1995
IEEE
5 h In e na ional Con e ence
on
Coiiduc ion and B eakdown
in
Solid Dielec ics Page
66
I
THE CYLINDRICAL CAPACITIVE MODEL FOR WATER TREEING DEGRADATION IN
EXTRUDED HV CABLES
M.
Acedo"),
F.
F u os"',
M.
To es"' and
J.
C.
Filippini'*'
INTRODUCTION
The ini ia ion and g ow h o oids and mic ochaunels illed wi h wa e andl di e en ions in
ex uded polyme ic insula ion o HV cables
is
usually known in li e a u e as 'wa e eeing',
due o hei an and/o bush-like appea ance. Wa e and elec ical ees ha e e ealed as some
o he mos impo an causes o cable b eakdown de ec ed om la e six ies. Since hen, many
pape s ha e been de o ed o s udy he p oblem. Ne e heless, al hough la ge amoun s o
expe imen al esul s ha e been p o ided, he e is a lack o physical-ma hema ical models
which explain why hose esul s a e p oduced. In o de o p opose an adequa e and speci ic
model o ex uded cables de e io a ed by wa e eeing, we mus ake in o conside a ion:
a)
b)
In e nal s uc u e o wa e ees.
Cylind ical geome y o
he
de ice (cable) in which wa e ees a e g owing.
INTERNAL STRUCTURE AND DIELECTRIC PROPERTIES
Inhomo~enei y
Chen and Filippini
111
imp o ed op ical obse a ion echniques o wa e ee oids,
SO
ha
hey could conclude e e y ee is made
up
o 'bouque s', which a e alignmen s o mic oca i ies
(see Fig.1). F om all his expe imen al e idence and complemen a y mic oscopic obse a ions,
i has been p o ed ha in e nal s uc u e o wa e ees is highly
non
uni io m. The e is an
e iden dec ease in he numbe o mic oca i ies as we mo e away om he base o he I ee.
Dielec ic beha iou
Se e al esea che s had p e iously shown a dielec ic beha iou o polye hylene (PE) a eic ed
by
wa e ees (pe mi i i y
E,-6
a equency
=ISOOHz
,Koo
e
al.
121).
Chien and Filippini
[I]
ga e a u he s ep by ob aining he dielec ic pe mi i i y,
e,,
as a unc ion o he oillme
ac ion o inclusions,
A.
I is absolu ely basic o any physical model o wa e eeing he ac
ha pe mi i i y
e,
inc eases wi h
A.
Besides,
A
g ows as we app oach he base o he wa e
ee. All his implies ha
e,
mus be ep esen ed by a dec easing unc ion e sus dis ance as
we mo e away om he base o he wa e ee
on
i s cen al axis owa ds he PE una ec ed
by deg ada ion. As he dis ance om he ip o he needle inc eases,
A
diminishes and
pe mi i i y dec eases g adually om i s maximum alue o he minimum alue co esponding
o PE
(e,=2.3).
We
will
see
below he impo ance o he dec easing law o a ia ion o
e"
because he dielec ic beha iou o wa e ees de e mines he elec ic ield dis ibu ion in
de e io a ed cables. Se e al hypo heses o such a dec easing unc ion will also be examined.
-
(I'
Depa ameu o de Fisica Aplicada. Uni e sidad de Se illa, Spain.
")
C.N.R.S., Labo a oi e d'Elec os a ique e de Ma e iaux Dielec iques, C enohle, F ance.
0-780;-2040-9/95/$4.00
0
1995
IEEE
Page 662
CYLINDRICAL CAPACITIVE MODEL FOR WATER TREEING DEGRADATION IN A
COAXIAL CABLE
Wa e eeing has been modelled in many ways. Summa izing, we will say ha wa e ees
we e p ima ily conside ed as conduc o s by Asbc a
131.
Then,
Koo
e al. [2] showed ha he
wa e ee beha iou was ha o a dielec ic, and inally Chen and Filippini
111
depic ed hem
as inhomogeneous dielec ic sphe es o linea ly dec easing pe mi i i y, g owing uni o mly and
adially om a small wa e sphe ical elec ode.
In
he p esen wo k, we conside a mul i ude
o
en ed wa e ees modelized as a dielec ic cylind ical zone (Fig.Za). Such a si ua ion can
be ound in cables ailed in se ice (41.
