Figures
Abstract
The reliability of the electrical distribution system is a contemporary research field due to diverse applications of electricity in everyday life and diverse industries. However a few research papers exist in literature. This paper proposes a methodology for assessing the reliability of 33/11 Kilovolt high-power stations based on average time between failures. The objective of this paper is to find the optimal fit for the failure data via time between failures. We determine the parameter estimation for all components of the station. We also estimate the reliability value of each component and the reliability value of the system as a whole. The best fitting distribution for the time between failures is a three parameter Dagum distribution with a scale parameter and shape parameters
and
. Our analysis reveals that the reliability value decreased by 38.2% in each 30 days. We believe that the current paper is the first to address this issue and its analysis. Thus, the results obtained in this research reflect its originality. We also suggest the practicality of using these results for power systems for both the maintenance of power systems models and preventive maintenance models.
Citation: Alwan FM, Baharum A, Hassan GS (2013) Reliability Measurement for Mixed Mode Failures of 33/11 Kilovolt Electric Power Distribution Stations. PLoS ONE 8(8): e69716. https://doi.org/10.1371/journal.pone.0069716
Editor: Derek Abbott, University of Adelaide, Australia
Received: April 17, 2013; Accepted: June 3, 2013; Published: August 1, 2013
Copyright: © 2013 Alwan et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was supported by the USM Incentive Grant: 1001/PMATHS/823005. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Reliability and Failure Functions
Electric power is transmitted through an electric circuit. The terms “high voltage” and “high power” indicate that the voltages of the electrical energy are high enough to inflict harm or death on living things. Electrical power systems are highly complex and extremely integrated. Reliability is one of the most important factors considered in the planning, design, operation, and maintenance of electric power systems [1], [2]. This factor is one of the most effective indicators of product quality that buyers take into account when choosing among different varieties [3]. Moreover, reliability generally becomes more important to consumers, as failure, repair, and maintenance entail expensive costs [3]. The reliability function is a mathematical and engineering indicator that is used to describe the state of the equipment in the system through the probability function. Many factors and definitions are related to reliability (e.g., mean time to failure [MTTF], mean time between failures [MTBF], and mean time to repair [MTTR]). The MTTF is the expected value representing the return period of equipment failure [4]–[7]. It can be expressed mathematically as [8], , where E(t) is the expected value of time, and f(t) is the probability density function (pdf) for variable t. The MTTF is also referred to as the expected life. The mean time between failures (MTBF) and the MTTR are defined in Section Availability. The term reliability can be defined in many ways. For example, for an electrical switch, reliability may be defined as the probability that it successfully functions under a stipulated load and at a specific temperature. An operational definition of reliability must be sufficiently precise to establish a clear distinction between reliable and unreliable items. In addition, this definition must be sufficiently general to account for the complexities that arise in making this determination [8]. Based on this definition of reliability, reliability analyses often involve the analysis of binary outcomes (0, 1) (i.e., success = 1, failure data = 0) [8].
Assume that the period of failure T is a continuous random variable with values in a positive real line. Many methods are available to specify the properties of a random variable [8]. The first method involves using the pdf, f(t), that satisfies, and
.
When T is a Dagum random variable, its pdf is [9]–[11](1)in which
,
is the scale parameter (
),
(
) and
(
) are the shape parameters.
A second method to specify the properties of T is the cumulative distribution function. Mathematically, this function is expressed as [12].
where f(s) is a pdf. The cumulative distribution function is the complement of the reliability function, and thus, it is called the unreliability function [8]. The cumulative distribution function for a Dagum random variable is
(2)A third method to specify the properties of a random variable is through its reliability function, also known as the survival function [8]. We define the reliability function as
where f(s) is a pdf. The reliability function for a Dagum random variable is [9], [13], [14].
(3)
The fourth method specifies the properties of a random variable as the hazard function, also called the instantaneous failure rate function (further details are provided in [8]).
