Failure Analysis and Replacement Strategies of Distribution

Transformers Using Proportional Hazard Modeling

P.V.N. Prasad •• Osmania University •• Hyderabad •• ••K.R.M. Rao •• Osmania University •• Hyderabad •• •• Key Words: Non-homogeneous poisson process (NHPP), Mean cumulative repair function (MCRF), Proportional hazard model

(PHM), Covariate, Weibull process, Cumulative hazard rate (CHR), Partial likelihood function (PLF)

SUMMARY & CONCLUSIONS

The paper presents a graphical method for plotting mean cumulative repair function (MCRF) for different capacities or types of distribution transformers. The plots provide information about their failure behavior with time. The intensity function graphs obtained using parameters of Weibull process proportional hazard model confirms the results obtained from non-parametric MCRF graphs. Proportional hazard modeling (PHM) technique is quite helpful to consider the effect of covariates on the failure performance of different transformers with time. The property of proportionality is validated with the help of graphical and analytical methods. The higher values of intensity function for 100 kVA transformers and rural environment transformers show their poor performance compared to less than 100 kVA transformers and urban environment transformers respectively. This information is quite helpful in evaluating maintenance/ replacement policies. It is considered that normal life of distribution transformer is about 25 years, for a capacity of 100 kVA or less. The present study indicates that in case of 100 kVA transformers and rural transformers an early replacement is required.

1. INTRODUCTION

In the study of reliability analysis involving repairable / non-repairable equipment or system, normally 'time' is chosen as the random variable. The failure data is processed and modeled to find the parameters of the model. If the data is not independently and identically distributed (i.i.d) and indicating any form of trend due to deterioration or improvement, a time dependent model or non-homogeneous Poisson process (NHPP) model like the Weibull process model may be an appropriate model. A reliability-based approach is used to estimate the optimum replacement time interval. The replacement interval is arrived at on the basis of the replacement costs and unplanned maintenance costs, considering the effects of covariates like capacity of transformer, operating environment etc.

Over the past hundred years, transformers played a major role in the growth of many industries in the world. In any distribution network, distribution transformers play an effective role in the successful operation. Due to the increased number of domestic and agricultural services, more and more transformers are being installed, thus causing expansion of

distribution networks. In the present study, the failure data of different group of distribution transformers is collected based on capacity and environment. The study of failure characteristics is based on estimation of mean cumulative repair function (MCRF) developed by Nelson (Ref. 1) with proportional hazard modeling (Ref. 2). Quite often, we come across some population of items, which experiences recurrent events (Ref. 3). These recurrent events may be failures of system or equipment. In such cases, we wish to study the pattern of rate of occurrence of such events, compare their failure characteristics and assess the effect of explanatory variables with the help of proportional hazard modeling. These models will be useful in predicting the future behavior and assists in making comparative studies. The reliability of the distribution system depends on failure-free operation of distribution

2. MEAN CUMULATIVE REPAIR FUNCTION

Suppose that a system 'i' in a group of 'k' is observed over a time period 0 to ti and let Ni(t) denote the number of events occurring during that period. The cumulative mean function for Ni(t) is given by,

Mi(t) = E[Ni(t)] (1)

The distribution of the mean M(t) is called mean cumulative repair function. In any system of recurrent events, the intensity function is the rate of occurrence of failures i.e. the number of repairs per unit time per system. The integration of this function gives mean cumulative repair function, which is a function of time. Thus, if Ri is the number of units under repair and Ni is the number of serving units during a time interval ti-1 to ti, where ti is global time, then MCRF is defined as, (Ref. 1)

MCRFi = Σik=1 (Rk / Nk) (2)

This equation can be modeled using the relation,

MCRFi = ( ti / α )ϒ (3)

where, α and ϒ are the scale and shape parameters of Weibull process (Ref. 1,4).

3. PROPORTIONAL HAZARD MODELING

In this model, the failure intensity of different groups of transformers is proportional to each other. It is possible to

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study the effect of covariates on the failure pattern of distribution transformers by introducing covariates.

