ps2103unit 5 new-part 1 lpep

7/26/2019 Ps2103unit 5 New-part 1 Lpep

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UNIT V - COMPUTATION OF POWER

SYSTEM TRANSIENTS - EMTP

PART - 1

Digital computation of line parameters: why lineparameter evaluation programs? salient featuresof mtline; constructional features of that affecttransmission line parameters; elimination ofground wires bundling of conductors

REFERENCE:R Ramanu!am" Computational ElectromagneticTransients Modelling, Solution Methods and

Simulation, #$ #nternational %ublishing &ouse"New Delhi" '()* R Ramanu!am


2/30

Why Line Parameter Evalati!n Pr!"ram#

+he series resistance and inductance and shuntcapacitance parameters of an overhead transmissionline are evenly distributed throughout the length ofthe line

Cannot be considered lumped elements; parametersare also functions of the fre,uency of the currents and

voltages on the line and cannot be regarded asconstants

-ine parameters needed for simple and detailedstudies of transmission line phenomena:.implest:

)constants are /( &0 1 2( &0 positive se,uenceseries impedance and shunt capacitance needed forpower flow studies

' For short3circuit 0ero se,uence parameters alsoneeded +hese parameters are available from tablesin handboo4s or can be calculated using simpleformulas

Detailed studies:

R Ramanu!am


3/30

5i6 .teady3state problems at power fre,uency:

Current and voltage unbalances on long anduntransposed lines

7ne,ual current distribution within bundledconductor

Calculation of induced voltages and currents in ade3energi0ed line which runs parallel with one ormore energi0ed lines 5very important for thesafety of maintenance crews6

5ii6 .teady3state problems at higher fre,uencies:

&armonics on &8DC lines

#nterferences in parallel communication lines orcarrier communication on power lines

5iii6 +ransients problems: %ropagation of switching and

lightning surges and means of reducing them withprotective gaps" surge arresters" insertion ofresistors in circuit brea4ers"etc

R Ramanu!am


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Note: -ine parameters can be measured after they havebeen built 9ut some of them are needed at the designstage itself Even after the line is built it is easier to

compute them

Salient Featre$ !% a ty&i'al line &arameter evalati!n

&r!"ram Can accept input data in 9ritish or metric units

Calculates the parameters of an overheadtransmission line based on the conductorcharacteristics and tower3conductor geometry

ccepts any combination of line circuits" phaseconductors" and ground wires up to a total of )((

conductors and /( phases

9undled conductors can be specified as individualsubconductors or by describing the bundlearrangement

+reats C.R conductor as a tubular conductor

Can handle continuous and segmented ground wires

R Ramanu!am


5/30

utputs the lineimaterepresentation for short lines 5eg" length )((((1f6.hort3line sections can then be cascaded to simulate

longer lines +he nominal3= model should not be usedfor transient simulations

C!n$tr'ti!nal Featre$!%That A%%e't Tran$mi$$i!n

Line Parameter$

5i6 .pacing between ad!acent towers:

+he conductor profile can be considered regarded asR Ramanu!am


6/30

@odified )A1)(1'()/

a parabola for tower spans B /(( metermetres

a catenary for /(( B spans B '((( metermetres

elastic string for spans '((( metermetres5+ypical spans: *(( 48 towers: *(( m" ''( 48: A/( m" )A' 48: AA/ m6

+he height that is used in the computation of seriesinductance and @a>well


7/30

5ii6 +ype of conductor: nonmagnetic or magnetic

RecallGMR

r =e

1

4

5/'6

holds good only for solid" round nonmagneticconductor at low fre,uency

concept of @R was originally developed fornonmagnetic conductors at power fre,uency at which

s4in effect can be ignored @R defined as geometric mean distance between

all elements in the conductor cross section area if thearea were divided into infinite number of e,ualinfinitesimal elements &ence" meaning purelygeometric

E,5/'6 is e>tended to magnetic conductor as

GMR

r =e

r4 5/A6

where ris the relative permeability of the conductor

&ence" when a magnetic conductor is considered andwhen s4in effect is ta4en into account the geometricmeaning is lost R Ramanu!am


8/30

Relationship between reactance at unit distance 5feetor metermetres6 which are is normally available from

handboo4s" and @R: E,5Aa6foot

1

XA' =

0

2 ln

5/*6

5iii6 round wire segmentation: continuous or segmented5a6 Continuous ground wire

+he e,uation for induced voltage in the ground wired Vgdx

=ZgA'

