A USER-FRIENDLY TOOL FOR SIMULATING THE TIME-DEPENDENT FIELD DISTRIBUTION IN PE INSULATION ON THE BASIS OF A PHYSICAL APPROACH

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1 A USER-FRIENLY TOOL FOR SIMULATING THE TIME-EPENENT FIEL ISTRIBUTION IN PE INSULATION ON THE BASIS OF A PHYSICAL APPROACH Séverine LE ROY, Laboratoire Plasma et Conversion Energie, (France), severine.leroy@laplace.univ-tlse.fr Fakher BOUFAYE, Laboratoire Plasma et Conversion Energie, (France), fakherbo@yahoo.fr Fulbert BAUOIN, Laboratoire Plasma et Conversion Energie, (France), fulbert.bauoin@laplace.univ-tlse.fr Gilbert TEYSSERE, Laboratoire Plasma et Conversion Energie, (France), gilbert.teyssere@laplace.univ-tlse.fr Christian LAURENT, Laboratoire Plasma et Conversion Energie, (France), christian.laurent@laplace.univ-tlse.fr ABSTRACT A bipolar moel of transport intene to escribe the behavior of polyethylene uner C stress has been recently evelope. Though the moel has not been pushe yet to the stage of using it routinely, it i lea to progress in the unerstaning of ielectric physics in the sense that various hypotheses for explaining a given experimental observation can be evaluate.the purpose of this communication is to give the general features of the moel an to introuce the graphic interface allowing simulation in polyethylene plaques. Simulation has been evelope with the purpose of giving outputs irectly comparable with experimental ata. The latter can be external current measurements or internal space charge istribution as obtaine using the Pulse Electro Acoustic technique. A emonstration of the software will be one at the conference. A comparison between simulation an experimental results is iscusse in a companion paper. KEYWORS Fiel istribution - Space charge - Graphic Interface - moelling INTROUCTION In recent years, a number of numerical moels have been evelope with the aim to reprouce the space charge behavior in organic insulators, mainly polyethylene, uner C or AC stress [-5]. Among the riving forces for the evelopment of these moels we can name: the trens towars more compact systems in power engineering, leaing to an increase in the power ensity - the trens towars higher reliability of electrical systems, ue to their use in critical applications - the evelopment of new materials for electrical application with tailore properties. Moreover, the increasing eman in solutions for preicting materials behaviour uner stress is to be aresse on one han with a better unerstaning of physical mechanisms unerlying such behaviour, an on the other with isposable tools for moelling what such mechanisms imply as macroscopic properties an to confront such preictions with the actual behaviour of materials in systems. These general principles are tackle here with regar to internal fiel istortions ue to charge generation an accumulation into polyethylene-base insulate high voltage C cables submitte to thermoelectric stress. State of the art numeric techniques have been applie to resolve the transport equations in non stationary conitions, proviing a prevision of the macroscopic behaviour in a reasonable computing time for C stress conitions. In this communication, we present the physical moel an the software that has been evelope for that purpose [6]. In the first part, we briefly present the general features of the physical moel inclue in the software core coe in Fortran language. In the secon part, we escribe the graphic interface which has been evelope uner Java environment. This language has unerlying avantages resorting to the open nature of the source (no issues with licence agreemen an to the large portability over ifferent operating systems, requiring only recompilation uner non- Winows OS. This woul permit a large iffusion of such tool in a near future. PHYSICAL MOEL Figure shows a schematic representation of system use for the simulations. The type of moel is bi-polar, an features injection of electronic charges at both electroes, charge trapping in traps istribute exponentially in trap epth, an hopping transport. For sake of simplification, recombination of charges an internal generation are not taken into account [5]. C XLPE ε 5 µm Figure : schematic representation of the oneimensional system use for the simulation. Trap istribution: The chemical structure of the material is taken into account by consiering an exponential istribution of trap levels (Figure ), i.e. a large amount of shallow traps corresponing to physical efects, an a smaller amount of eep traps corresponing to impurities. This exponential istribution of traps has a maximum limit in trap epth, an is of the form: e,h Nt(e,h) N' (e,h) exp e,h max(e,h) () kbt (e,h) where N t(e,h) is the trap ensity istribution, for electrons an A

