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What happens when load increases? What is the consequence of generating system failure? Almost every electricity utility computes reliability indices on an annual basis. The most important reliability indices involving decision-making criteria are given as follows [ 1 ]:. Application is done for a part of the distribution system of Algiers city Algeria. Considering the electrical characteristics network topology, section length, power value at load points and the fault search method and reliability parameters mentioned earlier, the overall system reliability indices are computed.
To improve the reliability level, technical and organizational measures are considered during system planning and operation. The actions currently carried out are as follows: In [ 11 ], some options are added such as load transfer between feeders, undergrounding circuits, and replacement of aging equipment. From a practical standpoint, this application allows to highlight the goodness of each measure to the system performances by a simple comparison of reliability indices. The results of reliability indices improvement are published in [ 1 ]. In several studies dealing with electrical distribution system reliability, the objective often sought by the energy distributor is the balance between the required reliability level and its cost.
In the following, we develop two important points of view in the reliability of the electrical systems: In the last decade, a novel vision of interruption modeling in power systems was developed and consists of the Weibull-Markov process. The purpose is to model the failure and operating data according to Weibull distribution proprieties, while retaining those assigned to the Markov model where the system occupies discrete states. This process was initially developed by Van Castaren [ 2 ] and was applied successfully by Pivatolo [ 13 ] and Medjoudj et al.
Applications were made to highlight maintenance policies gathered on three types of actions: This action, denoted 1 a as a minor maintenance, is characterized by an improvement factor m 1. A second action is considered and denoted 2 b and can touch some of the components of a system up to their replacement. To this action is associated an improvement factor m 2 , and the maintenance is a major one. The third and final proposed maintenance action is on the renewal of equipment, and it is assumed to be perfect, and after its implementation, the system is assumed as good as new, and it is denoted 2 p.
From a practical standpoint, this action is highlighted by taking m 1 and m 2 equal to the unity. This notion is introduced by Tsai et al. Starting from the expression of reliability function expressed by a desired threshold, the need of performing preventive maintenance action at time is decided regarding the behavior of this function at the coming stage of maintenance. Then, the choice of the type of action to perform is dictated by the value of the maximum benefit brought by this action.
Threshold reliability is allocated to the opposite risk of system failure occurrence. Considering the formulations of mean up time and mean down time of an item is given respectively by:. Subsequently, the PM interval for maximizing the availability can be derived by differentiating Eq. The obtained results show that, based on the Kolmogorov Smirnov KS test [ 15 ], d ks is lower than d n 0. In this study, it is assumed that the substation failures are due either to the transformer or to the internal cable connector failures.
Let X 12 , X 13 , X 1 , X 2 , X 3 be the random variables representing the duration of the operation until failure, the duration of the operation until maintenance, the duration of the operation state S 1 , the duration of the interruption state S 2 and the duration of the maintenance state S 3 , respectively. The improvement of maintenance to reliability is developed using two factors, and the selection of the action to do for the components on every PM stage is decided by maximizing system benefit in maintenance.
Depending on the percent of the survival parts of system when it is maintained, the reliability function is:. Considering periodical PM which interval is t m , the reliability of surviving parts is defined as:. To model the reliability of systems following PM , the effects of various actions on R 0 , j and R V , j must be evaluated.
Evolved from the lectures of a recognized pioneer in developing the theory of reliability, Mathematical Models for Systems Reliability provides a. Evolved from the lectures of a recognized pioneer in developing the theory of reliability, Mathematical Models for Systems Reliability provides a rigorous.
Action 1 b can improve the surviving parts of the system and also recover the failed parts. Generally, the impact of this action on the failed parts can be measured by an improvement factor m 2 , which is also set between 0 and 1 representing the restored level except the surviving parts. According to the definition, the initial reliability on the action 1 b can be expressed as:. The benefit of component maintenance on the j th stage is defined as [ 14 ]:. Once the action of maintenance is defined and retained, the availability of the system at any stage is processed as:. In the following are described the different types of PM actions in the case of the power transformer.
The obtained results are: The action 1 b is retained looking at the maximum value of the benefit. The risk management is highlighted by thresholds of reliability. Depending on the reliability level reached, or fixed a priori, maintenance operations can be decided. The objectives are the determination of maintenance frequencies on an item and consequently their costs. It will be remarked that a high level of reliability is required i. The states were defined using thresholds of degradation parameters.
To the degradation processes was associated a shock process highlighting the effects of short circuit arrivals on the HVOCB when defaults occur at the downstream feeder.
The novelty in this work is outlined by the use of three-dimensional matrix to show the possible states, where the HVOCB can sojourn. It is assumed that at the present time, the system is not at fault condition catastrophic state F. The operation can be described by the mathematical function formulated as follows:. The function f is defined by: The line at the top of the matrix H represents the states of the degradation process 1, the right column of the matrix represents the states of the degradation process 2, and the top page of the matrix H represents the degradation process 3.
We define time until failure by: It is important to know that the life of the system depends on a single process among the three degradations and of that of the shock.
However, the system failure is caused by the process that occurs first, exceeding its critical value corresponding to the level which can bring the system back to failure. Initially, the system is considered in good states of operation M 1 , M 2 , and M 3. The same process is repeated at each degradation stage with the exception of the states 0 1 0 2 0 3. With respect to time t , it corresponds to an irreversible accumulation of damage;. Random shock process the system passes to the condition of the catastrophic failure state F , if: The system subject to three displacement processes is defined by: The process of increasing degradation representing the wear of the contacts of the circuit breaker is denoted by Y 1 t ;.
The process of increasing degradation representing the aging of oil insulating circuit of the circuit breaker is denoted by Y 2 t ;. The degradation process of bus bars supports is denoted by Y 3 t ;. We obtain a system with four competing degradation processes. For the wear of the contacts: Consequently, the space of the system is defined as follows: Thus, the function f is defined as being the Cartesian product of three sets following f: The evolution of the probability of the system in degradation state 2 is complementary to that of the probability of the degradation state 3.
Then, it increases exponentially to reach its maximum value of 0. Recently researchers in electrical systems have proposed differentiated electricity service based on reliability and have shown some inconveniences to apply it into a real case. In the same location are connected both consumers with high reliability requirements, with an agreement to pay more and others who are not concerned. Because the technical measure proposed is to add a reliable feeder, the differentiation is not quite possible.
The differentiated service is directly related to the reliability of the substation where the improvement is a function of maintenance actions and the frequency of interventions. For statistical considerations and for interruption forced or scheduled outages modeling, we have applied the Weibull-Markov approach rather than the Markov method, which is usually used for the case of electrical systems. It has been proven that it is possible to maintain in another way than the classical one based on systematic preventive maintenance.
In this chapter, it is shown that the maintenance is decided on the reliability level and benefit bases.
Another critical component of electrical substation is studied using competing failure processes and consists a circuit breaker. The reliability aspects are formulated in the bases of oil aging, contacts wear and bus bars support degradation. Investigations conducted by Pham in a theoretical framework have been applied successfully to complement system such as electrical system. The models applied on simple numerical examples have been validated by application to a real case engineering.
During system operation, the results analysis of the network current state allows to the decision maker to reach better information and target the equipment that reduces the performances of the system and practicing suitable maintenance actions. Recent studies in energy sustainability and smart energy grid have revealed that reliability is the main criterion taken into account by decision makers in electricity market behavior and a performance index for electric utilities classifications.
In , a group of professors in mathematics and engineering A. From that manifestation was born the group of work in power system reliability FSE2. Over than various works were conducted in engineer, master and doctorate theses dealing with all the aspects of power systems reliability and with a large cooperation with other universities and various manufactures and services. A great number of applications were done around the power systems including production, transportation and distribution parts.
In recent years, a lot of novelties were developed compared to what is done over the word, such as the Weibull-Markov modeling in data analysis, nonparametric distributions in switching components behavior, Box and Jenkins models in blackouts forecasting and reliability aspects in smart grids development and multicriteria optimization.
The results were valorized in a great number of international conference proceedings and in valuable international journals. This chapter dealing with power system reliability constitutes an interesting opportunity to express our acknowledgments and tributes to these pioneers of reliability in Algeria for what they have given for research.
Embed this code snippet in the HTML of your website to show this chapter. Two-unit repairable system Example 5: One out of n repairable systems Example 6: First passage time distribution References Index A Problems and Comments section appears at the end of each chapter. Reviews … The material presented in the book is classic material, but is also timeless because the basic theory, probability and statistical rigor and applications to system reliability are still relevant today and important for any student or practitioner of reliability theory.
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