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ECE 476 Power System Analysis
Lecture 5:Transmission Line Parameters
Prof. Tom Overbye
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
overbye@illinois.edu
Special Guest Lecturer: TA Won Jang
Announcements
• Please read Chapter 4• HW 2 is 2.44, 2.48, 2.49, 2.51
• It does not need to be turned in, but will be covered by an in-class quiz on Sept 10
• San Diego Gas & Electric is on campus for the ECE Career Fair on 9/9) (ARC Gym) and then for interviews on 9/10
2
Development of Line Models
• Goals 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
3
Primary Methods for Power Transfer
• The most common methods for transfer of electric power are – Overhead ac– Underground ac– Overhead dc– Underground dc– other
4
Magnetics Review
• Ampere’s circuital law:
e
F = mmf = magnetomtive force (amp-turns)
= magnetic field intensity (amp-turns/meter)
d = Vector differential path length (meters)
= Line integral about closed path (d is tangent to path)
I =
eF d I
H l
H
l
l
Algebraic sum of current linked by
5
Line Integrals
•Line integrals are a generalization of traditional integration
Integration along thex-axis
Integration along ageneral path, whichmay be closed
Ampere’s law is most useful in cases of symmetry, such as with an infinitely long line
6
Magnetic Flux Density
Magnetic fields are usually measured in terms of flux density
0-7
0
= flux density (Tesla [T] or Gauss [G])(1T = 10,000G)
For a linear a linear magnetic material
= where is the called the permeability
=
= permeability of freespace = 4 10
= relative permea
r
r
H m
B
B H
bility 1 for air
7
Magnetic Flux
Total flux passing through a surface A is
=
= vector with direction normal to the surface
If flux density B is uniform and perpendicular to an area A then
=
Ad
d
BA
B a
a
8
Magnetic Fields from Single Wire
Assume we have an infinitely long wire with current of 1000A. How much magnetic flux passes through a 1 meter square, located between 4 and 5 meters from the wire?
Direction of H is givenby the “Right-hand” Rule
Easiest way to solve the problem is to take advantage of symmetry. For an integration path we’ll choose acircle with a radius of x.
9
Two Conductor Line Inductance
Key problem with the previous derivation is we assumed no return path for the current. Now consider the case of two wires, each carrying the same current I, but in opposite directions; assume the wires are separated by distance R.
R
Creates counter-
clockwise field
Creates a
clockwise field
To determine the
inductance of each
conductor we integrate
as before. However
now we get some
field cancellation
10
Two Conductor Case, cont’d
R R
Direction of integration
Rp
Key Point: As we integrate for the left line, at distance 2R from
the left line the net flux linked due to the Right line is zero!
Use superposition to get total flux linkage.
0 0left
For distance Rp, greater than 2R, from left line
ln ln2 ' 2
Rp Rp RI I
r R
Left Current Right Current11
Two Conductor Inductance
0left
0
0
0
0
Simplifying (with equal and opposite currents)
ln ln2 '
ln ln ' ln( ) ln2
ln ln2 '
ln as Rp2 '
ln H/m 2 'left
Rp Rp RI
r R
I Rp r Rp R R
R RpI
r Rp R
RI
r
RL
r
12
Many-Conductor Case
Now assume we now have k conductors, each with
current ik, arranged in some specified geometry.
We’d like to find flux linkages of each conductor.
Each conductor’s flux
linkage, k, depends upon
its own current and the
current in all the other
conductors.To derive 1 we’ll be integrating from conductor 1 (at origin)
to the right along the x-axis. 13
Many-Conductor Case, cont’d
At point b the net
contribution to l1from ik , l1k, is zero.
We’d like to integrate the flux crossing between b to c. But the flux crossing between a and c is easier to calculate and provides a very good approximation of l1k. Point a is at distance d1k from conductor k.
Rk is the
distance
from con-
ductor k
to point
c.
14
Many-Conductor Case, cont’d
0 1 21 1 2'
12 11
01 1 2'
12 11
01 1 2 2
1 1 2
0
1
ln ln ln2
1 1 1ln ln ln
2
ln ln ln2
As R goes to infinity R so the second
term from above can be written =2
nn
n
nn
n n
nn
jj
RR Ri i i
d dr
i i id dr
i R i R i R
R R
i
1ln R
15
Many-Conductor Case, cont’d
1
01 1 2'
12 11
1 11 1 12 2 1
Therefore if 0, which is true in a balanced
three phase system, then the second term is zero and
1 1 1ln ln ln
2
System has self and mutual inducta
n
jj
nn
n n
i
i i id dr
L i L i L i
nce. However
the mutual inductance can be canceled for
balanced 3 systems with symmetry.16
Symmetric Line Spacing – 69 kV
17
Birds Do Not Sit on the Conductors
18
Line Inductance Example
Calculate the reactance for a balanced 3, 60Hz
transmission line with a conductor geometry of an
equilateral triangle with D = 5m, r = 1.24cm (Rookconductor) and a length of 5 miles.
