1-4244-0023-6/06/$20.00 2006 IEEECIS 2006 Development of the Short-Circuit Capacities Curves for Distribution Systems Planning Using Matlab Wen-Tsuan Lai, Wei-Tzer Huang, Shiuan-Tai Chen Department of Electrical Engineering Chienkuo Technology University Changhua, Taiwan, Republic of China opplei@yahoo.com.tw, vichuang@ctu.edu.tw, chen6502@ctu.edu.tw Heng-Kuan Chao Department of power and energy business Aerospace Industrial Development Corporation Taichung, Taiwan, Republic of China widthchao@pchome.com.tw AbstractThecalculationsoftheshort-circuitcapacities (SCCs)alongaprimaryfeederisessentialfordeterminingthe ratings of protective devices of the distribution systems, especially intheplanningstage.Accordingtothecalculationresults,the interruptioncapacities(ICs)thatshallbeabletowithstandthe mechanical and thermal stresses resulting from maximum flow of short-circuit current through them of the protection devices were evaluated.Therefore,aneasy,straightforward,rapidand systematicmethod,thecomplexshort-circuitMVAmethodwas employed to estimate the SCCs in this paper, the SCCs along a primaryfeederdependentontherelatedparametersand networkstructurewererepresentedbysimpleformulasand illustrated by curves that plotted in Matlab. It is more convenient forthedistributionengineerstoselecttheratingsofprotective devicesandplanningthedistributionsystemsbythedeveloped SCCs curves. Keywordscomplexshort-circuitMVAmethod,distribution system,normallyclosedloop,primaryfeeder,radial,short-circuit capacity. I. INTRODUCTION TheSCCisdefinedastheproductsofthemagnitudeof totalfaultcurrentofathree-phaseshort-circuitfaultandthe busvoltagebeforefault.Itisausualmeasureofthestrength ofabus,andisusedfordeterminingtheinterruption capacitiesofover-currentprotectivedevicesaswellasthe dimensionofabusbar[1].Additionally,theSCCsalso implied that the value of three-phase short-circuit current, and thiscurrentgenerallyresultsinthemaximumshort-circuit current availableinasystem.Theboltedfaultcurrentswhich arenotthevaluesofthree-phaseshort-circuitwillusuallybe less.Wheretheboltedline-to-linecurrentsareabout87%of the three-phase value, while bolted line-to-ground currents can rangefromabout25~125%ofthethree-phasevalue, depending on the system parameters. Nevertheless, the line-to-groundcurrentsofmorethan100%ofthethree-phasevalue seldomoccurinpracticalsystems.Consequently,thecurrent ofthree-phaseshort-circuitfaultisfrequentlytheonlyone calculated in most systems. Forthesakeofthisreason,thecalculationresultsofthe three-phaseshort-circuitfaultplayedanimportantroleof determiningtheICsoftheprotectivedevices.Therefore,the ohmic and per unit methods are usually used in power system fault current calculations [2-4]; besides, an alternative method, called short-circuit MVA method was proposed by Yuen [5] to calculatethethree-phaseshort-circuitcurrentbythe representationofpowersystemelementsintermsofMVA. Furthermore,amodifiedmethod,complexshort-circuitMVA methodhasbeenproposedforpowersystemstudiesbyChen [6-7].Inthispaper,thecomplexshort-circuitmethodwas adoptedtoevaluatetheSCCsalongtheprimaryfeeders. Accordingly, the SCCs along a primary feeder of the common typesofthedistributionsystems,i.e.radialandnormally closedloop[8-11]wereallrepresentedbysimpleformulas andillustratedbycurvesthatmakeSCCsalongtheprimary feeders more easily be