Analysis of Unbalanced Faults

ANALYSIS ANO PROTETiON OF POI{ER sYSTS.rs couRsE

ANALYSIS

OF

UNEALAT{CED FAULTS

BY

c H or{c

ANALYSIS OF UNBALANCED FAULTS

I

1. INTRODUCTION \In a ba'lanced three-phase systemr €dch of the three phases of\*rpart_ of.,the system wl'11 have currents and voltages which are equaland 120" dlsplaced with respect to each other. To malntaln ba]ancedoperatlonr €dch item of system plant must be symmetrlcal: l.e. haveidentlcal impedances in each 'llne, equal mutual impedances betweenphases and ground' and equal shunt admittances to ground. This isthe case for machines and transformers, and lt is ilso va]ld for'l 'lnes i f they are fu1'ly transposed.

Three phase faults wlth symmetrica'l fault inrpedances leave the systemin ba'lanced operatlon. Such faults can be analysed using the simp'lesingle phase representatlon. Howeverr these faults are iare.

The majorlty of fau'lts occur between one I ine and groundr or betweentwo 'llnes and ground. These are asymm€trlcal or unba'lanced faults.They arlse from 'lightnlng dlscharges and other overvo'ltages whlchinltlate f'lashovers foJlowed by power arcs i or they may arlse frommechanlcal causes such as blrds on overhead llnes or mechanlcaldamage to cab'les, etc. Another type of unbalanced fau'lt which ls of.interest are the open clrcuit faults. They can arise from brokenconductorr ma]operatlon of slngle phase sw{tchgear or the operatlonof fuses.

During unba'lanced fau'ltsr the symmetry of the system is'lost and theslngle phase representatlon used for three phase ba'lanced fau'lts no'longer appl ies.

and introduced

n-phase systemssequence or

2.

2.L

SYMMETRICAL COMPONENTS METHOD

Fortescr.le dlscovered a property of unba'lanced phasorsthe method of symrntrical conponents.

n phasors may be resolved lnto (n-1) sets of ba1ancedof different phase sequence and one set of zero-phaseunl-dl retlonal phasor systan.

Conslder n-dlmenslonal system of phasors.

V =V-+V-+V^+aalalaJVb = Vbl * YOZ * Vb3 *

... * Van

... + V.DN

V = V - + V ^ + V - + ... + VnnInZnSnn

Where Vurr Vbl, etc.,

n-phase system.

YaZ, YbZ' etc.,

n-phase system.

are phasors of the flrst set of balanced

Phasors are slngle spaced.

are phasors of the second set of balanced

Phasors are double spaced.

And so on.

Vun, Vbn, etc.,

system.

are phasors of the uni-di retional phasor

Vzt

V<Va,

FIRST SETBALANCED

Vv+

t V".,A

POSITIVESEOUENCE

Vq+

SECOND SET OFBALANCED PHASORS

NEGATIVESEOUENCE

THIRD SET OFBALANCED PHASORS

Va'VO' V".2.2

FOURTH SET OF FIFTH SET OFBALANCED PHASORS ZERO SEOUEI\CE PHASORS

Now conslder an unbalanced three phase system.

v" = v.l + Yaz + va3

vb=vbr*Yoz*vu3

V =V.+V^+V-c cI -cz c5

Va.z

4/";'ZERO

SEOUENCE

Take for examp'le an unbalanced 5-phase systan.

Va, Vo-r-

Vd-

Vc,/

OF

PHASORS

vd+

Vnu

Vcr y's, {u> V.-

v" = var + Yaz * vuo

vb= nzvaL**v.2*vao

v" = *val + n2v u2 + v.0

where o< = I .O /L2Ao

It 1s convenient to deletE subscrlpt rar for the symrnetrlcalcomponents.

V.=VI+V2+VO

vb = *tr, **V2 * Vo

V" = *Vl **2vZ * vo 3

Add equatlons l, 2 and 3

V.*Vb+V"=3VO

.'. YO = L/, (V. + Vb + Vc)

Multlply equatlon ? Uy-o. and equatlon 3 by n2 and add the resultlngequatlons to equatlon 1,

Vu **Vb +o.zv" = 3Vl

Multlply equatlon ? oy_ o.2 and equatlon 3 by o. and add the resu'ltingequatlons to equatlon 1r

.,Va+o<'Vb**V"=3Yz

.'. Y, = Ilr(V. + n"r+c.V")

Three unba'lanced phasors have been

Choose tat phase as the reference p

reso1ved into nlne phasors.

hase and replace Va3 by Va'.,/

/

. . Y, = L/, (V. + *Vb * o.zv")

Equatlons I to 6 can be re-written ln matrlx form.

v0

vt

Yz

va

vu

vc

r11.2Isa

.2Ioeo<

I=J

vo

vt

Yz

ItII o.2

,2l*o{-

111,2I*a<

L J o<

lrtL J o'

I ot otz

va

V.D

Vc

-----------

and 8 respectlvely asRe-wrlte matrlx equatlons

[,,] = [ n ] [ ,,]

Iu,] =[ ^l-'[ur]Where Vp =

vs=

IO

{

Exampl e

Resol ve thesymmetrlcal

!=a

V.=b

[=c

A-I =

fo'llowlng 3-phaseconponents.