3.1 Weibull Process PHM

When the data exhibits trends and does not satisfy the independent and identical distribution (i.i.d) assumption, then the model is sometimes called a Proportional intensity model (Ref. 5). The covariate considered here is capacity of transformer or environment. Thus, MCRF model of equation (3) can be modified as,

MCRF = [ t / exp(a.z) ]ϒ (4)

where, a = [a0 a1 a2 ……..] is the regression coefficient vector and zT = [ 1, z1 z2 ……… ] is the covariate vector. Taking logarithm of equation (4), we get,

Ln (MCRF) =ϒ.Ln (t) –ϒ.(a.z) (5)

Regressing the above equation gives the constants of the model. The validity of the above model can be checked by the presence of parallel lines for different covariates in the plot of Ln(MCRF) vs. Ln(t). The reliability of the system which functions an additional time ‘d’ without repairs, given the system age of time ‘t’, is

R(t, z) = exp[-{MCRF (t + d, z) - MCRF(t, z)}] (6)

and the associated intensity function is,

u(t, z) = [ϒ / exp(a.z)] .[t / exp(a.z)] ϒ -1 (7)

The instantaneous MTBF is the inverse of intensity function, which is given by

MTBF (t, z) = 1/ u(t, z) (8)

3.2 Non-Parametric PHM

In proportional hazard modeling, the hazard rate of the equipment is assumed to be a function of time and system covariates. It is the product of an arbitrary and unspecified baseline hazard rate λ0(t) and an exponential function consisting of system covariates. It is given by,

λ(t,z) = λ0(t)exp(z.ββββ) (9)

where z is a vector consisting of covariates and ββββ is a vector of regression coefficients. The base-line hazard function is not fitted into a specific model and has a non-parametric form. It represents the hazard rate of a system when all the covariates are equal to zero. The model assumes that the covariates act multiplicatively on the hazard function, so that for different values of explanatory variables the hazard functions at each time are proportional to each other. To estimate the regression parameters of this model, the partial likelihood function (PLF) should be maximized (Ref. 2). This PLF is the product over all failure times of the conditional probability of failure of the item, which actually failed at time ti. That is, if NFi is the set of indices of items not failed just before time ti, then,

Li = λ(ti,zi) / ∑j∈NFi λ(ti,zj) (10)

is the conditional probability for time ti. The partial likelihood L is the product of terms of Li. The product is taken over all failure times. For the PH model,

L = ∏ {exp(ziββββ) / [∑j∈NFiexp(zjββββ)]di} (11)

where di is the number of tied failure times, which is assumed small compared to the number of items j in the risk set at ti. The estimated values of ββββ are tested for statistical significance so that it can be verified whether a particular covariate has any effect on the failure behavior of the system. The reliability function is given by,

R(t, z) = R0(t)exp(z.ββββ) (12)

where,

R0(t) = exp[-H0(t)] (13)

and R0(t) and H0(t) are baseline reliability and base-line cumulative hazard rate functions respectively.

3.2.1Goodness of Fit Tests

To test the validity of PHM assumption and the significance of covariates, graphical and analytical methods are utilized.

a. Cumulative Hazard Rate Plots The cumulative hazard rate (CHR) function, H(t,z) is

a product of baseline cumulative hazard rate and an exponential term consisting of covariates. It can be easily obtained from equation (12) and equation (13), which is given as

H(t,z) = H0(t)exp(z.ββββ) (14)

Taking logarithm on both sides, we get,

Ln[H(t,z)] = Ln[H0(t)] + z.ββββ (15)

Thus, for two covariates z1and z2 with regression coefficients β1 and β2 respectively, if plots of Ln(CHR) vs. time are drawn, they should be roughly parallel to each other. This confirms the multiplicative assumption that the ratio of two hazard rates is constant with time.

