9.1 OVERVIEW
Note that the ratios of the capacitances between phase A and the two other phases are close to unity, and therefore the
correcting factors in equation 8.25 for the B and C-phase voltages are small numbers, while the factor for the V
close to 3. If so, equation 8.25 takes a familiar simplified form:
The V
term in the operate equation can be either the neutral component in the bus voltages (one-third of the vectorial sum
0
of the phase voltages calculated by the relay), or directly measured neutral voltage component (open-corner-delta VT volt-
age).
c) SENSITIVITY
Now consider the consequences of a capacitor element failure in one leg, most conveniently leg C, making a small capaci-
tance change in leg-C capacitance. The effect on the operating signal can be calculated by taking the derivative of equation
8.25 with respect to C
the initial value is zero, the derivative of the absolute function is simply the absolute value of the derivative of its argument.
As such, the derivative is:
---------- - V
dC
Recall equation 8.29.
Differentiating this gives:
Substituting for V
from equation 8.34 gives:
X
This can be substituted into equation 8.33 to obtain:
9
For our purposes,
C
A
9-8
V
. In the general case, the derivative of the absolute value function is messy, but in our case where
C
d
-- - d
1
(
(
---------- - V
1
k
=
+
OP
X
AB
3
dC
C
C
1
(
)
=
-- - 1
+
k
+
k
AB
AC
3
C
+
C
⎛
⎞
B
C
V
1
3V
+
-------------------- -
–
⎝
⎠
X
C
A
dV
C
+
C
⎛
⎞
X
B
C
---------- - 1
V
+
---------------------
+
⎝
⎠
X
dC
C
C
A
3V
d
0
---------- - V
=
-------------------------------------------------------------------------------------
X
dC
C
d
1
(
---------- - V
-- - 1
=
OP
dC
3
C
3V
1
=
-- -
-------------------------------------------------------------------------------------
3
≅
≅
≅
C
C
, so
k
1
and
k
B
C
AB
d
---------- - V
OP
dC
C
C70 Capacitor Bank Protection and Control System
V
V
=
–
OP
X
0
) 3V
(
)
k
V
1 k
+
–
+
–
AC
0
B
AB
dV
X
---------- -
dC
C
C
C
⎛
⎞
⎛
B
C
V
1
V
1
+
–
------ -
+
–
-------
⎝
⎠
⎝
0
B
C
C
C
A
A
⎛
⎞ V
⎛
⎞
1
1
------ -
------ -
0
–
=
⎝
⎠
⎝
⎠
C
C
C
A
A
V
–
d
⇒
X
---------- - V
=
–
-------------------------------------------- -
X
dC
C
⎛
C
C
1
+
-------------------- -
⎝
A
V
–
d
⇒
X
---------- - V
=
–
----------------------------------------------- -
(
X
dC
C
1
k
+
C
A
(
)
(
)
–
V
1 k
–
–
V
2
+
k
B
AB
C
AB
2
(
)
C
1
k
k
+
+
A
AB
AC
dV
)
X
k
k
---------- -
+
+
AB
AC
dC
C
(
)
(
–
V
1 k
–
–
V
2
+
k
0
B
AB
C
AB
(
)
C
1
k
k
+
+
A
AB
AC
≅
1
. This results in:
AC
1
V
–
V
≅
C
0
-- -
------------------- -
3
C
C
1
1
×
×
-- -
-------
=
V
–
V
C
0
3
C
C
9 THEORY OF OPERATION
(
)
)
V
1 k
+
–
C
AC
⎞
0
=
⎠
V
C
+
C
⎞
B
C
⎠
C
A
V
C
)
k
+
AB
AC
)
voltage is
X
(EQ 9.32)
(EQ 9.33)
(EQ 9.34)
(EQ 9.35)
(EQ 9.36)
(EQ 9.37)
(EQ 9.38)
GE Multilin