49th European Organization for Quality Congress Topic: Quality Improvement Service Reliability in Electrical Distribution Networks José Mendonça Dias, Rogério Puga Leal and Zulema Lopes Pereira Department of Mechanical and Industrial Engineering, Faculty of Sciences and Technology, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 CAPARICA, Portugal. ABSTRACT For the past few years, reliability has been assuming an increasingly important role in quality improvement and customer satisfaction. Reliability can be defined as the ability of an item to perform a required function under stated conditions for a certain period of time. The item can be a manufactured product or an assembly system such as an electrical process system. In this last example, the quality of the service provided by the system can significantly improve if appropriate reliability techniques are used to improve the overall performance. An electrical system usually comprises three different areas, namely the power generation, transmission and distribution. This paper characterises the reliability of the distribution subsystem (the so-called repartition network). Electrical shortages may be caused by several parameters, which include internal and external network factors or a failure of a key constituent of the system, such as a major distribution line or a substation plant. Furthermore, there are also failures associated with the energy generation and transmission which can be responsible for the blackouts in repartition networks. These faults are addressed as external to the distribution system. For the purpose of this study, the term service reliability denotes the extent to which the electrical needs of the community served by a local 60 kv substation can be met, regardless the set of available equipment at each distribution point. A procedure to determine the service reliability is suggested and is applied to a repartition network. The proportional hazard model (PHM) is used to model the service reliability and evaluate the network substations with high risk of failure. Keywords: Service Reliability; Service systems; Electrical Networks; PHM. INTRODUCTION It is generally accepted that customer s needs and requirements are continuously increasing. Quality is very often defined as conformity to specifications. However, this definition does not specifically address one of the most important features of a product, which is its performance over time. Therefore, a time based quality concept, such as reliability, becomes of extreme importance. Nowadays, reliability is one of the most important quality characteristics for customers and it is applied to products, systems, processes and components of systems as an interdisciplinary concept. Reliability efforts started in some critical areas, such as nuclear facilities and space vehicles, and have rapidly extended to several other areas, such as software reliability, human reliability,
structure reliability and service reliability. Nevertheless, applications of reliability analys is in services are relatively scarce in the available literature (Gunes and Deveci, 2002). Service quality associated to the supply of electrical energy is usually regulated. The regulations establish quality standards such as service continuity (number and duration of breakdowns) and the quality of the voltage wave (amplitude, frequency, wave shape and symmetry of the three phase system). Symmetry is monitored through indicators that are periodically established and assessed. The most used indicators are the SAIFI - System Average Interruption Frequency Index (the number of times a customer experiences a service interruption during one year) and the SAIDI - System Average Interruption Duration Index (the total amount of time a customer does not have power during one year). However, these indicators lead to a service quality analysis that is solely based in average values, which does not provide an integrated analysis of service reliability, making it impossible to assess the probability of an interruption of energy supply be experienced by a group of customers. The electrical shortages are generally originated by a complex set of factors, which may be internal or external to the repartition network. The most important external factor is related to breakdowns in the transmission network that cause significant perturbations in the distribution system. This study intends to assess the service reliability of a 60 kv repartition network in order to identify conditional probabilities of failure in the network substation and its influence on system failures. The term service reliability denotes the extent to which the electrical needs of the community served by a local 60 kv substation can be met. A procedure to determine the service reliability is suggested and is applied to a repartition network in the next sections. The proportional hazard model (PHM) is used to model the service reliability and evaluate the network substations with high risk of failure. METHODOLOGY There is a large group of factors that might cause failures in a repartition network, thus making it difficult to evaluate the reliability of a substation. Figure 1 shows a repartition network composed by several substations and inter-connections. They constitute a set that must be carefully managed to provide adequate service reliability levels. The proposed methodology assumes that each substation corresponds to an energy delivery point directly connected to an injection point of 150kV in the energy transmission network.
Figure 1 Repartition network Figure 2 depicts the repartition network constituted by the injection point (SE1502) of the transportation network and the substations that are supplied by it. Figure 2 Radial distribution of substations If the sequence of failures in each substation is known, it is possible to identify those with higher probability of failure and therefore with lower reliability. This approach assumes that all substations are equal, regardless the available equipments in each one. In fact, from a customer s standpoint, service reliability is what is perceived, no matter the number and condition of the equipments in each substation. Some other techniques have been applied to model service reliability (Gunawardane, 2004), but the PHM (Proportional Hazard Modelling), which is often utilised in maintenance management, seems to be a valuable tool for modelling the behaviour of each substation (Dias, 2002).
