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International Journal of Advances in Engineering & Technology, May 2013.
©IJAET
ISSN: 2231-1963

TRANSFORMER LIFETIME MODELLING BASED ON
CONDITION MONITORING DATA
Dan Zhou
Beijing 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 of
power transformers based on condition monitoring data. Transformer’s 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 growth
periods, 1960s and 1970s, are approaching or have already exceeded their designed lifetime [2-3]. A
concern 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 of
the aged equipment. Traditional statistical lifetime modelling as the simplest means to build the model
however is usually hindered by the lack of actual failure data. The problem is even worsened as
electric utilities tend to proactively replace their transformers in order to guarantee the level of
operational reliability.
In the absence of actual failure data, lifetime modelling has to be approached by a thorough
understanding 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 of
information 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 and
quantitative index, referred as health index, reflecting the overall health of a transformer unit at the
moment of analysis [7-13]. Databases of health index for transformer fleets are starting to become
available 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 requiring
replacement can hence be modelled.
The paper then proposes an approach for functional lifetime modelling of power transformers based
on condition monitoring data. It is organised in the way that the two-parameter Weibull distribution
chosen to characterise the statistical property of transformer’s functional lifetime is presented in
Section II, together with the maximum likelihood estimation method recommended to be used for

613

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

International Journal of Advances in Engineering & Technology, May 2013.
©IJAET
ISSN: 2231-1963
Weibull parameter estimation. An application example is then presented in Section III to showcase the
application of the proposed approach. Verification and practical application of the derived lifetime
model are also discussed, based on which conclusion and future work are finally provided in Section
IV and Section V, respectively.

II. MAXIMUM LIKELIHOOD ESTIMATION FOR WEIBULL DISTRIBUTION
2.1. Two-Parameter Weibull Distribution
The cumulative distribution function (CDF) of the two-parameter Weibull distribution [14] is as
shown in (1). It’s corresponding survival function (SF), probability density function (PDF) and hazard
function (HF) are presented in (2)-(4), respectively.
  x
F ( x; ,  )  1  exp    
   







(1)

  x  
S ( x; ,  )  1  Fx ( x; ,  )  exp     
    

(2)


f ( x; ,  )  


  1 
  x  
  x 




exp
   
  

    
   

(3)


h( x; ,  )  


  1 
  x 



  

   

(4)

where
x is the failure time, expressed as a variable;
η is the scale parameter;
β is the shape parameter.
Two-parameter Weibull distribution is chosen mainly due to its flexibility in representing different
relationships of hazard rate versus age, that:
for β = 1, the hazard rate remains constant over time, representing the flat region of the
conventional bathtub curve;
for β > 1, the hazard rate increases with age, representing the backend phase of the bathtub curve;
for β < 1, the hazard rate decreases with age, representing early failures, often termed as ‘infant
mortalities’.
The scale parameter, η, represents the age by which 63.2% of the transformer units are expected to
have failed. For the extreme case where β = 1, the mean lifetime of the distribution equals to the value
of η.

2.2. Maximum Likelihood Estimation Method
The method of maximum likelihood estimation (MLE) [15] is chosen over the other parameter
estimation 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 likelihood
function 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).
k

L( ,  | x1 , x2 ...xk , ck 1 , ck  2 ...cN )  M   f ( xi ; ,  ) 
i 1

N

 S (c ;,  )

j  k 1

j

(5)

where
N is the total number of observed samples;
x1, x2…xk are ages of the k units whose failures are observed;
ck+1, ck+2 …cN are the running times of the remaining N-K survival units whose failures are not yet
observed but known to be operating beyond the current running time;

614

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

International Journal of Advances in Engineering & Technology, May 2013.
©IJAET
ISSN: 2231-1963
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 survival
data. For failure data, the information it carries is the probability that the event of failure occurring at
the time, x, which is approximately equal to the PDF of x at this time. For survival data, the
information it provides is that the unit has already survived up to the present running time, c, which is
approximately equal to the survival function evaluated at the time. The likelihood function is hence
constructed 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 rather
than the likelihood function itself as the approach simplifies the expression of product into
superposition. The log-likelihood function is obtained as shown in (6).


