Development Small Sample Verfication

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IEEE Transactions on Dielectrics and Electrical Insulation Vol. 15, No. 4; August 2008

1070-9878/08/$25.00 2008 IEEE

1131

A New Dielectric Response Model for Water Tree Degraded

XLPE Insulation Part A: Model Development

with Small Sample VerificationAndrew J. Thomas and Tapan K. Saha

University of Queensland

School of Information Technology and Electrical Engineering

Brisbane, QLD 4072, Australia

ABSTRACT

Water tree degradation in underground XLPE insulated cables is a growing, world-

wide problem. This form of degradation is ultimately fatal for affected cables, and

therefore the detection of damaging trees in power cable insulation is vital for

distribution companies to avoid catastrophic failure. Dielectric responsemeasurements, in both the time and frequency domains, can generate valuable

information about the condition of the cable. However, the interpretation of how these

dielectric response measurements relate to water tree density and length is a difficult

task. This paper will present a new dielectric response model for water tree degraded

XLPE insulation. The model is based on finite element analysis to determine the

electrical behaviour of water tree degraded insulation. Preliminary simulations will

verify the model development by comparing the results to small sample Pulsed Electro-

Acoustic (PEA) measurements performed by other researchers. The importance of a

strong non-linearity mechanism for accurate modelling will also be elucidated.

Index Terms water trees, cross linked polyethylene insulation, finite element

method, dielectric measurements, space charge, conductivity, dielectric loss,

nonlinearities.

1 INTRODUCTION

WATER tree degradation is one of the most serious

afflictions that can occur within underground medium voltage

cross-linked polyethylene (XLPE) cables. Because of this

fact, many studies have been performed concerning how water

trees grow, how they bring about failure and how they can be

detected. Perhaps the best way to fully understand the

previous points is through modelling, which can analytically

describe the physical situation. Water tree growth is somewhat

inevitable in XLPE cables under certain conditions. Therefore,

it is a useful exercise to develop a model which can relatedielectric response measurements to the water tree degraded

condition of a cable, which aids in the cable diagnosis. This

will result in a more efficient and accurate diagnostic process.

A number of electrical behaviour models of water tree

affected XLPE cables have been developed in the past. Many

papers [1, 2] concentrate on power frequency (i.e.

capacitively) graded fields, with an emphasis placed on field

enhancement above the norm at the tree tip, with the general

aim of deducing the cause of water tree induced failure, which

of course is an important area of study. However, a water tree

model which is to be used in interpreting dielectric response

measurements will need to be dynamic, as dielectric response

measurements usually cover many decades of time or

frequency. In addition to this, the model will need to consider

non-linear effects with charging voltage, as this is an often

observed measurement phenomena.

The authors of [3] developed a model investigating a

mechanical non-linearity mechanism, which involved theopening and closing of conducting channels between voids

under Maxwell stresses. However, this model was developed

to prove that the mechanical non-linearity mechanism could

be responsible for the observed non-linearity, and is therefore

only local i.e. its solution domain is of a microscopic scale

around a number of voids and channels.

Another series of papers considers the dynamics of water

tree channels, where perhaps the best example is [4]. These

papers consider channel dynamics in the form of a Maxwell-

Wagner type equivalent circuit. The total capacitance andManuscript received on 11 April 2007, in final form 23 January 2008.

Authorized licensed use limited to: M S RAMAIAH INSTITUTE OF TECHNOLOGY. Downloaded on April 24,2010 at 07:05:33 UTC from IEEE Xplore. Restrictions apply.


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A. J. Thomas and T. K. Saha: A New Dielectric Response Model for Water Tree Degraded XLPE Insulation Part A 1132resistance of the water treed region is calculated by allowing

the size of the water treed region to expand and contract to

simulate the filling of voids. The conduction in the region is

modelled by asinh dependence on the electric field, with the

strength of the sinh dependence governed by a value h. The

authors allow the size of the water treed region and the value h

to vary (along with the low field conductivity of the region,

which varies little) at different applied voltages in order to

match the simulated and measured data. However, allowing h

to vary along with the size of the water treed region with

voltage is perhaps physically unintuitive, as it is difficult to

find a physical reason as to why any change of this variable

should occur.

Therefore in order to expand beyond the scopes or

limitations of the previously mentioned papers, this paper will

detail the investigation towards a new electrical model for

water tree degraded XLPE. The goal of this model is to

macroscopically describe the electrical behaviour of water

trees in XLPE under certain applied electrical conditions, on

the scale of the water tree length itself. The overall goal of this

study is to enable a more refined understanding of the effect ofwater trees on dielectric response measurements, with the

actuation of this goal shown in the accompanying paper [5].

However, before such an application can be realised, an

electrical model for water tree degraded insulation must be

developed and verified, which is the content of this paper.

In Section 2, the details of a constructed one dimensional

finite element model will be given, which describes the time-

varying electrical behaviour of water tree degraded XLPE

insulation. Particular attention is payed to simulating the

space charge build up at the interface between the water tree

and the healthy XLPE, with this interface being modelled

diffusely in the investigation. In order to validate the model, in

Section 3 the results of the simulations will be compared tospace charge measurements performed by other researchers

[6] in small sample experiments. It will be seen that in order

for the model to accurately simulate the measurements, a

mechanical non-linearity mechanism based on Maxwell forces

is needed. Section 4 will contain discussion and conclusions

of the results of this study. This paper is a major expansion

of the preliminary/investigative work presented in [7].