recognized. The outcomes are ofvalue totheutilitiesthatareplanningtoselecttheICsofthe protectivedevicesdependingondistributionnetwork configurations. II.DESCRIPTION OF THE PROBLEMS AND SOLVING METHOD Inthissection,theproblemsandthreecommontypesof systemstructurewithnecessaryparametersforfaultanalysis areintroducedfirst,andfollowedbytheintroductionofthe adoptedsolvingmethod,thecomplexshort-circuitMVA method. A.Problem Description Ingeneral,thestrengthofabusbarisincreasinggradually fromdistributiontogenerationsystemsforaverticalpower systemstructure;hence,theSCCsatabusbarinthe distributionsystemsisusuallylowerthanthatofsub-transmissionsystems.Moreover,becausethelocationsand systemstructuresaredifferent,theSCCsaredifferenteven thoughatthesamevoltagelevel.Thatisthedifferencethe incomingline,whichisthedistributionsubstationsfedfrom sub-transmission systems, the SCCs at the primary side of the distributionsubstationsareusuallynotthesame. Consequently,theradialanddifferenttypesofnormally closed-loopprimaryfeederswill resultinvariedSCCs along distribution feeders. Moreover, the major factors that affect the SCCsalongthefeedersaretheincominglineshort-circuit

duty,theratedcapacitiesandimpedancesofthemain transformers,thebusbarschemes,theimpedancesofthe conductors,aswellasthefaultlocationontheprimary feeders. Hence, the SCCs curves along the primary feeders of five different types of distribution systems will be discussed in detailedinthispaperbyasystematicandstraightforward method, and will be implemented using Matlab. Figure1showstheschematicdiagramofprimaryfeeder arrangementsfortheevaluationoftheSCCsalongafeeder. In this paper, the feeder arrangements are classified as follows: Radial:ThefeederF#1isfedbythemaintransformerT1 and radiates from the distribution substation. Normally closed loop: This feeder arrangement is classified as follows: TypeI:Twofeeders,F#1andF#2arefedbythesame power transformer T1 and tied together normally at their ends. Type II: Two feeders are fed by two different transformers, T1andT2locatedinthesamesubstationandtiedtogether normallyattheirends.Thistypeisfurtherdividedintotwo subtypesbasedonwhetherthetiebreakerofthesecondary busesofthetwotransformersisnormallyclosedoropen,as follows: Type II.1: The tie breaker is normally open, Type II.2: The tie breaker is normally closed. TypeIII:Thetwofeeders,F#1andF#2arefedbytwo differenttransformers,T1andT2locatedindifferent substations and tied together normally at their ends. Additionally,inthispaper,theMVAsaandMVAsb representtheincominglineshort-circuitdutiesofthetwo distributionsubstationsAandB,respectively.Thetwo substations are fed from different subtransmission circuits that arefedfromdifferentbulkpowersubstations.Theratingsof threepowertransformersareassumedtobeST1,ST2andST3, and their corresponding impedances are ZT2, ZT2 and ZT3. Two mainfeedersarenamedasF#1andF#2,andthetielineare namedasTL.Becauseofthedifferenceinsystemtypes,the SCCs on the primary feeders are quite different. The formulas for evaluating the SCCs along a feeder will be derived in the followingsection.TheSCCsalongafeederarefunctionsof theincominglineshort-circuit duties,the ratedcapacitiesand impedancesofthepowertransformersandtheimpedancesof thefeeders.Finally,theSCCcurvesalongafeederofvarious system types were be derived. B.Solving Method