In repairable systems,Atwill be equal to or greater thanRt. A second action is considered and denoted 2band can touch some of the components of a system up to their replacement. For an MV/LV transformer which is a critical item of an electrical substation, the estimated parameters and the adequate probability distribution functions are listed in Table 1.  have generalized the use of LOLP as a reliability index. Subsystem 1 has a reliability of 99.5%, subsystem 2 has a reliability of 98.7% and subsystem 3 has a reliability of 97.3% for a mission of 100 hours. Availability = uptime ÷ (uptime + downtime) Here’s an example of the system availability formula in action: One of your top production assets ran for 100 hours last month. The failure rate of each of the cables is constant at 0.0007 if both cables are operating properly. System availability is calculated by dividing uptime by the total sum of uptime and downtime.Availability = Uptime ÷ (Uptime + downtime)For example, let’s say you’re trying to calculate the availability of a critical production asset. where R0,jis the initial reliability of the jthstage and m1is the improvement factor of action 1a. A common metric is to calculate the Mean Time Between Failures (MTBF). We are IntechOpen, the world's leading publisher of Open Access books. There is a wide variety of issues that should be addressed when talking about material handling system reliability. The reliability function for the exponential distributionis: R(t)=e−t╱θ=e−λt Setting θ to 50,000 hours and time, t, to 8,760 hours we find: R(t)=e−8,760╱50,000=0.839 Thus the reliability at one year is 83.9%. The equivalence relations between degradation states Ω1=M1,…,11,01; Ω2=M2,…,12,02and Ω3=M3,…,13,03,and their corresponding intervals are given as follows: The states’ space of the system is defined by Ωμ=M…10Fwith M+2states. Unfortunately, there is no way to directly observe or calculate the true score, so a variety of methods are used to estimate the reliability of a test. Help us write another book on this subject and reach those readers. which is very reliable. The results were valorized in a great number of international conference proceedings and in valuable international journals. As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access publisher that levels the playing field for scientists across the world. - 8 - Problem 16. Depending on the percent of the survival parts of system when it is maintained, the reliability function is: where R0,jis the initial reliability of the jthstage and RV,jtis the reliability degradation of surviving parts on this stage. 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. calculation of exact system reliability A representation of the structure of a system can be expressed in terms of a network, and some of the material presented in Chap. In some works, the correlation between maintenance policies and reliability aspects was highlighted. Elements are also screws and many other things. Each Reliability Prediction standard offers a set of mathematical formulas to model and calculate the failure rate of a variety of electromechanical components that make up a product or system. After that was introduced the notion of loss of load probability (LOLP), which has a lot of applications in load modeling and electrical parameters dimensioning. For a system state where the remaining generating capacity is Cj, the percentage of time tjduring which the load demand exceeds Cjcan be determined from the load curve L. The overall probability that the load demand will not be met is called the loss of load probability and is given by the Eq. There is a loss of load when the demand is greater than the supply. Parameters are needed to compute the benefits such as the distribution function parameters βηPMand CMtimes (taand tb), maintenance actions costs (C1a, C1b, and C2P) listed in Table 4, and the threshold value of reliability (Rth= 0.8). In the beginning, the methods used were classical to evaluate reliability indices of distribution systems such as failure frequency, mean failure times, mean time between failure and energy not supplied. Rtest associated with some available test time Ttest is then calculated and used in the above 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. The maintenance action to retain is based on the maximum benefit value, and the results at different maintenance stages are listed in Table 5 using the following notations: Rj+1Tm: The instantaneous reliability at j+1×Tm. The decreasing level of system reliability is highlighted by degrading state, corresponding to each decreasing in the SAC curve behavior. The evolution of the probability of the system in degradation state 2 is complementary to that of the probability of the degradation state 3. According to the definition, the initial reliability on the action 1bcan be expressed as: where R0denotes the initial reliability of the new system. Medjoudj et al. Brief introduction to this section that descibes Open Access especially from an IntechOpen