Electrical Power and Energy Systems 32 (2010) 518–523

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Modeling of breakdown voltage of White Minilex Paper in the presenceof voids under ac and dc conditions using fuzzy logic techniques

Sanjeeb Mohanty *, Saradindu GhoshDepartment of Electrical Engineering, National Institute of Technology, Rourkela 769 008, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 November 2008Received in revised form 19 June 2009Accepted 25 September 2009

Keywords:Insulation breakdownMicrodefectsModelingFuzzy logic

0142-0615/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijepes.2009.09.012

* Corresponding author. Tel.: +91 0661 2462405; faE-mail addresses: sanjeebmohanty@yahoo.com (S.

com (S. Ghosh).

Occluded gaseous cavities within the insulating materials are potential sources of electrical trees whichcan lead to continuous deterioration and breakdown of materials. To determine the suitability of use andto acquire the data for the dimensioning of electrical insulation systems breakdown voltage of insulatorsshould be determined. In this paper, fuzzy logic (FL) method is used to model breakdown voltages ofWhite Minilex Paper samples based on experimental data generated in the laboratory. Different modelsare proposed with different shapes of the membership functions for the FL under both dc and ac voltageconditions. The cavities are created artificially with different dimensions. Low values of mean absoluteerrors of the estimated breakdown voltage of the test data show the effectiveness of such models.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In industrial insulation systems, aging can be mainly related totechnological microdefects present in the bulk of insulation sincemanufacturing. The changes in microdefect numbers and sizesmakes the aging process critical and difficult to determine particu-larly as the insulation samples were both thermally and electricallyaged. Breakdown voltage tests are widely used to test for generaldegradation of insulation or insulation systems [1].

The breakdown of solid dielectrics is an event that is suddenand catastrophic. The insulation cannot withstand the service volt-age following it. It occurs in a time duration which is very muchless than a second. The breakdown can be due to various causessuch as intrinsic, electromechanical, thermal, microdischarges inthe cavities [2]. Due to the application of the voltage, while in ser-vice, the electrical stress experienced by the cavities entrappedinto the insulation initiate discharges when the stress value ex-ceeds a certain critical value. At a given voltage, these dischargesproduce a deterioration of the insulating properties in diverse waysdepending on geometrical factors and the nature of the dielectric.These eventually cause the material degradation and lead to finalbreakdown. Since, breakdown by discharges is so important indus-trially, it is worthwhile examining the factors which control thedischarges.

ll rights reserved.

x: +91 0661 2462999.Mohanty), sghosh.ee@gmail.-

In recent times, the modeling of breakdown voltage using softcomputing techniques, such as, artificial neural network (ANN) [3]and FL [4] has been gaining popularity. The advantage of using a softcomputing model is that it is highly flexible [5–10] and a model canbe improved simply by providing additional training data. More-over, this kind of model can be developed in a shorter time as wellas more accurately. In this paper, FL techniques have been proposedto model breakdown voltage for White Minilex Paper (potentiallyused solid dielectric materials in the industry) both under dc andac condition. As a diagnostic tool, FL techniques have been exploitedfor breakdown voltage estimation under artificially created air cav-ities of different sizes. Section 2 of the paper describes the experi-mental set up and the procedure for obtaining the experimentaldata on the breakdown voltage. Section 3 has described the FL mod-el proposed. Section 4 discusses on the results obtained by assum-ing six different shapes of the membership functions (MFs).

2. Experimental set up

2.1. Sample preparation

The samples are prepared from commercially available insula-tion sheets. Three different thicknesses of the insulation, 0.26,0.18 and 0.125 mm were used. Before testing, the conditioningprocedure was adopted to the test specimen in accordance withthat laid in ASTM Handbook [11]. This ensured that the surfacesof the insulating sample were cleaned and dry, since the contami-nation on the insulating specimen or absorption of moisture mayaffect the breakdown voltage.

S. Mohanty, S. Ghosh / Electrical Power and Energy Systems 32 (2010) 518–523 519

2.2. Creation of void

The voids are artificially created by means of a spacer made upof Kapton film, with a circular punched hole at the centre whosediameter values are 1.5, 2, 3, 4 and 5 mm. The thickness of the Kap-ton spacer used is of 0.125 mm. Thus, the volume of air space, thatis, the sizes of the void depends on a typical diameter of thepunched hole and thickness of the spacer.