N
 cj 
x 
ln L( ,  )  ln M  k (ln    ln  )  (   1) ln xi    i     

i 1
i 1 
j  k 1   

k

k



(6)

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


N
 cj  
 ln L( ,  )
k    k  xi 

          0

   i 1    j  k 1    


k
 k  x   x
 ln L( ,  ) k
  k ln    xi    i  ln  i


 i 1   
i 1


(7)


N
 cj 
 c j 


ln


   0


 j  k 1   
   

(8)

By transforming (7), η can be expressed as a function of β, as:
N
 1  k
 
    xi   c j  
j  k 1
 k  i 1
 

1



(9)

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

1





1 k
 xi 
k i 1

N

 x  ln x   c  ln c
i 1

i

i

k

j  k 1
N

i 1

j  k 1

 xi 



j

j

0

(10)

c j

With (10), the value of β can be iteratively solved with numerical methods, such as the NewtonRaphson method [16]. Once the value of β is determined, η can be obtained by taking the estimated β
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 Weibull
distribution.

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 of
the 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 classified
as ‘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.

615

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

International Journal of Advances in Engineering & Technology, May 2013.
©IJAET
ISSN: 2231-1963
The deterioration process of a transformer unit can hence be represented with a conceptual model as
shown in Figure 1.

Health Index

good state

normal state
poor state
state requiring
replacement

functional lifetime
x

Age (years)

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 the
unit entering the ‘state requiring replacement’. Different units may deteriorate at different rate, hence
resulting 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 ages
are 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 can
be extracted. Profiles of these extracted information are presented in Figure 2.
60

survival data
failure datafailure data
functional

number of transformer units

50

40

30

20

10

0
10

20

30

40

50

age (years)

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 distribution
with the extracted functional lifetime data, as presented in Figure 2. Weibull parameters are obtained
with MLE.
Figure 3 is a plot showing the non-parametric and the Weibull MLE estimates of the survival
probability. The non-parametric estimates are obtained with the Kaplan-Meier (K-M) estimator [17]

616

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

International Journal of Advances in Engineering & Technology, May 2013.
©IJAET
ISSN: 2231-1963
which is frequently used in survival analysis. The points in Figure 3 were plotted at each known
lifetime, i.e. functional failure time, and the point of K-M estimates.
1
maximum likelihood estimates
K-M estimates

0.9
0.8

Survival Probability

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

0

20

40

60
Age (years)

80

100

120

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 estimates
of the scale and shape parameters of the Weibull distribution together with some useful percentile
values. The corresponding hazard function of the derived Weibull lifetime model is presented in
Figure 4.
Table 1. Estimated Parameters and Percentile Values of the Weibull Distribution.

Shape Parameter

Scale Parameter
(years)

3.3

79

2.5% Percentile
Lifetime
(years)
26

50% Percentile
Lifetime
(years)
71

97.5% Percentile
Lifetime
(years)
117

0.12

0.1

Hazard Rate

0.08

0.06

0.04

0.02

0

0

20

40

60
Age (years)

80

100

120

Figure 4. Derived hazard rate versus age relationship

617

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

International Journal of Advances in Engineering & Technology, May 2013.
©IJAET
ISSN: 2231-1963
As shown in Figure 4, the derived hazard rate versus age relationship successfully captures the
deterioration process of transformer units that their hazard rate increases as the age increases. The
derived relationship, combining with the current age profile of the transformer fleet, can make
replacement volume projection into the future.
Percentile values as presented in Table 1 can provide some general guidance for replacement planning
of the transformer fleet. The 2.5% percentile lifetime representing the age by which 2.5% of the
population would require replacement is determined as 26 years. This value can be considered as the
earliest onset of significant unreliability of the transformer fleet. The 97.5% percentile lifetime
representing the age by which 97.5% of the population would require replacement is found to be 117
and is hence considered as the latest onset of significant unreliability. The 50% percentile lifetime
value 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 confidence
to say that it might be reasonable to allow transformers to be operated beyond the original assumed
design 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 to
gain knowledge on the future performance of the aged equipment. The traditional approach of lifetime
modelling, however, is usually hindered by the lack of actual failure data as transformers are normally
highly reliable equipment. The paper therefore proposes an approach for functional lifetime modelling
of 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 unit
enters the state requiring replacement, functional lifetime data of a transformer fleet are systematically
collected. Two-parameter Weibull distribution whose parameters estimated with the maximum
likelihood estimation method are then used to model the transformer fleet’s functional lifetime. The
derived model is verified with its corresponding non-parametric estimates. The derived lifetime model
gives the 50% percentile lifetime as 71 years which provides asset managers some confidence to
allow 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 their
transformer 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 be
extended 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 model
which is of great practical value to help optimize transformer asset management.