L /Si:

1.5 /-9oo

0.5 /+tZOo

phase components

sequence components

unbalanced vol

Va

tages lnto thel r

I3

2.3

v4

ltaa d{Y v J

Solution :

Vu = I + j0

Vb = 0 - jl.5

V" = 0.5 (-0.5 +j0.g6)

= -0.25 + j0.433

= L/3 (-0.549 +J0.31g) = 0.211 /L56o

V"O = L/3 (V. + Vb + Vc)

= L/3 [ (1 + J0) + (0 - Jt.S) + (-0.25 +j0.433) ]= L/3 (0.75 - J1.067) = 0.434 /_5So

vbr = Jt.l,

J = L /2400 x 0.e65 /If = 0.965 /zssovbz = *v.l

= L /L20o x 0.211 /LsOo = 0.211 /Z7Oo

vbo = vao

= 0.434 /-SSo

vcl = *val

= L /L20o x 0.965 /!So = 0.965 l35o

Ycz= Jr",= L /?4Qo x 0.21L /LSO1 = O.2LL nglo

v.o = Vao

= Q.434 /-55o

Figure I shors the sequence components of the phase vortages.

vul = 1/3 [ vu + ..vo + ..2v" J

= L/3 [ (1 + j0) + (-0.5 +J0.866)(0 - J

(_0.5 _j0.s66) (_0.25 +J0.433) l= L/3 (2.798 + JO.7S) = 0.965 E-

Y"Z= I/3 [ v. + o..2VO **V" ]= L/3 t (1) + (-0.5 -j0.866)(-J1.5) +

(_0.5 +j0.966) G0,25 +J0.433) l

(

L

1.5) +

V.zy't,./\./\

\\

\Vr\

\

V.-

I

Ur, /

\V^o "\

{

{(^z'\-

--'-.f-,4u9t z- \

\Vcr

t-2

V,.oVtoV-o

vo,.( FtG. I

SY/^^A€TRrcALCOly.PONE.rr-ITS\

Vuo

a

vu

Vc

Ia

I.D

I

3. SYMMETRICAL COFIPONENT TRANSFOFMATION

N

FIGURE 2

Take a set of symmetrrca'r three phase rmpedances (equaily spaced,I:lt,

transposedr etc. ) carryng unbal.n"!J-;t,;;" current-s r., -io ano

l,/e may write the followlng equatlons.

Vu=ZrIu*ZrIb*ZrI"vb=ZrIu*Z.Ib*ZrI"V"=ZrI.*ZrIb*ZrI"

Where Z, = sel f irpedance per phase

Z , = mutual lnpedance between anym pnaie ili;--0r, in matr lx form

zt z^ z^

z^ 7, z^

z^ z^ zt

V

Resol v lng V and I phasors into thelr symnetrlcal corponents.

c

llt.2Io(4

,2Ioad.

zzzsmmzzzmsmzzzmms

vo

vt

Yz

r11,2Io(

,2la(

Io

Itlz

Page ?

Re-a

;:/

vzl

I

I

zo

0

0

I]

rl

rrange

=li i:i'' i:':^[; "l L1 i:

zs

zm

zm

+[ ] llz

s

zm

zm

* 22^ Z,

'zn z,

-zn z^

'0

't

2

0

II

I

I,

0

0

, *ir,0

0

I,.,,

Itrz

Io

Itlz

zm

zm

zs

zm

zm

zs

zm

zs

zm

Vo

vt

vz

vo

vt

Yz

I=J

zs

z5

z5

I

t

+22m

+<z5

+*2+ nzz

m

I e,7S-m

+22m

* nzm * nzz,

* o-2Zn * o.Zs

io(

2o(

I24

(z

(zs - zn)

0 (zs - zn)

11

io

ItTz

0

zt

0

0

0

zz

l{here

Io

rt

1,2

z, = zr'z,Z2=Zr-Z^ZO=Zr*2Zn

Thereforer 1f the sv?tT rs symnetrrcal rn its normar state thesymmetrlcar comoonent impedance becomes oi.goi.i (equatron rr) and,therefore, tso'riteo sequlnce"n"tro"r.s ars o6iaineo wttn impedancbszL' z, and zo. These ih;;;-^;ilorrs wilr become interconnetedwhen an unbarance such as a faurt or unbaranced roadlng rsintroduced. The manner of interconnetron rlrl oepend on the naconstrarnts: r.e. the addrtl0nar system connectlons.

9t^^ ! n''5v .-

4, PLANT II,IPEDANCE DATA

4.L For statlc networks 1.e.negative sequence impede non-notatlng pl:ntr:. the positive andimieoanie,oi-in"-t.ulli.ont"s are tn" ruT:_.

