power system slide07

7/21/2019 Power System Slide07

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INTRODUCTION

All transmission lines in a power system exhibit the

electrical properties of resistance, inductance,capacitance andconductance.

Inductance and capacitance are due to the effectsof magnetic and electric fieldsaround theconductor.

These parameters are essential for thedevelopment of the transmission line modelsusedin power system analysis.

The shunt conductanceaccounts for leakagecurrents flowing across insulators and ionized

pathways in the air. The leakage currents are negligiblecompared to

the current flowing in the transmission lines andmay be neglected.


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RESISTANCE

Important in transmissionefficiency evaluation and

economic studies.

ignificant effect! "eneration ofI2Rloss in

transmission line.

! #roducesIR$type voltage dropwhich affect voltage regulation.


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RESISTANCE

The dc resistance of a solid roundconductorat a specified temperature is

%here &

' conductor resistivity ()$m*,

l ' conductor length (m* + andA ' conductor cross$sectional area (m*

dc

lR A

=


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RESISTANCE

-onductor resistance isaffected by three factors&$

re/uency (0skin effect1*

piraling

Temperature


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RESISTANCE

re/uency ! kin 2ffect %hen ac flows in a conductor, the current

distribution is not uniformover the

conductor cross$sectional area and the

current density is greatest at the surfaceof the conductor.

This causes the ac resistance to be

somewhat higher than the dc resistance.

The behavior is known as skin effect.


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RESISTANCE

The skin effect is where alternatingcurrent tends to avoid travel through

the center of a solid conductor, limiting

itself to conduction near the surface.

This effectively limits the cross$

sectional conductor area available to

carry alternating electron flow,

increasing the resistance of thatconductor above what it would

normally be for direct current


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RESISTANCE


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RESISTANCE

kin effect correction factoraredefined as

%here

3 ' A- resistance + and3o' 4- resistance.

O

R

R


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RESISTANCE

piraling or stranded conductors, alternate layers

of strands are spiraled in oppositedirections to hold the strands together.

piraling makes the strands 5 ! 6longer than the actual conductor length.

4- resistance of a stranded conductor is5 ! 6 larger than the calculated value.


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RESISTANCE

Temperature

The conductor resistance increases as temperatureincreases. This change can be considered linearover the range of temperature normally encounteredand may be calculated from &

%here&35' conductor resistances at t5in 7-

3' conductor resistances at t in 7-

T ' temperature constant (depends on theconductor material*

22 1

1

T tR R

T t

+=

+


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RESISTANCE

The conductor resistance is bestdetermined from manufacturer1s

data.

ome conversion used incalculating line resistance&$

5 cmil ' 8.9:;x59$


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Resistivity & Temparature

Constant of Conductor Metals

=aterial 20C T

3esistivity at 9>- Temperature -onstant

m10-8 cmil/ft >-

-opper

Annealed 5.; 59.?; ?


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RESISTANCE

2xample&$A solid cylindrical aluminum

conductor 8km long has an area of

??:,


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RESISTANCE

Answer (a*

( ) ( )

( )

25

8 3

4

6

2.8 10 25 10

336,400 5.076 10

4.0994 10

l km

l

R A

=

=

=

=


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RESISTANCE

Answer (b*

( )

50

50 2020

6

6

228 504.0994 10

228 204.5953 10

C

C CC

T t

R R T t

+

= ++

=

+=


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RESISTANCE

2xercise 5A transmission$line cable consists

of 5 identical strands of

aluminum, each ?mm in diameter.The resistivity of aluminum strand

at 97- is .x59$)$m. ind the

897- ac resistance per km of the

cable. Assume a skin$effectcorrection factor of 5.9 at 89@z.


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RESISTANCE

2xercise &$

A solid cylindrical aluminum conductor

558km long has an area of ??:,


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RESISTANCE

2xercise ?A transmission$line cable consists

of 58 identical strands of

aluminum, each .8mm indiameter. The resistivity of

aluminum strand at 97- is

.x59$)$m. ind the 897- ac

resistance per km of the cable.Assume a skin$effect correction

factor of 5.958 at 89@z.


