M. Acedo, F. F u os
Uni e sidad de Se illa
Depa amen o de Física Aplicada 1, ETS Ingenie ía In o má ica, A da. Reina Me cedes s/n
41012 Se illa, Spain
I. Radu
Ecole Poly echnique de Mon éal
2900 Boul. Edoua d-Mon pe i , C.P. 6079, Succ. Cen e-Ville
Mon éal, Québec, H3C 3A7, Canada
and J.C. Filippini
Labo a oi e d’Elec os a ique e Ma é iaux Diélec iques de G enoble, CNRS-UJF
25 A enue des Ma y s, B.P. 166
38042 G enoble, F ance
ABSTRACT
Dis ibu ion o elec ic ene gy by ex uded polyme insula ed cables con inues o be a
subjec o ou s anding ele ance in mode n indus ialized coun ies all o e he wo ld.
Dielec ic cha ac e iza ion, conduc ion modelling and inally diagnos ics o polyme ic
insula ions a e necessa y s eps owa ds he de elopmen o eliable and less expensi e
obus echnologies o elec ic powe dis ibu ion. This pape is de o ed o a de ailed
expe imen al / heo e ical s udy o he conduc i e p ope ies o LDPE a ec ed by
di e en le els o deg ada ion by wa e ees. Wa e ee laye s o di e en leng hs
we e g own in accele a ed condi ions and we e cha ac e ized by wa e ee kine ics,
ime-dependen pe mi i i y and pola iza ion cu en . The pola iza ion cu en was
ound o obey a Cu ie- on Schweidle law whose pa ame e s we e used o cha ac e ize
he e ec o ageing ime. A new conduc ion model ha akes in o accoun dipole
in e ac ions and was ob ained om a wo-wells Debye model is p esen ed which allows
us o gi e an in e p e a ion o he e ec o ageing. This labo a o y s udy was in ended
o imp o e he cha ac e iza ion o se ice powe cables aged by wa e ees.
Index Te ms — Pola iza ion cu en , conduc ion model, wa e eeing, accele a ed
ageing, pe mi i i y, low densi y polye hylene.
1 INTRODUCTION
ONE o he mos impo an applica ions o polyme
dielec ics (EPR, HDPE, XLPE, LDPE...) is hei ex ended use
as powe cable insula ions. The ypical deg ada ion known as
wa e eeing usually appea s in unde g ound / subma ine
cables and i is mainly due o elec ical s ess and humidi y in
we en i onmen . Tha is why indus ies o PE p oduc ion and
ab ica ion o cables as well as u ili ies o ene gy dis ibu ion
a e e y in e es ed in he s udy o wa e ees. Thei exis ence
has been known o he las hi y yea s ( o e iews, see
[1,2]). Wa e ees a e small-damaged zones ha can appea in
polyme ic insula ions o ac medium and high ol age (MV,
HV) cables [3,4]. They equen ly look like bushes o ees
and imp o ed op ical mic oscopy echniques pe mi ed o
conclude ha e e y wa e ee is made up o "bouque s",
which a e alignmen s o mic oca i ies [5]. Wa e ees g ow
du ing he se ice li e o powe cables om de ec s whe e he
elec ic ield is ampli ied [3,4,6,7,8]. The longe he wa e
ee, he mo e dange ous i becomes, because he insula ion
b eakdown ol age dec eases wi h he inc ease o he leng h o
wa e ees [4,9,10].
Ano he e y impo an phenomenon in powe cables
a ec ed by wa e ees is he conduc ion p ocess. This is also
o g ea in e es o bo h use s and manu ac u e s o cables
[4,11,12,13] because elec ical conduc ion is one o he ac o s
ha e eal he le el o deg ada ion o he cable. Conce ning
he polyme conduc i i y, he s udy o he mechanisms o
he mo-elec ical deg ada ion o he polye hylene insula ion is
essen ial in o de o de elop imp o ed assessmen s a egies
[14]. Al hough many wo ks ha e been de o ed o he
de elopmen and e iew o diagnos ic me hods o MV/HV
Dielec ic Cha ac e iza ion and Conduc ion Modelling
o a Wa e T ee Deg aded LDPE
powe cables a ec ed by he mal ageing o deg ada ion in he
p esence o wa e [15,16,17,4] in he las ew yea s, a ho ough
heo e ical / expe imen al s udy o he conduc ion p ocesses is
necessa y o de eloping new diagnos ic echnologies. They
should se e bo h o new ex a high ol age (EHV) cables as
well as o old-in-se ice cables which is o ou s anding
impo ance because u ili ies mus ensu e eliabili y o hei
p esen dis ibu ion sys ems wi hou making unnecessa y huge
in es men s.
“Mac oscopic” modelling o conduc ion p ocesses can be
done by conside ing he dielec ic unc ion esponse ([18,19,
20,21] o he polyme . Ne e heless, in o de o ha e a deepe
look in o ma e ial, we we e in e es ed in de eloping a
“mic oscopic” modelling o conduc ion in PE. Hence, on he
basis o a p e iously published Debye-like model [22] and
aking in o accoun he idea o dipole in e ac ion om he
‘Many Body Uni e sal Model o Dielec ic Relaxa ion’
[23,24,25], we p opose a new conduc ion model o wa e ee
deg aded LDPE ha ep oduces he Cu ie- on Schweidle
po en ial dependence o eal cu en measu emen s and pe mi s
an easy physical in e p e a ion o hei i ing pa ame e s.
Consequen ly, in his a icle, we cha ac e ize he conduc ion
p ocesses in new and aged polye hylene by using expe imen al
esul s and we p esen a new heo e ical model o he
conduc ion p ocesses.