2 MODEL DESCRIPTION

This section will address the technical details and

assumptions used to generate the model of electrical

behaviour of water treed insulation.

2.1 STRUCTURE OF WATER TREES

Water trees are generally considered to consist of water

filled micro-voids, which may or may not have

interconnecting, conducting channels. Numerous studies

[8-10] have been performed (to name only a few), using a

variety of techniques such as Transmission Electron

Microscopy (TEM) and Scanning Electron Microscopy

(SEM), to discover this. The largest voids generally found

within water tree channels are on the order of 5 m in

length [9]. The connecting channels between voids were

found to be on the order of >1 m in diameter at the foot of

a water tree, whereas the channels throughout the rest of the

tree are on the order of 10-100 nm in diameter [9].

The interface between a water tree and XLPE within an

insulation can easily be considered to be a rather

complicated structure. On a macroscopic scale (on the scale

of the water tree length itself), it is likely that there is a

decay of void and channel density near the tree tip, leading

to a change in conductivity and permittivity that is not

instantaneous, but diffuse. There is also the question of

field assisted opening and closing of water tree channels

through Maxwell forces, which will be investigated further

in later sections.

In addition to this, the electro-osmosis water tree growth

theory, reviewed in [11], sensibly proposes that it is likely

that ionic species will diffuse, in an electrically assisted

manner, into the polymer beyond the tree tip. This diffusion

of impurities will likely increase the concentration of

hopping sites located close to the conduction band,

increasing the conductivity and permittivity somewhat

beyond the tree tip. This effect is also likely to result in a

macroscopic conductivity and permittivity profile that is

relatively gradual.

In order to model the electrical behaviour of water tree

degraded insulation, it is vital that this interfacial region is

considered properly, as this region will dominate the

electrical characteristics of the local material. These

considerations will be addressed in the next section.

2.2 PERMITTIVITY AND CONDUCTIVITYREPRESENTATIONS

In this investigation, the insulation properties will be

macroscopically considered in one-dimension only, with an

assumption of uniformity in the remaining two dimensions.

The water tree permittivity and conductivity are considered

to be uniform up to a certain point depending on the length

of the water tree. Beyond this region of relative uniformity,

there will be a transition from water tree electrical

properties to healthy XLPE electrical properties (i.e. a

substantial decrease in conductivity, and a less substantial

decrease in permittivity). In some previous water tree

modelling exercises, this change of electrical properties has

been considered as an ideal boundary, with an infinitely

sharp profile. However, for reasons stated previously, this

ideally sharp boundary at a macroscopic level is unlikely to

exist. Experimental evidence for a more diffuse boundary

will be shown in later sections.

The exact form of the interface, with regards to the

decreasing profile of the conductivity and permittivity is

practically impossible to measure directly. However

inferences or postulations can be made from available

information and these inferences can be tested through

less direct means for verification. As mentioned

previously, a major proposed water tree growth

mechanism is electro-osmosis, whereby hydrated ions

diffuse into the insulation beyond the water tree proper

with electric field assistance [11, 12]. Diffusion of



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IEEE Transactions on Dielectrics and Electrical Insulation Vol. 15, No. 4; August 2008 1133

contaminants into XLPE has been studied previously [13,

14] with the contaminants in this case being sourced from

the semi-conductive layer. Figure 4 in [13] shows an

exponential-like decay in the FTIR absorbance level

(which is related to the concentration of species) of

contaminants away from the semi-conductor layer. It is a

reasonable assumption that the conductivity (and

permittivity) of an insulation are proportional to the

amount of contaminants it contains. In addition to this,the authors of [14] had some success in modelling the

space charge levels near the semi-conductive layer using

an exponential conductivity and permittivity profile.

It can also be noted at this point that according to the

electro-osmosis theory discussed in [12] a continual

recombination and resolvation process (the resolvation

owing to osmosis and perhaps dielectrophoresis) acting on

diffused ion pairs mechanically fatigues the insulation

around the ion locations. Therefore, it is also sensible to

propose that permanent mechanical damage of the polymer

is proportional to the ion/contaminant diffusion profile,

leading to the existence of proto-voids and channels in the

tree tip region, which may enable an electricalreconnection process through Maxwell stresses opening

electrically conductive paths up to these areas [3]. This

will be discussed further in Section 2.3.

While the contaminants and diffusion processes studied

in this literature is of a different form to the ionic

contaminants and electro-osmosis diffusion in the water

tree case, an exponential change of electrical properties

through space is nevertheless intuitive in light of this

information and a reasonably well informed hypothesis.

Therefore in this study, the conductivity and permittivity of

the dielectric mixture will change in an exponential manner

from the water tree electrical properties to the XLPE

electrical properties. This interfacial change will bemodelled by thesigmoidfunction, given by:

( )xe

xf+

=1

1(1)

Where is a shape parameter. In the limit of ,

equation (1) approaches the heavy-side step function. Using

equation (1) to express the conductivity and permittivity of

the modelled insulation (for a vented tree) is given by:

( ) ( )kxp

kx

w

eex

++

+=

11)(

(2)

( ))()( 11 kx

p

kx

w

eex

++

+=

(3)

In equations (2) and (3), k is a location parameter

determining the spatial location in the x-direction of

uniform water tree electrical properties. Also in equations

(2) and (3), w and w are the water tree channel

conductivity and permittivity respectively and p and p are

the XLPE conductivity and permittivity respectively. The

ranges of these values considered in this study are [15-17]:

Conductivity

Water tree: 1x10-11 1x10-7 S/m

XLPE: 1x10-18 1x10-16 S/m

Permittivity (relative)

Water tree: 2.3 5

XLPE: 2.3

It should be noted that the electrical behaviour of the

electrode/XLPE interface in this model has been neglected,

and therefore whatever limitations on accuracy that this

omission may make should be considered.