Wa e ee deg ada ion g owing om he inne elec ode
Geome ic and oh sical o ooe ies o he model. We conside ha he wa e - eed-
deg ada ed zone o a coaxial cable can be ep esen ed as a dielec ic cylinde ha g ows
uni o mly and adially om a small cylind ical wa e elec ode owa ds ano he elee ode, a
concen ic conduc o cylinde , as i is shown in Fig.2a. We supposed, in a i s le el model ha
he pe mi i i y o he wa e - eed- egion, ,( ) dec eases linea ly. The linea dependence o
he dec ease in
c,( )
can be w i en ma hema ically by equa ion
(l),
e,(')
=
pe mi i i y o deg ada ed cylinde a poin M
( 199J
l
=
adius
o
wa e elec ode
,
=
maximum adius o cylind ical deg ada ed zone
,( ,)
=
pe mi i i y a su ace o wa e elec ode
(
we assume
,( J=3 l)
,( J
E
pe mi i i y o homogeneous PE non a ec ed by eeing
(e,( 3= l=2.3)
Dis ibu ion o elec ic ield,
E,( ).
due o wa e eeing, We will conside he elec ic ield,
E,( ),
in e e y poin M o he dielec ic ma e ial, in
o
ou side he egion a ec ed by eeing
(i
,W2
+
qln,;
i
> 2
-*
= ouJ.
We will also e e he exis ing ield in any poin M o he
dielec ic ma e ial be o e he g owing o ees (be o e he deg ada ion o PE)
as
Ej( ).
F om
he dis ibu ion o pe mi i i y shown in Fig.2a and equa ion
(l),
we can calcula e he
a ia ions o he elec ic ield
E,(')
in e e y poin M, du ing he g owing o wa e ees.
Conside ing ha he poin M can be placed in o ou side he deg ada ed zone, he elec ic
ield
E,(M)
can
be
diminished
o
enhanced compa ed o i s alue
Ej(W
be o e he de elopmen
o ees, we de ine he a ia ion a e
k=E&
The capaci ance pe uni leng h o a new cable
is ep esen ed by
c,
while
cag
s ands o he capaci ance o he cable a e he g owing o wa e
ees;
c,
is he capaci ance o he deg ada ed egion and
c,
he capaci ance o he emaining
ex e nal zone.
Conside a ions abou he elec ic cha ge con ained
in
he wa e ee lead o exp essions o
k
ou side
(k,",)
and inside
(kin,)
he deg ada ed egion,
Page
663
In igu e 2b, we show he a ia ion o
k
as a unc ion o
/ l
o a deg ada ed egion o
leng h,
I= ,=O.S l
(*
/ I=I5)
and pe mi i i y
a
su ace o wa e elec ode,
( 3=3~,,
whe e
e,=2.3.
Explo ing igu e 2b, we ge :
a) Ou side he de e io a ed zone, he elec ic ield
E
is ampli ied wi h a a ia ion a e
k,
independen o he dis ance
and hence, o he dis ance o he deg ada ed egion.
b) Inside he de e io a ed zone,
k
diminishes when app oaching he inne elec ode and, in he
la ges pa o he deg ada ed egion,
E
is lowe han in he absence iD he ee
(&I),
especially nea he wa e elec ode. I can be poin ed ou , e en mo e p ecisely, ha im he
wa e ee zone nea i s bo de , he elec ic ield equals and ge s bigge han in he absence
o eeing
(E ( )>Ei( )).
O he unc ional dependencies we e p o ed a he dec easing
pe mi i i y,
,( )
. A dec easing exponen ial pe mi i i y (i) and ano he unc ion which we
will e e as modi ied exponen ial pe mi i i y (ii). Fo each one, aking in o accoun he
su ounding condi ions a bo h ends o he wa e eed egion,
(~,(' J=3~,=6.9)
and,
(e ( 3=eI=2.3),
we ge :
(ii)
q( ) =a-exp(b )
-
exp(b ,)-exp(b l)=2el
,
u= , +exp(b J
(4)
In igu es
3a
and 3b, we compa e he pe mi i i y dependencies and he ampli ica ion esul s
o he geome ic coe icien
/ l=l.S.
Wa e ee deg ada ion g owing om he ou e elec ode
Fea u es o he model.
A
simila s udy can be pe o med when supposing ha he
deg ada ed egion g ows om he ou e semiconduc o o he cable and when co esponiding
nea symme ical dependencies o pe mi i i y,
e ( ),
a e used i
,> > ,
In
Fig.4a we show
he pe mi i i y dependencies used. These dependencies would appea i he pe cen age alue
o deg ada ed wid h o PE we e
D.W.=75%.
