电力系统分析双语课件LineParameters.pdf

1 Instructor:Kai Sun Fall 2014 ECE 421/599 Electric Energy Systems 4 Transmission Line Parameters 2 Introduction Transmission lines Overhead lines Underground Cables(less than 1%)Properties Series Resistance(stranding and skin effect)Series Inductance(magnetic&electric fields;flux linkages within the conductor cross section and external flux linkages)Shunt Capacitance(magnetic&electric fields;charge and discharge due to potential difference between conductors)Shunt Conductance(due to leakage currents along insulators or corona discharge caused by ionization of air)Line-to-line voltage levels 69kV,115kV,138kV and 161kV(sub-transmission)230kV,345kV,and 500kV(EHV)765kV(UHV)Corona discharge on insulator string of a 500 kV line(source:wikipedia.org)3 Overhead Transmission Lines Shield wires(ground wires)are ground conductors used to protect the transmission lines from lightning strikes (Source:wikipedia.org and EPRI dynamic tutorial)4 Overhead Transmission Lines Materials AAC(All Aluminum Conductor),AAAC(All Aluminum Alloy Conductor)ACSR(Aluminum Conductor Steel Reinforced)ACAR(Aluminum Conductor Alloy Reinforced)ACCC(Aluminum Conductor Composite Core)Why not copper?Relative lower costs and higher strength-to-weight ratios than copper Bundle conductors Preferred for high voltages,e.g.2-conductor bundles for 230kV,3-4 for 345-500kV,and 6 for 765kV 24/7 ACSR and modern ACCC conductors A bundle of 4 conductor ACSR(7 steel and 24 aluminum strands)5 Line Resistance Consider a solid round conductor at a specific temperature:DC resistance =conductor resistivity l =conductor length A=conductor cross-sectional area AC resistance Current is not uniformly distributed over the cross-sectional area;current density is greatest at the surface(skin effect)2%higher than DC resistance at 60Hz Temperature impact t1 and t2 are in oC T228 for aluminum dclRA=Skin effect:circulating eddy current IW cancelling the current flow in the center of the conductor.(source:wikipedia.org)6 Inductance of a Single Conductor Inductance for nonmagnetic material:Amperes law Ix:current enclosed at radius x Hx:magnetic field intensity Bx:magnetic flux density 0=410-7H/m:permeability of free space /LI=Hx 002xxxIBHx=Hx(Ix=I)x intintextext/LLII=+=+7 r dx Ix 1 Inductance due to the internal flux linkage Assume uniform current density Differential flux and flux linkage:Total internal flux linkage and the inductance:22 xIIxr=222xxIIHxxr=0022xxIBHxr=The portion of I is linked to dx 8 D1 D2 Inductance due to the external flux linkage Ix=I for xr External flux linkage and inductance between two points:00022xxxIIBHxx=20212xxxIrddB dxdxrx=2170211210ln Wb/m2DextDIDdxIxD=721210ln H/mextDLD=dx 9 Inductance of Single-Phase Lines 1-meter length of a single-phase line with two solid round conductors I1=-I2 D1=r1 D2=D(why?)If two conductors are identical:GMR(Geometric mean radius):DS=r for a single conductor Phase current Return current 7711(int)1()11102 10ln H/m2extDLLLr=+=+7112 10ln H/mDLr=7222 10ln H/mDLr=72 10ln H/mDLr=1/4rre=0.2ln mH/kmsDLD=def1/411rre=72 10ln H/mextDLr=GMR is the radius of a fictitious conductor without internal flux but the same