Science Academy

Publisher

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE)

Vol. 1, No. 2, June 2011

ISSN: 2046-5149

Copyright © Science Academy Publisher, United Kingdom

www.sciacademypublisher.com

Reliability Evaluation of High Voltage Insulation Using Weibull

Distribution

Sunitha Anup, A S. Deshpande, A N. Cheeran, and H. A. Mangalvedekar

Electrical Department, VJTI, Mumbai, India

Email: anup.sunitha@gmail.com , asdamolsd@gmail.com , ancheeran@vjti.org.in , ham.vjti@gmail.com

Abstract – Partial Discharge (PD) is the phenomenon observed in High Voltage (HV) Insulation Systems. During the event

of a Partial Discharge, there are significant statistical variability in properties such as partial discharge pulse charge

amplitude, pulse shape and time of occurrence. Due to the random nature of pulse charge amplitude and occurrence time that

describe the process of Partial Discharge, these can be described in terms of probability distributions. PD from different

sources (i.e., corona discharges, as well as surface and voids discharges) has different PD pulse height distribution (PDHD)

patterns. Modeling of these patterns can be performed using Weibull distribution. It is shown that Partial Discharge Height

Distributions fit the two-parameter Weibull Function when Partial Discharges having the same origin occur, while deviations

from the Weibull plot are detected when various PD sources occur simultaneously. In the presence of two simultaneous PD

phenomena, the tested specimen is ideally divided into two elements, each affected by a single type of PD source. This

problem is approached by additive mixing of Weibull Distribution function. The different approaches used for parameter

estimation of Weibull Distribution are Least Square Regression and Maximum Likelihood Estimation (MLE). The empirical

cumulative probability density functions are estimated using Mean rank, Median rank, Bernard‟s rank and Hazard plotting.

Comparison of these methods in estimation of parameters and in model validation is shown in this paper. The data of pulse

charge amplitude is corresponding to the failure data of insulation of the system. The characteristics of reliability function

are plotted with respect to pulse charge amplitude. The pulse charge amplitude appears to be inversely proportional to the

reliability function. Therefore, it is desirable to have minimum pulse charge amplitude so as to achieve better reliability

characteristics.

Keywords – Partial Discharge Height Distribution, Weibull Distribution, Parameter Estimation, Reliability.

1. Introduction

Partial Discharge is an electrical discharge that occurs in

gaseous inclusions within electrical insulation or over

contaminated or damp sections of the surface of such

insulations. PD can cause solid insulation to age or

deteriorate [1]. PD signals are of different nature depending

on the type and source of the defect. Analysis of PD enables

identification of source and type, leading to the assessment of

deterioration rate and the reliability of high voltage insulation

systems. PD phenomena being stochastic in nature [2],

statistical inferences become inevitable. Since PD

measurement generates enormous amount of data, usage of

various analysis methods to organize data within manageable

proportions is imperative to derive the desired inferences on

PD source/type identification. PDHD is the three dimensional

pattern, with the three parameters being φ-q-n i.e. phase

angle-charge amplitude-pulse repetition rate. PDHD data

resulting from a single and multiple PD sources are often

characterized using the Weibull distribution [3]. In this paper,

the parameters of Weibull function have been estimated on

two sets of PDHD data obtained from measurement of

rotating machines. The results of PDHD modelling using

Weibull distribution for different approaches like Mean rank,

Median rank, Bernard‟s rank, Hazard Function and MLE is

compared. Also, the quality of fitting of the standard Weibull

model has been confirmed by the comparison

with probability density function (pdf) plots of each method.

The pdf is the function that describes the relative likelihood

for the random variable to occur at a given point and is given

by the integral of the variable‟s density over the region of

study[4]. Quality of fitting of two parameter Weibull function

can be also performed with the help of estimated parameters

using Kolmogorov-Smirnov (K-S) test [5]. In reliability

analysis, a probability distribution is the key element which

enables the appropriate data handling and manipulation.

