Formula Sheets: Symmetrical and Unsymmetrical Fault Analysis | Power Systems - Electrical Engineering (EE) PDF Download

 Page 1


P o w er Systems: Symmetrical and Unsymmetrical
F ault Analysis F orm ula Sheet for Electrical GA TE
Symmetrical Comp onen ts
• Sequence Comp onen ts : T ransform phase quan tities to sequence quan tities.
?
?
V
a
V
b
V
c
?
?
=
?
?
1 1 1
1 a
2
a
1 a a
2
?
?
?
?
V
0
V
1
V
2
?
?
,
?
?
V
0
V
1
V
2
?
?
=
1
3
?
?
1 1 1
1 a a
2
1 a
2
a
?
?
?
?
V
a
V
b
V
c
?
?
where a = e
j120
?
= -
1
2
+ j
v
3
2
, a
2
= e
j240
?
= -
1
2
- j
v
3
2
, V
0
, V
1
, V
2
are zero, p ositiv e,
and negativ e sequence v oltages, resp ectiv ely .
• Curren t T ransformation : Same transformation applies to curren ts (I
a
, I
b
, I
c
).
Symmetrical F aults (Three-Phase F ault)
• F ault Curren t : F or a three-phase fault at a bus.
I
f
=
V
pre-fault
Z
1
where V
pre-fault
is the pre-fault v oltage (usually 1.0 pu), Z
1
is the p ositiv e sequence
imp edance.
• F ault MV A :
MV A
fault
=
V
2
pre-fault
Z
1
× Base MV A
Unsymmetrical F aults
• Single Line-to-Ground (SLG) F ault :
I
f
= I
a
=
3V
pre-fault
Z
1
+Z
2
+Z
0
+3Z
f
where Z
0
, Z
1
, Z
2
are zero, p ositiv e, and negativ e sequence imp edances, Z
f
is fault
imp edance.
V
0
= V
1
= V
2
=-
(Z
1
+Z
2
)V
pre-fault
Z
1
+Z
2
+Z
0
+3Z
f
• Line-to-Line (LL) F ault :
I
b
=-I
c
=
V
pre-fault
Z
1
+Z
2
+Z
f
, I
a
= 0
V
1
=-V
2
=
(Z
2
+Z
f
)V
pre-fault
Z
1
+Z
2
+Z
f
, V
0
= 0
1
Page 2


P o w er Systems: Symmetrical and Unsymmetrical
F ault Analysis F orm ula Sheet for Electrical GA TE
Symmetrical Comp onen ts
• Sequence Comp onen ts : T ransform phase quan tities to sequence quan tities.
?
?
V
a
V
b
V
c
?
?
=
?
?
1 1 1
1 a
2
a
1 a a
2
?
?
?
?
V
0
V
1
V
2
?
?
,
?
?
V
0
V
1
V
2
?
?
=
1
3
?
?
1 1 1
1 a a
2
1 a
2
a
?
?
?
?
V
a
V
b
V
c
?
?
where a = e
j120
?
= -
1
2
+ j
v
3
2
, a
2
= e
j240
?
= -
1
2
- j
v
3
2
, V
0
, V
1
, V
2
are zero, p ositiv e,
and negativ e sequence v oltages, resp ectiv ely .
• Curren t T ransformation : Same transformation applies to curren ts (I
a
, I
b
, I
c
).
Symmetrical F aults (Three-Phase F ault)
• F ault Curren t : F or a three-phase fault at a bus.
I
f
=
V
pre-fault
Z
1
where V
pre-fault
is the pre-fault v oltage (usually 1.0 pu), Z
1
is the p ositiv e sequence
imp edance.
• F ault MV A :
MV A
fault
=
V
2
pre-fault
Z
1
× Base MV A
Unsymmetrical F aults
• Single Line-to-Ground (SLG) F ault :
I
f
= I
a
=
3V
pre-fault
Z
1
+Z
2
+Z
0
+3Z
f
where Z
0
, Z
1
, Z
2
are zero, p ositiv e, and negativ e sequence imp edances, Z
f
is fault
imp edance.
V
0
= V
1
= V
2
=-
(Z
1
+Z
2
)V
pre-fault
Z
1
+Z
2
+Z
0
+3Z
f
• Line-to-Line (LL) F ault :
I
b
=-I
c
=
V
pre-fault
Z
1
+Z
2
+Z
f
, I
a
= 0
V
1
=-V
2
=
(Z
2
+Z
f
)V
pre-fault
Z
1
+Z
2
+Z
f
, V
0
= 0
1
• Double Line-to-Ground (DLG) F ault :
I
f
= I
b
+I
c
, I
1
=
V
pre-fault
Z
1
+
Z
2
(Z
0
+3Z
f
)
Z
2
+Z
0
+3Z
f
V
0
=-
(Z
2
I
1
)(Z
0
+3Z
f
)
Z
2
+Z
0
+3Z
f
, V
1
= V
pre-fault
-Z
1
I
1
, V
2
=-Z
2
I
1
Sequence Imp edances
• P ositiv e Sequence Imp edance (Z
1
) : Deriv ed from sync hronous mac hine and line
parameters.
• Negativ e Sequence Imp edance (Z
2
) : Similar to Z
1
for lines, differs for rotating
mac hines.
• Zero Sequence Imp edance (Z
0
) : Dep ends on grounding and transformer connec-
tions.
Key Notes
• Base v alues: T ypically , 100 MV A and nominal v oltage (e.g., 11 k V, 33 k V) for p er-unit
calculations.
• P er-unit system: Z
pu
= Z
actual
·
Base MV A
( Base k V)
2
.
• F ault imp edance Z
f
is zero unless sp ecified.
• T ransformer zero-sequence imp edance dep ends on connection (e.g., Y-Y, ?-Y).
• Use Thev enin equiv alen t at fault p oin t for calculations.
2