Research ArticleStudy on the Fractal Dimension andGrowth Time of the ElectricalTreeing Degradation at Different Temperature and Moisture

Youping Fan Dai Zhang and Jingjiao Li

School of Electrical Engineering Wuhan University Wuhan Hubei China

Correspondence should be addressed to Youping Fan ypfanwhueducn

Received 23 June 2018 Revised 24 September 2018 Accepted 11 October 2018 Published 1 November 2018

Academic Editor Federica Bondioli

Copyright copy 2018 Youping Fan et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

+e paper aims to understand how the fractal dimension and growth time of electrical trees change with temperature andmoisture +e fractal dimension of final electrical trees was estimated using 2-D box-counting method Four groups of electricaltrees were grown at variable moisture and temperature +e relation between growth time and fractal dimension of electrical treeswere summarized +e results indicate the final electrical trees can have similar fractal dimensions via similar tree growth time atdifferent combinations of moisture level and temperature conditions

1 Introduction

Electrical treeing is a typical degradation mechanism inpolymers Insulation fault can be generated in differentprocesses as a result of operating at different electric fieldcondition and insulation failure time [1] Breakdown is anaccident in a short time while degradation is a process todecrease the insulation class over a period of time [2]

+e analysis of electrical trees is important due to thefaulty insulation in polymers Electrical treeing can lead tobreakdown of the polymers due to the electric field strengthElectrical treeing occurs in weak parts of the polymers [2]Electric charges around the weak parts could cause partialdischarge +e PD (partial discharge) activity around weakpart (as a result of void) can cause the initial electrical tree[3] +e insulation breakdown in electrical machines HVcable insulation switchgear and transformer bushings canbe triggered by electrical treeing [4] According to the qualitycontrol of manufacturing industry the insulator of a cablecan contain residual water contaminates and void [5] Atypical electrical tree in underground cable insulator couldgrow from the electrode asperities gradually until finalbreakdown +e faulty insulation in the polymers of thiscable could cause a short circuit between the inner and outerconductors

Fractal is a kind of shape parts which have self-similarproperties to the whole [6] Mandelbrot was the first to usefractal to describe complex images in [7 8] +e fractal canbe classified as regular and natural +e regular fractal ismathematics conception such as Koch curve [9] and Sier-pinski sieve [10] Fractal dimension can be used to describethe complexity of natural shapes In Euclidean geometrystraight lines and curves are one-dimensional and planesand circles are two-dimensional However the Euclideandimension could not well describe the mathematic con-ception fractal Hence a HausdorffndashBesicovitch Dimensionwas used to describe the dimension of fractal [11] +edimension is defined using minimum spheres to coverimages

+e analysis of fractal dimension could reveal charac-teristics of electrical trees Electrical trees growth has self-similar properties and the properties are the basic theory offractal [12] +e electrical trees often finally indicate byseveral of types which is completely similar on fractal di-mension by facts include temperature [13 14] voltagefrequency [15] composited voltage [16] and aged materials[17]+e fractal dimension describes the density of images insome way Electrical trees are generally classified as branchbush and bush-branch trees by fractal dimensions andthe fractal dimension is actually the branch density [18 19]

HindawiAdvances in Materials Science and EngineeringVolume 2018 Article ID 6019269 10 pageshttpsdoiorg10115520186019269

In this paper we aim to understand how the fractal di-mension and growth time of electrical trees changes withdifferent temperature and moisture

2 Sample Preparation

21 Needle-Plane Samples Electrical treeing test is used toanalyse electrical trees characteristics A typical electricaltree can grow from the electrode asperities+e test is usuallyprocessed in a needle-plane sample and the needle is used tosimulate an initial electrical treeing condition as show inFigure 1 A tungsten carbide needle is embedded in themiddle of the sample A high voltage can be applied to thisneedle+e sharp needle point can simulate the ldquoweak partsrdquoin polymers +e shank diameter and radius of curvature ofthis needle are 1mm and 3 μm respectively +e distancebetween needle point and the plane is 2mm +e electricaltrees can grow in between needle point and plane

Araldite CY1311 was selected to make block epoxy resinsamples +e 30 g epoxy resin CY1311 and 10 g hardenerwere measured into two beakers via a balance respectively+e 40 g solution was calculated a little bit more than mouldvolume multiplied by density of epoxy resin (1 gml) viataking an integer Hence mixed solution was just enough topull into the mould +e little amount of hardener was leftinside the beaker while pouring into the epoxy resin Hencethe hardener was measured little bit more than it wasdesigned Epoxy resin and hardener were mixed using bya stirring magnet for 10 minutes afterwards the two beakerswere placed inside the oven at 40degC for 10 minutes Waterand air bubbles were removed from the solutions At thesame time silicone oil was evenly applied adequate amountto the inside of the mould in order to avoid conglutinationbetween the sample and mould Seven needles were verti-cally placed in the hollow area of a concave mould +eneedle-plane distance was 2mm Mixed sample degassed inthe oven at 40degC for 10 minutes before pouring into theconcave mould After curing at room temperature for 48hours the samples were heated for 1 hour at 100degC forpostcuring Samples were stored in the vacuum oven for atleast 2 days in order to minimize the moisture level +ismoisture level was treated as 0 (initial weight of thesample) when calculating the final moisture level of thesesamples At last the long sample was cut into several one-needle blocks (Figure 2)

22 Water Conditions +e block samples were stored indifferent humidity-controlled containers before the elec-trical test in order to make different water conditions +einitial weight of each sampleM0 was measured +e sampleswere stored in four saturated salt solution containers +esefour-solution lithium chloride (LiCl) magnesium chloride(MgCl2) potassium carbonate (K2CO3) and sodiumchloride (NaCl) were used to create a constant relativehumidity and these relative moisture levels are 15 3044 and 75 respectively at 15degC All samples weremeasured every day in the first week and every Monday tillthe electrical treeing experiment Prepared samples were

measured with an electronic balance and recorded ina personal computer +e mass uptake ΔM is defined inEquation (1)

ΔM Mi minusM0

M0times 100 (1)

where Mi is measured weight and M0 is original weight ofsimple

+e average ΔM and experiment time t of each groupwas calculated +e graph of average ΔM vs log (t) wasplotted to calculate relative humidity (Figure 3)

3 Electrical Treeing Experiment

31 Experiment Apparatus +e center of the experimentalset-up for electrical treeing experiments is a Faraday cage(Figure 4) +e cage was used to exclude external noisearound and prevent experimenter getting an electric shock+e white box was used to exclude light noise around +ered line above the box is connected to the positive electrode+e block epoxy resin sample was tested in the box withelectric insulating oil +ere are four main kinds of equip-ment connected to the cage including a CCD system anoscilloscope system a voltage system and a temperaturesystem

+e CCD system was used to take two kinds of picturesaround the needle point+e CCD camera was controlled viaa computer +e expose time of the camera can be changedvia a camera program in this computer PD light andelectrical tree images were taken via changing the exposetime +e lens inside the cage takes pictures and then dis-plays them on a monitor

+e voltage systemwas a variac with a protective relayingdevice +e input voltage of the needle is controlled via thevariac Some special voltage levels were shown on a step-voltage reference table +e protective relaying device wasused to disconnect the power and variac when the electricaltrees break down

+e temperature system was a heater to create a constanttemperature in the test cell A small variac was used tochange the input voltage of the heater An output powercontroller made the temperature of heater can be controlled

50mm

20mm

5mm

Polymer

Plane (GND)

Needle (HV)

Figure 1 Needle-plane sample

2 Advances in Materials Science and Engineering

to the accuracy of 01degC +e temperature shown on thiscontroller was the heaterrsquos temperature A set-sample ref-erence table was used to achieve an accurate sample tem-perature when the variac was set at 60 units

32 Experiment Samples were prepared 2 months beforethe electrical treeing experiment After two months ofsample water conditions preparation each sample wastaken out from the containers with saturated salt solutionsand measured weight before it was going to do a test Finalmass uptakes can be calculated from the last weights whichwere used to describe the humidity of the samples +eprotruding needle was cut into 05 cm after being measuredin order to fix in the treeing test cell +e needle point wasfixed in the middle of the lens sight A camera was used totake photos around the needle point In order to get cleargrowth trees photos a sharpest needle image was taken viachanging the focus of the lens at the beginning A typicalelectrical treeing model can start from charge transport+e space charge accumulated at the cute point will resultin a partial discharge +is space charge combines the

partial discharge that can affect electric field around +enthe electrical trees can grow from this point +e lens focuscould be changed if necessary during the test +e tem-perature system then began to work for at least 1 hour inorder to make a right temperature for the sample A timerwas set to 5 seconds due to the timer has an approximate 5seconds delay to the power supply Other devices werepreset during this one hour