Among the different distributions, Weibull distribution aptly

represents various physical phenomena like partial discharge

[6]. Analysis of PD enables identification of source and type,

leading to the assessment of deterioration rate and the

reliability of high voltage insulation systems. The reliability

of the system being a crucial factor in the engineering design,

the asset management tasks with respect to reliability and

maintainability has to be coordinated throughout its life

cycle. This coordination demands better understanding and

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE) 56

utilization of the partial discharge measurements of various

insulation specimens in the system. The reliability of a

product depends on technical decisions made during the

design and manufacturing of the product. Thus, Weibull

models assist in decision making process in the asset

management of insulation system design. Similarly, attention

is paid to the pulse charge height distribution of the sample.

As brought out above, since the pulse charge height appears

to be inversely proportional to the reliability function, it is

desirable to have minimum pulse charge height so as to

achieve better reliability characteristics.

The subject has been detailed in the succeeding sections

as follows. Section 2 explains PDHD modeling using

Weibull distribution with emphasis on the methodology for

standard Weibull model and mixed Weibull model.

Parameter estimation using graphical method and analytical

method are explained for the standard Weibull model and

mixed Weibull model. Section 3 gives the result obtained

when the proposed statistical procedure has been applied to

standard Weibull model and additive-mixed Weibull model.

Reliability characteristics are plotted with respect to the pulse

charge amplitude for both standard Weibull model and

additive-mixed Weibull model.

2. PDHD Modeling using Weibull Distribution

Assuming that different PD sources can produce different

PD-pulse height distributions, the problem of PD separation

and identification can be approached resorting to the

stochastic analysis of PDHD Shape [7]. Partial Discharges

taking place in the tested insulation system are detected at the

measuring impedance Zm and fed into signal conditioning

circuit having correct bandwidth and gain [3].At this stage

pulses are stretched (in shape and time) to match the

instrumentation requirements for the input signals. The

analog-to-digital converter(A/D-C) turns the peak magnitude

of PD electric signal (mV or mA) into discrete digital form

Si, where i ranges from 1 to the maximum number of

quantization levels or channels available on the measurement

system. Hence the interpretation of PD measurements can be

approached by means of the shape analysis of partial

discharge height distribution (PDHD). The PDHD is obtained

by plotting the ratio of ni to the total number of pulses NT as a

function of their amplitude Si. The results of the PD

measurements is stored as an array of integers [ni, Si] where

ni is the number of discharge signals in the channel i that

constitutes the histogram to be processed. Due to the high

degree of randomness in the Partial Discharge data, it can be

very well modeled by a suitable probability model such as a

distribution function so that the data can be viewed as

observed outcomes (values) of random variables from the

distribution. The Black-Box approach to modeling [9] is

performed with the help of data, which is preceded by

determination of type of mathematical formulation

appropriate to model the data. The statistics inference of the

black box modeling is executed in such a way that the

available data makes statements about the probability model,

either in terms of probability distribution itself or in terms of

its parameters or some other characteristics. Among the many

standard probability distribution functions, Weibull

distribution is significant to model the data sets of widely

differing characteristics. PDHD histograms, that is, number

of pulses, ni, as a function of their pulse charge amplitude, qi,

derived from 3D pattern, were processed according to two

different stochastic models, that is: the two-parameter

standard Weibull model and the five-parameter additive

mixed Weibull function. From the studies [8], it is clear that

the data fits a two-parameter Weibull function when partial

discharges having the same nature i.e. generated by the same

type of defect, occur while an additive mixed-Weibull

distribution holds when partial discharge phenomena of two

different natures are superimposed.

2.1. Methodology for Standard Weibull Models

In a standard Weibull probability distribution function is

defined as

(

) (1)

where β>0 and α>0 are the parameters referred to as shape

and scale parameters of the distribution. The strength of the

Weibull model lies in its flexible shape as a model for many

different kinds of data. The parameter β plays the role in

determining, how the Weibull will appear. The methodology

for the standard Weibull model can be explained as follows.

2.1.1. Collecting the Data

Figure 1 shows the components of PD Analyzer [10] are

Pulse Height detector, Phase Detector and Interface Unit to

PC. The Pulse Height Detector identifies the PD Pulse,

sample and digitize the peak of the pulse(height). The phase

detector gives the phase of detected pulse with the zero

crossing of the applied voltage as the reference. The interface

unit synchronizes the pulse height and the phase data transfer

to the PC where the data is stored as PD Record.