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INDUCTANCE

A SIN!"E CONDUCTOR

A current$carrying conductor produces amagnetic field around the conductor.

The magnetic flux can be determined by

using the right hand rule.

or nonmagnetic material, the inductanceLis the ratio of its total magnetic flux linkage

to the currentI, given by

where'flux linkages, in %eber turns.LI

=


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INDUCTANCE

A SIN!"E CONDUCTOR

or illustrativeexample, consider along round conductorwith radius r, carryinga currentIas shown.

The magnetic fieldintensityHx, arounda circle of radiusx, isconstant and tangentto the circle.

2

xx

IH

x=


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INDUCTANCE

A SIN!"E CONDUCTOR

The inductance of the conductorcan be defined as the sum of

contributions from flux linkages

internal and external to theconductor.


23/89

#lu$ "in%ae


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INDUCTANCE

A SIN!"E CONDUCTOR


25/89

INDUCTANCE

A SIN!"E '(ASE "INES


26/89


27/89

INDUCTANCE

)*'(ASE TRANSMISSION "INES


28/89

What and How to Calculate:- Fint , Fext G FH

F5, FG FH

F55 , F5 G F H "=3H

"=4H


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INDUCTANCE

A SIN!"E CONDUCTOR

IT23AF I4J-TA-2! Internal inductance can be express as

follows&$

! %here

o' permeability of air (


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INDUCTANCE

A SIN!"E CONDUCTOR

I4J-TA-2 4J2TD 2KT23AF

FJK FILA"2

! 2xternal

inductance

between to point

4,and 4

5can be

express as

follows&

7 2

1

2 10 ln /extD

L H mD

=

INDUCTANCE


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INDUCTANCE


A single phase lines consist of asingle current carrying line with a

return line which is in opposite

direction. This can be illustrated as&

INDUCTANCE


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INDUCTANCE


Inductance of a single$phaselines can be expressed as

below with an assumption

that the radius of r5'r,'r.7 7 2

int

1

7 7 7

1

7 74

1

4

7

0.25

110 2 10 ln /

2

1 110 2 10 ln / 2 10 ln /

2 4

12 10 ln ln / 2 10 ln ln /

2 10 ln /

ext

DL L L H m

D

D DH m H m

r r

D De H m H m

r re

D

H mre

= + = +

= + = + = + = +

=

SE"# AND MUTUA"


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SE"# AND MUTUA"

INDUCTANCES

The series inductance per phase can be express in terms

of self$inductance of each conductor and their mutualinductance.

-onsider the one meter length single$phase circuit infigure below&$

! %here F55 and F are self$inductance and the mutual inductanceF5

SE"# AND MUTUA"


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SE"# AND MUTUA"

INDUCTANCES

( )

( )

( )

D

xD

xL

DxL

erxL

ILLID

xer

xIL

ILL

ILL

mHD

xer

xL

mHD

x

er

xL

1ln102

1

ln102

1ln102

1ln102

1ln102

1ln102

/1

ln1021

ln102

/

1

ln1021

ln102

77

12

7

12

25.01

7

11

112111

7

25.0

1

7

111

222212

112111

7

25.0

2

7

2

7

25.0

1

7

1

=

=

=

=

=

+==

+=

=

+=

+=

SE"# AND MUTUA"


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SE"# AND MUTUA"

INDUCTANCES

F55, F,,and F5,can be expressed asbelow&$

7

11 0.251

7

22 0.25

2

7

12 21

1

2 10 ln

12 10 ln

12 10 ln

L re

L

r e

L LD

=

=

= =

SE"# AND MUTUA"


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SE"# AND MUTUA"

INDUCTANCES

lux linkage of conductor i

ijDIerIx

n

j ijj

iii

+= =

1

ln

1

ln1021

25.0

7

INDUCTANCE


37/89

INDUCTANCE


ymmetrical pacing! -onsider 5 meter length of a three$phase

line with three conductors, each radius r,

symmetrically spaced in a triangular

configuration.