The above formulations for the conductivity and

permittivity are to be considered as the low-field

representations. As is well known, water tree degraded

cables exhibit dielectric response non-linearity. This non-

linearity must arise in changes with the charging voltage of

the electrical properties of the insulation. These changes

will be considered in the next section.

2.3 NON-LINEAR BEHAVIOUR OF ELECTRICALPROPERTIES

A change of the electrical properties in water tree

degraded insulation due to the electric field, at constant

temperature, can occur in two conceivable ways. The first

way is through electronic or hopping processes due to the

lowering of potential barriers between charge trapping

centres through high electric fields. The second way was

proposed in [3], and this is due to a proposed mechanical

alteration of the water tree degraded material through the

action of Maxwell forces. Both mechanisms will be

considered in this model.

Figure 1. Potential well model showing the hopping of an electron or

negative ion over a potential barrier, from site S1 to S2. Arrows show both

a classical jump over the potential barrier and a quantum mechanical

tunnelling through the barrier.

Figure 1 shows the mechanism of hopping or tunnelling

(electron) of a negative carrier over a potential barrier

lowered by an applied electric field. In solid insulation,

carriers do not move freely through the material, but rather

hop between localised trapping centres. In order for this

hop to occur, the carrier must gain thermal energy sufficient

to overcome the barrier height between adjacent trapping

centres. The energy required to traverse this barrier can be

significantly reduced by an applied electric field of

S1

W

S2

E



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A. J. Thomas and T. K. Saha: A New Dielectric Response Model for Water Tree Degraded XLPE Insulation Part A 1134sufficient magnitude. The number of charges released from

trap centres is a stochastic process based on Boltzmanns

statistics, therefore, an applied electric field increases the

probability of a site transition taking place and thus more

charges are released to contribute to charge flow.

Because conductivity is a function of the number of charge

carriers, it is easily seen that the prior explained process will

increase the conductivity of the material. A derivation of field

dependent conductivity from the jump probabilities in theforward and reverse directions (with respect to field) was

performed in [18], with the result being:

=

kT

eEa

eEa

kTE

2sinh

2)( 0 (4)

Where a is the distance between trap centres, e is the

charge of the carrier in Coulombs, k is Boltzmanns

constant and Tis the temperature in Kelvin.

A similar line of reasoning can be made to determine the

non-linearity with respect to applied voltage for the real and

imaginary parts of the permittivity. If, under an ac appliedvoltage, the carriers undergo a certain hop, or multiple hops

in the direction of the field, then a polarisation will be

created. The more carriers that undergo this process due to

field assisted de-trapping the higher the complex

permittivity will be. The derivation of the field dependent

permittivity expressions can be seen in Appendix I, and are

found as:

( ) 0''

2sinh

2 +

=

kT

eEa

eaE

kTE LF (5)

( )

= kT

eEaeaE

kTE LF2

sinh2'' '' (6)

Where the subscriptLFstands for low-field, with respect

to the permittivity. The above formulations complete the

proposed electronic or hopping non-linearity mechanism.

The second non-linear mechanism that will be

considered in this work is the mechanical mechanism. This

mechanism for non-linearity was proposed in [3]. The

mechanism is based upon the observed structure of water

trees consisting of voids with interlinking channels. The

authors of the referenced study consider these interlinking

channels to be closed, and therefore non-conducting under

low-field conditions, but with an opening of these channels,and conduction within them, under high field conditions.

The explanation given as to the source of the energy

required to open these channels is Maxwell forces due to

the electric field within the region. These Maxwell forces

are a pressure acting on a 2-dimensional surface in the

direction of the electric field, with this pressure being

related to the electric field and permittivity as:

( ) 22

1

2

1EnDEpr

r

rr

== (7)

Where nr

is the unit normal vector. If this pressure p is

considered to act on the many interfaces between the voids

and XLPE, it may act to open the channels between voids,

allowing them to fill with liquid and become conducting.

Over the tree cross section, this field induced pressure and

subsequent opening of channels should lead to an average

increase in conductivity and permittivity of the area.

Deducing the exact relationship between the Maxwell

forces (pressure) and the conductivity analytically (throughan additional model) is not within the scope of this study.

However, some experimental results can be examined to

give an indication of the empirical form of this relationship.

A number of Frequency Domain Spectroscopy (FDS)

measurements were performed on accelerated, wet aged

cable samples. Some of these cable samples were showing

typical LC (leakage current) responses as designated in

[19], which indicate long vented tree degradation. Figure 2

shows the loss response (imaginary part of the permittivity)

of one of these cables.

