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ELE-B7
TRANSMISSION LINES
Development of Line ModelsGoals of this section are
1) Develop a simple model for transmission lines
2) Gain an intuitive feel for how the geometry of the transmission line affects the model parameters
Primary Methods for Power TransferThe most common methods for transfer of
electric power are 1) Overhead AC2) Underground AC3) Overhead DC4) Underground DC
Transmission lines and cablesExtra-high-voltage lines
Voltage: 345 kV, 500 kV, 765 kV
High-voltage linesVoltage: 115 kV, 230 kV
Sub-transmission linesVoltage: 46 kV, 69 kV
Distribution linesVoltage: 2.4 kV to 46 kV, with 15 kV being the most commonly used
High-voltage DC linesVoltage: ±120 kV to ±600 kV
Transmission lines and cablesThree-phase conductors, which carry the electric current;Insulators, which support and electrically isolate the conductors;Tower, which holds the insulators and conductors;Foundation and grounding; andOptional shield conductors, which protect against lightning
Transmission lines and cables
Shieldconductor
Insulator
Phaseconductor
Tower
69kVLine
CompositeInsulatorCrossarm
Compositeinsulator
Steel tower
Twoconductor
bundle
Shield conductor
Transmission lines and cables
Double circuit69 kV line
Distribution line12.47kV
Wooden tower
Shieldconductor
Distribution Line
Transmission lines and cables
Distribution line
Transformer
240V/120Vinsulated line
Transformers
Fuse cutout
Surge arrester
Insulator
Transmission lines and cables
Span
SagInsulator
Tension Tower
SupportingTower Tension Tower
Definition of Parameters
Aluminum Conductor Steel Reinforced (ACSR);All Aluminum Conductor (AAC); andAll Aluminum Alloy Conductor (AAAC).
ACSR Coductor
Aluminum outer strands2 layers, 30 conductors
Steel core strands,7 conductors
Transmission lines and cables
Transmission lines and cables
Locking Key
Insulator's Head
Expansion Layer
Imbedded Sand
Skirt
Petticoats
Iron Cap
Ball Socket
Compression Loading
Cement
Insulating Glass or Porcelain
BallCorrosion Sleeve for DC Insulators
Steel Pin
Insulators
Transmission lines and cablesInsulator Chain
Line Voltage
Number of Insulators per String
69 kV 4–6
115 kV 7–9
138 kV 8–10
230 kV 12
345 kV 18
500 kV 24
765 kV 30–35
Transmission lines and cablesComposite insulator.
(1) Sheds of alternating diameters prevent bridging by ice, snow andcascading rain.(2) Fiberglass reinforced resin rod.(3) Injection molded rubber (EPDM or Silicone) weather sheds and rodcovering. (4) Forged steel end fitting, galvanized and joined to rod by swaging process.
1 432
Transmission lines and cables
(1) is the clevis ball,(2) is the socket for the clevis,(3) is the yoke plate, and (4) is the suspension clamp. (Source: Sediver)•Figure 4.15 Line post-composite
insulator with yoke holding two conductors.
