Dielectric Characterization and Conduction Modelling of a Water Tree Degraded LDPE

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. 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