INDUCTANCE


38/89

INDUCTANCE


Assume balance ?$phase currentIaM I

bM I

c' 9

The total flux linkage of phase a

conductor

ubstitute for IbM I

c'$I

a

++=

DI

DI

erIx

cb

a

aa

1ln

1ln

1ln102

25.0

7

25.0

7

25.0

7ln102

1ln

1ln102

=

=

er

DIx

DI

erIx

a

aa

a

aa

INDUCTANCE


39/89

INDUCTANCE


Because of symmetry, Na'Nb'Nc

The inductance per phase per

kilometer length

kmmHre

Dx

IL /ln102

25.0

7

==

INDUCTANCE


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INDUCTANCE


Asymmetrical pacing! #ractical transmission lines cannot maintainsymmetrical spacing of conductors because of

construction considerations.

! -onsider one meter length of three$phase line with

three conductors, each with radius r. The conductor

are asymmetrically spaced with distances as shown.

INDUCTANCE


41/89

INDUCTANCE


! The flux linkages are&$

++=

++=

++=

2313

25.0

7

2312

25.0

7

1312

25.0

7

1ln

1ln

1ln102

1ln

1ln

1ln102

1ln

1ln

1ln102

DI

DI

reI

DI

DI

reI

DI

DI

reI

bacc

cabb

cbaa

INDUCTANCE


42/89

INDUCTANCE


! or balanced three$phase currentwithI

aas reference, we have&$

a

o

ac

a

o

ab

aIII

IaII

== == 120

240 2

INDUCTANCE


43/89

INDUCTANCE


Thus La, Lband Lccan be foundusing the following e/uation&$

++==

2 3

2

2 5.0

1 2

7 1ln

1ln

1ln1 02

D

a

r eD

a

I

L

b

bb

++==

1312

2

25.0

7 1ln

1ln

1ln102

D

a

D

a

r eI

L

a

a

a

++==

25.0

2313

27 1ln

1ln

1ln102

reDa

Da

IL

c

c

c

INDUCTANCE


44/89

INDUCTANCE


Transpose Fine! Transposition is used to regain symmetry

in good measures and obtain a per$phaseanalysis.

INDUCTANCE


45/89

INDUCTANCE


This consists of interchanging the phaseconfiguration every one$third the length so thateach conductor is moved to occupy the nextphysical position in a regular se/uence.

Transposition arrangement are shown in the figure

INDUCTANCE


46/89

INDUCTANCE


ince in a transposed line eachphase takes all three positions,

the inductance per phase can be

obtained by finding the average

value.


47/89

0.25

12 13

7

0.25

23 12

0.25

13 23

7

0.25

12

3

1 1 1

ln 1 240 ln 1 120 ln

2 10 1 1 1ln 1 240 ln 1 120 ln

3

1 1 1ln 1 240 ln 1 120 ln

2 10 1 1 13ln ln ln

3

a b cL L LL

re D D

re D D

re D D

re D D

+ +=

+ + = + + +

+ + +

=

R R

R R

R R

23 13

312 23 137

0.25

1ln

2 10 ln

DD D D

re

=


48/89

ince in a transposed line each phase

takes all three positions, the

inductance per phase can be obtained

by finding the average value.

3

cba

a

LLL

L

++

=


49/89

oting a M a ' $5

Inductance per phase per kilometer length

( )

( )25.0

3

1

1323127

3

1

132312

25.0

7

132312

25.0

7

ln102

1ln1ln102

1ln

1ln

1ln

1ln3

3

102

=

=

=

re

DDD

DDDre

DDDreL

( )kmmH

re

DDDL /ln2.0

25.0

3

1

132312

=


50/89

What and How to Calculate:- Fint , Fext G FH

F5, FG FH

F55 , F5 G F H "=3H

"=4H

Inductance of Composite


51/89


Conductors

n !"#l$#ti%n %f in&$ct#nc!, '%li& (%$n&c%n&$ct%(' !(! c%n'i&!(!&. H%!"!(, in

*(#ctic#l t(#n'mi''i%n lin!', 't(#n&!&

c%n&$ct%(' #(! $'!&.

