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Electrical Power and Energy Systems 24 (2002) 859±865

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Interruption modelling in medium voltage electrical networks
? tse ? a,*, Paul Villeneuve b, Euge ? ne Patrice N. Ndong a, Franc Thomas Tamo Tatie ? ois Kenfack a
b

? , Cameroon Ecole Polytechnique, P.O. Box 8390, Yaounde ? Laval, 1636 pav. F. A. Savard, Que ? bec., Canada G1K 7P4 CRAD?Universite

a

Received 27 October 2000; received in revised form 27 June 2001; accepted 20 September 2001

Abstract Failures frequently occur in developing countries' electrical energy distribution networks. This paper proposes an approach to network reliability through modelling the interruptions on medium voltage lines. This modelling is based on a representative feeder sampling of a source station in an urban area. In order to determine the probability law governing these interruptions, statistical techniques were used: density estimation using the kernel method and approximation by the least squares. The results we obtained show that, from the quality of the equipment and their maintenance, interruptions for a given network follow a truncated and shifted gamma distribution or a truncated normal law. For the managers of such systems, these results would allow, amongst other things, reduction in the probability of failure, thus improving operational safety on electricity distribution with medium voltage lines. q 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Interruption; Medium voltage; Probability law

1. Introduction Despite the high level of electricity cover in developing cities, the technical and commercial performance of electricity boards is much lower than expected performance. This results in a mediocre quality electricity service. In a context where the electricity distribution service must cope with an increase in users as well as demands for quality and continuity, it is essential to know to what degree a cognitive approach to interruptions would allow optimum maintenance and electrical equipment renewal programming. Indeed, the electricity supply system safety problem represents a technological and socio-economic challenge. Failures are observed more in distribution than in production and electricity transmission systems. Indeed, nearly 90% of all failures are observed in distribution systems [1]. Several authors have shown that use of emergency energy sources to handle interruptions is very frequent for users. In this way, a study conducted in Canada in 1991 shows that individual customers, more than the company in charge of managing the network, tend to consider emergency electrical energy production sources in event
* Corresponding author. Tel.: 1237-77-93-99; fax: 1237-22-4547/234514. ? tse ? ), E-mail addresses: ttamo@polytech.uninet.cm (T.T. Tatie ? tse ? ). ttamo@uycdc.uninet.cm (T.T. Tatie Abbreviations: EHV, Extra high voltage; EM, Expectation-maximization; HV, High voltage; LV, Low voltage; MV, Medium voltage

of an interruption [2]. These failures mainly concern interruption and voltage ?uctuations. As far as voltage ?uctuations are concerned outside its tolerance margins, appropriated distribution network design methods allow considerable reduction [3,4]. In the case of the existing network, the literature provides devices to cope with overvoltages and voltage drops [5,6]. The more worrying failures thus concern interruptions. The ?nancial impact of interruptions is considerable for every category of customer. Costs in the administrative and of?ce sectors are of a level comparable to those observed in the industrial sector [7]. In order to better evaluate the economical consequences of interruptions, research has been conducted on the origin of network failures. The main indicators often chosen for the analysis are: interruption rate and length, the proportion of customers concerned and the undistributed energy [6,8±10]. Models established in order to control the interruptions have been the subject of many publications. Interruptions in the distribution system occur according to the exponential law or Poisson distribution according to the network con?guration [11]. However, these laws do not describe the case of young components, whose failure rate decreases in time, nor of old components, whose failure rate increases in time. The repair time complies with a continuous distribution law [12]. Other laws could also describe system reliability and are presented in the literature. Thus: the normal law, which suits old components (amongst

0142-0615/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(02)00010-8

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? tse ? et al. / Electrical Power and Energy Systems 24 (2002) 859±865 T.T. Tatie