Thecomplexshort-circuitMVAmethodisusedby separatingthecircuitintocomponents,andcalculatingthe short-circuitMVAofeachcomponentwithitsowninfinite bus.Therefore,thefirststepthismethodisconvertingallthe relatedcomponentsintotheircorrespondingshortcircuit MVAquantities.Asforthegenerators,motors,and transformers, the conversion formula is as follows ( )RatedSCpuS MVAMVAZ=JJJJJKJK(1) Besides, the conversion formula for the conductor is 2( )LSCKVMVA MVAZ=JJJJJKJK(2) Where subscripts sc, Rated, pu and refer to short circuit, rated capacity, per unit, and ohm, respectively. Substation AT1Primary sideSecondary sideLP1LP2LPn-1LPnF#1RadialF (a) Radial Substation AT1Primary sideSecondary sideF#1Normally Closed loop LP1LP2LPn -1LPnLP1LP2LPm-1LPmF#2Type IFTie line

(b) Normally closed loop: Type I Substation AT1F#1Normally Closed loopLP1LP2LPn-1LPnLP1LP2LPm- 1LPmT2Primary sideSecondary sideF#2 N.O. or N.C.Type II.1Type II.2FTie line (c) Normally closed loop: Type II Substation AT1Primary sideSecondary sideF#1Normally Closed loopType IIILP1LP2LPn-1LPnLP1LP2LPm- 1LPmSubstation BT3Primary sideSecondary sideF#2FTie line (d) Normally closed loop: Type III Fig. 1. The schematic diagram of primary feeder arrangements for studying AMVAdiagramisdevelopedafterallthecomponents wereconvertedintotheircorrespondingshortcircuitMVA quantities.ThesecondstepistoreducetheMVAdiagram. The parallel and series MVA combinations must be calculated by(3)and(4), respectively.Finally,theshort-circuitMVA at the fault point is calculated. 1nParallels iSC SCiMVA MVA== JJJJJK JJJJJK(3) 111nSeriesSCiiSCMVAMVA= (=( JJJJJKJJJJJK(4)

III.DERIVATION OF THE SCC CURVES Thecomplexshort-circuitMVAmethodwasemployedto evaluate the SCCs along a feeder. Therefore, the short-circuit MVAofeachequipment;suchasgenerators,transformers, motors,feeders,etc.needbefoundfirst.Forpractical purposes,thesamplesystemasshowninFig.1wasdesigned accordingtoanactualundergrounddistributionsystemof Taipower in an urban area with little modifications. TableIliststheparametersofthesampledistribution systemrequiredfortheevaluationoftheSCCs.The transformercapacities,thenominalvoltageoftheprimary system, the line impedance of the feeders, and the length of tie linewereallkeptconstantinthefollowingdiscussions.For the base case, incoming line short-circuit duties are assumed to be 5000 MVA. The lengths of the feeder cables are all assumed tobe10km,andtheperunitimpedanceofthethreemain transformers are assumed 0.1672 pu. Asconversionwasmade,TableIIandanMVAdiagram weredeveloped.ReferredtotheMVAdiagram,theshort circuit MVA quantities of all equipments were combined. The seriesMVAquantitieswerecombinedasresistancesin parallel,andtheparallelMVAquantitieswereadded arithmetically. Finally, the short circuit MVA quantities of the fault point were found. TABLEIPARAMETERS OF THE SAMPLE DISTRIBUTION SYSTEM ParametersVariablesBase CaseRanges Incoming line short-circuit duties MVAsa, MVAsb 5000 MVA20%~200% Transformer capacities ST1, ST2, ST3 60 MVA100% Transformer impedances ZT1, ZT2, ZT3 0.1672 pu, X/R=34.1 90%~110% Feeder lengths of F#1 and F#2 L1, L210 km20%~200% Tie line lengthLTL0.3 km100% Feeder impedancesZ 0.1075+j0.1437 /km 100% TABLE IICORRESPONDING SHORT-CIRCUIT MVA OF EACH EQUIPMENT EquipmentBase CaseRanges MVAsa, MVAsb 5000 MVA 1000 MVA~10000 MVA MVAT1, MVAT2, MVAT3 359 MVA326 