V Griseri
email: griseri@laplace.univ-tlse.fr

G Teyssèdre

N Saidi-Amroun

S Mouaci

C Mouchache

M Saidi

Dynamic of Space Charge in Electron-Beam Irradiated Cross-Linked PolyEthylene

Keywords: Space Charge, Pulse Electro-Acoustic Method, Electron Irradiation, Polarization

The electrical properties of Crosslinked PolyEthylene -XLPE-material have been studied for many years as it is used for cable applications. In this work, XLPE peelings, previously submitted to thermal ageing are investigated under electronic irradiation. FTIR spectra did not reveal specific thermal ageing signatures. The space charge distribution has been analyzed by Pulsed Electro-Acoustic method during and after application of a DC field. The space charge profiles appear to be very dependent on the electron beam energy that determines the electron penetration depth and irradiated volume. Most of the space charge features appear linked to charge redistribution in the bulk following irradiation.

I. INTRODUCTION

The Cross Linked Poly-Ethylene (XLPE) has been studied for a long time as it is commonly selected for the design of insulation in energy transmission systems. These cables are developed to be used under DC or AC stresses. However the XLPE electrical properties are still not fully understood. Insulated cables can be installed in radiative environment. In the present paper we propose to study XLPE behavior when submitted to an electronic irradiation. In this case, charges are deposited in the bulk due to the primary electron deposition. Besides, electron-hole pairs are created and constitute a further source of charges. We have chosen to follow the space charge distribution by the Pulse Electro Acoustic (PEA) method just after the irradiation to check the charge deposition then during a polarization under DC field and during the relaxation.

During the irradiation some structural modifications can happen to the polymer. They might be at the origin of some properties modifications. However, it is obvious that just after irradiation the presence of the charges in the bulk is modifying considerably the behavior of the charge injection and transport from the electrode during polarization and relaxation. The behavior of the charges will be described in details and analyzed. Complementary analysis by Fourier-Transform Infra-Red spectroscopy (FTIR) will be presented in this paper. The purpose is to determine if there is a link between the electrical behavior and the chemical structure of the sample. Indeed, thermal ageing mainly can produce structural evolution of the material with time.

II. EXPERIMENTAL SET-UP

A. Sample selection

XLPE samples were taken from rolls issued from HV cable peelings produced in the frame of the Artemis project [START_REF] Fothergill | Electrical,microstructural, physical and chemical characterization of HV XLPE cable peelings for an electrical aging diagnostic data base[END_REF][START_REF] Teyssedre | Semi-quantitative analysis of photoluminescence in thermoelectrically aged cables: II-Analysis of a population of cables[END_REF]. The peelings, 150 µm in thickness and 80 mm wide, were taken from a cable previously aged for 4600 h under a ac voltage of 225 kV under a thermal gradient, with maximum temperature of 90°C at the inner semiconducting layer. Details on samples source are given elsewhere [START_REF] Teyssedre | Semi-quantitative analysis of photoluminescence in thermoelectrically aged cables: II-Analysis of a population of cables[END_REF]. It was considered that the ageing induced to the material cable is negligible especially as its insulation properties were not downgraded. This is substantiated by presented FTIR results.

In this work space charge measurements were achieved by PEA method using a set-up adapted to the XLPE film thickness. Measurements are realized after electron deposition in the bulk by irradiation at various depths.

B. The irradiation chamber: Matspace

The irradiation chamber called Matspace, which is equipped with a 100 keV electron beam was used [START_REF] Griseri | Charge build-up and transport in electron beam irradiated polymers in a new irradiation chamber[END_REF]. The irradiation was performed under vacuum on the whole surface of the sample with a homogenous flux of about 1 nA/cm 2 . The irradiation time was limited to 10 minutes to avoid surface discharges. After the irradiation, the sample was taken out of the chamber at atmospheric pressure for further investigations.

C. Space charge measurements

The classic PEA [START_REF] Maeno | Measurement of spatial charge distribution in thick dielectrics using the Pulsed Electroacoustic method[END_REF][START_REF] Li | Pulsed Electroacoustic method for measurement of charge for accumulation in solid dielectrics[END_REF] device available in the laboratory, provided by FiveLab has been used as a polarization and relaxation protocol was applied. The measurements were performed at atmospheric pressure at a controlled temperature of 25°C. The polarization voltage was applied through the semiconducting film attached to the top electrode whereas the Aluminum bottom electrode was grounded.

The first set of space charge (SC) results was recorded on a virgin sample. Space charge density profiles were recorded for 1 h while the sample was polarized under a field of 40 kV/mm. Afterwards, the signal was recorded for an extra hour under Volt-off to probe the charge release. A second set of samples has been irradiated with an electron beam before polarization. Two energies have been selected: 40 keV and 100 keV. The idea was to inject charges at different depths. For an energy of 40 keV, electrons are expected to be injected just beneath the irradiated surface whereas the 100 keV electrons are expected to be deposited up to a depth approaching the back surface of the 150 µm thick sample.

D. Fourier-Transform Infra-Red Spectroscopy (FTIR)

To investigate the impact of the thermal aging on the chemical structure the FTIR method has been implemented. FTIR analyses were performed on a VERTEX 70 fully digital infra-red (IR) spectrometer designed for data acquisition in the mid IR region (400-4000 cm -1 ) with a resolution of 2 cm -1 . The transmission mode was adopted for the measurement of the spectrum.

III. RESULTS AND DISCUSSION

A. Space charge profiles on the virgin sample

The first measurements by PEA on XLPE films have been performed on non-irradiated sample. During the polarization under 40 kV/mm we can see on Fig. 1 that both peaks on the electrodes are not symmetrical. This is due to a broad attenuation of the signal in the bulk. However, we can observe clearly a negative charge injection from the ground electrode (in the region 30-60 µm) which is growing slowly with time. We also notice a small negative charge accumulation near the HV electrode (right) which tends to decay with time.

During the relaxation (Fig. 2) we can observe that the small amount of negative charges injected near the ground electrode is decaying with time. Therefore, the positive induced charges on the electrodes are also decreasing slowly during the relaxation. After one hour of relaxation a small amount of negative charges is still detected in the volume. One can expect that after a few hours of relaxation most of the charges have been extracted or recombined.

B. Space charge profiles on irradiated samples

Space charge measurements were performed after irradiation on samples just after removal from the vacuum chamber. The irradiation energy was either 40 keV or 100 keV. On Fig. 3 we can see that in both cases a broad peak of negative charges is detected in the bulk. In the case of the 40 keV irradiation the peak maximum is detected at about 37 µm from the irradiated surface whereas after the irradiation under 100 keV the peak maximum is observed at about 125 µm. Theoretically the maximum penetration depth (under continuous slowing down approximation -csda) of the 40 keV electrons is expected at about 30 µm. In the present case electrons seem to have been injected deeper than expected in the bulk as they are detected up to 50 µm from the surface. The difference can be explained in part by the transport of charges under the field induced by the charges. For electron energy of 100 keV, the charge peak is found at about 125 µm from the surface, which is less than the csda for this energy (approximately 140 µm [START_REF] Woods | Applied Radiation Chemistry: Radiation Processing[END_REF]). One has to realize first that the csda is the maximum penetration depth; hence the majority of electrons are stopped before that range. Second, the influence charge on the ground electrode superimposes to the deposited charge. Given the fact that the spatial resolution is not at the best for the relatively thin films treated here, it is not surprising to find the main peak at this position. C. Space charge profiles with polarization of irradiated samples Fig. 4 shows the results with 40 kV/mm field applied after sample irradiation under 40 keV. The applied field is such that the initially stored charge should be repelled towards the irradiated side of the sample (anode to the right). As soon as the polarizing field is applied the peak of previously stored negative charges starts to decrease. After the quick formation of a positive peak in the bulk near the ground electrode (left) a continuous decay is also observed. The peaks of influence charges on both electrodes are also decaying continuously during the polarization due to internal charges redistribution.