other things), and whose failure rate increases with time, Weibull distribution, normally suitable for mechanical components for whose failure rates are not constant [13]. The exponential law, usually used to describe failure, is not always 100% suitable for electricity distribution lines [14]. The extent of electrical supply failures suggests that the earlier laws are not adapted to a network supporting frequent interruptions. Indeed, in most developing countries, energy losses due to interruptions are very high: more than 10% compared with less than 6% in countries with well-maintained networks [4,10,13]. The average annual interruption rate per km is also very signi?cant: 1.906 [15] compared with between 0.298 [6] and 0.015 [1] in the case of well-maintained lines. The same applied for the average repair time, which is relatively high. In such a context, classical models used to describe failures are no longer relevant. Moreover, it is obvious that the average good operating time of electrical line components before failure is shorter than the theoretical value. This is good for the mean onset operation time before its ?rst failure (mean time to failure: MTTF) as well as the mean length between two consecutive failures of a feeder (mean time between failures: MTBF). Ensuring the continuity of the electrical supply service implies implementation of a maintenance policy mainly based on controlling network behaviour in the event of interruption. For this reason, a new approach to the interruptions modelling is necessary, considering the inaccuracies of the laws described earlier. The following paragraph describes the method of the research conducted by the authors since 1995. 2. Method In a context characterised by frequent power cuts, interruption modelling is required in order to reduce the occurrence of failures on power lines. This modelling uses formalisation of the electrical supply system and implementation of an evaluation method for the variables describing the interruptions. 2.1. Formalisation of the system at study (Fig. 1) The purpose of this formalisation is to emphasise the part of the network concerned by the study and vulnerability indicators, which are also called reliability factors. On the topological level, medium voltage (MV) network operations are represented by: nodes: transformer stations, breaking devices, connection points on the network, segments: lines or cables. Network topology is given by the list of relations existing between the nodes and segments.

Fig. 1. Formalisation of the system studied.

We now turn to part (a). Remember that the voltage in a distribution network varies from a few hundred volts to 50,000 V and that in medium voltage, one must include a voltage between 1 and 50 kV. Low voltage (LV) distribution is thus not directly concerned by this study. The main components of an MV distribution network are: the source station (High voltage (HV)/MV transformer), power lines (MV cables), MV/LV transformer stations. Each feeder (or electrical line) is likened to a `component' made up of three serial-connected elements. Weather conditions, technical problems and human acts are the factors explaining failures in the network. Their effects are measured by the following random variables: interruption rate, length of interruptions, proportion of customers affected by the failure, losses (?nancial, energy, time, etc.) resulting from failures. The aim of this paper is to de?ne a relevant evaluation approach to these failure indicators pertaining to the distribution lines. 2.2. Modelling process Evaluation of operational safety of MV networks requires collection of information on this network. This information relates to events such as failures or breakdown occurring on one or several feeders forming the network. In a very schematic way, the process consists of observing the network we are concerned with over a certain period of time (several months or years), in real operation conditions, and of classifying all failures or breakdowns of these feeders. Thus, basic information is obtained that allows us to quantify operational safety. Consequently, it is essential to search for data describing ef?cient operation of the network, including those pertaining to vulnerability factors described earlier. It is from these factual data that variables

? tse ? et al. / Electrical Power and Energy Systems 24 (2002) 859±865 T.T. Tatie

861

Fig. 2. Method ?owchart.

characterising operation safety (failure rate, repair rate, repair time, operation time before failure, etc.) in a given urban area will be determined. The modelling process included the following steps (Fig. 2): (a) evaluate the interruption on all the feeders (lines) of a source station from the actual operating history of these feeders, (b) choose the most representative medium voltage lines on which we will conduct further studies in order to model the interruptions, (c) amongst other things, carry out a qualitative analysis of these lines from the sample, (d) develop a mathematical model governing the interruptions, particularly by examining the shape of the histograms obtained, (e) sort out suggestions and recommendations for preventive maintenance of the system.

the different strategic situations of the two stations in terms of the urban zones they serve, the Ngousso station feeding, e.g. administrative nervous systems, while that of Melen concerns mainly popular areas. So, the rate of components renewal seems to be much lower at Melen than at Ngousso. There are two types of interruptions, grouping the four classes proposed by Willis [6]: intermittent interruptions, lasting 5 min (these are instant and temporary interruptions). permanent interruptions (also including temporary interruptions), lasting more than 5 min. We have determined the law governing interruptions for some very representative feeders. Four feeders are considered at Ngousso station and two at Melen station, according to homogeneity of feeder characteristics based on the rate of failure, average repair time and MTBF; this homogeneity is greater at the second station than the ?rst. 3.2. Sampling and observed data

