The mechanical effects of short-circuit currents in - Montefiore

With a constant acceleration, the final velocity v follows
from:
a 1 2 1 v
y x − = & y&
ta
= t
2 2 2 t
(3.18) ( ) 2 a
(3.19)
y
v = 2
a
−
2
t
( x)
The centrifugal force acting on a conductor element is
m′
d xv
d F =
y
(3.20)
( x)
− a / 2 y(
x)
m′
⎡
= 4 y
t ⎢
⎣
a
2
=
a ⎤
2⎥
⎦
a
( x)
− d x
a
⎡
y
m′
d x ⎢
⎢2
− a / 2 ⎢
⎢⎣
a ⎤
−
2 ⎥
⎥
ta
⎥
⎥⎦
( x)
The integration along half the conductor gives a first
approach of the bending force at the insulator:
(3.21)
F
v
K
2
K
2
m′
⎡
= ∫ d F = 4 y
t ∫ ⎢
⎣
0
a
0
( x)
a ⎤ K µ 0
− d x 4
2⎥
=
⎦ 6 2π
( I ′′ )
The forces calculated with equation (3.21) are too low
compared with the measured ones. The reason may be
the simplifications done in the model of the dropper. It
is empirically found that Fv is to be corrected by the
factors ld/K and 2,5 ld/b and the actual bending force on
the insulator becomes :
ld
ld
K µ 0
Fd
= Fv
2,
5 = 4
b K 6 2π
(3.22)
2
5 µ 0 ( I k′′
) ld
= ld
3 2π
a b
( I ′′ )
with a validity range of 1,4 ≤ ld/b ≤ 3,3.
k
a
2
k
a
ld
ld
2,
5
b K
In equation (3.22), Fd only depends on the geometry and
the short-circuit current but not the conductor mass and
the short-circuit duration, as it follows from the tests.
The calculation of Fd according to equation (3.22) is
compared with the tests [Ref 41]. Most of the values lie
within a range of ±25 %, some are slightly more than 25
% on the safe side. This result allows the application of
the simplified method given above.
From this the ESL follows according to Paragraph 3.5.2
and with this the design load.
Droppers with flexible upper ends
The upper ends of droppers in the middle of the span or
near to it move with the main conductor in the bus. If
2
2
52
the droppers are not stretched, there will be no
significant bending force on the apparatus or insulators
at the lower end caused by the movement of the busbar.
If the droppers are stretched, a high loading can occur.
Until now a simplified method for the calculation of this
force is not available. Therefore it is recommended to
make the droppers longer to avoid stretching. If this is
not possible, advanced methods should be used.
The droppers with current (current path C in Figure
3.22) swing out due to the electromagnetic forces
between the droppers. The maximum force at the lower
end takes place during swing-out of the busbar. A good
estimation of the loading at the apparatus and insulators
can be determined assuming fixed upper ends of the
droppers and using the simplified method stated above.
3.4. THE BUNDLE PINCH
3.4.1. Introduction
This phenomena has been well described for the first
time by Manuzio in 1967 [Ref 53] and briefly repeated
here :
From initial rest position, subconductors move towards
each other, remaining more or less parallel in most part
of the subspan, except close to the spacer. After first
impact (in the centre of subspan) there is a quick
propagation of the wave through the spacer end of the
subspan.
There is a struggle between electromagnetic force which
increase on the moving part (the distance is decreasing)
and would like to force contact between subconductors,
and tension increase in the subconductor which react
against the electromagnetic force. There is a maximum
pinch when the wave propagation stops towards the
spacer and must come back to the centre of the span,
this is followed by oscillations (back to the spacer, then
to the subspan centre, etc.) because electromagnetic
force is still on, these oscillations are damped slightly so
that first peak is the maximum peak. If short circuit is
long enough, the pinch oscillations tends to a permanent
force, considerably lower than peak value (typically 2
times less).
During this phenomenon, spacer is strongly compressed.
The compression is related to maximum pinch force in
the conductor and angle between the spacer and the
subconductor.
Despite the fact that bending stiffness locally change the
curvature of the conductors, the deformation energy in
bending remain small compared to tensile deformation
energy, so that simple or advanced method can neglect
bending stiffness in their approach. Only structural
damping can be related to bending, which could have
influence on pinch oscillations but certainly not on the
peak value, which is a design value for the spacer.