The Cylindrical Capacitive Model for Water Treeing Degradation in Extruded HV Cables

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.