Figure 2. Typical measured leakage current response at differing applied

voltage levels. Uo -> rated voltage

These degraded samples had loss slopes inversely

proportional to frequency, which strongly suggests that the

conduction of the samples was completely dominated by

long vented trees. Therefore, assuming that the Maxwell

forces non-linearity mechanism, as proposed in [20], is the

dominant non-linearity mechanism, the square of the

applied voltage (and therefore applied field) versus the 0.01

Hz, low frequency loss will give the relationship between

the Maxwell forces and the conductivity of the long water

tree/s. The square of the applied voltage and the low

frequency loss for the cable samples showing leakage

current behaviour were found to be exponentially related,

with the exponential fitting havingR2 values in the range of

0.9-0.98, therefore signifying a likely exponential

relationship.

Therefore, in this study, we assume the relationship of

conductivity to pressure to be:



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( )

>


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A. J. Thomas and T. K. Saha: A New Dielectric Response Model for Water Tree Degraded XLPE Insulation Part A 1136The function that governs the variance of the solution

variable is called an interpolation function. The brief

mathematical treatment of the FEM is based on explanations

given in the aforementioned references [21, 22].

If the general form of the differential equation is

considered, it can be represented by:

( ) 0= fL (13)

Where L is a general differential operator acting on

andfis the forcing function acting on (charge density in

equation (10) or equation(11)). This approximation of the

solution variable (the potential V in the solution of

Poissons equation) through the use of interpolation

functions can be considered mathematically by:

( ) ( ) ( )=

+=P

n

nn xFaxx1

(14)

Where P is the number of interpolation functions

defining the solution domain , an is a constant defining

the magnitude of the nth interpolation function, Fn is the

nth interpolation function and is a function that satisfies

the Dirichlet or essential boundary conditions. It is now

clear that the representation in equation (14) is no longer

exact for finite P, so now a residual, R, will exist in

equation (13), which is a measure of the inaccuracy of the

approximation, such as:

fLR

=

(15)

The objective of the finite element method is to minimizethis residualR. This minimization can be achieved through

the use of weighing functions, w, which are multiplied

throughout equation (13) to obtain an integerable solution

expression:

( ) 01

=

+

=

dfFaLLwP

n

nni (16)

The Garlekin method involves choosing the weighing

functions from the same set as the trial functions (while

keeping functions linearly independent) i.e.:

( ) ( )xFxw ii =

In the case of solving Poissons equation, this gives an

integrand of (from (16)):

00

=

+

dxdx

dV

dx

dw

L

(17)

Due to the linear nature of the interpolation functions

(and therefore the weighing functions in the Garlekin

method), equation (17) is now solvable forVthrough linear

algebraic methods, given appropriate boundary conditions

to make the solution unique. In this investigation, the

boundary conditions for each solution of equations (10) or

(11) are the potentials at the electrodes.

This ends a brief description of the method used to solve

for the potential, and also therefore the electric field. Thesolution domain , whose size is governed by the thickness

of the insulation, is segmented into 1000 elements for this

study. The valid solution domain can be considered to exist

within the proposed water tree structures, interface and

healthy XLPE segments in thex direction, and applicable in

the y and zdirections where approximate uniformity holds

in the electrical properties of the material.

3 SMALL SAMPLE INVESTIGATION

In order to access the accuracy of the model, or its ability

to reproduce physical situations faithfully, it was decided

that the model would be tested on a small sample, spacecharge measurement study performed in [6]. In this way,

the space charge profiles produced by the model could be

compared to measurement results on a small scale

experiment, before being extended for use onto large scale

cable specimen geometries.

3.1 MEASUREMENT DETAILS

This subsection will briefly address the measurement

details as given in [6]. The researchers used the Pulsed-

Electro Acoustic (PEA) method to measure volume space

charge densities within the bulk of water tree aged, small

sample specimens. The samples measured in the

aforementioned study were 1mm thick, laboratory aged

specimens. An FeSO4 solution, with concentration of 0.5

M/L, was applied to the sample to encourage initiation and

growth of water trees. After 800 h of 7 kV (peak), 5 kHz

voltage application, water trees were found uniformly over

the voltage application electrode area, with the trees being

of sufficient length to reach the centre of the sample.

During measurement of the space charge profile, a 7 kV

(peak) voltage was applied at varying frequencies, 50 Hz,

0.1 Hz, 0.01 Hz and dc. Two dc measurements of note

were made. One of these dc measurements consisted of the

application of the dc voltage for 1 h, followed by a shorting

of the sample for a further hour, with space charge

measurements being made sporadically during theseintervals. The other of these dc measurements consisted of

the application of a dc voltage, followed immediately by

the space charge measurement.

Seven different samples are measured in the referenced

study. Four of these samples are labelledA,B,CandD, and

were aged for 300, 400, 800 and 1200 h, respectively,

under the previously mentioned conditions. The other three

samples W, D1 and D2 were each aged for 800 hours, but

with moisture contents of 1310 (parts per million) ppm, 480

ppm and 50 ppm respectively.



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3.2 PEA MEASUREMENT DETAILS ANDIMPLICATIONS

This section will address briefly certain considerations

that need to be taken into account when interpreting

Pulsed-Electro Acoustic measurement results. The PEA

measurement method involves the application of short

voltage pulses to a measurement sample. These voltage

pulses exert a force on any charges within the sample,

and these charges in turn launch acoustic waves

throughout the sample to be measured by a pressure

transducer. Comprehensive descriptions of the PEA

method can be found in [23, 24].

