Development of Transmission Line Models

1
ECE 476POWER SYSTEM ANALYSIS

• Lecture 5
• Development of Transmission Line Models
• Professor Tom Overbye
• Department of Electrical andComputer Engineering

2
Reading and Homework

• For lectures 5 through 7 please be reading
  Chapter 4
• we will not be covering sections 4.7, 4.11, and
  4.12 in detail
• Go through Section 1.5, building the PowerWorld
  case
• HW 2 is 2.32, 43, 47
• (You can download the latest educational version
  of PowerWorld (version 13) at http//www.powerworl
  d.com/gloversarma.asp
• The Problem 2.32 case will also be on the website

3
Substation Bus
4
In the News

• 9/2/08 Kansas utilities agree to cooperate on
  major transmission system project

5
Special Guest Talk

• Linda Brown is the director of Transmission
  Planning with San Diego Gas and Electric (SDGE)

6
Inductance Example

• Calculate the inductance of an N turn coil wound
  tightly on a torodial iron core that has a radius
  of R and a cross-sectional area of A. Assume
• 1) all flux is within the coil
• 2) all flux links each turn

7
Inductance Example, contd
8
Inductance of a Single Wire

• To development models of transmission lines, we
  first need to determine the inductance of a
  single, infinitely long wire. To do this we need
  to determine the wires total flux linkage,
  including
• 1. flux linkages outside of the wire
• 2. flux linkages within the wire
• Well assume that the current density within the
  wire is uniform and that the wire has a radius of
  r.

9
Flux Linkages outside of the wire
10
Flux Linkages outside, contd
11
Flux linkages inside of wire
12
Flux linkages inside, contd
Wire cross section
13
Line Total Flux Inductance
14
Inductance Simplification
15
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.

To determine the inductance of each conductor we
integrate as before. However now we get
some field cancellation
Creates a clockwise field
Creates counter- clockwise field
16
Two Conductor Case, contd
R
R
Rp
Direction of integration
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.
Right Current
Left Current
17
Two Conductor Inductance
18
Many-Conductor Case
Now assume we now have k conductors, each with
current ik, arranged in some specified
geometry. Wed like to find flux linkages of each
conductor.
Each conductors flux linkage, lk, depends upon
its own current and the current in all the
other conductors.
To derive l1 well be integrating from conductor
1 (at origin) to the right along the x-axis.
19
Many-Conductor Case, contd
Rk is the distance from con- ductor k to point c.
Wed 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.
At point b the net contribution to l1 from ik ,
l1k, is zero.
20
Many-Conductor Case, contd
21
Many-Conductor Case, contd
22
Symmetric Line Spacing 69 kV
23
Birds Do Not Sit on the Conductors
24
Line Inductance Example
Calculate the reactance for a balanced 3f,
60Hz transmission line with a conductor geometry
of an equilateral triangle with D 5m, r
1.24cm (Rookconductor) and a length of 5 miles.

25
Line Inductance Example, contd
26
Line Inductance Example, contd
27
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.
Book coverhas a transmissionline withtwo
conductorbundling
28
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
29
Bundled Conductors, contd
30
Bundled Conductors, contd
31
Inductance of Bundle
32
Inductance of Bundle, contd
33
Bundle Inductance Example
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?
34
Transmission Tower Configurations

• The problem with the line analysis weve done so
  far is we have assumed a symmetrical tower
  configuration. Such a tower figuration is seldom
  practical.

Therefore in general Dab ? Dac ? Dbc
Unless something was done this would result in
unbalanced phases
Typical Transmission Tower Configuration
35
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.
36
Line Transposition Example
37
Line Transposition Example
38
Transposition Impact on Flux Linkages
a phase in position 1
a phase in position 3
a phase in position 2
39
Transposition Impact, contd
40
Inductance of Transposed Line
41
Inductance with Bundling
42
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