Symmetrical Components
25
Power Systems I Fault Analysis l Fault types: u balanced faults Percentage of total faults n three-phase <5% u unbalanced faults n single-line to ground 60-75% n double-line to ground 15-25% n line-to-line faults 5-15% l Unbalance fault analysis requires new tools u symmetrical components u augmented component models
-
Upload
josset-aldridge-aguila -
Category
Documents
-
view
230 -
download
3
Transcript of Symmetrical Components
Po
wer
Sys
tem
s I
Fau
lt A
nal
ysis
lF
ault
typ
es:
uba
lanc
ed fa
ults
Per
cent
age
of to
tal f
aults
nth
ree-
phas
e <
5%
uun
bala
nced
faul
tsn
sing
le-li
ne to
gro
und
60-7
5%
ndo
uble
-line
to g
roun
d15
-25%
nlin
e-to
-line
faul
ts
5-1
5%
lU
nb
alan
ce f
ault
an
alys
is r
equ
ires
new
to
ols
usy
mm
etric
al c
ompo
nent
su
augm
ente
d co
mpo
nent
mod
els
Po
wer
Sys
tem
s I
Sym
met
rica
l Co
mp
on
ents
lA
llow
un
bal
ance
d t
hre
e-p
has
e p
has
or
qu
anti
ties
to
be
rep
lace
d b
y th
e su
m o
f th
ree
sep
arat
e b
ut
bal
ance
dsy
mm
etri
cal c
om
po
nen
tsu
appl
icab
le to
cur
rent
and
vol
tage
su
perm
its m
odel
ing
of u
nbal
ance
d sy
stem
s an
d ne
twor
ks
lR
epre
sen
tati
ve s
ymm
etri
cal c
om
po
nen
ts
I a1
I b1
I c1
120°
120°
120°
0°
I a0
I b0
I c0
I a2
I b2 I c2
120°
120°
120°
I a
I cab
c se
quen
cepo
sitiv
e se
quen
ceac
b se
quen
cene
gativ
e se
quen
ce
zero
seq
uenc
e
Po
wer
Sys
tem
s I
Sym
met
rica
l Co
mp
on
ents
()
()
()
01
01
01
866
.05.0
240
1
866
.05.0
120
1
120
240
0
2
32
11
1
12
11
11
1
=+
++
=°
∠=
−−
=°∠
=+
−=°
∠=
=°
+∠
=
=°
+∠
=
=°
+∠
=
aa
ja
ja
ja
Ia
II
Ia
II
II
I
aa
c
aa
b
aa
a
δδδl
Po
siti
ve s
equ
ence
ph
aso
rs
lO
per
ato
r a
iden
titi
es
Po
wer
Sys
tem
s I
Sym
met
rica
l Co
mp
on
ents
()
()
()
()
()
()
00
0
00
0
00
0
22
22
22
2
22
2
000240
120
0
aa
c
aa
b
aa
a
aa
c
aa
b
aa
a
II
I
II
I
II
I
Ia
II
Ia
II
II
I
=°
+∠
=
=°
+∠
=
=°
+∠
=
=°
+∠
=
=°
+∠
=
=°
+∠
=
δδδδδδl
Neg
ativ
e se
qu
ence
ph
aso
rs
lZ
ero
seq
uen
ce p
has
ors
Po
wer
Sys
tem
s I
lR
elat
ing
un
bal
ance
d p
has
ors
to
sym
met
rica
l co
mp
on
ents
lIn
mat
rix
no
tati
on
Sym
met
rica
l Co
mp
on
ents
22
10
21
0
21
20
21
0
21
02
10