Transmission lines and cables
Transmission lines and cables
ConductorConductor shieldInsulation
Insulation shieldFiller
Copper screenPVC-sheet
Inductance of a Single WireThe inductance of a magnetic circuit has a constant permeabilitycan be obtained by determining the following:
a) Magnetic field intensity H, from Ampere’s law.b) Magnetic flux density B (B = μH)c) Flux linkage λd) Inductance from flux linkage per ampere
(L= λ/I)
Flux linkages within the wire :
mHI
LId
dxr
Ixdrxd
dxBd
r
xIHB
r
xIHIrxI
xIHIxH
IdlH
r
x
xx
xx
xxxx
enclosed
/102110
21
2
:flux by the linked iscurrent total theof (x/r)fraction only the Since
:islength unit per dflux aldifferenti The2
:conductor magnetic-nonFor 2
:ondistributicurrent uniform Assume2
)2(
7intint
7
0int
4
30
2
2
20
0
2
2
−− ×==⇒×∫ ==
=⎟⎠
⎞⎜⎝
⎛=
=
==
=⇒⎟⎠
⎞⎜⎝
⎛=
=⇒=
=∫ ⋅
λλλ
π
μφλ
φφ
π
μμ
π
ππ
Flux linkages outside of the wire :
mHDD
IL
DDId
dxxIdd
dxBd
xI
xIHB
xIHIxH
IdlH
D
D
x
xx
xx
enclosed
/21ln102
21ln102
102
:conductor theoutsideflux by the linked iscurrent entire theSince
:islength unit per dflux aldifferenti The
1022
104
2)2(
71212
72
112
7
770
⎟⎠
⎞⎜⎝
⎛×==⇒⎟⎠
⎞⎜⎝
⎛××∫ ==
×==
=
×=××==
=⇒=
=∫ ⋅
−−
−
−−
λλλ
φλ
φφ
ππμ
ππ
r x
D1 D2
Total flux linkage:
mHr
DI
L
re
r
DrDe
rD
pp
p
/ln102
r :where
lnI102 lnlnI102lnI102I1021
thenD, D andr D Since linkages.flux external and interanl theof sum theis
D distanceat Ppoint external out toconductor thelinking linkageflux Total
'7
4/1'
'7-4/17-7-7-
21
p
⎟⎟⎠
⎞⎜⎜⎝
⎛×==
=
××=⎟⎠
⎞⎜⎝
⎛+××=⎟
⎠
⎞⎜⎝
⎛××+×=
==
−
−
λ
λ
λ
Total flux linkage, cont.:
conductors M ofarray an in k conductor linkingflux total thegives :where
1ln102
:onmanipulati almathematic someAfter
lnI102
:is Imcurrent todue Ppoint out tok conductor links which and
lnI102
Icurrent todue Ppoint out tok conductor links which linkageflux the: thenzero, is currentsconductor theof sum theand Icurrent
carries mconductor each Assume .conductors solid M ofarray heconsider t Finally,
k
1
7-
m7-
kPm
'k7-
kkPk
m
λ
λ
λ
λ
λ
λ
km
M
mmk
km
Pmkpm
k
Pkkpk
DI
DD
r
D
×∑×=
××=
××=
=
Inductance of single-phase, two-wire
⎟⎟⎠
⎞⎜⎜⎝
⎛×=⇒==
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛×=⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛×=⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛+×=+=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛×==⎟
⎟⎠
⎞⎜⎜⎝
⎛×==
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛××−=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛××=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠
⎞⎜⎝
⎛×−⎟
⎟⎠
⎞⎜⎜⎝
⎛××=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛×+⎟⎟
⎠
⎞⎜⎜⎝
⎛××=
−
−−−
−−
−
'7'y'x'
''7
''
27
''7
'7
'7
'7-
4/1'
'7-
'7-
yx7-
ln104rrr If
ln104ln102ln102
ln102 and conductor,per /ln102
ln102
r :where
ln102
1lnI1lnI102
1lnI1lnI102
:hence and validisrelation previous thezero is currents two theof sum theSince
r
DL
rr
D
rr
D
r
D
r
DLLL
r
DI
LmHr
DI
L
r
DI
similarlyre
r
DI
Dr
DD
yxyxyxyx
yy
yy
xx
xx
yy
xx
xx
xx
xyxxx
λλ
λ
λ
λ
λ