IA+ZgB'

IB+ZgC'

IC+Z'Ig 5//6

#f tower and tower footing resistance are ignored then

V"( ( andd Vg

dx =0

Ig=ZgA

'IA+ZgB

'IB+ZgC

'IC

Z' 5/26

Note: #n general" the mutual impedance between thephases and the ground wire will not be e,ualnumerator of E,5/26 will not add up to 0ero givingrise to a non0ero I"produced bypositive se,uence

phase currentsa circulating current as shown inFigure /'

R Ramanu!am


9/30

I"will produce additional losses and it reflects as anincrease in positive se,uence resistance +heformulas found in handboo4s will not account for thisincrease+he effect of tower and tower footing resistance willnot be appreciable if they are e,ual at all towers

5b6 .egmented ground wire

.ee Figure /A .egmentation resorted to reduce the losses due to

circulating ground current +he ground wire provides shielding to the

transmission line against lightning bserve that within a segmentation interval the

ground wire is insulated at all towers e>cept atone tower 5mar4ed C in Figure /A6 where it isgrounded

+he segmented ground wire can become acontinuous ground wire when a lightning flash

I"

Figure /':Circulating current in the ground wire

I"

R Ramanu!am


10/30

bridges the segmentation gaps at the towers atthe end of the segmentation interval

5iv6 the line parameters are also influenced by:

type of conductor 5solid" single or bundled andstranded6

transposed line or flat line configuration

fre,uency of operation

)

The!ry *+ehin, Line Parameter Evalati!n Pr!"ram

Consider specific e>ample of a double3circuit three3phaseline with a twin3bundle phase conductor and a singleground conductor 5total )A conductors6 shown in Figure/*

.egmentation interval

C

Figure /A:.egmented ground wire; at tower C" the ground wire is grounded

R Ramanu!am


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@a!or steps involved in building the series impedance andshunt admittance matrices:

) Form the matri> 5series impedance1shunt admittanceor @a>well


12/30

* Compute the se,uence ,uantities by applyingsymmetrical component transformation to the matri>

obtained at the end of .tep A

%ictorial illustration:

Line Parameter$ %!r In,ivi,al Pha$e C!n,'t!r$

Serie$ Im&e,an'e Matri

E,5AA26 e>panded form for the )A3conductor line:

)A 22)'

22)')A

Figure //:#llustration of steps in computing impedance 1 potential coefficient matri>; 5a6:after .tep ) IJ physicalK or I% physicalK is obtained; 5b6: after .tep' ground wire is eliminated;5c6: after .tep A IJ phaseK or I% phaseK is obtained; 5d6 after .tep * IJsymK or I%symK is obtained

5d65c65b65a6

R Ramanu!am

R Ramanu!am


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Z11'

Z12'

Z1,13'

I1I2

I13

Z21'

Z22'

Z2,13'

Z13,1'

Z13,2'

Z13,13'

[

d V1

dx

d V2

dx

d V13dx

]

=

5/G6

r in compact form

[d Vdx]=[ Zph!si"a#' ][I]

where Vi is the voltage of conductor i measured fromground and Iiis the current in the conductor i

IJLphysicalK symmetric" comple> functions of fre,uency"ie""


14/30

IJLphysicalK M IRL56K O!I-L56K5/6

JLii M RLii O ! -Lii is the series self impedance perunit length of the loop formed by conductor and groundreturn

JLi4M RLi4 O ! -Li4is the series mutual impedanceper unit length between conductors i and 4 #t determines

the longitudinal voltage induced in conductor i when acurrent flows in conductor 4 and vice versa

+he resistive term RLi4 represents the phase anglebetween induced voltage and inducing current #t is due tothe presence of the ground

+he e>pressions for diagonal and off3diagonal elementsincluding Carson


15/30

Zi&' =$ R i&' +%(2 104 ln i&d i& +$ Xi&' ) ohms14m

5/)(6

whereRiis the ac resistance of conductor i in ohms14m

hi is the average height of the conductor i above groundin metermetres

Di4is the distance between conductor i and image ofconductor 4 in metermetres

di4is the distance between conductor i and conductor 4 in metermetres@Riis the geometric mean radius of conductor i in

metermetresPRL and PQL are the Carson


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whereD M 'hiin metermetres for self impedance and e,uals