- Tnese !."'in" reakaoetransmiision ;i;";;;;. r dr.r€rs and the normal ;;;;"_;ipJoun"" of the

i:i:ff:1ii !ru"::ff:,";.il";iTl",lill^::r,e ci rcuits isearthr earth wires o.-.uole sheaths. y6l Tlu"nce currents tnroughseneral ly sreater than

-;;;-p;;;i;:."^,ln:^1ero sequence rmpedance i sbeins,',ui1r ;i ti""".JJi #'i;l:"uihd,::d:iiu",'"q;Jice impeodrc€,sequence varue in the caie or overhead .,;;":1""r the positive4 '2 For transl:.:*I.r: i f ._zero sequence currents have an ava i rable path;i3:i:' I}"1; ;l;il :lllrin':; '""-ti"'ilunun" reactance rn eachpa rticula r windr.s,

_?i-i;:^ii::ri:r;:[:,irtii_ i; ;;3;1,;':iT",."

; :ff [". "il fi ' r" r'. i!"iJr T% jn il ;j']:":,'r Jo J

j.:j,:;. : Jr r r :,.consider the.transformer e1u]valent crrcull

:t^,.gure 3 0verreaf. Themagnetlsing imped.n"" i, is or ,.,"i.o"' of 200ffi, compared to theleakage impedance ,.r, * Z.l. of about l0%. Thereforer magnetisingfmpedance

:ln bg ignored and the transformlli :"'l'Ji], 1., n{iiiu"'iJo,i!il';:il:il:'oi.l

ll,.i:gTff:j:,..J,-lp '1s"

t

FIGURE 3. TRANSFORfVIER EOIJIVALENT OIR0UIT

Thereforer_consi!1r zeno sequence clrcuittmpedance Zr. The mode oi-".oinltion of

',, = iJJ[:# i,il3J:;""

Z.l, = secondary windingreak age .lmpedanc6

Z^ = magnetislngimpedance

_of transformer as a seriesZ, to the external ci rcuit

In the zero seo

]*l!i"g;'io'fitu"n"e networkr although the leakage rmpedance is

iff H :i#::;;;,gfi;;;:iui 1{ iii ril#;ijtil;riii'r; ;th re+ph ase banl

;! er r ;; il ; ;tx :,::,J I i :' i^ 3 ^:iir'iii*ji:ni;:l

; :ff , l;,:ff::,,rarse and can oe lgn-orJ'.r"in'ir.," pori;;;; H; negative sequencenetworks. In thre6-r iro .o.""ty?" i.anrro.rJr], howeverr the zerosequence flux must be ""rpi"ij'in"orgn-ti; ;ii or tanr. owinq tothe hlgh reluctance of the riux patr,,-r"io-rJqu"n"" rnagnetlsrnJrr,rpedance is of trre oroerii'"ir v, ryw];-4;#:. ,o*"u"r, this-is;lll'.nJ;f,l'ii;i,l; ?:j::i*i:; ,n ;;; ;;ff; studies, particularly

is determlned bv.taking account of eachconnetlon o,. oih"*i;: ii"iloJro. wlndlng arrangement and its

Page I I

Imaglnau Hnks ?ar and fbr (see Figure 4) are used to derrve theconnetlons. If zero sequence currents can frow rnto and out of awindlng' for example a solldly earthed star winotng, the wrndlng:iJ3Jl:r rs conncted to the external crrcuri,"that rs ilnk ,ar rs

i,a-H

Zero Sequence EqulvalentCl rcult Connetlons

The zero sequence lrpedance of a

3Zn. The reason for th ls can be

bel ow

a-

3lo

2,._

?-g

Transformer Connetlons

neutral earthlng lnpedance Zn lsreadlly understood from Flgure 5

If zero sequence currents can c1rc_urate rn the wlndrng wrthoutflowing ln the externar "r.uri, ror erarf r""J o"rta wrndrngr theiJ?ol:n"l:::Jltt ls dlrecttv-"oin*ted to the-zero bus, thai rs unk

Example I =

L

3Z'.'-Zero Sequence Clrcult

FIGURE

FIGURE 5. NEUTRAL EARTHING IMPEDANCE

At the neutral polnt the zero sequence currents I^ ln the threephases combine to glve 3In ln the netural earthin$ lnpedance. Thezero seguence voltage at the neutral polnt ls glv6n Oy

VO = L/, (V"n * Vbn * V"n) = Vn

But Y = 3I^Zn un.'. VO = 3IOZ.

.v^a_u_tO =i = 3Zn-0

\xample 2

I

Transformer Connetlons

3R.

Zero Seguence EquivalentCi rcult Connetions

The posltlve sequence {rpedance of synchronous machlnes ls the normalnrachlne reactance. There are three deflned values of posltlvesequence lmpedancesr name'ly the synchronous translent andsubtranslent lmpedances and they are used accordlng to whether steadystater translent or lnltlal short-clrcuit values of current arerequl red.

