissionswitch- ing(TS)isinvestigatedin[9]fortheTEPshowingthattheTScould improvethecapacityexpansionplanningmodelaswellasreduc- ingthetotalplanningcost.Ameta-heuristicandholisticapproachis

∗Correspondingauthor.Tel.:+9809125179374;fax:+982188462066.

E-mailaddresses:tohidakbari@yahoo.com,tohidakbari@ee.kntu.ac.ir(T.Akbari).

presentedin[10]fortheTEPwhichhasbeentestedonarealistic powersystem.In[11],ascenario-basedmulti-objectivemodelis presentedformulti-stageTEPwherethenon-dominatedsorting geneticalgorithm(NSGA-II)is usedtoovercomethedifficulties insolvingthenon-convexmixed-integeroptimizationproblem.A multi-objectiveframeworkispresentedin[12]fortheTEPindereg- ulatedenvironment.Recently,intermittentenergyresourceshave beenexperiencingarapidgrowthinpowergenerationaroundthe world.Therefore,newchallengesareintroducedtointegratethe renewableenergysources(RES)tothepowergrid.Therearemany publishedpaperswhichfocusonthisnewlymainissue[13–15].

Theplanner ofpowersystemshoulddeal withmany uncer- tainties during the planning process such as load uncertainty, uncertaintyinprices,marketrulesandetc.AmultistageTEPprob- lem includingavailable transfer capability (ATC) is modeled in [16]thattakesloaduncertaintyintoaccountbyconsideringsev- eralscenariosgeneratedbyMonteCarlosimulation. In[17],the TEP is studied by considering the load uncertainty using ben- dersdecomposition.Sincethispaperisfocusedontransforming a mixed-integer nonlinear programming (MINLP) probleminto a MILP, uncertainties have not been considered in this paper.

However,thepresentedmodelcaneasilybeextendedfortaking uncertaintiesintoaccount.

Thesurveyed literatures usetheDC-OPFfor solvingtheTEP problemwhichisnotcompletelysuitableduetoignoringreactive power.However,therehavebeensomepublishedpaperswhich usetheAC-OPFtosolvetheTEPproblem[18–21].Thispaperpro- posesanapproachfortransmissionplanningbasedontheAC-OPF, http://dx.doi.org/10.1016/j.epsr.2014.04.013

0378-7796/©2014ElsevierB.V.Allrightsreserved.

Nomenclature Indices

g indexofgenerators i,j indicesofbuses l indexoflines

m indexofblocksusedforpiecewiselinearization t indexofsub-periods(times)

Sets

B setofallbuses

CL setofallcandidatelines

CL_i setofallcandidatelinesconnectedtobusi EL setofallexistinglines

EL_i setofallexistinglinesconnectedtobusi G setofallgenerators

SP setofallsub-periods Constants

AP^max_L.I maximumapparentpowerflowoflinel I interestrate

IC_l investmentcostofcandidatelinel

M numberofblocksusedforpiecewiselinearization nc numberofcandidatelines

ng numberofgenerators

PD,it activepowerdemandatbusiintimet Q_D,it reactivepowerdemandatbusiintimet P_G,g^min minimumactivepowerofgeneratorg P_G,g^max maximumactivepowerofgeneratorg Q_G,g^min minimumreactivepowerofgeneratorg Q_G,g^max maximumreactivepowerofgeneratorg

|V_i^min| minimumofthevoltagemagnitudeatbusi

|V_i^max| maximumofthevoltagemagnitudeatbusi Y_ij⁰,Yij admittanceoflineijfortheexistingandcandidate

lines,respectively.(Y_ij⁰=G⁰_ij+jB⁰_ij,Y_ij=G_ij+jB_ij)

˛_ij,m slopeofthemthblockofthevoltageangledifference betweencorridoriandj

_ij maximumofeachblockwidthforcorridoriandj _ref voltagephaseanglefortheslackbus(_ref=0) aconstanttomakeinvestmentandoperationcost

comparable

ij,,ωij, ₁,ij, disjunctiveparameters Variables:

ICt investmentcostintimet OC_t operationcostintimet OF objectivefunction

PG,gt activepowerofgeneratorgintimet P_L⁰

i→j,lt activepowerflowofexistinglinelfrombusitobus jintimet

PL_i→j,lt activepowerflowofcandidatelinelfrombusito busjintimet

Q_G,gt reactivepowerofgeneratorgintimet Q_L⁰

i→j,lt reactivepowerflowofexistinglinelfrombusito busjintimet

QL_i→j,lt reactivepowerflowofcandidatelinelfrombusito busjintimet

u_lt binaryvariablerelatedtothecandidatelinesintime t

|V_it| voltagemagnitudeatbusiintimet

_ij,mt widthofthemthangleblockofcorridoriandjin

timet

⁺_ijt,⁻_ijt positivevariablesusedsoastoeliminatetheabso- lutefunction

_it voltageangleatbusiintimet

providing a more accurate picture of both active and reactive powerflowsintheexpandedpowernetworkinthefutureplanning horizon.ThenoveltyofthispaperistheintroductionofaMILPfor- mulationusingtheAC-OPFapproachsoastosolvetheexpansion planningproblemoftransmissiongrid.Inbrief,anAC-OPF-based TEPisformulated,and linearizedaround theoperating pointin ordertoderiveaMILPproblem.SolvingaMILPproblemisamature technology,wheretheMILPsolverscanbeembeddedinmanytools and applications.Moreover,somenumerical examplesare pre- sentedinwhichsimulationsarediscussedaccordingly.Thewhole linearizationprocessprovidedinsubsection“B”ofsection“II”that convertsthenon-linearACapproachtoaMIPproblemisnoveland hasnotbeenpreviouslypresented.Thispapercontributestothe TEPbyapproximatingthesineandcosinefunctionsinpowerflow equationsbytheirTaylor’sseries;then,thequadraticfunctionis modeledusingpiecewiselinearfunctions.Moreover,theinequal- ityconstraintsforapparentpowersofexistingandcandidatelines aretransformedintoasetoflinearconstraints.Numericalresults confirmthecontributionoftheproposedmethodincomparison withtheconventionalsolutions.Sincetheproposedoptimization problemforsolvingtheTEPislinear,theglobaloptimalsolution canbeobtainedeasilybytheavailablesoftware.Inaddition,out- comesobtainedbytheproposedmethodaremoreaccurate(due totakingreactivepowerintoaccount)thanthoseoftheavailable conventionalmethods(duetoignoringreactivepowerbyusing theDC-OPFforsolvingtheTEP).Itshouldbeemphasizedthatthe ISO(independentsystemoperator)isresponsiblefortransmission expansionplanning;theISOaimsatminimizingtheinvestment costplusthetotalpaymenttothegeneratingcompanies.

2. AnalysisandformulationoftheTEP

Heretheproposedmethodisformulated,presentingtheTEP basedontheAC-OPFusingaMINLPthatwillbetransformedintoa MILP.

2.1. TheAC-OPF-basedTEPformulation

Theobjectivefunctionistheinvestmentcost(IC)whichisthe constructioncostofnewlinesandtransformers(ifany)plusthe operationcost(OC).TheOCincludesthetotalcostofgenerationin thepowersystemunderstudy.UsinganACpowerflow,objective functionoftheTEPcanbeformulatedasfollows:

MinOF=

t∈SP

(1+I)^−t

⎛

⎜ ⎜

⎜ ⎜

⎝

nc

l=1

u_ltIC_l

Investmentcostintimet(ICt)

+

ng g=1

CgPG,gt

Operationcostintimet(OCt)

⎞

⎟ ⎟

⎟ ⎟

⎠

⁽¹⁾

Theobjectivefunction(1)issubjectedtothefollowingequalityand inequalityconstraints;

Equalityconstraints:

PG,it−PD,it=

l∈EL_iP_L⁰

i→j,lt+

l∈CL_iPL_i→j,lt ∀i,j∈B,∀t∈SP (2)

QG,it−QD,it=

l∈EL_iQ_L⁰

i→j,lt+

l∈CL_iQL_i→j,lt ∀i,j∈B,∀t∈SP (3)

P⁰_L

i→j,lt =|V_it|²G_ij⁰−|V_it||V_jt|×(G⁰_ijcos(_it−_jt)

+ B⁰_ijsin(_it−_jt)) ∀^i,j∈B,∀^l∈EL,∀^t∈SP (4)

PL_i→j,lt =ult(|Vit|²Gij−|Vit||Vjt|×(Gijcos(it−jt)

+ Bijsin(it−jt))) ∀i,j∈B,∀l∈CL,∀t∈SP (5)

Q_L⁰

i→j,lt =|Vit|²B⁰_ij−|Vit||Vjt|×(G⁰_ijsin(it−jt)

+ B⁰_ijcos(it−jt)) ∀i,j∈B,∀l∈EL,∀t∈SP (6)

QL_i→j,lt =ult(|Vit|²Bij−|Vit||Vjt|×(Gijsin(it−jt))