Wa e eeing was p oduced in labo a o y ma e ial
specimens and no in cables al hough ou expe imen al es
objec s (LEMD-CNRS) simula e e y closely his deg ada ion
p ocess in eal cables. Acco dingly, in he beginning o ou
pape we desc ibe he expe imen al p ocedu es o he g ow h
and cha ac e iza ion o wa e ees and he elec ical
measu emen echniques. A e wa ds, we p esen he
expe imen al esul s o he wa e ee kine ics, elec ical
capaci ance / pe mi i i y e olu ion and cu en measu emen s,
especially using specimens wi h di e en deg ees o eeing. A
special a en ion was paid o ensu e high le els o epea abili y
and ep oducibili y, and o p e en any ype o a i ac s and
pa asi ic e ec s in all he expe imen al p ocedu es used. We
hen p esen ou new elec ical conduc ion model ha akes
in o accoun bo h he cu en ime dependence and he
e olu ion o wa e ees and gi e an in e p e a ion o he
expe imen al esul s. Finally, using he simula ion esul s om
ou model, we discuss he co ela ion be ween bo h he deg ee
o ageing / le el o pola iza ion ol age, and he e olu ion o
he i ing pa ame e s o cu en expe imen al da a o he
po en ial Cu ie- on Schweidle law.
2 EXPERIMENTAL
Plane-plane LDPE specimens we e ob ained om shee s
p epa ed wi h a "Ca e " hea ing p ess a LEMD-CNRS. A
p ecise me hod o he ab ica ion o hose shee s was
de eloped which allowed us o cu disks o g ea homogenei y
whose hickness/diame e we e espec i ely L≅500 µm
(∆L≅10 µm) and Φ=35 mm. A g anula ed mass o LDPE (base
polye hylene o compounds o cable insula ion wi h
an ioxidan s and s abilize s om Bo ealis), m=1 g, was
inse ed in each o he 12 ci cula holes o a b ass ma ix and
exposed o successi ely selec ed condi ions o p essu e,
empe a u e and wai ing pe iods. In his way we could a ain
he necessa y homogenei y and quasi-uni o m wid h in ou
expe imen al specimens o ensu ing a good (i) epea abili y /
(ii) ep oducibili y in elec ic measu emen s which a e de ined
as: (i) quali y in he epe i ion (i e a ion) o a measu emen on
a unique specimen; (ii) quali y in he epe i ion o a
measu emen on se e al specimens −iden ical specimens in
p inciple−. F om he poin o iew o accele a ed ageing, he
ypical deg ada ion o cable polyme ic insula ions known as
"wa e eeing" was simula ed a labo a o y using a plane-
plane elec ode con igu a ion, a 0.1 M NaCl solu ion and an
ageing ol age o 5 kV ms a 1500 Hz. Once he labo a o y
specimens we e assembled om disks, and p e iously o he
measu emen o pola iza ion cu en wi h a Kei hley 6517A
elec ome e , I( ), he e olu ion o capaci ance, C, and
dielec ic losses, anδ, [4,10] was also con olled o he same
scheduled pe iods o accele a ed ageing. In pa allel wi h hese
expe iences, a des uc i e mic oscopic analysis o he wa e
ee aged specimens was also pe o med, which allowed us
(once he wa e ee kine ics o he LDPE unde s udy was
comple ed) o es ablish a con inued co ela ion be ween he
e olu ion o he abo e elec ical a iables and he ac ual eed
hickness. Such a co ela ion pe mi s he usage o he ela ed
a iables as an ins umen o deg ada ion diagnosis o any
specimen unde es . The desc ip ion o he samples used in
non-des uc i e measu emen s o pola iza ion cu en ,
capaci ance and dielec ic losses is summa ized in Table 1. All
he measu emen s we e aken a oom empe a u e.
Table 1. Summa y o accele a ed ageing condi ions and elemen a y samples
cha ac e is ics ( hickness, L, and capaci ance, C) o samples se ies K, M and
J used in non-des uc i e measu emen s.
AGEING CONDITIONS:………………………V=5 kV ms ; =1500 Hz
SAMPLES CHARACTERISTICS…………{ (L[µm] , ∆L[µm]) ; C[pF] }
Se ies K ( o ag=0h) Se ies M ( ag=0h) Se ies J ( ag=202h)
K4: (534 , 10) ;13.0 M1: (530 , 0) ; 12.9 J1: (514 , 10) ; 17.4
K5: (534 , 10) ; 13.0 M2: (532 , 10) ;12.6 J3: (514 , 10) ; 17.2
K6: (534 , 10) ; 13.0 M3: (532 , 10) ;12.9 J4: (518 , 10) ; 17.2
K7: (530 , 10) ; 12.7 M4: (528 , 10) ;12.8 J5: (520 , 0) ; 18.8
K9: (528 , 10) ; 13.0 M5: (524 , 10) ; 13.1 J6: (520 , 0) ; 16.9
2.1 GROWTH AND CHARACTERIZATION OF WATER
TREES
In o de o ob ain a mul i ude o wa e ees we p oduced a
wide dispe sion o ee incep ion poin s in one o he sides o
he cu ed disks o LDPE. We applied a p essu e o 372 ba ,
du ing an in e al o 2 min wi h an ab asi e pape P400 (used
and eplaced a e i s applica ion) and we epea ed his p ocess
wice o each disk [4,26]. This ab asi e-pape me hod, a leas
in LDPE, e ealed as less agg essi e and mo e ep oducible
han he me hod o sandblas ing [27], especially because we
could choose a g ain wid h which p oduces a kind o de ec
deep enough o ini ia ing wa e ees wi hou in ol ing a
di ec isk o dielec ic b eakdown when applying he ageing
ol age. The labo a o y specimens [4,10] (see Figu e 1) we e
placed on a me allic ecipien , which ac ed bo h as g ound
elec ode and as a con aine o a small quan i y o silicone oil.
Figu e 1. Specimen con igu a ion: (1) pla inum elec ode, (2) silicon cap, (3)
sal solu ion, (4) PE cylinde , (5) LDPE disk, (6) sil e pain .
I helps o a oid pa ial discha ges in he ai by illing he
occasional small ca i ies ha usually appea be ween he
LDPE disk and he me allic plane o he g ound elec ode. We
go se e al e y impo an ad an ages om he usage o he oil
echnique. (a) The e is a d as ic diminu ion o he isk o
dielec ic b eakdown o sho ageing pe iods o hin laye s o
wa e ees. I means no only a educ ion in he numbe o los
es objec s bu he ac ha he p ocess o ee g ow h o he
es o specimens is much mo e con inued and eliable, as
wa e ee e ac ions a ibu ed o pe iods o absence o
applied elec ic ield a e mainly a oided [28]. (b) The
possibili y o applying a s onge ageing elec ic ield
(Eoiled≅10 kV/mm s Enon-oiled≅3 kV/mm). Consequen ly,
deepe and hicke deg aded laye s a e de eloped o much
sho e ageing pe iods. I pe mi s us hei de ec ion h ough
e y signi ican changes o he alues o elec ical a iables in
measu emen s {C,I( ), anδ}. Once he ageing pe iod is
inished, he ac i e pa o he es objec (LDPE disk wi h
wa e ees) is ex ac ed and s ained wi h odhamine du ing 48
h a a empe a u e o 60ºC. A e wa ds, i e 200 µm wid h
slices a e cu om each disk wi h a mic o ome and hei
colou ed wa e ees can al eady be isualized by using an
op ical mic oscope. Wa e ee a e age leng h was de e mined
o each scheduled ageing ime om equa ion (1),
max
1
x
xl
l
N
i
ii
w
∑
=
= (1)
Whe e li and xi a e espec i ely he leng h and wid h o each
indi idual wa e ee, N is he o al numbe o ees pe slice
and xmax is he o al leng h o each slice.
2.2 EXPERIMENTAL PROCEDURES FOR ELECTRIC
MEASUREMENTS
2.2.1 POLARIZATION CURRENT TECHNIQUE
The LEMD expe imen al se -up o pola iza ion cu en
measu emen is shown in Figu e 2. Cu en measu emen and
DC s abilized ol age supply we e bo h pe o med by using a
Kei hley 6517A. Da a acquisi ion was done by means o a
Figu e 2. Expe imen al se -up o cu en measu emen s: (1) IEEE-488 ca d
inse ed in a PC586, (2) GPIB cable, (3) amme e (Kei hley 6517A), (4)
s abilized DC sou ce (Kei hley 6517A), (5) iaxial connec ions, (6)
p o ec ion esis ance, (7) specially designed es cell, (8) con inuous coppe
Fa aday cage.
so wa e de eloped om Kei hley Tes poin packe . A special
goal o his kind o e y low cu en measu emen s consis s in
minimizing noise. Elec omagne ic noise could be educed by
p o iding an especially designed es cell, iaxial connec ions
be ween Kei hley and es cell, a me al box o he p o ec ion
esis ance and an addi ional con inuous coppe Fa aday cage
which con ined all he p e ious elemen s. The physical
componen s o he es cell a e schema ically d awn in Figu e
5
6
7
8
9
1
2
3
4
Figu e 3. Tes cell: (1) PE cylinde , (2) gua d ing, (3) LDPE sample, (4)
lowe elec ode, (5) p o ec i e sc een, (6) PTFE insula o , (7) uppe
elec ode, (8) con ac sp ing, (9) coppe wi e.
3. A special a en ion mus be paid o he design o i s uppe
elec ode which mus make con ac a a e y well de ined a ea
on e e y LDPE specimen. A i s sc ewed e sion o his
uppe elec ode was subs i u ed by a mo e weigh ed one whose
con ac was simply done by i s weigh . In his way,
epea abili y and ep oducibili y we e g ea ly imp o ed
because o he cons an p essu e a he con ac and by a oiding
he possible gene a ion o cha ges by a kind o mechanical
s i ing on he su ace o he polyme [29].
Analy ical/g aphical p ocessing o expe imen al da a was
pe o med wi h Kaleidag aph 3.07 and Sigmaplo 5.0
p og ams.
5
6
1
4
2
3
12 3 4
5
6
7
8
2.2.2 CAPACITANCE
MEASUREMENTS
An IRLAB appa a us (model LDTRP-2) and a 1621 Gene al
Radio b idge successi ely swi ched o he specially designed
cell men ioned abo e we e used o ge he alues o
capaci ance, C, and dissipa ion ac o , anδ, o e e y
scheduled ageing pe iod.
2.2.3 PARASITIC EFFECTS
Be o e each measu emen , he sal solu ion is emo ed and
he es o solu ion on he sample su ace is elimina ed by a
p ocess o abso p ion wi h abso p ion pape [4]. Ne e heless
we obse ed ha solu ion con inues o e apo a e om he
sample a oom empe a u e o abou 20 minu es, which is
p o ed by a con inuous diminu ion o i s capaci ance and
dielec ic losses ac o . Consequen ly, a e abso p ion, a
wai ing pe iod, d y≅20 min, is absolu ely equi ed o a oiding
a ia ions in capaci ance (and in he es o elec ical
a iables) wi h ime −pa icula ly in hick eed laye s−. Then,
when his pe iod is o e , we conside ha he measu ed
capaci ance is ep esen a i e bo h o polye hylene and he
wa e ee wi h i s wa e . Conce ning pola iza ion cu en
measu emen s, esidual cu en s can be measu ed in he
absence o applied pola iza ion ol age, VDC , when he ageing
pe iod is o e . They can endu e o pe iods anging om 10
min up o 1 h depending on he LDPE olume a ec ed by
wa e ees. An adequa e me hod o minimizing hese esidual
cu en s was applied by sho -ci cui ing he en i e es objec
suppo −once he ageing pe iod is inished− du ing a pe iod
o abou , sho ≅30 min.
(a)
(b)
Figu e 4. Wa e ee e olu ion in LDPE unde accele a ed condi ions −ageing
ol age o 5 kV ms a 1500 Hz wi h solu ion [NaCl]=0.1 M−: (a) h ee
pho og aphs aken a e ageing imes ( om op o bo om) o 4 h, 31 h and
272 h; (b) a e age wa e ee leng h dependence on ageing ime, lw e sus
ag1/2.
3 EXPERIMENTAL RESULTS
3.1 THE GROWTH KINETICS AND THE
CAPACITANCE OF WATER TREES
In pa allel wi h he non-des uc i e measu emen s o
capaci ance and cu en wi hin ou expe imen al p ocedu es,
we also unde ook des uc i e measu emen s in o de o ob ain
he hickness o ee deg aded polyme −wa e ee leng h lw−
o di e en ageing pe iods as we explained in sec ion 2.1.
Des uc i e measu emen s a e necessa y o wo main easons :
i s ly hey enable us o es ablish a con inued co ela ion
be ween elec ical a iables (cu en / capaci ance) and
deg ada ion hickness which ep esen s a diagnos ic pu pose
and secondly and mos impo an , wa e ee leng h
measu emen s a e necessa y o e alua e wa e ee pe mi i i y
[30] which is essen ial o ma hema ical simula ions om ou
new p oposed conduc ion model.
The e olu ion o wa e ees wi h he ageing ime in a LDPE
sample is shown bo h g aphically wi h pic u es and om he
a ia ions o he a e age wa e ee leng h −wa e ee
kine ics− in Figu e 4.
F om Figu e 4b we conclude ha he wa e ee leng h a ies
e y app oxima ely wi h ageing ime as ag1/2, in ag eemen
wi h he e y ecen heo e ical model published by C ine and
Jow [31].
Figu e 5. A e age elec ic capaci ance as a unc ion o ageing ime.
O he wise, a quasi-linea dependence o he a ia ions o
he a e age capaci ance o he samples wi h he ageing ime
was also ob ained om di ec measu emen s (Se ies K). This
esul is plo ed in Figu e 5.
Using an equi alen plane capaci o model o he wa e
eed samples and aking in o accoun he a ia ions o
capaci ance, C, and wa e ee leng h, lw, wi h he ageing ime,
ag, he a e age ela i e pe mi i i y o wa e ees, ε1, can be
de i ed om ,
[]
iagagagw
agwag
C CL C l
l C
−−
=)()()(
)()(
)( 2
1
ε
ε
(2)
whe e ε2=2.3 is he ela i e pe mi i i y o non-deg aded PE
and Ci is he non-deg aded capaci ance o he sample. A
maximum alue, ε1max=4.1, was a ained.
ag
1/2(h)1/2
024681012141618
lw (µm)
0
100
200
300
400
T ee leng h, lw(
µ
m)
Ageing ime,
en (h)
0 50 100 150 200 250 300
C(pF)
12
13
14
15
16
17
18
19
ag (h)
Capaci ance, C (pF)
Ageing ime,
3.2.1 EVOLUTION OF I( ) WITH DEGRADATION
Figu e 6 shows he pola iza ion cu en expe imen al da a o a
measu ing pe iod o 10 min, a pola iza ion ol age VDC=300 V
and di e en pe iods o accele a ed ageing. The expe imen al
da a we e i ed o he po en ial Cu ie- on Schweidle law,
I=I0 −m . Thei co esponding pa ame e s {I0,m} and deg aded
wid h pe cen age, g(%)=(lw/L)·100, we e summa ized in Table
2. We can ob ain a i s e sion o diagnos ic me hod
−es ima ion o he deg ee o deg ada ion om a unique I( )
measu emen − by simply plo ing and adjus ing he
dependence g=g(m), which is shown in Figu e 7.
Figu e 6. Pola iza ion cu en measu emen s o di e en wa e ee ageing
pe iods and a pola iza ion ol age o VDC=300V (a e aged alues om
samples o Se ies K): (1) 0h, (2) 15h, (3) 34 h, (4) 64 h, (5) 146 h, (6) 250 h.
Table 2. Fi ing pa ame e s {I0,m} o each ageing ime and co esponding
deg aded wid h pe cen age.
ag(h) g(%) I0(A) m
0 0
6.251·10−12 0.666
15 14.8
1.1477·10−10 0.8106
34 22.3
4.1352·10−10 1.0566
64 34
3.5889·10−10 1.0893
146 56.3
1.2905·10−9 1.2491
250 70.7
6.1215·10−10 1.3001
Figu e 7. The dependence g=g(m) as a i s e sion o diagnos ic ool o
LDPE eed samples.
3.2.2 EVOLUTION OF I( ) WITH POLARIZATION
VOLTAGE
Figu e 8 shows he pola iza ion cu en measu emen s in aged
samples (Se ies J) as a unc ion o he pola iza ion
ol age, VDC. A simila s udy was pe o med o new samples
(Se ies M) and hei co esponding i ing pa ame e s {I0,m}
a e summa ized in Table 3. A p e ailing end o sa u a ion in
he e olu ion o exponen m wi h pola iza ion ol age, VDC, o
bo h aged (dec easing a ia ion) and new (inc easing
a ia ion) LDPE samples is obse ed and depic ed in Figu e 9.
Consequen ly, a second ool o deg ada ion diagnosis o his
kind o es objec s could be de eloped on he basis o he
a ia ions o exponen m as a unc ion o he pola iza ion
ol age le el.
Figu e 8. Pola iza ion cu en measu emen s and hei linea cu e i ing o
di e en alues o he pola iza ion ol age, VDC (a e aged alues om aged
samples o Se ies J wi h deg aded wid h pe cen age, g=63%): (1) 100 V, (2)
300 V, (3) 500 V, (4) 1000 V.
Table 3. Pola iza ion cu en i ing pa ame e s {I0,m} o new samples
(Se ies M) and aged samples (Se ies J) as a unc ion o VDC.
VDC(V) I0(A) m
Se ies M Se ies J Se ies M Se ies J
100 4.7826·10-13 1.3465·10-9 0.42736 1.6603
300 2.5357·10-12 1.6406·10-9 0.61002 1.3546
500 4.0321·10-12 2.4369·10-9 0.63351 1.3102
1000 1.0702·10-11 4.0661·10-9 0.69758 1.2419
Figu e 9. De ail o he endency o sa u a ion in he e olu ion o he exponen
m in new (1) and aged (2) LDPE samples as a unc ion o pola iza ion
ol age.
4 CONDUCTION PROCESS MODELLING
A Debye- ype conduc ion model based on he dielec ic
elaxa ion o independen dipoles o polye hylene was
p esen ed in [22]. A mo e ealis ic new app oach o he
modelling o hese p ocesses is p oposed in his wo k [32]. I
is based on he assump ion o in e ac ion be ween dipoles
ollowing he "Many-Body Uni e sal Model o Dielec ic
m
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
g%
0
20
40
60
80
Deg aded wid h pe cen age, g%
Exponen , m
Se ies K , VDC=300 V
(s)
10 100 1000
I(A)
10-13
10-12
10-11
10-10
(1)
(2)
(3)
(4)
(5)
(6)
Pola iza ion cu en , I(A)
Time, (s)
VDC(V)
0 200 400 600 800 1000 1200
m
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
(1)
(2)
Exponen , m
Pola iza ion ol age, VDC(V)
10-13
10-12
10-11
10-10
10-9
10 100 1000
SERIE J
(1)
(2)
(3)
(4)
(s)
Se ies J
I(A)
Pola iza ion cu en , I(A)
Time, (s)
3.2 CURRENT MEASUREMENTS

Relaxa ion” [23,24,25]. We ha e ied o alida e he p oposed
model in a LDPE a ec ed by di e en le els o wa e ee
deg ada ion. The model is ounded on se e al hypo heses: one
o hem akes as a basis he a ia ions o pe mi i i y wi h he
deg ee o wa e eeing damage and o he s ake in o accoun
he mic oscopic s uc u e o he ma e ial. Wi h hese
simpli ying hypo heses he p oposed model can accoun o he
la ge pa o ou expe imen al esul s. The applica ion o he
"Many Body Uni e sal Model o Dielec ic Relaxa ion"
implies ha o a long enough measu ing ime, a "slow down"
p ocess in pola iza ion is ac i a ed: i is due o he mechanism
o local edis ibu ion o ene gy ( lip- lop ansi ions) which
esul s in a powe dependence o pola iza ion cu en , wi h a
nega i e exponen whose absolu e alue is m∈(0,2). Because
o he ma hema ical complexi y o his heo y, we p opose in
his pape a modi ica ion o he model o non-in e ac ing
dipoles p esen ed in [22], in o de o ake in o accoun
in e ac ions. Wi h he aid o he new model we ha e ied o
explain he a ia ions o pa ame e s in he powe cu en law
(bes - i ed dependence o he pola iza ion cu en
measu emen s) as a unc ion o (a) he deg ee o deg ada ion
by wa e ees and (b) he pola iza ion ol age. Finally, a
discussion abou he alida ion o he model and i s ange o
applicabili y is pe o med.
4.1 MODEL
4.1.1 TWO POTENTIAL WELL MODEL
The mechanism o dipole o ien a ion is o en conside ed as
esponsible o pola iza ion cu en s in solid dielec ics.
Ne e heless, in a dielec ic ma e ial in which no impo an
dipola g oups a e p esen , pola iza ion cu en s could be
a ibu ed o ion jumps be ween equilib ium nea by-posi ions.
Bo h mechanisms can be explained by means o a unique
simpli ied model in which a cha ged pa icle can jump
be ween wo close posi ions sepa a ed by a po en ial ene gy
ba ie − wo po en ial well (Figu e 10)− [23,33]. Fo his
Figu e 10. Elemen a y scheme o wo po en ial well model: E, applied
elec ic ield; 2a, dis ance be ween wo well si es.
model and conside ing a plane-plane sample o ela i ely low
applied elec ic ields, he a ia ion o local pola iza ion P
wi h ime is gi en by,
),(
)(
),(
),( xE
Tk
xb
xP
xP
B
α
α
+−=
∂
∂ (3)
whe e E(x, ) is he elec ic ield, α is he p obabili y pe uni
ime ha one ion changes i s equilib ium posi ion in he
absence o applied elec ic ield and due o he he mal ene gy,







−= Tk
W
B
exp2
να
(4)
whe e ν is he ib a ion equency o ions inside hei wells, W
is he heigh o he po en ial ene gy ba ie which sepa a es he
wo espec i e well si es and b(x) is gi en by,
b(x) = p2n(x) (5)
whe e he concen a ion o ions is ep esen ed by n(x) and he
dipola momen associa ed o one ion is ep esen ed by p.
F om (3) and conside ing ha he con ibu ion o ohmic
conduc ion is i ele an , we ge a simpli ied exp ession o he
cu en densi y,
xP
xE
J ∂
∂
+
∂
∂
=),(),(
)( 0
ε
(6)
We will s udy he dielec ic esponse o a s ep ol age gi en
by,
)(·)( uV V DC
= (7)
4.1.2 ALTERNATIVE APPROACH FOR DIPOLE
INTERACTIONS
Due o he di icul y o he in e p e a ion o he physical
meaning in he coe icien s o he powe cu en esponse
supplied by he "Many Body" heo y −especially om an
expe imen al/applied poin o iew−, we p opose a simpli ied
al e na i e app oach: depa ing om he non-in e ac ing
dipoles model (ma hema ical equa ions, nume ical calcula ions
and simula ions de eloped in [22]), we in oduce a simple
ma hema ical algo i hm ha accoun s o dipole in e ac ions.
This new model also p edic s a powe cu en law o he
esponse o he dielec ic ma e ial (simila dependence o he
expe imen al Cu ie- on Schweidle law, I( )=I0 -m) bu his
ime, coe icien s {I0,m} can be eadily de e mined as easy
unc ions o he mic oscopic/mac oscopic pa ame e s o he
ma e ial. Dipole in e ac ions will be desc ibed by an
"e ec i e" po en ial ene gy ba ie , which g ows wi h he
measu ing ime as,
)()( 0 WW W ∆+= (8)
whe e W0 is he po en ial ene gy ba ie o he non-in e ac ing
dipoles model. Consequen ly, i implies ha pa ame e α o he
new model mus depend on ime . Such a ime dependence
mus be a dec easing one and i s ini ial alue mus coincide
Dis ance
-3-2-10123
Po en ial
-2
0
2
4
6
8
10
12
2a
2Ea
wi h ha o he non-in e ac ing model, α
0
. We p opose nex
exp ession o α( ),
τ
α
α
+
=
1
)(
0
(9)
4.2 HOMOGENEOUS MATERIAL
In he case o a homogeneous ma e ial, all he a iables in
equa ion (3) a e independen om posi ion, x. Fu he mo e,
o > 0 elec ic ield is independen om ime and is gi en by
E=V
DC
/L , hen equa ion (3) becomes,
()
S
P P
d
dP −−= )()(
)(
α
(10)
whe e,
TLk
bV
P
B
DC
S
=
(11)
By in eg a ion o (10) and making use o equa ions (9) and
(6) we ge ,
1
1)()(
−−
⎟
⎠
⎞
⎜
⎝
⎛++=
τα
τ
αδε
o
P
L
V
J
Soo
DC
(12)
Finally, i >> τ , cu en is
m
o
I I
−
=)( (13)
whe e,
(i)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−+=+= Tk
W
m
B
o
0
exp211
νττα
(14)
(ii) m
B
DC
m
Soo
TLk
bV
SPSI
τατα
0
==
which means ha we ge exp essions ha di ec ly connec he
mic oscopic/mac oscopic a iables o ou LDPE sample wi h
he ac ual alues o ou measu ed expe imen al cu en
pa ame e s {I
0
,m} [34]. Mo eo e , we will be able o discuss
he e olu ions o he pa ame e s (Cu ie- on Schweidle law)
wi h he a ia ions o pola iza ion ol age and wa e ee
deg ada ion, as we will see la e on.
4.3 INHOMOGENEOUS MATERIAL
In he case o a wa e ee deg aded LDPE −inhomogeneous
ma e ial (see schema ic d awing in Figu e 11)− we canno
ob ain, in gene al, an analy ical solu ion om (3).
Consequen ly we used nume ical me hods o he calcula ion
and simula ion o pola iza ion, P(x, ), and pola iza ion cu en ,
I( ). The hypo heses used o he nume ical calcula ions a e:
1.
Pa ame e α( ) is independen o posi ion, x.
2.
Pa ame e b(x) which is ela ed o he pe mi i i y o he
ma e ial depends on he posi ion as,
(15)
Figu e 11. Model o he a ia ion o pa ame e b=b(x) wi hin he LDPE
plane-plane sample om he hypo heses o he a ia ion o pe mi i i y in a
wa e eed specimen [30,10]: l
w
=gL/100 ≡ deg aded wid h co esponding o
a con inuous laye o wa e ees.
whe e g ep esen s he pe cen age alue o deg aded ma e ial,
b
max
is he alue o b(x) ha co esponds o a maximum
ela i e pe mi i i y whose ac ual alue was chosen as 4.1 ( o
ag
=272 h), b
PE
is he alue o b in he non-deg aded egion,
whose ela i e pe mi i i y was chosen as 2.3 and u is he
slope o he assumed linea dependence. The ela ed
pe mi i i y alues a e aken om eal pe mi i i y
measu emen s pe o med on polye hylene specimens deg aded
by wa e ees [34,4]. We also ook in o accoun o ou
calcula ions he ela ionship be ween he mac oscopic a iable
ela i e pe mi i i y, ε
, and he mic oscopic pa ame e b=np
2
,
Tk
xb
x
B
0
)(
1)(
ε
ε
+=
(16)
4.4 SIMULATION RESULTS
Expe imen al esul s a e in e p e ed om he p oposed
model o in e ac ing dipoles by using equa ion (13).
Ne e heless, o small wa e - ee deg aded laye s, an
in e p e a ion o he exponen pa ame e m o expe imen al
pola iza ion cu es is no possible as m<1. Acco ding o
equa ion (14i), his model can only explain he cu en cu es
co esponding o a highe le el o deg ada ion which gi es
expe imen al exponen s m g ea e han uni y (m>1): i.e. he
expe imen al esul s o Figu e 6 ha a e ela ed o ageing
imes:
ag
= 34 h, 64 h, 146 h, 250 h. F om nume ical
in eg a ion, we can ob ain he alues o he cha ac e is ic
mic oscopic pa ame e s o ou conduc ion modelling (τ, α
0
,
⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎥
⎦
⎤
⎜
⎝
⎛
∈
⎥
⎦
⎤
⎢
⎣
⎡
∈−
=
LL
g
xb
L
g
xuxb
xb
PE
,
100
,
100
,0,
)(
max
x
x
b
(
x
)
L
l
w
l
w
b
PE
b
max
Table 4. Mic oscopic pa ame e s om nume ical in eg a ion (τ, α0, W0) as a
unc ion o he deg ee o wa e ee deg ada ion, g.
g(%) τ
ττ
τ (s) α
αα
α0 (s-1) W0 (eV)
22.3 0.14 0.124 0.768
34 0.12 0.37 0.74
56.3 0.52 0.454 0.735
70.7 0.27 1.0385 0.713
W0) as a unc ion o he le el o deg ada ion in he insula ing
ma e ial, as shown in Table 4. Figu e 12 is jus an example o
he deg ee o adjus men be ween he cu en esul s I( ),
ob ained om nume ical in eg a ion (cu e (2)) and om he
expe imen al da a (cu e (1)) co esponding o a sample o
deg aded wid h g=70.7% ( ag= 250 h). F om Table 4 we
obse e an inc ease o he alue o α0 wi h deg ada ion which
co esponds o a diminu ion in he po en ial ene gy ba ie ,
W0.
Time, (s)
10 100 1000
Pola iza ion cu en , I(A)
10-13
10-12
10-11
10-10
o (1)
(2)
Figu e 12. Cu en esul s om nume ical simula ion (2) and om
measu emen s (1) o a LDPE eed sample o deg aded wid h, g=70.7%
( ag=250 h) and mic oscopic pa ame e s: τ=0.27 s, α0=1.0385 s-1 , W0=0.713
eV.
5 DISCUSSION
The ela i e pe mi i i y o wa e ees in LDPE
con inuously inc eased om i s alue o non-deg aded PE,
ε2=2.3, up o a maximum alue ε1max=4.1, o an ageing pe iod
o 272 h and an a e age wa e ee leng h o 373 µm
(g=74.6%); i supplies a pe mi i i y ampli ica ion ac o
Aε=1.8, a li le bi highe han he alue ound o XLPE in
[10], Aε=1.6 . F om pola iza ion cu en measu emen s we
could de e mine he e olu ion o coe icien s {I0,m} o he
Cu ie- on Schweidle law wi h he ageing ime. A p og essi e
inc ease was ob ained o bo h coe icien s: 9897% o I0 and
95% o m, when he deg aded wid h g ew om g=0% (new
sample) up o g=70.7% (sample wi h a high le el o
deg ada ion). O he wise, we could es ablish, as VDC go om
100 V up o 1000 V: (I0) 2137% inc ease in new samples and
202% inc ease in aged samples; (m) 63% inc ease in new
samples, 25% dec ease in aged samples and, in bo h cases, i
appea s a ema kably end o sa u a ion wi h VDC.
5.1 VARIATIONS OF {I0,m} WITH g(%)
The diminu ion o ene gy ba ie W0 wi h deg ada ion is a
consequence o he inc easing acili y in he mo emen o
cha ge ca ie s, which is due o he des uc ion o he in e nal
s uc u e o he polyme known as "wa e eeing" [2,4,35].
The inc ease o α0 as an e ec o he diminu ion o W0 is
gi en by equa ion (4). Then, i equa ions (14) a e supposed o
be alid in a quali a i e sense o he case o a wa e - eed
ma e ial, he inc ease o α0 explains he inc easing dependence
o I0 and m wi h deg ada ion (Figu e 6, Table 2 and Table 3).
Conce ning nume ical calcula ions, he e was a good
ag eemen be ween measu emen s and simula ion esul s o
pola iza ion cu en s in aged samples o ageing pe iods: 34 h,
64 h, 146 h and 250 h (Figu e 12).
5.2 VARIATIONS OF {I0,m} WITH VDC
(I0) F om equa ion (14ii) we conside ha , basically,
pa ame e I0 is linea ly dependen on VDC (i.e. dependencies
on VDC h ough pa ame e s α0 and τ a e supposed o be less
impo an o i ele an ). Consequen ly I0 should inc ease wi h
VDC bo h in aged samples −in a quali a i e sense− and in new
samples −in an analy ical sense−, as we ound in ou cu en
measu emen s (see Table 3, Figu e 8).
(m) New samples: F om equa ion (14i) we can explain he
enhanced alues o m as VDC g ows (Table 3 −Se ies M−,
Figu e 9) i we show he physical eason o he dec ease o
W0 wi h VDC: a highe applied ol age, VDC, p oduces a highe
in e nal ene gy o he sys em which means again an inc eased
acili y o mo emen inside he a omic-molecula sys em (bo h
o he cha ge ca ie jumps and o he o a ions o molecula
chains ha cons i u e he polyme ). The esul an e ec is "jus
as i " he e ec i e wo-wells ene gy ba ie , W0, had a
diminu ion. Na u ally and u he mo e, he supe posi ion o he
cu e ep esen ing he wo wells po en ial ene gy o he cu e
o he applied po en ial ene gy (applica ion o VDC) is
equi alen o a physical diminu ion in he heigh o he
po en ial ene gy ba ie .
Aged samples: We should emphasize he ac ha he mo e
deg aded he wa e eed LDPE, he mo e inhomogeneous i
becomes and a la ge numbe o possible pola iza ion
mechanisms i acqui es, i.e., he deg aded LDPE can adop a
whole spec um o possible di e en ene gy ba ie s, W0i, one
o each ype o a omic-molecula species in ol ed and/o
p ocess o dielec ic elaxa ion. Such di e en mechanisms a e
a ailable due o he s a e o deg ada ion o he polyme bu
hey will only be e ec i e mechanisms − hey will be
ac i a ed− depending on he le el o pola iza ion ol age ha
is applied o each expe ience. I a highe DC ol age is
applied, new pola iza ion mechanisms wi h a g ea e ba ie
heigh , W0i, a e ac i a ed, which implies om equa ions
(4,14i) espec i ely ha he co esponding jumping
equencies α0i=2νexp(−W0i/kT) a e smalle in hese p ocesses
and hen ha he dec easing slopes in he log-log
ep esen a ion, mi, a e smalle . The inco po a ion o hose new
mechanisms esul s in an a e age slope, m, ha diminishes as
VDC g ows (see Table 3 −Se ies J−, and also ha e a look a
Figu es 8-9). Al hough he p ocess o e ec i e diminu ion o
each pa icula ba ie heigh wi h he applied ol age
desc ibed o new samples also applies o aged samples, i is
clea ha he e ec o ac i a ion o g ea e ba ie s in aged
samples due o inc easing deg ada ion mus be a p edominan
one.
5.3 FURTHER CONDUCTION MODELLING
The new p oposed model explains sa is ac o ily he po en ial
−Cu ie- on Schweidle − dependence o pola iza ion cu en s
in LDPE specimens new and aged wi h di e en deg ees o
wa e - ee damage up o a e y high pe cen age deg aded
wid h o abou 70%. In aged samples, he new model can also
accoun o he a ia ions o pa ame e s {I0,m} as a unc ion o
deg ada ion (g%) and I0 as a unc ion o pola iza ion ol age.
The inc ease o he wa e ee hickness seems o diminish he
po en ial ene gy ba ie which is compa ible wi h he
inc easing alues o I0 and dec easing alues o W0 om
Tables 2, 3 and 4. The explana ions o he a ia ions o
exponen m as a unc ion o VDC a e much mo e con o e sial
as discussed in p e ious sec ion and need mo e cla i ica ion.
In new samples, expe imen al cu en measu emen s show
exponen s m<1 and hen ou model (see equa ion 14.i) canno
explain hem, he e o e o he models should be e ised
[36,37,38] and e en an expe imen al s udy as a unc ion o
empe a u e would be o in e es o con i m he p edic ed
empe a u e dependences in equa ions 14.
In o de o elabo a e mo e de ailed models o pola iza ion
cu en s in PE i is necessa y o inco po a e he he mo-
elec ical deg ada ion mechanisms o simula ions. These
mechanisms should ake in o accoun he p ocesses o space
cha ge accumula ion, cha ge anspo and he dis ibu ions
and concen a ions o apping cen e s [39]. Recen
in es iga ions (“ARTEMIS” Eu opean p ojec [14,40]) ha e
al eady cha ac e ized PE insula ion by using “ageing ma ke s”.
This cha ac e iza ion is pe o med by means o sophis ica ed
expe imen al echniques (PEA, FTIR, SEM, TEM, TSM,
DEA, DSC,...), some o which ou esea ch g oup has also
begin o use. These s udies will ha e many indus ial
applica ions o cable manu ac u e s and powe dis ibu ion
u ili ies. Pa icula ly, he de elopmen o unde g ound ex a
high ol age XLPE cables (EHV: 400-500 kV) whe e elec ic
ield is app oxima ely, E∼16 kV/mm, qui e highe han elec ic
ield in adi ional mode a ely-s essed powe cables, E∼6
kV/mm. Secondly, he de elopmen o “ obus ” diagnos ic
echnologies which would also se e o e alua e and imp o e
he eliabili y o adi ional powe cables ha ha e been in
se ice o abou 30 yea s. This is o ou s anding ele ance o
dis ibu ion companies in he p esen de egula ed ma ke ,
whe e u ili ies a e “ ie cely” compe ing o o e ing clien s a
mo e eliable and non-expensi e p oduc .
6 CONCLUSIONS
Accele a ed-aged LDPE specimens wi h di e en le els o
wa e ee deg ada ion we e cha ac e ized by means o
measu emen s o wa e ee kine ics, capaci ance and
pola iza ion cu en . Combining wa e ee kine ics and
capaci ance measu emen s, he maximum a e age alue o he
ela i e pe mi i i y o wa e ees in LDPE was calcula ed,
ε1max=4.1 . Pola iza ion cu en measu emen s we e i ed o
he po en ial Cu ie- on Schweidle law in o de o s udy he
e olu ion o i s pa ame e s {I0,m} wi h ageing ime and
pola iza ion ol age.
An in e p e a ion o he expe imen al esul s has been gi en
om a new conduc ion model in LDPE ha akes in o accoun
dipole in e ac ions and was ob ained om a modi ica ion o a
p e iously published wo-wells Debye model. The p oposed
model p o ides easy analy ical exp essions o he
dependencies o he expe imen al Cu ie- on Schweidle
pa ame e s on bo h mic oscopic and mac oscopic a iables o
he LDPE plane-plane samples unde es . F om he e olu ion
o hese a iables wi h wa e ee deg ada ion wid h, g(%), and
pola iza ion ol age, VDC, we could simula e pola iza ion
cu en s ha had a good deal o acco d o measu emen s in
accele a ed-aged LDPE specimens.
This wo k ep esen s a con ibu ion o imp o e he
knowledge o conduc i e p ocesses in wa e - ee deg aded
polye hylene which can ce ainly allow a be e
cha ac e iza ion o powe cable insula ion unde se ice
condi ions.
ACKNOWLEDGMENT
Au ho s a e g a e ul o indus ial pa ne s GENERAL
CABLE (Manlleu-Ba celona, Spain) –cable manu ac u e - and
Se illana-ENDESA (Se illa, Spain) –dis ibu ion u ili y- o
supplying cable samples and collabo a ion du ing he pe iod o
de elopmen o he p esen wo k.
REFERENCES
[1] L. A. Dissado and J.C. Fo he gill, Elec ical Deg ada ion and
B eakdown in Polyme s, ed. G.C. S e ens P. Pe eg inus o IEE
London, U.K. (ISBN 0 86341 186 7), 1992.
[2] E.F. S eennis and F.H. K euge , ″Wa e eeing in polye hylene powe
cables″, IEEE T ans. Elec . Insul., Vol. 25, pp. 989-1028, 1990.
[3] I. Radu, P. No inghe and J.C. Filippini, ″The in luence o an indi idual
wa e ee on he elec ic ield dis ibu ion in poin -poin geome y″, J.
Elec os a ics, Vol. 48 (3-4), pp. 165-178, 2000.
[4] M. Acedo, “Rela ions be ween he wa e ee deg ada ion o
polye hylene and he elec ical p ope ies o he ma e ial. Applica ion o
he insula ion diagnos ics o powe cables”. PhD Thesis, Uni e si y o
Se illa, 2004.
[5] J.L. Chen and J.C. Filippini, “The Mo phology and Beha io o he
Wa e T ee”, IEEE T ans. Elec . Insul., Vol. 28, pp.271-286, 1993.
[6] M. Acedo, I. Radu, F. F u os, J.C. Filippini and P. No inghe , ″Wa e
T ees in Unde g ound MV Cables: Modelling and Elec ic Field
Pe u ba ion″, J. Elec os a ics, 53 (4), pp. 267-294, 2001.
[7] I. Radu, ″Elec ic ield calcula ion and he in luence o wa e ees on
insula ion b eakdown in needle–plane geome y″, J. Elec os a ics, Vol.
60, pp. 49-67, 2004.
[8] M. Acedo, F. F u os, M. To es and J.C. Filippini, "The Cylind ical
Capaci i e Model o Wa e T eeing Deg ada ion in Ex uded HV
Cables", 5 h In e na ional Con e ence on Conduc ion and B eakdown in
Solid Dielec ics, Leices e , IEEE Publica ion 95CH3476-9, pp. 661-
666, 1995.
[9] L.A. Dissado, ″P edic ing Elec ical B eakdown in Polyme ic
Insula o s″, IEEE T ans. Dielec . Elec . Insul., Vol. 9, pp. 860-875,
2002.
[10] I. Radu, M. Acedo, J.C. Filippini, P. No inghe and F. F u os, ″In luence
o Wa e T ees on he Elec ic Field Dis ibu ion in XLPE and i s
Consequences on Insula ion Dielec ic S eng h″, IEEE T ans. Dielec .
Elec . Insul., Vol. 7, pp. 860-868, 2000.