b. Significance of Individual Coefficients The estimation of regression coefficients β should be

tested for their significance on the basis of t-statistics. The t-statistic is calculated by taking the ratio of the estimate of ββββ to the standard deviation of the estimate and the corresponding p-value is obtained from the table of N(0,1) distribution. A p-value of the t-statistic corresponding to the estimate of β may be interpreted as equal to the probability of obtaining such extreme value for the estimate of β, if it is not equal to zero (Ref. 6). Normally a p-value of 10% is considered as the upper limit to check the significance of covariates.

c. Log-Likelihood Ratio Test Let L0 be the log-likelihood function when the

covariate vector is null and L be the corresponding function with estimated covariate vector. Then, the test statistic (Ref. 2,7)

Λ = 2.Ln(L/L0) (16)

is asymptotically distributed as χ2(k), where k is the number of

parameters being estimated. This is a reliable and convenient method for an overall test of the significance of the model. After evaluating the regression coefficients and verifying the proportionality assumption of hazard rates at different

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covariates, the reliability and CHR functions are obtained. The following section present one of the techniques of estimating these functions (Ref. 5,6).

This method proposed by Breslow is simple to evaluate, but approximates to true value only when the number of tied failures are small, compared to the total number of failures. The baseline CHR can be estimated as,

H0(t) = ∑ti<t[di / ∑j∈NFiexp(ziβ)] (17)

where di is the total number of tied failures at time ti. Once the CHR is obtained using equation (14), the reliability function can be calculated with equation (12).

4. REPLACEMENT POLICIES

If any equipment is to be minimally repaired, then a decision should be made to determine when to replace the equipment. If the equipment under consideration has an increasing intensity function, then at some point of time, it will be no longer economical to continue to maintain it. In any Weibull process with increasing intensity function, the optimum replacement interval T* is given by the following equation (Ref. 4). This process is also called the Power Law process.

T* = exp(a.z).[ (Cr /Cf) / (ϒ -1)] 1/ϒ (18)

where Cr and Cf are cost of replacement and cost of failure respectively and the ratio Cr/Cf is called the cost ratio.

5. ANALYSIS OF THE DATA

The failure data and MCRF calculations of 100 kVA transformer is shown in table I. Two different groups of transformers based on capacity (100 kVA or < 100 kVA) and environment (Rural or Urban) are analyzed separately. Since there are only two covariates in each group, zT = [ 1 z′ ], where, z′ equal to 1 or 0, is assumed in the two cases. Using equation (2), MCRF is calculated. Performing multiple regression analysis using equation (5), the parameters of the Weibull PHM values are estimated. The covariates and model parameters are summarized in table II. The survival function and intensity function can be obtained using equation (6) and equation (7) respectively.

If a graph of Ln(MCRF) vs. Ln(t) is plotted for different covariates, the presence of parallel lines should confirm the assumption of Weibull process PHM. These are shown in Figure 1 and Figure 2 for the two cases and the lines are approximately parallel to each other. The plots of reliability function (case i) and intensity function (case ii) are shown in Figure 3, Figure 4 respectively. Using equation 7, the ratio of intensity function for two different covariates z1 and z2, can be obtained easily as follows.

u(t,z′1)/u(t,z′2)=[exp(a0+a1z′2)/exp(a0+a1z′1)]β (19)

The MTBF of each transformer at the end of the period of observation can be obtained using equation (8). For a given value of the cost ratio, the optimum replacement interval can be found using equation (18), if the value of shape parameter is more than unity. The details of ratios of intensity function, optimum replacement times and MTBFs are given in table III.

The intensity function ratios can also be obtained

using non-parametric methods. Since the results cannot be extrapolated, T* and MTBFs cannot be obtained. Using the covariate z = [ 1 ln(Ri/Ni) ] for less than 100 kVA / Urban transformers and z = [ 0 ln(Ri/Ni) ] for 100 kVA / Rural