PROPORTIONAL HAZARD MODEL Cox (1972) introduced a non-parametric free distribution model known as PHM (Proportional Hazards Model). The model was structured according to the traditional hazard function defined by h ( t z) ( t < T t + t T t, z ) P ; = lim (1) t 0 t or, generally, h ( t ; z ) = h0( t) g( z, ß) (2) The not specified baseline function ( t ) h 0 is multiplied by the effect of the covariates (load factors or others that can significantly influence the probability of failure of the system). The coefficients vector ß is estimated by maximising the maximum partial likelihood function. The hazard function is equal to the baseline function ( t ) h 0 when z = 0. The PHM constitutes a class of models in which different fixed covariates must have proportional hazard function, that is, the ratio h ( t; z1 ) h( t; z2 ) of the hazard functions is constant over time. Under these circumstances the PHM is given by ( t; z) = h0 ( t ) exp( z ß) h (3) T The model contains two unknown elements: the regression coefficients vector ß = ( β,..., β ) related to the p covariates considered in the model and the baseline hazard function ( t ) example, if the covariate corresponding to coefficient β 2 is significant, the value given by exp z β represents the hazard ratio relatively to the baseline covariate. ( ) 2 2 To estimate ß, Cox (1972) proposed the use of the maximum partial likelihood function L ( ) = k z e i ß i= 1 e z l ß l R( ) ß (4) t i where k is the observed number of failure times and R ( t i ) the number of elements at risk t, that is, in R ( t i ) = { i, ( i + 1),..., k }. immediately before i The estimate of ß maximum likelihood can be obtained from equations ln ( ) [ L( ß )] U j ß = (5.a) ß j and h 0 1 p. For
[ L( ß )] 2 ln Ihj ( ß ) = (5.b) ß h ß j by using the iterative procedure of Newton-Raphson (Kalbfleisch and Prentice, 1980). There are some software packages able to deal with the PHM. One of the most used is SAS (SAS Institute Inc.), which is employed to model the case study presented in this article. CASE STUDY The repartition network presented in Figure 2, where 247 failures had occurred, will be modelled by PHM. The injection network SE1502 is constituted by eight substations and seven of them (SE210, SE212, SE213, SE214, SE215, SE216 e SE217) are considered to be the covariates of the model. Substation SE211 is excluded because it is very recent and the record of failures is not available. Table 1 presents the estimates of the coefficients for the seven covariates. The elimination through the backward procedure (Table 2) reveals as significant the covariates corresponding to substations SE210, SE212 and SE215. The data show that substation SE212, on its own, increases the network s failure risk by 73%, while substation SE 210 raises it by 62%. Table 1. Model with SE1502 substations as covariates Covariate Test Without With Covariates Covariates Chi-Square Pr > ChiSq -2LnL( b ) Statist score Wald test 1918.299 ------ ------ 1887.838 ------ ------ 30.461 w/ 7 d.f. 26.874 w/ 7 d.f. 25.013 w/ 7 d.f. d.f. Testing Global Null Hypothesis: BETA=0 Analysis of Maximum Likelihood Estimates Parameter Standard Wald Hazard Pr > ChiSq Estimate Error Chi-Square Ratio SE210 1 0.578 0.195 8.747 0.003 1.782 SE212 1 0.647 0.283 5.241 0.022 1.909 SE213 1 0.344 0.346 0.990 0.320 1.411 SE214 1-0.201 0.416 0.234 0.629 0.818 SE215 1-0.870 0.394 4.882 0.027 0.419 SE216 1-0.169 0.414 0.167 0.683 0.845 SE217 1 0.473 0.298 2.529 0.112 1.605
Covariate Test d.f. Table 2. Final model for SE1502 substations Testing Global Null Hypothesis: BETA=0 Without Covariates Covariates -2LnL( b ) 1918.299 1892.100 26.200 c/ 3 g.l. Score Estatístico ------ ------ 23.285 c/ 3 g.l. Teste de Wald ------ ------ 21.754 c/ 3 g.l. Pr > ChiSq Parameter Standard Wald Hazard Pr > ChiSq Estimate Error Chi-Square Ratio SE210 1 0.480 0.149 10.336 0.001 1.616 SE212 1 0.549 0.253 4.726 0.030 1.732 SE215 1-0.942 0.378 6.206 0.013 0.390 With Analysis of Maximum Likelihood Estimates Chi-Square SERVICE RELIABILITY As mentioned before, the analysis here made considers that the reliability of the electrical energy