74电力系统分析英文课件LineModel

1Instructor:KaiSunFall2014ECE421/599ElectricEnergySystems5–LineModelandPerformance2LineModels•ShortLineModel–80km(50miles)orless,69kVorlower–Ignoringcapacitance•MediumLineModel–80km(50miles)~250km(150miles)–Lumpedlineparameters•LongLineModel–250km(150miles)orlonger–Distributedlineparameters•Example:L=1mH/km,C=0.01µF/kmandr=0.01Ω/kmω=2π×60=377Hzl=80kml=250kmR=r×l(Ω)XL=ωL×l(Ω)XC=1/(ωC)×l(Ω)0.8=0.027XL2.5=0.027XL30.294.33315.6=109.8XL1061.0=11.3XL3ShortLineModel•Capacitanceisignored.()ZrjLlRjXSRSRVVABICDI=101ABZCD====*(3)3SSSSVIφ=(3)(3)(3)LSRSSSφφφ=−(3)(3)RSPPφφη=101SRRRSRRVVZIVZIII+==4•VoltageRegulation(VR):–Itisameasureoflinevoltagedrop–Dependsontheloadpowerfactor:VRispooreratlowlaggingpowerfactor–Perhaps,VR0foraleadingpowerfactor(i.e.|VS||VR|).SeeExample5.1()()()||||Percent100%||RNLRFLRFLVVVRV−=×ZIRZIRZIR()()||||100%||SRFLRFLVVV−=×5MediumLineModel•Modelthetotalshuntadmittanceofthelineby–g,theshuntconductanceperunitlength,representstheleakagecurrentovertheinsulatorsisnegligibleundernormalcondition.–Cisthelinetoneutralcapacitanceperunitlength•Nominalπmodel:–HalfofCisconsideredtobelumpedateachendoftheline()YgjCjCωω=+≈62LRRYIIV=+()2SRLRRRVVZIYVZIV=+=++(1)2SRRZYVVZI=++2()(1)222SLSRRRRYIIVYYZYIVVZI=+=++++12(1)142SRRSRRZYZVVVABICDIZYZYIY+==++12ZYABZ=+=(1)142ZYZYCYD=+=+(1)(1)42SRRZYZYIYVI=+++det1ABADBCCD=−=SRRSVVDBICAI−=−Linear,passive,bilateraltwo-portnetwork(nosource)•FindVS,IS↔VR,IR7LongLineModel•Seriesimpedanceperunitlength•Shuntadmittanceperunitlength•Considerasmallsegmentof∆xatdistancexfromthereceivingendzrjLygjC()()()VxxVxzxIx+∆=+∆()()()VxxVxzIxx+∆−=∆()()dVxzIxdx=()()()IxxIxyxVxx+∆=+∆+∆()()()IxxIxyVxxx+∆−=+∆∆()()dIxyVxdx=22()()()dVxdIxzzyVxdxdx==8Principalsquareroot:22()()dVxzyVxdx=2zyγ=222()()0dVxVxdxγ−=12()xxVxAeAeγγ−=+()()jzyrjLgjCγαβωω∆=+==++γ-Propagationconstantα-Attenuationconstant(≥0)β-Phaseconstant(≥0)12121()()()()xxxxdVxyIxAeAeAeAezdxzzγγγγγ−−==−=−ZC=𝑧/𝑦-CharacteristicimpedanceIflinelossesareneglected,i.e.r=0andg=02jLCjLCγαβωω=+=−=0,LCαβω==withjzreϕπϕπ=−/2jzreϕ∆=121=()xxCAeAeZγγ−−9•FindA1andA2:atthereceivingend,x=0,V(x)=VRandI(x)=IR12(0)RVVAA==+121(0)()RCIIAAZ==−12RCRVZIA+=22RCRVZIA−=()2222xxxxxxRcRRcRRcRVZIVZIeeeeVxeeVZIγγγγγγ−−−+−+−=+=+1()2222RRRRxxxxxxccRRcVVIIeeeeZZIxeeVIZγγγγγγ−−−+−−+=−=+12()xxVxAeAeγγ−=+121()()xxCIxAeAeZγγ−=−cosh2xxeexsinh2xxeex|A1||A2|or|A1||A2|?ZC=𝑧/𝑦-Characteristicimpedance10()coshsinhRcRVxxVZxIγγ=+1