ELECTRIC POWER ENGINEERING 2009

Wear-Out characteristics of underground cables by one- and two- mode models Pavel Praks1, Tadeusz Sikora1, Pierre-Etienne Labeau2, Radim Briš1 1

VŠB – Technical University of Ostrava, Faculty of Electrical Engineering and Computer Science, 17. listopadu 15, 708 33 Ostrava, Czech Republic 2 Service de Métrologie Nucléaire, Université Libre de Bruxelles (ULB), Av. F.D. Roosevelt 50, 1050 Bruxelles, Belgium E-mails: {pavel.praks, tadeusz.sikora, radim.bris}@vsb.cz, [email protected]

ABSTRACT The aim of this work is to compare wear-out characteristics of underground cables of an electric distribution network. The empirical failure intensity obtained from a real failure database is compared with two ageing models: the traditional two-parameter Weibull model and an alternative bi-mode model with three parameters. The least squares linear regression is used for the parameter estimation. The quality of fitting is analyzed by the analysis of variance. Results indicate that the novel two-mode model overperforms the two-parameter Weibull model as for its ability to approximate the empirical failure intensity. Keywords: Modeling of Ageing, Failure Rate, Failure Database, Wear-Out, Electric Distribution System

The paper is organized as follows. We introduce the age distribution of medium voltage cables and the empirical failure rate in Sections 2 and 3. In Section 4 we present the ageing models considered in the estimation procedure. In Section 5, a study based on parameter estimations of onemode and two-mode ageing models is described. Finally, conclusions are summarized.

2 AGE DISTRIBUTION OF MEDIUM VOLTAGE CABLES

The age distribution of medium voltage (MV) cables of an investigated part of the electrical distribution network consists of cable lengths for each specific production 5-year period. An example of an age distribution (also called the age histogram of cables) is shown in Figure 1:

1 INTRODUCTION

400 350 Cable length (km)

The failure prediction and description of electric distribution component’s reliable life is a very important part of the distribution network operation plans. A well-filled-out database is a suitable source for determining wear-out characteristics of components - if information on the failed component’s age and the age distribution of components working in the network are available. Medium voltage underground cables are considered in this paper, because for these we have the failure data required for analysis [2-4]. The second reason is that underground cables are not maintainable, so the influence on reliability of their wear-out is more significant than that of other electric distribution system components. Methods described here can be (with some adjustments) adapted also for the other components. The Weibull probability distribution function is very often used to describe the failure density of various products all over the world. Although lots of Weibull models are studied in the literature (for detailed information, consult e.g. [1]), there is a trend in industry to use models with a reduced number of parameters, because of the scarcity of industrial failure data. The aim of this contribution is to show how suitable is the traditionally used 1-mode Weibull model with 2 parameters and an alternative 2-mode model with three parameters [6] to approximate the given empirical reliability model of the underground cables.

450

300 250 200 150 100 50 0 2001 - 1996 - 1991 - 1986 - 1981 - 1976 - 1971 - 1966 - 1961 - 1956 - 1951 - until 2005 2000 1995 1990 1985 1980 1975 1970 1965 1960 1955 1950 Production year period

Fig.1. Cable age histogram It is possible to see from Fig. 1 that only a small part of the network uses cables produced before the year 1960. Because the quality of old failure data records remains questionable, these old failure data are not accounted in calculations. The period distribution has not a significant influence on calculations. We performed at VŠB-TU of Ostrava calculations using both one- and five-year period studies and the results were similar [2].