MVA~399 MVAMVAL1, MVAL2 290 MVA 145 MVA~1448 MVA MVATL9653 MVA9653 MVA A.Radial Arrangement ForthesystemshowninFig.1(a),theMVAdiagramof this radial arrangement was shown in Fig. 2. Assumed a three-phase fault occurs at a point on the main feeder at a distance of (1-k)L1mfromthefeedpoint,theSCCatthefaultpointcan be expressed as 1 11,(1 )1 1 1 FSCRadialsa T KLMVAMVA MVA MVA (= + + ( JJJJJKJJJJJK JJJJJK JJJJJK(5) Where sa MVAJJJJJKistheincominglineshort-circuitdutyatthe primary side of the substation transformer A. 1T MVAJJJJJKis the corresponding short-circuit MVA of the main transformer #1. 1(1 ) KL MVAJJJJJKistheshort-circuitMVAofthefeedersection betweenthefaultpoint(anylocationon themainfeeder)and feed point (the secondary busbar of the main transformer) K: variable from 1 to 0, for indicating the fault location Substituting (1) and (2) into (5), the SCC of the fault point on the radial feeder can be formulated as 1 111( ) ( ),121(1 ) ( )T pu T puFsa TSC RadialLR jXjMVA SMVAKL R jXKV +(+ ( (= ( ++ ( ( JJJJJK(6) Basedon(6),theSCCcurvesweredevelopedasshown in Fig.3.TherearefivecurvesinFig.3.Thesolidcurveshow theSCCsalongthefeederF#1undertheconditionsofbase case.Theotherfourcurvesshowthedeviationsofthe corresponding SCCs from the base case for two values of the incominglineshort-circuitdutiesandtwovaluesofthe transformer impedances.Thesmaller theincominglineshort-circuitdutyorthegreaterthetransformerimpedanceis,the smallerthecorrespondingSCCsobtain,asshowninFig.3. ThesetwofactorspredominantlyaffecttheSCCsalongthe feeder. ,FSCRadial MVAJJJJJKF1T MVAJJJJJKsa MVAJJJJJK1(1 ) KL MVAJJJJJKF Fig. 2. MVA diagram of radial arrangement

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100150200250300350400KS.C.C.(MVA)20%*MVAsaMVAsa=5000 MVA, ZT1=0.1672pu, L1=10km200%*MVAsa90%*ZT1110%*ZT1 Fig. 3. SCCs along the primary feeder for radial arrangement B.Normally closed Loop Arrangement TypeI:TheMVAdiagramofanormallyclosedloop arrangementofTypeIwasshowninFig.4.Assumedfault occurs at a point on the main feeder at a distance of (1-k)L1 m fromthefeedpoint,theSCCatthefaultpointcanbe expressed as 1112111,(1 )1 111 1TLsa TFSC TypeIKLKLL LMVA MVAMVAMVAMVAMVA MVA + ( | |= +( ` | ( |+ + ( |+ ( |\ . ( )JJJJJK JJJJJKJJJJJKJJJJJKJJJJJKJJJJJK JJJJJK(7) Where ,FSC TypeI MVAJJJJJK is the SCCs along the primary feeder of Type I arrangement. Similarly, Substituting (1) and (2) into (7), the SCCs along theprimaryfeederofTypeIarrangementcanbeformulated as: ( )11( ) 1( )112,121 21(1 ) ( )( )T pu T pusa TFLSC TypeILTLR jXjMVA SKVMVAKL R jXKVKL L L R jX + + ( = + `( + (+ ( (+ + + ( )JJJJJK(8) Based on (8), seven SCC curves, relating the incoming line short-circuit duty, transformer impedance, the length of feeder F#2andthefaultlocation,havebeendeveloped,asshownin Fig.5.ThesolidcurveshowtheSCCsalongthefeederF#1 undertheconditionsofbasecase.Theothersixcurvesshow thedeviationsofthecorrespondingSCCsfromthebasecase fortwovaluesoftheincominglineshort-circuitduties,two valuesofthetransformerimpedancesandtwovaluesofthe lengthsoffeederF#2. Theeffectsoftheincoming lineshort-circuitdutyandthetransformerimpedancearethesameas thatofthefeederwithradialarrangement.Theeffectsofthe incominglineshort-circuitdutyandthetransformer impedance on the SCCs along the feeder F#1 is the largest in the feed point and the smallest in the end of the feeder. On the contrary,theeffectsofthe lengthoffeederF#2ontheSCCs alongthefeederF#1isthesmallestinthefeedpointandthe largest in the end of the feeder. 