()sinhcoshRRcIxxVxIZγγ=+11•Atthesendingend,x=l,V(l)=VS,I(l)=IScoshsinhsRcRVVZIγγ=+1sinhcoshsRRcIVxIZγγ=+SRSRVVABICDI=coshsinhCABZγγ==1sinhcoshCCZDγγ==12ZYABZ=+=(1)142ZYZYCYD=+=+Comparedtothemediumlineπmodel:12ZYZ′′=+′=(1)412ZYYZY′′′=+′′=+(1)2SRRZYVVZI=++(1)(1)42SRRZYZYIYVI=+++sinhsinhsinhCzZZzyyZγγγγγ′===tanthcosh1cosh122sinh2anh22CYyzyZZzYγγγγγγγ′−−===′=A=D,AD-BC=1Linear,passive,bilateraltwo-portnetwork(nosource)12EquivalentπModelforLongLengthLines()()zyrjLgjCγωω==++()ZzlrjLl()YylgjCl|γl|(pu)Z’/Z=sinh(γl)/(γl)Y’/Y=tanh(γl/2)/(γl/2)13Example4.1inBergenandVittal’sBook•A60-Hz138kV3-phasetransmissionlineis225milong.Thedistributedlineparametersarer=0.169Ω/mi,L=2.093mH/mi,C=0.01427µF/mi,g=0.Thetransmissionlinedelivers40MWat132kVwith95%powerfactorlagging.–Findthesending-endvoltageandcurrent.–FindthetransmissionlineefficiencySolution:ω=2π×60=377rad/sz=r+jωL=0.169+j377×2.093×10-3=0.169+j0.789=0.807∠77.9oΩ/miy=jωC=j377×0.01427×10-6=j5.38×10-6=5.38×10-6∠90oS/miZC=𝑧/𝑦=387.3∠-6.05oΩ/mi≈Realnumberγ=α+jβ=𝑧𝑦=0.136×10-6+j1.29×10-6γl=225𝑧𝑦=0.4688∠83.95o=0.0494+j0.466≈Imaginarynumber2sinhγl=eγl-e-γl=e0.0494ej0.466-e-0.0494e-j0.466=1.051∠0.466rad-0.952∠-0.466radsinhγl=0.452∠84.4o.Similarly,coshγl=0.8950∠1.42oLet∠VR=0.VR=132×103/3=76.2kVPload=0.95|VR||IR|=40/3=13.33MWθ=cos-1(0.95)=18.195oIR=184.1∠-18.195oAcoshsinh89.2819.39kVsRcRVVZIγγ=+=∠1sinhcosh162.4214.76AsRRcIVxIZγγ=+=∠°*13.3392%Re()14.45loadSSPVIη===|A1|=72.1kV|A2|=1.57kV14VoltageandCurrentWaves•InstantaneousvoltageasafunctionoftandxIncidentwave:amplitude↑whenx↑Reflectedwave:amplitude↓whenx↑1212()xxxjxxjxVxAeAeAeeAeeγγαβαβ−−−=+=+()()1212(,)2Re2Re()),(,xjtxxjtxvtxAeeAvtxeevtxαωβαωβ+−−=+=+111(,)2||cos()xvtxAetxAαωβ=++∠222(,)2||cos()xvtxAetxAαωβ−=−+∠()2||cos()vvtVtωθ=+||vVVθ=∠15t(s)x(mi)v(t,x)(V)ReceivingSendingt(s)x(mi)v2(t,x)(V)ReceivingSendingt(s)x(mi)v1(t,x)(V)ReceivingSending16VelocityandWavelengthofPropagation•Consider–Forapointonthetravelingwave:–Itsmovingspeed(velocityofpropagation)andthewavelength–Ifα=0,–GMRL≈GMRC22(,)2||cos()vtxAtxωβ=−constanttxωβ−=2dxfvdtωπββ===2/vfπλβ==LCβω=CLZC=Surgeimpedance1vLC=1fLCλ=87120011310m/4108.8510vsµεπ−−==××××0015000km60λµε=1334For3bundledconductors://'1.09CLGMRGMRrre×===001ln2CcGMDZGMRµπε60lnCGMDGMR0.2lnmH/kmLGMDLGMR=0.0556F/kmlnCCGMDGMRµ=comingfromthereceivingend(x=0)17LosslessLinescoshcoshcos2jxjxeexjxxsinhsinhsin2jxjxeexjxjx()cossinRCRVxxVjZxIββ=+1()sincosRRCIxjxVxIZββ=+()coshsin