Of particular interest in this study is the attainable

resolution of the PEA systems whose results are being

analysed in this paper. The theoretical space resolution

attainable within a PEA system is given by [23]:

tvl = (18)

Where l is the space resolution, v is the velocity of

propagation of the acoustic wave (approximately 2000

m/s in polyethylene [23]) and t is the time-width of the

applied voltage pulse. This finite resolution will act to

broaden in space any actual space charge profile in the

measurement results. In addition to this, attenuation and

dispersion effects can also act to broaden the space

charge profiles, although in [25] it was found that LDPE

(and presumably by extension XLPE) was a non-

dispersive material. In [23], PEA measurements were

performed on a 2 mm thick LDPE specimen, using a

voltage pulse width of 30 ns. The theoretical space

resolution, using equation (18), is therefore

approximately 60 m. However, the authors of the study

found a resolution of 100 m was the minimum

achievable, which is a 67% increase on the theoretical

resolution.

The implications for this on interpreting PEA

measurement results is that the actual space charge

distributions may be somewhat narrower in space than

the measured distribution. Within the resolution range

however, only the net charge density can be obtained

[23]. Therefore, in order to effectively compare model

simulations (which of course produce actual space

charge distributions) with those obtained by PEA

measurements, the finite resolution must be applied to

the simulated results. This can be estimated by applying

a moving average filter to the simulated distribution,with a window size equal to the resolution. This is

supported by a mathematical study of the PEA

measurement system performed in [26], which found that

the PEA signal is a weighted running average of the

space charge distribution.

A demonstration of this adjustment of the simulated

space charge profile can be observed in Figure 3,

whereby the model is used to produce a very localised

space charge profile in the same geometry and applied

voltage as that in [6], by creating a very sharp change in

conductivity and permittivity between the water

tree/XLPE interface. The resolution in this case is set to

100 m.

Figure 3. Comparison between simulated space charge profile andresolution adjusted profile

It can be observed in Figure 3 that the resolution

adjustment can have a very significant effect on the charge

profile; however the effect isnt quite as significant for

broader simulated profiles. The area under both displayed

profiles is the same (net charge density), so it can be seen

that the resolution simulation operation manipulates the

profile in the correct fashion. All simulated space charge

profiles to be compared with PEA measurements in the

following sections will be resolution corrected.

3.3 COMPARISON OF RESULTS

This section will compare the measurements of the

previously described experimental setup with the

simulations from the model. However, first it is pertinent to

examine the experimental evidence for the hypothesis that

the water tree/XLPE interface is macroscopically diffuse,

instead of an ideal boundary. If the interface does

approximate an ideal boundary, the space charge

distribution due to the interfacial polarization should ideally

be a surface charge and approach an impulse function. A

possible example of an approximation of such a distribution

can be observed in Figure 3, with regards to the actual

space charge profile. After resolution is taken into account,it can be observed that the high and narrow space charge

density is reduced in magnitude and broadened. The spatial

extent of the resolution corrected distribution is 100-130

m. Therefore, it can be said that any space charge

distribution due to a water tree/XLPE interface that is

significantly greater than the PEA measurement system

resolution has some spatial extent. A space charge profile

with spatial extent at such an interface strongly suggests a

relatively gradual electrical properties profile if equation

(12) is considered.



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A. J. Thomas and T. K. Saha: A New Dielectric Response Model for Water Tree Degraded XLPE Insulation Part A 1138If Figure 4 is carefully examined along with Figure 6

from [6], it can be observed that the spatial extent of the

space charge profile due to the water tree/XLPE interface

is approximately 330-350 m. The theoretical resolution

for the PEA measurement system used in [6] is 60 m

(pulse width of 30 ns), so assuming that the same

mechanisms occur as in [23], the resolution can be

assumed to be equal to 100 m. Therefore, it is clear that

the water tree/XLPE interface in [6] cannot be consideredto be an ideal boundary. To give further support to this

generally, the results of a higher resolution system [27]

can be analysed.

Figure 4 Time resolved space charge profiles under 7 kV 50 Hz ac voltage

application, sample W.

The resolution given in [27] is 10 m, however this

corresponds to the theoretical resolution of the PEA

system using equation (18) (pulse width of 5 ns).

Therefore, in order to be conservative, 40 m is added to

give an assumed resolution of 50 m (in [23], the actual

resolution is 40 m greater than the theoretical

resolution). The sample to be analysed in [27] is a 0.5 mm

thick XLPE specimen, with water trees grown to the

centre of the sample. If Figure 5 (a) in [27] is carefully

examined, it can be observed that the space charge profile

due to the water tree/XLPE interface has a spatial extent

of approximately 170 m. This again supports the

hypothesis that the water tree/XLPE interface is gradual

in its electrical properties. The two examples given

previously are for samples with likely numerous trees

uniformly reaching the centre of the samples, however the

same diffuse interface can be observed for single trees,

see Figure 4 in [28].

From the previous analysis, it is clear that the diffuse

water tree/XLPE interface hypothesis has been confirmed.

Therefore, the rest of this section will be dedicated to

comparing the model results and space charge

measurements in detail. Figures 4 and 5 show the time

resolved space charge profile under ac voltage excitation

for sample W, and the maximum space charge density of

samples W, D1 and D2 for differing applied frequencies,

respectively [6].

Figure 5 Maximum space charge profiles for varying frequency applied

voltages (50 Hz, 0.1 Hz, 0.001 Hz and dc) for samples W, D1 and D2.