2.3. Electrode geometry

The cylinder-plane electrode system as shown in Fig. 1 is usedfor breakdown voltage measurements. The electrodes were madeof brass. They were polished, buffed and cleaned with ethanol be-fore the start of the experiment. Further, the electrodes contactsurfaces are cleaned by ethanol between two consecutive applica-tions of voltage. Sufficient care had been taken to keep the elec-trode surfaces untouched and free from scratches, dust and otherimpurities. The insulation sample is sandwiched between the elec-trodes as shown in Fig. 1 with the help of insulating supports.

2.4. Measurement of dc breakdown voltage

The dc voltage applied to the set-up was obtained from a 40 kVac/dc Series Hipot Tester (MODEL HD 100) manufactured by Hipo-tronics, USA. The voltage is raised in steps of 1 kV and held con-stant for a period of 30 s in each level until the breakdownoccurs. The total time from the application of voltage to the instantof breakdown were noted down. Five data points were obtained fora particular sample and the mean value of the voltage is taken formodeling. All the tests were carried out in air at room temperatureand atmospheric pressure. The breakdown data obtained are thencorrected for atmospheric condition before being used for model-ing. It is observed that the samples break in the middle in each casewhich was due to maximum stress concentration in the middleand presence of cavity at the centre.

2.5. Measurement of ac breakdown voltage

In this case an ac voltage of 50 Hz is applied from the Hipot Tes-ter to the insulating sample. The voltage was raised in steps of

6

Fig. 1. Cylinder-plane electrode system used for breakdown voltage measurement.

200 V. Rest of the procedure is identical to that presented in Sec-tion 2.4 earlier. It is observed that the breakdown occurs at muchlower ac voltage than dc condition for the identical samples andcavity.

3. Proposed fuzzy logic models

The breakdown voltage of White Minilex Paper under dc and acconditions has been modeled using FL Technique. Forty-five sets ofinput–output data are used for modeling purpose for both dc andac conditions, out of which 38 sets are used for framing the rulebase and remaining seven sets for the testing purpose. The break-down voltage, V is a function of the thickness of the paper, t and thediameter of the void, d, that is V = f(t, d). The relationship betweenthe linguistic values and the actual values for t, d and V are pre-sented in Tables 1 and 2.

The set of linguistic values assigned to t, d and V are given by

Ł¼ lowðLÞ;medium lowðMLÞ;mediumðMÞ;medium highðMHÞ;highðHÞð1Þ

The membership functions (MFs) for t, d and V are lt, ld and lV,respectively. Since, t and d can have five linguistic values; the rulebase can be created with a maximum of 25 rules from the experi-mentally generated data. Also, lt, ld and lV would be having fivecomponents corresponding to each linguistic value as

lt ¼ ltL; ltML; ltM; ltMH; ltH ð2Þld ¼ ldL; ldML; ldM; ldMH; ldH ð3ÞlV ¼ lVL; lVML; lVM; lVMH; lVH ð4Þ

The Mamdani rule based inferencing (MRBI) is computationallyvery efficient and saves a lot of memory. Hence, it is a very popularmethod and has been used here to evaluate the modeled value ofthe breakdown voltage.

3.1. Breakdown voltage modeling

3.1.1. Under dc conditionCorresponding to the 38 sets, 38 ‘if then’ rules are formulated

under dc conditions. Out of these 38 rules, 22 rules have been usedto form the rule base and the rest 16 rules could not be considered.This is because these 16 rules satisfied the inconsistency propertyof the ‘if then’ rules [12]. Since seven sets of input output data havebeen used for testing purpose, the number of crisp input outputpairs are 7 in number. The 22 rules have been presented in Table 3.The seven sets of crisp input for the thickness of the paper and thediameter of the void, fire each of the 22 rules given in Table 3.

A typical clipped fuzzified MFs obtained by firing a rule is asfollows:

lVMH1 ¼minimum3ðl�t ; l�d; lVMHÞ ð5Þ

Where, l�t , l�d are the MFs corresponding to the crisp inputs for thethickness of the paper and the diameter of the void, respectively.

Similarly the other fuzzified MFs obtained by firing the rest 21rules are lVH1, lVMH2, lVL1,. . .,lVL3, lVMH4. All the 22 clipped fuzzified

Table 1Relationship between the linguistic values and the actual values for t and d.

Linguistic values t (mm) d (mm)

Low 0–0.13 1.0–3.0Medium low 0.05–0.18 1.7–3.7Medium 0.10–0.23 2.4–4.4Medium high 0.15–0.28 3.1–5.1High 0.20–0.33 3.8–5.8

Table 2Relationship between the linguistic values and the actual values for Vdc and Vac.