Figure 1. Schematic of PD Analyzer [10].

2.1.2. Analysis of Data

The analysis of data assists in determining whether a

particular model is appropriate or not to model a given data

set. The basic quantities of PD Analysis are the PD Pulse

Magnitude and Phase Data. In this paper, statistical approach

is used for the probabilistic model for PD Analysis and

distribution of charge magnitude and phase position is used

for representation of Analysis.

2.1.3. Model Selection

This step involves choosing an appropriate model

formulation i.e., distribution function. Due to the significant

variability of the data set (q1, q2….qn) it is viewed as random

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE) 57

variable (Q1,Q2,…..Qn), each Q is modelled by the

Probability Distribution function

F (q, θ) = P(Q ≤ q) (2)

where θ denotes the set of parameters for the distribution.

The reliability function of standard Weibull model is given

by

(

)

(3)

Weibull Distribution can be used to model failures due to

fatigue extremely well. The distribution function is given as

(

)

(4)

The parameters of the distribution are given by set θ = {α, β}

with α>0, β>0.

2.1.4. Parameter Estimation

The parameters in the Standard Weibull model i.e. 2-

parameter Weibull Distribution viz., shape (β) and scale (α)

parameters are estimated graphically and analytically. In this

paper, the graphical methods used are Least Square

Regression using the Rank Regression Techniques and

Hazard Plotting. The analytical method used is Maximum

likelihood estimation.

a) Graphical Methods

The data q (1) ≤q (2) ≤……… q (n) is reordered and the

empirical distribution F^q (i) is computed using the ranks. The

three different approaches are Mean rank, Median rank and

Bernard‟s rank. Hazard function is also a graphical method of

standard Weibull model parameter estimation. The ranking

method is exclusively associated with how to approximate

the probability associated with a particular observation. The

philosophy of rank methods is such that, it estimates the

probability of observing an event non-parametrically. In other

words, it is the probability of observing the ith

event out of N. This implies that, there is an associated probability without

any model assumption. However to interpolate and

sometimes extrapolate, it is essential to perform model

fitting. Using regression, this distribution is fitted to these

points. Weibull Probability Plot (WPP) verify the linearity of

a specific distribution. Figure 2 shows the linear regression of

cumulative distribution of WPP [11].

Figure 2: Linear regression of cumulative distribution in a WPP[11]

Thus, distribution line (cumulative distribution function)

is a straight line on a WPP. If the data follows the standard

Weibull distribution, then that data is linear on that

distribution‟s WPP.

i) Mean Rank Approach

A method of handling data which has the same observed

frequency occurring at two or more consecutive ranks; it

consists of assigning the average of the ranks as the rank for

the common frequency. The empirical distribution function is

given as

(5)

ii) Median Rank Approach

Median ranks are used to obtain an estimate of the

unreliability, F(q(i)), for each failure. The cumulative

binomial gives you the probability of occurrence of the ith

observation at a particular confidence level. The choice of the

confidence level for probability plotting is usually 50% CL.

The empirical distribution function is given as

(6)

iii) Bernard’s rank approach

For a sample size n, the ith

rank, as obtained from the

Bernard‟s Approximation is given as

(7)

iv) Hazard Plot:

Failure Rate/Hazard Function of Weibull distribution is given

as

(8)

Cumulative Hazard Function is

∫

(9)

The hazard function h (q) is estimated from the inverse of

the reverse rank of the ordered failures; and the cumulative

hazard function, H (q), is the cumulative of the values of h(q).

As seen in Figure 2, WPP is plotted for each empirical

distribution F^ q(i), and the best straight line fit using

regression is obtained. The slope of the line yields the shape

parameter β^, the estimate of β. From the y-intercept of the

fitted line, the scale parameter α^ the estimate of α is

obtained.

b) Analytical Methods

Maximum likelihood estimation (MLE) is a

popular analytical method used for fitting a statistical

model to data, and providing estimates for the model's

parameters. Maximum likelihood estimates depend on the

type of data available. Maximum likelihood estimation seeks

the parameter values that are most likely to have produced

the observed distribution The likelihood function is given by

α β ∏ β

β

αβ

β

αβ ] (10)

The maximum-likelihood estimates are obtained by

solving the equations resulting from setting the two partial

derivatives of L(α, β) to zero. The maximum likelihood

estimate of {α, β}(the unknown parameter in the model) is

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE) 58

that value that maximizes the log-likelihood, given the data.