3 EMPIRICAL FAILURE RATE The failure rate can be defined as the ratio of the number of failures per unit time and unit length in the studied file of components over the file size. In our case the file of components contains all MV cables in the studied network of a given age (or also a given type) and the failure number we

can obtain from the failure database. The empirical failure rate function λ(t) (lambda) is then a set of point values of particular failure rates calculated according to equation (1):

λi =

ni ⋅100 li ⋅ p

(1)

Here ni is the number of failures of i-year-old cables (-) li is the total length of i-year-old cables in network (km) p is the duration of the studied period (years), so λi is given in [1/year/100 km]. The failure rate value is expressed in failures per year per 100 km of a cable length because it is the usual unit for power lines in the Czech Republic. The following moving average procedure was used to smooth out short-term fluctuations and highlight longer-term trends: hi = (λi-1 + λi + λi+1) / 3 (2) The obtained filtered empirical failure rate is denoted by the symbol lambdaData, see Figure 2. The origin of the horizontal axis (Years) represents the year 2005. The main objective of this study is to compare the empirical failure rate of MV cables versus failure rates given by two ageing models. The ageing models assumed for comparisons are presented in the following section.

wear-out, is modeled by β1 > 1. Moreover, the expected shape of the bathtub curve is obtained with values of β1 > 2. The scale parameter, η1 (eta), represents the characteristic life of the unit. In other words, it is the time, when approximately 63 % units will fail. The failure rate of the 2-parameter Weibull model writes as follows: β1 −1 (3)

λ1 (t ) =

β1 ⎛ t ⎞ .⎜ ⎟ η1 ⎜⎝ η1 ⎟⎠

. H (t )

Here the symbol H(t) denotes the Heaviside step function. 4.2 Two-mode ageing model We will also use an alternative bi-mode model with three parameters to approximate the given empirical failure rate model of the underground cables. The failure rate is modeled by

β ⎛ t ⎞ λ2 (t ) = λo + 2 .⎜⎜ ⎟⎟ η2 ⎝ η2 ⎠

β 2 −1

.H (t ) .

(4)

The bi-mode model relies on an exponential distribution (constant failure rate) to describe the random failure of the system, and on a two-parameter Weibull distribution to describe its ageing [6]. When λo = 0, the ageing model (4) reduces to the 2-parameter Weibull law (3).

5 PARAMETER ESTIMATION OF WEAR-OUT Failure Rate per Year

CHARACTERISTICS

25

The least squares linear regression is used for the parameter estimation of wear-out characteristics of MV cables. Two regression models were employed, one for each of the ageing models from eqs. (3) and (4). Results of parameter estimations are summarized in Table 1.

Failure Rate

20 15 10 5 0 0

10

20

30

40

Years lambdaData

lambdaEst3P

lambdaEst2P

Fig.2.: Empirical failure rate per year and 100km (denoted by lambdaData) and failure rate given by the two ageing models.

4 AGEING MODELS

SSE SST η β Model λ0 error error Est2P 0,00 3,05 7,48 134,60 661,49 Est3P 2,49 5,88 18,88 68,24 661,49 Table 1. Estimated wear-our characteristics

Here the symbol Est2P denotes results for the twoparameter ageing model of eq. (3) and the symbol Est3P denotes results corresponding to the three-parameter ageing model of eq. (4), respectively. The quality of fitting the empirical failure rate is measured by the analysis of variance (ANOVA) [7]. The "total (corrected) sum of squares" (SST) is

In this section, two ageing models considered for comparison with the empirical failure rate are presented. 4.1 One-mode ageing model The 2-parameter Weibull model is one of the most widely used lifetime distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter β1. Depending on the values of the shape parameter, we can describe all three life stages of the bathtub curve [1]. For instance, an increasing failure rate, which represents

R2 0,80 0,90

SST =

n

∑ (h − h) i

i =1

2

,

(5)

where h=

1 n

n

∑h

i

(6) and hi represents the empirical failure rate of i-year-old cables in the network computed by eq. (2) with i = 1, 2, …, n. i =1

The error (or "unexplained") sum of squares SSE, which is the sum of square of residuals, is given by SSE =

n

~

∑ (h − h ) i

i

2

.