C%n'i&!( # 'in+l!-*#'! lin! c%n'i'tin+ %f

t% c%m*%'it! c%n&$ct%('x#n& #' '%n

in i+$(! 1. ! c$((!nt inxi'I(!f!(!nc!&

int% t! *#+!, #n& t! (!t$(n in i' I.



52/89


Conductors

C%n&$ct%(x c%n'i't %f ni&!ntic#l 't(#n&' %('$c%n&$ct%(', !#c it (#&i$' (x.

C%n&$ct%(y c%n'i't %f mi&!ntic#l 't(#n&' %(

'$c%n&$ct%(', !#c it (#&i$' (.

! c$((!nt i' #''$m!& t% ! !$#ll &i"i&!&

#m%n t! '$c%n&$ct%('. ! c$((!nt *!(

't(#n&' i'I/ninx#n&I/miny.



53/89


Conductors

&

#

c

&

n

#

c

m

x y


54/89

nncnbnax

mnmncnbnan

n

nanacabx

mamacabaaa

a

nanacabx

mamacabaa

a

amacabaa

anacabx

a

DDDr

DDDDn

nIL

DDDr

DDDD

nnIL

DDDr

DDDDI

or

DDDDm

I

DDDrn

I

...

...ln102

/

...

...ln102

/

...

...ln102

1ln...

1ln

1ln

1ln102

1ln...

1ln

1ln

1ln102

7

7

7

7

7

==

==

=

++++

++++=


55/89

...

)...)......

)...)......

/ln102

2

7

xnnbbaa

nnnnbnaanabaax

mnnmnbnaamabaa

x

x

rDDD

where

DDDDDD!R

DDDDDD!Dwhere

mH!R

!DL

===

=

=

=


56/89

!MR of +undled Conductors

&

& &

&

&

&

&

&

5xt(# i+ "%lt#+! t(#n'mi''i%n lin!' #(!$'$#ll c%n't($ct!& it $n&l!& c%n&$ct%('.

6$n&lin+ (!&$c!' t! lin! (!#ct#nc!, ic

im*(%"!' t! lin! *!(f%(m#nc! #n& inc(!#'!'

t! *%!( c#*#ilit %f t! lin!.


57/89

!MR of +undled Conductors

4 316 42/1

3 29 3

09.1)2

)

dDdddDD

b"ndleor#"bcond"ct$o"rthe$or

dDddDD

b"ndleor#"bcond"ctthreethe$or

##

b

#

##

b

#

==

==

dDdDD

b"ndleor#"bcond"cttwothe$or

DDDDDD!R

##

b

#

nnnnbnaanabaax

==

=

4 2)

)...)......2

Inductance of T,ree*p,ase


58/89

Inductance of T,ree p,ase

Dou-le Circuit "ines

7 t(!!-*#'! &%$l!-ci(c$it t(#n'mi''i%nlin! c%n'i't' %f t% i&!ntic#l t(!!-*#'!

ci(c$it'. % #ci!"! #l#nc!, !#c *#'!

c%n&$ct%( m$'t ! t(#n'*%'!& itin it +(%$*

#n& it (!'*!ct t% t! *#(#ll!l t(!!-*#'!lin!.

C%n'i&!( # t(!!-*#'! &%$l!-ci(c$it lin!

it (!l#ti"! *#'! *%'iti%n' #11c1-c22#2.



