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1 Cooper, E. S., Dssado, L. A. & Fothergll, J. (2005). Applcaton of thermoelectrc agng models to polymerc nsulaton n cable geometry. IEEE Transactons on Delectrcs and Electrcal Insulaton, 12(1), pp do: /TDEI Cty Research Onlne Orgnal ctaton: Cooper, E. S., Dssado, L. A. & Fothergll, J. (2005). Applcaton of thermoelectrc agng models to polymerc nsulaton n cable geometry. IEEE Transactons on Delectrcs and Electrcal Insulaton, 12(1), pp do: /TDEI Permanent Cty Research Onlne URL: Copyrght & reuse Cty Unversty London has developed Cty Research Onlne so that ts users may access the research outputs of Cty Unversty London's staff. Copyrght and Moral Rghts for ths paper are retaned by the ndvdual author(s) and/ or other copyrght holders. All materal n Cty Research Onlne s checked for elgblty for copyrght before beng made avalable n the lve archve. URLs from Cty Research Onlne may be freely dstrbuted and lnked to from other web pages. Versons of research The verson n Cty Research Onlne may dffer from the fnal publshed verson. Users are advsed to check the Permanent Cty Research Onlne URL above for the status of the paper. Enqures If you have any enqures about any aspect of Cty Research Onlne, or f you wsh to make contact wth the author(s) of ths paper, please emal the team at publcatons@cty.ac.uk.

2 Applcaton of thermoelectrc ageng models to polymerc nsulaton n cable geometry E. S. Cooper, L. A. Dssado*, J. C. Fothergll Unversty of Lecester Lecester LE1 7RH U.K. *lad4@le.ac.uk Abstract The lfe expressons of models of nsulaton ageng are functons of temperature and feld as well as materal parameters. A methodology s presented that allows these models to be appled to a cable geometry n whch there s a radal varaton of both feld and temperature. In ths way materal parameters can be extracted from cable data. The methodology s llustrated usng one such model and the parameters deduced from cable falure dstrbutons are compared wth those obtaned for thn flms. Ths comparson allows conclusons to be drawn about how the ageng process affects specmens of the same materal wth dfferent volumes. Keywords Modellng, Ageng, Falure, Cable Geometry, Analyss of Cable Lfetmes 1

3 1. Introducton Polymerc materals are used as electrcal nsulators n a wde range of ndustral applcatons, from thn flms n capactors to thck nsulaton layers n hgh voltage power cables. In all cases the servce lfe of the delectrc s of major commercal nterest and consequently a number of theoretcal models have been developed wth the am of relatng the workng lfetmes of delectrc polymers to the electrcal and thermal stress experenced. Current nterest has focussed on physcal theores proposed by: by L.A. Dssado, G.C. Montanar and G. Mazzant [1-4], T.J. Lews, P.J. Llewellyn, C.L. Grffths, P.W. Sayers and S. Betterdge [5-13], J.P Crne and J. Parpal [14,15], L. Smon [16], and J. Artbauer [e.g. 17]. Ageng models such as those mentoned above generally result n a mathematcal expresson that descrbes the lfetme of a specmen as a functon of the electrcal stress, E, and temperature, T t experences. Other factors related to materal propertes are also nvolved. When the specmens under nvestgaton are thn flms aged under spatally constant feld and temperature condtons, fttng such expressons to lfetme data and conversely predctng lfetmes usng them s relatvely straghtforward. However, n systems such as power cable nsulaton the stuaton s more complex. The nsulaton of a power cable under load experences a radally varyng temperature dstrbuton due to Joule heatng of the conductor [18,19], as well as a radally varyng electrcal stress dstrbuton, whch wll be dfferent for AC and DC appled voltages [18,19,20]. The dfference n AC and DC electrcal stress profles arses from the fact that n the AC case the stress profle s controlled by the permttvty of the nsulaton, whch depends only very weakly on 2

4 temperature at least for the range of temperatures typcally experenced by cable nsulaton. In the DC case the electrcal stress s controlled by the conductvty, whch s strongly dependent on both temperature and stress, and ths can lead to a stuaton n whch the feld stress experenced by the nsulaton s not necessarly largest close to the cable core. Ths effect s descrbed brefly n the followng sectons. The radal varaton n E and T makes usng the ageng models to ether ft or predct lfetme data n cable ageng experments more dffcult than n the thn flm case. In the frst nstance t may seem that all that s requred to convert the lfe models to cable geometry s to calculate the regon where the electrc feld and temperature s largest and then apply them to that regon as a cylndrcal shell. However such s not the general case. For DC power cables n partcular t cannot always be assumed that the regon where the temperature s hghest s the regon where the electrc feld s hghest [21]. In the second place the regon at rsk depends upon the physcs of the ageng process, for example n [1-4] ths s assumed to be the regon of hghest space charge concentraton, n [5-13] t s assumed to be a layer of hgh electro-mechancal stress, and n [15, 16] regons of free volume that allow hgh local currents. Fnally there are the materal factors that are nvolved n ageng, such as the actvaton energy for the process, the susceptblty of the local regons to the acton of the feld, and the amount of local damage needed for mmnent falure to be ntated. All of these may be dstrbuted n value [22]. There may also be radal dfferences n morphology that wll affect the factors controllng ageng. Sample ageng wll take place most rapdly n the regon where the combned factors of temperature, feld, and materal propertes are worse. It s therefore necessary to develop a method that can take account of such 3

5 varatons, and ths cannot be restrcted to a sngle shell but wll have to nvolve the whole of the cable. In the followng sectons we descrbe a general methodology that allows ageng models to be appled to cable geometry. In prncple ths methodology allows predcton of cable lfetme usng model parametersaton from thn flms, however the unknown volume dependence of the thn flm parameters makes ths procedure mpossble at present. Instead the method s used to derve parameter values approprate to the cable volume from cable falure data. Ths approach s llustrated usng the DMM [1-4] lfetme expresson. In ths work the method s appled to data from ac ageng as ths smplfes the calculaton and was the only data avalable to the authors n suffcent quantty to make an analyss feasble. The method s however, equally applcable to ageng n a dc feld provded that the dfference n the radal varaton of the temperature and feld s taken nto account. The calculaton s also smplfed by focussng upon the characterstc values of the parameters of the lfe expresson. These are derved and compared to those obtaned for thn flms and the dfferences commented upon. Calculatons that take account of the dstrbuton n value of the materal parameters n order to ft the expermental lfe dstrbutons [22] wll be reported at a later date. 2. Method All theoretcal models yeld an expresson for the lfetme n terms of temperature, electrc feld and mechanstc parameters that may be dependent upon the materal and/or the electrode-materal nterface. In the case of cable nsulaton one or both of the temperature and electrc feld wll vary radally from the conductor to the outer 4

6 electrode. The theoretcal lfe expressons wll therefore yeld dfferent thermoelectrc lfetmes dependng on the radal locaton of the regon consdered. Some regons wll therefore reach the endpont of ageng before others. In order to relate the theoretcal models to the servce lfe of the cable, t s therefore necessary to decde upon the condton under whch the cable fals.e. whether t s necessary for the whole cable to age to a defned end-pont, or whether t s suffcent for only one regon to reach ths degree of ageng. In the latter case cable falure wll be ntated n the regon that has reached a crtcal level of ageng and rapdly proceed to completon. The phlosophy of the current ageng theores s n accord wth the latter vewpont and t s therefore the one that we shall adopt here. 2.1 Shell model The radal varaton of temperature and electrc feld s allowed for by dvdng the cable nsulaton nto a seres of thn flms wthn whch temperature and feld can be consdered constant. Each shell can therefore be assgned a lfetme usng the theores approprate to thn flms under unform feld and constant temperature. Fgure 1 shows a cable wth a typcal smple desgn, comprsng a cylndrcal core covered n a layer of nsulaton. An example shell at radus r s shown. Such shells can be of equal thckness, or equal volume. None of the ageng models mentoned n the ntroducton takes account of any of the spatal dmensons of the polymer specmen, or those of the test electrodes. In fact, dependng on the mechansms of ageng, at least one of these factors s lkely to be mportant to the lfetme of a polymer specmen. It s commonly assumed that ageng s a bulk process, n whch case t s often argued that the volume of a specmen must affect ts lfetme (e.g.[23]) 5

7 and ths s dscussed further n secton 4.2. Consderng shells of equal volume means that the effect of volume on ageng wll be the same for each, and ths opton s therefore used here. 2.2 Radal dependence of E and T The feld and temperature experenced by each of the shells descrbed above s a functon of the shell s radal poston, r. The temperature of a shell at radus r can be wrtten as (see [18,19]): T W Th R O ( r ) T1 ln (1) 2 r where W s the power dsspated per unt length by the core under load and Th s the combned thermal resstvty of the nsulaton and any outer layers. R O s the cross sectonal radus of the cable as shown n fgure 1, and T 1 s the temperature of the outsde of the cable.e. the ambent temperature. Under AC condtons, the RMS electrcal stress experenced by the same shell s e.g. [20]: V E( r ) (2) RO r ln R I V s the voltage appled to the cable core, R O s as defned above and R I s the cross sectonal radus of the cable core. The above expresson shows that n the AC case, the electrcal feld strength n cable nsulaton s always largest close to the cable core. 6

8 If a DC voltage s appled to the cable core, the electrcal stress profle s more complcated, and s gven by [18,20]: E( r ) R V O 1 r R O R R I O 1 (3) where all symbols have ther prevous meanngs, and δ s gven by W Th a 2 mv R O R I R O mv 1 R I (4) In equaton 4, a and m are constants n equaton (5) descrbng the resstvty, ρ of the nsulaton n terms of electrcal feld, E and temperature T: 0 exp( at )exp( me ) (5) In equaton (5) ρ 0 s resstvty at T=0 and for vanshngly small E. The expresson for E(r) n the DC case leads to a stuaton n whch the electrcal feld strength may actually be largest at the outer edge of cable nsulaton systems for some values of current [18,20,21]. 2.3 Predctng Cable Lfetmes The parameters determned for the lfe expressons of current models are values approprate to the characterstc lfe n a lfetme dstrbuton [22]. Substtutng for the 7

9 temperature T(r ) and feld E(r ) wll therefore gve the characterstc lfe of the shell f ts volume s the same as that of the specmens for whch the parameters are derved. The probablty of survval to tme t of the th shell s thus gven by t PS ( ) exp (6) L where β s the tme exponent of the Webull dstrbuton [23,24] that fts the thn sample data. Of course an alternatve dstrbuton could be used f applcable. The probablty of survval of the whole cable s gven by the jont probablty of survval of all the shells, under the assumpton that a falure ntated n any one of them s suffcent to fal the whole cable. Ths gves equaton (7), P F PS 1 PS ( ) 1 (7) The resultng falure dstrbuton P F can be analysed to determne the characterstc lfe of the whole cable and the falure tme dstrbuton [22]. There s however a drawback to carryng through ths approach. In general t wll be dffcult to equate the shell volume to that of the specmens used n the parametersaton of the lfe expresson. If ths s not possble the thn flm parametersaton cannot be assumed to apply to the cable shell, and n the absence of a measured or theoretcal sze dependence the parameters cannot be modfed approprately. Of course equaton (7) could be taken to refer to a cable length small enough that the shell volumes are the same as the thn flms. The whole cable survval probablty would then be gven by 8

10 l( cable) ( ) (sec ) l(sec ton) S cable PS ton P (8) where l(cable) s the cable length and l(secton) s the secton length as defned above. Even ths approach s only possble f the volume of the thn flm samples s known and we assume that the sze effect s n fact a volume effect rather than one related to electrode area or sample thckness. In the lght of these dffcultes we have adopted a dfferent approach descrbed n the next secton. 2.4 Parametersng lfe expressons from cable data In ths approach we relate observed cable data to lfe expresson parameters approprate to the complete nsulaton volume of the cable. In ths secton we shall denote the characterstc lfetme of the cable by B63, whch s defned as the tme at whch a fracton (1-e -1 ) equal to 63.2% of the samples have faled. The value of B63 for cables can be obtaned from ther lfetme dstrbuton n exactly the same way as for thn flms. However fttng t to the theoretcal lfe expressons s more complex than for thn flms. In the thn flm case t s only necessary to ft the expressons to a set of lnes gvng the T and E dependence of the characterstc lfe (e.g. [1-4]). In the case of cables each of the shell lfe expressons s a functon of the model parameters and a dfferent E and T value dependng on ts radal poston. To fnd values for the parameters relevant to a partcular set of cable specmens, the shell expressons must be combned, and ftted to expermental ageng data. A method for combnng N shell lfetme expressons n order to ft them to the characterstc lfetme at each expermental condton s descrbed here. 9

11 It s assumed that n cable nsulaton of volume VC made up of N shells, falure n any one of the shells wll cause the whole nsulaton to fal. In ths case, the followng equaton lnks the probablty of survval of the whole nsulaton to the probablty of survval of N consttuent shells. S N P ( C) P ( S) (9) 1 S Here, P S (C) s the probablty of survval at a gven tme of the whole nsulaton. P S (S) s the probablty of survval of a consttuent shell. Each shell has a volume VS=VC/N. The tme to falure dstrbutons resultng from ageng tests on polymer specmens are commonly assumed to be Webull dstrbutons wth a shape parameter,, whch s characterstc of the ageng process e.g. [22-24]. Ths assumpton s reasonable f a falure n polymerc specmens can be assgned to the weakest regon of polymer, where ageng proceeds faster than n any other. Assumng, therefore, that P S (C) and P S (S) are Webull dstrbutons wth the same β value, they are gven by e.g.[24]. ( C) exp t B63 PS (10) t PS( S) exp (11) L Here t s tme, B63 s the characterstc lfetme of a set of cables aged under the same expermental condtons and L s the characterstc lfetme of a set of shells of nsulaton all aged under one partcular condton. n equaton (10) s the shape 10

12 parameter of the tme-to-falure dstrbuton from the cable ageng experments, and n equaton (11) s the shape parameter of the dstrbuton of the shell tmes-to-falure. It has been assumed that the values of n the above equatons are the same. Ths may not necessarly be the case dependng upon the orgn of β [22], but the ntroducton of a dfference between ts value for the shell and the cable would requre more knowledge than we have at present and hence s not justfed. By substtutng equatons (10) and (11) nto equaton (9), an expresson can be derved for the characterstc lfetme of cable nsulaton, B63 n terms of the characterstc lfetmes, L, of a set of nsulaton shells. 1 B63 1 L (12) In ths case, B63 s the characterstc lfetme of a cable set, and L can be replaced wth an expresson for the lfetme of the th shell. Equaton (10) can therefore be used to ft the chosen expresson to expermental B63 values n order to obtan parameter values. The parameter values obtaned from fttng equaton (12) wll necessarly depend on the volume of the cable nsulaton through B63, just as n the case of thn flms the parameters depend upon the flm volume [24]. However, usng equaton (12) means that the parameter values must also have a dependence on the shell volume (or equvalently a dependence on N), snce the probabltes P S (S) n equaton (9) are volume dependent. Parameters that depend on both VC and N have the dsadvantage that drect comparsons between cable and flm experments are then dffcult, snce 11

13 the parameters from flm experments wll only depend on the total flm nsulaton volume equvalent to VC for cables. To get parameter values from cable experments that only depend on VC, t s necessary to scale up the probablty of falure of each shell to the total nsulaton volume. In other words, t s necessary to determne an expresson for the probablty of falure that each shell would have f t had the volume of the whole nsulaton. Ths s equvalent to the probablty of falure of a shell, wth volume VC, comprsng N shells each experencng the same E and T condtons. Ths can be obtaned usng an expresson of the same form as equaton (9): S N P ( SS) P ( S) (13) 1 S Here P S (SS) s the probablty of survval of the scaled up shell wth volume VC, and P S (S) s the probablty of survval of the orgnal shell. Snce each value of P S (S) s the same n ths case, ths gves N S ( SS) PS ( S (14) P ) Takng the product of the PS(SS) values over all the shells now gves the probablty of survval of a volume of nsulaton N tmes bgger than VC.e. S N P ( NC ) P ( SS) (15) 1 S Substtutng for P S (SS) from equaton (14) then gves P N N S ( NC ) PS ( S) (16) 1 12

14 P S (NC) s the probablty of survval of a cable specmen wth a volume N tmes bgger than VC. To get the probablty of survval of cable nsulaton of volume VC (.e. of the total cable nsulaton), equaton (14) can be used together wth equaton (16) to gve P 1 1 N N N S ( C) PS ( NC ) PS ( SS) (17) 1 Here P S (C) s the probablty of survval of the cable. Usng ths equaton, and assumng agan that the probabltes of survval are all Webull dstrbutons wth the same shape parameter, the followng equaton s derved 1 B63 1 N 1 L (18) L s now an expresson for the lfetme of a scaled up shell.e. an expresson for the lfetme that a shell would have f t had volume VC. It s mportant to note that L n expresson (18) has a dfferent meanng to L n equaton (12), despte the fact that they both relate to the same lfe expresson. Fttng of expresson (12) to data results n model parameter values that depend on N, whereas usng equaton (18) gves parameter values that are ndependent of N and depend only on the total volume of nsulaton VC, through B63. 13

15 3. Applcaton to data In order to llustrate the methodology descrbed n secton2 we wll apply t to a specfc lfe expresson, namely that of the DMM model [1-4]. Ths model gves the lfe, L, of the th shell, at temperature T(r ) and feld E(r ) n the form h 2kT( r exp ) Sd k exp H dk Cd T( r E( r 2 ) ) 4b ln Aeq Aeq A* L cosh Kd Cd 2T ( r E( r ) ) 4b (19) The factor A eq s defned through expresson (20), A eq 1 exp[ ( K d C 1 d E( r ) 4b ) / T( r )] (20) There are therefore only sx nomnally ndependent parameters n the lfe expresson, b, A*, H dk, S d, K d and C d. Ths expresson s based on the concept that the energy stored n local concentratons of space charge causes a local deformaton of the polymer to exceed a crtcal level at whch free volume generaton and nano-vod coalescence occurs. Falure s then rapdly brought about by partal dschargng leadng to electrcal trees and connecton of the vod populaton. The polymer chans are conceved as possessng alternatve confguratons wth the one correspondng to the deformaton beng energetcally unfavourable wth respect to the other. The energy dfference s K d (n unts of Kelvn). The reacton from one confguraton to the other requres a free energy barrer to be exceeded, composed of an actvaton enthalpy H dk (unts of Kelvn) and an actvaton entropy S d (unts of Kelvn). In the 14

16 absence of the space charge produced by the appled feld E, the fracton of local confguratons n the deformed state wll reach an equlbrum value A eq. The space charge concentraton has been assumed to be proportonal to a power b of the appled electrc feld, (.e. local charge = q loc E b ) [1-4] and to modfy the energy barrer and energy dfference between the alternatve confguratons va an electro-mechancal energy leadng to the energy term C d E 4b n the above expressons. It s assumed that lfe s termnated when the fracton of confguratons n the deformed state reaches a level suffcent for coalescence nto vods, startng from an ntal non-equlbrum state correspondng to an unaged materal. Ths crtcal fracton s denoted by A*. The reader s referred to references [1-4] for more detal. Each of the parameters are expected to be essentally ndependent of temperature, and the role played by the local temperature n the lfe expresson s explctly defned va equaton (19). In AC felds S d and H dk become frequency dependent [4], whereas n DC felds S d can be taken to be zero [4] thereby reducng the number of parameters to fve. Ths expresson was chosen here because the parameter values for a number of materals are avalable n the lterature [1-4]. In addton the expresson exhbts all the basc features present n the other models.e. an actvaton free energy that must n general nvolve two parameters, a feld effect term nvolvng a composte parameter, here C d, and a feld power term usually assumed to have the value b=0.5. The other features A* and K d lead to a feld threshold, whch s a controversal feature that s also found n [5]. The choce of expresson encompasses all possble features and also poses a challenge to the method, and for ths reason we have chosen to use t as an example to llustrate the applcaton of the method. However n the context of cable geometry the chosen lfe model can be smplfed by usng ether a known or calculated radal dstrbuton of space charge. Ths wll elmnate the assumpton 15

17 relatng space charge concentraton to the local feld and hence remove b from the lst of parameters. It should be noted however, that our choce of model was made strctly for convenence n testng and that the method can be appled to any lfe expresson and s not restrcted to the model chosen. 3.1 Detals of fttng method Data used The method descrbed n secton 2 was used to ft the DMM lfe expresson to expermental data from cable ageng experments carred out for BICC Cables Ltd (now owned by Prell Cables Ltd.) [26]. Cables nsulated wth extruded XLPE of thckness 4.4mm were aged under nne dfferent expermental condtons. Twelve cables were aged under each condton, and the tests were stopped after eght cables had faled. A Webull analyss applcable to sngly censored data was therefore carred out [23], resultng n B63 and β values for each of the nne expermental condtons. The cables were 15kV medum rated cables, wth alumnum cores and values of R I and R O of 5.9mm and 10.3 mm respectvely. They were all 9.14m long. Cables were aged at temperatures of 60 C, 75 C and 90 C and appled AC r.m.s. voltages of 34.6kV, 26kV and 17.3kV. The cables also contaned thn semcon layers between the core and the nsulaton, though these were gnored for the purposes of the fttng. Ths s justfable here, snce the only effect of an extra layer would be to change the mean thermal and electrcal resstvtes of the nsulaton/semcon layer and n all cases the cables were aged under AC voltage, no current was appled to the cable cores, and the 16

18 temperature was assumed constant across the nsulaton. There was therefore no temperature gradent across the nsulaton only an AC-type feld gradent, whch s ndependent of both the electrcal and thermal resstvty of the nsulatng layer as shown n equaton (2). Error functon In order to ft expermental data to the DMM model, the followng error functon, representng the dfference between expermental data and the model predctons, was mnmsed to fnd optmal DMM parameter values n L 1 2 J N ln( B63J ) ln 1 (21) L( r ) J Above, J s the number of B63 values avalable. For each B63 value, the error functon takes the dfference between the log of the B63 value and the log of the hypothessed cable lfetme expresson as n equaton (19). The squares of the dfferences are summed over all expermental condtons to gve the fnal error value for the whole data set. Natural logarthms are used n the error functon due to the extreme non-lnearty of the DMM equaton. The square of the dfferences s used to avod fts where the ft s good for most B63 values but very poor n one or two cases. Equaton (21) was mnmsed usng a grd search method mplemented usng a FORTRAN computer program. 17

19 Values of N and β The number of shells used n ths fttng, N, was 100. Ths value was chosen for several reasons. Frstly, N=100 corresponds to a stuaton where the thckest shells n the cable are roughly the same thckness as the PET flms for whch the DMM model has been prevously shown to gve a good ft [3]. Snce the models were orgnally appled to thn flms, ensurng that the shells are of a smlar thckness ensures that the applcablty of the expresson demonstrated on thn flms s retaned. Secondly, N was chosen to be hgh enough to gve as good a ft as possble to the data. Snce the N dependence of the model parameters s elmnated n the fttng functon, the only effect of ncreasng N should be to ncrease the qualty of the fts due smply to an mproved accuracy n the dscrete representaton of a contnuous system. Ths effect was found to reach saturaton at a value of N of approxmately 100. The thrd crteron for a value of N s that t cannot be too large that the computaton takes too much tme. A value for β also had to be chosen, snce the error functon requres only one value of beta. Each expermental condton yelds ts own value of β, and for the data used here the β values ranged from 2.4 to 8.5 each wth farly wde confdence lmts. We have assumed that each of these values s the same for each cable set so long as the ageng process s the same, so an average of all the β values was used. Ths average was weghted towards the smaller end, snce the hghest β was much hgher than the other values and was therefore deemed atypcal. 3.2 Results 18

20 The results obtaned from fttng the DMM model to cable data as outlned above are shown n fgure 2. In fgure 2, the y-axs represents tme n seconds, and the x-axs shows appled RMS voltage n kv. B63 values from each of the nne condtons under whch cables were aged are shown as crosses, crcles and trangles correspondng to tests at 363K, 348K and 333K respectvely. The 90% confdence lmts for each B63 are shown as error bars. Each of the lnes n fgure 2 represents the lfetme predcted by the DMM model usng the method descrbed above. Each lne shows predcted lfetme as a functon of appled voltage at a temperature correspondng to one of the ageng temperatures. The lnes show voltage threshold behavour.e. below a threshold voltage, cables are predcted to have an nfnte lfe. However, the cable lfetme only becomes nfnte f the feld and temperature experenced everywhere n the nsulaton.e. by each of the consttuent shells - s below the threshold for the materal. The parameter values used to obtan the fts shown n fgure 2 are gven n table 1, along wth the parameters from fttng to lfe data from other types of polymer specmen. In the FORTRAN grd search used here, the DMM parameters were allowed to vary over wde ranges. The magntudes of the model parameters obtaned are nevertheless all smlar n magntude to those obtaned n prevous fttngs to AC ageng data of XLPE mn-cables and PET thn flms. 4. Dscusson 4.1 Ft to data 19

21 The ft n fgure 2 can be seen to be good, wth the predcted lfetmes beng wthn the 90% confdence lmts of the expermental data for four out of the nne condtons. Ths s a good ft consderng the assumptons nvolved n the dervaton of the fttng method. The most sgnfcant of these assumptons s the way that the electrcal feld strength s used n the method frstly the electrc feld n each shell s calculated on the bass of smplfyng assumptons gvng equaton (2) and then ths value s used n the DMM model n another assumed relatonshp descrbng the amount of charge n the materal n terms of the local macroscopc feld. 4.2 Parameter values volume consderatons The magntudes of the model parameters obtaned are smlar n magntude to those obtaned n prevous fttngs to AC ageng data. Ths suggests that the cable fttng method works well, and supports the contenton that the ageng process s the same for each of the materals studed. The parameters n table 1 were obtaned from fts to AC ageng data nvolvng very dfferent specmen types. In ths nvestgaton, the cables were nsulated wth XLPE wth a volume of 2x10-3 m 3 and a thckness of 4.4mm. The mn-cables for whch parameter values are quoted had an nsulaton thckness of 1.5mm, and a volume of approxmately 3x10-6 m 3. The volume of the PET flms s not known, but the thckness of each was 50x10-6 mm mplyng a volume many tmes smaller than n ether of the cable cases. Any dependence of specmen lfetme on volume must be reflected n the magntudes of the DMM parameter values obtaned from fttng to data nvolvng specmens of 20

22 dfferent volumes. The queston as to how the specmen volume affects parameter values n table 1, however, s not clear, snce the parameters obtaned are all for dfferent materals, as well as for dfferent volumes. Ths s true even of the XLPE cables evaluated here and the XLPE nsulated mn-cables, snce the XLPE was made by dfferent cable manufacturers n each case, and there are therefore lkely to be sgnfcant dfferences n composton between the two. It s therefore not possble to separate out dfferences n parameter values due to volume, from dfferences due to materal morphology and chemcal composton. The volume of nsulaton of the cables used here, however, s consderably larger than n the other two cases almost 700 tmes larger than the mn-cable nsulaton, and lkely to be much larger agan than the PET flms. In spte of the materal dfferences t s possble to use the parameter sets of table 1 to speculate as to whch of the DMM model parameter values mght be affected by volume. The common assumpton that a larger volume of nsulaton wll fal faster than a smaller volume under the same condtons s essentally based on a statstcal argument (see for example [22,23]),.e. bgger volumes gve a greater preponderance of stes susceptble to ageng and hence a bgger lkelhood of the exstence of hghly susceptble stes. In the DMM models these stes are charactersed by polymer moetes that can trap charge and respond to the trapped charge by surmountng an energy barrer to an alternatve conformaton correspondng to a local deformaton. The nfluence of the local space charge s to both accelerate the local changes and to stablse a more extreme local dstorton than that whch would have been produced at thermal 21

23 equlbrum n ts absence. When the dstorton exceeds a crtcal level t s assumed to ntate a rapd falure process. Dfferences n the parameter sets obtaned for cable data as compared to the other two sets, partcularly the mn-cables, may therefore relate to a greater severty of the most susceptble stes. Snce the temperature s constant across the radus of the cable n the samples whose falure data has been analysed (see secton 3.1), a characterstc free energy barrer #G (=H dk -TS d ) can be defned as for the thn flm samples. As shown n Table 1 ts value s actually very smlar for all of the three systems, however t s clear that the component of the barrer arsng from the actvaton enthalpy s greater n the fullscale cables than n the other two cases. Ths means that ageng s much more senstve to temperature n the present cable data than for the other two systems. The changes n barrer factors H dk and S d cannot be regarded as a volume effect as we would expect larger volumes to lead to more susceptble stes, wth smaller #G, over the temperature range experenced by the cable. It should be noted that the value of the actvaton entropy, S d, s negatve. Ths corresponds to a reacton n whch the ground state s more confguratonally dsordered than the barrer state through whch the reactants move to the product state. In ths knd of reacton we can pcture the reactng moetes havng to adopt specfc orentatons and bond angles n order to acheve passage through the barrer. Ths requres work to be done on the group of enttes that have to pass through the barrer. The smlarty n #G values appears to be an nstance of a compensaton law and s also found for the ageng of PET flms [1-4], where #G s found to be the same n DC and AC ageng over the temperature range measured. It s possble to speculate that ths result ndcates that the basc features of the ageng mechansm are the same n all cases, but that the free energy surface 22

24 changes wth frequency and morphology. More specfcally t would appear that as the amount of orderng requred to enter the barrer state becomes smaller (.e. S d moves closer to zero) the system s forced to surmount a hgher enthalpy barrer,.e. the dsordered state has a large enthalpy barrer but orderng allows reacton va a smaller entropy barrer at the expense of the free energy requred for the orderng. It s possble that the parameter A* may be affected by volume. A* s the fracton of moetes that must be nvolved n deformaton for breakdown to occur n any localsed area. It seems lkely that ths fracton mght vary from regon to regon of the specmen. Ths means that n a larger volume of polymer there may be an ncreased lkelhood of fndng regons where fewer moetes need to be nvolved for breakdown to be ntated. As a result, a specmen wth a larger volume wll requre the converson of fewer moetes to ntate falure, and consequently a smaller local energy concentraton wll be requred. The dfferences n characterstc A* shown n table 1 seem to support the hypothess that the specmens wth larger volumes requre fewer moetes to be converted, and therefore lower energy concentraton, to ntate breakdown. Larger volumes would therefore experence a reducton n lfetme under a gven condton. C d and b descrbe the effect of a feld on the barrer to ageng, #G. On the applcaton of an electrcal feld of magntude E, #G s reduced by an amount equal to C d E 4b, and ths acts to accelerate the ageng reacton. Large values of C d and b for a set of specmens therefore ndcate that the ageng reacton s accelerated strongly by the electrcal feld. In addton, ths feld dependent energy helps to stablse the state of the moety correspondng to the deformed polymer and hence facltates the 23

25 achevement of suffcent deformaton to ntate falure. A greater volume of polymer s more lkely to contan stes at whch ths s the case.e. stes at whch the feld can have a strong nfluence on the ageng process. In the DMM model such stes wll be those that have greater ablty to trap charge and store electro-mechancal energy. They may therefore be stes that have a bgger electrostrcton coeffcent than the average for the specmen. Such stes may also (or nstead) have a smaller bulk modulus or relatve permttvty than average. Mcroscopc varatons n macroscopc materal characterstcs such as these seem very lkely, whch makes these two parameters lkely to have a volume dependency. The data n table 1 seems to support ths to some extent, wth the largest polymer volume showng by far the largest values of C d. The values of b are all qute smlar, however, wth no observable pattern wth volume. Overall, the C d E 4b term for felds from 0 to 20kV/mm s always largest for the XLPE cable parameters. The same term s larger for the mn-cables than for the thn flms for all felds above about 3kV/mm, but below ths feld the values are very smlar. These parameters are also lkely to be strongly materal dependent, however, so ths s by no means conclusve. 4.3 Relatonshp to Space Charge The DMM ageng theory dffers from the other models [5-14] n ascrbng the ageng to energy (electromechancal) concentraton produced by trapped space charges. The local energy s proportonal to ether the 4 th power of the space charge feld or ts square dependng upon whether or not the local centre was assumed to behave as a regon wth macroscopc propertes or as an rreducble volume element wth atomstc/molecular propertes [27]. The relatonshp to the known appled feld was made va the assumpton that the trapped charge q E b where E s the appled feld. 24

26 The physcs behnd the model [1] therefore allows the possblty that a drect measurement of space charge could be used nstead of estmaton of E(r ). Ths has the beneft of elmnatng the exponent b as a parameter, snce the local space charge feld can be taken to be proportonal to the space charge densty. The effect of local feld would stll be expressed through a parameter lke C d. One drawback to ths course of acton s that trapped charges may be present though ther net value may be zero. The energy concentraton and local feld would stll exst on the atomstc scale even though the measured space charge would be zero. Secondly, the dvergence of the appled Laplacan feld n cables would also yeld an electromechancal energy concentraton [5] that would have to be added to the atomstc value. In order to pursue ths approach space charge measurements would have to be taken durng ageng and ths nformaton s not yet avalable. 5. Conclusons The radal varaton of temperature and electrc feld experenced n cable geometry can be ncluded n an ageng theory by means of a shell approach. The methodology can be appled to any ageng theory and can, n prncple, be used to predct cable lfetmes from thn flm experments. In practce t s better suted to nvestgatng the volume dependence of ageng parameters. The methodology has been appled to the lfe expresson of the DMM model, whch was found to ft the data very well. It was shown that values of the A* and C d parameters correspond to a characterstc centre that s more susceptble to ageng, as would be expected for a system whose nsulaton had a bgger volume. The free energy barrer to ageng had almost the same value as for mn-cables, but the 25

27 actvaton enthalpy component was bgger, It therefore seems possble to conclude that the larger cable contans centres that requre less energy concentraton to acheve the ntaton of falure, that the centres are more susceptble to the affect of an electrcal feld, but that the barrer to the ageng reacton nvolves dfferent routes across the free energy surface correspondng to dfferences n the local morphology. 26

28 REFERENCES [1] L.A. Dssado, G. Mazzant, G.C. Montanar, The Incorporaton of Space Charge Degradaton n the Lfe Model for Electrcal Insulatng Materals, IEEE Transactons on Delectrcs and Electrcal Insulaton, Vol. 2, No 6, pp , [2] L.A. Dssado, G. Mazzant, G.C. Montanar, The Role of Trapped Space Charges n the Electrcal Agng of Insulatng Materals, IEEE Transactons on Delectrcs and Electrcal Insulaton Vol. 4, No 5, pp , 1997 [3] L.A. Dssado, G. Mazzant, G.C. Montanar, Dscusson of space-charge lfe model features n dc and ac electrcal agng of polymerc materals, Annual Report CEIDP, pp36-40, 1997 [4] G. Mazzant, G.C Montanar, L.A. Dssado, A Space-charge Lfe Model for ac Electrcal Agng of Polymers, IEEE Transactons on Delectrcs and Electrcal Insulaton, Vol. 6, No 6, pp , [5] T.J. Lews, J.P. Llewellyn, M.J. van der Slujs, J. Freestone, R.N. Hampton, A new model for Electrcal Ageng and Breakdown n Delectrcs, IEE DMMA, Conf Pub No 430, pp , 1996 [6] T.J. Lews, Ageng A Perspectve, IEEE Electrcal Insulaton Magazne, vol. 17, pp 6-16,

29 [7] T.J. Lews, J.P. Llewellyn, M.J. van der Slujs, Electroknetc propertes of metaldelectrc nterfaces, IEE Proceedngs-A, Vol. 140, No 5, pp , September 1993 [8] T.J. Lews, J.P. Llewellyn, M.J. van der Slujs, Electrcally nduced mechancal Stran n Insulatng Delectrcs, IEEE Annual report CEIDP, pp , [9] T.J. Lews, J.P. Llewellyn, M.J. van der Slujs, J. Freestone, R.N. Hampton, Electromechancal Effects n XLPE Cable Models, Proc.5th ICSD, (IEEE Pub. 95CH3476-9), pp , 1995 [10] P. Connor, J.P. Jones, J.P Llewellyn and T.J. Lews, Electrc Feld-Induced Vscoelastc Changes n Insulatng Polymer Flms, Ann. Rep. CEIDP, pp27-30, 1998 [11] C.L. Grffths, J.Freestone and R.N. Hampton, Thermoelectrc Agng of Cable Grade XLPE, Proc. IEEE ISEI, pp , [12] P.W. Sayers, T.J. Lews, J.P Llewellyn and C.L. Grffths Investgaton of the Structural Changes n LDPE and XLPE Induced by hgh Electrcal Stress, IEE DMMA, Conf Pub No 473, pp , 2000 [13] C.L. Grffths, S. Betterdge, J.P. Llewellyn and T.J. Lews, The Importance of Mechancal Propertes for Increasng the Electrcal Endurance of Polymerc Insulaton, IEE DMMA, Conf Pub No 473, pp ,

30 [14] J.L. Parpal, J.P. Crne and C. Dang, Electrcal Ageng of Extruded Delectrc Cables a physcal model, IEEE Transactons on Delectrcs and Electrcal Insulaton, Vol. 4, No 2, pp , 1997 [15] J.P. Crne, A Molecular Model to Evaluate the Impact of Agng on Space Charges n Polymer Delectrcs, IEEE Transactons on Delectrcs and Electrcal Insulaton, Vol. 4, No 5, pp , 1997 [16] L. Smon, A general approach to the endurance of electrcal nsulaton under temperature and voltage, IEEE Transactons on Electrcal Insulaton, Vol. EI-16, No 4, pp , 1981 [17] J. Artbauer, Electrc strength of Polymers, J.Phys. D: Appl. Phys., vol. 29, pp , 1996 [18] C.K. Eoll, Theory of Stress Dstrbuton n Insulaton of Hgh-Voltage DC Cables: Part 1, IEEE Trans EI Vol. E1-10, No. 1, pp.27-35, 1975 [19] The Development of a Hgh Voltage DC Cable Prepared by The Okonte Company. EPRI EL-606 Project [20] Edted by G.F. Moore, BICC Electrcal Cables Handbook, Blackwell Scence Ltd ISBN:

31 [21] S.A.Boggs, D.H.Damon, J.Hjerrld, J.Holboll, and M.Henrksen, Effect of nsulaton propertes on the feld gradng of sold delectrc DC Cable, IEEE Trans. PD., vol. 16, pp , 2001 [22] L. A. Dssado, Predctng electrcal breakdown n Polymerc Insulators, From Determnstc mechansms to Falure Statstcs, IEEE Trans DEI, vol.5, pp , 2002 [23] J.C. Fothergll, IEEE Draft Standard P930: 'IEEE Gude for the statstcal analyss of electrcal nsulaton breakdown data' [24] J.C. Fothergll and L.A. Dssado, Electrcal degradaton and breakdown n polymers P. Peregrnus for IEE, London, ISBN: , 1992 [25] V. M. Morton and A Stannett, Volume dependence of electrc feld strength of polymers. Proc. IEEE, vol 115, pp1857, 1968 [26] BICC company report Analyss of Accelerated Ageng Tests On Extruded XLPE and EPR Power Cables Carred Out By EPRI at the Marshall Technology Centre of BICC by F. Chang, 1998 EPRI report RP

32 [27] L.A.Dssado, G.Mazzant, G.C.Montanar, Elemental Stran and trapped space charge n thermoelectrc ageng of nsulatng materals. Lfe modellng, IEEE Trans DEI., vol. 8, pp ,

33 BIOGRAPHIES Elzabeth Cooper Elzabeth Cooper was born n the UK n 1976 and ganed a degree n Physcs wth Astrophyscs from the Unversty of Lecester n In 2002 she receved a PhD, also from the Unversty of Lecester. She currently works wth Professors Dssado and Fothergll n the hgh voltage laboratory of the engneerng department at Lecester, researchng the physcal processes relevant to charge movement n the nsulaton of hgh voltage DC power cables. Len A Dssado (Senor Member snce 1996) Len Dssado graduated from Unversty College London wth a 1 st Class degree n Chemstry n 1963 and was awarded a PhD n Theoretcal Chemstry n 1966 and DSc n After rotatng between Australa and England twce he settled n at Chelsea College n 1977 to carry out research nto delectrcs. Hs nterest n breakdown and assocated topcs began wth a consultancy wth STL began n Snce then he has publshed many papers and one book, together wth John Fothergll, n ths area. In 1995 he moved to The Unversty of Lecester, and was promoted to Professor n He has been a vstng Professor at The Unversty Perre and Mare Cure n Pars, Paul Sabater Unversty n Toulouse, and Nagoya Unversty, and has gven numerous nvted lectures, the most recent of whch was the E.O.Forster Memoral lecture at ICSD 7 n Endhoven. Currently he s an Assocate Edtor of IEEE Transactons DEI, and co-char of the Multfactor Agng Commttee. 32

34 John C Fothergll (Senor Member snce 1995) Professor Fothergll was born n Malta n He graduated from the Unversty of Wales, Bangor, n 1975 wth a Batchelor s degree n Electroncs. He contnued at the same nsttuton, workng wth Pethg and Lews, ganng a Master s degree n Electrcal Materals and Devces n 1976 and doctorate n the Electronc Propertes of Bopolymers n Followng ths he worked as a senor research engneer leadng research n electrcal power cables at STL, Harlow, UK. In 1984 he moved to the Unversty of Lecester as a lecturer. He now has a personal char n Engneerng and s currently Pro-Vce-Chancellor. 33

35 Table and Fgure captons Table 1 Parameter values from fttng the DMM model to data. #G s the actvaton free energy (=H dk -TS d ) at the temperatures quoted. Fgure 1 Typcal coaxal cable geometry wth an example shell shown at radus r Fgure 2 Expermental lfetme data wth lfelnes predcted by the DMM model 34

36 Table 1 Parameters from Parameters for AC Parameters for AC ths nvestgaton ageng of thn flm ageng of XLPE PET [4] nsulated mn-cables [4] Sd (J/K) -3.8E E E-22 Hdk (K) Kd (K) C d (J(mm/kV) 4b ) A* b #G (J) T=20 C 2.0E E E-19 T=100 C 2.3E E E Applcaton of thermoelectrc ageng models to polymerc nsulaton n cable geometry ES Cooper, LA Dssado, JC Fothergll 27 th May 2003 Table 1 35

37 Thckness of cylndrcal shell ~0 R O R I r 2 2 Applcaton of thermoelectrc ageng models to polymerc nsulaton n cable geometry ES Cooper, LA Dssado, JC Fothergll 27 th May 2003 Fgure 1 36

38 Lfe (s) 1.E+09 1.E+08 1.E+07 Lfelnes wth Expermental Data Lfelne at 363K Lfelne at 348K Lfelne at 333K Data ponts at 363K Data ponts at 348K Data ponts at 333K 1.E Voltage (kv) 3 3 Applcaton of thermoelectrc ageng models to polymerc nsulaton n cable geometry ES Cooper, LA Dssado, JC Fothergll 27 th May 2003 Fgure 2 37