Table I: MCRF Calculation of 100 kVA Transformers

Month Ni Ri Ri/Ni MCRF

1 2642 70 0.0265 0.0265

2 2669 90 0.0337 0.0602

3 2703 74 0.0274 0.0876

4 2738 66 0.0241 0.1117

5 2772 84 0.0303 0.1420

6 2801 77 0.0275 0.1695

7 2852 84 0.0295 0.1989

8 2894 71 0.0245 0.2235

9 2928 67 0.0229 0.2464

10 2962 68 0.0230 0.2693

11 3003 89 0.0296 0.2990

12 3098 115 0.0371 0.3361

13 3163 105 0.0332 0.3693

14 3203 120 0.0375 0.4067

15 3249 159 0.0489 0.4557

16 3284 176 0.0536 0.5093

17 3308 182 0.0550 0.5643

18 3330 171 0.0514 0.6156

19 3444 168 0.0488 0.6644

20 3434 117 0.0341 0.6985

21 3469 172 0.0496 0.7481

22 3521 194 0.0551 0.8032

23 3598 215 0.0598 0.8629

24 3709 142 0.0383 0.9012

25 3779 140 0.0370 0.9383

26 3817 84 0.0220 0.9603

27 3908 82 0.0210 0.9812

28 3953 136 0.0344 1.0156

29 3975 245 0.0616 1.0773

30 4020 164 0.0408 1.1181

31 4065 116 0.0285 1.1466

32 4116 181 0.0440 1.1906

33 4284 147 0.0343 1.2249

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transformers and maximizing the likelihood function using equation (11), the regression parameters ββββ can be obtained. The log-likelihood ratio test yielded a p-value of less than 10% for both cases. The details of non-parametric analysis are shown in table IV. Thus the PHM model can be assumed and the results in general are in agreement with the results obtained with parametric approach.

CONCLUDING REMARKS

Two groups of distribution transformers are studied separately using Proportional hazard modeling to evaluate their failure rates. The first group is based on the capacity of transformer (100 kVA and less than 100 kVA) and the second group is based on the operating environment (Rural and Urban). In non-parametric PHM, the cumulative hazard rate (CHR) is the product of a base-line CHR, a function of time only and an exponential term, which is a function of covariates and regression coefficients. The regression coefficients are evaluated by maximizing the likelihood function. The PH model assumes that the failure rate of one transformer is proportional to another transformer in each group. The MCRF is modeled by Weibull process PHM and the parameters are evaluated to study the failure trend of these transformers. A shape parameter of greater than unity indicates an increase in failure rate and this necessitates the replacement policy to be adopted for a given group of

transformers. An effective replacement policy is the optimum time at which the overall cost is minimum. The ratio of cost of replacement to the cost of scheduled & unscheduled repairs and operational costs is the deciding factor in evaluating the optimum replacement time. Of the two capacities of transformers under study, ‘100 kVA’ transformers exhibited higher failure rate (1.9 times) than those of ‘less than 100 kVA’. The replacement policies of these two capacities are found to be about 25 years and 13 years respectively. In the second group, the failure rate in rural environment is 2.1 times than in urban environment and the optimal replacement times are 8 years and 17 years in respective areas. The cost ratio in all the above cases is assumed to be unity.

ACKNOWLEDGEMENTS

The authors would like to thank the Head of Electrical Engineering Department, Osmaina University for the support and facilities provided to pursue the present work.

Table II: Weibull Process PHM Parameters

Case z′ ϒ a0 a1

< 100 kVA 1 i

=100 kVA 0

1.137 3.331 0.632

Rural 1 ii Urban 0

1.190 3.162 0.739

Table III: Results of Parametric PHM

Optimum Replacement

Interval (Yrs.)

Case

Intensity

Ratio Cost ratio =1.0

Cost ratio =2.0

MTBF (Yrs.)