During Volt-off (Fig. 5) the same decaying processes are observed. However, after 1 h of relaxation the position of the initial negative charge cloud deposited by the electron beam is still clearly observed. This peak is remaining at the same location without spreading. Only the amplitude has been modified during the polarization step.

The behavior of the sample irradiated under 100 keV is very different from that of the sample irradiated under 40 keV. During polarization, cf. Fig. 6, the formation of a prominent positive peak at 40 to 50 µm from the ground electrode is observed. Concomitantly, a negative charge peak grows up adjacent to the right electrode. With these features, the position of the charge cloud deposited initially shifts to the ground electrode. This corresponds most probably to an apparent shift, linked to the growing up of the positive peak and the partial superposition of the positive and negative charge clouds. The influence charge on the ground electrode grows progressively. It is weak in the first instants as the (positive) image charge of the previously stored negative charge is annihilated by the (negative) capacitive charge of the cathode. On the irradiated face (anode), the positive influence charge does not appear distinctly, owing to the presence of the negative charge in the bulk. A probable mechanism occurring in the irradiated region is the motion of the charge pairs created during the irradiation due to deposited dose. Positive charge would be driven towards the non-irradiated regions and negative ones towards the irradiated face. The amounts here are roughly of the same amplitude.The region which was initially neutral lets then appear a net charge.

During Volt-off, Fig. 7, both positive and negative charge peaks detected in the bulk slowly decay. The positive peak at 35 µm shifts slightly to the left; the negative one at 140 µm slightly to the right. Presumably, charges are repelled and recombined, or at least neutralized so as to form zero net charge regions. It is not clear whether the so called radiation induced conductivity in irradiated materials is linked to the formation of electronic carriers which are then involved in transport or to the creation of defects that assist transport of injected carriers. The space charge signal could be interpreted in the first case by the dissociation of pairs and the redistribution of ionic or electronic carriers. In the second case it may reflect conductivity gradient in the insulation.

D. Discussion

It is worth commenting on the state of samples used for the present characterizations, first because they were previously electro-thermally aged, second because they were stored under ambient conditions for about 15 years prior to irradiation. The FTIR spectrum of the investigated samples (before irradiation) is shown in Fig. 8. Compared to typical spectra reported in the literature [START_REF] Fothergill | Electrical,microstructural, physical and chemical characterization of HV XLPE cable peelings for an electrical aging diagnostic data base[END_REF][START_REF] Boukezzi | Observations on structural changes under thermal ageing of cross-linked polyethylene used as power cables insulation[END_REF] there is not significant change in the characteristic peaks of XLPE. The most important bands are found, as the shear-type vibration of C-H bond of the CH2 group at 720 cm -1 and at 1468-1472 cm -1 (for the crystalline phase), and the 2852 and 2916 cm -1 bands due to C-H bond of symmetric and antisymmetric elongation of the CH2 group [START_REF] Gulmine | Correlations between structure and accelerated artificial ageing of XLPE[END_REF].

The ageing (at T = 90°C) and storage of the XLPE sample does not have an effect on the disappearance or the creation of new bonds, such as the carbonyl groups. The ATR mode spectrum did not reveal either specific signature at the surface. There are at least 4 possible contributing processes to charge build-up in our conditions of irradiation and measurement that are: i/ primary electron deposition, at positions that are predictable and depend mainly on electron beam energy; ii/ electron-hole pair generation during irradiation followed by separation and transport under field application; iii/ positive charge injection at the ground electrode during irradiation (due to field created by implanted charges) and then negative charge injection under field when ground electrode is cathode; iv/ positive charge injection from irradiated face when top electrode is anode; v/ enhanced carrier mobility in the irradiated region producing space charge due to nonhomogeneous conductivity. According to the present results, charges are relatively deeply trapped in the bulk and do not relax in a period of one hour. It seems relatively clear that dissociation of created electron-hole pairs is at the origin of the main features obtained for 100 keV electrons, besides primary electron deposition. However, the charge amount created by dissociation is not easy to anticipate [START_REF] Banda | Polarization of electron-beam irradiated LDPE films: contribution to charge generation and transport[END_REF] since most of the created pairs may recombine quickly.

IV. CONCLUSIONS

The effects of the irradiation on charge injection and transport in XLPE films are highly dependent on the initial charge deposition depth. Bulk transport was observed on the sample irradiated with high energy electron beam. Besides, the accumulation of charges at the sample-electrode interface seems to confirm that we are dealing with bulk effects more that interface effects. No specific structural changes were detected by FTIR. Complementary physico-chemical analyses are necessary to reveal if there is a direct link between spacecharge behavior in post-irradiation and material structure. 1465 cm -1 2840 cm -1 2940cm -1 720 cm -1 1380 cm -1 Absorbance (a.u.)

Fig. 3

 3 Fig. 3 Charge density profiles recorded on a 150 µm thick XLPE film after irradation under 40 and 100 keV electron beam for 10 min with a flux of 1 nA/cm 2 in the irradiation chamber MATSPACE.

Fig. 1

 1 Fig. 1 Charge density profiles recorded by PEA on a 150 µm thick-XLPE film polarised under 40 kV/mm for 1 h. Left: ground electrode (cathode); Right: HV electrode (anode).

Fig. 2

 2 Fig. 2 Charge density profiles recorded by PEA during relaxation on a XLPE 150 µm-thick after 1 h polarisation under 40 kV/mm.

Fig. 4

 4 Fig. 4 Charge density profile on a 150 µm-thick XLPE film during 1 h polarisation after irradiation under 40 keV for 10 min. Applied field=40 kV/mm. Irradiated face to the right.

Fig. 5 Fig. 6

 56 Fig. 5 Charge density profiles recorded on 150 µm-thick XLPE film during the relaxation after the irradiation under an electron beam of 40 keV and 1 h polarisation under 40 kV/mm.

Fig. 7

 7 Fig. 7 Charge density profiles recorded on 150 µm-thick XLPE film during the relaxation after the irradiation under an electron beam of 100 keV and 1 h polarisation under 40 kV/mm.

6 F

 6 T IR -X L P E A b s o rb a n c e (a ,U ) W a v e n u m b e r (1 /c m )

ACKNOWLEDGMENT

The work is supported by French-Algerian Tassili research program in the frame of Partenariat Hubert Curien (PHC).


M E Banda

V Griseri

G Teyssedre

S Le Roy

Modelling Charge Generation and Transport in Low Density Polyethylene Irradiated by an Electron-Beam

Keywords: LDPE, charge generation and transport, fluid model, electron-beam irradiation, ageing

One way to bypass charge generation due to injection in an insulator sandwiched between parallel electrodes and submitted to an applied voltage is to implant charges in the material with the help of an electron beam. The electrons position and quantity is theoretically known as long as the beam energy and beam current are known. Low density polyethylene (LDPE) has been characterized with in-situ space charge measurements by pulsed electroacoustic method during irradiation, and with exsitu measurements while a DC voltage is applied. A fluid charge transport model has been developed using a commercial software, to reproduce the space charge behaviour during and after irradiation. Simulated results during irradiation are first compared to in-situ space charge measurements, in order to validate the model parameters related to e-beam irradiation. Simulations are then performed on post-irradiated samples, polarized under different electric fields. Space charge measurements and current measurements are available for comparison. Simulated results are in relatively good agreement with experimental ones as long as the model parameters are adapted to irradiated low density polyethylene, compared to a best set of parameters adapted uniquely for non-irradiated polyethylene.

