The mechanical effects of short-circuit currents in - Montefiore

8.4. INFLUENCE OF THE RECLOSURE
The following annex gives a simplified method
without dropping after the first fault. The motion of
the equivalent pendulum is on a circle.
After the first short-circuit
π
Let us examine the most simple case when: δ m ≤
2
without dropping.
This connection can be done following two methods:
- chapter 4.2 (volume 1 of [Ref 1]) gives the
maximum angle after the first short-circuit:
δ m = arccos χ .
- the IEC 60865 gives the following formula (*31).
This swing -out angle has been chosen to take into
account the "worst case" which is a short-circuit
duration less than or equal to the stated short-circuit
duration Tkl.
At the end of dead time Tu, the angle of the
pendulum is called δ i . Its values :
⎛ Tu ⎞
δ i = δ m cos⎜2π
+ α 0 ⎟
⎝ T ⎠
0
⎛ Tu ⎞ T . δ ′ k ⎛ Tu ⎞
δ i = δ k cos⎜2π
⎟ + sin
0
⎜2π
0 ⎟
⎝ T ⎠ 2π
⎝ T ⎠
0.
8bc
2π
o g
where T =
is the period of the free
2
2
π ⎛ δ ⎞ m 1−
64 ⎜
90 ⎟
⎝ ⎠
motion (without current) and δ m is the maximum
swing-out angle (4.9 volume 1 of [Ref 1]).
The speed before the reclosure is given by :
2π
′ =−
⎛ Tu
⎜
⎞
⎟ + ′
⎛ Tu
δ
⎜
⎞
i δ k.sin 2π δ k.cos2π
⎟
o
T ⎝ o
T ⎠ ⎝ o
T ⎠
Let us consider the time
0
T α
t 0 =−
2π
for
which the δ angle is equal to δ m . If Tu is higher
than to t0 , the angle δ is effectively equal to
δ m . If not done δ angle and subsequently δ i
always lower than δ m .
is
128
Linkage to the second short-circuit
During the second short-circuit, the movement is
ruled by equation :
2
T
δ
″
= r
2
2
4π
cos ( δ) − sin ( δ)
=
−
2
+ r −
1 2
( δ δ12)
sin ,
in which r 2 is the ratio of LAPLACE force due to
the second current I2 to gravitational force per unit
length. The first range integral can be expressed by :
2
T 2
′ −
2
′ = r .sin + cos − r .sin
[ ]
2
8π
δ δ i 2 δ δ 2 δ i
− cosδ
leading to the relation :
2
2 8π
δ ′ = 2
T
2
1 + r2cos − , − cos
with cos( δ )
2
=
defined by r2 i
i
[ ( δ δ12) ( δ 2)
]
χ
i
2
2
1 + r
.sinδ + χ .
The period is given by :
08 . b c
2π
g
Tres2
=
⎛ 2
4 2 π ⎛ δ 2 ⎞
1+ r ⎜ 2 ⎜
1−
⎜ ⎟
⎝ 64 ⎝ 90⎠
and χχχχ i parameter
At first approximation, the movement is given by :
⎛ t ⎞
δ = δ12 , + δ 2sin⎜2π
+ ϕ⎟
=
⎝ T ⎠
res2
⎛ t
δ12 , + ( δ i −δ12
, ) cos⎜2π
⎝ T
Tres2δ ′ i ⎛ t ⎞
sin⎜2π
⎟
2π
⎝ T ⎠
res2
res2
2
⎞
⎟
⎠
⎞
⎟ +
⎠
At the end of the short-circuit t=Tk2, the angle is
δ = δ T
( )
k 2 k 2
The radial force during this short-circuit is given by
the (4.11 in volume 1 of [Ref 1]) formula type which
is transformed as follows :
R′
G′
( ) ( )
= 3 cos δ + 3 sin δ − 2χ
r2 i