Figure 6 shows the model space charge densitysimulation for a tree bridging half of the 1 mm thick

insulation, with an applied voltage of 7 kVpeak, 50 Hz at

the phases of 108o and 288o, which are the maximum

space charge profiles. The space charge profiles are

resolution corrected, and the water tree conductivity is

equal to 5x10-8 S/m.

By examining Figure 6, it can be observed that the

simulated space charge density corresponds very well to

the space charge density as shown in Figure 4, at its

maximum values of 108 and 288 degrees. It can be

observed in Figure 4 that the spatial extent of the space

charge density is approximately 350 m, with a maximum

space charge density of 1.6 C/m3. The maximum spacecharge density follows the applied voltage with a slight

lag, as in Figure 5, with the maximums occurring at 108

and 288 degrees. Figure 7 shows the total charge versus

the applied voltage, with the total charge being calculated

by integrating the charge density in Figure 6 and

multiplying it by the electrode area given in [6]. It can be

observed that the total charge behaves linearly with

applied voltage, and is practically identical to (A-1),

Figure 5 in [6].



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Figure 6. Simulated space charge density for a water tree bridging half the

insulation thickness, at phases of 108o and 288o.

Figure 7. Total integrated charge at 108 degrees versus applied voltage

The nature of the space charge build up at the interface of

the water tree and XLPE is due to interfacial polarisation, asargued for convincingly in [6]. This interfacial polarisation is

due mainly to the large decrease in conductivity at the water

tree/XLPE interface. As the relatively large conductivity of

the water tree is likely due to the presence of ionic carriers

[16], the space charge distribution at the tree tip is almost

certainly comprised of ions. It is also possible that beyond

the tree tip, due to the electro-osmosis hypothesis of water

tree growth, the diffusion of ionic impurities and therefore

the likely existence of impurity states relatively close to the

conduction band, that ionic hopping between traps/impurity

sites occurs.

The dc behavior of the water trees can also be

examined with the model. Figure 8 shows the spacecharge measurements for samples A, B, Cand D in [6]

under 7 kV dc voltage application. The measurements

were made immediately after the dc voltage application

[6]. By examining Figure 7 in [6], it can be observed that

trees bridging increasingly greater percentages of the

thickness of the insulation moving from sample A to D.

Figure 9 shows the simulated space charge density after 1

s of voltage application (in an attempt to replicate the

measurement being made immediately after the dc

voltage application).

It can be observed that the simulated space charge profiles

in Figure 9 match well with the measurements shown in

Figure 8. The conductivity of the water treed region in Figure

9 is 5x10-11 S/m, which is within the range of reported

conductivities in the literature. As can be observed, the water

tree conductivity used in Figure 9 is much below that of

Figure 6. This reduction is supported by the results shown in

Figure 5. It can be observed in Figure 5 that the samples that

were dried (D1 and D2) have a reduced maximum spacecharge magnitude (compared to sample W), and that this

magnitude takes considerably longer to saturate (if it does at

all within the applied voltage time frame, this is hard to tell

visually) under dc voltage conditions.

Figure 8. 7 kV dc space charge measurement on four aged samples, A, B,

CandD [6].

Figure 9. Simulated space charge profiles for 7 kV dc voltage application

(one second after voltage application) with 4 different water tree lengths.

If the space charge process is considered via a simple

Maxwell-Wagner interfacial model, the time constant for

the build up of the interfacial space charge can be

determined essentially by the ratio of the dual permittivity

and the water tree conductivity (XLPE conductivity can be

neglected in this simple model due to its small magnitude).

If the charge build up process can be roughly approximated



10/13

A. J. Thomas and T. K. Saha: A New Dielectric Response Model for Water Tree Degraded XLPE Insulation Part A 1140by this mechanism, than a slower build up of charge (longer

time constant) can be explained through a reduced water

tree conductivity. Because the magnitude of maximum

space charge density in Figure 8 is equal to approximately 1

C/m3, which is somewhat close to the magnitude of samples

D1 and D2 in Figure 5 (which almost certainly have a

reduced water tree conductivity due to drying), a lower

conductivity for the simulation of Figure 8 was deemed

necessary. According to the model results, which clearlycan reproduce the profiles of Figure 8 accurately, it is

unlikely that the space charge profiles of Figure 8 have

reached a steady state value.

The previous discussion is an effective segue to the

discussion of the clear steady state behaviour of sample

W under dc voltage application as shown in Figure 5. For

sample W, within 60 s the maximum space charge density

has reached its highest value, approximately 1.5 C/m3. It

was found that, in replicating this saturation of the

maximum space charge density with the model that the

Maxwell forces based non-linearity mechanism plays an

important part. When no non-linear mechanisms are being

considered, the only explanation for a steady state spacecharge density for sample W is a close to ideal boundary

between the water tree and XLPE interface. Because of

the very slow time constant of such an interface, the

movement of charge into a close to ideal boundary is

likely to be beyond any practical measurement time.

However, as was discussed previously, the measured

space charge profiles in the literature, and the successful

simulations shown in this section, strongly suggest a

diffuse boundary, with a relatively gradual change of

electrical properties. The simulation results of a diffuse

interface, with electrical properties the same as that which

produced Figure 6, without the inclusion of a non-linear

mechanism, under 7 kV dc voltage application can beobserved in Figure 10.