I. INTRODUCTION

Applying a DC voltage on an insulator sandwiched between parallel electrodes is a common way to investigate space charge generation and transport in such materials. The physical processes occurring at the interface remain however complex to understand, as charge injection is not only dependent upon the insulating material, but also upon the contact type, its roughness, the chemicals at the surface and the way the interface has been processed, etc [START_REF] Taleb | IEEE Int. Conf. on Electrical Insulation and Dielectric Phenomena[END_REF]. One way to bypass injection processes is to use an electron beam, where the quantity and position of charges are normally known given the electron beam energy, its flux density and the properties of material under study. However, other processes, linked to charge generation due to the beam, are also not trivial to understand. Low density polyethylene (LDPE) has been used to characterize the behaviour of space charge during irradiation and post-irradiation. In the particular case of postirradiated samples where a voltage is applied, a peculiar behaviour, with strong charge build-up, has been observed [START_REF] Banda | [END_REF].

In this contribution, a bipolar charge transport model has been used to predict the space charge behaviour in nonirradiated LDPE as a reference, and in electron beam irradiated LDPE, for most of the available experiments under irradiation and post-irradiation. A comparison of the results obtained between experiments and simulations under irradiation and post-irradiation is proposed. The model is globally able to predict the experimental behaviour, provided changes in the material parameters are achieved. These changes traduce the modifications in the chemical structure of the material under irradiation.

II. MATERIAL, MEASUREMENTS PROCEDURE AND MODEL

A. Material and experimental procedure LDPE samples of thickness 300 µm were prepared from pellets using a hydraulic press in our laboratory, to ensure sample reproducibility. LDPE samples are then inserted in an irradiation chamber, under a vacuum of ~10 -7 mbar, and submitted to a 80 keV electron-beam for 8 minutes with a current flux of 1 nA/cm 2 . Charge density profile distributions could be determined in situ, i.e. under vacuum conditions, using the in-situ Pulsed Electro-Acoustic (PEA) space charge set-up under open circuit conditions [3]. Samples could also be removed from the irradiation chamber, i.e. they return to atmospheric pressure, and characterized outside the chamber, using different set-ups (space charge measurements, current measurements, luminescence techniques, and physico-chemical characterizations, such as FTIR or DSC). Space charge measurements will be presented under vacuum conditions, and post-irradiation current measurements will also be presented and compared to simulation results.

B. Model description

A numerical model, based on the one proposed earlier [4], has been developed using a commercial software to reproduce the space charge behavior in an electron-beam irradiated LDPE, for configurations encountered in the experiments, i.e. under vacuum conditions during irradiation, and at atmospheric pressure after electron beam irradiation, under applied voltage. The resolution of the model using COMSOL Multiphysics® is performed with the help of the mathematical module, for each equation. The model features bipolar injection of carriers at the electrodes, conduction using an electric field and temperature dependent mobility (i.e. of the hopping type), for each kind of carriers. Charge of each polarity can be trapped into a single level of deep trap, from which they can escape, via a thermally activated process. Langevin recombination is also taken into account [4]. The physical hypotheses linked to the electron beam irradiation have been adapted from Sessler et al [5], and are linked to electron deposition, and electron/hole pair generation. The equations to solve are:

-The Poisson equation:

∂E(x, t) ∂x = ρ(x, t) ε 0 ε r (1)
where E(x,t) is the electric field, x is the direction perpendicular to the electrodes, ε 0 is the vacuum permittivity, ε r is the dielectric permittivity (2.3. in the case of LDPE), and ρ is the net charge density, which can be written as:

ρ(x, t) = e -n eµ (x, t) -n et (x, t) + n hµ (x, t) + n ht (x, t) ( ) (2)
where n is the particles density (m -3 ), e is the elementary charge, the indexes e and h refer to the type of charge, i.e. electron or hole, and µ and t refer to mobile or trapped charges respectively.

-The continuity equation:

∂n e,h (x, t) ∂t + 1 e ∂j e,h (x, t) ∂x = s i (x, t) ( 3
)
where t is the time, and s i (x,t) are the source terms, which encompass the changes in local density not linked to transport.

-The transport equation, where j e,h is the conduction current density for each kind of carrier, neglecting diffusion:

j e,h (x, t) = ±en eµ,hµ (x, t)µ e,h (x, t)E(x, t) (4)
where µ e,h refers to the mobility, which is of the hopping type:

µ e,h (x, t) = 2νd E(x, t) exp - ew hop(e,h) k B T         sinh eE(x, t)d 2k B T       ( 5
)
where ν is the attempt to escape frequency, d the distance between traps, w hop(e,h) is the hopping barrier height for electrons and holes, k B the Boltzmann constant, and T the temperature.

An example of source term for mobile electrons is given here: (6) where S 0 to S 3 refer to the recombination coefficients, B e,h, are the trapping coefficients, and D e,h are the detrapping coefficients. G(x) is the electron/hole pair generation, and is of the form [5]:

G( x) = G 0 .D( x) (7)
where G 0 is the electron/hole pair generation coefficient and D(x) the normalized dose rate deposited by the electron beam. G(x) is void in the non irradiated region (NIR). J 0 (x) in (6) refers to the beam current density in the dielectric. Note here that once thermalized, primary electrons are considered as a source term for mobile electrons in the model. Also, carriers issued from pair generation are initially mobile.

Generation of positive or negative charges is possible through a modified Schottky law [6], the nature of injected carriers being function of the polarity of the electrode:

j inj (x, t) = AT 2 exp -e.w inj k B T       exp e k B T e.E(x, t) 4.π.ε 0 .ε r       -1           (8)
where A is the Richardson constant, and w inj the injection barrier height. In the case of vacuum conditions, i.e. under irradiation, there is no possible injection at the irradiated face, and only charge injection is effective at the ground electrode. In the case of post-irradiation conditions, where the LDPE samples are sandwiched between two electrodes, injection of positive/negative charges is possible at each electrode, depending on the sign of the electric field.

Simulations have been performed, using an open access code called PENELOPE [7] to calculate the electron path length distribution as well as the dose rate deposited inside LDPE for an electron beam of energy 80 keV and a flux of 1 nA/cm 2 . This code allows simulating the trajectory of a large number of particles over a wide energy range, accounting for the matter-radiation interactions. Profiles of the normalized beam current density and the normalized dose are given in Fig. 1 as a function of the sample thickness. The irradiated region extends over 125 µm from the irradiated face.

III. SIMULATION RESULTS

A. In-Situ conditions

Space charge measurements have been performed using the in-situ PEA set-up, on a 300 µm thick LDPE submitted to the irradiation conditions previously described for 8 min. Fig. 2 presents the space charge profiles function of the position in the sample for different irradiation times. It is to note that irradiation comes from the right side. A negative peak is observed with a peak maximum located at around 100 µm from the irradiated face. A positive peak, located at the ground electrode, shows the positive image charge due to the negative charge inside the bulk. Note that as PEA measurements are carried out in open circuit conditions, there are no image charges on the right side of the sample. The maximal value of the negative peak inside the bulk seems to be reached after around 4 min of irradiation. After this time, the peak amplitude seems to stabilize or even to decrease. This is certainly due to the fact that the electron beam does not irradiate the entire sample thickness, so charges accumulate, and enhance the surface potential at the irradiated face. Surface flashovers may have happened during the irradiation time, thus also discharging the sample in the bulk. Most of the parameters for the simulations can be found in the literature, or have already been adapted for this model for nonirradiated LDPE [8]. However, the coefficient related to generation of electron/hole pair due to the dose, G 0 , remains difficult to evaluate [4]. A sensitivity analysis has been performed on this parameter, letting the other parameters as in Table 1, to be as close as possible to the experimental results (Fig. 2).

TABLE I.