Linguistic values Vdc (kV) Vac (kV)

Low 17–21 1.6–2.0Medium low 19–23 1.8–2.2Medium 21–25 2.0–2.4Medium high 23–27 2.2–2.6High 26–30 2.4–2.8

Table 3Rule base under dc condition.

Thickness, t If input parameters Then output parametersDiameter of the void, d Breakdown voltage, V

MH L MLH ML MMH ML MHH L MLML M ML ML MML L LM ML LL M LM MH MHML ML MHH H MMH MH MML MH MLM H MHLM H LML H MML L MHMH H MHH MH MM L MMH M H

Table 4Rule base under ac condition.

Thickness, t If input parameters Then output parametersDiameter of the void, d Breakdown voltage, V

MH L MH ML MHMH ML MLH L MHML M ML ML MLML L MM ML MLL M MM MH MHML ML MH H MLMH MH MHML MH MM H MLML H ML H MML L MMH H MHH MH MM L MMH M MH

Table 5Comparison of the crisp (Vb1) and modeled values (Vb2) of the breakdown voltagewith generalized bell MF for White Minilex under dc condition.

t(mm)

d(mm)

Breakdown voltage Vb1

(kV)Breakdown voltage Vb2

(kV)MAE(%)

0.26 3 24 23.5000 2.05800.26 2 22 22.98220.125 3 23 22.99300.18 5 23 23.74570.18 3 24 23.50000.26 5 24 23.41810.26 2 23 22.9822

520 S. Mohanty, S. Ghosh / Electrical Power and Energy Systems 32 (2010) 518–523

MFs are aggregated to form the aggregated fuzzified MFs.Theaggregated fuzzified MFs is given by

lA1ðVÞ ¼ maximum22ðlVMH1; lVH1; lVMH2; . . . ;lVL3;lVMH4Þ ð6Þ

Eqs. (5) and (6) have been implemented in MATLAB 7.1 environ-ment by writing suitable codes for it. The defuzz function in thefuzzy toolbox was used to compute the defuzzified or the modeledvalue of the breakdown voltage Vb2 from lA1(V).

The mean absolute error (MAE) is a good performance measurefor judging the accuracy of the fuzzy system. The MAE under dcconditions expressed in percentage is given by

MAEdc ¼ ð1=7Þ �X7

s¼1

ðVb1ðsÞ � Vb2ðsÞ !�����

,Vb1ðsÞð Þ

������ 100 ð7Þ

where Vb1 is the experimental or the crisp value for the breakdownvoltage under dc condition and s in this case varies from 1 to 7.

3.1.2. Under ac conditionSimilarly under ac conditions corresponding to the 38 training

sets, 38 ‘if then’ rules are formulated. Out of these 38 rules, 22 ruleswere used to form the rule base and the rest 16 rules could not beconsidered as explained in Section 3.1. The 22 rules have been pre-sented in Table 4. The seven sets of crisp input for the thickness ofthe paper and the diameter of the void, fire each of the 22 rules gi-ven in Table 4.

The procedure followed to calculate the aggregated fuzzy MFsand the MAE under ac conditions is identical to the dc conditionsand hence Eqs. (5)–(7) can be used for this case also.

For computational efficiency, efficient use of memory, andperformance analysis needs, a uniform representation of the MFs

is required. This uniform representation can be achieved byemploying MFs with uniform shape and parametric, functional def-inition. The most popular choices for the shapes of the MFs includetriangular, trapezoidal, Gaussian, generalized bell, PI shaped. In ourwork the MFs lt, ld and lV have assumed these shapes.

4. Results and discussions

The lt, ld and lV given in Eqs. (2)–(4) have assumed six differ-ent closed shapes which are defined as given below.

(A) Triangular MFs: a triangular MFs is defined as

lx ¼ 0; x < a

¼ ðx� aÞ=ðb� aÞ; a � x � b

¼ ðc � xÞ=ðc � bÞ; b � x � c

¼ 0; x > c

ð8Þ

(B) Trapezoidal MFs: a trapezoidal MFs is defined as

lx ¼ 0; x < a

¼ ðx� aÞ=ðb� aÞ; a � x � b

¼ 1; b � x � c

¼ ðd� xÞ=ðd� cÞ; c � x � d¼ 0; x > d

ð9Þ

(C) Gaussian MFs: a Gaussian MFs is specified by two parametersc and r and is defined as

10 20 30-1

0

1

2

10 20 300

0.5

1

10 20 300

0.1

0.2

10 20 300.2

0.3

0.4

0.5

10 20 30-1

0

1

2

10 20 300.2

0.3

0.4

0.5

10 20 300

0.5

1

Breakdown Voltage(dc condition)

Aggr

egat

ed F

uzzy

MFs

Fig. 2. Aggregated fuzzy MFs for all the seven crisp inputs with MAE = 2.0580% (generalized bell MFs for t, d and V under dc conditions).

