analysis of unbalanced faults
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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.
Page 2
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
Page 3
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.-
Page 4
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.
Page 5
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, /
Page 7
\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
Page 8
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
Page 12
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-
Page 14
(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
Page 15
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
Page 23
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
Page 24
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.
Page 29
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 )
Page 32
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 *
Page 33
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
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