OPTIMIZED PARAMETERS USED FOR ALL THE SIMULATIONS Fig. 3 shows the simulated space charge distribution after 10 min of irradiation for different values of G 0 . In this figure, no image charge has been added to the simulated profiles. When G 0 increases, i.e. when the generation of positive/negative charges increases inside the bulk, the net charge density remains negative but the peak position shifts towards the ground electrode and its magnitude increases, with a decrease of the spread of charges. A parameter G 0 of 8 C.m -3 .s -1 seems the best to fit the experimental data.

Simulations have then been performed with the parameters of Table 1 and for the same conditions as for Fig. 2. The top PEA electrode is supposed to be at 1 mm from the surface. The simulated space charge profiles are presented in Fig. 4 for different irradiation times. Image charges are added to the simulated results, allowing a better comparison with the experimental profiles. As in the experiment, the net charge density remains negative during irradiation, even if positive charges are generated by the electron beam and injected at the ground electrode, due to a favorable electric field. The peak maximum is located at 90 µm from the irradiated face, which is consistent compared to the experimental results. The value of the peak maximum is of the order of 40 C.m -3 , which is twice the one observed experimentally after 8 min of irradiation. However, the model does not take into account the possible discharges that occur during irradiation due to a too high floating potential. Globally, the simulated net charge density profiles are comparable to the experimental ones.

B.

Atmospheric (ex-situ) conditions Conduction current measurements and related simulations were then performed on non-irradiated and post-irradiated LDPE samples at atmospheric pressure (Fig. 5). The protocol of application of the voltage consists in increasing the applied electric field by steps of 5 kV/mm with a resting time of 10 min at each step, up to 100 kV/mm. The set of parameters used for the simulations are given in Table I, using the Schottky injection barriers referring to 'MIM' conditions. Each point of the experiment/simulation is an average of current over the last minute in each field step.

The shape of the current-field characteristic is of SCLCspace charge limited current-type, for a non-irradiated and an irradiated sample. At low fields, there is more than one decade difference in current between the irradiated and the nonirradiated sample. For electric fields above 40 kV/mm, this difference decreases and it seems that the shape of the characteristic for the irradiated sample draws closer to the one of the non-irradiated one. This behavior seems consistent with space charge measurements performed on post-irradiated samples, where more charges are measured compared to nonirradiated samples [START_REF] Banda | [END_REF]. At higher electric fields, the charge injection becomes dominant compared to the charge generated by the electron beam, and the sample behavior comes back to a non-irradiated behavior. Simulated currents for non-irradiated and irradiated samples are in good agreement with the experiment for fields below 70 kV/mm. The difference in current density between the irradiated and the non-irradiated sample could be reproduced.

The simulated results also reproduce the SCLC shape, even if the model is not based on this hypothesis, with a first slope of 1 for fields below 10 kV/mm, and a slope of 2 for fields between 10 and 60 kV/mm. Above this field, the model predicts a large increase of the slope in current density, which is not observed experimentally. However, the model also predicts a current density for the irradiated sample that comes closer to the non-irradiated behavior, as the difference is less than a decade for an electric field of 100 kV/mm. These simulated results have been obtained with the optimized set of parameters of Table I, which are not the same as the one optimized for a non irradiated LDPE based on space charge and current data for MIM structures [8]. Irradiating the LDPE sample, even for a so short time, can change its chemical structure irreversibly. On the basis of physico-chemical measurements (FTIR, RMN and DSC), this is what has been concluded, even if the changes are in a small range. Changes in the physico-chemical structure imply also changes in the trapping, and detrapping behavior, and hence in the parameters linked to these processes. Step-wise increase protocol. Optimized parameters of Table I.

The main changes in the parameters of the model are the decrease of the trapping coefficients, by one decade, and the value of the detrapping barrier, for holes mainly.

IV. CONCLUSIONS

Various experiments have been performed on electron-beam irradiated LDPE samples, in situ and in post-irradiation, in order to comprehend the specific behavior of charge generation and transport within the material. A bipolar fluid charge transport model has then been used to predict the space charge behavior in the same experimental conditions, for most of the in-situ and post-irradiation experiments. This model is globally able to reproduce the experimental data, i.e. the hypotheses of the model seem enough to describe the space charge behavior. However, this goal has been reached by changing some of the optimized parameters of the model, mainly the trapping and detrapping parameters. These parameters being linked to the chemical structure of the sample, one could think that the experimental protocol used to characterize the LDPE samples is enough to age the material, and that the changes in the simulation parameters is a consequence of these chemical changes. However, more work needs to be done, mainly on the time of irradiation, in order to validate these conclusions.

Fig. 1 .

 1 Fig. 1. Normalized electron density and normalized dose as a function of the position in the sample calculated by PENELOPE for a 300 µm thick LDPE irradiated with an electron beam of 1 nA/m 2 of energy 80 keV.

Fig. 2 .

 2 Fig.2. Experimental space charge profiles measured on a 300 µm LDPE with the 'in situ' PEA under vacuum conditions for 10 min of irradiation with a 80 keV electron beam.

Fig. 3 .

 3 Fig. 3. Simulated net charge density function of the position, for different values of the electron/hole pair coefficient G 0 , for 10 min of irradiation.

Fig. 5 .

 5 Fig. 5. Experimental and simulated conduction current density as a function of the applied electric field. Step-wise increase protocol. Optimized parameters of TableI.


Y Kemari

A Mekhaldi

M Teguar

G Teyssèdre

Unthresholded Recurrence Plots Analysis for Degradation Assessment of Power Cables Insulation

Keywords: Electrical insulation, dielectrics, thermal aging, cross-linked polyethylene, polyvinyl chloride, power cables, recurrence plot, diagnosis. I

Underground power cables have been widely used in transmission and distribution power systems, especially in urban areas. During their operating conditions, power cables are subjected to thermal, mechanical and electrical stresses. These stresses lead to accelerate the degradation process of the insulation material. In this paper, we present a novel methodology in the field that could be used to detect degradation level of power cables insulation due to thermal aging. In light of this, cross-linked polyethylene (XLPE) and polyvinyl chloride based compound (PVC, class B or PVC/B) were chosen as the studied materials. Thermal aging tests were performed at different aging temperatures (80, 100, 120 and 140 °C).

Thereafter, the alternative bulk current through the aged samples was recorded at different aging stages and under AC applied voltage of 12 kV. For the aim to better analyze in-depth the obtained current waveforms, unthresholded recurrence plots (UTRPs) technique is introduced in order to extract features from the AC bulk current and reveal its characteristics. The obtained results show that the proposed method could be used as an assessment tool for current analysis and thermal degradation monitoring of insulation materials used in extruded power cables.

INTRODUCTION

The deterioration of extruded power cables insulation could be attributed to a variety of stresses from the surrounding environment (i.e external excess temperature, external radiation, humidity) [START_REF] Aras | Aging of 154 kV underground power cable insulation under combined thermal and electrical stresses[END_REF]. In fact, the thermal constraint, which is mainly due to the heating effect from the conductor carrying load current [START_REF] Aras | Aging of 154 kV underground power cable insulation under combined thermal and electrical stresses[END_REF], is often considered as the most harmful factor causing not only to the full deterioration of the power cable, but also to environmental and financial damage [START_REF] Wald | Evolution of Medium Voltage Cable Technology in Europe[END_REF]. For this reason, many researches are focusing on the development of new nondestructive diagnostic methods for the aim to assess whether such cables have been electrically degraded due to thermal aging [START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF][START_REF] Teyssedre | Advances in High-Field Insulating Polymeric Materials Over the Past 50 Years[END_REF][START_REF] Werelius | Development and Application of High Voltage Dielectric Spectroscopy for Diagnosis of Medium Voltage XLPE Cables[END_REF].