Un'l lke the non-rotatlng netrorksr the negatlve sequence lrpedance ofthe rotatlng plants ls not equal to the posltlve sequence lmpedance.It relates to mmf at synchronous speed travelllng ln the opposltedirectlon to the rotor. Its value ls usually less than that of thepositlve sequence lnpedance.

In the zero sequence netrork, the wlndlng connetlon and earthlngarrangement rust be consldered as for transformers. Any earthlngimpedance wll'l be seen by each phase and therefore the corretvoltages will be obtalned lf three tlmes the lmpedance value lsinc'luded ln the zero sequence netrork.

4.3

PaEe 13

Typ ica'l tu rbo-generator seguence

synchronous reactance

transl ent rectance

subtranslent reactance

negatlve sequence lrnpedance

zero sequence lmpedance

reactances are :

= I.0 p.u.

= 0.15 p.u.

= 0.10 p. u.

= 0.13 p.u.

= 0.04 p. u.

5. \

5.1

\CoNNECTIoN OF SEOUENCE NETWORKS TO REPRESENT UNBALANCED FAULTS(a) For any.grven faurt there..are srx quantrtres to be conslderedat the fau.tt polnt ; Va, vo,

-v",^Il;"1;, Ic, If any

three are knorn (provrded they are not aIr vortages or a'currents) on ff nny tvo are knqrn .no-tro others knorn to havea spectf tc re.lattonshtpr then . ."i.if"lvo and rr, r;;;;l;""5i u" estab.t rshed.shrp betreen vr, v, and

These reratronshrps are ca'ted the crrcult constralnts. :Frcm the

"1rc!rt constrarnts we can detennrne the manner rnwhich the isorated sequence netvorrs can oe lnterconnected.(b) The reratronstrlps are derJled rith phase f a, as the referencephase and the faurts are,setecteo t6"iI-out.n"ed reratlve tothe reference phase. Thrs vreiJ, ii"-Irrptert rnterconnetionof the sequence netror*s, ir tnirl; ;; done theinterconnetrons of the sequence netrorki requrre additionartransformatrons whrch are.achr?y9c dt-il; rntroductron of phaseshrftrng transforrners. ih!s wrti-oe'.ppir"nt rn the case ofsrmurtaneous faurts rhere_':; i;'l"ii""liiore for both thefaults to be symnretrtcal about th;-.!;;;errce phase.

Shunt Faults

L lne-to-ground faultsr 'llne-to-lJne faultsr .l lne-to-l lne to groundIil;*.and three phase rauits-arr raii-i;; ir,e catesory or shunt

(a) Figure 6 shors a systan rrth a faurt at F, The posrtrve,negatrve and zero lequence netrorrs oiltre system are shorn rnFtgure 7. The faurt 'terminar; ;";H; p""rtrve sequencenetrork are F, and N' r aod the corresoo'njing faurt tErmrnalsfor the negattve anOrzero-sequence netrorkTp-i3ll-Tliverv. ii i;'Jt tn"r" terminarr"rili ffr;

*t and Fo'rnterconnectron of the networks wril occur. In the derrvatronof sequence netrork rnterconnetronsr rt-is convenrent to showthe sequence netrorks as brocks ,i;;'r.rri termrna.rs F and Nfor external connetloni irrgu.e ei. 'rr'e

L-

(b) To derlve the system constrarnts. at the faurt termrna.rs, rt isconvenrent to imag.rne three snort "onJuctJlr'oi .""o rmpedanceconnected to the three trne coiouctori-;;-l;"";ornt of faurt(Ftgure 9). The t-erminur

"oiji,i1onp. irpor#-oi, ,^" dt f ferentiJ!=,ril, t'i;J;l'"?;;l!ilil; to ir,"s5-ii.giil..y reaos, in"tr.."nt, r.I io'.nd r". t be vu' vo ino v", Jn-i-ir,"""

=FIGURE 9

Faquf Porttf

5rxGLE Lr,rt€. p1aQf.44a o? T,+so A/d.CHIN1 SYST<=n

Pos,n/e S€C"€ryeg N o

z€eo s€QU€^Jce N6fdoek- oF srst€r\

Ftca.l $.Qq4J\JcE /.jie-,h,,<Ks oF- ag,EreD SysTa:V.

trro

No

N€GArrv6. S€4ru6nlcg rJefrpaek. ae SySf€'u.

ha .8 SeAuglc2 f,QurvAq$tYf NeT,;oR--a BUo*- s

f rom section (2,2) that

=Vl*V2*VO

=Q

., Vl*VZ+V0=0

Ground Fault 0n Phase

point:

=Q

=t =Q

= L/^ TJA

= L/^ (IJA

D:^a -1 (qJv ! J

They suggest that the

5.1.1 L ine To

At fault

V =0a

I. = f0c

We know

v.. it

But Va

We know

Io

But IO

io

Also, It

ItEquat'lon ssequence

fAt

I

2

from section,(2,2) that

= L/^ (I + I. + I )JADC

+< I, + o<.2I ) = I/^ IDCJA

?+ e<-Ib * *I.) = I./, I"I, = L/, (I"

= IZ= IO = 1/, I"

3 & 4 are the CIRCUITnetrorks are conneted

CONSTMINTS.in series.