59/89



c1

#1

#2

1 2

c211

22

33

9:; !t!!n !#c *#'! +(%$*

4 22122111

422122111

422122111

cacacacaAC

cbcbcbcb%C

babababaA%

DDDDD

DDDDD

DDDDD

=

=

=



60/89



! !$i"#l!nt 9:; *!( *#'! i' t!n

3AC%CA% DDD!D =

imil#(l, 9:< %f !#c *#'! +(%$* i'

214 2

21

214 2

21

214 2

21

)

)

)

ccb

ccb

&C

bb

b

bb

b

&%

aa

b

aa

b

&A

DDDDD

DDDDD

DDDDD

##

##

##

==

==

==

!(! i' t! +!%m!t(ic m!#n (#&i$' %f

$n&l!& c%n&$ct%('.

b

#D



61/89



! !$i"#l!nt 9:< *!( *#'! i' t!n

3&C&%&AL DDD!R =

! in&$ct#nc! *!(-*#'! i'

mH!R

!DL

L

x /ln102 7

=

INDUCTANCE


62/89


Ouestion ! !$i"#l!nt 9:; *!( *#'! i' t!n

3AC%CA% DDD!D =

The "=3-of each phase is similar to

the "=3F, with the exception that rbisused instead of

b

#D

This will results in the following e/uS

21

21

21

cc

b

C

bb

b

%

aa

b

A

Drr

Drr

Drr

=

=

= 3C%AC rrr!R =

E##ECT O# EART( ON T(E


80/89

CA'ACITANCE

or isolated charged conductor the

electric flux lines are radial andorthogonal to cylindrical e/uipotentialsurfaces, which will change the effectivecapacitance of the line.

The earth level is an e/uipotentialsurface. Therefore flux lines are forcedto cut the surface of the earthorthogonally.

The effect of the earth is to increase thecapacitance.

E##ECT O# EART( ON T(E


81/89

CA'ACITANCE

But, normally, the height of theconductor is large compared to thedistance between the conductors, andthe earth effect is negligible.

Therefore, for all line models used forbalanced steady$state analysis, theeffect of earth on the capacitance canbe negligible.

@owever, for unbalance analysis suchas unbalance faults, the earth1s effectand shield wires should be considered.

MA!NETIC #IE"D


82/89

INDUCTION

Transmission line magnetic fieldsaffect obUects in the proximity of the

line.

#roduced by the currents in the line.

It induces voltage in obUects that have

a considerable length parallel to the

line (2x& telephone wires, pipelines

etc.*.

MA!NETIC #IE"D


83/89

INDUCTION

The magnetic field is effected bythe presence of earth return

currents.

There are general concernsregarding the biological effects of

electromagnetic and electrostatic

fields on people.

E"ECTROSTATIC


84/89

INDUCTION

Transmission line electric fields affectobUects in the proximity of the line.

It produced by high voltage in thelines.

2lectric field induces current inobUects which are in the area of theelectric fields.

The effect of electric fields becomes

more concern at higher voltages.

E"ECTROSTATIC


85/89

INDUCTION

#rimary cause of induction to vehicles,

buildings, and obUect of comparable size.

@uman body is effected to electric

discharges from charged obUects in the

field of the line. The current densities in human cause by

electric fields of transmission lines are

much higher than those induced by

magnetic fieldsV

CORONA


86/89

CORONA

%hen surface potential gradientexceeds the dielectric strength ofsurrounding air, ionization occurs inthe area close to conductor surface.

This partial ionization is known ascorona.

-orona generate by atmosphericconditions (i.e. air density, humidity,

wind*

CORONA


87/89

CORONA

-orona produces power loss andaudible noise (2x& radio

interference*.

-orona can be reduced by&! Increase the conductor size.

! Jse of conductor bundling.

Revie.


88/89

Revie.

Transmission Fine #arameters&! 3esistance

kin effect

! Inductance ingle phase line

? phase line e/ual W une/ual spacing

! -apacitance ingle phase line

? phase line e/ual W une/ual spacing

! -onductance eglected -orona

Revie.


89/89

Revie.

2ffect of 2arth on the-apacitance

=agnetic ield Induction

2lectrostatic Induction -orona

power system slide07

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