aa
ac
cc
c
aa
ab
bb
b
aa
aa
aa
a
Ia
Ia
II
II
I
Ia
Ia
II
II
I
II
II
II
I
++
=+
+=
++
=+
+=
++
=+
+=
=
210
2
2
11
11
1
aaa
cba
III
aa
aa
III
Po
wer
Sys
tem
s I
l[A
] is
kn
ow
n a
s th
e sy
mm
etri
cal c
om
po
nen
tstr
ansf
orm
atio
n m
atri
x
lS
olv
ing
fo
r th
e sy
mm
etri
cal c
om
po
nen
ts le
ads
to
Sym
met
rica
l Co
mp
on
ents
=
=2
201
2
11
11
1
aa
aa
abc
AI
AI
*
2
21
101
2
31
11
11
1
31A
A
IA
I
=
==
−
−
aa
aa
abc
Po
wer
Sys
tem
s I
Sym
met
rica
l Co
mp
on
ents
lIn
co
mp
on
ent
form
, th
e ca
lcu
lati
on
fo
r sy
mm
etri
cal
com
po
nen
ts a
re
()
()
()
cb
aa
cb
aa
cb
aa
aII
aI
I
Ia
aII
I
II
II
++
=
++
=
++
=
231
2
231
1
310
Po
wer
Sys
tem
s I
Sym
met
rica
l Co
mp
on
ents
lS
imila
r ex
pre
ssio
ns
exis
t fo
r vo
ltag
es
lT
he
app
aren
t p
ow
er m
ay a
lso
be
exp
ress
ed in
ter
ms
of
sym
met
rica
l co
mp
on
ents
abc
abc
VA
V
VA
V1
012
012
−== (
)()
* 22
* 11
* 00
* 012
012
3
** 01
2*
012
3
*01
201
23
*3
33
33
3
aa
aa
aa
abc
abc
IV
IV
IV
SSSS
++
==
====
IV
AA
IA
AV
AI
AV
IV
T
TT
T
T
T
φφφφ
Po
wer
Sys
tem
s I
lO
bta
in t
he
sym
met
rica
l co
mp
on
ents
of
a se
t o
fu
nb
alan
ced
cu
rren
ts
lS
olu
tio
n
°∠
=°
∠=
°∠
=
132
9.0
180
0.1
256.1
cba III
Exa
mp
le
()
()
()
()
()
()
()
()
()
°∠
=°
∠+
°∠
+°
∠=
°−
∠=
°∠
+°
∠+
°∠
=
°∠
=°
∠+
°∠
+°
∠=
3.22
60.03
132
9.018
00.1
256.1
1.094.0
3
132
9.018
00.1
256.1
5.96
45.03
132
9.018
00.1
256.1
2
2
2
10
aa
I
aa
II aaa
Po
wer
Sys
tem
s I
Exa
mp
le
I a1
I b1
I c1
I a2
I b2
I c2
I a0,
I b0,
I c0
I a
I b
I c
posi
tive
sequ
ence
set
nega
tive
sequ
ence
set
zero
seq
uenc
e se
t
abc
set
Po
wer
Sys
tem
s I
Exa
mp
le
lT
he
sym
met
rica
l co
mp
on
ents
of
a se
t o
f u
nb
alan
ced
volt
ages
are
Ob
tain
th
e o
rig
inal
un
bal
ance
d v
olt
ages
:
°−
∠=
°∠
=°
∠=
308.0
300.1
906.0
210
aaa
VVV
()
()
()
()
()
()
()
()
()
°∠
=°
−∠
+°
∠+
°∠
=
°∠
=°
−∠
+°
∠+
°∠
=
°∠
=°
−∠
+°
∠+
°∠
=
8.15
570
88.1
308.0
300.1
906.0
904.0
308.0
300.1
906.0
2.24
7088
.130
8.030
0.190
6.0
2
2
aa
V
aa
VV
cba
Po
wer
Sys
tem
s I
Exa
mp
le
Va1
Vb1
Vc1
Va2
Vb2
Vc2
Va0
, Vb0
, Vc0
Va
Vb
Vc
posi
tive
sequ
ence
set
nega
tive
sequ
ence
set
zero
seq
uenc
e se
t
abc
set
Po
wer
Sys
tem
s I
Seq
uen
ce Im
ped
ance
s
lT
he
imp
edan
ce o
ffer
ed t
o t
he
flo
w o
f a
seq
uen
ce c
urr
ent
crea
tin
g s
equ
ence
vo
ltag
esu
posi
tive,
neg
ativ
e, a
nd z
ero
sequ
ence
impe
danc
es
lA
ug
men
ted
net
wo
rk m
od
els
uw
ye-c
onne
cted
bal
ance
d lo
ads
utr
ansm
issi
on li
neu
3-ph
ase
tran
sfor
mer
su
gene
rato
rs
Po
wer
Sys
tem
s I
Bal
ance
d L
oad
s
Va
Vb
Vc
I n
I a
I b
I c
Zs Z
s
Zs
ZM
ZM
ZM
Zn
abc
abc
abc
cba
nS
nM
nM
nM
nS
nM
nM
nM
nS
cba
cb
an
nn
cS
bM
aM
c
nn
cM
bS
aM
b
nn
cM
bM
aS
a
III
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
VVV
II
II
IZ
IZ
IZ
IZ
V
IZ
IZ
IZ
IZ
V
IZ
IZ
IZ
IZ
V
IZ
V=
++
++
++
++
+=
++
=+
++
=+
++
=+
++
=
Mo
del
an
d
go
vern
ing
eq
uat
ion
s
Po
wer
Sys
tem
s I
Bal
ance
d L
oad
s
()
()
[]
[]
−−
++
=
++
++
++
++
+
=
=
=→
=
=→
=
−−
MS
MS
Mn
S
nS
nM
nM
nM
nS
nM
nM
nM
nS
abc
abc
abc
abc
abc
abc
ZZ
ZZ
ZZ
Z
aa
aa
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
aa
aa
00
00
00
23
11
11
1
11
11
1
31
2
2
2
2
101
2
012
012
012
012
101
2
012
012
AZ
AZ
IZ
VI
AZ
AV
IA
ZV
AI
ZV
Po
wer
Sys
tem
s I
Tra
nsm
issi
on
Lin
e
21
222
111
21
21
21
0
0
abc
abc
abc
abc
cba
cba
nS
nn
nn
Sn
nn
nS
cba
cb
an
nn
n
cn
nc
Sc
bn
nb
Sb
an
na
Sa
VVV
III
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
VVV
II
II
IZ
V
VI
ZI
ZV
VI
ZI
ZV
VI
ZI
ZV
VI
ZV
+=
+
++
+=
=+
++
+=
+−
=+
−=
+−
=
Va1
Vb1
Vc1
I nI a
I b
I c
Zs Z
s
Zs
Zn
Va2
Vb2
Vc2
Vn
Po
wer
Sys
tem
s I
Tra
nsm
issi
on
Lin
e
+
=
++
+
=
=
+=
+=
+=
→+
=
−
−−
−−
−−
S
S
nS
nS
nn
nn
Sn
nn
nS
abcab
c
abc
abc
abc
abc
abc
Z
Z
ZZ
aa
aa
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
aa
aa
00
00
00
3
11
11
1
11
11
1
31
2
2
2
2
101
2
201
201
201
22
012
012
11
012
201
201
21
012
21
AZ
AZ
VI
ZV
IA
ZA
V
VA
IA
ZV
AV
IZ
V
Po
wer
Sys
tem
s I
Gen
erat
ors
lS
imila
r m
od
elin
g o
f im
ped
ance
s to
seq
uen
ce im
ped
ance
sl
Typ
ical
val
ues
fo
r co
mm
on
gen
erat
ors
ure
mem
ber
that
the
tran
sien
t fau
lt im
peda
nce
is a
func
tion
of ti
me
upo
sitiv
e se
quen
ce v
alue
s ar
e th
e sa
me
as X
d, X
d’, a
nd X
d”
une
gativ
e se
quen
ce v
alue
s ar
e af
fect
ed b
y th
e ro
tatio
n of
the
roto
rn
X2
~ X
d”
uze
ro s
eque
nce
valu
es a
re is
olat
ed fr
om th
e ai
rgap
of t
he m
achi
nen
the
zero
seq
uenc
e re
acta
nce
is a
ppro
xim
ated
to th
e le
akag
ere
acta
nce
nX
0 ~
XL
Po
wer
Sys
tem
s I
Gen
erat
or
Mo
del
X1
E1
VT
1
X2
VT
2
X0
VT
0Z
ero
Seq
uenc
e
Posi
tive
Sequ
ence
Neg
ativ
e Se
quen
ce
Po
wer
Sys
tem
s I
Imp
edan
ce G
rou
nd
ed G
ener
ato
rs
Ea
Eb
Ec
Zn
+
=
S
S
nS
Z
Z
ZZ
Z
00
00
00
3
012 Z
S
ZS
ZS
Po
wer
Sys
tem
s I
lS
erie
s L
eaka
ge
Imp
edan
ceu
the
mag
netiz
atio
n cu
rren
t and
cor
e lo
sses
rep
rese
nted
by
the
shun
t bra
nch
are
negl
ecte
d (t
hey
repr
esen
t onl
y 1%
of t
he to
tal
load
cur
rent
)u
the
tran
sfor
mer
is m
odel
ed w
ith th
e eq
uiva
lent
ser
ies
leak
age
impe
danc
e
lT
hre
e si
ng
le-p
has
e u
nit
s &
fiv
e-le
gg
ed c
ore
th
ree-
ph
ase
un
its
uth
e se
ries
leak
age
impe
danc
eis
the
sam
e fo
r al
l the
seq
uenc
es
lT
hre
e-le
gg
ed c
ore
th
ree-
ph
ase
un
its
uth
e se
ries
leak
age
impe
danc
e is
the
sam
efo
r th
e po
sitiv
e an
d ne
gativ
e se
quen
ce o
nly
Tra
nsf
orm
ers
lZ
ZZ
Z=
==
21
0
lZ
ZZ
==
21
Po
wer
Sys
tem
s I
Tra
nsf
orm
ers
lW
ye-d
elta
tra
nsf
orm
ers
crea
te a
ph
ase
shif
tin
g p
atte
rn f
or
the
vari
ou
s se
qu
ence
su
the
posi
tive
sequ
ence
qua
ntiti
es r
otat
e by
+30
deg
rees
uth
e ne
gativ
e se
quen
ce q
uant
ities
rot
ate
by -
30 d
egre
esu
the
zero
seq
uenc
e qu
antit
ies
can
not p
ass
thro
ugh
the
tran
sfor
mer
lU
SA
sta
nd
ard
uin
depe
nden
t of t
he w
indi
ng o
rder
(∆-
Y o
r Y
- ∆)
uth
e po
sitiv
e se
quen
ce li
ne v
olta
ge o
n th
e H
V s
ide
lead
s th
eco
rres
pond
ing
line
volta
ge o
n th
e LV
sid
e by
30°
uco
nseq
uent
ly, f
or th
e ne
gativ
e se
quen
ce v
olta
ges
the
corr
espo
ndin
g ph
ase
shift
is -
30°
Po
wer
Sys
tem
s I
Tra
nsf
orm
ers
lZ
ero
-seq
uen
ce n
etw
ork
co
nn
ecti
on
s o
f th
e tr
ansf
orm
erd
epen
ds
on
th
e w
ind
ing
co
nn
ecti
on
upr
imar
y w
indi
ng -
wye
/ w
ye-g
roun
ded
/ del
tau
seco
ndar
y w
indi
ng -
wye
/ w
ye-g
roun
ded
/ del
ta
Po
wer
Sys
tem
s I
Tra
nsf
orm
ers
wye
-gro
unde
d w
ye-g
roun
ded
wye
-gro
unde
dde
lta
wye
-gro
unde
d w
ye
Po
wer
Sys
tem
s I
Tra
nsf
orm
ers
wye
de
lta
delta
de
lta
