transformer lifetime modelling based on condition monitoring data

7/30/2019 TRANSFORMER LIFETIME MODELLING BASED ON CONDITION MONITORING DATA

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International Journal of Advances in Engineering & Technology, May 2013.

IJAET ISSN: 2231-1963

613 Vol. 6, Issue 2, pp. 613-619

TRANSFORMERLIFETIME MODELLING BASED ON

CONDITION MONITORING DATA

Dan ZhouBeijing Key Laboratory of High Voltage and EMC, North China Electric Power University,

Beijing, 102206, P.R. China

ABSTRACT

Effective transformer lifetime models are of paramount importance for asset management but are hindered by

the lack of actual failure data. The paper hence proposes an approach for functional lifetime modelling ofpower transformers based on condition monitoring data. Transformers functional lifetime is proposed to be

modelled as following a two-parameter Weibull distribution whose parameters are recommended to be

estimated with the maximum likelihood estimation method. An example is presented to show how the proposed

approach is applied on a set of field collected transformer data. The derived Weibull distribution is verified with

its corresponding non-parametric estimates. The earliest and latest onsets of significant unreliability for the

transformer fleet are then obtained from the derived model so as to as general guidance for asset managers to

make replacement planning.

KEYWORDS: Condition Monitoring, Lifetime Estimation, Maximum Likelihood Estimation, Weibull

Distribution

I.INTRODUCTION

Power transformers are generally very reliable equipment with designed lifetime of 40 years or more[1]. However, as a large proportion of the transformers which are installed in the major load growthperiods, 1960s and 1970s, are approaching or have already exceeded their designed lifetime [2-3]. Aconcern is aroused on the large group of aged power transformer and their impact on the reliability of

the network [4]. Lifetime models are hence desired to gain knowledge on the future performance ofthe aged equipment. Traditional statistical lifetime modelling as the simplest means to build the modelhowever is usually hindered by the lack of actual failure data. The problem is even worsened aselectric utilities tend to proactively replace their transformers in order to guarantee the level ofoperational reliability.In the absence of actual failure data, lifetime modelling has to be approached by a thoroughunderstanding of the physical process leading to failure as well as the relevant information relating to

equipment and component conditions [5-6]. This is becoming increasingly possible as the adoption ofinformation system in most utilities enables the central collection and management of all the relevant

condition monitoring data. Attempts have also been made to convert all these information, e.g.inspection data, site and laboratory testing results, operational condition, etc., into an objective andquantitative index, referred as health index, reflecting the overall health of a transformer unit at themoment of analysis [7-13]. Databases of health index for transformer fleets are starting to becomeavailable and will soon become more comprehensive and widespread. By tracking the changes of

health indexes, degradation processes of transformer units can be analysed. Functional lifetime,defined as the time when the health condition of a transformer unit deteriorates to a state requiringreplacement can hence be modelled.The paper then proposes an approach for functional lifetime modelling of power transformers basedon condition monitoring data. It is organised in the way that the two-parameter Weibull distributionchosen to characterise the statistical property of transformers functional lifetime is presented inSection II, together with the maximum likelihood estimation method recommended to be used for


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Weibull parameter estimation. An application example is then presented in Section III to showcase theapplication of the proposed approach. Verification and practical application of the derived lifetimemodel are also discussed, based on which conclusion and future work are finally provided in SectionIV and Section V, respectively.

II.MAXIMUM LIKELIHOOD ESTIMATION FORWEIBULL DISTRIBUTION2.1. Two-Parameter Weibull Distribution

The cumulative distribution function (CDF) of the two-parameter Weibull distribution [14] is asshown in (1). Its corresponding survival function (SF), probability density function (PDF) and hazardfunction (HF) are presented in (2)-(4), respectively.

( ; , ) 1 expx

F x

(1)

( ; , ) 1 ( ; , ) expx

xS x F x

(2)

1

( ; , ) expx x

f x

(3)

1

( ; , )x

h x

(4)

where

x is the failure time, expressed as a variable; is the scale parameter;is the shape parameter.

Two-parameter Weibull distribution is chosen mainly due to its flexibility in representing differentrelationships of hazard rate versus age, that:

for = 1, the hazard rate remains constant over time, representing the flat region of theconventional bathtub curve;

for > 1, the hazard rate increases with age, representing the backend phase of the bathtub curve;

for < 1, the hazard rate decreases with age, representing early failures, often termed as infant

mortalities.The scale parameter, , represents the age by which 63.2% of the transformer units are expected to

have failed. For the extreme case where = 1, the mean lifetime of the distribution equals to the valueof.

2.2. Maximum Likelihood Estimation Method

The method of maximum likelihood estimation (MLE) [15] is chosen over the other parameterestimation methods mainly due to its superior versatility and ability to deal with different data types.For a fixed set of lifetime data assumed as following a specific statistical distribution, MLE finds

parameter values of the distribution as the ones maximize the likelihood function. The likelihoodfunction is a function of the parameters given all the observation data. It is assumed as equal to the

probability of observing all the samples given the parameters, as shown in (5).

1 2 1 2

1 1

( , | , ... , , ... ) ( ; , ) ( ; , )k N

k k k N i j

i j k

L x x x c c c M f x S c

(5)

whereN is the total number of observed samples;x1,x2xkare ages of the k units whose failures are observed;

ck+1, ck+2cNare the running times of the remaining N-K survival units whose failures are not yetobserved but known to be operating beyond the current running time;


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M is a combinatory constant representing the number of ways the sample might have been observed.As shown in (5), the two types of data considered in the present paper are failure data and survivaldata. For failure data, the information it carries is the probability that the event of failure occurring atthe time, x, which is approximately equal to the PDF of x at this time. For survival data, theinformation it provides is that the unit has already survived up to the present running time, c, which isapproximately equal to the survival function evaluated at the time. The likelihood function is henceconstructed by combining all the information each observation carries.To calculate the Weibull parameters, the PDFs and SDs as shown in (3) and (2) are inserted into (5).A technique usually applied is to work with the natural logarithm of the likelihood function ratherthan the likelihood function itself as the approach simplifies the expression of product intosuperposition. The log-likelihood function is obtained as shown in (6).

1 1 1

ln ( , ) ln (ln ln ) ( 1) lnk k N

ji

i

i i j k

cxL M k x

(6)

As a one-to-one relationship exists between a number and its corresponding logarithm, the parametersmaximize likelihood function are hence the same ones maximize their log-likelihood function.Weibull parameters are then calculated as the parameters that make their corresponding partialderivatives equal zero, as shown in (7) and (8).

1 1

ln ( , )0

k Nji

i j k

cxL k

(7)

1 1 1

ln ( , )ln ln ln 0

k k Nj ji i

i

i i j k

c cx xL kk x

(8)

By transforming (7), can be expressed as a function of, as:1

1 1

1k N

i j

i j k

x ck

(9)

And by eliminating between (7) and (8), the following equation is obtained.

1 1

1

1 1

ln ln1 1

0

k N

i i j jki j k

i k Ni

i j

i j k

x x c c

xk

x c

(10)

With (10), the value of can be iteratively solved with numerical methods, such as the Newton-Raphson method [16]. Once the value of is determined, can be obtained by taking the estimated

into (9).Up to this point, Weibull distribution can be modelled with MLE once a set of lifetime data, including

failures as well as survivors, is collected.

III.APPLICATION OF WEIBULL MODELLING TO FIELD COLLECTED DATA

An application example is presented to showcase how could information of functional lifetime for a

transformer fleet being extracted and hence further modelled with a two-parameter Weibulldistribution.

3.1. Data Collection and Pre-processing

A collection of health index data of more than 800 transformers, covering a time span of 10 years is

collected from an electric utility company. In the utility, the health index is adopted as an indication ofthe overall health condition of a transformer unit. It is classified into four discrete states to aid

decision making for effective asset management of the transformer fleet. The four states are classifiedas good condition, normal condition, poor condition and the condition requiring replacement;with the underlying assumption that the condition of a transformer unit can only change from good

state to bad state but not otherwise.


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The deterioration process of a transformer unit can hence be represented with a conceptual model asshown in Figure 1.

Age (years)

good state

normal state

poor state

state requiring

replacement

x

HealthIndex

functional lifetime

Figure 1. A conceptual model of the deterioration process of a transformer unit

As shown in Figure 1, the functional lifetime of a transformer unit, x, is defined as the time that theunit entering the state requiring replacement. Different units may deteriorate at different rate, henceresulting in a spreading range of functional lifetimes.By adopting this end-of-life criterion, functional lifetimes of transformer units can be extracted in a

systematic way. For units whose functional lifetimes are actually observed, their corresponding agesare extracted as functional failure data. For units whose functional failures are not yet observed within

the observation time their corresponding current running times are recorded as survival data.In this way, the functional failure data as well as survival data of the collected transformer dataset canbe extracted. Profiles of these extracted information are presented in Figure 2.

10 20 30 40 50

0

10

20

30

40

50

60

numberoftransformerunits

age (years)

survival data

failure data

10 20 30 40 50

0

10

20

30

40

50

60

numberoftransformerunits

age (years)

survival data

functional failure data

Figure 2. Extracted functional failure data and survival data for the collected dataset

3.2. Results and Practical Implications

Functional lifetime of the transformer fleet is then modelled as a two-parameter Weibull distributionwith the extracted functional lifetime data, as presented in Figure 2. Weibull parameters are obtainedwith MLE.