2 holes, an is characterize by the parameters N, T an the maximum limit in trap epth max. This exponential istribution of trap levels hols for each kin of carrier (electrons an holes). Traps are consiere to be fille from the eepest level upwars. Trap epth (ev). N/N' Max fille trap level.4.8 Mobility: Charge transport within insulating polymers is often escribe by a hopping mechanism in which carriers move from site to site by getting over a potential barrier. In our case the hopping charges essentially come from the highest fille trap state at a epth f, an the resulting mobility µ( is then function of the trappe charge ensity t (): a. υ. t e.. µ. + exp max sinh () N'. k T k T B. B.. kb. T T with a (3) T 3 max 3 B exp k BT (4) an ( N'k T ) max } f eepest trap level Fille t Figure : istribution of trap levels for one kin of carriers. f efines here the upper fille level, which is variable as a function of time an space. where ν is the attempt to escape frequency, is the average istance between traps, is the electric fiel, an e is the elementary charge. If the electric fiel is small ( e.e. k B. T ), equation () can be reuce to: a e. υ. t x t x t (, ) µ (, ). + exp max (5) kb. T N'. kb. T kb. T When T >> T, only a fraction of charge f from the trappe charge t is available for conuction: t f (6) + a Generation of carriers: The only source of charge is the injection of electronic carriers at both electroes. The injection for each kin of carrier follows a moifie-schottky law (7): e. w e, h e e. j, AT. ( x).exp exp (7) e h kb. T k. T 4. π. ε B where j h ( an j e ( are the injecte fluxes of holes an electrons at the anoe an cathoe respectively, A is the Richarson constant, w e an w h are the injection barriers. ε refers to the permittivity of the ielectric. The time an space epenent equations escribing the behaviour of charge carriers are the following, neglecting iffusion, an in Cartesian coorinates: n( j( + t r continuity (8) (t ε Poisson (9) j( µ (. (. transport () f NUMERICAL TECHNIQUES The thickness of the system is iscretise using a nonuniform gri of elements of size x (typical orer of magnitue being.4 µm for a 5 µm-thick sample), being tightene next to the electroes, in orer to follow the penetration of charge in the ielectric (See Figure ). At each time of the simulation, the time step is calculate to be less than the quickest phenomenon occurring in the ielectric. It must also satisfy the Courant-Frierichs-Lewy conition, for each mobile carrier. It means that the charge isplacement within one time step (typical value of t - s) is less that the size of an element x. The time an space epenent electric fiel an potential are calculate by iscretization of the Poisson equation. The numerical metho use to resolve steep front problems ue to charge penetration in the ielectric is base on a scheme first evelope by Leonar [7], avoiing numerical iffusion. To avoi prouction of negative ensities of species, a flux limiter has been also inclue in the coe. Further etails on the numerical resolution of the equations in the moel can be foun in [8]. TREATMENT OF SIMULATION OUTPUTS Results prouce by the moel cannot be irectly compare to experimental ata, especially space charge profiles obtaine by the PEA metho, for the following reasons. First, only internal charges are prouce an not image charge on the electroes. To account for image charges, surface charge ensities have been ae at the electroes accoring to the following equation(s): cathoe anoe x x ( x) x εv x. εv x. () where V is the applie voltage an x is the with of the cell on the electroe. Cathoe is at abscissa. Secon, profiles obtaine by the PEA metho are, like with any other technique, accompanie by incertitue on the position of the charges. This results from a combination of the with of the excitation pulse, of the ban pass of the piezoelectric sensor, of relate amplification an of the oscilloscope (which also as iscretization), an of filtering of the signal uring its processing. As a result, capacitive charges measure on a space-charge free material uner voltage appear as Gaussian curves instea of a single out point.

