EE%20369%20POWER%20SYSTEM%20ANALYSIS

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EE 369POWER SYSTEM ANALYSIS

• Lecture 5
• Development of Transmission Line Models
• Tom Overbye and Ross Baldick

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Reading

• For lectures 5 through 7 read Chapter 4
• we will not be covering sections 4.7, 4.11, and
  4.12 in detail.
• HW 4 is 2.31, 2.48, 4.8, 4.10, 4.12, 4.13, 4.15,
  4.19, 4.20, 4.22, 4.24, 4.25 (assume Cardinal
  conductor and look up GMR in Table A.4), due
  Thursday 9/22.
• HW 5 is Problems 4.1, 4.3, 4.6, 4.26, 4.33, 4.36,
  4.38, 4.49, 5.2, 5.4, 5.7, 5.9 due Thursday
  9/29.
• Mid-term I is Thursday, October 13, covering up
  to and including material in HW 5.

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Substation Bus
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Inductance Example

• Calculate the inductance of an N turn coil wound
  tightly on a toroidal 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
• 3) Radius of each turn is negligible compared to R

Circular path G of radius R within the iron core
encloses all N turns of the coil and hence links
total enclosed current of Ie NI. Since the
radius of each turn is negligible compared to R,
all circular paths within the iron core have
radius approximately equal to R.
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Inductance Example, contd
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Inductance of a Single Wire

• To develop 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 is solid with a
  radius of r.
• In fact, current density is non-uniform, and
  conductor is stranded, so our calculations will
  be approximate.

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Flux Linkages outside of the wire
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Flux Linkages outside, contd
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Flux linkages inside of wire
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Flux linkages inside, contd
Wire cross section
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Line Total Flux Inductance
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Inductance Simplification
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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 D.

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
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Two Conductor Case, contd
Key Point Flux linkage due to currents in each
conductor tend to cancel out. Use superposition
to get total flux linkage.
Left Current
Right Current
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Two Conductor Inductance
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Many-Conductor Case
Now assume we now have n conductors, with the
k-th conductor having current ik, and 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.
For example, to derive the flux linkage for
conductor 1, l1, well be integrating from
conductor 1 (at origin) to the right along the
x-axis.
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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.
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Many-Conductor Case, contd
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Many-Conductor Case, contd
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Symmetric Line Spacing 69 kV
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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.

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Line Inductance Example, contd
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Line Inductance Example, contd
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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.
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Bundled Conductor Flux Linkages

• For the line shown on the left,
• define dij as the distance between
• conductors i and j.
• We can then determine lk for conductor k.
• Assuming ¼ of the phase current flows
• in each of the four conductors in
• a given phase bundle, then for conductor 1

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Bundled Conductors, contd
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Bundled Conductors, contd
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Inductance of Bundle
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Inductance of Bundle, contd
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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?
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Transmission Tower Configurations

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

• Therefore in
• general Dab ?
• Dac ? Dbc

• Unless something
• was done this would
• result in unbalanced
• Phases.

Typical Transmission Tower Configuration
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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.
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Line Transposition Example
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Line Transposition Example
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Transposition Impact on Flux Linkages
a phase in position 1
a phase in position 3
a phase in position 2
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Transposition Impact, contd
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Inductance of Transposed Line
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Inductance with Bundling
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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.

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Inductance Example