Theprocedureforcomputingthephaseimpedancematrixfortwooverhead
parallellinesispresentedinSection4�1�8�Figure4�18showstwoconcentric
neutralparallellineseachwithaseparategroundedneutralconductor�
The process for computing the 6 × 6 phase impedance matrix follows
exactlythesameprocedureasfortheoverheadlines�Inthiscase,thereare
a total of 14 conductors (6 phase conductors, 6 equivalent concentric neu-tral conductors, and 2 grounded neuneu-tral conductors)� Applying Carson’s
equationswillresultina14×14primitiveimpedancematrix�Thismatrix
ispartitionedbetweenthesixthandseventhrowsandcolumns�TheKron
reductionisappliedtoformthefinal6×6phaseimpedancematrix�
Example 4.5
Two concentric neutral three-phase underground parallel lines are
showninFigure4�19�
Cables(bothlines):250kcmil,1/3neutral Extraneutral:4/0copper
Determinethe6×6phaseimpedancematrix�
Solution
FromAppendixBforthecables, Outsidediameter:dod=1�29in�
Neutralstrands:k=13#14copperstrands FromAppendixAfortheconductors,
250kcmilAl: GMRc=0�0171ft, rc=0�41Ω/mile, dc=0�567in�
#14Copper: GMRs=0�00208ft, rs=14�8722Ω/mile, ds=0�0641in�
4/0Copper: GMRn=0�1579ft, rn=0�303Ω/mile, dn=0�522in�
D1–2 D1–3 D2–3 D1–13
D2–13
D3–13 9
3 8
1 2 7
D4–5 D5–6
D4–6 D4–14
D5–14
D6–14
4 5 6
Dnm
10 11 12
14 13
FIGURE 4.18
Parallelconcentricneutralundergroundlines�
10 in.
4 in. 4 in. 2 in.
a b
4 in. 4 in.
c
5 in.
b a c
FIGURE 4.19
Parallelconcentricneutralthree-phaselines�
Theradiusofthecircletothecenterofthestrandsis
R d d
b= od− s = − =
24
1 29 0 0641
24 0 0511
. . . ft
TheequivalentGMRoftheconcentricneutralstrandsiscomputedas
GMReq=kGMR k Rs⋅ ⋅ bk−1 =130 00208 13 0 05111. ⋅ ⋅ . 12 =0 0486. ft ThepositionsofthesixcablesandextraneutralusingCartesiancoordi-nateswiththephaseacableinline1(topline)astheordinate�Notethe
phasinginbothlines�
Phasea,
line1: d1=0+j0 Phaseb,
line1: d2 4 j 12 0
= + Phasec,
line1: d3 8 j 12 0
= +
Phasea,
line2: d4 4 j 12
10
= − 12 Phaseb,
line1: d5 0 j10
= − 12 Phasec,
line1: d6 8 j 12
10
= − 12 Equivalent neutrals
Phasea,
line1: d7=d1+jRb Phaseb,
line1: d8=d2+jRb Phasec,
line1: d9=d3+jRb Phasea,
line2: d10=d4+jRb Phaseb,
line2: d11=d5+jRb Phasec,
line2: d12=d6+jRb Extra neutral
d13 10 j
12 5
= − 12
Thespacingmatrixdefiningthedistancesbetweenconductorscanbe
computedby
i j
Di j di dj
= =
= −
1 13– 1–13
,
Thediagonaltermsofthespacingmatrixaredefinedastheappropriate
GMR:
D1 1, =D2 2, =D3 3, =D4 4, =D5 5, =D6 6, =GMRc=0 0171. ft
D7 7, =D8 8, =D9 9, =D10 10, =D11 11, =D12 12, =GMReq=0 0486. ft
D13 13, =GMRn=0 01579. ft
Theresistancematrixisdefinedas
7 93402
Once the primitive impedance matrix is developed, it is partitioned
4.3 Summary
This chapter is devoted to presenting methods for computing the phase
impedancesandsequenceimpedancesofoverheadlinesandunderground
cables�Carson’sequationshavebeenmodifiedinordertosimplifythecom- putationofthephaseimpedances�WhenusingthemodifiedCarson’sequa-tions,thereisnoneedtomakeanyassumptions,suchastranspositionof