(7)

i =1

~

Here h i is given for the two-parameter and the threeparameter ageing model by eqs. (3) and (4), respectively. Pearson's coefficient of regression is defined as SSE . (8) R2 = 1 − SST In regression, the R2 coefficient of determination is a statistical measure of how well the regression model approximates the real data points. If R2 = 1, the regression curve perfectly fits the observed data. From Table 1 we can see that the alternative bi-mode model with three parameters provides a suitable fitting of the empirical failure intensity obtained from the industrial failure database. Sum of squares of residuals (SSE) per year 50,0

SSE

40,0 30,0 20,0 10,0 0,0 0

10

20

30

40

Years SSE3P

SSE2P

Fig. 3: Sum of squares of residuals per year. The sum of squares of residuals (SSE) per year is summarized in Fig. 3. Symbols SSE2P and SSE3P represent SSEs for the two-parameter and the three-parameter ageing models, respectively. We can see that the bi-mode model with three parameters has a lower residual error than the twoparameter Weibull model in almost all analyzed years. The maximum value of the residual errors is observed for the year 1963 (represented in X-axis of the Fig. 3 as Year 41). This may be due to the fact that only a small part of the network uses cables from this year. Of course, also the quality of these old failure data records remains questionable.

6 CONCLUSIONS The aim of the paper was to compare wear-out characteristics of underground cables of an electric distribution network. For this reason, the empirical failure intensity taken from a real failure database was compared with two ageing models: the traditional Weibull model with two parameters and a novel bi-mode model with three parameters. The quality of fitting was analyzed by ANOVA. Results indicate that the two-mode model overperforms the two-parameter Weibull model. There are at least two reasons for this: • In the 2-parameter Weibull model, the failure rate vanishes at the origin, see eq. (3) and Fig. 2. This strong

constraint of the model is in contradiction with observations given by the empirical failure rate for t ~ 0. • One-mode Weibull models do not map well more than exactly one part of the bathtub curve reliability model. The ability of classically used one-mode approximations to model the wear-out effects given by a bathtub reference is limited [5, 6]. The main problem of calculations was the quality of the source failure database. Only a small part of the network uses cables produced before the year 1960, so measurements have different uncertainties. Moreover, also the production technology of cables was changed: The unreliable paper-based isolations have been continuously replaced by PVC. It is expected that these structural changes of components will be accounted for in future modelings. It could be interested to repeat tests contained in this contribution using a weighted regression, where each weight would be equal to the reciprocal of the variance of the measurement. In future research, we would like to also add the maintenance cost models and to provide the maintenance modeling and optimization. Acknowledgment This work is supported by The Ministry of Education, Youth and Sports of the Czech Republic - Project no. CEZ MSM6198910007. P.Praks’s postdoctoral stay at ULB during 2007-2008 was supported by FNRS Belgium (Project “Comparison of ageing models for industrial equipments”) and by the ARC project ”Advanced supervision and dependability of complex processes: application to power systems”. REFERENCES

[1] Murthy, D. N. P., Xie M., Jiang R.. Weibull Models. Hoboken, New Jersey, USA. John Wiley & Sons, 2004. [2] Sikora T., Mathematical methods for calculating the reliability and reliability indicators. Ph.D. thesis, VŠB-TU Ostrava, 2007 (Matematické metody výpočtu spolehlivosti a spolehlivostních ukazatelů, in Czech) [3] Sikora T., Goňo R., Monte Carlo Simulation of Electrical Power Distribution Network with Exponential and Weibull Parameters. Proceedings of 8th International Scientific Conference Electric Power Engineering 2007, Kouty nad Desnou 2007 [4] Sikora T: Wear-Out Characteristics of Electric Distribution System Components. Proceedings of 9th International Scientific Conference Electric Power Engineering 2007, Brno 2008 [5] Praks P., Bacarizo H. F., Labeau P.E. On the modeling of ageing using Weibull models: Case studies. In S. Martorell, C. Guedes Soares, and J. Barnett (Eds.), "Safety, Reliability and Risk Analysis: Theory, Methods and Applications. Vol 1, pp. 559–565. ISBN 978-0-415-48513-5. Taylor and Francis Group, London, 2008. [6] Praks P., Labeau P.E.: One- and two-mode approximations of the bathtub model: A comparative study. In preparation. [7] Montgomery, D. C. and G. C. Runger. Applied Statistics and Probability for Engineers. John Wiley & Sons, New York, 2003