3. Model elaboration The proposed interruption modelling approach in the electrical energy distribution network was the subject of ? an experiment on actual electrical networks: the Yaounde urban area in Cameroon. This application enabled us to determine the interruption distribution laws. 3.1. Networks studied and basic variables ? and its suburbs are supplied with The city of Yaounde electricity by the source stations of Ngousso and Melen (or BRGM). The feeders making up these source stations (16 for the Ngousso station and 15 for the Melen station) are subject to a large number of interruptions. The poor quality of the maintenance is the cause of such a situation. Indeed, maintenance program completion rates are quite low (between 8 and 62% for every rubric, see Table 4), which results in an increase of the number of incidents on the network. However, the frequency and length of interruptions are much higher at the Melen feeders than at those of Ngousso, indicating an even more mediocre maintenance. Empirical evidence suggests that this is due to

Following a comprehensive evaluation of all feeders on the MV network in the considered urban area, the most representative ones are kept for further analysis. However, a preliminary exploration study was conducted in 1995± 1996 by the same team, allowing collection ?les to be de?ned on the information relevant to network operational safety. Laws are de?ned on the basis of data collected monthly over a period of two years (1996±1997) (see attachment). These data were balanced using the following equation: z ? x 1 3y ; ? 1?

where x is the number of interruptions #5 min, and y . 5 min. This balance arose from a preliminary study based on the data from all the 31 feeders at the two stations [10,15]. This study reveals a strong linear correlation (coef?cient of correlation between 0.7 and 0.9) between the respective numbers of intermittent interruptions and of permanent interruptions at the same feeder; and the poorer the quality of the maintenance at the feeder, the stronger this correlation appears to be. An empirical diagnostic then suggests that this balance

862 Table 1 Weighted interruptions (1996±1997) Stations Onsets Ngousso D110 D15 6 7 6 14 4 0 3 0 6 1 0 3 0 6 3 9 12 18 3 0 18 4 3 9 5.21 4.96

? tse ? et al. / Electrical Power and Energy Systems 24 (2002) 859±865 T.T. Tatie

with
Melen (BRGM) D13 D113 D31 15 10 14 8 11 9 11 5 0 15 2 6 15 0 0 6 3 3 3 9 3 14 10 19 7.96 5.45 D13 12 8 9 2 8 6 1 0 12 9 4 5 0 9 9 18 9 18 3 0 0 11 3 2 6.58 5.22
2 1 K ?x? ? p???? e2x =2 : 2p

?3?

January 97 0 February 15 March 3 April 0 May 0 June 3 July 0 August 3 September 9 October 3 November 6 December 97 3 January 96 3 February 0 March 0 April 6 May 3 June 12 July 0 August 3 September 3 October 6 November 6 December 96 3 Average 3.75 Standard deviation 3.80

6 3 9 12 7 3 5 3 0 0 0 0 0 6 0 0 0 3 10 6 3 0 0 1 3 6 3 0 3 6 0 12 0 0 6 9 0 6 3 0 5 0 0 3 1 3 0 3 2.67 3.54 3.08 3.61

The kernel K(x) used in this case is very common (note: it must be a symmetrical probability density with respect to 0). The choice of window width h has been experimentally set by trial and error in order not to alter the nature of the observations in the sample (i.e. results close to the empirical distribution but smoother). Let us note that the kernel estimator can affect a non-null probability, including in the real axis zone where there is no observation, unlike the empirical distribution. This allows evaluation of the law, including from truncated samples, as we will show is the case here. 3.3.1. Probability laws at the Ngousso station An observation of the curve on IR of the f~n;h kernel density evaluator for interruptions at each feeder of this station leads us to make the following fundamental statement: if the curve is shifted to the right of a certain constant c . 0 (which is a function of the considered feeder), a gamma law probability density pattern is obtained. In other words, there is a real number c . 0, such that the random variable Z 1 c follows a g (a, l ) law truncated to p the left of c. The g (a, l ) law is the probability law on IR1 of density ga;l ?t? ? ?la =G?a??t a21 e2lt ?t . 0?; ?4? where a and l are two real constants .0 characterising the law. For each feeder at this station, the constant c has been read on the f~n;h graph on IR. As far as constant a and l evaluation mode are concerned, we will use a least squares principle on the ?tk 1 c; ln f~n;h ?tk ?? pairs, where t1,?,tp are chosen as regularly spaced values in the real axis zone including z1,?,zn (with a slight spillover to the left of 0 and the right of max zi). According to our observation earlier, shifting the f~n;h curve to the right by a distance c brings it to that of a function in the form of