It is well known that the dielectric response of a cable insulation is a result of its capacitive and conductive properties. In fact, in conventional power cables, the voltage is applied between a central metallic conductor and an earthed metallic screen that surrounds it. The two components are separated by the insulation system [START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF]. Consequently, we suggest that the analysis of the bulk current signal through insulation materials could be applied for the degradation diagnosis of power cables. Moreover, several signal processing methods, known also as feature extraction techniques, could be used to study the characteristics of the current signal. Some of them privileged to extract features in the time domain, while others suggest the frequency one [START_REF] Chaou | Recurrence quantification analysis as a novel LC feature extraction technique for the classification of pollution severity on HV insulator model[END_REF].

In the present work, we propose an experimental-signal processing methodology for evaluating the degradation level of the studied insulation materials. The methodology involves the record of the AC current waveforms in unaged and thermally aged XLPE and PVC/B samples. Unthresholded Recurrence Plots (UTRPs) method has been introduced for the first time in this area for the sake to characterize and identify the behavior of the AC current versus aging process. Indeed, UTRPs reflects the current shape, and thus the insulating state. By means of graphical structures, UTRPs is an effective technique to monitor and detect the critical degradation of cable insulation.

II. EXPERIMENTAL TECHNIQUES AND PROCEDURE

A. Material and samples preparation XLPE and PVC/B compounds used in power cables are intended for voltages up to 36 kV and 3.6/6 kV respectively, according to IEC 60502-2 standard [START_REF]Power cables with extruded insulation and their accessories for rated voltages from 1 kV up to 30 kV[END_REF]. Crosslinked Polyethylene XLPE was supplied by General Cable, ENICAB factory of Biskra, Algeria. The insulation material, LE4201R compound, intended for voltages up to 36 kV was delivered by Borealis, Austria. The melt temperature of the material is approximately 130 °C, and it contains dicumyl peroxide (DCP) as crosslinking agent and Di-tert-butyl peroxide as antioxidant. On the other hand, PVC/B is manufactured by the electric cables factory CABEL in Algiers, Algeria, where Polyvinyl chloride resins (SE-1200) are blended with di-isodecyl phthalate (DIDP) plasticizer and Calcium-Zinc Stabilizer. XLPE and PVC/B plates were cut into circularly shaped samples using a cutting press. The samples have a thickness of 2 mm and a diameter of 46 mm.

B. Thermal aging experiments

Thermal aging was performed according to the IEC 60216 standard [START_REF]Guide for the Determination of Thermal Endurance Properties of Electrical Insulating Materials[END_REF]. For both materials, the circular samples were aged at three temperatures; 80, 100 and 120 °C using air-circulating ovens. For PVC/B, the aging time was 5040 h (210 days) at 80 and 100 °C against 1008 h (42 days) at 120 °C, corresponding to the mechanical properties deterioration of PVC/B. Due to its excellent thermal endurance, XLPE samples are exposed to an additional aging temperature of 140 °C for 2520 h (105 days).

C. Experimental arrangement

Current measurements were carried out in the high voltage laboratory of Ecole Nationale Polytechnique, Algiers, Algeria. The experimental setup is schematically presented in Fig. 1. The experimental setup consists of a high voltage test transformer (50 kVA, 300 kV, 50 Hz) supplied by a regulating transformer (50 kVA, 220/500 V, 50 Hz) and a voltage capacitive divider constituted by two capacitances in series (C1=400 pF and C2=1000.C1).

As shown in Fig. 2, the prepared samples were mounted in an in-house made test cell between two electrodes of 25 mm in diameter with a grounder guard ring to prevent leakage current and provide homogenous electric field. The test cell is filled up with transformer oil in order to prevent external discharges and flashover. Acquisition of the AC bulk current waveforms was performed using Instek GDS-3504 4-CH Digital Storage Oscilloscope of 500 MHz bandwidth. These waveforms are recorded across a shunt resistor of Rm = 1 kΩ.

III. UNTHRESHOLDED RECURRENCE PLOTS (UTRPS)

Recurrence plot (RP) technique is a graphical tool used to visualize the recurrence behavior of phase space trajectory of a given time series. There are two distinct types of recurrence plots: unthresholded (UTRPs) and thresholded (TRPs) recurrence plots [START_REF] Yang | Multiscale Recurrence Quantification Analysis of Spatial Cardiac Vectorcardiogram Signals[END_REF][START_REF] Cazares-Ibáñez | Application of recurrence plots as a new tool in the analysis of electrochemical oscillations of copper[END_REF]. In a given m-dimensional phase space, the RPs are formed by comparing all embedded vectors with each other. If two state vectors lie in a neighborhood characterized by a threshold ε, the points are considered to be recurrent [START_REF] Chaou | Recurrence quantification analysis as a novel LC feature extraction technique for the classification of pollution severity on HV insulator model[END_REF]. The embedding dimension m has been chosen m=1 using false nearest neighbor algorithm [START_REF] Kennel | Determining embedding dimension for phase-space reconstruction using a geometrical construction[END_REF]. The delay time τ was determined by the method of mutual information [START_REF] Fraser | Independent coordinates for strange attractors from mutual information[END_REF], where τ=20. In UTRPs, points (i, j) get a colored dot according to the distance between points i and j in the phase space [START_REF] Yang | Multiscale Recurrence Quantification Analysis of Spatial Cardiac Vectorcardiogram Signals[END_REF][START_REF] Cazares-Ibáñez | Application of recurrence plots as a new tool in the analysis of electrochemical oscillations of copper[END_REF]. First, the vector i x  can be reconstructed from a time series x(i) as follows [START_REF] Chaou | Recurrence quantification analysis as a novel LC feature extraction technique for the classification of pollution severity on HV insulator model[END_REF]:

[ ] (1) ,..., 1 , ) 1 ( ( ),..., ( ), ( N i m i x i x i x x i = - + + = τ τ 
Where N is the length of a time series, m is the embedding dimension and τ is the delay time. Mathematically, for a given trajectory xi (i = 1, . . ., N, x ∈ R m ), the recurrence matrix Ri,j is defined as [START_REF] Chaou | Recurrence quantification analysis as a novel LC feature extraction technique for the classification of pollution severity on HV insulator model[END_REF] :

(2) ) ,..., 1 ( , , ) ( ) ( , N j i x x R j i j i = - - =   ε Θ ε
Where . is the norm (Euclidian norm is chosen is this work), Θ is the Heaviside function, ε is threshold value and x is the studied time series. As far as the threshold value ε is concerned, many researchers concluded that ε can be selected as 0.25σ [START_REF] Chaou | Recurrence quantification analysis as a novel LC feature extraction technique for the classification of pollution severity on HV insulator model[END_REF], where σ is the standard deviation [START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF].

IV. RESULTS AND DISCUSSION

This section is dedicated to presenting the experimental results of the obtained AC current waveforms. In addition, UTRPs is employed in order to extract further information about the current signals for XLPE and PVC/B materials. In this investigation, UTRPs is used for qualitative analysis of the current waveforms. The 2-D representation has distinct topologies and texture patterns. This offers the possibility to detect changes in the graphic patterns and structures with thermal aging. In the UTRPs, each color indicates a range of distances between states, i.e. the Euclidean distances between all vectors are mapped to colors as shown on the color bars of each UTRPs diagram. If the color code at the point is blue, then the points are located close to each other, and if the color code is red, the points are located farther apart.