inductance as the actual conductor Compared to r1=r2=r 10 Self-and Mutual Inductances Consider the flux linkages for 1-meter length of the single-phase circuit I1+I2=0 Compare to 111 1112211 1211()LL IL IL ILI+=221 12222221222()L IIILLL IL=+=+=771211217112 10l12 10n2 10ln H/mlnrrDDL=771222212712 102 10lnln12 10ln H/mDLrrD=+77172222211211112 10ln12 1012 10lnlnLLLLrrD=LI11211112117221222212211lnln21011lnlnrDLLIILLIIDr=11 Consider n conductors 71111210(lnln)nniiiiijjijjjiijj ij iL IL IIIrD=+=+11217122212111lnlnln111lnlnln2 10111lnlnlnnnnnnrDDDrDDDr=L=LIiiiDr=1112171222212111lnlnln111lnlnln2 10111lnlnlnnnnnnnDDDDDDDDD=71117222712211212 10ln12 10ln12 10lnLrLrLLD=I1+I2=0 120inIIII+=7712 10ln12 10lniiiijjiijLrLLD=12 Inductance of 3-Phase Transmission Lines(Asymmetric Spacing)7121371223111210(lnlnln)111210(lnlnln)aabcbabcIIIrDDIIIDrD=+=+71323111210(lnlnln)cabcIIIDDr=+=LI1213712231323111lnlnln111210lnlnln111lnlnlnrDDDrDDDr=L2120240baaaIIIa I=240120caaaIIIaI=1 120a=721213721223111210(lnlnln)111210(lnlnln)aaabbbLaaIrDDLaaIDrD=+=+721323111210(lnlnln)cccLaaIDDr=+For balanced three-phase current:Note:La,Lb and Lc may have imaginary terms Consider a three-phase line with 3 identical conductors 21 1201 2401aa+=+=(D11=D22=D33=r)13 Transpose Line:Mitigation of Asymmetry 3abcLLLL+=71223132101111(3lnlnlnln)3rDDD=312313722 10lnD D Dr=0.2ln mH/kmsGMDLD=0.2ln mH/kmsDLD=GMD GMD GMD For symmetric spacing:D12=D23=D13=D GMD:geometric mean distance=GMD 14 A three-phase line has three conductors with r=1.345 in.Determine the inductance per phase.What if transposition is adopted?Ds=r=1.345e-1/4=1.0475in=0.0266m D12=D23=0.889m D13=1.778m With transposition:13122313()0.2ln0.7480 mH/km 3abcsLLLD D DLD+=21213212231110.2(lnlnln)0.77110.1201=0.7804-8.85 mH/km1110.2(lnlnln)0.7018 mH/kmabLaajrDDLaaDrD=+=+=213231110.2(lnlnln)0.77110.1201=0.7804 8.85 mH/kmcLaajDDr=+=+15 Inductance of Bundled Conductors Single-phase with two bundled conductors n conductors m conductors 71111210(lnlnlnln)axabacanInrDDD=+711112 10(lnlnlnln)aaabacamImDDDD+GMD GMRx 72 10ln H/mxxGMDLGMR=,mniji x j yGMDD=222,nxiji xj xniiiji xi j x ijGMRDDD=16 7210lnmaaabacamanxabacanDDDDIr D DD=7210ln/maaabacamaaanaxabacanD D DDLnII nr D DD=abcnavLLLLLn+=2avabcnxLLLLLLnn+=72 10ln H/mxxGMDLGMR=()()mnaaabacamnanbnmGMDDDDDDDD=2()()nxaaabannanbnnGMRD DDD DD=,aabbnnxs xDDDrD=Inductance of bundle x:Since all conductors are connected in parallel,Average inductance of each conductor:7210lnmnanbncnmnnxnbncncD D DDLnr D DD=GMD GMR for m=1,n=1:GMR=r,GMD=D 17 GMR of Typical Bundled Conductors 24()ssGMRDdDd=3293()ssGMRDddDd=1164324(2)1.09ssGMRDdddDd=2()()naaabannanbnnD DDD DD=2nxiji xj xGMRD=18 Inductance of Stranded Conductors Example 4.1:Determine GMR A special case of the bundled conductors:r 1 2 3 4 5 6 7 121617142213151445242 3DDDrDrDDDDr=649661766774(22 342 322)()(2)(3)(3)(2)2.1767GMRrrrrrrrrer=19 Line Capacitance Consider a long round conductor carrying a charge of q(c/m)Capacitance:Electric flux density at a cylinder of radius x:Electric field intensity Electric potential difference between cylinders at D1 and D2 Defined as the work done in moving a unit charge from D1 to D2 (Voltage drop from D1 to D2)qCV=q