The hazard function for a Dagum random variable is [14](4)
The functions f(t), F(t), R(t), and h(t) are called “failure functions.”
Availability
At first, we have to define several factors that are closely associated with availability (e.g., failure, availability, and so on). Failure is defined as the incapability of the system (subsystem or one of its components) to perform its job [15], [16] or the “inability of the item to meet the requirements of the work”[17], [18]. The term “available” is defined as the state of an item such that it can perform its function under stated conditions of use and maintenance in the required location [19]. Most researchers define availability as the probability that an item will be available [19], [20] or the probability that the system will operate satisfactorily at any point in time when operating under a specified condition [20], [21].
where UT is the uptime or operating time, DT is downtime (excluding free time), MTBF is the mean time between failures, and MTTR is the mean time to repair. For a more accurate quantity, “inherent availability” is defined as [19], [21]:where ART is the active repair time. The MTBF is defined by Frankel, Dinesh and Bryant [15], [22], [23] as a parameter of basic reliability of the repairable components. It is the ratio of the total number of life units for components of the total number of failures. MTTR is the mean time to repair, it is defined as the whole time required to manage the failure, including factors: the way in which the fault is detected and the response speed of the maintenance team with the repair time [24]. The mean corrective maintenance time is defined by Dhillon [25] as the main criterion of the maintainability of repairable items. It represents the average (mean) time required to repair failed equipment. This criterion can be observed based on the inherent availability equation wherein availability is integrated between reliability and the times for maintenance and repair [26]. This relationship is very important because it is used to express the probability that the system will be operating according to the mission time without failure [26].
Description of the Problem
The electric power distribution station in Iraq was designed to have two power transformers (T and T
). Each transformer has a circuit breaker with limited capacity (1,200 A), denoted as CBT
functions as a main circuit breaker of the transformer. These two transformers are connected to the communication bus situated between them (termed as Bus-Bar). The Bus-Bar feeds group of feeders (10 feeders). The Bus-Bar was divided into two parts separated by circuit breaker with a limited capacity of 800 A, called the Bus-Bar circuit breaker (CBB). Each one of the ten feeders has a circuit breaker with a limited capacity of 400 A, called as (CBF
). The main circuit breakers must be switched ON and the CBB must be switched OFF if the transformers are in normal operation. However, if one of these transformers fails, the CBT of the failed transformer must be switched OFF, and the CBB is switched ON to provide power to the failed transformer feeders.
Data Collection
Five years data of the electricity distribution company in Baghdad, Iraq, were collected for time between failures (TBFs) of the electric power distribution station. The failure data were recorded manually. To deal with this problem, we reported the number of breakdowns within five years, and also the time between them. This is shown in Table 1. The first column in Table 1 represents failures numbers, and second column mentioned to the number of days require for the station to step-down. For example, the first TBF value was calculated between 12 am on 1 January and 11:50 pm on 14
April. Accordingly, the first step-down was occurred after (104.895833).
Each component of the station can be failed in random manner. The component average time between failures was modeled to be random variable following certain distribution [2]. EasyFit is the distribution fitting software that can be used to fit the appropriate statistical distribution for the TBFs. In the next section we will focus on the goodness of fit to find the best fitting statistical distribution for each component of TBFs.
Goodness of Fit
In this paper a goodness of fit for the TBFs statistical distribution was tested. Such test can be done using many tools. EasyFit software was used to perform this task. It includes the using of the Kolmogorov-Smirnov, Anderson-Darling and Chi-square test. The idea behind the goodness of fit tests is to have the “distance,” critical values, measured between the data and the distribution being tested. Then that critical value is compared to some threshold value. The goodness of fit reports includes the test statistics and the critical values calculated for various significance levels ( = 0.2, 0.1, 0.05, 0.02, 0.01). Furthermore, the goodness of fit test statistics indicates the distance between the data and the provided distributions [27]. The P-value can be helpful specifically when the null hypothesis is rejected at all selected significance levels, where the P-value is criteria uniformity between the results actually obtained in the experiment and the random chance explanation for those results [28]–[30]. It is required to know at which level it could be accepted [31]. EasyFit deals with data using histogram based on TBFs samples. The number of vertical bars was based on the total number of observations (27 values). The equation
, was used to find the number of bins (histogram), where N is the total number of TBFs and Q is the resulting number of classes [32]. The height of each histogram bar indicates how many data points fall into that class. To obtain the best fitting model, we chose various distributions. Our analysis reveals that the distribution with the lowest statistical value is the best-fitting model.Similar conclusion is also drawn by Fakhraei [33], This support the validity of our analysis. Based on this fact, each distribution is ranked (1 = the best model, 2 = the next best model and so on). The data was analyzed and tested under several nonnegative distributions using the EasyFit software. Dagum distribution is the optimal analysis of the TBFs, with scale parameter
= 90.001 and shape parameters
= 0.33998 and
= 2.4011. Table 2 shows the summary of the goodness of fit of TBFs for the (39) nonnegative distributions. Table 3 shows the goodness of fit details of TBFs for Dagum distribution. Figure 1 shows only the six much closer distributions from all 39 nonnegative distributions. Figure 2 shows the fitting result of TBFs histogram with the Dagum distribution while Figure 3 (a, b, c and d) shows the failure functions (Pdf, CDF, Reliability function and Hazard function) respectively, of Dagum distribution (
= 0.33998,
= 2.4011,
= 90.001).
,
and
, (a) Pdf, (b) CDF, (c) Reliability function and (d) Hazard function.
Maximum Likelihood Estimation
The best result of goodness of fitting to the TBFs under many distributions using EasyFit software is the Dagum distribution. Alwan et al. [34] provides a more detailed treatment of the fitting method. The maximum likelihood method is used to estimate the parameters ,
,
of the Dagum distribution. The likelihood function,
, from a generic distribution with density and reliability functions
and
, respectively, can be written as [9].
where is the parameter(s) of the distribution and
. Consider a sample of size n (which is 27 samples in this paper). The log-likelihood function for the estimate of the parameters
of the Dagum distribution is given by Domma et al. [9]. That is, the log-likelihood function,
, based on data from Eq.1 is [9].
(5)The MLEs
are obtained from the numerical maximization of Eq.5, since the solution of the maximum likelihood equations is not in closed form [9]. Using Eq.5 the values of the estimated
parameters for each component of the station relying on the maximum likelihood method are presented in Table 4.
Reliability Assessment
Figure 4 shows the reliability block diagram for the electric power distribution station. It also represent the visualization of the components working. The reliability function for a Dagum random variable was provided in Eq.3. In the section Problem Statement, the CBB does not function if, and only if, one of the transformers do not operate. Fourteen different components exist, excluding the CBB. Based on Figure 4, the following classifications of the block diagram reliability of the system for the electric power distribution station have been described as
First Group(FG)
Transformers 1 and 2 as well as the main circuit breaker are connected together in a series. At the same time, the two transformers (Transformers 1 and 2) and their main circuit breaker are connected in a parallel manner (see Figure 4). The reliability function of this group is expressed as(6)
where and
are the reliability of transformer i and of the main circuit breaker of transformer i, respectively, during the period t.
Second Group (SG)
The feeders are connected in a parallel manner, indicating that(7)
where is the reliability of feeder i during the period of t.
The group of transformers and the group of feeders are connected in a series. The reliability function of the system is(8)
Based on the values presented in Table 4 and by using Eqs. 3, 6, 7, and 8, we can calculate the system reliability for the times imposed from t = 1 to t = 30, where t is expressed in days. The data are presented in Table 5.
Limitations of the Study, Open Questions, and Future Work
We believed that the limitations are:
- For the sake of brevity, we restrict our investigation to one electric power distribution station.
- We have used the data for five consecutive years.
- We used EasyFit software for our investigation.
- The study focused in details inside the electric power distribution station, without return to the source. Note that if the source feeds the electric power distribution station by low energy (less than 33 kilovolt), this leads to a high temperature in the transformer which will cause a sudden stop of power station.
The present paper deals with the electric power distribution station as independent and separate components. If one take the data of failure rate () for each components of this station and deal with
by using the Markov model “Hidden Markov model.” Then one may get better performance of the electric power distribution station. It is know that the Markov model dependent on the current state of the failure, rather returning to the history of the data [35].
The current paper may be extended to scrutinize preventive maintenance modelling and to estimate its effects on the components of the station. This might improve the supply of electrical energy and will reduce the operating cost of the power station.
Conclusion
The time between failures was analyzed to determine the best-fitting distribution. Using the distribution fitting software EasyFit, we determined that the most valid distribution is the Dagum distribution with a scale parameter , shape parameters
and
. The reliability value for the system on the first day was 0.99. If the station works for 30 days, the reliability value of station was decreased to 0.61. The value of the reliability function was declined by 38.2% in 30 days. This percentage indicates that the electric power distribution station studied in this paper exposed to fail close together in time, even in the same part. This leads to two possibilities; the first is that, the maintenance staff and engineers are not doing their work at best performance, the second is that parts for the electric power distribution station, consisting of 14 components are not good and are exposed to crash shortly after repairs. The first reason can be ignored, because, the field visits by researcher can emphasize the expertise of engineers and maintenance workers in the completion of repairs and maintenance in record time and dynamically good as mentioned in the records. The main problem is that the sum of component is not good. With reference to the case of calculating the Eqs. 6 and 7 for the reliability value for each component, it can be seen that the items CBT
, CBF
, CBF
, CBF
, CBF
, CBF
, CBF
and CBF
have the largest percentage in the value of reliability. Their reliability values after 30 days become as, 0.42, 0.45, 0.40, 0.37, 0.47, 0.41, 0.40, and 0.38 respectively. These values show that the components of the electric power distribution station continuously deteriorate because of aging. Furthermore, these components CBT
, CBF
, CBF
, CBF
, CBF
, CBF
, CBF
and CBF
had the highest critical value and must be changed with new items as soon as possible.
Author Contributions
Conceived and designed the experiments: FMA. Performed the experiments: FMA. Analyzed the data: FMA GSH. Contributed reagents/materials/analysis tools: FMA GSH. Wrote the paper: FMA AB GSH. Data collected from the electricity company: FMA.
References
- 1.
Billinton R, Allan RN (1996) Reliability Evaluation of Power Systems. New York, USA: Plenum Press.
- 2.
Alwan FM, Baharum A, Hasson ST (2012) Reliability and failure analysis for high power station based on operation time. In: International Conference on Statistics in Science, Business, and Engineering (ICSSBE). IEEE, 370–373.
- 3. Garvin DA (1996) Competing on the eight dimensions of quality. IEEE Engineering Management Review 24: 15–23.
- 4.
Zio E (2007) An introduction to the basics of reliability and risk analysis, volume 13. Danvers, USA: World Scientific Publishing Company.
- 5. Kullstam PA (1981) Availability, MTBF and MTTR for repairable M out of N system. IEEE Transactions on Reliability 30: 393–394.
- 6. Schneeweiss W (1981) Computing failure frequency, MTBF & MTTR via mixed products of availabilities and unavailabilities. IEEE Transactions on Reliability R-30: 362–363.
- 7. Holtz J, Werner KH (1990) Multi-inverter ups system with redundant load sharing control. IEEE Transactions on Industrial Electronics 37: 506–513.
- 8.
Hamada MS, Wilson AG, Reese CS, Martz HF (2008) Bayesian Reliability. New York, USA: Springer.
- 9. Domma F, Giordano S, Zenga M (2011) Maximum likelihood estimation in dagum distribution with censored samples. Journal of Applied Statistics 38: 2971–2985.
- 10. Dagum C (2008) A new model of personal income distribution: specification and estimation. Modeling Income Distributions and Lorenz Curves 5: 3–25.
- 11. Dagum C (1980) The generation and distribution of income, the lorenz curve and the gini ratio. Economie Applique 33: 327–367.
- 12. Meyer J, Ormiston MB (1983) The comparative statics of cumulative distribution function changes for the class of risk averse agents. Journal of Economic Theory 31: 153–169.
- 13.
Domma F, Giordano S, Zenga M (2009) The fisher information matrix in doubly censored data from the Dagum distribution. Technical report, Università Della Calabria.
- 14. Domma F (2002) Landamento della hazard function nel modello di dagum a tre parametri. Quaderni di Statistica 4: 1–12.
- 15.
Frankel EG (1988) Systems Reliability and Risk Analysis. New York, USA: Springer.
- 16. Purcell IF, Poole-Wilson PA (1999) Heart failure: why and how to define it? European journal of heart failure 1: 7–10.
- 17.
Carter ADS (1997) Mechanical Reliability and Design. New York: John Wiley & Sons, third edition.
- 18. Davison A, Kantor RN (1982) On the failure of readability formulas to define readable texts: A case study from adaptations. Reading Research Quarterly 17: 187–209.
- 19.
Carter ADS (1986) Mechanical Reliability. London: Macmilan Education Ltd, second edition.
- 20. Audin G (2002) Reality check on five-nines. Business Communications Review 5: 22–27.
- 21.
Martz H, Waller R (1982) Bayesian Reliability Analysis. New York, USA: John Wiley & Sons, Inc.
- 22. Dinesh Kumar U, Knezevic J, Crocker J (1999) Maintenance free operating periodan alternative measure to mtbf and failure rate for specifying reliability? Reliability Engineering & System Safety 64: 127–131.
- 23. Bryant CM, Schmee J (1979) Con_dence limits on mtbf for sequential test plans of mil-std 781. Technometrics 21: 33–42.
- 24.
Gaj F, Germani GU (2008) Dealing with availability in an international service management sce-nario, Springer. 235–244.
- 25.
Dhillon BS (2006) Maintainability, Maintenance, and Reliability for Engineers. New York, USA: CRC Press.
- 26.
Paul Barringer P (2004) Analyzer reliability. In: 18th International Forum for Process Analytical Chemistry. Virginia, USA, 2–9.
- 27. Alwan F, Baharum A, Hasson S (2012) The pdf fitting to time between failure for high power stations. Applied Mathematical Sciences 6: 6327–6339.
- 28. Goodman S (2008) A dirty dozen: Twelve p-value misconceptions. Seminars in Hematology 45: 135–140.
- 29. Chaubey YP (1993) Resampling-based multiple testing: Examples and methods for p-value ad-justment. Technometrics 35: 450–451.
- 30. Hedges SB (1992) The number of replications needed for accurate estimation of the bootstrap p value in phylogenetic studies. Molecular Biology and Evolution 9: 366–369.
- 31. Jha SK, Dwivedi A, Tiwari A (2011) Necessity of goodness of fit tests in research and development. International Journal of Computer Science and Technology 2: 135–141.
- 32.
EASY-FIT (2013) Software for parameter estimation. University of London, UK. URL departments/math/kschittkowski/easyfit.htm.
- 33.
Fakhraei Roudsari F (2011) Spatial, temporal and size distribution of freight train time delay in Sweden. Ph.D. thesis, KTH, Sweden.
- 34. Alwan FM, Baharum A, Hasson ST (2013) The performance of high-power station based on time between failures (TBF). Research Journal of Applied Sciences, Engineering and Technology 5: 3489–3498.
- 35.
Ibe O (2008) Markov Processes for Stochastic Modeling. California, USA: Academic press.