1T MVAJJJJJK1(1 ) K L MVA JJJJJK1KL MVAJJJJJK2L MVAJJJJJKTLL MVAJJJJJKFsa MVAJJJJJK,FSC TypeI MVAJJJJJKF Fig. 4. MVA diagram of a normally-closed feeder arrangement of type I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1180200220240260280300320340360380KS.C.C.(MVA)20%*MVAsaMVAsa=5000 MVA,ZT1=0.1672pu, L1=10km200%*MVAsa90%*ZT1110%*ZT120%L2200%L2 Fig. 5. SCCs along the primary feeder for a normally-closed feeder arrangement of Type I TypeII.1:TheMVAdiagramofanormallyclosedloop arrangementofTypeII.1wasshowninFig.6.Accordingto thesystemstructure,theSCCsalongtheFeeder#1canbe represented as 1 12 21111(1 )1, .11 111 11 1TLT KLFSC TypeIIsaT LL KLMVA MVAMVAMVAMVA MVAMVA MVA (| | (+ | | (\ . ( | |(= + `+ | ( | ( + | ( + + | ( \ . )JJJJJK JJJJJKJJJJJKJJJJJKJJJJJK JJJJJKJJJJJK JJJJJK

(9) Substituting (1) and (2) into (9) yields ( )1 112 2211( ) ( )1, .121( ) ( )2 121(1 ) ( )( )saT pu T puTFSC TypeIILT pu T puTTLLjMVAR jXSKL R jXMVAKVR jXSL L KL R jXKV + (+ | | (+ | ( |+ ( | + ( |= ` | (\ . (+ | |(+|(|(|(+ + +|( | (\ . JJJJJK1 )(10) Based on (10), seven SCC curves were developed as shown in Fig. 7. The curves in Fig. 7 are similar to the corresponding curvesshowninFig.7.However,thelevelsoftheSCCsare all going up and the effects of the factors are enlarged. 1T MVAJJJJJK1(1 ) KL MVAJJJJJK1KL MVAJJJJJK2L MVAJJJJJKTLL MVAJJJJJKsa MVAJJJJJK2T MVAJJJJJK, .1FSC TypeII MVAJJJJJK Fig. 6. MVA diagram of a normally-closed feeder arrangement of type II.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1250300350400450500KS.C.C.(MVA)20%*MVAsaMVAsa=5000 MVA, ZT1=0.1672pu, L1=L2=10 km200%*MVAsa90%*ZT1110%*ZT120%*L2200%*L2 Fig. 7. SCCs along the primary feeder for a normally-closed feeder arrangement of Type II.1 TypeII.2:TheMVAdiagramofanormallyclosedloop arrangementofTypeII.2wasshowninFig.8.Accordingto thesystemstructure,theSCCsalongtheFeeder#1canbe represented as ( ) 1 22111111, .2(1 )111 1TLT TsaFSC TypeIILKLL KLMVA MVAMVAMVAMVAMVAMVA MVA + + (| | +( = `| ( |+ + ( |+ ( | ( \ . )JJJJJK JJJJJKJJJJJKJJJJJKJJJJJKJJJJJKJJJJJK JJJJJK (11) Equation (11) can be rewritten as ( )11 122 211( ) ( )( ) ( ), .212122 11(1 ) ( )( )TT pu T puTsaF T pu T puSC TypeIILLTLSR jXjSMVAR jXMVAKVKL R jXKVL L KL R jX | | | + |+ | | + |+ \ .= `( + ( +( +( ( + + + ( )JJJJJK(12) Based on (12), seven SCC curves were developed as shown in Fig. 9. The curves in Fig. 9 are similar to the corresponding curvesshowninFig.7.However,thelevelsoftheSCCsare allgoingupbecausethetwotransformerswereoperated parallel in this arrangement. 1T MVAJJJJJK1(1 ) KL MVAJJJJJK1KL MVAJJJJJK2L MVAJJJJJKTLL MVAJJJJJKFsa MVAJJJJJK2T MVAJJJJJK, .2FSC TypeII MVAJJJJJKF Fig. 8. MVA diagram of a normally-closed feeder arrangement of type II.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1250300350400450500550600650700KS.C.C.(MVA)20%*MVAsaMVAsa=5000 MVA, ZT1=0.1672pu, L1=L2=10 km200%*MVAsa90%*ZT1110%*ZT120%L2200%L2 Fig. 9. SCCs along the primary feeder for a normally-closed feeder arrangement of Type II.2 TypeIII:TheMVAdiagramofanormallyclosedloop