Di4for mutual impedance and S 5M )1T6 is the earth

resistivity in hm3metermetre

#nfinite series for various values of a:5i6 a UV" ie" very low earth resistivity whichimplies a perfectly conducting earth:

For this case PR and PQ become 0ero

5ii6 a B /:

$ R'=4 (104

{

8*1 acos++*2 [("2lna ) a2cos2+++ a2 sin2+ ]

+*3a3cos3 +d 4 a

4cos 4+

*5a5cos5 ++*6[ ( "6ln a) a6cos6+++ a6sin6+ ]

+*7 a7

cos7+d8 a8

cos8+( ((

}

5/)'6

$ X'=4 ( 104

{

1

2( 0.6159315ln a )

+*1 a cos +d 2 a2

cos 2+

+*3 a3

cos3 +*4[ ( "4ln a ) a4 cos 4 +++ a4 sin 4 + ]+*5 a

5cos5 +d6 a

6cos6 +

+*7 a7

cos7 +8

[ ("8ln a

)a

8cos8 +++ a8 sin8 +

](( (

}

5/)A6

R Ramanu!a


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*i=*i2 signi(i+2) with the starting values:

b)MW'12 for odd subscriptsb'M)1)2 for even subscripts

"i="i2+1

i+

1

i+2 with the starting valuec'M )A2/HA)/

d i=

4( *i

5/)*6with sign M X ) changing after each * successiveterms" ie"sign M O) for i M )"'"A"* and sign M 3)for i M /"2"G" etc

5iii6 For a / the following finite series is used

$ R'=

4 104

2 (cos +

a

2cos2 +

a2 +

cos3 +

a3 +

3cos5 +

a5

45cos7 +

a7 )

5/)/6

$ X'=

4 104

2 (cos +

a

cos3 +

a3 +

3cos5 +

a5 +

45cos7 +

a7 )

R Ramanu!am

R Ramanu!am


18/30

5/)26For mutual impedance correction terms thetrigonometric functions are calculated from the

geometrycos + i&=

hi+h&i&

, sin +i&=x i&

i& 5/)G6

Following points are worth noting regarding thecorrection terms:

5i6 t power fre,uency only few terms are needed inthe infinite series of e,uations 5/)'6 and 5/)A65ii6 t higher fre,uencies and wider spacings such as

those encountered power3communication lineinterference problems" the parameter YaZbecomeslarger and more terms must be ta4en into account

5iii6 nce Carson is the inverse of @a>well 5@%C@6

+he latter is computed first in line parameterevaluation program 5mt-ine in @icrotran pac4age6since the e>pressions for @%C@ elements are

R Ramanu!am


19/30

similar to the reactive part of the impedance matri>elements e>cept for Carson is of the form

R Ramanu!am


20/30

-11'

-12'

-1,13'

.1. 2

.13

-21'

-22'

-2,13'

-13,1'

-13,2'

-13,13'

[V1V2

V13

]=

[ V]=

[-ph!si"a#

'

][ . ]

5/''6

)Line Parameter$ %!r Eivalent Pha$e C!n,'t!r$

btained from series impmatri>1 @%C@ by eliminating[ and bundling:

)Eliminati!n !% 2r!n, Wire$

Following assumptions are to be made :

d V13

dx =0 5/'A6

R Ramanu!am


21/30

V13( ( 5/'*6

E,5/'A6 used in E,5/G6 and E,5/'*6used in E,5/''6

8alidity of the assumptions: E,5/'A6" is valid if [ grounded at distances

wavelength so that there is no potential build upbetween the grounded points t power fre,uencyand with normal spacings 5'/( \ A(( metermetres6"

this is valid 5wavelength is around /((( \ 2((( 4m6

For higher fre,uencies which are encountered fore>ample" in lightning surge studies this assumptionmay not be valid and we cannot eliminateshouldincludethe ground wire

E,5/'*6 [ is continuously grounded 3 notstrictly true [ grounded at towers and 0ero onlyat those points &owever" continuous groundinggives substantially accurate results for fre,uenciesless than that which ma4es the spacing between thetowers e,ual to ,uarter wavelength For normal

spacings and power fre,uency" this fre,uency is '/(4&0

E,5/'A6 in the last entry of E,5/G6

R Ramanu!am


22/30

I13=1

Z13,13' (Z13,1

'I1+Z13,2

'I2+( ((+Z13,12

'I12 )

5/'/6

+he first entry of E,5/G6:

d V1dx

=Z11'

I1+Z12'

I2+( ((+Z1,13'

I13 5/'26

E,5/'/6 in E,5/'26 d V1

dx =Z11,red/"ed

'I1+Z12, red/"ed

'I2+( ((+Z1,13,red/"ed

'I13

5/'G6

where

Z11,red/"ed'

=Z11'

Z1,13

'Z13,1

'

Z13,13'

Z12,red/"ed

' =Z12'

Z1,13'

Z13,2'

Z13,13'

5/'6etc

[e can similarly reduce the other entries in E,5/G6 and

we will get a matri> e,uation of the form [d Vdx]=[ Zgred/"ed' ][I] 5/'H6where


23/30

[ d Vdx]=[d V1

dx

d V2

dx ( ((

d V12

dx]0

[I]=[I1I2 ( ( ( I 12]0

and [ Zgred/"ed' ] is a )'>)' series impedance matri> whoseelements are given by e>pressions similar to E,5/'6

#f there are YpZ phase conductors and YgZ groundconductors"

[[d Vpdx]

[d Vg

dx

]]=[[Zpp

' ] [ Zpg' ] [ Zgp' ][ Z' ] ] [

[Ip ] [Ig ] ]

5/A(6

.etting [ d Vgdx] to 0ero and eliminating ground wires wewill get the reduced set of e,uations of the form [d Vpdx]=[ Zgred/"ed' ][Ip ] 5/A)6

Elimination of [ in the @%C@ is similar[ V]=[-gred/"ed' ][ . ] 5/A'6

R Ramanu!am

R Ramanu!am


24/30

)+n,lin" !% Pha$e C!n,'t!r$

Consider a simpler e>ample of tower3conductorconfiguration shown in Figure /2

#f we bundle the conductors say" ) and ' of phase " thenthe following conditions are satisfied: I1O I4 M IA 5/AA6

d V1

dx =

d V2

dx=

d VA

dx 5/A*6

.imilar relations hold good for conductors of phases 9

and C

2/C

A') *

9

Figure /2:.i>3conductor e>ample to illustrate bundling


25/30

Following steps are to be followed to bundle theconductors

)[rite down the voltage e,uations of individualconductors of phase :

d V1dx

=Z11'

I1+Z12'

I2+( ((+Z16'

I6

d V2dx =Z21

'

I1+Z22'

I2+(( (+Z26'

I6 5/A/6

'Replace I1 by IA in the above e,uations and correctfor the additional terms introduced due to I4:

Z

Z11'

12'I

d V1dx

=Z11'

IA+

Z

( 22'Z21' )I2+ (( (+Z26

'I6

d V2dx

=Z21'

IA+

+he above e,uations can be rewritten asd V1dx

=Z11'

IA+Z13'

I3+( ((+Z16'

I6+Z12' '

I2 d V2

dx =Z21

'IA+Z23

'I3+ (( (+Z26

'I6+Z22

''I2

5/A26


26/30

whereZ

Z12' '=

6;Z

Z22

' ' =

6 5/AG6

+he remaining four e,uations after replacing I1by IAin them becomed V3

dx =Z31

'IA+Z33

'I3+ (( (+Z36

'I6+Z32

''I2

d V6

dx =Z61

'IA+Z63

'I3+(( (+Z66

'I6+Z62

''I2

5/A6

whereZ

Z32' '=

6 etc

AReplace I3by I+and I)by ICin e,uations 5/A26 and5/A6

d V1dx

=Z11'

IA+Z13'

IB+Z15'

IC+Z12' '

I2+Z '14'

I4+Z '16'

I6 d V2

dx =Z21

'IA+Z23

'IB+Z25

'IC+Z22

' 'I2+Z '24

'I4+Z '26

'I6

d V3dx

=Z31'

IA+Z33'

IB+Z35'

IC+Z32''

I2+Z '34'

I4+Z '36'