A. XLPE material

The evolution of the current waveforms for XLPE before and after aging at 80, 100, 120 and 140 °C, under 12 kV is illustrated in Fig. 3. Besides, Fig. 3 presents also UTRPs generated from the different current waveforms. It can be observed that aging has a very weak influence on the current magnitude at 80, 100 and 120 °C. However, at 140 °C, the current magnitude significantly increases. On the other hand, it has been found that the reason behind the shape of symmetrically distorted "M" is the existence of odd order harmonics, essentially the 5 th (250 Hz) and 7 th (350 Hz) ones [START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF]. In UTRPs diagram, warm colors (red) can be associated with short distances between vectors, and contrarily cool colors (blue) can be associated with long distances. The diagonally oriented lines and the repeated structural elements in the plot indicate the periodicity of the current. Moreover, the system is perfectly predictable since the diagonal lines in the plot are long [START_REF] Mayorga-Cruz | Analysis of Michelson Optical Interferometry Using Recurrence Plots During Corrosion of Aluminium in NaCl Solution[END_REF]. Furthermore, interesting "window" shape structures could be found in the vicinity of the main diagonal line (the Line of Identity LoI) as indicated with circles in Fig. 3e. The appearance of these shape structures is related to the "M" shape observed in the current waveforms which has been already explained by the existence of odd harmonics. It can be noticed that the topology is practically the same for the five cases. However, the intensity of the warm colors (red and yellow) increases with aging as the distances between points increases. This can be clearly noticed after 2520 h at 140 °C (Fig. 3e). In fact, the color bar that corresponds to the case of the most damaged sample (aged for 2520 h at 140 °C, Fig. 3e) has been chosen as a reference color-bar. In another word, for the sake of better comparison, the color scale of Fig. 3e has been used for unaged and aged samples at 80, 100 and 120 °C. The distances are scaled to the value range [0, 1.29x10 -4 ].

B. PVC/B material

As far as PVC/B is concerned, on the right left side of Fig. 4, the evolution of current waveforms before and after aging at 80, 100 and 120 °C under 12 kV is portrayed. On the right side of Fig. 4, UTRPs topologies of the current waveforms in PVC/B material are presented. The aging effect is significant on the current magnitude at 100 and 120 °C. Furthermore, in the case of PVC/B samples aged at 120 °C after 504 h (Fig. 4d and4e), the degradation is more accentuated. Indeed, it has been found that the 3 rd harmonic rises and its existence is considered to be more important with regard to 5 th and 7 th components. Therefore, the 3 rd harmonic becomes the most important harmonic component that could be used in order to characterize the critical degradation of the insulation [START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF]. We propose that the breakdown of PVC/B becomes imminent when the 3 rd harmonic content increases [START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF][START_REF] Suda | Frequency characteristics of leakage current waveforms of an artificially polluted suspension insulator[END_REF]. Moreover, in unaged and aged PVC/B samples at 80 and 100 °C, the obtained graphic structures are the same obtained previously for XLPE material. These structures consist mainly of diagonally oriented lines and "window" shape structures (Fig. 4a, 4b and4c). With regard to XLPE material, a significant change can be noticed in the UTRPs diagram color of PVC/B at 100 °C after 5040 h of aging.

Nevertheless, at 120 °C, the "window" shaped patterns have been disappeared and new structures have been obtained as indicated with circles in Fig. 4c and4e. This recurrence structure could be attributed to the middle peak obtained in the current waveform. This peak is related to the appearance of the 3 rd harmonic component in the current, whereas 5 th and 7 th harmonics are always existing. A similar recurrence structure has been found by other researchers [START_REF] Zhou | Recurrence Plot Based Damage Detection Method by Integrating Control Chart[END_REF]. Additionally, the increase of the Euclidean distances between all recurrences points leads to the increase of the intensity of the warm colors. This last result contributes to the appearance of the "Octagram" shape which can be easily distinguished at 120 °C. In fact, the increase of the distances between points indicates that the process is becoming more and more deterministic [START_REF] Zhou | Recurrence Plot Based Damage Detection Method by Integrating Control Chart[END_REF]. As a result, "Octagram" structure could be a good indicator of the advanced aging process. Similarly to XLPE, the color-bar corresponds to aged sample at 120 °C (Fig. 4e) has been used as a reference for better comparison. The distances are scaled to the value range [0, 3.85x10 -4 ]. It can be noticed that for unaged and aged PVC/B sample at 80 °C, UTRPs show only the blue color with regard to unaged and aged XLPE sample at the same temperature. Moreover, at 100 and 120 °C, the distance between points, for PVC/B material, is higher than that of XLPE for a fixed position in the UTRPs diagram. This observation denotes that the degradation mechanism of PVC/B versus thermal constraint is more rapid and significant than that of XLPE material.

The extension of UTRPs method from samples to cables would require recording the bulk current signatures in the cables as a function of time. Adaptation of the system would be possible using the central conductor as high voltage electrode and a coaxial cable connected to the grounded screen as the measurement electrode. A guard electrode could be used to avoid surface leakage current and any undesirable partial discharge phenomena to occur.

V. CONCLUSION

Unthresholded recurrence plot analysis was carried out on the AC bulk current. UTRPs diagram was evaluated for unaged and aged XLPE and PVC/B samples. The results show that the increase of the current magnitude is characterized by the appearance of red and black zones for both materials in UTRPs. Nevertheless, for PVC/B samples aged at 120 °C, new shaped structures appeared that might reflect the possibility of breakdown occurrence. The different geometries and topologies of UTRPs diagram could allow us to use these plots as a means of determining when the degradation of insulation materials becomes serious and intense. Thus, UTRPs analysis appears to be a sensitive tool to evaluate the insulation material quality. Future investigation will deal with recurrence quantification analysis method (RQA) to quantify the evolution of the AC bulk current dynamics versus thermal aging.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Schematic diagram of the experimental set-up

Fig. 3 .Fig. 4 .

 34 Fig. 3. From left to right, current waveforms and UTRP of the current for XLPE material under 12 kV: (a) before aging, (b) at 80 °C for 5040 h, (c) at 100 °C for 5040 h, (d) at 120 °C for 5040 h, (e) at 140 °C for 2520 h


Y Kemari

A Mekhaldi

M Teguar

G Teyssèdre

Nonlinear Regression Modeling to Predict Thermal Endurance of XLPE Material under Thermal Aging

Keywords: Electrical insulation, dielectrics, thermal aging, cross-linked polyethylene, power cables, nonlinear regression, thermal endurance. I

In this work, the mechanical behavior of XLPE under accelerated thermal aging is investigated. The tensile strength and elongation at break are used as aging markers in order to assess the degradation level of the insulation. In the aim to describe the evolution of the mechanical properties with aging and to estimate the insulation lifetime, a nonlinear regression model is proposed. The model parameters have been estimated using the method of least squares. The experimental results show that thermal aging leads to mechanical degradation of XLPE by decreasing its tensile strength and elongation at break. The critical deterioration of the material was reached at 120 and 140 °C. The validity of the regression model was verified and the results show that one-phase exponential decay expression could well fit the experimental data. In addition, thermal endurance of XLPE was investigated using the Arrhenius law. On the other hand, permittivity and AC conductivity have been monitored in order to study their correlation with thermal aging.

INTRODUCTION

Cross linked polyethylene (XLPE), as all other cable insulators are subjected to a diversity of stresses. The thermal stress is one of the most severe service constraints that can lead to the reduction of the cable lifetime [START_REF] Zhang | Residual life assessment of 110 kV XLPE cable[END_REF]. In fact, the electrical and mechanical properties of XLPE can be deteriorated by long-term thermo-oxidation reactions [START_REF] Boubakeur | Observations on structural changes under thermal ageing of cross-linked polyethylene used as power cables insulation[END_REF]. Some researchers [START_REF] Boubakeur | Observations on structural changes under thermal ageing of cross-linked polyethylene used as power cables insulation[END_REF][START_REF] Xu | Investigation on insulation material morphological structure of 110 and 220 kV XLPE retired cables for reusing[END_REF] have reported that the changes of the chemical structure of XLPE are attributed to chains scission reactions and to the diminution of the crystalline fraction of the insulation material. Therefore, these changes can result in the deterioration of XLPE mechanical characteristics leading to a significant drop of its elongation at break (E aB) and tensile strength (TS).