5.L.2 L ine to Ground

At fault point

Fau'lt Impedance Z,Faul t Th rough

th at

I)c

n2rc ) = L/,

* *I") = L/,

V =17a -a-fT-r-A'b - tc - '

f/e know from sectJon (2.2)

TO=I/r(Iu+IO+

I

2

IO = L/, Iu,

S\mila11y,

since IO = I" = 0

* vz r vo

from constraint

* y2* VO = I.Zf3IO from equation 3

*YZ*VO=IOBZf)

I, = U, (Iu +ocIO +

T, = I/, (Iu + nrrO

II = 12 = fO = L/3 Iu

Ia

Ia

l'/e know

Va

ButV =a

V.IButI =a

vt

=Vl

I Z-at

Equations 3 and 4 suggests therl

fol 1ow ing interconnections.

3 Zf-

5 .1.3 L ine to

At fault

V,o

ia

I,D

Line Fault on phases rBt

Po int

-v c

-0+I =Qc

and rc r

l'/e know IO = L/3 (Iu +

Substituting equat ions

Io=oSimilarly,

into equation 4r

I,D

2

+J

and

)c

3

\I, = I/, (I. + *IUI, = I/, (Iu +aa21o

It+Ir=o

= L/3 ,o.-o.2) Ib

-L/S ( o<. -*2) Iu

* <2r )c

+o(J ) =

l'/e know V, = L/, (Vu + *Vb **2V")Substitutlng equation I into equation 7,

YL = I/3 (Va _ Vb)

simi'l arlv v, = r/3 (v. +o42yb + o<v") = l/, (va _ vb)

vl=v2From equations 5, 6 andnetrorks are in para.llelunconnected.

8, the positlve and negative sequencebut the zero sequ"n"""n"t ;d-;,

FiL

,\Jz

lt

---Eof c-l2s4 |

fsc4,.c.rcel T V"l*t€t.v.ok, | '. -,V=

\- ----/

= L/^ (IJA

= L/^ (IJA

= I/^ (I5a

+L +D

**Ib

* n?r.D

I") = Q

+ d,zrc)

+ o< Ic)

J: -6 J

5.1.4 L ine to L ine Fau]t on

At point of fault,

I =0a

I' + I = QbcV. -V =l.Z-'b 'c -b-f

Phases tBt and fCf Through Fault Impedance Z,

i2

3

Io

itlz

L/ z l*,-az) Io-r/z to<-<z) ro

Io=oI1 + 12 = o

'r{e know Ib = IO *Jtt +a1I2)

Substituting equation 4 in 5

Ib = (o<2 -o<)

Vb=VO*n'r,V" = VO +a<V, +

It* o<y2

n",

-oe) V, - (*2 -o<) YZ

and 6 lnto 7,

= (o<2 -*) vl - (o<2 -

VD

Substitutec

( o<'

ata

Equatlons 4

v" = b<'2

equatlon 3

-s<) ItZf x,) Y,

interconnections.

Fr-

Yt- VZ= ILZ.

and 8 suggest the followlng

tu,?o

NoN L

5.I.5 L ine to

At iau'l tV

I

=\r/=SL

-u

Line to Ci ound Fauit on

pc intPhases rB r and tCl

6b

A

VE

Ta-

Vo-

----/t/ -. Yl -

Yz =

l/ -v0-

L/^ (VJA

T/. (VJA

L/^ (VJa

?' o.-Vc) - I/3

**V") - I/3

V^) = I/^u5

+a<V,0

+ Jv.b

+v, +D

Va

va

va

V,=I V2 = VO = L/3 Va

Ir+Io=oIu=Ir+

From equaticn 3 and 4, it canare connQted in par-al.lel .be concluded that the sequence netv/orl:s

FD

r.lo,{,

a3 ! L _

5 . 1.6 L ine to L ine to Grounti Fau I tTh rough Fau I t IrnPedance Z,

At fault po'int :

I =0

v, = ! = (L + J ) Z_0cDct

on Phases fBl and tC'

a1

2

La-

Iu = IL* IZ * IO

I^=L/^ (I +I. +UJAD

IO*I"=3IO

V^=I/^ (V +V. +V)-UJADC

Y., = I/, (Vu +c<Vo +&,2V.)

Yr= I/, (Vu **tUO o*V.)

' ,, - t,rl - '2V^-V.=L/^(2V,fV,)=UTJDD

=

Subsitute equation 4 in 6

VO-Vt=3TOZ,

0

I)3

I)(^

h

= I/^ (I. +JD

t'=t (Va

1t-,3

L/z

+o<)

+o<)

5

L/z

L/z

V)D

D

+

IVa

fltLYa

2V. )b

+ (o-2

* (*2

(Va

(Va

V, ] :D

tr "!v, J -D

tv.-o-tJo

vo

(LD

+ I ) Z-ct

.. Vl=V0-TO3Zf

Equations 3r 5 and 7 suggest the fol'lowing jnterconnections.