+ Bijcos(it−jt))) ∀i,j∈B,∀l∈CL,∀t∈SP (7)

Inequalityconstraints:

P^min_G,g ≤PG,gt≤P_G,g^max ∀^g∈G,∀^t∈SP (8) Q_G,g^min≤Q_G,gt≤Q_G,g^max ∀^g∈G,∀^t∈SP (9) (P_L⁰

i→j,lt)²+(Q_L⁰

i→j,lt)²≤(AP_L,l^max)² ∀l∈EL,∀t∈SP (10)

(P_L

i→j,lt)²+(Q_L

i→j,lt)²≤u_lt(AP_L,l^max)² ∀^l∈CL,∀^t∈SP (11)

|V_i^min|≤|V_it|≤|V_i^max| ∀ⁱ∈B,∀^t∈SP (12) u_1(t+1)≥u1t ∀^l∈CL,∀^t∈SP (13) While(1)showstheobjectivefunction,i.e.ICplusOC,(2)and (3)representactiveandreactivepowerbalanceateachbus,respec- tively.Constraints(4)and(5)indicateactivepowerflowingthrough existingandcandidatelines,respectively.Similarly,constraints(6) and(7)introducereactivepowerflowsthroughexistingandcan- didatelines,respectively. Superscriptindex 0isused todenote theexistinglines.Activeandreactivepowergenerationlimitsof thegeneratorsarerepresentedby(8)and(9).Transmissionpower limitsareshownby(10)and(11)forboththeexistingandcan- didatelines,respectively.Voltagemagnitudelimitsareshownby (12).Eq.(13)impliesthatifacandidatelineisconstructedwithin thesub-periodt,it isconsideredasanexistinglineinthenext sub-period.

Theabove-formulatedproblemisaMINLPproblemduetothe nonlinearitiesin(4)–(7)aswellasin(10)–(11).Itshouldbenoted thattheobtainedsolutionforthemodel(1)–(12)isapracticalfeasi- blelocaloptimumsolutionthatmeetstherequirementsofISO;but, notaglobalone.Thisisduetothenon-convexitynatureofthefor- mulatedproblem.However,whenonetriessolvinganon-convex MINLPoptimizationproblem,therewillbenoguaranteetoobtain theglobaloptimumsolution.Thisrestrictionremainsunresolved forthemostpracticaloptimizationmodelsinacomplexsystem [22,23].

Hence,theproposed MINLPproblemis transformed intoan MILPtoavoidanylocaloptimalsolutions.ThisobtainedMILPfor- mulationisanapproximatefortheoriginalACformulation.Further, theglobaloptimal solutionofthe MILPformulation caneffica- ciouslybeobtainedbyexistingsolvers.Whiletheresultantsolution isnotnecessarilythesameastheglobaloptimalsolutionofthe

MINLPformulation(hardlyattainableandnoguaranteetoreach it),theMILPsolutionremainsclosetotheglobalsolutionoforigi- nalMINLPproblemdependingonthenumberofblocksinmodeling thecircleorquadraticterms(Morn).

2.2. LinearizationofthepresentedMINLP

HereitisintroducedthelinearizationoftheMINLP,presented by (1)–(12),around thenormal operating point. Let usassume thephasedifferencebetweenthevoltagesatbothendsofevery existing orconstructedlines issmall enough;this implies vali- dationoffirstorderapproximationofsineand cosinefunctions intheirTaylorseries.Moreover,assumethevoltagemagnitudes arenearly1p.u.forallbuses.Theseassumptionsarepractically trueundernormaloperatingcondition,maintainingthesystemfar frominstabilityandothersecuritylimits.Basedontheaforemen- tionedassumptions,(4)–(7)canbere-arrangedbysubstitutingsine andcosinefunctionsintheirTaylorseriesexpansionaboutzero.

In thefollowing formulation,just forthesake ofsimplicity,we havedroppedthesubscriptt.Itcanbeaddedtothecorresponding variablesorparametersifreferringtosub-periodt.