Taking in o accoun he co esponding
su ounding condi ions used o he inne deg ada ion, now we ha e,
,( 3=3eI=6.9,
and
e,( J=c,=2.3;
hen, we ind:
exp(b ,)
-exp(b ,) =:le,
a= l -exp(b ,)
2
a)
( )=a.log(b )
-a=----!--,
b=
;
b) e,( )=a+exp(b )
-
log( 3/ ,)
exp(b ,)
-exp(b ,) =:le,
a= l -exp(b ,)
a)
( )=a.log(b )
;
b) e,( )=a+exp(b )
-
2 ,( 3-3 2)
c)
,( )=a +b
-
a=L,
b=-
,
- 2
( , - 2)
Elec ic ield ampli ica ion.
As
in he inne case, i can be shown ha ampli ica ion ac o ,
k=(E,E$
is he bigges o he modi ied pe mi i i y and he lowes o he dec easing
exponen ial pe mi i i y, bu o all o hem, ampli ica ion is now di ec ed owa ds he inne
elec ode. Howe e , he mos signi ican ea u e can be ound in he ac ha , due
o
he
Page 664
ABLE
I
-
Field ampli ica ions o di e en cases o eeing cable deg ada ion
k
/F.A.
k
/EA.
k
/F.A.
pe mi i i ),
pe mi i i
),
cylind ical shape o he de ice, inne deg ada ion zones (and hence inne wa e ees) could
be mo e dange ous han ou e deg ada ion Lones (ou e wa e ees). In ac , ampli ica ion
ac o
k
is bigge when he wa e ee zone depa s om
,
owa ds
,
(do ed line in Fig.4b)
han when he wa e ee zone depa s om
j
owa ds
,
(con inuous line in Fig.4b), o he
same deg ada ed wid h
(D.W.=75%).
This e alua ion has been done o he en i ely
symme ical caw, he linea case. In Table.1, ampli ica ion ac o ,
k,
and i s pe cen age alue
o ield ampli ica ion,
EA.(%),
a e p in ed o di e en deg ada ion wid hs and di e en
pe mi i i y dependencies o bo h, in e nal and ex e nal deg ada ion cases.
FINAL CONCLUSIONS
AND
UTILITY
OF
THE
MODEL
The cylind ical capaci i e model o
HV
cable de e io a ion shows in an easy way ha wa e
ees ha e an impo an in luence on he dis ibu ion o elec ic ield in a dielec ic ma e ial
a ec ed
by
he ype o deg ada ion known as 'wa e eeing'. In a mo e ex ended wo k, we
will show ha an e en mo e ealis ic model, conside ing he exis ence o wa e ees g owing
om bo h he inne and ou e elec odes, lead o nex conclusion: i is mo e dange ous an
in e nal deg ada ion zone o double wid h
(50%
o PE wid h) han
WO
smalle deg ada ion
egions, in e nal
(25%
o
PE wid h) and ex e nal
(25%
o PE wid h), which sum o wid hs is
he same,
50%=25%+25%,
ha he i s one.
LIST OF REFERENCES
I.
Chen, J.L., and Filippini, J.C., 1993, "The Mo phology and Beha io o he Wa e
T ee", T ans. on Elec .Insul.,
28,
271-286.
Koo,
,J.Y.,
C oss,
J.D.,
El-kahel, M., Me e C.T., and Filippini, J.C., 1983, "Elec ical 2.
..
Beha io and S uc u e o Wa e T ees in Rela ion o hei P opaga ion", Annual
Repo Con e ence on Elec ical Insula ion and Dielec ic Phenomena,
301-306.
Ashc a , A.C., 1977, "Wa e eeing in Polyme ic Diele ics". Wo ld Elec o echnical
Cong ess, Moscow.
We elius,
P.,
Tha ning, P., Holmg en,
B.
and Ga e ,
U.,
1994, "High Vol age Dielec ic
Response as a Tool o Diagnos ic o XLPE Cables", No dic Insul. Con e ence,
217-225.
3.
4.
Fiel e
1:
(a) Wa e ee s uc u e.
(b)
Mic oc. pe cen age
in
c oss mic osec ion.
Figu e
3:
Di e en pe mi . dependencies
(a) and co esponding
k
cu es
(b).
InIe naI
elec ode
Figu e
2:
(a) Inne deg ada ion m,odel.
(b)
Ampli ica ion cw e
k( / J.
Figu e
4:
(a) Pe mi .dep.
o
ou e model.
(b)
Compa ing inne aind ou e models.