of 1 against 3 re?ects quite well the relative importance to be given to these two types of interruptions. The monthly observations balanced in this way are presented in Table 1. 3.3. Probability law of the interruptions at each source station and feeder Let z1,?,zn, be observations of a random variable Z (in this case, the number of monthly interruptions at the same feeder). The problem is to evaluate the probability law describing Z, based on observations in z1,?,zn. There is no initial information on the subject; as similar studies rely on data collected in very different environmental conditions. The most natural idea is then to use the empirical distribution associated with the observed z1,?,zn sample. However, the number of observations (in this case, n ? 24) is too small for the empirical distribution to provide suf?cient information on the type of probability law followed by interruptions on a feeder. It is for this reason that we were obliged to use a probability density technique evaluation based on the kernel method [16]. This assumes that the unknown law is continuous and evaluates its probability density by the following function:   n 1 X x 2 zi f~n;h ?x? ? K ; ?2? nh i?1 h

w?t? ? eA ta e2lt ;
where we have put A ? ln?la =G?a??; and

?5? ?6? ?7?

a ? a 2 1:

This quite intuitive approach has been preferred to sophisticated evaluation techniques for incomplete data situations (e.g. the EM algorithm) whose use would be a possibility here [17,18]. In order to obtain function w , points t1,?,tp are ?rst chosen. Then A, a , l are found so that, for any k, value ~ k ? ln f~n;h ?tk ? y ?8?

? tse ? et al. / Electrical Power and Energy Systems 24 (2002) 859±865 T.T. Tatie Table 2 Evaluated laws of the interruptions in the Ngousso feeders Feeders D110 D15 D13 D113

863

h 4 5 4 4 c 15 17 15 15 22(1)15 24(1)18 23(1)13 21.5(1)14.5 t1,?,tp A 224.01 223.02 224.78 223.95 Evaluated g (12.54, 0.66) g (11.04, 0.49) g (13.24, 0.73) g (12.50, 0.66) law r (as a %) 1.3 1.08 0.78 0.18

Fig. 4. Number of monthly interruptions at Ngousso D15.

is close to yk ? A 1 a ln?tk 1 c? 2 l?tk 1 c?: ? 9?

This is done by choosing A, a , l so as to minimise the quantity: D?A; a; l? ?
p X k?1

~k 2 ?A 1 a ln?tk 1 c? 2 l?tk 1 c???2 ; ?y ?11?

3.3.2. Probability laws at the Melen station On each feeder of this station, an observation of the kernel density estimator curve f~n;h on IR reveals a density rhythm of a normal law. This leads us to assume that in this case the number of monthly interruptions Z follows a law of the N(m, s 2) type, truncated to the left of 0. The N(m, s 2) law is the probability law on IR of density: p???? 2 2 ?14? w m;s 2 ?x? ? e2?x2m? =?2s ? =?s 2p? where m and s 2 are, respectively, the average and variance of this law and characterise it. In this case, the value of m is taken to be the point where f~ n;h reaches its maximum on IR, the latter being numerically determined. In order to evaluate s 2, a least squares principle is used on pairs ?t k ; ln f~n;h ?tk ??; where t1,?,tp are chosen as earlier. In this case, once m has been calculated, all we have to do is to approach the curve from f~n;h via a function in the form of ?16? p???? where A and B are given by A ? 2ln?s 2p? and B ? 1=?2s 2 ?: After choosing points t1,?,tp, A, B are calculated ~k ? ln f~n;h ?tk ? is close to yk ? so that for any k, the value y A 2 B?tk 2 m?2 : In order to achieve this, A and B are computed so as to minimise the quantity (17): D ? A; B? ?
p X k?1

which is a totally classic linear least squares problem. The quality of this approximation is measured in terms of least squares by the following ratio (expressed as a percentage): v?????????? p p???????????? u uX ~k ?2 ; r ? D?A; a; l?=t ?y ?12?
k ?1

where A, a , l are in this case the calculated values, minimising D. Naturally, the choice of a continuous law model can be surprising at ?rst. However, it can be considered that z1,?,zn are rounded observations stemming from such a model; where each zi is in fact z i 1 1i ?13?

w?t? ? eA2B?t2m? ;

2

where 1 i is an unknown random perturbation between 0 and 1. The results at the four Ngousso feeders are reported in Table 2. Note: The real law is the one displayed, shifted by c units to the left and truncated to the left at 0. Since the value of r is very small for each feeder, the evaluation of the law obtained can be considered to be very acceptable (see attached Figs. 3±6).