P_L⁰

i→j,l=G⁰_ij(1−cos(_i−_j))−B⁰_ijsin(_i−_j)

∼=G⁰_ij

(_i−_j)² 2

−B⁰_ij(_i−_j) ∀^i,j∈B,∀^l∈EL (14)

P_L

i→j,lt=u₁(G_ij(1−cos(_i−_j))−B_ijsin(_i−_j))

∼=u₁

G_ij

(_i−_j)² 2

−B_ij(_i−_j)

∀^i,j∈B,∀^l∈EL (15)

Q_L⁰

i→j,lt=−B⁰_ij(1−cos(_i−_j))−G⁰_ijsin(_i−_j)

∼= −B⁰_ij

(_i−_j)² 2

−G_ij⁰(i−j) ∀i,j∈B,∀l∈EL (16)

QL_i→j,l=u1(−Bij(1−cos(i−j))−Gijsin(i−j))

∼=u1

−Bij

(i−j) 2

2

−Gij(i−j)

∀i,j∈B,∀l∈CL (17)

Then,thesecondorderfunctionsin(14)–(17)canbelinearized by considering 2M piecewise linear blocks as shown in Fig.1, approximatingtheexistinglinelbetweencorridorsiandjasbelow:

(_i−_j)²=

M m=1

˛_ij,m⁰_ij,m, ∀^i,j∈B (18)

|i−j|=

M m=1

_ij,M⁰ , ∀i,j∈B (19)

0≤_ij,m⁰ ≤ ij, ∀i,j∈B (20)

˛ij,m=(2m−1) ij m=1,...,M∀i,j∈B (21) where˛_ij,mand⁰_ij,maretheslopeandthevalueofthemthblockof phasedifferencebetweencorridorsiandj,respectively.Theslope canbecalculatedaccordingto(21).Thesameconstraintswillbe appliedtothecandidatelines.However,(20)shouldberewritten forcandidatelinesasfollow:

0≤_ij,m≤ ij+(1−u₁)_ij ∀^i,j∈B (22)

Fig.1.Piecewiselinearmodelingoflossfunction.

Thedisjunctiveparameter_ijshouldbelargeenoughtoavoid restrictionofthewidthofthemthphasedifferencebetweenthe twonodesiandj.In(22),ifacandidatelinelisconstructed,the blockwidthwillbeboundedaboveby_ij;otherwise,forasuffi- cientbignumber_ij,theaboveconstraintwillbenonbinding.An appropriatevalueforijcanbe2.

Thenonlinearity in(19)(the absolutefunction)canbeelim- inatedbyintroducingtwo extrapositive variables_ij⁺and ⁻_ij as follows:

|_i−_j|=_ij⁺+_ij⁻ ∀^i,j∈B (23) _i−_j=_ij⁺−_ij⁻ ∀^i,j∈B (24) ⁺_ij ≥0and⁻_ij ≥0, ∀^i,j∈B (25) Applying the above linearization method to (14)–(17) will removethenonlinearitiesinboth(14)and(16).However,thenon- linearitiesin (15)and (17)willstill remaininplace duetothe productofdiscreteandcontinuousvariables(u₁byi).Thesenon- linearitiescanbeeliminatedbyintroducingconstantlargeenough numbersω_ijand_ijasfollows:

−(1−u1)ωij≤PL_i→j,l+Bij×(i−j)

−G_ij

M m=1

˛_ij,mi_j,m

2 ≤(1−u₁)ω_ij, ∀^i,j∈B,∀^l∈CL (26)

−(1−u₁)_ij≤Q_L

i→j,l+G_ij×(_i−_j)

−B_ij

M m=1

˛_ij,mi_j,m

2 ≤(1−u₁)_ij, ∀^i,j∈B,∀^l∈CL (27) Appropriatevaluesforω_ijand_ijcanbeselectedasbelow:

ωij=2²Gij−2Bij (28)

ij=2Gij−2²Bij (29)

It should be emphasized that the above linearization tech- nique adds three sets of continuous variables to the problem, ⁰_ij,m(_ij,m),⁺_ijand⁻_ij (M+2variables foreach linebetweencor- ridorsiandj).Further,nonlinearityin (10)canberemovedby

introducing the following n equations (see Appendix for more detailsandexplanations):

sin

_360◦k

n

−sin

_360◦

n (k−1)

P_L⁰

i→j,l

−

cos

_360◦k

n

−cos

_360◦

n (k−1)

Q_L⁰

i→j,l

−AP_L,l^max×sin

_360◦

n

≤0 k=1,...,n,∀l∈EL (30)

Thecorrespondingequationforcandidatelinescanalsobelike:

u₁

sin

_360◦k

n

−sin

_360◦

n (k−1)

P_L

i→j,l

−

cos

_360◦k

n

−cos

_360◦

n (k−1)

Q_L

i→j,l

−AP_L,l^max×sin

_360◦

n

≤0, k=1,...,n,∀^l∈CL (31)

Itcanbeseenthat(31)isstillnonlinearduetothemultiplication ofdiscreteandcontinuousvariables(productofu₁ byPL_i→jland QL_i→jl).Inordertomake(31)linear,asecondstepshouldbetaken furtherbyusingthefollowingthreerelationships:

−u1× 1≤PL_i→j≤u1× 1, ∀l∈CL (32)

−u₁× 1≤Q_L

i→j,l≤u₁× 1, ∀^l∈CL (33)

sin

_360◦k

n

−sin

_360◦

n (k−1)

PL_i→j,l

−

cos

_360◦k

n

−cos

_360◦

n (k−1)

Q_L

i→j,l

−AP_L,l^max×sin

_360◦

n

≤0 k=1,...,n,∀l∈CL (34)

Asuitableamountfor ₁canbethemaximumapparentpower forlinel,i.e.AP^max_L,I .Ifcandidatelinelisselected,(34)restrictsactive andreactivepowerflowsuchthatconstraints(32)and(33)willbe nonbinding.Otherwise,constraints(32)and(33)willlimitactive andreactivepowerstozerosothat(34)willbenonbinding.Note thatsin(360^◦/n)isalwayspositive(n≥4).Thus,linearizationof(10) and(11)requires usingnlinearequationsforeach existingline alongwithn+2equationsforeachcandidatelineinsteadofusing oneequation.Therefore,asthenumbernbecomeslarger,thesearch spacegetslargeraswell.Asasolutionforthiscase,firstthemethod canbeappliedtothehigh-voltagetransmissionlinesinorderto reducethesearchspace;second,theplanningcanbeconducted forlowervoltageleveltransmissionlines.

3. Numericalexample

Theproposedmodelhasbeensuccessfullyappliedtofourdiffer- entpowersystems;asimple3-buspowersystem,Garver’s6-bus testsystem,IEEE24-busreliabilitytestsystems(RTS)andIranian 400-kVnetwork.ThisproblemwassolvedonaPCrunningwith Core2 Duo CPU, clockingat2.00GHzand 1GB RAM. Theused softwareisCPLEX12.2.0.2underGAMS23.6.3(GeneralAlgebraic ModelingSystem)[24].Cplexisacommonlyusedsolvertosolve theMIPproblemswhichcanefficaciouslygettotheglobaloptimal solution.Thissolverusesthebranchandcutalgorithm.

Fig.2.Asimple3-bustestsystem.

3.1. Firstcasestudy:asimple3-bustestsystem

First,averysimple 3-buspowersystemisassumed toshow theeffectiveness ofthemethod asshown in Fig.2.The capac- ity ofall existingand candidate transmissionlines is supposed tobe220MVA.Admittanceofallexistingandcandidatelinesis assumedtobeY_ij⁰=Y_ij=15+j(−60)Investmentcostofcandidate lines1,2and3are7×10⁶,8×10⁶and9×10⁶,respectively.Offer coefficientsofgenerators1and2areequalto20$/MWhwhichare locatedatbuses1and2,respectively.Demandatbus3isconsid- eredtobePD+jQD=600+j100.Themaximumactiveandreactive powerofgeneratorsare320MWand64MVAr.

Hereit iscompared thesimulationsfortheDC-TEP, theAC- TEPandthelinearizedAC-TEP(LAC-TEP)approach.Forthesake ofsimplicity,theobjectivefunctionisassumedtobetheinvest- mentcostoflines,excludingtheoperationcostofgenerators.Inthe AC-basedapproach,thevoltagemagnitudesofbusesarelimited to0.95and 1.05p.u.FortheAC-TEP,theobjectivefunction(OF) willbe1.7×10⁷andcandidatelines2and3areselectedasthe optimalsolution.Theactivepowergenerationofunits1and2are 320MWand283.4MW,respectively.FortheLAC-TEPmethod,the OFisthesameasthatoftheAC-TEPandactivepowergeneration ofunits1and2are320MWand293.5MW,respectively.However, theresultsfortheDC-TEParedifferentfromthoseofboththeAC- TEPandtheLAC-TEP;theOFis1.5×10⁷andcandidatelines1and 2isselectedasoptimalsolution.Units1and2generate320MW and280MW,respectively.Table1comparesthesimulationsforall thethreemethods.

3.2. Secondcasestudy:Garver’s6-bustestsystem

TheGarvertestsystemisdepictedinFig.3[4].Ithassixbuses, 15candidatebranches,andatotaldemandof760MW.Generators, loadsandlinesdatacanbefoundin[4].Reactivepowerdemandis assumedtobe20%ofactivepowerdemandforeachbus.Maximum

Table1

Simulationdataforthe3-bustestsystem.