The bilateral confidence limits [5] at probability 95% of the

mean values of the parameters were estimated according to

the expression

√ (11)

where M is the arithmetic mean of the parameter values

obtained from a series of N observations, s is the estimation

of the standard deviation of this series and t is the Student

variable (t=1.96).

2.1.5. Model Validation

Calculation of shape and scale parameters are proceeded

by control of the quality of fitting of the two-parameter

Weibull function .The good quality of fitting of the two

parameter Weibull function is confirmed [12] by the

probability density graphs which are relevant to the

respective data. The probability density function, f(x),

describes the behavior of the random variable „q‟ [6]. Figure

3 shows the bell shaped Probability Density curve [4]. The

area below the probability density function to the left of a

given value, q, is equal to the probability of the random

variable represented on the x-axis being less than the given

value q.

Figure 3. .Bell shaped Probability density curve following the data

histogram[4].

2.2. Methodology for mixed Weibull Models

Deviation from the standard two parameter Weibull

model is detected in the presence of simultaneous PD

phenomena, which superimposes each pattern with the

previous one [8,12]. This scenario prompts the investigation

of mixed distributions, based on the Weibull function, which

are able to fit the recorded PDH distribution. In the presence

of two simultaneous PD phenomena, it can be assumed that

the tested specimen is ideally divided into two elements, each

affected by different PD phenomena. The five parameter

Weibull function permits the separation of the two sub

distributions relevant to each PD source, so that each source

can be described by its characteristic statistics [13].

2.2.1. Model structure of mixed Weibull model

Each sub population of the mixed model are standard two

parameter Weibull distribution given by (2). The density

function is given by

g (q) = Σpi fi (q) (12)

where fi(q) is the density function associated with F(i)

When there are two subpopulations, the above mixture model

is a two-fold mixture model with cumulative distribution can

be given as

G(q)= pF1(q)+(1-p)F2(q) (13)

The model is characterized by five parameters –the shape

and scale parameters of two subpopulations and the mixing

parameter p(0<p<1).The density function of the two –fold

mixture model is given as

g (q) =pf1 (q) + (1-p) f2 (q) (14)

The reliability function of two-fold mixed Weibull

distribution is defined by

R(q) = pR1(q) + (1− p)R2 (q) (15)

2.3. Model Analysis

For small q (very close to zero) G (q) can be approximated as

G (q) ~cF1 (q) (16)

where c= Σ p (

)

β

For large q, G(q) can be approximated by

G(q) = 1- p1[1- F1(q)] (17)

Similarly the density function can be approximated as

g (q)~ cf1(q) for small q and by g (q)~ p1f1(q) for large q.

This implies that density function is increasing for small q if

β1 >1. For a mixed Weibull model, the WPP plot is smooth

curve in the x-y plane. Figure 4 gives the WPP plot of

mixture model [9].The shape of the curve is significantly

influenced by the two shape parameters of the

subpopulations.

Figure 4. WPP plot of Weibull mixture model (n=2,α1=2,α2=10,β1=

β2=2,p=0.4) [9]

2.3.1. Parameter Estimation

For the two-fold mixed Weibull distribution, there are

totally five unknown parameters ( p , α1, α2, β1, β2). Both

graphical and analytical approaches can been used for the

parameter estimation [15].

Graphical Method: Jensen Peterson method

In the graphical method, the estimates are obtained from

plotting the data; it is popular due to its simplicity and

visibility [14] .A shown in fig 5,the sample data is plotted on

the WPP. The cumulative distribution function is empirically

estimated using Bernard‟s rank. The plot follows a smooth

curve. The coordinates (x1, y1) with smallest slope is

obtained. The ordinate value corresponds to ln(-ln(1-F))

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE) 59

=y1.The value of F is equivalent to p, the mixing weight of

the sub population. Thereafter, α1 and α2 are obtained from

the abscissa which corresponds to (0.632 p) and (p+0.632p)

level horizontally. The slopes of the tangent lines at the two

ends of the cumulative distribution graph estimates the shape

parameters β1and β2. The approximation is valid only when

the scale parameters differ significantly, and the shape

parameter for subpopulation 1 is bigger than that for

subpopulation 2. These are the basic features of the graphical

method of parameter estimation of a 2-Weibull mixture,

when two subpopulations are well separated.

Figure 5. WPP plot of a 2-Weibull Mixture, estimating the Weibull

parameters graphically.[14]

2.3.2. Modeling the Data Set

The WPP plot serves as a good starting point to determine

if a Weibull mixture model is appropriate to model a given

data set. If a smooth fit to the plotted data has a shape similar

to that shown in Figure 4, one can tentatively assume that the

two-fold Weibull mixture model can be an appropriate

model.

3. Results and Discussion The proposed statistical procedure has been applied to the

shape analysis of PDHD obtained from measurements

performed on rotating machine‟s stator bars.

3.1. Two-Parameter Weibull Function

Using the method of standard Weibull modelling

mentioned in section 2.1.4, cumulative function F^ has been

estimated. This F^ has been used to get the following PDHD

plots (using ln(-ln(1-F)) versus. ln(q) for two types of

discharges i.e. surface discharge and delamination. Figure 6

shows this PDHD plot for different approaches viz. Mean

rank, Median rank, Bernard‟s rank, Hazard plotting and

MLE.

The linearity observed in these plots shows that the given

data fits fairly well to the two parameter approximation of

Weibull function which corresponds to single type of

discharge. This implies that empirical calculation of

cumulative distribution function using rank regression

approaches are matching with the analytical estimation using

MLE, though the MLE curve stands out from the rank

regression WPPs. Table 1 shows the comparison of these

approaches for the parameter estimation of the standard

model Weibull distribution. The scale parameter α follows

the unit of pulse charge amplitude, while the shape parameter

β is a dimensionless quantity. For the delamination discharge

in the sample, numerical value of β is a positive quantity that

lies between 1 and 2 (1<β<2).

Figure 6. PDHD plotted in Weibull plot for delamination at 3.78kV

Table 1. Standard model Weibull distribution parameters

of delamination discharge at 3.78 kV

Method α(μC) β

Mean Rank 37.8 1.269

Median Rank 37.7 1.296

Bernard‟s

Rank

37.7 1.284

Hazard Plot 37.68 1.288

MLE 35.7 1.635

Figure 7. Comparison of probability density functions of each mentioned

methods with the pulse rate Vs. charge curve.

As explained in section 2.1.5, the standard Weibull model

validation of the considered PDHD data is performed

graphically using the probability density function graphs.

Pulse Charge amplitude is the random variable and it is seen

that the pdf plot follows the pulse rate vs charge curve which

implies that the probability of occurrence of discharge (pdf)

is a bell shaped curve with a single mode and it has the same

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE) 60

behaviour of the pulse rate variation with pulse charge

amplitude.

All the rank regression pdfs are overlapping, while the pdf

of MLE stands out. This implies that maximum likelihood

parameter estimation determines the distribution parameters

that maximize the probability (likelihood) of the sample data.

From a statistical point of view, the method of maximum

likelihood is considered to be more robust and yields

estimators with good statistical properties.

3.2. Reliability function Characteristics

The most frequently used function in life data analysis

and reliability engineering is the reliability function or the

survival function. This function gives the probability of an

item operating for a certain amount of time without failure

[5]. It is dependent on the distribution parameters and the

failure data. Once the Weibull distribution (with a particular

set of parameters) has been fit to the data, computation of

cumulative distributive function (F(q) given in equation (4))

is possible. The reliability function is also sometimes referred

to as the survivorship or survival function (since it describes

the probability of not failing or of surviving until a certain

time q. Reliability theory and reliability engineering make

extensive use of Weibull distribution. The reliability function

is expressed in terms of the random variable, „q‟(pulse charge

height). The distribution parameters enable to estimate the

reliability function (equation 3) and it is calculated as

compliment of cumulative distribution function. Since the

result in Figure 6 shows that linearity of MLE is more than

that in regression approaches, the parameters of MLE is used

in estimating the reliability characteristics using equation 3.

Figure 8 shows the reliability plot of 2 parameter model

and Figure 11 shows the reliability plot of mixed Weibull

model. Reliability function is a decreasing curve, which

implies that as the pulse charge amplitude increases, the

survival or the reliability of the insulation is decreased. This

implies that the pulse charge amplitude is a vital quantity to

be monitored in order to maintain the reliability Table 2 and

table 4 shows the pulse charge amplitudes at various critical

values of reliability percentages for standard Weibull model

and mixed model respectively. Various percentiles of

reliability function at 1%, 50%, 90% and 99 % can be

estimated and corresponding charges can be obtained. There

has been drastic decrease in reliability from 90% to 50%

when pulse charge increased from 0.09µC to 0.2843µC. The

reliability function of a surface discharge at 5.75kV is also

plotted and it is observed that the increase of pulse charge

amplitude has resulted in the dip of reliability function. The

reliability characteristics at 3.78kV have steeper descent than

that at 5.75kV. These estimates are particularly valuable for

determining the percentage of insulation that can be expected

to have failed at particular points of time corresponding to

respective pulse charge amplitude.

3.3. 5-parameter Weibull Function

For PDHD plot in case of simultaneous PD phenomena,

where one type of discharge superimposes with the other, two

parameter Weibull function shows a non linear behaviour.

Each individual distributions(F1 and F2 corresponding to

slot and end winding) follows standard Weibull model; hence

each has two distribution parameters(one shape and one scale

parameter). The mixing proportion parameter(p) along with

two pairs of shape and scale parameters corresponds to five

parameters in the mixed Weibull model. Hence the number of

parameters are increased and the estimation is complicated.

Hence Graphical method is adopted for simplicity of

estimation of parameters. The additively mixed Weibull

mixture modeling is performed as explained in section 2.2.3.

Fig 9 shows the PDHD plotted in Weibull graph of mixture

discharges such as end winding and slot discharges at 6 kV.

As seen, F represents the Weibull graph of mixture

dishcharges. It has a deviation with some vestiges of

linearity. F1 and F2 represent the standard Weibull graph

corresponding to individual discharge (slot winding and end

winding discharges). It is seen that F lies between the F1 and

F2.

Figure 8. Plot of reliability functions for delamination at 3.78 kV and for

surface discharges at 5.75 kV.

Table 2. Critical values of reliability function

of Weibull distribution at 3.78 kV

Critical values of

Reliability

α(μC)

R (0.1) 0.5793

R (0.5) 0.2843

R (0.9) 0.09

R (0.99) 0.0018

Figure 9. PDHD plotted in Weibull graph for mixture discharges.

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE) 61

Table 3. Weibull Parameter estimation for mixture discharges at 6 kV

Method p α1(µC) β1 α2(µC) β2

Graphical

method

0.607 16.39 0.4714 42.038 2.1761

Standard Weibull

23.52 1.6359

Table 3 gives the comparison of distribution parameters

of mixture discharges with standard Weibull modeling of the

same data. As seen, the shape parameter of standard Weibull

estimated using MLE, for slot and end winding discharge (α

=23.52µC) lies in between the scale parameters of mixed

model(α1=16.39µC and α2= 42.038 µC). Similarly shape

parameter of standard Weibull for slot and end winding

discharge (β=1.6359). The mixing parameter value (p=0.607)

corresponds to the fact that there is 60.7% of one of the

discharges(slot or end winding dishcharge) and 39.3 % of the

other.Once the parameters are estimated, the probability

density function is plotted with pulse charge height as

random variable using the equation 14.

Figure 10 shows the model validation using the

probability density function graph. It follows the same shape

as pulse rate data. This implies that the probability of

occurrence of discharge (pdf) is a bell shaped curve with a

single mode and it has the same behaviour of the pulse rate

variation with pulse charge amplitude.

Figure 10. Comparison of probability density functions of mixed Weibull

function with the pulse rate Vs. charge amplitude curve.

Figure 11 shows the reliability function characteristics of

the mixture discharge and the standard Weibull model for

end winding and slot discharges at 6 kV.

Figure 11 Comparison of reliability plot of a mixture model with standard

Weibull model for slot and mixture discharges at 6kV.

As seen, both the curves are decreasing with increase in

pulse charge height. If the data is processed using the

standard Weibull model, the parameters can be estimated as

explained in section 2.1.4 (one shape parameter and one scale

parameter). From the fig 11, it is seen that up to a certain

pulse charge amplitude, the reliability characteristics of

standard Weibull model is above the characteristics of

mixture model, and beyond that reliability characteristics of

mixture model is above the standard model. This implies that

at higher pulse charge amplitudes, for more precision in the

analysis it is advisable to follow the additive–mixed Weibull

model, in case of simultaneous occurrence of partial

discharges. Thus, mixture model ensures better verification

than the standard model though standard Weibull modelling

is also possible with the same data.

Table 4 (a)

Reliability (Using

Mixture Model)

α(μC)

R (0.5) 21.92

R (0.9) 0.353

R (0.99) 0.17

Table 4 (b)

Reliability (Using

Mixture Model)

α(μC)

R (0.5) 18.75

R (0.9) 5.8

R (0.99) 0.353

Table 4(a) and 4(b) gives the comparison of critical

values of reliability of a standard Weibull model and mixture

model for same data of slot winding and end winding

discharges at 6 kV.

4. Conclusion

PD measurement is based on measurement of charge

transfer taking place at the discharge site. Thus, the measured

charge is proportional to the PD event. Study of statistical

behaviour of PD phenomena by considering pulse charge

amplitude as a random variable is presented in this paper.

This charge is responsible for degradation of the insulating

material in high voltage systems. Thus data of pulse charge

amplitude is corresponding to the failure data of insulation of

the system. Therefore failure analysis modeling is essential.

Failure data can be best modeled using Weibull Distribution.

Weibull distribution parameters are used to proceed towards

the reliability analysis. Rank regression (Mean rank, Median

rank, and Bernard‟s rank), hazard plotting and MLE method

are used for parameter estimation of two parameter Weibull

function while graphical method is used in case of five

parameter Weibull function. The PDHD plotted in the

Weibull graph (WPP) follows the linear relation in case of 2-

parameter standard Weibull model, while there is deviation in

the additively mixed Weibull model with some vestiges of

linearity. It was observed that the probability density graphs

of the Rank Regression method and MLE method matches

the experimental plot, with MLE having a better closeness.

International Journal of Research and Reviews in Electrical and Computer Engineering (IJRRECE) 62

The WPP of the additively mixed Weibull model is compared

with the standard 2-parameter Weibull model and it is seen

that the non-linear Weibull curve of mixed model lies within

the linear curves of standard Weibull model. This implies that

partial discharge analysis of the given data can be proceeded

to do the standard Weibull model analysis in case of single

discharge while in case of simultaneous discharges, PD

analysis of additively mixed Weibull model gives more

accurate result. The estimated parameters of the Weibull

distribution give the overview of the discharges. Evaluation

of insulation can be performed using Reliability function by

considering PD pulse charge height (q) as a random variable.

As the pulse amplitude increases, the reliability follows a

decreasing path. Thus the measured charge, which is

proportional to the PD event, determines the

reliability/survival of the life of insulation. The pulse charge

amplitude can be taken as an index to evaluate the percentiles

of reliability. The work can be extended to find the life

expectancy characteristics of an insulation taking into

account the rate of change of applied voltage and behavior /

physical parameters which are obtained from the pulse

sequence analysis.

References

[1] F, H, Krueger, Partial Discharge Detection in HV equipment, pp, 19-

21, London Butterworth, 1989.

[2] R. J. Van Brunt, “Stochastic Properties of Partial-discharge Phenomena”, IEEE Transactions on Electrical Insulation Vol. 28 No.

5, pg.902-948, Oct. 1991.

[3] M. Cacciari, A. Contin, G. Rabach and G. C. Montanari, „„An Approach to Partial Discharge Investigation by Height Distribution

Analysis‟‟, IEE Proc.: Sci., Meas. Technol., Vol. 142, pp. 102-108.

[4] Athanasious Papoulis, S. Unnikrishna Pillai,” Probability, Random variable and Stochastic Processes, Tata-McGraw Hill 2002.

[5] B.S.Dhillon, “Design reliability, Fundamentals and Applications”,

CRC Press, 1999. [6] L. A. Dissado, J. C. Fothergill and S. V. Wolfe, R. M. Hill,” Weibull

statistics in dielectric breakdown; theoretical basis, applications and

implications”, IEEE Transactions in Electrical Insulation Vol .EI-19, No. 3, June 1984.

[7] A. Contin, G. Contessotto, G. C. Montanari, M. Cacciari, “Comparing

Different Stochastic Models for the Identification and Separation Of concurrent partial discharge phenomena,” IEEE conf. Dielectric

Materials, Measurements and Applications, No. 473, pp.374-379,

2000. [8] A.Contin, G.C.Montanari, C. Ferraro, “PD source recognition by

Weibull processing of Pulse Height Distributions”, IEEE transactions

on dielectrics and electrical insulation, Vol 7 No. 1, Feb 2000. [9] D. N. Prabhakar Murthy, Min Xie, Renyan Jiang-“Weibull Models;” A

John Wiley & Sons, INC., Publication, 2004.

[10] S Senthil Kumar,” Enhancing Partial Discharge Data with Extended Resolution Analyzer,” XI International Symposium on High Voltage

Engineering, 1999, London. [11] Robert B Abernethy, „The New Weibull Handbook‟, Reliability and

Statistical Analysis for Predicting Life, Safety, Failures, and Test

Substantiation, ISBN 0965306224, 9780965306225, 2005. [12] M. Cacciari, A.Contin, G.Rabach, G.C.Montanari.” Use of Mixed-

Weibull Distributions for the Identification of Partial Discharges

Phenomena”, IEEE Trans. On D.E.I, Vol.2, N.4, August 1995. [13] A. Contin, E. Gulski, M. Cacciari, G. C. Montanari, “Applications of

the Weibull Function to Partial Discharge Data Coming from Different

Sources Typologies”, IEEE-CEIDP 95 Annual Report, pp335-338, 1995.

[14] Siyuan Jiang, Dmitri Kececioglu,” Graphical Representation of Two

Mixed-Weibull Distributions‟IEEE Transactions on reliability, vol. 41, no. 2, 1992 June .

[15] Dan Ling, Hong-Zhong Huang, Yu Liu,” A Method for Parameter

Estimation of Mixed Weibull Distribution” IEEE Trans on reliability 1-4244-2509-9/09,2009.

Sunitha Anup received B Tech degree from National

Institute of Technology(NIT), Calicut, Kerala, India in 2002 and currently pursuing

M Tech degree in Power Systems from Veermata

Jeejabhai Technological Institute (VJTI), Mumbai, India. Her fields of interest are high voltage

engineering and partial discharge phenomenon.

Amol Shripad Deshpande received the B.E. degree in 2006 and the M. Tech degree in 2008 from Dr.

Babasaheb Ambedkar University, Maharashtra, India

and now pursuing his Doctoral Programme at VJTI, Mumbai, India. He is currently working in the area of

Partial Discharge.

Dr. Alice Cheeran has completed B. E in Electrical

Engineering in 1984 from Kerala University. She joined

V J T I, Mumbai as a lecturer in 1987. She completed Masters in Electrical Engg with specialization in

Control Systems and in Electronics respectively in 1994

and 1996 from Mumbai University. Further she did Ph. D. from IIT Bombay in 2005 in the topic of “Speech

Processing for the Hearing Impaired. Currently she is

working as an Associate Professor in the Electrical Engineering department, V J T I, Mumbai.

Harivittal A. Mangalvedekar received the BE degree

in 1979, the ME degree in 1984, and the Ph.D degree in 1995 in electrical engineering from Bombay University,

India. He has been with V.J. technological Institute for

the last 22 years and is currently a Professor in the Elec. Engg. Dept. He has developed the high voltage

laboratory at VJTI and presently conducting research in the areas of power systems and high voltage.