REFERENCES
[1].P. Jarman, Z.D. Wang, Q. Zhong and T. Ishak (2009), “End-of-life modelling for power transformers in
aged power system networks”, Cigre 2009 6th Southern Africa Regional Conference, Somerset West, South
Africa, Paper C105.
[2].M. Belanger (1999), “Transformer diagnosis: part 1 - a statistical justification for preventive
maintenance,” Electricity today, vol. 11(6), pp. 5–8.
[3].E. Duarte,et al. (2010), "A practical approach to condition and risk based power transformer asset
replacement", Conference Record of the 2010 IEEE International Symposium on Electrical Insulation
(ISEI), San Diego, CA.
[4].H.L. Willis, G.V. Welch, R.R. Schrieber (2001), Aging Power Delivery Infrastructures. New York:
Marcel Dekker.
[5].R.P. Hopskins, A. T. Brint and G. Strbac (1997), “The use of condition information and the physical
processes of failure as an aid to asset management”, Proceedings of the 32nd Universities Power
Engineering conference (UPEC’ 97), Manchester, UK.
[6].R.P. Hopskins, G. Strbac and A. T. Brint (1999), “Modelling the Degradation of Condition Indices”,
IEE Proceedings on Generation, Transmission and Distribution, Vol. 146, No. 4, pp. 386-392.

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

International Journal of Advances in Engineering & Technology, May 2013.
©IJAET
ISSN: 2231-1963
[7]. A. Naderian, S. Cress, R. Piercy, F. Wang, J. Service (2008), “An Approach to Determine the Health
Index of Power Transformers”, Conference Record of the 2008 IEEE International symposium on Electrical
Insulation, Vancouver, BC.
[8].A. Jahromi, R. Piercy, S. Cress, J. Service, W. Fan (2009), “An approach to power transformer asset
management using health index”, Electrical Insulation Magazine, Vol. 25, No. 2, pp. 20-34.
[9].J.A. Lapworth and A. Wilson (2008), “The Asset Health Review for Managing Reliability Risks
Associated with Ongoing Use of Ageing System Power Transformers”, 2008 International Conference on
Condition Monitoring and Diagnosis, Beijing, China.
[10].G. Ofualagba (2012), “Methods to Determine the Overall Health and Condition of Large Power
Transformers”, Innovative Systems Design and Engineering, vol. 3, pp.55-60.
[11]. A. D. Ashkezari, H. MA, C. Ekanayake, T. K. Saha (2012), "Multivariate analysis for correlations
among different transformer oil parameters to determine transformer health index", 2012 IEEE Power and
Energy Society General Meeting, San Diego, CA.
[12].H. Malik, A. Azeem, R. K. Jarial (2012), "Application research based on modern-technology for
transformer health index estimation", 9th International Multi-Conference on Systems, Signals and Devices
(SSD), Chemnitz.
[13].A. E. B. Abu-Elanien, M. M. A. Salama and M. Ibrahim (2012), “Calculation of a Health Index for
Oil-Immersed Transformers Rated Under 69kV Using Fuzzy Logic”, IEEE Transactions on Power
Delivery, Vol. 27, No. 4, pp. 2029-2036.
[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 from North China Electric Power University (NCEPU), China in 2007. Currently
she is a PhD student of NCEPU majored in electrical engineering. Her current research
interests include transformer condition monitoring and lifetime modelling.

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