0 1 1 1ln( ) ln( ) ln( )
2 'a a b ci i ir D D
a
Since system is assumed
balanced
i b ci i
19
Line Inductance Example, cont’d
a
0a
0
70
3
6
Substituting
i
Hence
1 1ln ln
2 '
ln2 '
4 10 5ln ln
2 ' 2 9.67 10
1.25 10 H/m
b c
a a
a
a
i i
i ir D
Di
r
DL
r
20
Conductor Bundling
To increase the capacity of high voltage transmission
lines 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.
21
Bundled Conductor Pictures
The AEP Wyoming-JacksonFerry 765 kV line uses
6-bundle conductors.Conductors in a bundle areat the same voltage!
Photo Source: BPA and American Electric Power22
Bundled Conductor Flux Linkages
For the line shown on the left,
define dij as the distance bet-
ween conductors i and j. We
can then determine l for each
18
12 13 14
01
15 16 17
19 1,10 1,11 1,12
1 1 1 1ln ln ln ln
4 '
1 1 1 1ln ln ln ln
2 4
1 1 1 1ln ln ln ln
4
a
b
c
ir d d d
id d d d
id d d d
23
Bundled Conductors, cont’d
14
12 13 14
01 1
415 16 17 18
14
19 1,10 1,11 1,12
Simplifying
1ln
( ' )
1ln
2 ( )
1ln
( )
a
b
c
ir d d d
id d d d
id d d d
24
Bundled Conductors, cont’d
14
12 13 14
1
12 1
1
14
15 16 17 18 2 3 4
1 19 1
geometric mean radius (GMR) of bundle
( ' ) for our example
( ' ) in general
geometric mean distance (GMD) of
conductor 1 to phase b.
( )
(
b
bb
b
b b b ab
c
R
r d d d
r d d
D
d d d d D D D D
D d d
14
,10 1,11 1,12 2 3 4) c c c acd d D D D D
25
Inductance of Bundle
a
01
0 01
01
If D and i
Then
1 1ln ln
2
ln 4 ln2 2
4 ln2
ab ac bc b c
a ab
ab b
b
D D D i i
i iR D
D DI I
R R
DL
R
26
Inductance of Bundle, cont’d
0a 1
But remember each bundle has b conductors
in parallel (4 in this example). So
L / ln2 b
DL b
R
27
Bundle Inductance Example
0.25 M0.25 M
0.25 M
Consider the previous example of the three phases
symmetrically spaced 5 meters apart using wire
with a radius of r = 1.24 cm. Except now assume
each phase has 4 conductors in a square bundle,
spaced 0.25 meters apart. What is the new inductance
per meter?
2 3
13 4
b
70a
1.24 10 m ' 9.67 10 m
R 9.67 10 0.25 0.25 2 0.25
0.12 m (ten times bigger!)
5L ln 7.46 10 H/m
2 0.12
r r
28
Transmission Tower Configurations
•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 in
general Dab
Dac Dbc
Unless something
was done this would
result in unbalanced
phases
29
Transposition
• 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.
Aerial or side view of conductor positions over the length
of the transmission line.
30
Line Transposition Example
31
Line Transposition Example
32
Transposition Impact on Flux Linkages
0a
12 13
0
13 23
0
23 12
For a uniformly transposed line we can
calculate the flux linkage for phase "a"
1 1 1 1ln ln ln
3 2 '
1 1 1 1ln ln ln
3 2 '
1 1 1 1ln ln ln
3 2 '
a b c
a b c
a b c
I I Ir d d
I I Ir d d
I I Ir d d
“a” phase in
position “1”
“a” phase in
position “3”
“a” phase in
position “2”
33
Transposition Impact, cont’d
13
13
12 13 230a
13
12 13 23
Recognizing that
1(ln ln ln ) ln( )
3We can simplify so
1 1ln ln
'
2 1ln
a b
c
a b c abc
I Ir d d d
Id d d
34
Inductance of Transposed Line
13
m 12 13 23
0 0a
70
Define the geometric mean distance (GMD)
D
Then for a balanced 3 system ( - - )
1 1ln ln ln
2 ' 2 '
Hence
ln 2 10 ln H/m2 ' '
a b c
ma a a
m
m ma
d d d
I I I
DI I I
r D r
D DL
r r
35
Inductance with Bundling
b
0a
70
If the line is bundled with a geometric mean
radius, R , then
ln2
ln 2 10 ln H/m2
ma
b
m ma
b b
DI
R
D DL
R R
Inductance Example
Calculate the per phase inductance and reactance of a balanced 3, 60 Hz, line with horizontal phase spacing of 10m using three conductor bundling with a spacing between conductors in the bundle of 0.3m. Assume the line is uniformly transposed and the conductors have a 1cm radius.
Answer: Dm = 12.6 m, Rb= 0.0889 m Inductance = 9.9 x 10-7 H/m, Reactance = 0.6 /Mile
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