Inductance of three-phase, three-wire with equal phase spacing
⎟⎟⎠
⎞⎜⎜⎝
⎛×=
⎟⎟⎠
⎞⎜⎜⎝
⎛××=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛×−⎟⎟
⎠
⎞⎜⎜⎝
⎛××=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛×++⎟⎟
⎠
⎞⎜⎜⎝
⎛××=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛×+⎟
⎠
⎞⎜⎝
⎛×+⎟⎟
⎠
⎞⎜⎜⎝
⎛××=
'7-
'7-
a'a7-
b'a7-
cb'a7-
ln102
ln102
1lnI1lnI102
1ln)I(1lnI102
1lnI1lnI1lnI102
:hence and validisrelation previous thezero is currents three theof sum theSince
r
DL
r
DI
Dr
DI
r
DDr
a
aa
a
ca
a
λ
λ
λ
λ
Inductance of composite conductors
GMRDGMDDDD
IL
D
DI
DNMDNI
DNMDNI
N
DMI
DNI
NN
k
N
mkmMN
N
k
M
mkm
xx
xyxx
N
k NN
mkm
NMM
mkm
N
K
M
m km
N
m km
N
k
M
m km
N
m km
k
M
m km
N
m kmk
=∏ ∏==∏ ∏=×==
∏
⎟⎟⎠
⎞⎜⎜⎝
⎛∏
⎟⎟⎠
⎞⎜⎜⎝
⎛∏
××=
∑ ⎥⎦
⎤⎢⎣
⎡∑∑ −××∑ ==
⎥⎦
⎤⎢⎣
⎡∑∑ −××==
⎥⎦
⎤⎢⎣
⎡∑∑ −×=
= == =
−
=
=
=−
= ==
−
=
==
−
==
−
2
2
1 1xx
1 '1xy
7
1 /1
1
/1
'17x
1 '1127
1kx
'1127
k
k
'11
7
k
D and D : whereln102
ln102
1ln11ln1102
:isx conductor of linkageflux totalThe
1ln11ln1102
:isk or subconduct of linkageflux theflux, by this linked is Icurrent total theof (1/N)fraction only the Since
1ln1ln102
:is Xconductor ofk or subconduct linking flux totalThe
λ
λ
λλ
φλ
λ
φ
φ
Inductance of unequal phase spacing The problem with the line analysis we’ve done
so far is we have assumed a symmetrical tower configuration. Such a tower figuration is seldom practical.
Typical Transmission Tower Configuration
Therefore ingeneral Dab ≠Dac ≠ Dbc
Unless something was done this wouldresult in unbalancedphases
Inductance of unequal phase spacing
To keep system balanced, over the length of a transmission line the conductors are rotated so each phase occupies each position on tower for an equal distance. This is known as transposition.
D
: whereln102
3 312312eq
7
DDD
DD
LS
eq
=
×= −
D12 D31
D23
Conductor BundlingTo increase the capacity of high voltage transmissionlines it is very common to use a number of conductors per phase. This is known as conductor bundling. Typical values are two conductors for 345 kV lines, three for 500 kV and four for 765 kV.
Inductance of bundled conductors
:follows as calculated is D and D
: whereln102
SL3 312312eq
7
DDD
DD
LSL
eq
=
×= −
( ) ( )dDdD SS ×=×= D 4 2SL
d
d
dd
( ) ( )3 29 3SL D dDddD SS ×=××=
d
d ( ) ( )4 316 4SL 091.1 2D dDdddD SS ×=×××=
Electric field and voltage: Solid cylindrical conductor
The capacitance between conductors in a medium with constant permittivity can be obtained by finding the following:
a) Electric field from Gauss’s law.
b) Voltage between conductors.
c) Capacitance from charge per unit volt (C = q/V)
Gauss law: the total electric flux leaving a closed surface equals the total charge within the volume enclosed by the surface.
Electric field and voltage: Solid cylindrical conductor
1
22
1
2
112 ln
22
2))(2(
DDqdx
xqdxEV
xqEqLLxE
QEds
D
D
D
Dx
xx
enclosed
πεπε
πεεπ
ε
∫ =∫ ==
=⇒=
∫∫ =
Electric field and voltage: Solid cylindrical conductor
∑=
=
=
M
m km
immki
km
immkim
DDqV
DDqV
1
ki
ln2
1
:bygiven is charges all todue V voltage theiion,superposit Using