On the other hand, the prediction of cables remaining lifetime is nowadays a major challenge. Accelerated thermal aging and Arrhenius model have been widely used by many researchers [START_REF] Marzinotto | Investigation on thermal endurance of PVC compounds for low voltage cable insulation[END_REF][START_REF] Montanari | Thermal endurance evaluation of XLPE insulated cables[END_REF] to evaluate the insulation lifetime since the estimation of degradation lifetime at service temperature requires time consuming tests. In addition, the choice of end of life criterion is a very important aspect of applying the Arrhenius model. According to IEC 60216 standard [START_REF]Guide for the determination of thermal endurance properties of electrical insulating materials. Part 1: General guidelines for aging procedures and evaluation of test result[END_REF], a loss of 50% of the initial value of EaB or/and TS is considered as end-of-life criterion. Nevertheless, the lifetime threshold at low aging temperatures could not be achieved and an adequate prediction of mechanical properties evolution is needed. This paper is focused on three investigations; the first one aims to study the effect of the accelerated thermal aging on EaB and TS of XLPE material. The second part is dedicated to establishing a nonlinear regression model which could help to describe the evolution of mechanical properties with aging and to predict the outcome of a lifetime test at low stress levels. Meanwhile, XLPE thermal endurance characteristics have been determined using the Arrhenius law. Finally, the third part presents some dielectric properties characterization results on XLPE. In order to investigate their correlation with thermal degradation, permittivity εr' (the real part of the complex dielectric constant) and AC conductivity σ' (the real part of AC complex conductivity) have been monitored at the end of thermal aging tests at each temperature -i.e. after 5040 hours at 80 and 100 °C, 4032 hours at 120 °C and after 1512 hours at 140 °C.

II. TESTS AND EXPERIMENTAL METHODS

A. Material and samples preparation

In this work, crosslinked Polyethylene XLPE was supplied by ENICAB factory of Biskra, Algeria. The compounded cross-linkable material was of commercial name LE4201R, provided by BOREALIS Company (Denmark). The material contains dicumyl peroxide (DCP) as crosslinking agent and Ditert-butyl peroxide as antioxidant. Pellets were preheated for 5 minutes at 125 °C then molded and crosslinked at 185 °C under a pressure of 300 MPa using a pressurized heat press. According to IEC 60216 publication [START_REF]Guide for the determination of thermal endurance properties of electrical insulating materials. Part 1: General guidelines for aging procedures and evaluation of test result[END_REF], dumbbell samples of 7.5 cm in length were used for the mechanical tests. On the other hand, electrical measurements were carried out on circular shaped samples of nominal thickness of 2 mm and a diameter of 46 mm.

B. Accelerated thermal aging

Thermal aging was performed according to the IEC 60216 standard [START_REF]Guide for the determination of thermal endurance properties of electrical insulating materials. Part 1: General guidelines for aging procedures and evaluation of test result[END_REF] on vertically arranged XLPE samples inside aircirculating ovens. Three dumbbell samples were taken out every 504 h (21 days), at temperatures of 80, 100, 120 and 140 °C. The aging times were 5040 h (210 days) at 80 and 100 °C, 4032 h (168 days) at 120 °C and 1512 h (63 days) at 140 °C corresponding to the mechanical deterioration of the XLPE material as will be discussed later on.

C. Mechanical properties tests

Tensile tests were performed for XLPE samples according to IEC 60811 standard [START_REF]Common test methods for insulating and sheathing materials of electric cables and optical cables[END_REF] via universal testing machine Zwick/Roell ProLine Z050, as shown in Fig. 1, with a load capacity of 50 kN (available at General Cable laboratory, ENICAB, Biskra, Algeria). All the measurements were carried out at room temperature.

D. Dielectric spectroscopy

Prior to the dielectric measurements, gold electrodes were deposited onto both surfaces of the XLPE circular samples by sputtering. The diameter and the thickness of the sputtered electrodes are respectively 35 mm and 30 nm. Permittivity ε' and AC conductivity σ' were measured using Novocontrol Alfa-A High Performance Analyzer dielectric spectrometer with a ZGS active sample cell. The measurements were carried out under N2 atmosphere at 20 °C. The frequency varied from 10 -2 to 10 4 Hz. In addition, the control and the evaluation of dielectric measurements were realized using WinDETA® software.

III. NONLINEAR REGRESSION ANALYSIS

Nonlinear Regression is used in this work to describe the experimental data of tensile experiments as a mathematical equation. Hence, EaB and TS will be defined as a function of aging time and other fixed parameters. This will lead to estimate the best fitting of the XLPE mechanical behavior on the long-term. Z. Zhang et al [START_REF] Zhang | Residual life assessment of 110 kV XLPE cable[END_REF] investigated the thermal life curve of aged XLPE samples from 110 kV power cable. They proposed a fifth-degree polynomial equation to calculate elongation at break, although, this model contains six parameters to be determined. Besides, polynomial models can give a good fit, but can also substantially diverge outside the defined range of the experimental data. In this study, a single exponential decay function has been chosen. EaB and TS can be expressed by equations ( 1) and ( 2) respectively as follows:

) / ( 0 1 . τ t aB e A E E - ≈ (1) ) / ( 0 2 . τ t S e B T TS - ≈ (2)
where, E0, TS0, A, B, τ1 and τ2 are the regression parameters. A major aim of the investigation is to estimate the parameters as precisely as possible. For this purpose, least-squares regression technique has been adopted under MATLAB® 2009 environment. This technique consists of selecting the parameters that would minimize the difference, or residual, between each data point and the calculated values. Moreover, the criteria that have been used to assess the proposed regression model are: the coefficient of determination R², the residual sum of squares (RSS), the standard deviation of the model (RMSE) and the analysis of variance (ANOVA) [START_REF] Zhang | Residual life assessment of 110 kV XLPE cable[END_REF][START_REF] Sahli | Prediction parameters of dimensioning of insulators under non-uniform contaminated conditions by multiple regression analysis[END_REF].

The coefficient of determination R², is defined as the ratio of the residual sum of squares (RSS) and total sum of squares (TSS). The closer R² to 1 is, the more accurate the modelling function is. In addition, the RMSE is a measure of the average deviation of the etimated values from the observed ones.

IV. RESULTS AND DISCUSSION

A. Tensile experiments results

Fig. 2 and Fig. 3 present respectively the variation of elongation at break and tensile strength of XLPE samples over aging time at 80, 100, 120 and 140 °C. Each point is represented by the arithmetic average of three experimental values and their standard deviation. In addition, the fitting curves, obtained by the regression method, have been presented with the experimental data by continuous lines.

It can be noticed that the higher the aging temperature is, the more the degradation of mechanical properties is accentuated. For instance, at 80 °C, EaB and TS remain practically unchanged at the beginning of aging (up to 2520 h), then they slightly decrease with aging time. A decrease of 8.42 % from the initial value has been obtained after 5040 h at 80 °C for elongation at break and of 18.90 % for tensile strength. At temperature 100 °C, it is found that the drop of EaB and TS with aging time is fast than at 80 °C. EaB decreased from 581 % to 414 % after 5040 h of aging at 100 °C (a loss of 28.74 % from the initial value). For TS, it decreased from 20.26 to 13.9 MPa after 5040 h. However, XLPE samples need to be aged over a longer period to reach the end of life criterion at these relatively low temperatures.