3 As it is not possible in practice to correct experimental ata for incertitue associate to position of the charge, we have applie a numerical filter to our ata in orer to lose part of the resolution reache by the simulate profiles. This has been one by applying a Gaussian filter of the form: nf g ( x) exp a x () where a efines the with of the Gaussian (to be aapte to fit to measure profiles) an nf/ is the centre of the Gaussian. The numerical filter is applie to both electroe an internal charges. Figure 3 shows an example of the spreaing of the influence charge introuce by the filter in the case of capacitive charge (no internal charge an hence no image charge). using the space charge mapping techniques, or to take into account an initial istribution with non zero net charge, which coul be the case for example when the ensity is below the sensibility limit of roughly. C.m -3 in case of PEA measurements.. Injection barrier for holes (ev) 5e45 N' for holes (m -3.J - ) To for holes (K).8 max for holes (ev) 5e- Holes initial ensity (C.m -3 ). Injection barrier for electrons (ev) 5e44 N' for electrons (m -3.J - ) To for electrons (K).83 max for electrons (ev) 5e- Electrons initial ensity (C.m -3 ).3 Relative permittivity 37 Temperature (K) 4e6 Fiel (V/m) 5e-6 Thickness (m) 5 Charging time (s) 5 echarging time (s) Gri size ratio 3 Cell number.5 Gaussian filter factor Figure 4: Example of parameters text file Figure 3: Introuction of capacitive charge on both electroes (no charges in the bulk) an treatment by a Gaussian filter. In orer to use the software quite efficiently, a graphic interface has been evelope. Its general features are given in the following part. When running, a winow of the form of that shown in Figure 5 appears. The program can be stoppe at any time. The running time for a given cycle epens on the uration of the polarization/epolarization steps, on the velocity of carriers (the time step becomes shorter when this is high, an hence the computation time increases), an on the machine on which it runs. For 3h/3h cycle this can give typically min of computation, but it can be also much more (ays). GRAPHIC INTERFACE The program is constitute by the following files (size of 6kb in total) in a Java run-time environment: o the Fortran-base executable coe, referre to as Injection in the following; o the Interface program which constitutes the user-frienly part of the package; o an ASCII type file containing parameter moel parameter values. The Injection program The Injection program calculates space charge ensity, current, electric fiel without recourse to the Interface itself. Parameters signification is self-explanatory as shown in Figure 4. The use of this program is limite to single ielectric of plaque geometry (an extension of the program has been evelope for multiple-ielectric an co-axial geometry-see companion paper in this conference [9]) an only one polarization / epolarization cycle can be programme, for ajustable times. An initial charge ensity for positive an negative carriers can be set, being homogeneous in the volume to simulate a charge istribution with zero net charge that cannot be measure Figure 5: The winows of Injection appearing when simulation is running. Output ata are store in a set of 8 text files, as etaile in Table. 3 files are mae of 3 columns, containing respectively time (in s.), position (in m), an the quantity. An exact calculation of the fiel at the both interfaces is provie in one of these output text files. Of course, the time epenence of the external current, which is one of the variables accessible experimentally, is prouce by the moel. At this stage, we have not incorporate contributions that coul arise from orientation polarization, which woul require an expression for the time epenence of the permittivity.

4 File Content Format rho3.at Net charge ensity vs. 3 3columns rh3.at Hole charge ensity vs. 3 3columns re3.at Electron charge ensity vs. position an 3 3columns time mu3h.at Hole mobility vs. 3 3columns mu3e.at Electron mobility vs. 3 3columns fiel3.at Fiel vs. position an time 3 3columns fielak.at Anoe an Cathoe fiel vs. time 3columns current.at Total current vs. time columns Table : Output files prouce by Injection Figure 6 an Figure 7 show an example of output ata obtaine by this graphic interface. For a better unerstaning, the curve are plotte in. They correspon respectively to fiel an space charge ensity profiles obtaine after h an 3h uner polarization, an.5h uner short circuit. Note that in this example, the Gaussian filter is has not been applie to the sampece charge ensity profiles. Figure 6: Electric fiel profiles at ifferent times The Interface program Running simulations The parameters associate to a given simulation, the list of which being given in Table 4, are ispatche in a set of 3 winows resorting to: -Physical parameters of the material an interfaces; -Experimental conitions (fiels, time, temperature ); -Numeric parameters. Concerning the later, the cell number, i.e. the number of points along the thickness is not an ajustable: it is fixe to 3 in Injection. The gri size ratio efines the ratio of the with of the cell in the mile of the ielectric to that of the cells ajacent to the electroes (cells are tightene close to the electroes to improve resolution). Finally, the Gaussian filter factor efines the with of the filter to be use for posttreatment of the ata. They are aapte consiering experimental profiles (a calibration profile for example). The Interface program calls the Injection program for running a simulation, making the later transparent to the user. Exploiting results Results from simulation being in progress can be visualise while the program Injection is still running. Within a given set of parameters, a number of them are use by Interface to procee to post-treatment of ata. This proceure has a sense only for space charge profiles (especially net space charge). It consists in aing influence charge (capacitive an image charge) an applying the Gaussian filter. Hence, the visualization of a given output file goes with the selection of the relevant parameter text file. Once a ata file has been selecte, an prior to visualization, one nees to specify the format of the file (an hence of the plo: or 3, an if the post-treatment aition of influence charge an filtering must be one. An example of 3 plot of fiel vs. time an thickness is shown in Figure 8. Note that no units are provie so that the tool is very general (3 plots of mobility can be visualize the same way). Figure 7: Net charge ensity taken at three ifferent times.5h, 3h of charging, an.5h of ischarging. Figure 8: 3 plot of fiel vs. time an epth an sie menu. X-axis correspons to time in a non-linear scale; Y-axis is the position in the insulation (anoe to the bottom, cathoe to the top) an colors represent the quantity being plotte with scale given in the color bar.