Table 6Summary of minimum value of MAE for six different shapes of MFs for t, d and V for

S. Mohanty, S. Ghosh / Electrical Power and Energy Systems 32 (2010) 518–523 521

lx ¼ e�0:5 � ððx� cÞ=ðrÞÞ2 ð10Þ

where c controls the MFs center and r controls the MFs width.(D) Generalized bell MFs: the generalized bell MFs is defined as

lx ¼ 1=ð1þ jðx� cÞ=ðaÞj2bÞ ð11Þ

where b is usually positive. a is the width of MF, b controls the slopeat crossover points and c controls the center of MF.

(E) Pi shaped curve MFs: a PI shaped curve is specified by fourparameters a, b, c and d. It is defined as

lx ¼ Smooth Transition from 0 at a to 1 at b if a < b

¼ Smooth Transition from 1 at c to 0 at d if c < dð12Þ

a and d control the feet of the curve while b and c control the shoul-der of the curve.

(F) Difference of two sigmoidal MFs: The difference of two sig-moidal functions MF is defined as

lx ¼ ð1=ð1þ e�a1ðx�c1ÞÞÞ � ð1=ð1þ e�a2ðx�c2ÞÞÞ ð13Þ

White Minilex Paper under dc condition.

Shape of MFs MAE (%)

Triangular 2.3644Trapezoidal 2.3530Gaussian 2.4180Generalized bell 2.0580PI shaped curve 2.5739Difference of two sigmoidal functions 7.4589

Table 7Comparison of the crisp (Vb3) and modeled values (Vb4) of the breakdown voltagewith trapezoidal MF for White Minilex under ac condition.

t(mm)

d(mm)

Breakdown voltage Vb3

(kV)Breakdown voltage Vb4

(kV)MAE(%)

0.26 3 2.2 2.2000 1.65780.26 2 2.3 2.20000.125 3 2.3 2.13630.18 5 2.2 2.20300.18 3 2.2 2.20000.26 5 2.2 2.20000.26 2 2.2 2.2000

4.1. Results under dc conditions

4.1.1. MAE with triangular MFsThe MAE turned out to be minimum at 2.3644% when btL (cor-

responding to ltL) = 0.06, bdL (corresponding to ldL) = 1.6 and bVL

(corresponding to lVL) = 20. The b’s stand for b in Eq. (8).

4.1.2. MAE with trapezoidal MFsA combination of btL (corresponding to ltL) = 0.04, ctL = 0.08, bdL

(corresponding to ldL) = 1.6 and cdL = 2.0 gave the minimum valueof MAE as 2.3530%. The b’s and c’s stand for b and c in Eq. (9).

4.1.3. MAE with Gaussian MFsHere, the MAE was minimum at 2.4180% when rt (correspond-

ing to t) = 0.02, rd (corresponding to d) = 0.2 and rV (correspondingto V) = 0.5. The r’s stand for r in Eq. (10).

4.1.4. MAE with Generalized bell MFsThe MAE turned out to be minimum at 2.0580% when at = 0.02,

bt (corresponding to t) = 0.75, ad = 0.2, bd (corresponding to d) = 2.0,

aV = 1.0 and bV (corresponding to V) = 1.5. Table 5 shows the valuesof Vb1, Vb2 (corresponding to MAE of 2.0580%) for White MinilexPaper under dc conditions. Fig. 2 shows the aggregated MFs plotfor all the seven crisp inputs with this value of MAE. The a’s andb’s stand for a and b in Eq. (11).

4.1.5. MAE with PI shaped MFsIn this case, the MAE was at it’s minimum at 2.5739% when btL

(corresponding to ltL) = 0.05, ctL = 0.08, bdL (corresponding toldL) = 1.7 and cdL = 2.3. The b’s and c’s stand for b and c in Eq. (12).