Statisticdata Method

DC-TEP AC-TEP LAC-TEP

Numberofblocksofequations 11 15 26

Numberofblocksofvariables 6 10 14

Numberofnon-zeroelements 325 346 4087

Numberofsingleequations 143 129 1788

Numberofsinglevariables 45 86 464

Numberofdiscretevariables 3 3 3

Executiontime(s) 0.015 0.016 0.187

Fig.3. Garver’s6-bustestsystem.

reactivepowerforeachgeneratorisassumedtobeonethirdand onefifthofitsgeneratedmaximumactivepowerunderlaggingand leadingoperatingconditions,respectively.

First,itisassumedthatn=16fortheLAC-TEP(i.e.a16-sidedcon- vexregularpolygonisconsideredtomodelthecircleasdescribed inAppendix)and=0(i.e.theobjectivefunctiontakesnooper- atingcostintoaccount).Theexecutiontimewas0.234s.Table2 showssimulationdatareportedbytheGAMS.Thevalueofobjec- tivefunction,i.e.theinvestmentcost,is$5.14×10⁸.Simulation resultsshowlines1,2,3,5,6,7,9,10,11,12,14and15should bebuiltwithintheplanninghorizon.Generators1,2and3will produce150MW,360MWand291MWtogetherwith7.6MVAr,

−24.0MVArand −29.2MVAr,respectively.Thismeansthetotal powerlossesforthesystemisabout41MWor5.12%ofthetotal activepowergenerations.

Second,thenumberofsidesoftheregularpolygonwassettobe equalto8.Theproblemwillbecomeintegerinfeasibleinthiscase.

Thisoccursbecausewhenthenumberofsidesoftheregularpoly- gon(n)becomessmaller,transmissionofbothactiveandreactive powersbecomemorelimitedthroughthelines.Thus,thenumber ofsideswassettobeequalto32(n=32),resultingin$4.36×10⁸ fortheobjectivefunctionandcandidatelinestobebuiltwere3,5, 6,7,9,10,11,12,14and15.Thisshowsthatsometransmission Table2

SimulationdatafortheGarver’s6-bussystem.

Numberofblocksof equations

26 Numberofsingle equations

19,483 Numberofblocksof

variables

14 Numberofsingle variables

7018 Numberofnon-zero

elements

41,872 Numberofdiscrete variables

15

Table3

LMPofbusesofGarver’s6-bussystem.

Bus LMP($/MWh)

1 33.3

2 34.4

3 30.8

4 34.9

5 34.4

6 30.0

Table4

Candidatelinesdata.

Candidatelines Capacity(MVA) Reactance(p.u.) Investmentcost($10⁶US)

Name From To

CL1 1 4 175 0.015 7.72

CL2 2 7 175 0.021 10.82

CL3 7 10 175 0.020 10.29

CL4 9 10 175 0.016 8.24

CL5 11 15 500 0.022 11.13

CL6 11 20 500 0.024 12.35

CL7 11 24 500 0.011 5.66

CL8 13 20 500 0.011 5.66

CL9 14 19 500 0.017 8.76

CL10 19 21 500 0.014 7.21

CL11 20 22 500 0.014 7.21

Table5

Resultsfordifferentvalueof.

(h) IC($) Candidatelinestobebuilt OC($/h) OF($)

5×8760 7.21×10⁶ CL10 123,892.876 6.1475×10⁷

15×8760 1.442×10⁷ CL10,CL11 118,128.524 1.6964×10⁸

25×8760 2.008×10⁷ CL8,CL10,CL11 113,884.071 2.6949×10⁸

linesareoperatingattheircapacitylimits.Thenumberofnon-zero elementsandsingleequationsare60,304and28,699,respectively.

OthersimulationdatafortheAC-OPFandtheDC-OPFarethesame asthoseoflistedinTable2.ExecutiontimereportedbytheGAMS wasabout0.296s.

Third,for n=64 in the LAC-OPF, the objective function will become$4.34×10⁸andcandidatelinestobebuiltwere2,3,5,6, 9,10,11,12,14and15.Forn>64,simulationsremainunchanged.

Therefore,oneshouldbecarefulindeterminingthenumberofseg- mentsusedtomodelthecircle.UsingtheAC-TEPwillresultinthe constructionofthesamelinesasthoseoftheLAC-TEPundern=64 withtheexceptionofbuildingline13insteadofline12;theobjec- tivefunctionfortheAC-OPFbecomes$5.29×10⁸thatisbiggerthan thatoftheLAC-OPF.

Assuming=10×8760,¹theICandOCareequalto$4.34×10⁸ and $17,398.7per hourfor n=64. This meansthesame candi- datelinesareconstructedwhenconsideringoperationcost.The Lagrangianmultiplierofactivepowerbalanceequationorloca- tional marginal price (LMP) for bus numbers 1–6 are listed in Table3.² ItshouldbehighlightedthattheseLMPsareobtained afterfixingthebinaryvariablesofcandidatelinestotheircorre- spondingoptimalvaluesoftheoptimizationproblem. Notethat thegeneratorconnectedacrossbus6isthemarginalgenerator.

Table3showsLMPsaredifferentatthosebusessincepowerlosses arenotignored.Itshouldalsobenotedthatthereisnocongestion intransmissionlines.

3.3. Thirdcasestudy:theIEEE24-busRTS

Thewell-known 24-busRTS is used to applythe presented method.ParametersofthecandidatelinesarelistedinTable4, whileresistancesoflinesareassumedtobeonefifthoftheircor- respondingreactances.DetaileddataandtopologyoftheRTScan befoundin[25].Inordertoputthetransmissionlinesundermore stress,activepowerdemandsandmaximumactivepowergener- ationcapacitiesaremultipliedby2foreachgenerator.A32-sided

1 Pleasenotethat8760isthenumberofhoursperyear.

2 ItistobenotedthattocalculateLMPs,theobjectivefunctionhasbeenmodified toincludeonlythegenerationoffers(orequivalentlyoperatingcost)i.e.investment costmustberemovedfromtheaforementionedobjectivefunction.

regularpolygonisusedtomodelthecircle(seeAppendixformore details).

Theparameterwassettobe25×8760.Theexecutiontime was3.32seconds.SimulationsshowfortheLAC-OPFthatlines8, 10and11shouldbebuiltwithintheplanninghorizon.Generating units2,10,11,14,19,20,31and32willproduceactivepowers of29.4MW, 145.5MW, 0MW, 12.7MW, 15.6, 0MW, 216.9MW and 0MW, respectively;otherunitswilloperateat theirmaxi- mumcapacities.Table5indicatestheresultsforvarious,showing considerablechanges in resultswhen varies. Using a DC-TEP approachforthiscaseshowsthatlines8and10areconstructed.

This can be justified as reactive poweris neglected in the DC approach, and therefore more transmission line capacities are available;resultinginlowerrequiredinvestmentcost.Also,inan AC-TEPapproachthecandidatelines5,8,9and10areselectedto bebuiltwhichexplicitlyshowsthattheobtainedsolutionissub- optimalduetothenon-convexitynatureoftheAC-OPFequations.

3.4. Fourthcasestudy:Iran400-kVsimplifiednetwork

Asthelasttest,arealisticpowersystemischosen;i.e.,Iranian 400-kVpowersystem.Thetopologyanddataofthenetworkcanbe foundin[11].Powerfactorofloadsissettobe0.9asatypicalvalue.

Insteadofconsideringasingleperiodasinpreviouscasestudies,a dynamicapproachisselectedforthiscase.Todoso,aninterestrate of8%isregarded.Impedancesofthree-bundledandtwo-bundled 400-kVoverheadlinesareconsideredtobe0.0192+j0.288/km and0.0288+j0.3296/km,respectively.Also,annualloadgrowth isconsideredtobeequalto5%inaverage.Thesystemiscomprised of52buses,28generatingunitsand27candidatelines.Aplanning horizonof16yearsisconsideredwhichisdividedintotwosub- periods.

Theproblemwasrunfordifferentvaluesofn.Simulationshave beenreportedinTable6.Asitcanbeseenfromtheobtainedresults, theconsumedtimestoreachtheoptimalsolutionwillincreasesig- nificantlybyincreasingthenumbern.Toreducethesolutiontime, thenumberncanbedifferentforvariouslines.Itclearlydependson theoperatingpointofactiveandreactivepowerforeachlineinthe correspondingP–Qplane.Thenumbernhavetobebigforhighly congestedand heavilyloadedlines;but,ncanbesmallwithout losinganygeneralityfor thelinesoperatingfarfromtheirlim- its.SolvingtheTEPusingsimpleDCpowerflowmightbehelpful toidentifythemostcongestedlinesorareas.Inthemeanwhile,

Table6

ResultsfromdifferentcasesforIran400-kVpowersystem.