< 100 kVA 1.00 25.3 46.5 4.1 =100 kVA 1.88 13.4 24.7 2.0

Rural 2.10 8.0 14.3 1.7 Urban 1.00 16.7 29.9 3.7

Table IV: Results of Non-Parametric PHM

Case i Case ii Intensity Ratio 1.269 1.811 Parameters β1&β2 (p-value)

-0.687 (0.0260) -1.478 (0.0002)

-1.638 (0.0000) -1.483 (0.0005)

Test Statistic (p-value)

14.3 (0.0008) 20.0 (0.0005)

-5

-4

-3

-2

-1

0

1

0 0.5 1 1.5 2 2.5 3 3.5

Ln(Time) Fig. 1: Weibull process PHM (capacity)

Log(

MC

RF)

< 100 kVA

100 kVA

-5

-4

-3

-2

-1

00 0.5 1 1.5 2 2.5 3 3.5

Ln(Time)

Fig. 2: Weibull process PHM (Environment)

Log(

MC

RF)

Urban Rural

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REFERENCES

1. Wayne Nelson, (1988), “Graphical Analysis of System Repair Data”, Journal of Quality Technology, 20, 24 -35 2. J.D.Kalbfleish, R.L.Prentice, (1980), “The Statistical Analysis of Failure Time Data”, John Wiley & Sons 3. J.F.Lawless, C.Nadaeu, (1995), “Some Simple Robust Methods for Analysis of Recurrent Events”, Techno metrics, 37, 158 – 168 4. Charles E.Ebeling, (1997), “An Introduction to Reliability and Maintainability Engineering”, McGraw-Hill International Editions 5. Dhananjay Kumar, (1995), “Proportional Hazards Modeling of Repairable Systems”, Quality and Reliability International, 11, 361 – 369 6. D.Kumar & B.Klefsjö, "Proportional Hazard Model: A Review”, Reliability Engineering & System Safety, vol. 44, no.2, pp 177-178, 1994. 7. C.E.Love & R.Guo, “Application of Weibull Proportional Hazards Modeling to Bad-as-Old Failure Data”, Quality and Reliability Engineering International, vol.7, 1991

BIOGRAPHIES P.V.N.Prasad, B.Tech, M.E Asst. Professor, Dept. of Electrical Engg. University College of Engg., Osmania University, Hydearbad – 500 007, India. E-mail: polaki@rediffmail.com

P.V.N.Prasad graduated in Electrical & Electronics Engineering

from Jawaharlal Nehru Technological University, Hyderabad in 1983 and received M.E in Industrial Drives & Control from Osmania University, Hyderabad in 1986. He served as faculty member in Kothagudem School of Mines (1987-95) and at presently serving as Assistant Professor in the Department of Electrical Engineering, University College of Engineering, Osmania University, Hyderabad. His areas of interest are Reliability Engineering & Maintenance Management and Computer Simulation of Electrical Machines & Power Electronic Drives. He is a member of Institution of Engineers (India) and Indian Society for Technical Education. He is recipient of Dr.Rajendra Prasad Memorial Prize, Institution of Engineers (India), 1993-94 for best paper. He has completed Ph.D in the field of Reliability of Repairable Systems.

K.R.M.Rao, B.E., M.Sc., (Engg.), Dr. Inz Professor (Retd.), Dept. of Electrical Engg. University College of Engg., Osmania University Hyderabad – 500 007, India E-mail: drkrmr@yahoo.com

K.R.M.Rao graduated in Electrical Engineering from Andhra

University in 1962, received M.Sc. (Engg.) from Banaras Hindu University in 1964 and Doctor of Technical Sciences degree from Wroclaw Technical University, Poland, 1968. He served as Assistant Professor in Electrical Engineering during 1969–74 at H.B Technological Institute, Kanpur. Later, he served as Professor in Department of Engineering and Mining Machinery, Indian School of Mines, Dhanbad (1974–78), Professor, Kothagudem School of Mines (1978 - 95) and Dean, Faculty of Engineering, at Osmania University, Hyderabad (1997–99). His areas of interest are Mining Machinery, Material Handling, Mining Electrical Engineering, Reliability Engineering and Maintenance Management. He is a Fellow, Institution of Engineers (India), Member, ISTE, Member, MGMI and Member, SRE (India).

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 Time (Months) Fig. 3: Reliability Plot (Capacity)

Rel

iabi

lity

< 100 kVA 100 kVA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 Time (Months) Fig. 4: Intensity Function Plot (Environment)

Inte

nsity

Fun

ctio

n Urban

Rural

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