+e experiment devices were double checked before thetest in order to ensure safety Electrical tree experiment wastesting under a dark condition Photographs around theneedle point were monitored via the charged-coupleddevice (CCD) camera +e camera was explored everysecond At the same time a clock was stated to record thetree growth time+e explored time was changed to 10 s viathe camera program in order to acquire PD light Initial PDpoints position and main electrical tree branches could befound via appropriate white level of PD light images asshown in Figure 5 Electrical tree images were taken whilethe electrical trees were grown A tree is defined as a fastgrowth tree while the tree crosses the 14 in the first minuteotherwise it is defined as a slow growth tree +e electrical

Mould Pin Polymersample

(a) (b)

Figure 2 Epoxy resin samples preparation (a) sample prepared by using a concave mould (b) a block sample

21

20

25

15

10

05

00

3

M (

)

Log (t)

Lithium chlorideMagnesium chloride

Potassium carbonateSodium chloride

Figure 3 Plot log (t)ndashΔM of samples

Advances in Materials Science and Engineering 3

tree images were taken every one minute for fast growthtrees and every two minutes for slow growth trees Cor-responding PD light images was taken 11 seconds delaybecause 1 second was used to change the program and 10seconds were used to expose High voltage supply was cutoff when the tree was grown at 23 needle-plane distanceGrowth time distance at this needle-plane distance wastreated as tree growth time

4 Estimate Fractal Dimensions

41 Methods to Obtain Electrical Tree Images 2-D electricaltrees images are required to estimate fractal dimension Amicroscope was used to make 2-D superimposed images Acharged-coupled device (CCD) camera mounted on a mi-croscope was used to take digital pictures from two electricaltree samples +e pictures were taken after every 40 finefocus steps from the first clear branch to the last +ecomposed images were compiled by software called Mon-tage +e distance between branches and camera lens isdifferent Various branches will display on the differentpoints+e images can be taken from the top branch (branch1) of an electrical tree to the bottom (branch k) by changingthe lens-sample distance (Figure 6) +en a superimposedimage can be composed by extracting the clear parts ofdifferent camera lens images (red cycle parts)

+e software named montage is used to compose thoseimages into superimposed images Several methods are used

to estimate the fractal dimension of 2-D images in differentscientific researches For electrical trees the fractal di-mension describes the density of branches +e super-imposed images (Figure 7) were used to estimate the fractaldimension of electrical trees

42 Fractal Dimension Using Box-Counting Method AMinkowskindashBouligand dimension (box-counting dimension)was used to estimate fractal images in nature +e box-counting dimension is using grids to cover images Unifor-mity grids are superposed on the fractal images +e amountof covering boxes is used to estimate the box-counting di-mension +e box-counting dimension is greater than orequal to the Hausdorff dimension because the minimumfreedom spheres are used to cover a fractal image inHausdorffdimension +e box-counting dimension and Hausdorff di-mension are equal while all the boundaries of the fractalimages are coincident with the boundaries of grids+e fractaldimension using box-counting method has been utilized instudying electrical treeing behavior in XLPE cable insulationsamples [20] +e box-counting dimension can be defined asin Equation (2)

Dbox limε⟶0

logN(ε)log(1ε)

1113888 1113889 (2)

where N(ε) is the amount of boxes and ε is the size ofboxes

CCDcamera

High voltagetransformer

Heater

(a)

CCD cameraHigh voltage

Heater

(b)

Figure 4 +e Faraday cage used for electrical treeing experiment (a) diagram of the set-up (b) photograph of the cage

(a) (b) (c)

Figure 5 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) initial PD point(b) clear electrical branches (c) complex electrical branches

4 Advances in Materials Science and Engineering

Infinitesimal grids cannot be used to cover naturalimages (nonconvergence) Fractal sharps in nature mushhave upper and lower limits +e fractal dimension can onlybe estimated between these two limits +e upper limitmeans the size of the grids scale must be larger than the sizeof the image+e lower limit means boxes scale must containat least 1 point of the image Groups of the amount ofcovering boxes can be measured with different size gridsbetween the upper limit and lower limit +e grids are halfsize of the group next to it +e sizes of grids are 2nminus1 unitwhile the size of the minimum size is 1 unit (n is a positiveinteger) +e box-counting dimension can be estimated viainfinite approximation as in Equation (3)

Dfprop1n

1113944

n

nminus1

logN rn( 1113857

log 1rn( 1113857 (3)

+e groups of results satisfy the power law +e negativeexponent of the logN(r)minuslog r fitting curve gives the box-counting dimension Local scaling exponent (local fractaldimension) can be defined as in Equation (4)

D dInN(r)

dIn(1r) (4)

+e fractal dimension of digital images is estimated usingbox-counting method in MATLAB One pixel is the min-imum size of boxes Hence one pixel is the boundary oflower limits +is process can be described as follows

(1) A color image is converted into a grayscale image(Figure 8)

(2) +e image is binarized (translate the grayscale im-ages into white and black images) by an appropriate

1

2

k

1 2 k

Superimposed image

Figure 6 Schematic diagram of composing tree images

Figure 7 Superimposed images from samples

Advances in Materials Science and Engineering 5

light value (the value can be set in the code) for eachpixel (white pixels are binarized ldquoelectrical treesrdquo)

(3) +e total number of boxes N(r) containing whitepixels is calculated by different size of r

(4) A log-log plot of N(r) curve as a function ofr is indicated +e slope gives the fractiondimension

+ree groups of images are selected by the fractal di-mension from low to high linearly All binarized images canfit the original images well

+e fractal dimension increases when the fitting spacedecreases (Figure 9)

Local fractal dimension describes the differentiation offractal dimension distribution curve at each r (the size ofgrid) point (Figure 10) Local fractal dimension curve is notsmooth because local fractal dimension cannot accuratelydescribe the self-similarity from a large size grid At small rpoints the local fractal dimension is similar +e result offractal dimension will decrease due to these points How-ever the box-counting results still describe a fractal char-acteristic in statistics Hence the slope of the logN(r)minuslog rcan give the box-counting dimension

5 Results and Discussion

Four groups of samples A (Table 1) B (Table 2) C (Ta-ble 3) and D (Table 4) were stored in four saturatedsolution conditions (LiCl RH15 MgCl2 RH30 K2CO3RH 44 NaCl RH 75) for two months Samples weretesting at a different moisture level of 003 05 125 and different temperature at 15degC 20degC 25degC and30degC Trees were grown at 23 needle-plane distance +etree growth time was decreasing while the temperature wasgrowing up +e degradation mechanism has change tothermal breakdown at temperature 30degC and moisture levelof 25

+e tree growth time was decreasing while the moisturelevel was rising +e lower tree growth time at moisture levelof 003 was greater than the upper tree growth time atmoisture level of 05 +e upper tree growth time atmoisture level of 25 was lower than lower tree growth timeat moisture level of 05 Hence the moisture level can beone of the factors to affect the tree growth time

+e variable temperature and moisture level can affectthe electrical tree growth time and electrical tree typesObjectively electrical tree growth time changes by bothdifferent absorbed moisture and temperature Either in-creasing moisture level andor temperature could maketree growth faster +e density of electrical tree was de-creased while the moisture level andor temperature in-creasing +e tree growth time and tree density arenegative correlation (Figure 11) In Figure 12 the whitelight describes the different intensity and types of PDactivities +e PD activities of medium-density trees werelower than the low-density trees +e PD activities in-dicated that how the electric charges are affected by theelectric field strength +e power of electrical tree growthcame from the electric strength +e electrical tree wasgrown fast due to the high needle-to-plane directionstrength +e needle-to-plane direction strength of high-density trees can decrease significantly due the complexelectric field strength

+e fractal dimensions of electrical trees were estimatedusing Matlab box-counting method from the superimposedimages (Figure 13) +e relationship between tree growthtime and fractal dimension were positively correlated +emoisture level and temperature conditions of electricaltreeing were used to change the electric field strength Bothtree growth time and the fractal dimension may reflect thedirect influences of the electric field strength +e re-lationship between fractal dimension and growth time canwork with electrical trees growth at the same moisture leveland temperature conditions +e fractal dimension of fastergrowth tree was larger than slower growth trees at someenvironment conditions +e differences at some environ-ment can be caused by the small differences on the internalstructure and the chaos electric treeing processes +e chaosthings can be described by the fractal dimension +e fractaldimension can reveal the nature attributes of electrical treesHence the fractal dimension and the growth time of elec-trical trees can have close relationships

+e fractal dimension was deceasing while the moisturelevel and temperature were growing up A final fractal di-mension vs tree growth time image was plotted to find thetrend of them (Figure 14) In general the fractal dimensionof the electrical trees increases while the growth time goesdown

100200300400500600700800900

200 400 600 800 10001000

(a)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(b)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(c)

Figure 8 Brinarized images (a) Df 136 (b) Df 149 (c) Df 174

6 Advances in Materials Science and Engineering

Figure 14 describes the relationship between the rate(growth time) and the extent (fractal dimension) ofelectrical tree growth affected by different environmentconditions As expected the increasing rate of the fractaldimension drops while the growth time grows up Al-though the general relation between the fractal dimensionand growth time is not clear the results also show the treegrowth time and the fractal dimension of final electricaltrees may trend synchronous changes at variable tem-perature and moisture level conditions +e final electricaltree can have similar fractal dimensions via similar treegrowth time at different combinations of moisture leveland temperature conditions Temperature and absorbedmoisture affect the rate and the extent of electrical treegrowth time via the partial discharges Similar partialdischarge activity can lead to similar trees growth time

22D box-count

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(a)

2D box-count2

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(b)

2D box-count2

18161412

108060402

0100 101 102

r box sizendashd

ln n

d ln

r lo

cal d

imen

sion

103 104

(c)

Figure 10 Local fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

107

106

105

104

n (r

)

r

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(a)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(b)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(c)

Figure 9 Fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

Table 1 Tree growth at variable temperature of group A

Sample Moisture level Temperature Growth time Fractaldimension

A1 003 15degC 39min 17332A2 003 20degC 20min 16180A3 003 25degC 16min 15739A4 003 30degC 7min 14821

Table 2 Tree growth at variable temperature of group B

Sample Moisture level Temperature Growth time Fractaldimension

B1 05 15degC 28min 16981B2 05 20degC 17min 16271B3 05 25degC 9min 15646B4 05 30degC 45min 14767

Table 3 Tree growth at variable temperature of group C

Sample Moisture level Temperature Growth time Fractaldimension

C1 1 15degC 16min 15739C2 1 20degC 85min 15257C3 1 25degC 5min 15002C4 1 30degC 3min 14667

Table 4 Tree growth at variable temperature of group D

Sample Moisture level Temperature Growth time Fractaldimension

D1 25 15degC 12min 15388D2 25 20degC 5min 14981D3 25 25degC 25min 12691D4 25 30degC 30 s break Break

Advances in Materials Science and Engineering 7

and trees growth extend Same growth time trees couldgrow at different combinations of moisture level andtemperature conditions +ese different environmentconditions combinations could lead to similar electricaltrees fractal dimensions

6 Conclusion

+e aims of this paper were to study how the temperatureand absorbed moisture affect the rate and the extent ofelectrical tree growth and estimate the extent of electrical

30 degC10

(a)

25 degC05

(b)

20 degC003

(c)

Figure 12 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) a low-density treewith fast growth time (b) a medium-density tree with medium growth time (c) a high-density tree with slow growth time

0

14 16 18 20 22Temperature (degC)

Gro

wth

tim

e (m

in)

24 26 28 30 32

5

10

15

20

25

30

35

40

Moisture level003050

100250

Figure 11 Growth time of electrical trees at different temperature and moisture

8 Advances in Materials Science and Engineering

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

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to the accuracy of 01degC +e temperature shown on thiscontroller was the heaterrsquos temperature A set-sample ref-erence table was used to achieve an accurate sample tem-perature when the variac was set at 60 units

32 Experiment Samples were prepared 2 months beforethe electrical treeing experiment After two months ofsample water conditions preparation each sample wastaken out from the containers with saturated salt solutionsand measured weight before it was going to do a test Finalmass uptakes can be calculated from the last weights whichwere used to describe the humidity of the samples +eprotruding needle was cut into 05 cm after being measuredin order to fix in the treeing test cell +e needle point wasfixed in the middle of the lens sight A camera was used totake photos around the needle point In order to get cleargrowth trees photos a sharpest needle image was taken viachanging the focus of the lens at the beginning A typicalelectrical treeing model can start from charge transport+e space charge accumulated at the cute point will resultin a partial discharge +is space charge combines the

partial discharge that can affect electric field around +enthe electrical trees can grow from this point +e lens focuscould be changed if necessary during the test +e tem-perature system then began to work for at least 1 hour inorder to make a right temperature for the sample A timerwas set to 5 seconds due to the timer has an approximate 5seconds delay to the power supply Other devices werepreset during this one hour

+e experiment devices were double checked before thetest in order to ensure safety Electrical tree experiment wastesting under a dark condition Photographs around theneedle point were monitored via the charged-coupleddevice (CCD) camera +e camera was explored everysecond At the same time a clock was stated to record thetree growth time+e explored time was changed to 10 s viathe camera program in order to acquire PD light Initial PDpoints position and main electrical tree branches could befound via appropriate white level of PD light images asshown in Figure 5 Electrical tree images were taken whilethe electrical trees were grown A tree is defined as a fastgrowth tree while the tree crosses the 14 in the first minuteotherwise it is defined as a slow growth tree +e electrical

Mould Pin Polymersample

(a) (b)

Figure 2 Epoxy resin samples preparation (a) sample prepared by using a concave mould (b) a block sample

21

20

25

15

10

05

00

3

M (

)

Log (t)

Lithium chlorideMagnesium chloride

Potassium carbonateSodium chloride

Figure 3 Plot log (t)ndashΔM of samples

Advances in Materials Science and Engineering 3

tree images were taken every one minute for fast growthtrees and every two minutes for slow growth trees Cor-responding PD light images was taken 11 seconds delaybecause 1 second was used to change the program and 10seconds were used to expose High voltage supply was cutoff when the tree was grown at 23 needle-plane distanceGrowth time distance at this needle-plane distance wastreated as tree growth time

4 Estimate Fractal Dimensions

41 Methods to Obtain Electrical Tree Images 2-D electricaltrees images are required to estimate fractal dimension Amicroscope was used to make 2-D superimposed images Acharged-coupled device (CCD) camera mounted on a mi-croscope was used to take digital pictures from two electricaltree samples +e pictures were taken after every 40 finefocus steps from the first clear branch to the last +ecomposed images were compiled by software called Mon-tage +e distance between branches and camera lens isdifferent Various branches will display on the differentpoints+e images can be taken from the top branch (branch1) of an electrical tree to the bottom (branch k) by changingthe lens-sample distance (Figure 6) +en a superimposedimage can be composed by extracting the clear parts ofdifferent camera lens images (red cycle parts)

+e software named montage is used to compose thoseimages into superimposed images Several methods are used

to estimate the fractal dimension of 2-D images in differentscientific researches For electrical trees the fractal di-mension describes the density of branches +e super-imposed images (Figure 7) were used to estimate the fractaldimension of electrical trees

42 Fractal Dimension Using Box-Counting Method AMinkowskindashBouligand dimension (box-counting dimension)was used to estimate fractal images in nature +e box-counting dimension is using grids to cover images Unifor-mity grids are superposed on the fractal images +e amountof covering boxes is used to estimate the box-counting di-mension +e box-counting dimension is greater than orequal to the Hausdorff dimension because the minimumfreedom spheres are used to cover a fractal image inHausdorffdimension +e box-counting dimension and Hausdorff di-mension are equal while all the boundaries of the fractalimages are coincident with the boundaries of grids+e fractaldimension using box-counting method has been utilized instudying electrical treeing behavior in XLPE cable insulationsamples [20] +e box-counting dimension can be defined asin Equation (2)

Dbox limε⟶0

logN(ε)log(1ε)

1113888 1113889 (2)

where N(ε) is the amount of boxes and ε is the size ofboxes

CCDcamera

High voltagetransformer

Heater

(a)

CCD cameraHigh voltage

Heater

(b)

Figure 4 +e Faraday cage used for electrical treeing experiment (a) diagram of the set-up (b) photograph of the cage

(a) (b) (c)

Figure 5 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) initial PD point(b) clear electrical branches (c) complex electrical branches

4 Advances in Materials Science and Engineering

Infinitesimal grids cannot be used to cover naturalimages (nonconvergence) Fractal sharps in nature mushhave upper and lower limits +e fractal dimension can onlybe estimated between these two limits +e upper limitmeans the size of the grids scale must be larger than the sizeof the image+e lower limit means boxes scale must containat least 1 point of the image Groups of the amount ofcovering boxes can be measured with different size gridsbetween the upper limit and lower limit +e grids are halfsize of the group next to it +e sizes of grids are 2nminus1 unitwhile the size of the minimum size is 1 unit (n is a positiveinteger) +e box-counting dimension can be estimated viainfinite approximation as in Equation (3)

Dfprop1n

1113944

n

nminus1

logN rn( 1113857

log 1rn( 1113857 (3)

+e groups of results satisfy the power law +e negativeexponent of the logN(r)minuslog r fitting curve gives the box-counting dimension Local scaling exponent (local fractaldimension) can be defined as in Equation (4)

D dInN(r)

dIn(1r) (4)

+e fractal dimension of digital images is estimated usingbox-counting method in MATLAB One pixel is the min-imum size of boxes Hence one pixel is the boundary oflower limits +is process can be described as follows

(1) A color image is converted into a grayscale image(Figure 8)

(2) +e image is binarized (translate the grayscale im-ages into white and black images) by an appropriate

1

2

k

1 2 k

Superimposed image

Figure 6 Schematic diagram of composing tree images

Figure 7 Superimposed images from samples

Advances in Materials Science and Engineering 5

light value (the value can be set in the code) for eachpixel (white pixels are binarized ldquoelectrical treesrdquo)

(3) +e total number of boxes N(r) containing whitepixels is calculated by different size of r

(4) A log-log plot of N(r) curve as a function ofr is indicated +e slope gives the fractiondimension

+ree groups of images are selected by the fractal di-mension from low to high linearly All binarized images canfit the original images well

+e fractal dimension increases when the fitting spacedecreases (Figure 9)

Local fractal dimension describes the differentiation offractal dimension distribution curve at each r (the size ofgrid) point (Figure 10) Local fractal dimension curve is notsmooth because local fractal dimension cannot accuratelydescribe the self-similarity from a large size grid At small rpoints the local fractal dimension is similar +e result offractal dimension will decrease due to these points How-ever the box-counting results still describe a fractal char-acteristic in statistics Hence the slope of the logN(r)minuslog rcan give the box-counting dimension

5 Results and Discussion

Four groups of samples A (Table 1) B (Table 2) C (Ta-ble 3) and D (Table 4) were stored in four saturatedsolution conditions (LiCl RH15 MgCl2 RH30 K2CO3RH 44 NaCl RH 75) for two months Samples weretesting at a different moisture level of 003 05 125 and different temperature at 15degC 20degC 25degC and30degC Trees were grown at 23 needle-plane distance +etree growth time was decreasing while the temperature wasgrowing up +e degradation mechanism has change tothermal breakdown at temperature 30degC and moisture levelof 25

+e tree growth time was decreasing while the moisturelevel was rising +e lower tree growth time at moisture levelof 003 was greater than the upper tree growth time atmoisture level of 05 +e upper tree growth time atmoisture level of 25 was lower than lower tree growth timeat moisture level of 05 Hence the moisture level can beone of the factors to affect the tree growth time

+e variable temperature and moisture level can affectthe electrical tree growth time and electrical tree typesObjectively electrical tree growth time changes by bothdifferent absorbed moisture and temperature Either in-creasing moisture level andor temperature could maketree growth faster +e density of electrical tree was de-creased while the moisture level andor temperature in-creasing +e tree growth time and tree density arenegative correlation (Figure 11) In Figure 12 the whitelight describes the different intensity and types of PDactivities +e PD activities of medium-density trees werelower than the low-density trees +e PD activities in-dicated that how the electric charges are affected by theelectric field strength +e power of electrical tree growthcame from the electric strength +e electrical tree wasgrown fast due to the high needle-to-plane directionstrength +e needle-to-plane direction strength of high-density trees can decrease significantly due the complexelectric field strength

+e fractal dimensions of electrical trees were estimatedusing Matlab box-counting method from the superimposedimages (Figure 13) +e relationship between tree growthtime and fractal dimension were positively correlated +emoisture level and temperature conditions of electricaltreeing were used to change the electric field strength Bothtree growth time and the fractal dimension may reflect thedirect influences of the electric field strength +e re-lationship between fractal dimension and growth time canwork with electrical trees growth at the same moisture leveland temperature conditions +e fractal dimension of fastergrowth tree was larger than slower growth trees at someenvironment conditions +e differences at some environ-ment can be caused by the small differences on the internalstructure and the chaos electric treeing processes +e chaosthings can be described by the fractal dimension +e fractaldimension can reveal the nature attributes of electrical treesHence the fractal dimension and the growth time of elec-trical trees can have close relationships

+e fractal dimension was deceasing while the moisturelevel and temperature were growing up A final fractal di-mension vs tree growth time image was plotted to find thetrend of them (Figure 14) In general the fractal dimensionof the electrical trees increases while the growth time goesdown

100200300400500600700800900

200 400 600 800 10001000

(a)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(b)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(c)

Figure 8 Brinarized images (a) Df 136 (b) Df 149 (c) Df 174

6 Advances in Materials Science and Engineering

Figure 14 describes the relationship between the rate(growth time) and the extent (fractal dimension) ofelectrical tree growth affected by different environmentconditions As expected the increasing rate of the fractaldimension drops while the growth time grows up Al-though the general relation between the fractal dimensionand growth time is not clear the results also show the treegrowth time and the fractal dimension of final electricaltrees may trend synchronous changes at variable tem-perature and moisture level conditions +e final electricaltree can have similar fractal dimensions via similar treegrowth time at different combinations of moisture leveland temperature conditions Temperature and absorbedmoisture affect the rate and the extent of electrical treegrowth time via the partial discharges Similar partialdischarge activity can lead to similar trees growth time

22D box-count

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(a)

2D box-count2

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(b)

2D box-count2

18161412

108060402

0100 101 102

r box sizendashd

ln n

d ln

r lo

cal d

imen

sion

103 104

(c)

Figure 10 Local fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

107

106

105

104

n (r

)

r

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(a)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(b)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(c)

Figure 9 Fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

Table 1 Tree growth at variable temperature of group A

Sample Moisture level Temperature Growth time Fractaldimension

A1 003 15degC 39min 17332A2 003 20degC 20min 16180A3 003 25degC 16min 15739A4 003 30degC 7min 14821

Table 2 Tree growth at variable temperature of group B

Sample Moisture level Temperature Growth time Fractaldimension

B1 05 15degC 28min 16981B2 05 20degC 17min 16271B3 05 25degC 9min 15646B4 05 30degC 45min 14767

Table 3 Tree growth at variable temperature of group C

Sample Moisture level Temperature Growth time Fractaldimension

C1 1 15degC 16min 15739C2 1 20degC 85min 15257C3 1 25degC 5min 15002C4 1 30degC 3min 14667

Table 4 Tree growth at variable temperature of group D

Sample Moisture level Temperature Growth time Fractaldimension

D1 25 15degC 12min 15388D2 25 20degC 5min 14981D3 25 25degC 25min 12691D4 25 30degC 30 s break Break

Advances in Materials Science and Engineering 7

and trees growth extend Same growth time trees couldgrow at different combinations of moisture level andtemperature conditions +ese different environmentconditions combinations could lead to similar electricaltrees fractal dimensions

6 Conclusion

+e aims of this paper were to study how the temperatureand absorbed moisture affect the rate and the extent ofelectrical tree growth and estimate the extent of electrical

30 degC10

(a)

25 degC05

(b)

20 degC003

(c)

Figure 12 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) a low-density treewith fast growth time (b) a medium-density tree with medium growth time (c) a high-density tree with slow growth time

0

14 16 18 20 22Temperature (degC)

Gro

wth

tim

e (m

in)

24 26 28 30 32

5

10

15

20

25

30

35

40

Moisture level003050

100250

Figure 11 Growth time of electrical trees at different temperature and moisture

8 Advances in Materials Science and Engineering

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

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ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

tree images were taken every one minute for fast growthtrees and every two minutes for slow growth trees Cor-responding PD light images was taken 11 seconds delaybecause 1 second was used to change the program and 10seconds were used to expose High voltage supply was cutoff when the tree was grown at 23 needle-plane distanceGrowth time distance at this needle-plane distance wastreated as tree growth time

4 Estimate Fractal Dimensions

41 Methods to Obtain Electrical Tree Images 2-D electricaltrees images are required to estimate fractal dimension Amicroscope was used to make 2-D superimposed images Acharged-coupled device (CCD) camera mounted on a mi-croscope was used to take digital pictures from two electricaltree samples +e pictures were taken after every 40 finefocus steps from the first clear branch to the last +ecomposed images were compiled by software called Mon-tage +e distance between branches and camera lens isdifferent Various branches will display on the differentpoints+e images can be taken from the top branch (branch1) of an electrical tree to the bottom (branch k) by changingthe lens-sample distance (Figure 6) +en a superimposedimage can be composed by extracting the clear parts ofdifferent camera lens images (red cycle parts)

+e software named montage is used to compose thoseimages into superimposed images Several methods are used

to estimate the fractal dimension of 2-D images in differentscientific researches For electrical trees the fractal di-mension describes the density of branches +e super-imposed images (Figure 7) were used to estimate the fractaldimension of electrical trees

42 Fractal Dimension Using Box-Counting Method AMinkowskindashBouligand dimension (box-counting dimension)was used to estimate fractal images in nature +e box-counting dimension is using grids to cover images Unifor-mity grids are superposed on the fractal images +e amountof covering boxes is used to estimate the box-counting di-mension +e box-counting dimension is greater than orequal to the Hausdorff dimension because the minimumfreedom spheres are used to cover a fractal image inHausdorffdimension +e box-counting dimension and Hausdorff di-mension are equal while all the boundaries of the fractalimages are coincident with the boundaries of grids+e fractaldimension using box-counting method has been utilized instudying electrical treeing behavior in XLPE cable insulationsamples [20] +e box-counting dimension can be defined asin Equation (2)

Dbox limε⟶0

logN(ε)log(1ε)

1113888 1113889 (2)

where N(ε) is the amount of boxes and ε is the size ofboxes

CCDcamera

High voltagetransformer

Heater

(a)

CCD cameraHigh voltage

Heater

(b)

Figure 4 +e Faraday cage used for electrical treeing experiment (a) diagram of the set-up (b) photograph of the cage

(a) (b) (c)

Figure 5 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) initial PD point(b) clear electrical branches (c) complex electrical branches

4 Advances in Materials Science and Engineering

Infinitesimal grids cannot be used to cover naturalimages (nonconvergence) Fractal sharps in nature mushhave upper and lower limits +e fractal dimension can onlybe estimated between these two limits +e upper limitmeans the size of the grids scale must be larger than the sizeof the image+e lower limit means boxes scale must containat least 1 point of the image Groups of the amount ofcovering boxes can be measured with different size gridsbetween the upper limit and lower limit +e grids are halfsize of the group next to it +e sizes of grids are 2nminus1 unitwhile the size of the minimum size is 1 unit (n is a positiveinteger) +e box-counting dimension can be estimated viainfinite approximation as in Equation (3)

Dfprop1n

1113944

n

nminus1

logN rn( 1113857

log 1rn( 1113857 (3)

+e groups of results satisfy the power law +e negativeexponent of the logN(r)minuslog r fitting curve gives the box-counting dimension Local scaling exponent (local fractaldimension) can be defined as in Equation (4)

D dInN(r)

dIn(1r) (4)

+e fractal dimension of digital images is estimated usingbox-counting method in MATLAB One pixel is the min-imum size of boxes Hence one pixel is the boundary oflower limits +is process can be described as follows

(1) A color image is converted into a grayscale image(Figure 8)

(2) +e image is binarized (translate the grayscale im-ages into white and black images) by an appropriate

1

2

k

1 2 k

Superimposed image

Figure 6 Schematic diagram of composing tree images

Figure 7 Superimposed images from samples

Advances in Materials Science and Engineering 5

light value (the value can be set in the code) for eachpixel (white pixels are binarized ldquoelectrical treesrdquo)

(3) +e total number of boxes N(r) containing whitepixels is calculated by different size of r

(4) A log-log plot of N(r) curve as a function ofr is indicated +e slope gives the fractiondimension

+ree groups of images are selected by the fractal di-mension from low to high linearly All binarized images canfit the original images well

+e fractal dimension increases when the fitting spacedecreases (Figure 9)

Local fractal dimension describes the differentiation offractal dimension distribution curve at each r (the size ofgrid) point (Figure 10) Local fractal dimension curve is notsmooth because local fractal dimension cannot accuratelydescribe the self-similarity from a large size grid At small rpoints the local fractal dimension is similar +e result offractal dimension will decrease due to these points How-ever the box-counting results still describe a fractal char-acteristic in statistics Hence the slope of the logN(r)minuslog rcan give the box-counting dimension

5 Results and Discussion

Four groups of samples A (Table 1) B (Table 2) C (Ta-ble 3) and D (Table 4) were stored in four saturatedsolution conditions (LiCl RH15 MgCl2 RH30 K2CO3RH 44 NaCl RH 75) for two months Samples weretesting at a different moisture level of 003 05 125 and different temperature at 15degC 20degC 25degC and30degC Trees were grown at 23 needle-plane distance +etree growth time was decreasing while the temperature wasgrowing up +e degradation mechanism has change tothermal breakdown at temperature 30degC and moisture levelof 25

+e tree growth time was decreasing while the moisturelevel was rising +e lower tree growth time at moisture levelof 003 was greater than the upper tree growth time atmoisture level of 05 +e upper tree growth time atmoisture level of 25 was lower than lower tree growth timeat moisture level of 05 Hence the moisture level can beone of the factors to affect the tree growth time

+e variable temperature and moisture level can affectthe electrical tree growth time and electrical tree typesObjectively electrical tree growth time changes by bothdifferent absorbed moisture and temperature Either in-creasing moisture level andor temperature could maketree growth faster +e density of electrical tree was de-creased while the moisture level andor temperature in-creasing +e tree growth time and tree density arenegative correlation (Figure 11) In Figure 12 the whitelight describes the different intensity and types of PDactivities +e PD activities of medium-density trees werelower than the low-density trees +e PD activities in-dicated that how the electric charges are affected by theelectric field strength +e power of electrical tree growthcame from the electric strength +e electrical tree wasgrown fast due to the high needle-to-plane directionstrength +e needle-to-plane direction strength of high-density trees can decrease significantly due the complexelectric field strength

+e fractal dimensions of electrical trees were estimatedusing Matlab box-counting method from the superimposedimages (Figure 13) +e relationship between tree growthtime and fractal dimension were positively correlated +emoisture level and temperature conditions of electricaltreeing were used to change the electric field strength Bothtree growth time and the fractal dimension may reflect thedirect influences of the electric field strength +e re-lationship between fractal dimension and growth time canwork with electrical trees growth at the same moisture leveland temperature conditions +e fractal dimension of fastergrowth tree was larger than slower growth trees at someenvironment conditions +e differences at some environ-ment can be caused by the small differences on the internalstructure and the chaos electric treeing processes +e chaosthings can be described by the fractal dimension +e fractaldimension can reveal the nature attributes of electrical treesHence the fractal dimension and the growth time of elec-trical trees can have close relationships

+e fractal dimension was deceasing while the moisturelevel and temperature were growing up A final fractal di-mension vs tree growth time image was plotted to find thetrend of them (Figure 14) In general the fractal dimensionof the electrical trees increases while the growth time goesdown

100200300400500600700800900

200 400 600 800 10001000

(a)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(b)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(c)

Figure 8 Brinarized images (a) Df 136 (b) Df 149 (c) Df 174

6 Advances in Materials Science and Engineering

Figure 14 describes the relationship between the rate(growth time) and the extent (fractal dimension) ofelectrical tree growth affected by different environmentconditions As expected the increasing rate of the fractaldimension drops while the growth time grows up Al-though the general relation between the fractal dimensionand growth time is not clear the results also show the treegrowth time and the fractal dimension of final electricaltrees may trend synchronous changes at variable tem-perature and moisture level conditions +e final electricaltree can have similar fractal dimensions via similar treegrowth time at different combinations of moisture leveland temperature conditions Temperature and absorbedmoisture affect the rate and the extent of electrical treegrowth time via the partial discharges Similar partialdischarge activity can lead to similar trees growth time

22D box-count

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(a)

2D box-count2

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(b)

2D box-count2

18161412

108060402

0100 101 102

r box sizendashd

ln n

d ln

r lo

cal d

imen

sion

103 104

(c)

Figure 10 Local fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

107

106

105

104

n (r

)

r

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(a)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(b)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(c)

Figure 9 Fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

Table 1 Tree growth at variable temperature of group A

Sample Moisture level Temperature Growth time Fractaldimension

A1 003 15degC 39min 17332A2 003 20degC 20min 16180A3 003 25degC 16min 15739A4 003 30degC 7min 14821

Table 2 Tree growth at variable temperature of group B

Sample Moisture level Temperature Growth time Fractaldimension

B1 05 15degC 28min 16981B2 05 20degC 17min 16271B3 05 25degC 9min 15646B4 05 30degC 45min 14767

Table 3 Tree growth at variable temperature of group C

Sample Moisture level Temperature Growth time Fractaldimension

C1 1 15degC 16min 15739C2 1 20degC 85min 15257C3 1 25degC 5min 15002C4 1 30degC 3min 14667

Table 4 Tree growth at variable temperature of group D

Sample Moisture level Temperature Growth time Fractaldimension

D1 25 15degC 12min 15388D2 25 20degC 5min 14981D3 25 25degC 25min 12691D4 25 30degC 30 s break Break

Advances in Materials Science and Engineering 7

and trees growth extend Same growth time trees couldgrow at different combinations of moisture level andtemperature conditions +ese different environmentconditions combinations could lead to similar electricaltrees fractal dimensions

6 Conclusion

+e aims of this paper were to study how the temperatureand absorbed moisture affect the rate and the extent ofelectrical tree growth and estimate the extent of electrical

30 degC10

(a)

25 degC05

(b)

20 degC003

(c)

Figure 12 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) a low-density treewith fast growth time (b) a medium-density tree with medium growth time (c) a high-density tree with slow growth time

0

14 16 18 20 22Temperature (degC)

Gro

wth

tim

e (m

in)

24 26 28 30 32

5

10

15

20

25

30

35

40

Moisture level003050

100250

Figure 11 Growth time of electrical trees at different temperature and moisture

8 Advances in Materials Science and Engineering

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Infinitesimal grids cannot be used to cover naturalimages (nonconvergence) Fractal sharps in nature mushhave upper and lower limits +e fractal dimension can onlybe estimated between these two limits +e upper limitmeans the size of the grids scale must be larger than the sizeof the image+e lower limit means boxes scale must containat least 1 point of the image Groups of the amount ofcovering boxes can be measured with different size gridsbetween the upper limit and lower limit +e grids are halfsize of the group next to it +e sizes of grids are 2nminus1 unitwhile the size of the minimum size is 1 unit (n is a positiveinteger) +e box-counting dimension can be estimated viainfinite approximation as in Equation (3)

Dfprop1n

1113944

n

nminus1

logN rn( 1113857

log 1rn( 1113857 (3)

+e groups of results satisfy the power law +e negativeexponent of the logN(r)minuslog r fitting curve gives the box-counting dimension Local scaling exponent (local fractaldimension) can be defined as in Equation (4)

D dInN(r)

dIn(1r) (4)

+e fractal dimension of digital images is estimated usingbox-counting method in MATLAB One pixel is the min-imum size of boxes Hence one pixel is the boundary oflower limits +is process can be described as follows

(1) A color image is converted into a grayscale image(Figure 8)

(2) +e image is binarized (translate the grayscale im-ages into white and black images) by an appropriate

1

2

k

1 2 k

Superimposed image

Figure 6 Schematic diagram of composing tree images

Figure 7 Superimposed images from samples

Advances in Materials Science and Engineering 5

light value (the value can be set in the code) for eachpixel (white pixels are binarized ldquoelectrical treesrdquo)

(3) +e total number of boxes N(r) containing whitepixels is calculated by different size of r

(4) A log-log plot of N(r) curve as a function ofr is indicated +e slope gives the fractiondimension

+ree groups of images are selected by the fractal di-mension from low to high linearly All binarized images canfit the original images well

+e fractal dimension increases when the fitting spacedecreases (Figure 9)

Local fractal dimension describes the differentiation offractal dimension distribution curve at each r (the size ofgrid) point (Figure 10) Local fractal dimension curve is notsmooth because local fractal dimension cannot accuratelydescribe the self-similarity from a large size grid At small rpoints the local fractal dimension is similar +e result offractal dimension will decrease due to these points How-ever the box-counting results still describe a fractal char-acteristic in statistics Hence the slope of the logN(r)minuslog rcan give the box-counting dimension

5 Results and Discussion

Four groups of samples A (Table 1) B (Table 2) C (Ta-ble 3) and D (Table 4) were stored in four saturatedsolution conditions (LiCl RH15 MgCl2 RH30 K2CO3RH 44 NaCl RH 75) for two months Samples weretesting at a different moisture level of 003 05 125 and different temperature at 15degC 20degC 25degC and30degC Trees were grown at 23 needle-plane distance +etree growth time was decreasing while the temperature wasgrowing up +e degradation mechanism has change tothermal breakdown at temperature 30degC and moisture levelof 25

+e tree growth time was decreasing while the moisturelevel was rising +e lower tree growth time at moisture levelof 003 was greater than the upper tree growth time atmoisture level of 05 +e upper tree growth time atmoisture level of 25 was lower than lower tree growth timeat moisture level of 05 Hence the moisture level can beone of the factors to affect the tree growth time

+e variable temperature and moisture level can affectthe electrical tree growth time and electrical tree typesObjectively electrical tree growth time changes by bothdifferent absorbed moisture and temperature Either in-creasing moisture level andor temperature could maketree growth faster +e density of electrical tree was de-creased while the moisture level andor temperature in-creasing +e tree growth time and tree density arenegative correlation (Figure 11) In Figure 12 the whitelight describes the different intensity and types of PDactivities +e PD activities of medium-density trees werelower than the low-density trees +e PD activities in-dicated that how the electric charges are affected by theelectric field strength +e power of electrical tree growthcame from the electric strength +e electrical tree wasgrown fast due to the high needle-to-plane directionstrength +e needle-to-plane direction strength of high-density trees can decrease significantly due the complexelectric field strength

+e fractal dimensions of electrical trees were estimatedusing Matlab box-counting method from the superimposedimages (Figure 13) +e relationship between tree growthtime and fractal dimension were positively correlated +emoisture level and temperature conditions of electricaltreeing were used to change the electric field strength Bothtree growth time and the fractal dimension may reflect thedirect influences of the electric field strength +e re-lationship between fractal dimension and growth time canwork with electrical trees growth at the same moisture leveland temperature conditions +e fractal dimension of fastergrowth tree was larger than slower growth trees at someenvironment conditions +e differences at some environ-ment can be caused by the small differences on the internalstructure and the chaos electric treeing processes +e chaosthings can be described by the fractal dimension +e fractaldimension can reveal the nature attributes of electrical treesHence the fractal dimension and the growth time of elec-trical trees can have close relationships

+e fractal dimension was deceasing while the moisturelevel and temperature were growing up A final fractal di-mension vs tree growth time image was plotted to find thetrend of them (Figure 14) In general the fractal dimensionof the electrical trees increases while the growth time goesdown

100200300400500600700800900

200 400 600 800 10001000

(a)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(b)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(c)

Figure 8 Brinarized images (a) Df 136 (b) Df 149 (c) Df 174

6 Advances in Materials Science and Engineering

Figure 14 describes the relationship between the rate(growth time) and the extent (fractal dimension) ofelectrical tree growth affected by different environmentconditions As expected the increasing rate of the fractaldimension drops while the growth time grows up Al-though the general relation between the fractal dimensionand growth time is not clear the results also show the treegrowth time and the fractal dimension of final electricaltrees may trend synchronous changes at variable tem-perature and moisture level conditions +e final electricaltree can have similar fractal dimensions via similar treegrowth time at different combinations of moisture leveland temperature conditions Temperature and absorbedmoisture affect the rate and the extent of electrical treegrowth time via the partial discharges Similar partialdischarge activity can lead to similar trees growth time

22D box-count

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(a)

2D box-count2

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(b)

2D box-count2

18161412

108060402

0100 101 102

r box sizendashd

ln n

d ln

r lo

cal d

imen

sion

103 104

(c)

Figure 10 Local fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

107

106

105

104

n (r

)

r

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(a)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(b)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(c)

Figure 9 Fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

Table 1 Tree growth at variable temperature of group A

Sample Moisture level Temperature Growth time Fractaldimension

A1 003 15degC 39min 17332A2 003 20degC 20min 16180A3 003 25degC 16min 15739A4 003 30degC 7min 14821

Table 2 Tree growth at variable temperature of group B

Sample Moisture level Temperature Growth time Fractaldimension

B1 05 15degC 28min 16981B2 05 20degC 17min 16271B3 05 25degC 9min 15646B4 05 30degC 45min 14767

Table 3 Tree growth at variable temperature of group C

Sample Moisture level Temperature Growth time Fractaldimension

C1 1 15degC 16min 15739C2 1 20degC 85min 15257C3 1 25degC 5min 15002C4 1 30degC 3min 14667

Table 4 Tree growth at variable temperature of group D

Sample Moisture level Temperature Growth time Fractaldimension

D1 25 15degC 12min 15388D2 25 20degC 5min 14981D3 25 25degC 25min 12691D4 25 30degC 30 s break Break

Advances in Materials Science and Engineering 7

and trees growth extend Same growth time trees couldgrow at different combinations of moisture level andtemperature conditions +ese different environmentconditions combinations could lead to similar electricaltrees fractal dimensions

6 Conclusion

+e aims of this paper were to study how the temperatureand absorbed moisture affect the rate and the extent ofelectrical tree growth and estimate the extent of electrical

30 degC10

(a)

25 degC05

(b)

20 degC003

(c)

Figure 12 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) a low-density treewith fast growth time (b) a medium-density tree with medium growth time (c) a high-density tree with slow growth time

0

14 16 18 20 22Temperature (degC)

Gro

wth

tim

e (m

in)

24 26 28 30 32

5

10

15

20

25

30

35

40

Moisture level003050

100250

Figure 11 Growth time of electrical trees at different temperature and moisture

8 Advances in Materials Science and Engineering

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

light value (the value can be set in the code) for eachpixel (white pixels are binarized ldquoelectrical treesrdquo)

(3) +e total number of boxes N(r) containing whitepixels is calculated by different size of r

(4) A log-log plot of N(r) curve as a function ofr is indicated +e slope gives the fractiondimension

+ree groups of images are selected by the fractal di-mension from low to high linearly All binarized images canfit the original images well

+e fractal dimension increases when the fitting spacedecreases (Figure 9)

Local fractal dimension describes the differentiation offractal dimension distribution curve at each r (the size ofgrid) point (Figure 10) Local fractal dimension curve is notsmooth because local fractal dimension cannot accuratelydescribe the self-similarity from a large size grid At small rpoints the local fractal dimension is similar +e result offractal dimension will decrease due to these points How-ever the box-counting results still describe a fractal char-acteristic in statistics Hence the slope of the logN(r)minuslog rcan give the box-counting dimension

5 Results and Discussion

Four groups of samples A (Table 1) B (Table 2) C (Ta-ble 3) and D (Table 4) were stored in four saturatedsolution conditions (LiCl RH15 MgCl2 RH30 K2CO3RH 44 NaCl RH 75) for two months Samples weretesting at a different moisture level of 003 05 125 and different temperature at 15degC 20degC 25degC and30degC Trees were grown at 23 needle-plane distance +etree growth time was decreasing while the temperature wasgrowing up +e degradation mechanism has change tothermal breakdown at temperature 30degC and moisture levelof 25

+e tree growth time was decreasing while the moisturelevel was rising +e lower tree growth time at moisture levelof 003 was greater than the upper tree growth time atmoisture level of 05 +e upper tree growth time atmoisture level of 25 was lower than lower tree growth timeat moisture level of 05 Hence the moisture level can beone of the factors to affect the tree growth time

+e variable temperature and moisture level can affectthe electrical tree growth time and electrical tree typesObjectively electrical tree growth time changes by bothdifferent absorbed moisture and temperature Either in-creasing moisture level andor temperature could maketree growth faster +e density of electrical tree was de-creased while the moisture level andor temperature in-creasing +e tree growth time and tree density arenegative correlation (Figure 11) In Figure 12 the whitelight describes the different intensity and types of PDactivities +e PD activities of medium-density trees werelower than the low-density trees +e PD activities in-dicated that how the electric charges are affected by theelectric field strength +e power of electrical tree growthcame from the electric strength +e electrical tree wasgrown fast due to the high needle-to-plane directionstrength +e needle-to-plane direction strength of high-density trees can decrease significantly due the complexelectric field strength

+e fractal dimensions of electrical trees were estimatedusing Matlab box-counting method from the superimposedimages (Figure 13) +e relationship between tree growthtime and fractal dimension were positively correlated +emoisture level and temperature conditions of electricaltreeing were used to change the electric field strength Bothtree growth time and the fractal dimension may reflect thedirect influences of the electric field strength +e re-lationship between fractal dimension and growth time canwork with electrical trees growth at the same moisture leveland temperature conditions +e fractal dimension of fastergrowth tree was larger than slower growth trees at someenvironment conditions +e differences at some environ-ment can be caused by the small differences on the internalstructure and the chaos electric treeing processes +e chaosthings can be described by the fractal dimension +e fractaldimension can reveal the nature attributes of electrical treesHence the fractal dimension and the growth time of elec-trical trees can have close relationships

+e fractal dimension was deceasing while the moisturelevel and temperature were growing up A final fractal di-mension vs tree growth time image was plotted to find thetrend of them (Figure 14) In general the fractal dimensionof the electrical trees increases while the growth time goesdown

100200300400500600700800900

200 400 600 800 10001000

(a)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(b)

1000900

200 400 600 800 1000 1200

800700600500400300200100

(c)

Figure 8 Brinarized images (a) Df 136 (b) Df 149 (c) Df 174

6 Advances in Materials Science and Engineering

Figure 14 describes the relationship between the rate(growth time) and the extent (fractal dimension) ofelectrical tree growth affected by different environmentconditions As expected the increasing rate of the fractaldimension drops while the growth time grows up Al-though the general relation between the fractal dimensionand growth time is not clear the results also show the treegrowth time and the fractal dimension of final electricaltrees may trend synchronous changes at variable tem-perature and moisture level conditions +e final electricaltree can have similar fractal dimensions via similar treegrowth time at different combinations of moisture leveland temperature conditions Temperature and absorbedmoisture affect the rate and the extent of electrical treegrowth time via the partial discharges Similar partialdischarge activity can lead to similar trees growth time

22D box-count

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(a)

2D box-count2

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(b)

2D box-count2

18161412

108060402

0100 101 102

r box sizendashd

ln n

d ln

r lo

cal d

imen

sion

103 104

(c)

Figure 10 Local fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

107

106

105

104

n (r

)

r

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(a)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(b)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(c)

Figure 9 Fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

Table 1 Tree growth at variable temperature of group A

Sample Moisture level Temperature Growth time Fractaldimension

A1 003 15degC 39min 17332A2 003 20degC 20min 16180A3 003 25degC 16min 15739A4 003 30degC 7min 14821

Table 2 Tree growth at variable temperature of group B

Sample Moisture level Temperature Growth time Fractaldimension

B1 05 15degC 28min 16981B2 05 20degC 17min 16271B3 05 25degC 9min 15646B4 05 30degC 45min 14767

Table 3 Tree growth at variable temperature of group C

Sample Moisture level Temperature Growth time Fractaldimension

C1 1 15degC 16min 15739C2 1 20degC 85min 15257C3 1 25degC 5min 15002C4 1 30degC 3min 14667

Table 4 Tree growth at variable temperature of group D

Sample Moisture level Temperature Growth time Fractaldimension

D1 25 15degC 12min 15388D2 25 20degC 5min 14981D3 25 25degC 25min 12691D4 25 30degC 30 s break Break

Advances in Materials Science and Engineering 7

and trees growth extend Same growth time trees couldgrow at different combinations of moisture level andtemperature conditions +ese different environmentconditions combinations could lead to similar electricaltrees fractal dimensions

6 Conclusion

+e aims of this paper were to study how the temperatureand absorbed moisture affect the rate and the extent ofelectrical tree growth and estimate the extent of electrical

30 degC10

(a)

25 degC05

(b)

20 degC003

(c)

Figure 12 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) a low-density treewith fast growth time (b) a medium-density tree with medium growth time (c) a high-density tree with slow growth time

0

14 16 18 20 22Temperature (degC)

Gro

wth

tim

e (m

in)

24 26 28 30 32

5

10

15

20

25

30

35

40

Moisture level003050

100250

Figure 11 Growth time of electrical trees at different temperature and moisture

8 Advances in Materials Science and Engineering

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Figure 14 describes the relationship between the rate(growth time) and the extent (fractal dimension) ofelectrical tree growth affected by different environmentconditions As expected the increasing rate of the fractaldimension drops while the growth time grows up Al-though the general relation between the fractal dimensionand growth time is not clear the results also show the treegrowth time and the fractal dimension of final electricaltrees may trend synchronous changes at variable tem-perature and moisture level conditions +e final electricaltree can have similar fractal dimensions via similar treegrowth time at different combinations of moisture leveland temperature conditions Temperature and absorbedmoisture affect the rate and the extent of electrical treegrowth time via the partial discharges Similar partialdischarge activity can lead to similar trees growth time

22D box-count

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(a)

2D box-count2

18161412

108060402

0100 101 102

r box size

ndashd ln

nd

ln r

loca

l dim

ensio

n

103 104

(b)

2D box-count2

18161412

108060402

0100 101 102

r box sizendashd

ln n

d ln

r lo

cal d

imen

sion

103 104

(c)

Figure 10 Local fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

107

106

105

104

n (r

)

r

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(a)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(b)

n (r

)

r

107

106

105

104

103

102

101

100

100 101 102 103 104

Actual box-countSpace-filling box-count

(c)

Figure 9 Fractal dimension distribution curve (a) Df 136 (b) Df 149 (c) Df 174

Table 1 Tree growth at variable temperature of group A

Sample Moisture level Temperature Growth time Fractaldimension

A1 003 15degC 39min 17332A2 003 20degC 20min 16180A3 003 25degC 16min 15739A4 003 30degC 7min 14821

Table 2 Tree growth at variable temperature of group B

Sample Moisture level Temperature Growth time Fractaldimension

B1 05 15degC 28min 16981B2 05 20degC 17min 16271B3 05 25degC 9min 15646B4 05 30degC 45min 14767

Table 3 Tree growth at variable temperature of group C

Sample Moisture level Temperature Growth time Fractaldimension

C1 1 15degC 16min 15739C2 1 20degC 85min 15257C3 1 25degC 5min 15002C4 1 30degC 3min 14667

Table 4 Tree growth at variable temperature of group D

Sample Moisture level Temperature Growth time Fractaldimension

D1 25 15degC 12min 15388D2 25 20degC 5min 14981D3 25 25degC 25min 12691D4 25 30degC 30 s break Break

Advances in Materials Science and Engineering 7

and trees growth extend Same growth time trees couldgrow at different combinations of moisture level andtemperature conditions +ese different environmentconditions combinations could lead to similar electricaltrees fractal dimensions

6 Conclusion

+e aims of this paper were to study how the temperatureand absorbed moisture affect the rate and the extent ofelectrical tree growth and estimate the extent of electrical

30 degC10

(a)

25 degC05

(b)

20 degC003

(c)

Figure 12 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) a low-density treewith fast growth time (b) a medium-density tree with medium growth time (c) a high-density tree with slow growth time

0

14 16 18 20 22Temperature (degC)

Gro

wth

tim

e (m

in)

24 26 28 30 32

5

10

15

20

25

30

35

40

Moisture level003050

100250

Figure 11 Growth time of electrical trees at different temperature and moisture

8 Advances in Materials Science and Engineering

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

and trees growth extend Same growth time trees couldgrow at different combinations of moisture level andtemperature conditions +ese different environmentconditions combinations could lead to similar electricaltrees fractal dimensions

6 Conclusion

+e aims of this paper were to study how the temperatureand absorbed moisture affect the rate and the extent ofelectrical tree growth and estimate the extent of electrical

30 degC10

(a)

25 degC05

(b)

20 degC003

(c)

Figure 12 Superimposed red-white-black images of PD light and tree images with a same red and white light level (a) a low-density treewith fast growth time (b) a medium-density tree with medium growth time (c) a high-density tree with slow growth time

0

14 16 18 20 22Temperature (degC)

Gro

wth

tim

e (m

in)

24 26 28 30 32

5

10

15

20

25

30

35

40

Moisture level003050

100250

Figure 11 Growth time of electrical trees at different temperature and moisture

8 Advances in Materials Science and Engineering

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

tree growth using fractal dimension +e work has carriedout a relationship between fractal dimension and growthtime of electrical tree at different temperature and moisturelevel +e fractal dimension of the electrical trees decreases

and the growth time goes up while the moisture level andtemperature increases Tree growth time and the fractaldimension of final electrical trees may trend synchronouschanges at variable temperature and moisture level

Figure 13 Superimposed images of final electrical trees

Advances in Materials Science and Engineering 9

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

conditions On the basis of current study further work buildon this paper may imply improved methods to acquire 3-Delectrical trees images

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Youping Fan proposed the project and supervised the paperDai Zhang wrote the Matlab codes analysed the data anddrafted the manuscript Jingjiao Li coordinated the studyand helped to draft the manuscript

Acknowledgments

+e authors would like to thank the financial support fromthe Weng Hongwu Original Research Fund of PekingUniversity

References

[1] L A Dissado and J C Fothergill Electrical Degradation andBreakdown in Polymers ISBN 9780863411960 IEEE LondonUK ISBN 9780863411960 1992

[2] J C Fothergill ldquoAgeing space charge and nanodielectrics tenthings we donrsquot know about dielectricsrdquo in Proceedings of2007 IEEE International Conference on Solid Dielectricspp 1ndash10 2007

[3] P H Morshuis ldquoDegradation of solid dielectrics due to in-ternal partial discharge some thoughts on progress made andwhere to go nowrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 12 p 1275 2005

[4] K Kudo ldquoFractal analysis of electrical treesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 5 no 5pp 713ndash727 1998

[5] M Kosaki M Nagao N Shimizu and Y Mizuno ldquoSolidinsulation and its deteriorationrdquo Cryogenics vol 38 no 11pp 1095ndash1104 1998

[6] J Feder Fractals (Physics of Solids and Liquids) PlennumNew York NY USA 1988 ISBN 9780306428517

[7] S L Eerm Crmd and B B Mandelbrot ldquoFractal properties ofrain and a fractal modelrdquo Tellus A Dynamic Meteorology andOceanography vol 37 no 3 pp 209ndash232 1985

[8] B B Mandelbrot 9e Fractal Geometry of Nature W HFreeman New York NY USA 1983 ISBN 9780716711865

[9] G A Edgar Classics on Fractals Addison-esley Boston MAUSA 1993 ISBN 9780813341538

[10] A Fraenkel and A Kontorovich 9e Sierpinski sieve of Nim-varieties and binomial coefficients Integers 2006

[11] C T Brown and L S Liebovitch Fractal analysis Quanti-tative applications in the social sciences SAGE Publications+ousand Oaks CA USA 2010 ISBN9781412971652

[12] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[13] D Boxue Z Miaomiao J Huilan L Hongjing F Mingli andH Shuai ldquoGrowth characteristics of electrical tree in epoxyresin under low temperaturerdquo High Voltage Engineeringvol 42 pp 478ndash484 2016

[14] X Chen Y Xu X Cao and SM Gubanski ldquoElectrical treeingbehavior at high temperature in XLPE cable insulationsamplesrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 22 no 5 pp 2841ndash2851 2015

[15] G Chen and C +am ldquoElectrical treeing characteristics inXLPE power cable insulation in frequency range between 20and 500 Hzrdquo IEEE Transactions on Dielectrics and ElectricalInsulation vol 16 no 1 pp 179ndash188 2009

[16] M Liu Y Liu Y Li P Zheng and H Rui ldquoGrowth andpartial discharge characteristics of electrical tree in XLPEunder AC-DC composite voltagerdquo IEEE Transactions onDielectrics and Electrical Insulation vol 24 no 4 pp 2282ndash2290 2009

[17] J Champion and S Dodd ldquo+e effect of voltage and materialage on the electrical tree growth and breakdown character-istics of epoxy resinsrdquo Journal of Physics D Applied Physicsvol 28 no 2 pp 398ndash407 1995

[18] H Uehara and K Kudo ldquo+e 3-D fractal analysis of electricaltrees using a serial sectioning method and a CT methodrdquo inProceedings of 1998 IEEE 6th International Conference onConduction and Breakdown in Solid Dielectrics pp 309ndash312Vasteras Sweden June 1998

[19] S Kobayashi S Maruyama H Kawai and K Kudo ldquoEsti-mation of 3D fractal dimension of real electrical tree pat-ternsrdquo in Proceedings of 4th International Conference onProperties and Applications of Dielectric Materials pp 359ndash362 University of Queensland Brisbane Australia July 1994

[20] X Chen Y Xu X Cao S J Dodd and L A Dissado ldquoEffectof tree channel conductivity on electrical tree shape andbreakdown in XLPE cable insulation samplesrdquo IEEE Trans-actions on Dielectrics and Electrical Insulation vol 18 no 3pp 847ndash860 2011

45

40

35

30

25

20

15

10

5

0

12 14 16 18Fractal dimension

Gro

wth

tim

e (m

in)

Growth timeFitting curve

Figure 14 Fractal dimension vs growth time

10 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom