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

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EE 369POWER SYSTEM ANALYSIS

• Lecture 7
• Transmission Line Models
• Tom Overbye and Ross Baldick

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Announcements

• For lectures 7 to 10 read Chapters 5 and 3.
• HW 6 is problems 5.14, 5.16, 5.19, 5.26, 5.31,
  5.32, 5.33, 5.36 case study questions chapter 5
  a, b, c, d, is due Thursday, 10/6.
• Power plant tour is 10/6.
• Instead of coming to class, go to UT power plant.
  Turn in homework at beginning of tour.
• Homework 7 is 5.8, 5.15, 5.17, 5.24, 5.27, 5.28,
  5.29, 5.34, 5.37, 5.38, 5.43, 5.45 due 10/20.

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Transmission Line Models

• Previous lectures have covered how to calculate
  the distributed series inductance, shunt
  capacitance, and series resistance of
  transmission lines
• That is, we have calculated the inductance L,
  capacitance C, and resistance r per unit length,
• We can also think of the shunt conductance g per
  unit length,
• Each infinitesimal length dx of transmission line
  consists of a series impedance rdx j?Ldx and a
  shunt admittance gdx j?Cdx,
• In this section we will use these distributed
  parameters to develop the transmission line
  models used in power system analysis.

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Transmission Line Equivalent Circuit

• Our model of an infinitesimal length of
  transmission line is shown below

L
Units on z and y are per unit length!
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Derivation of V, I Relationships
L
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Setting up a Second Order Equation
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V, I Relationships, contd
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Equation for Voltage
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Real Hyperbolic Functions
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Complex Hyperbolic Functions
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Determining Line Voltage
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Determining Line Voltage, contd
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Determining Line Current
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Transmission Line Example
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Transmission Line Example, contd
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Transmission Line Example, contd
Squares and crosses show real and reactive power
flow, where a positive value of flow means flow
to the left.
Receiving end
Sending end
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Lossless Transmission Lines
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Lossless Transmission Lines
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Lossless Transmission Lines
If load power P gt SIL then line consumes VArs
otherwise, the line generates VArs.
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Transmission Matrix Model

• Often we are only interested in the terminal
  characteristics of the transmission line.
  Therefore we can model it as a black box

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Transmission Matrix Model, contd
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Equivalent Circuit Model
To do this, well use the T matrix values to
derive the parameters Z' and Y' that match the
behavior of the equivalent circuit to that of the
T matrix. We do this by first finding the
relationship between sending and receiving end
for the equivalent circuit.
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Equivalent Circuit Parameters
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Equivalent circuit parameters
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Simplified Parameters
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Simplified Parameters
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Three Line Models
The long line model is always correct. The
other models are usually good approximations for
the conditions described.
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Power Transfer in Short Lines

• Often we'd like to know the maximum power that
  could be transferred through a short transmission
  line

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Power Transfer in Lossless Lines
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Limits Affecting Max. Power Transfer

• Thermal limits
• limit is due to heating of conductor and hence
  depends heavily on ambient conditions.
• For many lines, sagging is the limiting
  constraint.
• Newer conductors/materials limit can limit sag.
• Trees grow, and will eventually hit lines if they
  are planted under the line,
• Note that thermal limit is different to the
  steady-state stability limit that we just
  calculated
• Thermal limits due to losses,
• Steady-state stability limit applies even for
  lossless line!

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Tree Trimming Before
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Tree Trimming After
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Other Limits Affecting Power Transfer

• Angle limits
• while the maximum power transfer (steady-state
  stability limit) occurs when the line angle
  difference is 90 degrees, actual limit is
  substantially less due to interaction of multiple
  lines in the system
• Voltage stability limits
• as power transfers increases, reactive losses
  increase as I2X. As reactive power increases the
  voltage falls, resulting in a potentially
  cascading voltage collapse.