By examining Figure 10, it can be observed that the

space charge profile, while initially resembling those in

Figure 4 at short times, continues to grow in magnitude and

shift to the right (towards to the negative electrode). It is

clear that this behaviour of the space charge is in gross

contradiction with the dc space charge behaviour of sample

W, shown in Figure 5. This behaviour can be understood in

two ways. First, the build up in magnitude of the total

charge (positive and negative) can be explained through the

simple, parallel plate, charge-capacitance relationship. As

the distance between the plates (in this case, the distance

between the active space charge region and the opposing

electrode) decreases, in order to keep the voltage constant,

the total charge on the plates must increase.

The shift to the right over time of the charge grouping,

can be explained through an analysis of Poissons and the

continuity equation. By examining equation (12) it can be

seen that for a steady state charge density to prevail in a

conductively inhomogeneous medium, the electric field

must redistribute in such a way as to cancel the rate of

change, in space, of the conductivity.

Figure 10. Simulated space charge profiles for 7 kV dc voltage

application, non-linearity mechanism omitted.

This redistribution of electric field is performed by a

redistribution, or general build up, of charge. This build up

of charge takes time, depending on the time constant of the

dielectric region, therefore giving a different chargedistribution at different times as seen in Figure 10. By

including equation (12) in Poissons equation (10), the

following expression can be derived (see Appendix II):

tgradJ

=

r

(19)

It can be seen in equation (19) that at steady-state (with

regards to charge density), the static charge density will

depend on the spatially uniform current density and the

spatial gradient of the ratio of the permittivity to the

conductivity. Therefore, after a steady state develops, at

any gradient of the ratio of the permittivity to the

conductivity will yield a time-independent charge density.

In order to cease this movement of charge through the

interface, and therefore cease its build up in magnitude so

that it reaches a steady state on the order of 1.5 C/m3 as in

Figure 5, it is required that there is a negligible gradient

of the ratio of the permittivity to the conductivity beyond

the observed charge magnitude in the centre of the

sample.

The introduction of the mechanical non-linearity

mechanism solves this problem. Figure 11 shows the

evolution of the space charge density in the sample when

the mechanical non-linearity mechanism is enabled, with an

A value of 350 Pa in equation (8). It should be noted that all

simulations shown previously, apart from Figure 10, have

had the non-linear mechanism enabled with an A value of

350 Pa. Figure 12 shows the corresponding simulated

change in the conductivity profile over time.

A steady-state condition in terms of the space charge

profile is achieved after approximately 50 ms. This result

can be compared to that seen in Figure 10, whereby the

steady state condition has not developed after 5 s and with a

space charge magnitude in gross contradiction with the



11/13


results seen in Figure 5. The behaviour seen in Figure 11 is

emergent after the introduction of a mechanical non-

linearity mechanism. This mechanism acts to flatten the

conductivity (and therefore permittivity) profile in the

region beyond the space charge peak, as can be observed in

Figure 12. It can be observed that the maximum space

charge in Figure 11 reaches a steady state value on the

order of 1.5 C/m3, which agrees with the maximum dc

space charge values shown in Figure 5 for sample W.

Figure 11. Space charge evolution in small sample simulation, with a tree

bridging roughly 50% of the insulation.

Figure 12. Simulated conductivity profile evolution over time under the

action of the proposed mechanical non-linearity mechanism.

The enhanced, or non-zero, space charge profile beyond

the space charge peak in Figure 11, towards the negative

electrode, is due to the proposed mechanical non-linearity

mechanism resulting in a slight slope of the conductivity in

the x~0.6-0.9x10-3 region, instead of a completely flat

response. Unfortunately, the static dc space charge

distribution due to water tree degradation which has been

simulated in Figure 11 has not been displayed in [6] or

elsewhere in the available literature to the authors

knowledge. It has already been discussed that the dc space

charge distributions shown in Figure 8 are unlikely to be

static distributions, being measured immediately after the dc

voltage application and having the characteristics of a low

water tree conductivity. Therefore, the non-linearity

mechanism which results in Figure 11 is difficult to confirm

directly. However, it was found that without such a

mechanism, the steady state maximum charge density under

dc voltage application for sample W in Figure 5 was

impossible to replicate. This and the successfulness of the

non-linear enabled model in reproducing the measured space

charge profiles in Figures 6 and 9 give confidence to the

postulated mechanism. In addition to this, the accompanying

paper [5] extends this model and enables it to predict the

dielectric response of water tree degraded XLPE cables. In

that paper, it is shown that the model can accuratelyreproduce the non-linear behaviour of degraded cable

samples measured using Frequency Domain Spectroscopy.

4 DISCUSSION AND CONCLUSION

A point of note can be made at this point about the

previous simulations and the mechanical non-linearity

mechanism. As was stated previously, it was found through

extensive simulations that without the proposed

mathematical form of the mechanical non-linearity

mechanism, a steady state maximum space charge density

as displayed for sample W in Figure 5 was not possible to

obtain. Therefore, the preceding results give supportingevidence to the general, non-analytical, mechanical non-

linearity mechanism as proposed in [3]. The electrical

mechanism of non-linearity described by equations (5) and

(6) was also considered. Without the mechanical non-

linearity mechanism, it was found that the electrical

mechanism caused some very slight increases in

conductivity and permittivity. However, when coupled with

the proposed mechanical non-linearity mechanism, the

effects of the electrical mechanism was negligible.