Numberofsingle equations/continuous variables/discretevariables

Elapsedtime (min)

M=10 n=8

3,910,760/1,828,279/54 3.38

M=10 n=16

5,122,152/1,828,279/54 3.72

M=10 n=32

7,544,936/1,828,279/54 6.43

M=10 n=64

12,390,504/1,828,279/54 43.14

Table7

OptimalsolutionsforIran400-kVpowersystem.

Candidatelinestobebuilt

Sub-period1 1–49,4–50,7–10,7–23,7–24,24–29,49–50,23–32 Sub-period2 4–50,7–10,7–12,7–24,10–12,13–14,18–19,39–40,42–43

selectingthenumbernwouldbecomemoreeffectivebyexperi- ence.Thesamecandidatelinesareselectedtobebuiltforn=32and n=64,whiletheresultswillbedifferentforsmallervalueofnwhich verifythatthequalityofthemethodologydependsuponthenum- bern.Table7showsthatwhichcandidatelinesshouldbeaddedto thepowersystemduringeachsub-period.Asitcanbeseenfromthe table,themostlinesconnectedtobus7areneededtobereinforced orconstructedbecausethemainelectricitydemandislocatedat bus7.Alsoitistobenotedthattheselinesarehighlycongested andthereforeoneshouldcareaboutselectingthenumbernfor theselines.Inaddition,anintegratedframeworkforsimultaneous reactivepowerplanningandtransmissionexpansionplanningis suggested.Thismightleadtomoreoptimalplaninpowersystem andlesspowerlosses.Moreover,theuncertaintyrelatedtothegen- erationcapacityexpansionplanninganditseffectonTEPhaveto beaddressedpreciselyforsuchareallarge-scalepowernetwork.

4. Conclusion

Inthispaperanewexpansionplanningmodelwasformulated andappliedtoseveralpowersystems.Thepresentedmethoduses theAC-OPFforsolvingtheTEPprobleminorder togeta more accurateandrealpictureofbothactiveandreactivepowerflow inthenetwork.TheAC-OPF–basedTEPisanMINLPthatistrans- formedintoanMILPbytheproposedmethod.Anewlinearization methodispresentedtotransformthenonlinearACmodeltoalinear one.Theformulatedproblemthencanbesolvedbytheavailable commercialMILPsoftware.Moreover,duetothelinearityofthe proposedformulatedproblem,theglobaloptimalsolutionisguar- anteedtobefoundbymanyexistingalgorithmsandsoftwarewhich arereliableandefficient.

The main difficulties and disadvantages of the presented methodologyaretodealwiththedimensionalityandthemath- ematicalcomputationburdenoftheproblem,inparticularwhen thesizeofthepowersystemgetslarger.Thisrequiressufficient memoryandhigh-speedprocessorstosolvetheMIPproblem.With increasingsystem size,the execution time of theproblemwill increaseaswell.However,thisdependsconsiderablyonthenum- berofbinaryvariablesratherthanthenumberofblocksorsegment usedinpiecewiselinearizationapproach.Asasolutionforthiscase, inordertoreducethesearchspacethemethodcanbeappliedto thehigh-voltagetransmissionlinesinitially,andthentheplanning canbeconductedforlowervoltageleveltransmissionlines.

Fig.A1.AcirclewithradiusR.

Appendix.

Thisappendixdescribesasimpleproofoftransformationofthe nonlinearequationintheformofx²+y²≤R²(acirclewithradiusR intheCartesiancoordinates)intonlinearequationslike(A2).Con- sideracirclewithradiusRasshowninFig.A1.Thiscircumscribed circle³orcircumcirclecanbeestimatedbyan-sidedconvexregular polygon.

ThelineequationthatdirectlyconnectspointAtopointBcan bewrittenasbelow:

y−R×sin

_360◦k

n

= sin(360^◦k/n)−sin(360^◦/n(k−1)) cos(360^◦k/n)−cos(360^◦/n(k−1))

x−R×cos

_360◦k

n

(A1)

ManipulatingEq.(A1)willleadtothefollowingequation:

sin

_360◦k

n

−sin

_360◦

n (k−1)

x−

cos

_360◦k

n

−cos

₃₆₀◦

n (k−1)

y−R×sin

₃₆₀◦

n

=0 (A2)

Inasimpleformatassuming˛=360^◦k/nandˇ=360^◦k/n(k−1) wewillobtain:

(sin ˛−sinˇ)x−(cos ˛−cosˇ)y−R×sin(˛−ˇ)=0 (A3) Thehigherthenumberofsegmentsisusedtomodelthecircle themoreaccurateresultswillbeobtainedbutattheexpenseof morecomputationaltime.

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