Download Lecture 7: Transmission Line Parameters

ECE 476
Power System Analysis
Lecture 7: Transmission Line Parameters
Prof. Tom Overbye
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
[email protected]
Announcements
• Please read Chapters 4 and 5 (skim 4.7, 4.11, 4.12)
• HW 3 is 4.9 (use lecture results for 4.8
comparison), 4.12, 4.19 (just compare 4.19a to
4.19b), 4.25
•
•
•
It does not need to be turned in, but will be covered by an
in-class quiz on Thursday Sept 15
Positive sequence is same as per phase; it will be covered
in Chapter 8
Use Table A.4 values to determine the Geometric Mean
Radius of the wires (i.e., the ninth column).
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Birds Do Not Sit on High Voltage Lines
2
Voltage Difference
The voltage difference between any two
points P and P is defined as an integral
V

P
P
E dl
In previous example the voltage difference between
points P and P , located radial distance R and R 
from the wire is (assuming  =  o )
V
 
R
R
R
dR 
ln
2 o R
2 o R
q
q
3
Voltage Difference, cont’d
With
V
 
R
R
R
dR 
ln
2 o R
2 o R
q
q
if q is positive then those points closer in have
a higher voltage. Voltage is defined as the energy
(in Joules) required to move a 1 coulomb charge
against an electric field (Joules/Coulomb). Voltage
is infinite if we pick infinity as the reference point
4
Multi-Conductor Case
Now assume we have n parallel conductors,
each with a charge density of q i coulombs/m.
The voltage difference between our two points,
P and P , is now determined by superposition
V
n
R i

qi ln

2 i 1
R i
1
where R i is the radial distance from point P
to the center of conductor i, and R i the
distance from P to the center of conductor i.
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Multi-Conductor Case, cont’d
n
If we assume that
 qi  0 then rewriting
i=1
V
1
1 n

qi ln

qi ln R i


2 i 1
R i 2 i 1
1
n
n
We then subtract
 qi ln R1  0
i 1
V
R i
1
1 n

qi ln

qi ln


2 i 1
R i 2 i 1
R 1
1
n
R i
As we more P to infinity, ln
0
R 1
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Absolute Voltage Defined
Since the second term goes to zero as P goes to
infinity, we can now define the voltage of a
point w.r.t. a reference voltage at infinity:
V
1
n
1

qi ln

2 i 1
R i
This equation holds for any point as long as
it is not inside one of the wires!
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Three Conductor Case
A
C
B
Assume we have three infinitely
long conductors, A, B, & C, each
with radius r and distance D from
the other two conductors.
Assume charge densities such
that qa + qb + qc = 0
1 
1
1
1
Va 
q
ln

q
ln

q
ln
a
b
c
2 
r
D
D 
qa
D
Va 
ln
2 r
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Line Capacitance
For a single line capacitance is defined as
qi  CiVi
But for a multiple conductor case we need to
use matrix relationships since the charge on
conductor i may be a function of Vj
 q1 
 C11
   
 

 qn 
Cn1
q  CV
C1n  V1 
 
 
Cnn  Vn 
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Line Capacitance, cont’d
To eliminate mutual capacitance we'll again
assume we have a uniformly transposed line.
For the previous three conductor example:
Va  V
Since q a = C Va

qa
2
C 

Va
ln D
r
10
Bundled Conductor Capacitance
Similar to what we did for determining line
inductance when there are n bundled conductors,
we use the original capacitance equation just
substituting an equivalent radius
c
Rb
 (rd12
1
d1n )
n
Note for the capacitance equation we use r rather
than r' which was used for R b in the inductance
equation
11
Line Capacitance, cont’d
For the case of uniformly transposed lines we
use the same GMR, D m , as before.
ln
2
Dm

 d ab d ac dbc 
C 
Rbc
where
Dm
R cb
 (rd12
1
d1n )
n
1
3
(note r NOT r')
ε in air   o  8.854  10-12 F/m
12
Line Capacitance Example
Calculate the per phase capacitance and susceptance
of a balanced 3, 60 Hz, transmission 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 a 1cm radius.
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Line Capacitance Example, cont’d
Rbc
Dm
C
Xc

1
(0.01  0.3  0.3) 3

1
(10  10  20) 3
 0.0963 m
 12.6 m
2  8.854  1012

 1.141  1011 F/m
12.6
ln
0.0963
1
1


C
2 60  1.141  1011 F/m
 2.33  10 -m (not  / m)
8
14
ACSR Table Data (Similar to Table A.4)
GMR is equivalent to r’
Inductance and Capacitance
assume a Dm of 1 ft.
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ACSR Data, cont’d
Dm
X L  2 f L  4 f  10 ln
 1609 /mile
GMR
1
3 
 2.02  10 f ln
 ln Dm 
 GMR

1
3
 2.02  10 f ln
 2.02  103 f ln Dm
GMR
7
Term from table assuming
a one foot spacing
Term independent
of conductor with
Dm in feet.
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ACSR Data, Cont.
To use the phase to neutral capacitance from table
2 0
1
XC 
-m where C 
Dm
2 f C
ln
r
Dm
1
6

 1.779  10 ln
-mile (table is in M-mile)
f
r
1
1 1

 1.779  ln   1.779  ln Dm M-mile
f
r f
Term independent
Term from table assuming
of conductor with
a one foot spacing
Dm in feet.
17
Dove Example
GMR  0.0313 feet
Outside Diameter = 0.07725 feet (radius = 0.03863)
Assuming a one foot spacing at 60 Hz
1
X a  2 60  2  10  1609  ln
Ω/mile
0.0313
X a  0.420 Ω/mile, which matches the table
7
For the capacitance
1
1
6
X C   1.779  10 ln  9.65  104 Ω-mile
f
r
18
Line Conductors
• Typical transmission lines use multi-strand
conductors
• ACSR (aluminum conductor steel reinforced)
conductors are most common. A typical Al. to St.
ratio is about 4 to 1.
19
Line Conductors, cont’d
• Total conductor area is given in circular mils. One
circular mil is the area of a circle with a diameter
of 0.001 =   0.00052 square inches
• Example: what is the area of a solid, 1” diameter
circular wire?
Answer: 1000 kcmil (kilo circular mils)
• Because conductors are stranded, the equivalent
radius must be provided by the manufacturer. In
tables this value is known as the GMR and is
usually expressed in feet.
20
Line Resistance
Line resistance per unit length is given by
R =

where  is the resistivity
A
Resistivity of Copper = 1.68  10-8 Ω-m
Resistivity of Aluminum = 2.65  10-8 Ω-m
Example: What is the resistance in Ω / mile of a
1" diameter solid aluminum wire (at dc)?
2.65  10-8 Ω-m
m

R 
1609
 0.084
2
mile
mile
  0.0127m
21
Line Resistance, cont’d
• Because 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
10% between 25C and 50C, 0.4% per degree C)
• Because ACSR conductors are stranded, actual
resistance, inductance and capacitance needs to be
determined from tables.
22
Variation in Line Resistance Example
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345 kV+ Transmission Growth at a
Glance
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345 kV+ Transmission Growth at a
Glance
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345 kV+ Transmission Growth at a
Glance
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345 kV+ Transmission Growth at a
Glance
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345 kV+ Transmission Growth at a
Glance
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Ameren Illinois Rivers 345 kV Project
• Ameren is in the process of building a number of 345
kV transmission lines across Central Illinois.
•
Locally this includes a line between Sidney and Rising
in Champaign County
http://www.ilriverstransmission.com/maps
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Sidney to Bunsonville 345 kV
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Sidney to Kansas (IL) 345
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Sidney to Rising 345 kV
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Champaign-Urbana Part of Grid
Additional Transmission Topics
• Multi-circuit lines: Multiple lines often share a
common transmission right-of-way. This DOES
cause mutual inductance and capacitance, but is
often ignored in system analysis.
• Cables: There are about 3000 miles of
underground ac cables in U.S. Cables are primarily
used in urban areas. In a cable the conductors are
tightly spaced, (< 1ft) with oil impregnated paper
commonly used to provide insulation
–
–
inductance is lower
capacitance is higher, limiting cable length
34
Additional Transmission Topics
• Ground wires: Transmission lines are usually
protected from lightning strikes with a ground
wire. This topmost wire (or wires) helps to
attenuate the transient voltages/currents that arise
during a lighting strike. The ground wire is
typically grounded at each pole.
• Corona discharge: Due to high electric fields
around lines, the air molecules become ionized.
This causes a crackling sound and may cause the
line to glow!
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Additional Transmission Topics
• Shunt conductance: Usually ignored. A small
current may flow through contaminants on
insulators.
• DC Transmission: Because of the large fixed cost
necessary to convert ac to dc and then back to ac,
dc transmission is only practical for several
specialized applications
–
–
–
long distance overhead power transfer (> 400 miles)
long cable power transfer such as underwater
providing an asynchronous means of joining different
power systems (such as the Eastern and Western grids).
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HVDC Lines in North America
http://www.grainbeltexpresscleanline.com/site/page/history_of_hvdc_transmission
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