arrangementofTypeIIIwasshowninFig.10.Accordingto thesystemstructure,theSCCsalongtheFeeder#1canbe represented as 1 132 11,(1 )11 1 11 11 1 1TLFSC TypeIIIsa T KLsb TL L KLMVAMVA MVA MVAMVA MVAMVA MVA MVA| |= + + | |\ .| |+ + | |+ |+ + |\ .JJJJJKJJJJJK JJJJJK JJJJJKJJJJJK JJJJJKJJJJJK JJJJJK JJJJJK(13) Rewritten (13) as ( )1 113 331( ) ( ),121( ) ( )2 121(1 ) ( )1( )T pu T puFsa TSC TypeIIILT pu T pusb TTLLR jXjMVA SMVAKL R jXKVR jXjMVA SL L KL R jXKV + | |+ | |= | ++ | |\ .+ | |+ +| |+ |+ + + | |\ .JJJJJK(14) Based on (14), seven SCC curves were developed as shown inFig.11.ThecurvesinFig.11aresimilartothe corresponding curves shown in Fig. 7. 1T MVAJJJJJK1(1 ) KL MVA JJJJJK1KL MVAJJJJJK2L MVAJJJJJKTLL MVAJJJJJKFsa MVAJJJJJK2T MVAJJJJJKsb MVAJJJJJK,FSC TypeIII MVAJJJJJKF Fig. 10. MVA diagram of a normally-closed feeder arrangement of type III 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1250300350400450500KS.C.C.(MVA)20%*MVAsaMVAsa=MVAsb=5000MVA, ZT1=0.1672pu, L1=L2=10km200%*MVAsa90%*ZT1110%*ZT120%L2200%L2 Fig. 11. SCCs along the primary feeder for a normally-closed feeder arrangement of Type III IV.CONCLUSIONS In this paper, the SCC curves were developed in Matlab by aeasy,straightforwardandplanning-orientedmethod,the complexshort-circuitMVAmethod.TheSCCcurves representingtheSCCsalongafeederforvariousnetwork arrangementswereplottedeasilybasedontheformulas obtained. Some major factors of the short-circuit capacity, such astheincominglineshort-circuitduty,thefeederlength,the transformerimpedancesareconsidered.Hence,thederived formulasareallfunctionsofthesethreemajorfactors.The SCCevaluationformulasandtheSCCcurvesmakeclearthe SCCsalongtheprimaryfeedersandareofvaluetothe distributionengineersfordeterminingtheICsofprotective devices, especially during the planning stage. ACKNOWLEDGMENT Oneofus(Wei-TzerHuang)wantstoheartilythankthe Chienkuo Technology University, for this article was extension form the project CTU-94-RP-EE-014-030. REFERENCES [1]Hadi Saadat, Power System Analysis, McGraw-Hill, New York, 1999. [2]Brown,H.E.,Person,C.E.,Kirchmayer,L.K.,andStagg,G.W., Digtalcalculationofthree-phaseshortcircuitsbymatrixmethod, Trans. AIEE, Feb., 1961, Part 111, 79, pp. 1277-1282. [3]Roy,L.,Generalizedpolyphasefault-analysisprogram:calculationof cross-country fault,Proc. IEE. Oct. 1979, 126(10), pp. 995-1001. [4]Stagg, G. W., and El-Abiad,A. H., Computermethods inpowersystem analysis,McGraw-Hill, New York, 1968. [5]Yuen,M.H.,ShortcircuitABC,GasIndustriesAssociation,San Francisco, 1975. [6]Chen,T.H.,ComplexshortcircuitMVAmethodforpowersystem studies,IEEProceedingsonGeneration,Transmissionand Distribution, Vol. 141, pp. 81-84, Mar, 1994. [7]Chen,T.H.andChuang,H.J.,Applicationsofthecomplexshort- circuitMVAmethodtopowerflowstudies,ElectricPowerSystems Research,pp. 135-143, Mar, 1996. [8]TuranGnen,ElectricPowerDistributionSystemEngineering, McGraw-Hill, New York, 1986. [9]Tsai-HsiangChen,Wei-TzerHuang,Jyh-CherngGu,Guan-ChihPu, Yen-FengHsu,andTzong-YihGuo,FeasibilityStudyofUpgrading Primary FeedersfromRadial andOpen-Loop to NormallyClosed-Loop Arrangement,IEEETransactionsonPowerSystems,Vol.19,No.3, pp. 1308-1316, August, 2004. [10]W. T. Huang and T.H. Chen, "Assessment of upgrading existing primary feeders from radial to normally closed loop arrangement," in Proc. 2002 IEEEPowerEngineeringSocietyTransmissionandDistributionConf., pp. 2123-2128. [11]B. Pagel, Energizing International Drive, Transmission & Distribution World, pp.18-34, April 2000.