F"

T5

FI,

x.o vzf

[,\_

!2^a / /gY v :4

5.I.7 fhevenln Equiva'lent Method

One method of reduclng or slnpllfylng a compllcated sequence netvorkis to derlve the Therrenln equlvalent ci rcuit. The rharenlnequ I va'l ent vo1tage of the posl ti ve sequence netrork i s the p re-f au 1tvoltage at the fault 'location F.,. The positlve sequence equlva1entinpedance ls the lnpedance seen'across the fault termlnals F, andN, when all thE sources arE de-actlvated.

It should be reallsed, however, that lf the Thevenin method ls usedto ca'lculate the varlous branch currentsr these values wi'l 'l representon]y the current changes ln each branch due to the fau'lt. Thepre'fau]t currents of the system has to be added to the fau'tt currentchanges to derive the total current for the fau]t condltion.

5.2 SERIES FAULTS (or Open C i rcu it Fau lts )

(a) F igure l0 shows a systen with an open cl rcuit pO. Thepositive' negatrve ind zero, sequ"n"" n"t*orks of theopen-cfrcuited systan are, shown-i"-FrgJ"" rr. un.rike the caseof shunt faurts,-the fauit t""niii"rr'ii"=rnr"rconnetron are pand e, therefore not lnvo.lyj.g ir,"'r"rI".f . The sequenceequivalent network blocki rriiure-rii"Jiir have terminais p ando for interconnectron, Termriar r,r-is aiso rnolcated in theblocks although it is-not-used for fnt!rJonnettons.

(b) The termlna'r. conditl0ns inposed by different open crrcuitfaurts w'rr be applied-u"rorr_points p ino Q on the three rineconductors (see Frgure 13). -The."io.L ii,'" ruu.rt termrnarcurrents w'l be r., ro and r" ;i;;;;;;'JJm p to o on the

three conductors, and the termlnal potentrars wrr be the

potentlal across p and O i.e. tv. - v.', vo _ vot

and n" - n"t.They wl'll be represented byVu,WO and { respecttvely.

u

FIGURE 13

Tra to 3'rlGL1- LrvL DTAG€r'./v\ oF Two y\AcH/^r€- Sysr614w,TH O?EN Ct?CutT huuT

44o g4u€;Nc.a Ne-TNoF*- oF sYsrau\

5,2,I Open Circuit Fau'lt on phase ,Ar

At fault point :

I^ = 0cl

Fb=%=o

vo

V1

u2

L/^ (fJA

L/ z tr4L/z (%

o

--Y*i Lv- l. un T.t t. , rr,

V<- t -trrE- I ._z Il'' -lta- |

?:,-s .!' ''J\' c-

the sequence networks

luQ"

..otIu=Ir

From equationsare conneted

tz12*

=

+

3in

* -b * f.) = I/:U"* vb * Jo") = L/rtf^*n\*1G)=L/r\

= % = L/3V"

Io=o

and 4 jt can be concluded thatpa ral 1el .

PoNl

.I-daAA,lui c

frPtfqt

5 ,?,2 Tr.ro

At

0pen

fault

L =I =Qt, \-

V=Q

lO = L/, (I. + Ib * I") = 1/, Iu

I, = L/, (Iu +alo + 4Ic) = L/,

I, = l/a (Iu + *r, **I" ) = L/,

II=TZ=fO=L/3Ia

fu=fl*tfr+tfr=Q

and rC I

Pov- , f.- ,1, '

vrl-- ls lV6'l- |

V- | YL +. itZ-'

-'+-lr<.

Ci rcuit Faults

po int :

on Phases tBt

q:! 4:

the sequence netrzorks

I

2

Ia

Ia

3

A

Frcn equations 3 and 4 it can be conc'luded thatare conneted in series.

4.e4ScQ*erJceNgf'rael

PaEe 27

5.3 SIMULTANEOUS FAULTS

The-range of faults we have consldered so far lnvolves only a slnglefau'lt at one fault location. symmetrlcal components can be used toanalyse two (or more) faults either ln the saine'locatlon or atdlf ferent 'locatlons ln a system.

when derlvlng the sequence netuork interconnetions for singlefau]tsr the sequence currents and vo'ltages are a'|1 sequencecomponents of the refererce phaser raf phase belng selected to be thereference phase. s ince the sequence corponents oi tne other twophases were not lnvo]ved, the phase subscrlpt raf was omitted withoutcauslng confuslon. In the derlvatlon of sequence netrork connectionsfor simu'ltaneo.ls faults, esp€clal1y when the faults are on di f ferentphases' sequence conponents of more than one phase are employed. Theomlsslon of phase subscripts wl'll cause confuslon. Thereiore, theseguence corponents wl l'l be phase subscr{pted accordingl y. It isessentlal r hotever, to f lnal ly express the constralnts-oi a'l'l fau'ltswlth respect to the same reference phase.