Figure 3 is a plot showing the non-parametric and the Weibull MLE estimates of the survivalprobability. The non-parametric estimates are obtained with the Kaplan-Meier (K-M) estimator [17]


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which is frequently used in survival analysis. The points in Figure 3 were plotted at each knownlifetime, i.e. functional failure time, and the point of K-M estimates.

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Age (years)

SurvivalProbability

maximum likelihood estimates

K-M estimates

Figure 3. Non-parametric estimates and MLE estimates of the survival probability

The nonparametric and the parametric estimates as shown in Figure 3 agree well with each other,which, in a way, proves the effectiveness of the MLE estimates. Table 1 presents the MLE estimatesof the scale and shape parameters of the Weibull distribution together with some useful percentilevalues. The corresponding hazard function of the derived Weibull lifetime model is presented inFigure 4.

Table 1. Estimated Parameters and Percentile Values of the Weibull Distribution.

Shape ParameterScale Parameter

(years)

2.5% Percentile

Lifetime

(years)

50% Percentile

Lifetime

(years)

97.5% Percentile

Lifetime

(years)

3.3 79 26 71 117

0 20 40 60 80 100 1200

0.02

0.04

0.06

0.08

0.1

0.12

Age (years)

HazardRate

Figure 4. Derived hazard rate versus age relationship


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As shown in Figure 4, the derived hazard rate versus age relationship successfully captures thedeterioration process of transformer units that their hazard rate increases as the age increases. Thederived relationship, combining with the current age profile of the transformer fleet, can makereplacement volume projection into the future.Percentile values as presented in Table 1 can provide some general guidance for replacement planningof the transformer fleet. The 2.5% percentile lifetime representing the age by which 2.5% of thepopulation would require replacement is determined as 26 years. This value can be considered as theearliest onset of significant unreliability of the transformer fleet. The 97.5% percentile lifetimerepresenting the age by which 97.5% of the population would require replacement is found to be 117and is hence considered as the latest onset of significant unreliability. The 50% percentile lifetimevalue then represents the age by which 50% of the population would require replacement. The value,as listed in Table 1, is found to be 71 years. This would then provide asset managers some confidenceto say that it might be reasonable to allow transformers to be operated beyond the original assumeddesign lifetime of 40 years, although more failure data are still needed for further verification.

IV.CONCLUSION

Statistical lifetime models of power transformers are recognised as important to asset managers togain knowledge on the future performance of the aged equipment. The traditional approach of lifetimemodelling, however, is usually hindered by the lack of actual failure data as transformers are normallyhighly reliable equipment. The paper therefore proposes an approach for functional lifetime modellingof power transformers based on condition monitoring data. It is demonstrated with an application

example that with the functional lifetime of a transformer unit being defined as the age when a unitenters the state requiring replacement, functional lifetime data of a transformer fleet are systematically

collected. Two-parameter Weibull distribution whose parameters estimated with the maximumlikelihood estimation method are then used to model the transformer fleets functional lifetime. The

derived model is verified with its corresponding non-parametric estimates. The derived lifetime modelgives the 50% percentile lifetime as 71 years which provides asset managers some confidence toallow transformers to be operated beyond the originally assumed design lifetime of 40 years.

V.FUTURE WORK

Functional lifetime modelling is of practical interest for electric utilities to effectively manage theirtransformer fleets. The current paper mainly focuses on the part of mathematical modelling after the

functional lifetime of a specific transformer fleet being collected. The on-going research will beextended to also discuss the effect of data collection/classification method (i.e. from diversified

condition monitoring results to the final integrated health index) on the final derived lifetime modelwhich is of great practical value to help optimize transformer asset management.

REFERENCES

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[14].Weibull Analysis (2008), International Standard IEC 61649:2008.[15].R. B. Abernethy (2006), The New Weibull Handbook, 5th ed., Florida: Robert B. Abernethy.[16].W. Nelson (1982),Applied Life Data Analysis. New York; Chichester: Wiley.

[17].J. Klein (2003), Survival analysis: techniques for censored and truncated data , 2nd ed. New York;

London: Springer.

AUTHORS

Dan Zhou was born in Hunan, China in 1986. She received BEng degree in electrical

engineering fromNorth China Electric Power University (NCEPU), China in 2007. Currentlyshe is a PhD student of NCEPU majored in electrical engineering. Her current researchinterests include transformer condition monitoring and lifetime modelling.

transformer lifetime modelling based on condition monitoring data

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