indir.pdf

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EPMN 302: ELEMENTS OF

POWER SYSTEMS

Dr. Mostafa Elshahed1

OBJECTIVES

Learn a brief history of the electrical power systems construction

List and describe different components of electrical power systems

Learn the per-unit system calculations

Draw the single line and impedance diagram

Define the transformers and synchronous machines representations

Calculate the transmission lines parameters

Formulate the various transmission lines models

Establish the transmission lines performance

Calculate the symmetrical faults variables

Learn the principles of symmetrical components transformations

Calculate the unsymmetrical faults variables

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TRANSMISSION LINES

PARARMATERS3

To develop models of transmission lines, we first need to determine the TL parameters.

TYPES OF CONDUCTORS

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ACSRAAACAAC

ACAR

ACSR conductors are most common.

A typical Al. to St. ratio is about 4 to 1.

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LINE RESISTANCE

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We’ll assume that the current density within the wire is uniform and that the wire is solid with a radius of r.

We’ll assume that the current density within the wire is uniform and that the wire is solid with a radius of r.

2rA

The variation of resistance of metallic conductors with temperature is practically linear over the normal range of operation.

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LINE RESISTANCE, CONT’D

Changes is about 8% between 25C and 50C

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LINE RESISTANCE, CONT’D

Uniform distribution of current throughout the cross

section of a conductor exists only for DC.

Because ac current tends to flow towards the surface of

a conductor, the resistance of a line at 50 Hz is slightly

higher than at dc.

An increase in frequency causes non uniform current

density.

This phenomenon is called skin effect.

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INDUCTANCE OF A SINGLE WIRE

The lines of magnetic flux form closed loops linking the

circuit, and the lines of electric flux originate on the

positive charges on one conductor and terminate on the

negative charges the other conductor.

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INDUCTANCE OF A SINGLE WIRE

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Variations of the current in the conductors causes a

change in the number of lines of magnetic flux linking

the circuit.

Any change in the flux linking a circuit induces a

voltage in the circuit which is proportional to the rate of

change of flux.

The inductance of the circuit relates the voltage induced

by changing flux to the rate of change of current.

INDUCTANCE OF A SINGLE WIRE

To do this we need to determine the wire’s total flux

linkage, including:

Flux linkages within the wire (Internal)

Flux linkages outside of the wire (External)

We’ll assume that the current density within the wire is

uniform and that the wire is solid with a radius of r.

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INDUCTANCE OF A CONDUCTOR DUE TO FLUX

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Internal

External

SINGLE PHASE TWO CONDUCTOR LINE

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r‘ = Geometric mean radius

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SINGLE PHASE TWO CONDUCTOR LINE

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INDUCTANCE OF COMPOSITE CONDUCTOR LINES

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INDUCTANCE OF COMPOSITE CONDUCTOR LINES

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Numerator ≡ Geometric Mean Distance (GMD) ≡ Dm

Denominator ≡ Geometric Mean Radius (GMR) ≡ Ds

INDUCTANCE OF COMPOSITE CONDUCTOR LINES

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EXAMPLE

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EXAMPLE

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EXAMPLE

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1 mile = 1.6 km = 1600 m

INDUCTANCE OF 3 PHASE LINES WITH SYMMETRICAL

SPACING

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Ds ≡ r’ for single conductor

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INDUCTANCE OF 3 PHASE LINES WITH UNSYMMETRICAL

SPACING

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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.

INDUCTANCE OF 3 PHASE LINES WITH UNSYMMETRICAL

SPACING

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Ds ≡ r’ for single conductor

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CONDUCTOR BUNDLING

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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 220 kV lines, three for 500 kV and four for 765 kV.

Two Bundles Three Bundles Four Bundles

Three Phase TL with Four Bundles

CONDUCTOR BUNDLING

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Reduced reactance is the other equally important

advantage of bundling.

Increasing the number of conductors in a bundle reduces

the effects of corona and reduces the reactance.

The reduction of reactance results from the increased

GMR of the bundle.

Increase the capacity of high voltage transmission lines.

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CONDUCTOR BUNDLING

25Dsb ≡ GMR of a Bundled Conductor (Bundles)

Ds ≡ GMR of a Single Conductor (a Bundle) ≡ r’

LINE INDUCTANCE EXAMPLE

Calculate the reactance for a balanced 3, 60Hz, transmission line with a conductor geometry of an equilateral triangle withD = 5m, r = 1.24cm = 0.0124 m.

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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

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1.24 10 m ' 9.67 10 m

9.67 10 0.25 0.25 ( 2 0.25)

0.12 m (ten times bigger than !)

5ln 7.46 10 H/m

2 0.12

Bundling reduces inductance.

b

a

r r

R

r

L

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LINE INDUCTANCE EXAMPLE

Dsb

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

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Dsb

Dsb

Deq

ACSR TABLES Because ACSR conductors are stranded, actual resistance,

inductance, and capacitance needs to be determined from tables.

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Inductance and Capacitance assume a geometric mean distance Deq of 1 ft.

GMR is equivalent toeffective radius r’