ln2
πε
πε
Assume each conductor m has a charge qm C/m, the voltage Vkim between conductors k and i due to the charge qm acting alone is:
Capacitance for single phase two wire line
⎟⎠⎞
⎜⎝⎛
=
==
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛==
====
=⎥⎥⎦
⎤
⎢⎢⎣
⎡−=
rD
Cxy
A
rrDVxy
qCxy
DDDDq
DD
qDD
qVxy
yx
yyxx
xyyx
xy
yy
xx
yx
ln
then r,r r if nd
ln
then ,r D and rD D,DD Using
ln2
lnln2
1
yx
yyyxxxyxxy
πε
πε
πεπε
Assume conductor x has a uniform charge q C/m and conductor y has –q. Using the previous last equation with k = x, i = y and m = x, y
X Cxyy
Capacitance for three phase with equal phase spacing
)/ln(2
ln2
1
qqq with and
ln)(ln22
131
3
lnln2
1imilarly
lnln2
1lnlnln2
1
then ,r DD D,DDD Using
lnlnln2
1
acb
yyaacbcaab
rDC
rDqV
Drqq
rDqV
VVVDrq
rDqVac
sDrq
rDq
DDq
Drq
rDqVab
DDq
DDq
DDqVab
an
aan
cbaan
anacab
ca
bacba
ac
bcc
ab
bbb
aa
baa
πεπε
πε
πε
πεπε
πε
=
⎥⎦
⎤⎢⎣
⎡=
−=+
⎥⎦
⎤⎢⎣
⎡++⎟
⎠
⎞⎜⎝
⎛=
=+
⎥⎦
⎤⎢⎣
⎡+=
⎥⎦
⎤⎢⎣
⎡+=⎥
⎦
⎤⎢⎣
⎡++=
=====
⎥⎦
⎤⎢⎣
⎡++=
Capacitance for stranded, unequal phase spacing and bundled conductors
bundleconductor -fourfor 091.1
bundleconductor -for three
bundleconductor -for two
)/ln(2
4 3
3 2
3
rdD
rdD
rdD
DDDD
DDC
sc
sc
sc
acbcabeq
sceq
=
=
=
=
=πε
Line Resistance
-8
-8
Line resistance per unit length is given by
R = where is the resistivityA
Resistivity of Copper = 1.68 10 Ω-m
Resistivity of Aluminum = 2.65 10 Ω-mExample: What is the resistance in Ω / mile of a
ρ ρ
×
×
-8
2
1" diameter solid aluminum wire (at dc)?
2.65 10 Ω-m1609 0.0840.0127m
mRmile mileπ
× Ω= =
×
Line Resistance, cont’dBecause ac current tends to flow towards the surface of a conductor, the resistance of a line at 60 Hz is slightly higher than at dc.Resistivity and hence line resistance increase as conductor temperature increases (changes is about 8% between 25°C and 50°C)Because ACSR conductors are stranded, actual resistance, inductance and capacitance needs to be determined from tables.
Tow-Port Network Model
Vs
Is
VR
IR
Two-port
network
1=−+=+=
BCADDICVIBIAVV
RRs
RRs
Short transmission lines
Vs
Is
VR
IR
Z
oCZBDAII
ZIVV
Rs
RRs
==
===
+=
1
Medium transmission lines
⎟⎠
⎞⎜⎝
⎛ +=
=
+==
⎟⎠
⎞⎜⎝
⎛ ++⎟⎠
⎞⎜⎝
⎛ +=
++=
+⎟⎠
⎞⎜⎝
⎛ +=⎟⎠
⎞⎜⎝
⎛ ++=
41
21
21
41
V of value thesubsitute ,22
21
2
s
YZYC
ZB
YZDA
IYZVYZYI
YVYVII
ZIVYZYVIZVV
RRs
sRRs
RRR
RRs
Long transmission lines
S/m /m
CjGyLjRzωω
+=Ω+=
Long transmission lines, cont.
0)()(
)()()(
)()(:zero approachesx aslimit theTaking
)()()()()()()(
)()(:zero approachesx aslimit theTaking
)()()()()()()(
2
2
2
2
=−
==
=
Δ
=Δ
−Δ+Δ+Δ+=Δ+
=
Δ
=Δ
−Δ+Δ+=Δ+
xzyVdx
xVd
xzyVdx
xdIzdx
xVd
xyVdx
xdI
xyVx
xIxxIxxVxyxIxxI
xzIdx
xdV
xzIx
xVxxVxIxzxVxxV
RRc
RcR
RcRRcR
c
c
xx
xx
xx
IxVxZ
xI
IxZVxxVso
IZVAIZVA
yzZ
ZeAeAxI
xzIeAeAdx
xdVzy
eAeAxV
)cosh()sinh(1)(
)sinh()cosh()(:
2 and
2
ZA-AI(0) I and A A V(0) V Since
impedance. sticcharacteri thecalled is
)(
)()(constantn propagatio thecalled is
)(
21
c
21R21R
21
21
21
γγ
γγ
γγ
γ
γγ
γγ
γγ
+=
+=
−=
+=
==+==
=
−=
=−=
=
+=
−
−
−
Home WorkQ1:
Home WorkQ2: