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8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
1/45
C
F
C 10
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
2/45
F
D
L
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
3/45
C
C.L F, 1918
B
B
,
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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1
aI
1
bI
1
cI
0
aI
0
bI
0cI
2
aI
2
bI
2cI
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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2
aI
2
bI
2
cI
1
aI
1
bI
1
cI
0
aI
0
bI
0
cI
=
+ +
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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C
210 aaaa IIII ++=
210
bbbb IIII ++=
210
cccc IIII ++=
000
acb III ==
1211 240 aab IaII ==
111 120 aac aIII ==
222
120 aab aIII ==
1212 240 aac IaII ==
I
= 1201a
= 24012a
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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012 012
=
2
1
0
2
2
11
111
a
a
a
c
b
a
II
I
aaaa
II
I
=
c
b
a
a
a
a
II
I
aaaa
II
I
2
2
2
1
0
11
111
3
1
012AIIabc
= abc
IAI1012
=
012
AVVabc
=
abc
VAV1012
=
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
8/45
=== 120200;120230;0200 cba VVV
EG ,
F .
=
c
b
a
a
a
a
VV
V
aaaa
VV
V
2
2
2
1
0
11
111
3
1
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
9/45
=
c
b
a
a
a
a
V
V
V
aa
aa
V
V
V
2
2
2
1
0
1
1
111
3
1
=++=
12010)(3
10cbaa
VVVV
=++= 12010)(3
1 21cbaa VaaVVV
=++= 0210)(3
1 22cbaa aVVaVV
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
10/45
=== 180100;00;0150 cba III
EG , ,
,
F .
=
c
b
a
a
a
a
II
I
aaaa
II
I
2
2
2
1
0
11
111
3
1
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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=
c
b
a
a
a
a
I
I
I
aa
aa
I
I
I
2
2
2
1
0
1
1
111
3
1
=++= 067.16)(
3
10cbaa IIII
=++= 41.2365.72)(3
1 21cbaa IaaIII
=++= 41.2365.72)(3
1 22cbaa aIIaII
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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I
1 : I
.
2 : I
.
0 : I .
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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I C
L
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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+++
+++
+++
=
c
b
a
nSnmnm
nmnSnm
nmnmnS
c
b
a
I
I
I
ZZZZZZ
ZZZZZZ
ZZZZZZ
V
V
V
abcabcabcIZV =
I ,012012
a
abc
a AIZAV =
012012
0121012
a
a
abc
a
IZ
AIZAV
=
=
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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AZAZ
abc1012 =
F
++
=
mS
mS
mnS
ZZ
ZZZZZ
Z
00
00003
012
0 12
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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L
21
ZZ =
nXXX 310+=
)ln2.0(2D
DfX n
n
=
,
>3
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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I
M
D
''1dXX = '1 dXX = dXX =1
''2
dXX =
lXX =0
=
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
18/45
I
021
ZZZ ==
,
=
=
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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0 = + 3
=0
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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0 = + 3
=
I0=0I0=0
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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0 = + 3
I0 I0=0
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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0 = + 3
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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0 = + 3
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
24/45
Zs
ZsZs
Ea
EbEc
+
+
+
Va
Vb
Vc
Ia
Ib
Ic
In
Zn
+
+
+
=
c
b
a
nsnn
nnsn
nnns
c
b
a
c
b
a
I
I
I
ZZZZ
ZZZZ
ZZZZ
E
E
E
V
V
V
012012012012
0121012012
012012012
aaa
a
abc
aa
a
abc
aa
abcabcabcabc
IZEV
AIZAEV
AIZAEAV
IZEV
=
=
=
=
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
25/45
=
+
=2
1
0
012
0000
00
0000
003
ZZ
Z
ZZ
ZZ
Z
s
s
ns
=
0
01012
aa EE
G ,
=
2
1
0
2
1
0
1
2
1
0
0000
00
0
0
a
a
a
a
a
a
a
II
I
ZZ
Z
EVV
V
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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Z0 0
aI
+
-
0
aV
2
aI
2a
V
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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I 1. 2.
3. . ,
4.
5.
6. , . ,
7. 3.
8. . .
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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L G F
Zs
ZsZs
Ea
EbEc
+
+
+
Va
Vb
Vc
Ia
Ib
Ic
In
Zn
I = 0
I = 0
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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=
0
0
1
1
111
3
1
2
2
2
1
0
a
a
a
a I
aa
aa
I
I
I
0==
=
cb
afa
II
IZV
fa ZZZZ
E
I 302100
+++
=
f
aaaf
ZZZZEIII
333
021
0
+++===
F C
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
30/45
L L F
Zs
ZsZs
Ea
EbEc
+
+
+
Va
Vb
Vc
Ia
Ib
Ic
In
ZnI
I
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
31/45
cb
a
bfcb
III
IZVV
=
=
=
0
=
b
b
a
a
a
I
I
aa
aa
I
I
I 0
1
1
111
3
1
2
2
2
1
0
( )
ba
ba
a
IaaI
IaaI
I
)(3
1
3
1
0
22
21
0
=
=
=
21
aa II =
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
32/45
( ) ( )( )aaaaIZIZZE afaa
=
22
1121 3
f
aa
ZZZEI
++=
21
1
112
1
1
2
2
3)(
0
1
1
111
aacb
a
a
c
b
a
IjIaaII
I
I
aa
aa
I
I
I
===
=
KL
F
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
33/45
D L G F
Zs
ZsZs
Ea
EbEc
+
+
+
Va
Vb
Vc
Ia
Ib
Ic
In
Zn
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
34/45
0)(
=
+==
a
cbfcb
I
IIZVV
f
aaa
ZZIZEI
30
110
+
=
2
11
2
ZIZEI aaa =
( )f
f
aa
ZZZZZZZ
EI
33
02
02
1
1
++
++
=
03 acbf IIII =+=
F
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
35/45
F 3
.
Generators G1 and G2: 1.021 jXX == p.u; 05.00
jX = p.u; Voltage: 11 kV
Transformers T1 and T2: 15.0021 jXXX === p.u; Voltage: 11/275 kV
Line: 05.021 jZZ == p.u, 1.00 jZ = p.u ; Voltage: 275 kV
All per unit values are on a base of 100 MVA, and pre-fault voltage is 1.0 per unit.
C B
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
36/45
D ,
.
.
j0 1 j0 1j0 15 j0 15j0 05
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
37/45
j0.1 j0.1j0.15 j0.15j0.05
j0.05 j0.05j0.15 j0.15j0.1
A B
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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D
B.
fBBB
a
aaa ZZZZ
E
III 3021210
+++===
)1.015.0//()05.015.01.0(1 jjjjjZB +++= = j0.1364
12BB ZZ = = j0.1364
)15.0//()1.015.0(0 jjjZB += = j0.09375
009375.01364.01364.0
0.1210
+++=== jjjIII aaa = -j2.728 .pu
1844.83)(0
jIFI aB == p.u
B
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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D
B.
A B,
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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)55.0
25.0)((*)15.01.0( 11 FIjjV BA += = 454.0*728.2*25.0 jj = -0.31
111 )0()( AAA VVFV += = 1.0 0.31 = 0.69 p.u
)55.0
25.0
)((*)05.015.01.0(11
FIjjjV BB ++= = -0.372
111 )0()( BBB VVFV += = 1.0 0.372 = 0.628 p.u
Negative sequence voltage buses A and B,
222 )0()( AAA VVFV += = 0.0 0.31 = - 0.31
222 )0()( BBB VVFV += = 0.0 0.372 = -0.372
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
41/45
Zero sequence voltage buses A
and B,
)55.0
25.0)((*)15.0(
00
FIjV BA = = - 0.186
000 )0()( AAA VVFV += = 0.0 0.186 = -0.186 p.u
)55.0
25.0
)((*)1.015.0(00
FIjjV BB += = - 0.31 p.u000 )0()( BBB VVFV += = 0.0 0.31 = -0.31 p.u
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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=
2
22
1
11
0
00
012
)()(
)()(
)()(
AB
BA
AB
BA
AB
BA
AB
z
FVFV
z
FVFV
z
FVFV
I =
05.0
)372.0(31.0
05.0
628.069.0
1.0
)31.0(186.0
j
j
j
=
24.1
24.1
24.1
j
j
j
p.u
)()( 012 FAIFI ABabcAB = =
24.1
24.1
24.1
1
1
111
2
2
j
j
j
aa
aa
)24.1(3 jIaAB = = -j3.72 p.u
++++= 1209024.12409024.19024.1bABI = 0.0 p.u
++++= 2409024.11209024.19024.1cABI = 0.0 p.u
C
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
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8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
44/45
F 3
.
Generators G1 and G2: 1.021 jXX == p.u; 05.00
jX = p.u; Voltage: 11 kV
Transformers T1 and T2: 15.0021 jXXX === p.u; Voltage: 11/275 kV
Line: 05.021 jZZ == p.u, 1.00 jZ = p.u ; Voltage: 275 kV
All per unit values are on a base of 100 MVA, and pre-fault voltage is 1.0 per unit.
C B
8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)
45/45
D
B.
F ,
12 )( acb IaaII ==ZfZZ
EIa
++=
21
01
upjjj
jIjII acb .725.4)09375.01364.01364.0(
133 1 =
++===