distribution only depends of the breakouts sequence and does not take into account their duration. The previous section led to the important conclusion that only SE210, SE212 and SE215 are significant, so the assessment will just focus on these substations. Figure 3 depicts the service reliability of the three substations. SE210 SE212 SE215 BASE 1.0 Service Reliability 0.8 0.6 0.4 0.2 0.0 0 50 100 150 200 250 300 350 400 450 500 Time since last failure (Days) Figure 3 Service reliability by substation The baseline is obtained for the entire set of significant covariates, revealing a 70% reliability value for a forty day period of energy supply. The reliability of substations SE210 and SE212 is about 60% for the same period. However, the most relevant situation occurs in substation SE215 with a negative coefficient corresponding to a decrease of about 61% in its hazard function. This situation is revealed by an increase in reliability of approximately 90% for the aforementioned
period. So, a deep study should be conducted to compare this last substation with the others and determine the best way to explore the substations with lower reliability values. MODEL VALIDATION Models similar to PHM can be validated through the analysis of different sorts of residuals (Collett, 1994). The Cox-Snell s residuals, which are quite useful and widely applied, are defined by eˆ i = ln Rˆ t i ( z ) i (6) The ê i must have an exponential distribution withλ = 1 (if t i is a censored o bservation, then ê i is also considered as censored) if the model is correctly adjusted. The graphical representation of ln [ R( ˆ )] e i (which are obtained through the Kaplan-Meier method) against ê i must be adjusted to a straight line of unitary slope (Figure 4). ) lnr e i [ ( )] 5.0 4.0 3.0 2.0 1.0 0.0 0.0 1.0 2.0 3.0 4.0 5,0 Observed residuals Figure 4 - Cox-Snell residuals The residual analysis reflects an adequate behaviour of the model, despite the deviation of some observations, which is justified by the censored data (9%). CONCLUSION This article makes use of the Proportional hazards Model to assess the reliability of electrical energy distribution service. A general conclusion is that it is possible to make use of modelling techniques usually associated to product reliability. The model identified the three more significant substations among the seven initially considered, regardless the local atmospheric
conditio ns and the equipment used by each one. The SE215 substatio n was found to be the one with higher reliability, thus assuring a better service quality to the customers. As regards substations SE210 and SE212, one can see from Figure 1 that their configurations are quite distinctive. Therefore, it is advisable to carry out a detailed study of each substation, so that the critical points may be identified and a revision of the maintenance programme and/or the equipment replacement can be undertaken. REFERENCES Collett, D. (1994), Modelling Survival Data in Medical Research, Chapman & Hall, London. Cox, D.R. (1972), Regression Models and Life Tables (with Discussion), Journal of the Royal Statistical Society (B), Vol.34, No.1, pp. 187-220. Dias, J.M. (2002), Fiabilidade em Redes de Distribuição de Energia Eléctrica (Reliability of Distribution Electrical Networks), PhD thesis, FCT/Universidade Nova de Lisboa, Portugal. Five-Year Electric Service Reliability Study 1998-2002. Electrical Safety and Reliability Section. Oregon Public Utility Commission. http://www.puc.state.or.us. Gunawardane, G. (2004), Measuring reliability of service systems using failure rates: variations and extensions, International Journal of Quality & Reliability Management, Vol. 21 No. 5, 2004, pp. 578-590. Gunes, M. and Deveci, I. (2002), Reliability of service systems and an application in student office, International Journal of Quality and Reliability, Vol. 19 No. 2, pp. 206-14. Kalbfleisch, J. D. and Prentice, R. L. (1980), The Statistical Analysis of Failure Time Data, John Wiley & Sons, Inc., New York. Lawless, J.F. (1982), Statistical Models and Methods for Lifetime Data, John Wiley & Sons, Inc., New York.