5 Simulate ata are in a non-linear scale in time. This has been set so to conform to experimental measurements carrie out in the project in which the frequency of space charge profiles acquisition is higher just after polarization an epolarization. Hence images themselves are in nonlinear scales. The interface can be use to visualize any 3- pictures an to extract - curves such as profiles at a given time or even to look at the evolution in time of the charge ensity for a given position. Finally, Figure 9 shows examples of jpeg images obtaine with the 3 plot tool, before an after post-treatment of the ata. These results were obtaine for 4kV/mm, at 37K an have been prouce for a matter of illustration only. However, we can see a charge packet which is mae up of injecte charges. These ones move across the sample. In experiments, this kin of charge packets can be repeate only a few times, an the magnitue of packet ecrease fast uring the packet movement. the unerstaning of ielectric physics in the sense that various hypotheses for explaining a given experimental observation can be evaluate. A comparison between simulation an experimental results is iscusse in this conference a companion contribution [9]. Acknowlegments This work has been performe in the 5th European Framework Research an evelopment Program "Benefits of HVC Links in the European Power Electrical System an Improve HVC Technology" (contract N ENK6-CT-- 67). We thank the HVC consortium for permission to publish. REFERENCES [] J.M. Alison an R.M. Hill, 994, "A moel for bipolar charge transport, trapping an recombination in egasse crosslinke polyethylene" J. Phys. : Appl. Phys., vol. 7, pp [] M. Fukuma, M. Nagao an M. Kosaki, 994, "Computer Analysis on Transient Space Charge istribution in Polymer", proc. International Conference on Properties an Applications of ielectric Materials, Brisbane (Australia), pp (a) [3] K. Kaneko, T. Mizutani an Y. Suzuoki, 999, "Computer simulation on formation of space charge packets in XLPE films", IEEE Trans. ielectr. Electr. Insul., Vol. 6, pp [4] S. Le Roy, G. Teyssère, C. Laurent, G.C. Montanari an F. Palmieri, 6, "escription of charge transport in polyethylene using a flui moel with a constant mobility: fitting moel an experiments", J. Phys. : Appl. Phys., Vol. 39, pp (b) Figure 9: Example of output space charge patterns. (a): net internal charge prouce by the coe; (b): after aition of influence charges an filtering. Fiel was on for the whole perio of time CONCLUSION We have introuce a version of a simulation moel esigne for escribing charge transport in a geometry an plaque samples. It is base on physical grouns that have been shortly recalle. Through this interface, it is possible to input simulation variables (temperature, stress cycle, geometry an intrinsic features of the sample for both kin of carriers ), to run the moule for transport in a transparent way, an finally to visualise an hanle simulation results, with the possibility to achieve a posttreatment to space charge profile in a form irectly comparable to experimental results resorting to the pulse electro-acoustic metho. The tool, which is still in a stage of consoliation as regars its ability to approach the behaviour of ifferent kins of polyethylene, oes lea to progress in [5] F. Boufaye, G. Teyssère, C. Laurent, S. Le Roy, L.A. issao, P. Segur, G.C. Montanari, 6, "Moels of bipolar transport in polyethylene", J. Appl. Phys., Vol., pp [6] N. Barbey, G. Teyssère, 5, Création une interface graphique pour un moèle étue e charge espace, Rapport e Stage, LGET, 3p. [7] B.P. Leonar, 99, "The ULTIMATE Conservative ifference Scheme Applie to Unsteay One- imensional Avection", Computer methos in applie mechanics an engineering, Vol. 88, p.7-74 [8] S. Le Roy, G. Teyssere an C. Laurent, 6, Numerical methos in the simulation of charge transport in soli ielectrics IEEE Trans. ielectr. Electr. Insul., Vol. 3, pp [9] S. Le Roy, F. Bauoin, G. Teyssere, C. Laurent, L.A. issao an G.C. Montanari, 7, Tools for unerstaning the thermo-electrical behaviour of XLPE insulation in power cables an accessories, to appear in Proc. 7 JCable Conference.