I6

d V4

dx =Z41

'IA+Z43

'IB+Z45

'IC+Z42

' 'I2+Z '44

'I4+Z '46

'I6

d V

5

dx =Z51' IA+Z53' IB+Z55' IC+Z52'' I2+Z '54' I4+Z '56' I6

d V6

dx =Z61

'IA+Z63

'IB+Z65

'IC+Z62

' 'I2+Z '64

'I4+Z '66

'I6

5/AH6

R Ramanu!am


27/30

whereZ

Z14' '=

6 "Z

Z24' '=

6 etc

and

Z

Z16' '= 6 "

Z

Z26' '= 6 etc

* Replace Row ' by Row ' \ Row )" Row * by Row *\ Row A and Row 2 by Row 2 \ Row / and push thenew rows to the end

d VAdx

=Z11'

IA+Z13'

IB+Z15'

IC+Z12' '

I2+Z '14'

I4+Z '16'

I6 d VB

dx =Z31'

IA+Z33'

IB+Z35'

IC+Z32' '

I2+Z '34'

I4+Z '36'

I6

d VC

dx =Z51

'IA+Z53

'IB+Z55

'IC+Z52

''I2+Z '54

'I4+Z '56

'I6

( Z21' ' IA+Z23'' IB+Z25' ' IC+Z22' ' 'I2+Z '24' ' I4+Z26' ' 'I6 ( Z41' ' IA+Z43' ' IB+Z45' ' IC+Z42' ' 'I2+Z '44' ' I4+Z46' ' 'I6 ( Z61' ' IA+Z63'' IB+Z65' ' IC+Z62' ' 'I2+Z '64' ' I4+Z66' ' 'I6

5/*(6

where we have used E,5/A*6 and

Z

Z21' ' =

6"Z

Z23

' '=

6 etc

andZ

Z22

' ' '=

6 "Z

Z24

' ' '=

6 etc

E,5/*(6

[[d V

dx][0 ]]=[ [ZRR

' ][ Z' ] [ Z1R' ] [ Z11' ]][

[IR ] [I1]]


28/30

5/*)6

where the subscript YRZ denotes ,uantities to be retained"

YEZ denotes ,uantities to be eliminated and[d Vdx]=[

d VA

dx

d VB

dx

d VC

dx]0

; [IR ]= [IAIBIC]0 [I1 ]= [I2I4I6 ]

0

Z11'

Z13'

Z15'

Z31'

Z33'

Z35'

Z51'

Z53'

Z55'

[ ZRR

'

]=

Z12

''Z14

' 'Z16

' '

Z32' '

Z34''

Z36' '

Z52' '

Z54''

Z56' '

[ Z' ]=

Z21' '

Z23''

Z25' '

Z41' '

Z43' '

Z45' '

Z61' '

Z63' '

Z65' '

[ Z1R' ]=

Z22

'' 'Z24

' ' 'Z26

' ' '

Z42' ' '

Z44' ' '

Z46' '

Z62' ' 'Z64

' ' 'Z66' ' '

[Z1' ]=


29/30

/ Eliminate the currents IIEK from the first set ofe,uations" E,5/*)6:From the second set of e,uations of E,5/*)6

IIEK M 3 IJ EEK3)IJ ERKIIRK 5/*'6

E,5/*'6 in first set of e,uations of E,5/*)6

[d Vdx]=[ ZRR 2 red/"ed' ][IR ] 5/*A6where

IJ RR" reducedK M IJ RRK 3 IJ REK IJ EEK3)

IJ ERK5/**6 is the desired series impedance matri> for the

e,uivalent phase conductors

Note:

) #t is not be numerically efficient to computedirectly IJ RR" reducedK as per E,5/**6 since itinvolves matri> inversion Either triangularfactori0ation or auss3]ordan elimination methodcan be used to compute the reduced matri> auss3]ordan method can also be used for matri>

inversion and for screening factor matri>calculation +he screening factor matri> ise>plained in the ne>t section and auss3]ordanmethod is e>plained in Chapter )(


30/30

' +he reduced impedance matri> where only thee,uivalent phase conductors are retained" IJ RR"reducedK" is denoted by IJ phaseK in Figure // 5c6

A +he method e>plained for obtaining the seriesimpedance is valid for obtaining the @%C@ fore,uivalent phase conductors if IJ phaseK is replaced

by I% phaseK" 3IdV1d>K by IVK and IIK by I5K

ps2103unit 5 new-part 1 lpep

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