The degradation of mechanical properties at 120 °C is more significant. EaB decreased abruptly and sharply, as well as TS, after 4032 h of aging. At this stage, the loss in EaB was 92.71 % against 66.73 % in TS. Meanwhile, the degradation of XLPE samples due to thermal aging at 140 °C was very rapid and intense. Elongation at break and tensile strength reached respectively values of 22 % and 5 MPa within 1512 h. In fact, the described evolution above of mechanical properties for XLPE depends on structural changes in the material [START_REF] Boubakeur | Observations on structural changes under thermal ageing of cross-linked polyethylene used as power cables insulation[END_REF][START_REF] Xu | Investigation on insulation material morphological structure of 110 and 220 kV XLPE retired cables for reusing[END_REF]. At lower temperatures, 80 and 100 °C, some combined effects of thermal oxidation and cross-linking occur simultaneously at the beginning of aging (up to 2520 h at 80 °C and 3024 h at 100 °C). The oxidation degradation takes place, but it seems still to have a limited effect [START_REF] Boubakeur | Observations on structural changes under thermal ageing of cross-linked polyethylene used as power cables insulation[END_REF]. Thereafter, the oxygen diffusion process takes place across the dense crystalline domains, which is a consequence of the continuous but slow decrease of the mechanical properties with aging time. Moreover, the degradation process at 120 °C becomes faster and more intense due to the auto-accelerated oxidation process. The oxidation process is followed by chains scission. The formation of shorter chain segments leads, in relatively short aging time (4032 h), to an abrupt decrease of the mechanical characteristics. Finally, at 140 °C, the oxidation process is dominant and after a relatively short time of aging, EaB and TS drops very substantially. A significant decrease of the crystalline fraction occurs, which makes oxygen diffusion easier leading to chain scission [START_REF] Boubakeur | Observations on structural changes under thermal ageing of cross-linked polyethylene used as power cables insulation[END_REF].

B. Nonlinear regression results:

In this part, we present the calculated parameters of the proposed model in addition to the statistical results of the regression method for the studied mechanical properties of XLPE. The obtained results for EaB model (equation 1) are shown in Table I, whereas Table II illustrates the results of TS model (equation 2). It can be noticed that the parameters E0 and A change with aging temperature as their sum represents the initial value of EaB before aging. The same observation could be made for TS0 and B in the case of TS. Besides, the time constants τ1 and τ2 changed slightly at 80, 100 and 120 °C, then a significant variation is obtained at 140 °C meaning that τ1 and τ2 are well correlated with the degradation process. For the statistical analysis, the R² values close to unity and the low RMSE values indicate that a strong correlation exists between the experimental data and expected ones [START_REF] Shirkovets | Accelerating dielectric response measurements on power transformers-Part II: a regression approach[END_REF]. Additionally, a high F ratio and values of P that are less than the threshold of 0.05 (or 95% confidence) mean that the model is statistically significant [START_REF] Shirkovets | Accelerating dielectric response measurements on power transformers-Part II: a regression approach[END_REF]. Therefore, the proposed model for EaB and TS prove to be significant for the estimation of end of life of XLPE due to thermal degradation.

C. Thermal endurance by Arrhenius model:

This section aims at estimating XLPE insulation lifetime using mechanical test results and the proposed regression model above. Besides, the Arrhenius model is used for the determination of the thermal endurance curve, which is for expression [START_REF] Marzinotto | Investigation on thermal endurance of PVC compounds for low voltage cable insulation[END_REF][START_REF] Montanari | Thermal endurance evaluation of XLPE insulated cables[END_REF]:

) . exp( . T R E A t = (3)
where t is the lifetime, A is a constant, E is the activation energy of the aging process (J.mol -1 ), T is the absolute temperature (K) and R is Boltzmann constant equal to 8.314 J.mol -1 . K -1 . First, the lifetime was estimated at each aging temperature by using the regression model. In fact, lifetime criterion has been achieved for EaB before TS. Then, natural logarithm of lifetime has been plotted versus reciprocal temperature (1000/T) as illustrated in Fig. 4. The parameters appearing in the equation (3) are determined using linear regression method. Furthermore, the temperature index of XLPE (TI) was obtained by extrapolation of thermal endurance straight line to a lifetime of 20000 hours. The ordinate intercept A is equal to 8.8×10 -4 h the calculated activation energy E is 11.47 kcal.mol -1 (0.46 eV) and TI was found 71.0 °C. The Arrhenius parameters present a good linear regression with R 2 = 0.91 as presented in Fig. 4.

D. Dielectric spectroscopy results

The variations of permittivity εr' and AC conductivity σ' with thermal aging are illustrated in Fig. 5 for 10 -2 to 10 4 Hz frequency range. As far as εr' is concerned, the dielectric constant slightly decreases after 5040 h of aging at 80 °C, compared to reference material, which could be attributed to cross-linking of the material. After aging for 5040 h at 100 °C and for 4032 h at 120°C, the permittivity values slightly exceed the ones of unaged sample.

The increase of εr' is highly significant at 140 °C due to the release of carbonyl groups and low molecular weight acid as thermal oxidation products [START_REF] Boubakeur | Observations on structural changes under thermal ageing of cross-linked polyethylene used as power cables insulation[END_REF][START_REF] Montanari | Thermal endurance evaluation of XLPE insulated cables[END_REF]. Besides, εr' is almost unchanged with frequency variations for unaged and aged XLPE samples at 80, 100 and 120 °C. A significantly larger dependence has been observed at 140 °C as the frequency decreases. For the AC conductivity σ', it can be said that thermal aging yields the greatest influence only at 140 °C as shown in Fig. 5b which could be explained by chains scission and polar oxidation products. These results are in good accordance with the increase of the AC bulk current which has been investigated in a previous work [START_REF] Kemari | Experimental investigation and signal processing techniques for degradation assessment of XLPE and PVC/B materials under thermal aging[END_REF].

V. CONCLUSION

A regression method has been successfully implemented in this work to define an aging model and estimate the time needed to reach the end of life criterion for XLPE based on mechanical properties. Additionally, XLPE thermal endurance characteristics have been determined using regression method. The present paper could be extended to develop a phenomenological aging model that can be used for better estimation of the remaining cable lifetime.

On the other hand, the dielectric properties of XLPE are highly affected at 140 °C (above the melting point) due to oxidation reactions which lead to the decrease of crystallinity and to the release of polar molecules that make raise both polarization and conductivity of the XLPE material.

Fig. 1 .

 1 Fig. 1. View of the tensile test machine and the arrangement of the sample.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Variation of elongation at break (EaB) of XLPE versus aging time. The dashed line defines the end of life.

Fig. 5 .

 5 Fig. 5. (a) Permittivity εr' and (b) AC conductivity σ' of XLPE versus frequency at different aging stages.

Fig. 4 .

 4 Fig. 4. Thermal endurance graph for XLPE.

TABLE I .

 I NONLINEAR REGRESSION RESULTS FOR EAB

	Elongation at break		Aging temperatures (°C)
	model coefficients	80	100	120	140
	E0	Value Standard error	596 4.2	604. 7.9	739 22.7	1779 133
	A	Value Standard error	3.73 0.03	29.2 23	143 30	1198 133
	τ1	Value Standard error	2385 1.5	2666 64.3	2827 525	3946 363
	Statistics	R² RSS	0.9994 9.92	0.9472 5.64	0.8339 188.5	0.9999 0.0164
	&	RMSE	0.99	0.79	5.18	0.10
	ANOVA	F value P	10289 2.2×10 -16 8.2×10 -1( 3.4×10 -6 2.43×10 -3 8980 173 84371

TABLE II

 II

		.	NONLINEAR REGRESSION RESULTS FOR TS
	Tensile strength model		Aging temperatures (°C)
	coefficients		80	100	120	140
	TS0	Value Standard error	27.5 0.20	30.4 0.71	72.7 6.6	311 24
	B	Value Standard error	7.22 0.07	11.2 0.39	52.2 5.91	291 24.13
	τ2	Value Standard error	14970 64.0	16204 262	19168 1270	32394 1450
	Statistics	R² RSS		0.9987 6.78	0.9876 69.8	0.9171 144	0.9864 9.10
	&	RMSE		0.82	2.64	4.25	2.13
	ANOVA	F value P		10084 2.22×10 -16 6.58×10 -12 4.02×10 -7 4.63×10 -4 1292 741 2158