thelines�Byassuminganuntransposedlineandincludingtheactualphas-ingoftheline,themostaccuratevaluesofthephaseimpedances,selfand
mutual,aredetermined�Itishighlyrecommendedthatnoassumptionsbe
made in the computation of the impedances� Since voltage drop is a pri-maryconcernonadistributionline,theimpedancesusedforthelinemust
beasaccurateaspossible�Thischapteralsoincludedtheprocessofapply-ingCarson’sequationstotwodistributionlinesthatarephysicallyparallel�
Thissameapproachwouldbetakenwhentherearemorethantwolines
physicallyparallel�
Problems
4.1 The configuration and conductors of a three-phase overhead line are
showninFigure4�20�
Phaseconductors:556,50026/7ACSR Neutralconductor:4/0ACSR
FIGURE 4.20
Three-phaseconfigurationfor
Problem4�1�
4.0 ft
2.5 ft 4.5 ft
3.0 ft
a c b
n
25.0 ft
Determine
a� Thephaseimpedancematrix[zabc]inΩ/mile
b� Thesequenceimpedancematrix[z012]inΩ/mile
c� Theneutraltransformationmatrix[tn]
4.2 Determinethephaseimpedance[zabc]matrixinΩ/mileforthetwo-phase
configurationinFigure4�21�
Phaseconductors:336,40026/7ACSR Neutralconductor:4/06/1ACSR
4.3 Determinethephaseimpedance[zabc]matrixin Ω/mileforthesingle-phaseconfigurationshowninFigure4�22�
Phaseandneutralconductors:1/06/1ACSR
4.0 ft 3.0 ft
a c
n
25.0 ft 7.0 ft
FIGURE 4.21
Two-phaseconfigurationforProblem4�2�
5.0 ft
n
25.0 ft b
0.5 ft
FIGURE 4.22
Single-phase pole configuration for
Problem4�3�
4.4 CreatethespacingsandconfigurationsofProblem4�1through4�3in
theWindmilprogram�Comparethephaseimpedancematricestothose
computedinthepreviousproblems�
4.5 Determinethephaseimpedancematrix[zabc]andsequenceimpedance
matrix[z012]inΩ/mileforthethree-phasepoleconfigurationinFigure
4�23�Thephaseandneutralconductorsare250,000AA�
4.6 Compute the positive, negative, and zero sequence impedances in
Ω/1000ftusingtheGMDmethodforthepoleconfigurationshownin
Figure4�23�
4.7 Determine the [zabc] and [z012] matrices inΩ/mileforthethree-phase
configurationshowninFigure4�24�Thephaseconductorsare350,000
AAandtheneutralconductoris250,000AA�
4.8 Compute the positive, negative, and zero sequence impedances in
Ω/1000ftforthelineofFigure4�24usingtheaverageself-andmutual
impedancesdefinedinEquations4�70and4�71�
4.9 A 4/0 aluminum concentric neutral cable is to be used for a single-phaselateral�Thecablehasafullneutral(seeAppendixB)�Determine
theimpedanceofthecableandtheresultingphaseimpedancematrix
inΩ/mileassumingthecableisconnectedtophaseb�
4.10 Three250,000CMaluminumconcentriccableswithone-thirdneutrals
are buried in a trench in a horizontal configuration (see Figure 4�14)�
Determinethe[zabc]and[z012]matricesinΩ/1000ftassumingphasing
ofc-a-b�
FIGURE 4.23
Three-phasepoleconfigurationforProblem4�5�
2 ft 2 ft 2 ft
25 ft 4 ft
a b
c n
4.11 Create the spacings and configurations of Problems 4�9 and 4�10 in
Windmil�Comparethevaluesofthephaseimpedancematricestothose
computedinthepreviousproblems�Inordertocheckthephaseimped-ancematrix,itwillbenecessaryforyoutoconnectthelinetobalanced
three-phasesource�Asourceof12�47kVworksfine�
4.12 Asingle-phaseundergroundlineiscomposedofa350,000CMalumi- numtape-shieldedcable�A4/0copperconductorisusedastheneu-tral�Thecableandneutralareseparatedby4in�Determinethephase
impedancematrixinΩ/mileforthissingle-phasecablelineassuming
phasec�
4.13 Three one-third neutral 2/0 aluminum jacketed concentric neutral
cablesareinstalledina6in�conduit�Assumethecablejackethasathickness
of0�2in�andthecableslieinatriangularconfigurationinsidetheconduit�
ComputethephaseimpedancematrixinΩ/mileforthiscabledline�
4.14 Create the spacing and configuration of Problem 4�13 in Windmil�
Connecta12�47kVsourcetothelineandcompareresultstothoseof
4�13�
4.15 Twothree-phasedistributionlinesarephysicallyparallelasshownin
Figure4�25�
Line#1(leftside) Phaseconductors=266,80026/7ACSR
Neutralconductor=3/06/1ACSR
Line#2(rightside) Phaseconductors=300,000CONLAYaluminum
Neutralconductor=4/0CLASSAaluminum
a b
c 2 ft
2 ft n
2 ft
2 ft
25 ft
FIGURE 4.24
Three-phasepoleconfigurationforProblem4�5�
a�Determinethe6×6phaseimpedancematrix�
b�Determinetheneutraltransformmatrix�
4.16 Twoconcentricneutralundergroundthree-phaselinesarephysically
parallelasshowninFigure4�26�
Line#1(top) Cable=250kcmil,1/3neutral
Additionalneutral:4/06/1ACSR
Line#2(bottom) Cable=2/0kcmil,1/3neutral
Additionalneutral:2/0ACSR
a�Determinethe6×6phaseimpedancematrix�
b�Determinetheneutraltransformmatrix�
n
18 ft
a b c
n
c a
2.5 ft 4.5 ft b 3.0 ft
4.0 ft
25.0 ft
2.5 ft 4.5 ft 3.0 ft
4.0 ft
25.0 ft
FIGURE 4.25 ParallelOHlines�
24 in.
6 in. 6 in. 4 in.
a b
6 in. 6 in. 4 in.
c a
c
b
FIGURE 4.26
Parallelconcentricneutralthree-phaselines�
Windmil Assignment
FollowthemethodoutlinedintheUser’sManualtobuildasystemcalled
“System1”inWindmilthatwillhavethefollowingcomponents:
• 12�47kVline-to-linesource�The“BusVoltage”shouldbesetto120V�
• Connecttothenodeandcallitnode1�
• A5,000ftlongoverheadthreedistributionlineasdefinedinProblem
4�1�CallthislineOH-1�
• Connectanodetotheendofthelineandcallitnode2�
• Awyeconnectedunbalancedthree-phaseloadisconnectedtonode
2andismodeledasconstantPQloadwithvaluesof
• Phasea–g:1000kW,powerfactor=90%lagging
• Phaseb–g:800kW,powerfactor=85%lagging
• Phasec–g:1200kW,powerfactor=95%lagging
Determinethevoltagesona120Vbaseatnode2andthecurrentflowingon
theOH-1line�
References
1� Glover,J�D�andSarma,M�,Power System Analysis and Design,2ndedn�,PWS
PublishingCo�,Boston,MA,1994�
2� Carson, J�R�, Wave propagation in overhead wires with ground return, Bell System Technical Journal,5,539,1926�
3� Kersting, W�H� and Green, R�K�, Application of Carson’s equations to the
steady-stateanalysisofdistributionfeeders,IEEE Power System Conference and Exposition,Phoenix,AZ,March2011�
4� Kron,G�,Tensorialanalysisofintegratedtransmissionsystems,partI,thesix
basicreferenceframes,AIEE Transactions,71,1952�
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