perspective, Want to get in touch? Our team is growing all the time, so we’re always on the lookout for smart people who want to help us reshape the world of scientific publishing. A great number of applications were done around the power systems including production, transportation and distribution parts. In a recent work, considering the importance of new technologies integration and renewable energy insertion to power systems, it was revealed that reliability aspects and energy sustainability are two fundamental issues of progress in a given society. Threshold reliability is allocated to the opposite risk of system failure occurrence. Then, the choice of the type of action to perform is dictated by the value of the maximum benefit brought by this action. Consequently, the space of the system is defined as follows: Ω=3210F. The benefit of component maintenance on the jthstage is defined as : where i,kdenote, respectively, the ithsubsystem or component and the maintenance action considered and Ci,k, the action cost. That asset ran for 200 hours in a single month. Parameters estimation of the random variables following Weibull distributions. Load parallel systems. Average Uptime Availability (or Mean Availability) 3. Superimposition of the system available capacity and the load model. Licensee IntechOpen. In this part, we develop a function that generates the relation between the states’ space of the system Ωμand degradation states’ space Ω1Ω2Ω3F. The Reliability and Confidence Sample Size Calculator This calculator works by selecting a reliability target value and a confidence value an engineer wishes to obtain in the reliability calculation. This notion is introduced by Tsai et al. RE: How to Calculate System Reliability VenkatV (Mechanical) 21 Jul 03 08:32 25362 is correct, how ever to add some more details for the system mentioned by you i.e. Therefore, we have the following equations for calculating the system reliability: Using the above formula and setting the reliability of each element at 0.9, we find. In the optimistic case, the availability is greater than the reliability. As this state is transient, its probability function decreases to 0 after 5 years. If at time tj, runits have failed from a total of nidentical and independent installed units in the generating system, the probability pjis given by: A case of unequal size of the units can appear. Thecombined system is operational only if both Part X and Part Y are available.From this it follows that the combined availability is a product ofthe availability of the two parts. A variety of indices have been developed to measure reliability and its cost in power systems area such as loss of load probability (LOLP), loss of load expectation (LOLE), expected frequency of load curtailment (EFLC), expected duration of load curtailment (EDLC), expected duration of a curtailment (EDC), and expected energy not supplied (EENS) . Conversely, given a fixed number of samples, Method 2B solves for test time required.. Toolkit Home 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 context, as advancement in reliability applications, Taboada et al. 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 2p. Typical reliability index values for US utilities are SAIFI, SAIDI, and CAIDI. I T reliability indices in its’ standard number P1366, “Guide for Electric Distribution Reliability Indices”. The results listed in Table 5 can be interpreted as follows: at every maintenance stage, verify for each component if its reliability for the coming stage is greater or equal to the Rth. In this context, two relevant questions are to be asked and the answers were given in [9, 10] with a case study application, such as: What happens when load increases? reliabilityanalytics.com. In several cases, it is defined as an objective to attempt in terms of indices directly related to the customer. 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. MTBF values are usually provided by hardware manufacturers and MTTR will be determined by the processes you have in place for your system. reliability probabilistic approaches have been introduced in some cases for the calculation of the reliability of structures or components. Binomial Distribution Handbook for Scientists and Engineers. Three subsystems are reliability-wise in series and make up a system. Instead of ΩμandΩ1Ω2Ω3F. The LOLP, number of days on which capacity is insufficient, is obtained by adding the probability that the amount of capacity on forced outage, on day iis greater than or equal to the reserve on day i, for all days of the period being studied. A three states diagram (working, failure and maintenance) is dressed for the life cycle modeling of the substation as shown in Figure 2. Reliability Calculation Methodology for k-out-of-n Systems Periodically Maintained In the Table 6.2.1-2: Redundancy Equation for Calculation Reliability, System Reliability Toolkit The novelty in this work is outlined by the use of three-dimensional matrix to show the possible states, where the HVOCB can sojourn. Following the Li and Pham theory , we consider the three state spaces Ω1,Ω2,andΩ3corresponding to the degradation processes Y1t,Y2t,andY3t,respectively. The reliability function is defined as follows: The system will fail if any of the degradation rates exceeds the critical level Gi; i=1,2,3or the process of shock also exceeds the critical level S. 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 Y1t; The process of increasing degradation representing the aging of oil insulating circuit of the circuit breaker is denoted by Y2t; The degradation process of bus bars supports is denoted by Y3t; A random process of cumulative shock damages is given by Dt=∑t=1NtXt. These works were vulgarized with applications in power system reliability by Van Casteren et al. Publishing on IntechOpen allows authors to earn citations and find new collaborators, meaning more people see your work not only from your own field of study, but from other related fields too. Method 2. Comments/Questions/Consulting: Calculate how much downtime should be permitted in your Service Level Agreement or Objective. Sol.) The estimated parameters of the random variables following the Weibull distributions are listed in Table 3. When the system reaches the first state of degradation, it can either remain in this state or go to the second degradation state M−21M−22M−23, or it passes to stateF. The first method is the calculation of the LOLP; there the load is given by the load duration curve. For the system reliability, we note that during the first 8 years, it is expected to decrease by 20% due to the random shock process that governs the system during this period. It has been proven that it is possible to maintain in another way than the classical one based on systematic preventive maintenance. The first method is the calculation of the LOLP; there the load is given by the load duration curve. They are also taken out of service from time to time for preventive maintenance. equation to calculate the number of test samples needed. In the last decade, a novel vision of interruption modeling in power systems was developed and consists of the Weibull-Markov process. The reliability estimates are incorrect if you have missing data.  and Medjoudj et al. The operation can be described by the mathematical function formulated as follows: The function fis defined by: Ω1×Ω2×Ω3⇒f⇒H. Considering Sand Das the supply and the load demand respectively, they compute the reliability of a multistate system (MSS) as: Using the well-known formulation of the LOLP given in several publications and discussed in the following section, they generalized the MSS reliability index R. where the operation period T is divided into M intervals, and each interval has duration Tjand a required demand level Dj. In life data analysis and accelerated life testing data analysis, as well as other testing activities, one of the primary objectives is to obtain a life distribution that describes the times-to-failure of a component, subassembly, assembly or system. The advantageous one will correspond to the maximum of the benefit, that is, Bi∗=MaxBi,k. 2. This issue is also treated in the case of a differentiated service of reliability in the case of customers with different requirements of the reliability level. 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 components constituting a high voltage oil circuit breaker (HVOCB) are subject to various degradations, namely the aging of the insulating oil in the arc’s extinguishing chamber, the contacts wear out and the sharp breakdown of bus bars supports. The objectives are the determination of maintenance frequencies on an item and consequently their costs. It is significant for any power enterprise to analyze customer satisfaction. Once the action of maintenance is defined and retained, the availability of the system at any stage is processed as: where nis the number of components or subsystems and ti,k,ais the time of the PMactions 1a, 1band 2pand T, the cycle time. Depending on the reliability level reached, or fixed a priori, maintenance operations can be decided. 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. The most important reliability indices involving decision-making criteria are given as follows : The Expected Frequency of Load Curtailment in (fault/yr): The Expected Duration of Load Curtailment in (hrs/yr): The Expected Energy Not Supplied in (kWh/yr): whereλk,Tkare failure rate and failure duration of an item kand Lis the load curtailed at a considered load point, respectively. The reliability aspects are formulated in the bases of oil aging, contacts wear and bus bars support degradation. The nonacceptation of the exponential distribution is comforted by the review results of reference , where the authors state that the exponential law, usually used to describe failures, is not always 100% suitable for electricity distribution systems. We note that in the matrix H, some elements are zeros and it can be assumed that when degradation 1 is in a certain advanced state I101