~k 2 ?A 2 B?tk 2 m?2 ??2 ; ?y

?17?

The quality of the approximation obtained is measured using a method similar to the one earlier. The value of s 2 is deduced from B. Table 3 shows the results at the two Melen (BRGM) feeders.

Fig. 3. Number of monthly interruptions at Ngousso D110.

Fig. 5. Number of monthly interruptions at Ngousso D13.

864

? tse ? et al. / Electrical Power and Energy Systems 24 (2002) 859±865 T.T. Tatie

Fig. 6. Number of monthly interruptions at Ngousso D113. Fig. 7. Number of monthly interruptions at Melen D32.

4. Discussion and conclusion In order to improve the degree of reliability of the electric energy distribution network, an interruption modelling approach at MV electrical network level is recommended. The approach proposed was the subject of an experimental study on the actual distribution networks operation. Data ? analysis on distribution line interruptions in Yaounde (Cameroon) has allowed us to determine the probability laws that they obey. At Ngousso station, these interruptions follow a shifted and truncated gamma distribution and at Melen station, a truncated normal law. The difference between these two types of law observed in the two stations seems to stem from the difference between the qualities of maintenance at these stations. At Ngousso station, apparently located in a more strategic position by the urban zones it serves (e.g. administrative nervous systems), a type of law quite usual for this category of statistical data is obtained. However, at Melen station, maintenance quality is much poorer and thus presents a greater accumulation of random factors (human, material or environmental) affecting the quality of the service provided by the station. Interruptions then follow a truncated normal law. This may be regarded in a sense, as an illustration of the usual practical interpretation of the central limit theorem of probability theory, in the general version of Lindeberg, that for a sum of independent random variables but which need not be identically distributed [19]. According to that interpretation [20], an additive accumulation of a large number of fairly balanced and independent random factors should result in a normal probability distribution for their global effect. Thus, the observed law at Melen station may serve as a serious warning to its managers that things are really going wrong at that station, and something needs to be done to change the situation. In a more general way, the different interruption
Table 3 Estimated laws for the interruptions at the Melen feeders. (Note: The real law is the one displayed truncated to the left of 0) Feeders D31 D13 h 5 5 t1,?,tp 0(1)22 0(1)21 Estimated law N(7.40,66.60) N(5.72,62.62) r a (in %) 0.90 0.44

laws obtained at the two stations are not those that usually describe the classical phenomenon studied. This illustrates a maintenance fault in the network, with quite low maintenance program completion rates reported in Table 4. Now, it is obvious that when a network is not maintained, maintenance fees turn into repair fees (breakdown service) with more serious ?nancial consequences, not to mention loss of money in the region's economy, due to electricity failures. For the managers of such systems, this study should enable them, through a better quantitative knowledge of their network, to signi?cantly reduce the probability of failure in MV electrical network. This study can equally be useful for the management of other electrical networks as well, the method used being obviously transposable outside its experimentation zone. However, for each of these networks, in order to achieve a signi?cant reduction in the probability of failure, a maintenance policy based on fundamental malfunction factors, ?rst, of which interruptions, must be implemented. The next step of this research consists in re?ning the analysis of the maintenance for a representative feeder at a station in order to elaborate proposals to improve the situation. Appendix A Figs. 3±8 show the density curves obtained at the different feeders considered in our study. For each feeder, the kernel f~ n;h density evaluator graph appears in solid lines, while the dotted line represents the probability density of the interruption law as evaluated by the least squares method. p To obtain the probability laws on IR1 , each of these two

a r is very small here as well for the two onsets. The evaluation of the resulting law is therefore acceptable (see Figs. 7 and 8).

Fig. 8. Number of monthly interruptions at Melen D13.

? tse ? et al. / Electrical Power and Energy Systems 24 (2002) 859±865 T.T. Tatie Table 4 Non-performance factors: maintenance insuf?ciencies Activities Support legs cleaning Tree trimming Systematic device maintenance (IACM, etc.) Replacement of rotten wooden poles Earthling (any form) Visit of transformer sites Systematic station maintenance Systematic maintenance Company visit Completion rate (%) 22 24.7 25 61.6 8.7 62.5 32.2 10.4 56.7 [8]

865

[5]

[6] [7]

[9]

[10]

densities of the non-truncated law has been divided by the total probability that it affects to the half axis of real numbers .0. Appendix B See Table 4. References
[1] Meeuwsen JJ, Kling WL, Ploem WA. The in?uence of protection system failures and preventive maintenance on protection systems in distribution systems. IEEE Trans Power Deliv 1997;12(1):125±31. [2] Doucet J-A, Oren S. Distributed backup generation and interruption insurance for electricity distribution. Cahier de Recherche du ? Laval, No. 91-21; 1991. p. 30. GREEN, Universite [3] Persoz H, Santucci G, Lemoine J-C, Sapet P. La Plani?cation des ? seaux e ? lectriques. Collection de la Direction des Etudes et re ? lectricite ? de France, Paris; 1984. Recherches d'E ? le ? tse ? T. E ? ments pour une prise en compte de la participa[4] Tamo Tatie ? nages au de ? veloppement des re ? seaux d'eau potables et tion des me

[11]

[12]

[13] [14] [15]

[16] [17] [18] [19] [20]

? lectricite ? dans les villes des pays en de ? veloppement: le cas du d'e ? se de Doctorat. INSA, Lyon; 1995. Cameroun. The Electricity Training Association. Power system protection, vol. 3 applications. Six Hills Ways, UK: The Institution of electrical engineer, 1995. Willis H-L. Power distribution planning. Reference book. New York: Marcel Dekker, 1997. Gates J, Billinton R, Wacker G. Electric service reliability worth evaluation for government, institutions and of?ce buildings. IEEE Trans Power Syst 1999;14(1):43±8. Billinton R, Wojczynski E. Distributional variation of distribution system reliability indices. IEEE Trans Power Appar Syst 1985;104(11):3152±60. ? visionnelle des risques lie ?s a ? l'exEDF. Approche sociologique pre ? seaux moyenne tension. Collection des notes internes ploitation des re ? tudes et recherches, Paris; 1992. de la direction des e ? seaux e ? lectriques: e ? tude de la ?abilite ? Noutat GL. Plani?cation des re ? seaux de distribution de Yaounde ? . Me ? moire de ?n d'e ? tude des re ? nieur, ENSP, Yaounde ? ; 1986. d'inge Billinton R, Goel R. An analytical approach to evaluate probability distributions associated with the reliability indices of electric distribution systems. IEEE Trans Power Deliv 1986;1(3):245±51. Billinton R, Pan J. Optimal maintenance scheduling in a two identical component parallel redundant system. Reliab Engng Syst Safety 1998;59(3):309±16. ? de fonctionnement des syste ? mes industriels. ? rete Villemeur A. Su Paris: Eyrolles, 1988. Milanovic J-V. On unreliability of exponential load models. Electric Power Syst Res 1999;49(1):1±9. ? tude ?abiliste du re ? seau de distribution MT d'e ? nergie Ndjanga TSP. E ? lectrique de la ville de Yaounde ? . Me ? moire de ?n d'e ? tude d'inge ? nieur, e ? ; 1998. ENSP, Yaounde Silverman BW. Density estimation for statistics and data analysis. London: Chapman & Hall, 1986. Dempster AP, Laird NM, Rubin DB. Maximum likelihood from incomplete data via EM algorithm. J R Stat Soc, Ser B 1977;39:1±38. Meilijson I. A fast improvement to the EM algorithm on its own terms. J R Stat Soc, Ser B 1989;51:127±38. ? s et statistiques vol. 2 Dacunha-Castelle D, Du?o M. Probabilite ? mes a ? temps mobile. Paris: Masson, 1990. proble ? thodes statistiques. Paris: Dunod, 1992. Grais B. Me


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