Another polnt to watch out for ls that when connetlng the sequencenetrot*sr lt rust be ensured that no addltlonal fault constraints-that cannot bE proved ls lntroduced. This is generally achieved bymaklng dlrect connectlon at one fault ]ocatlon-and empl oy L/L ratiotransformer coupl lng at the other, lf neessary (setion- 5.3.1), whenthe fau'lt constraints lnvolve phase shlfted sequence quantities,there wil'l be a need for phase shifting transformer coupl ing (setion5,3.2)

O:-^ ao, qVV av

5.3.1 Two Earth Faults on Phase rAr

Fat Dlfferent Locations

F

At F,

I.D

Va

Connectlons ---constralnts :

IYu2= Yaz'

?-?-?tal-ta2-rao

vul + Yaz+ v.o - o

-?.I

=Q

t

2

IAtFrttIu =I"IV =0a

I

2

,t-Tl-tt'aI ''a2 -ta0

ttrval +vuz *vuo =Q

:fI

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I.IROI€ INTErcOI{NECTIONS !

are not correct because thls vrould assume fo'llowing

IIval = val ' vuo = vao

ra,r ,ol[,_

Nr

vn, -ca

oec

ly^: (L ru.'

F

IN

I-.r.

Va,ucaG ,ln^'

Nr- -

A.o 4 rfo;

NoVa-o

0O-G-90o-

I t'a;Ml ^

I

Interposlng 1/I transformers must be used.

CORRrcT INTErcONNECTIONSUSING r/1 INTERPOSING rNNr,rirONUENS

5,3,2 Cross Country Faults

fAf Phase to Ground at F and rBr PhaseF

to Ground at F

At F,

I,D

Va

L

-0

al

V+AI

T-?taZ - taO

Yu?+ vuo = o

6,-a- a L!c-

F'

A+c,tt I t

ftT-T-^tu =ta -u-----------i

Iv. -o -----------zD

.ttlT-?tot =rb2 =Ibo

Convert to tat phase sequencec u r rents,

?trr*-IaI = 1l u2 = ia't?ftor IaI =4,'!aZ ={Iuo' -

ttfVOt *Vb2 *VbO =Q

Conver'r: to f al phase sequcncevo l taEe s,

?trro(-Val *<Vu2 * VuO = il?rror VaL + {Va2 **V.o - 0

I

2

since the fault constrainis invoJve plrase shifted sequencequantitiesr the sequence netlork connetions require phase shifiingtransforr,rers as shoyln be'low.

IF,

-,Ja,rtt9\..o-./O0\

) tu^, :I

-gAQt=OOg-

tv*i C

I

I

*r;.=o"l

N'I

Ni

r, I lo4o€cc,eec

{.

to'ro.i-;:'-l ffi Frarr-Yi,.

< 'lNu&ro {t Nj

=. Ir<)t

Ej-rd!--,

V^o

l,

Id

qer

:\+40

fxv,l

rJ7rug--z-yry

tr*o 1 1

tNo AJd

5.3,3 Open Ci rcuit and L ine to Ground

(a) 0pen Circujt Fau'lt

At fau'lt point :

=Q

+\A V5 Iu.'

be shorvn as 'in section (5.1.1)

Fault on Phase rAf

P

LL

74 (-lHx5 VEI va-

q! un'

',uu' r+t

I *____2

a

V,=1/=00c

I

2

uut =\z=40

Iul+IuZ+IuO=0

Fault(b) Line to Ground

At fau'lt point

Vt=oarIb.Ib =0-----

,tI +I =0-----ccFrom equation 3r

Vult * Vurt -,- Vuot = Q

From equations 4 ani 5' it canth at

3

4

5

ttt(Iui * Iul ) - (Iuz * I^2 ) - (Iuo * Iuo )

The sequence network interconnetions are shown be.lor

Pt Q,

A,o,

I

var

Jo.,o +tnl

,r"fL

PHASE SHIFTS IN DELTA.STAR TRANSFORMERS

In fault calculations uslng the per unit systsn and invo'lvlng only a slngleli,tlr dn! phase shlfts produced by de'lta-ltar transformers do not affectttre inipedance seen by the system at the termina'ls of the transformer. It istlrerefore va'l id to neglect initially the phase shifts invo'lved at thesetransformers when-ca'lculating tl" pg:itivb, negative and zero sequencecor'ponents of fault currents and voltages for the entire system. Theappropriate phase shifts must however be applied to the seiuence quantitiesbefore they are summated vectorially to obtain the phase quantities.If the positive sequence vo1tage and current quantitles are phase shiftedfgftll.q 9y angle 9, ,rn? correspondlng negatlve sequence quantities wi'l'l besh i fted by ang'le / backwardsr and v iie v6rsa.