x 21qqDAx=002DqEx=2211212001ln22DDDDDqqVEdxdxxD=20 Capacitance of Single Phase Lines Consider 1-meter length of a single-phase line Conductor 1 carries a charge of q1(c/m)Conductor 2 carries a charge of q2(c/m)Line-to-line capacitance between the conductors 1112()0ln2qqDVr=2221()0ln2qqDVr=2212()0ln2qqDVr=12121212()12()000lnlnln22qqqqDDqDVVVrrr=+=+=21()qqq=012 F/mlnCDr=72 10ln H/mDLr=Compare to the inductance per conductor:21201ln2DqVD=21 Define C,the capacitance per conductor,as the capacitance between each conductor and a neutral Compared to For an all aluminum conductor r=1.345 in(0.0342m)and D=35 in(0.889m)Al(20oC)=2.82108 m C=0.0171 F/km 1/(C)=0.155M/km L=0.7018 mH/km L =0.2646/km 34.4R R=Al(20oC)1000/(r2)=0.00769/km 0121222 F/m/2lnqCCDVr=0.0556 F/kmlnCDr=0.2ln 0.050.2ln mH/kmDDLrr=+22 Multi-Conductors Consider n parallel long conductors with charges of qk c/m Since for one single conductor with qk:Dii=Djj=r for n conductors:120nqqq+=101ln2nkjijkkkiDVqD=0ln2kjkijkiDqVD=23 Capacitance of Three-Phase Lines 0abcqqq+=2312()012131(lnlnln)2ab IabcDDrVqqqrDD=+2313()023121(lnlnln)2ab IIabcDDrVqqqrDD=+1312()013231(lnlnln)2ab IIIabcDDrVqqqrDD=+3122313122313301223131223131(lnlnln)32ababcD D DD D DrVqqqrD D DD D D=+131223131003122313()11(lnln)(lnln)22()abababD D DrGMDrVqqqqrrGMDD D D=+=+101ln2nkjijkkkiDVqD=24 01(lnln)2ababGMDrVqqrGMD=+01(lnln)2acacGMDrVqqrGMD=+031 2ln()ln21 (2lnln)23 ln2abacanabcaaaVVVGMDrqqqrGMDGMDrqqrGMDqGMDr+=+=Vca bcaqqq+=0.0556 F/kmlnCDr=Compared to 0.2ln mH/kmGMDLr=How to calculate the capacitance per phase?020.0556 F/m F/kmlnlnaanqCGMDGMDVrr=(for single-phase line)(for 3-phase line)25 Effect of Bundling 0.0556 F/kmlnbCGMDr=brrd=32brrd=341.09brrd=bssDDd=23bssDDd=341.09bssDDd=0.2ln mH/kmbsGMDLD=GMRL GMRC 26 A Summary:GMD,GMRL and GMRC L and C 1 11 21.n mn mnma ba ba ba bGMDDDDD=21 2111,(.).(.)nnn nbbnL Asa aa aa aa asGMRD DDDDD=21 2111,(.).(.)nnn nbbnC Aa aa aa aa aGMRr DDDDr=,0.0556 F/kmlnAC ACGMDGMR=,0.2ln mH/kmAL AGMDLGMR=a1 an b1 bm rb Dbs Da1,b1 Dan,bm Da1,bm Dan,b1 Da1,an Db1,bm A B rb=r and Dbs=Ds for single conductors L if GMRL.C if GMRC 27 GMRC of each phase group Equivalent GMRC Three-Phase Double-Circuit Lines GMD between 3 phase groups GMD per phase(consider transposition)GMRL of each phase group Equivalent GMRL 1 11 22 12 21 11 22 12 244ABa ba ba ba bBCb cb cb cb cDDDDDDDDDD=1 11 22 12 24ACa ca ca ca cDDDDD=A B C 3ABBCACGMDDDD=1 21 21 21 22424()()bbSAsa asa abbSBsb bsb bDD DD DDD DD D=1 21 224()bbSCsc csc cDD DD D=3LSASBSCGMRD D D=1 2bAa arr D=1 2bBb brr D=1 2bCc crr D=0.2ln mH/kmLGMDLGMR=0.0556 F/kmlnCCGMDGMR=3CA B CGMRr r r=Inductance and Capacitance:28 Effect of Earth on the Capacitance The presence of earth alters the distribution of electric flux lines and equipotential surfaces The earth level is an equipotential surface Image Charges Method The effect of the earth is to increase the capacitance Negligible for balanced steady-state analysis if conductors are high Problem 4.15 Ground 29 Example 4.2 1 mil=0.001 inch=0.0254 mm 1 cmil(circular mil,i.e.the area of a circle with a diameter of 1 mil)=/4 x mil2=5.067 x 10-10 m2=5.067 x 10-4 mm2 30 Example 4.3 ACSR(7 steel and 24 aluminum strands)