A further point to be made is regarding the acceptable

range of the mechanical non-linearity turn on constant A.

This constant was varied through various values, however it

was found that only within the range of 300-500 Pa was asteady state maximum space charge density attainable that

still allowed a good comparison between the space charge

profile simulations and Figures 4 and 8. In order to be

conservative with the magnitude of the mechanical non-

linearity mechanism, the highest value in this range, 500 Pa,

has been chosen to be used in the second part of this study

[5]. The relatively low value of this constant may bring into

question the physicality of the model. However, it should be

noted that this is a macroscopic simulation, and that stresses

due to microscopic features such as geometrically sharp

voids, may act to enhance these pressures.

In conclusion, this paper has detailed the development of a

dynamic electrical finite element model for water treedegraded insulation. The model uses a diffuse boundary

between the water tree and XLPE, along with an analytical

mechanical non-linearity mechanism based on the developed

Maxwell forces within the material. By doing this, the

simulated results incorporating the mechanical non-linearity

mechanism can attain a close match to the small sample PEA

measurements performed in [6], without the need for any

applied physical variables which change with applied

voltage. Due to the success of the model, a second half of the

study was undertaken to extend the model in a practical role



12/13

A. J. Thomas and T. K. Saha: A New Dielectric Response Model for Water Tree Degraded XLPE Insulation Part A 1142of interpreting diagnostic response measurements. This work

is presented in the accompanying paper [5].

APPENDIX I

Polarisation can be expressed both macroscopically and

microscopically:

uEP == )(0

'(20)

Where u is the number of dipoles operating at fieldEand

is the average dipole moment. If the simplistic 2-

dimensional potential well case is considered, as illustrated

in Figure 1, then the average dipole moment from the

hopping of carriers can be expressed as:

=+

==

=

kT

eEa

kT

eEaea

eafff

eafff

u

dq

rfe

rfe

N

i

ii

2cosh

2sinh

)(

)(1 (21)

Whereff is the probability of a hop in the direction ofthe field for an negatively charged carrier, given

thermal emission of a carrier, fr is the probability of an

hop in the opposite direction, given thermal emission of

a carrier, fe is the probability of therm-e of the carrier

and d is the distance between the dipolar charges. As

can be seen from equation (21), the simplification has

been made that all potential wells are a distance a apart

(hence d = a), and all the carriers are single charged

(hence q = e). From equations (20) and (21) then, the

polarisation for the orientation processes described

above is:

( )EkT

eEaea

kTWN

P 0'

2sinh

2

exp

=

= (22)

WhereNis the total number of possible dipoles in a dielectric

(Nis independent ofE, whereas u is dependent). Rearranging

(22), an expression for the permittivity can be obtained:

( ) 0'

2

2sinhexp

+

=E

kT

eEaea

kT

WN

E (23)

Equation (23) can be expressed without the N term byusing the low field permittivity:

( ) 0''

2sinh

2 +

=

kT

eEa

eaE

kTE LF (24)

Where the low field permittivity is:

kT

aekT

WN

LF4

exp 22

'

(25)

If the real part of the permittivity is converted to

susceptibility:

10

'' =

(26)

The constancy between the real and imaginary parts of

the susceptibility, through the Universal Relaxation Law

[29], can be evoked to give equation (6).

APPENDIX II

Poissons equation can be expressed as:

EDrr

== (27)

The conductive current density can be defined as:

EJr

= (28)

Therefore, equation (28) can be re-expressed as:

Jr

= (29)

Using the identity:

( ) ( ) ( ) AfAfAfrr

+=

Equation (29) can be re-expressed as:

JJrr

+=

(30)

Substituting the continuity equation (12) into equation(30) gives equation (19).

ACKNOWLEDGEMENTS

The authors would like to acknowledge Ergon Energy

and the Australia Research Council for the funding that

made this study possible. The authors would also like to

acknowledge the project members from the Queensland

University of Technology (QUT) for their support. In

particular, the authors would like to acknowledge Frith

Foottit, of QUT, for the development and continual

maintenance of the accelerated ageing experiment, Bolarin

Oyegoke, of QUT, for his experimental advice and support

and Prasanna Wickramasuriya, of Ergon Energy, for hissupport during the measurements.

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Dielectrics Press, 1996.

Andrew J. Thomas (S04) is a Ph.D. candidate at

the University of Queenslands School of

Information Technology and Electrical

Engineering. He graduated from the Queensland

University of Technology in 2003 with honors and

worked in Connell Wagners power systems group

for one year. During that year, 10 months were

spent at Connell Wagners Advanced Technology

Centre in Newcastle, Australia within the high

voltage testing division. His current research

interests include insulation condition assessment

techniques, specialising in the diagnosis of water tree deteriorated XLPEcables.

Tapan K. Saha (M93-SM97) was born in

Bangladesh and immigrated to Australia in 1989. Dr.

Saha is a Professor of Electrical Engineering in the

school of Information Technology and Electrical

Engineering, University of Queensland, Australia.

Before joining the University of Queensland in 1996,

he taught at the Bangladesh University of

Engineering and Technology, Dhaka, Bangladesh for

three and half years and then at James Cook

University, Townsville, Australia for two and half

years. He is a Fellow of the Institution of Engineers,

Australia. His research interests include power systems, power quality and

condition monitoring of electrical plants.

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