Example:

Star secondary I ine currents :

Sequence quantitles

T =T =T- sl 's2 's0Phase quantities

Iru = Irl * rr2+ Iro = 3lrlIrb=Ir"=o

De'lta primary I ine currents

Sequence quantltlesr

Ipl = IsI /30o,

Ipo = o

= Irl i30o-* rrz /4oo =6.IrlIpo = Isl Q /2700 + 1 /9oo) = o

!p2 = Isz /4oo

Phase quantltiesr

tou = Ipl -t Ipz * Ipo

Ipb = *"r, * n|pz *

Io. =*I pt *?Tpz * Ip' - Irl (l /1500 + L /zLoo) = - F.rrr.

(r ) STAef Wrf+{ NaUTRAI- Pa-ITST

- Att Geht.dfAT?€- AAJD laADNaOtetUS Ae.€- @)J'$4aeD -fO Nl

tNcrr.l (Z ,tIJ- sou,e.c4 aUF'ts

- H+Lle,-N€r) TeAl- vo lfit{Ge'

l^P@Nure NgTta)ate-

E_ "+1O66

Dtre4lq FrNtStlE.g y'r'T

(z)

c3)

(+)

F-

;*:1I

-|.,q

:'11

AAAI/V.4-E

G€hl -TRA^ISF

SY.STE,iA SlN6[-e. LJ$e DtAer;r',L\

Za, Z7 Zut

fv

RaeTt,/e SeQu€Nc€ Dt*..e,",,'(rut)

V = ?osrTr(b SEeu€rucg pH-N

AT raiug; fut'ttTL, = FosrTtvZ S€qu€Nce pry;& c-ua.€€rrlT

FtOvSt,.l4 tNTo Ft

Vt = Er - Ir (Z-f + Zr,+/',)

EI(T7

v'ot$Ge-

urxt

,a_

NeGAT\ve s,Eq'y eN4 pAGe\^^

Nz- zz F2o k

(f ) STAF-T w'TH ^)F-Ju,Ter'.L

tarNT

- y'\ , CW /,6p L-OAD NaUTCAT-S

AFe C,^}N€eTED -TD Nz-

(z) t{o elnlrF's lA}<J-uDeD>

-- NO N€'r24tlVA S&QUM- VqAGetSGt@:

(3) lAlPeE\AN<.e, |€ru;oe,r<-

- Ne:G*T\VL 4uet4 6^ @AAt.h78f' PYsg

(+) Df16r?A^^ FrnllsftEg, Af r4'u|.,r ParNT Fz

t>1AlulrPue:.

Teans? urrl&

sYSrem *srru6, e uru€ DtAGez.t*

(

NEcartvL -3E4u€^rc&

Fz-

V2

(u')DIA€z%iv\

VZ = NiAGATtVe .S€e u&+f 2. FH-NAT Fp*)LT' PotuT.

3a = N&,frtv? Sequ€^rce ?*att{4 Jr\Jt^tG /NTo Fz_

VOLTAGE

CrJeP.g$T

V2- = - T2- (=o1 l- /.rz-+ ar.)

a

'zeeo.58

(f ) fue- "h) 44sa', (zeeo t4*..sEr{u€^,.e)4rEe€$rs; Tb P ro (A! 4a ?t+Asa oF rHeSrsre*t A.^sr ae' A hoetu &N,.*,'.toN(TUS CONN€ZNO^r ., T(PIA,*Y THE, NAJTPAL&.Acrn aNN+-noA)

T4o*rbo +I=o -'glrc

(z) QE<srntz. g'.'ftleD .Szs fa|

H

lrao Z*o SEqvoJce vo4%e gen lec^tN &e Gvat €f

Vo = 3T*. R.

ZeFo squ4re. r@o?N€OrcAU Tb *a PAT|',i

Ug-q.

UEACE- Op.AR.AA

\-)

N

Tao

auo

z,o = = 3R-

ZIEPo -sEf,. vFNr E DAGfuavaa

e> s?&tAL Q^}sl1P4Ar(o|\J ?&?'Jt@tr> 6e-TeJg..rsfoeJ^Hzs

EG. @l0stD=- 4X rEANs(acrc

IINO Zeo.9.r4,u4t.-zl,N LrNe AuWnaN'oN .A sDe,

frus,64t/ltlAleproAGa.$,^

.srN6uE PJ+Asa 3F1A SE<?u€Afc€

a StoelEfitattl/A'L Too AsrDe

No(E")

Ia,".

TEgt^r$AL

')

lPnrtl'F t;**

=SYgfEA,t SirN4l,F, {UNe Du\€,eAv\\.

\

Z+o TcNe

\v 7?

Eo

\ic

J-6

?g-o €erqueAJcE,

= z#.o 4&Q.oepce.

F/to,5 ?owT

= WD 9€QvAoc€,

rr\)T'o Fo

NeTu.btK-

P+{-€ vdL%g AT

aUZeAy\ Flpt;ttQ

V = I-o(zr" +Zc.)

TPAtose u-t^EL FN

€,

- *a- -

b-

?osrcve sEe

NUaaYtv&. S€rq

(N,)

3uz- T-_1 E1-,

1 ur-

(rua)

.S-(ST€yV\ .stf.r6ae Uru6. DlAeoaq

(

zaeo 5€q

Analysis of Unbalanced Faults

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