EE369POWER SYSTEM ANALYSIS

Lecture 4Power System Operation, Transmission Line

ModelingTom Overbye and Ross Baldick

1

Reading and Homework• For lectures 4 through 6 read Chapter 4

– We will not be covering sections 4.7, 4.11, and 4.12 in detail,– We will return to chapter 3 later.

• HW 3 is Problems 2.43, 2.45, 2.46, 2.47, 2.49, 2.50, 2.51, 2.52, 4.2, 4.3, 4.5, 4.7 and Chapter 4 case study questions A through D; due Thursday 9/17.

• HW 4 is 2.31, 2.41, 2.48, 4.8, 4.10, 4.12, 4.13, 4.15, 4.19, 4.20, 4.22, due Thursday 9/24.

• Mid-term I is Thursday, October 1, covering up to and including material in HW 4.

2

Development of Line Models

• Goals of this section are:

1) develop a simple model for transmission lines, and

2) gain an intuitive feel for how the geometry of the transmission line affects the model parameters.

3

Primary Methods for Power Transfer

The most common methods for transfer of electric power are:

1) Overhead ac2) Underground ac3) Overhead dc4) Underground dcThe analysis will be developed for ac lines.

4

Magnetics Review

Magnetomotive force: symbol F, measured in ampere-turns, which is the current enclosed by a closed path,

Magnetic field intensity: symbol H, measured in ampere-turns/meter:– The existence of a current in a wire gives rise to an

associated magnetic field. – The stronger the current, the more intense is the

magnetic field H.Flux density: symbol B, measured in webers/m2

or teslas or gauss (1 Wb /m2 = 1T = 10,000G):– Magnetic field intensity is associated with a magnetic

flux density.5

Magnetics Review

Magnetic flux: symbol measured in webers, which is the integral of flux density over a surface.

Flux linkages measured in weber-turns.– If the magnetic flux is varying (due to a changing

current) then a voltage will be induced in a conductor that depends on how much magnetic flux is enclosed (“linked”) by the loops of the conductor, according to Faraday’s law.

Inductance: symbol L, measured in henrys:– The ratio of flux linkages to the current in a coil.

,

,

6

Magnetics Review• Ampere’s circuital law relates magnetomotive

force (the enclosed current in amps or amp-turns) and magnetic field intensity (in amp-turns/meter):

d

= mmf = magnetomotive force (amp-turns)

= magnetic field intensity (amp-turns/meter)

d = Vector differential path length (meters)

= Line integral about closed path (d is tangent to path)

e

e

F I

F

I

H l

H

l

l

= Algebraic sum of current linked by 7

Line Integrals•Line integrals are a generalization of “standard” integration along, for example, the x-axis.

Integration along thex-axis

Integration along ageneral path, whichmay be closed

Ampere’s law is most useful in cases of symmetry, such as a circular path of radius x around an infinitelylong wire, so that H and dl are parallel, |H|= H is constant,and |dl| integrates to equal the circumference 2πx.

8

Flux Density•Assuming no permanent magnetism, magnetic field intensity and flux density are related by the permeability of the medium.

0

0

= magnetic field intensity (amp-turns/meter)

= flux density (Tesla [T] or Gauss [G])(1T = 10,000G)

For a linear magnetic material:

= where is the called the permeability

=

= permeability of freesr

H

B

B H

-7pace = 4 10 H m

= relative permeability 1 for airr

9

Magnetic Flux

2

Magnetic flux and flux density

magnetic flux (webers)

= flux density (webers/m or tesla)

Definition of flux passing through a surface is

=

= vector with direction normal to the surface

If flux

A

A

d

d

B

B a

a

density B is uniform and perpendicular to an area A then

= BA10

Magnetic Fields from Single Wire

• Assume we have an infinitely long wire with current of I =1000A.

• Consider a square, located between 4 and 5 meters from the wire and such that the square and the wire are in the same plane.

• How much magnetic flux passes through the square?

11

Magnetic Fields from Single Wire• Magnetic flux passing through the square?

• Easiest way to solve the problem is to take advantage of symmetry.

• As an integration path, we’ll choose a circle with radius x, with x varying from 4 to 5 meters, with the wire at the center, so the path encloses the current I.

12

Direction of H is givenby the “Right-hand” Rule

Single Line Example, cont’d

4

0 0

5 04

70

5

22

2 10 2T Gauss

2

(1 meter)2

5 5ln 2 10 ln

2 4 4

4.46 10 Wb

A

Id xH I H

x

IB H

x x xI

dA dxx

II

H l

B

For reference, the earth’s

magnetic field is about 0.6 Gauss

(Central US)

13

H is perpendicularto surface of square

Flux linkages and Faraday’s law

i=1

Flux linkages are defined from Faraday's law

d= , where = voltage, = flux linkages

dThe flux linkages tell how much flux is linking an

turn coil:

=

If flux links every coil then

N

i

V Vt

N

N

14

Inductance

• For a linear magnetic system; that is, one where B = H,

• we can define the inductance, L, to be the constant of proportionality relating the current and the flux linkage: = L I,

• where L has units of Henrys (H).

15

Summary of magnetics.

16

d (enclosed current in multiple turns)

(permeability times magnetic field intensity)

(surface integral of flux density)

(total flux li

(c

nked by tur

urrent in a conductor)

e

A

F I

dA

I

N N

H l

B H

B

n coil)

/ (inductance)L I