4.1.6. MAE with difference of two sigmoidal MFsThe MAE turned out to be minimum at 7.4589% when a1t = 300,

a2t (corresponding to t) = 300, a1d (corresponding to d) = 10,

Fig. 3. Aggregated fuzzy MFs for all the seven crisp inputs with MAE = 1.6578% (trapezoidal MFs for t, d and V under ac conditions).

Table 8Summary of minimum value of MAE for six different shapes of MFs for t, d and V forWhite Minilex Paper under ac condition.

Shape of MF MAE %

Triangular 1.6660Trapezoidal 1.6578Gaussian 1.8078Generalized bell 1.9452Pi shaped curve 1.6812Difference of two sigmoidal functions 6.6603

522 S. Mohanty, S. Ghosh / Electrical Power and Energy Systems 32 (2010) 518–523

a2d = 10, a1V = 6 and a2V = 6 (corresponding to V). The a’s stand for ain Eq. (13).

Table 6 clearly shows that the generalized bell gave the best re-sults out of all the six MFs with MAE of 2.0580%. The triangular, thetrapezoidal, the Gaussian and Pi shaped curve also gave quietencouraging results. Only the difference of two sigmoidal functionsgave poor results. Hence, the generalized bell shape of MF is thebest candidate for modeling the breakdown by discharges underdc conditions and the triangular, trapezoidal, Gaussian, Pi shapedMFs can be also considered to be very good candidates for model-ing under dc conditions.

4.2. Results under ac conditions

4.2.1. MAE with triangular MFsIn this case, the MAE turned out to be minimum at 1.6660%

when btL = 0.07, bdL = 1.4 and bVL = 1.9.

4.2.2. MAE with trapezoidal MFsThe MAE was minimum at 1.6578% when btL = 0.06, ctL = 0.11,

bdL = 1.6 and cdL = 2.0. Table 7 shows the values of Vb3, Vb4 (corre-sponding to MAE of 1.6578%) for White Minilex Paper under acconditions. Fig. 3 shows the aggregated MFs plot for all the sevencrisp inputs for this value of MAE.

4.2.3. MAE with Gaussian MFsHere we find that the MAE turned out to be minimum at

1.8038% when rt = 0.025, rd = 0.3 and rV = 0.03.

4.2.4. MAE with generalized bell MFsHere a combination of at = 0.01, bt = 0.65, ad = 0.2, bd = 2.0,

aV = 0.05 and bV = 3.0 gave minimum MAE = 1.9452%.

4.2.5. MAE with Pi shaped MFsThe MAE turned out to be minimum at 1.6812% when btL = 0.04,

ctL = 0.11, bdL = 1.5 and cdL = 2.5.

4.2.6. MAE with difference of two sigmoidal MFsThe MAE was minimum at 6.6030% when a1t = 300, a2t = 300,

a1d = 10, a2d = 10, a1V = 50 and a2V = 50.

Table 8 summarizes the minimum value of MAE obtained withsix different shapes of MF under ac conditions. The Trapezoidalshape of the MF gave the best results in terms of MAE (1.6578%)out of all the six shapes and the difference of two sigmoidal func-tions gave the worst results at 6.6603%. The triangular, Gaussian,generalized bell and Pi shaped curve gave also very good results.Hence the trapezoidal shape is the best candidate for modelingthe breakdown by discharges of White Minilex Paper under ac con-ditions and the triangular, the Gaussian, the generalized bell, the Pishaped curve can also be used for modeling under ac conditions.

Also from Tables 6 and 8, it is very clear that the MAE for all theMFs under ac conditions is less than the MAE for all the MFs underdc conditions. This is because breakdown voltage data under acconditions is very close to each other compared to the breakdownvoltage data under dc conditions.

On the whole it can be inferred that the FL modeling is a veryeffective way of modeling the breakdown by discharges of any so-lid dielectric material under both dc and ac conditions as it predictsthe breakdown voltage quiet accurately and also requires less com-putational burden. Thus, the dielectric behaviour can be analyzedat negligible computing cost.

5. Conclusion

The breakdown voltage of insulating samples of White MinilexPaper of various thicknesses with artificially created void wasmodeled using six different shapes of the MFs under both ac anddc conditions. The MAE with five out of the six used shapes ofMF turns out to be very encouraging. The results suggest theeffectiveness of FL in modeling the breakdown voltage of insulating

S. Mohanty, S. Ghosh / Electrical Power and Energy Systems 32 (2010) 518–523 523

samples. An immediate advantage of this work is that the dielectricbehaviour can be analyzed at a virtually negligible computing cost.

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