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Minimal Cut Sets Determination for Renewable Systems with Limited Repairs
Bohdan Yu.Volochiу, Leonid D. Ozirkovskyi, Andriy Mashchak,
Oleksandr Shkiliuk.
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TCSET'2014, February 25 - March 1, 2014, Lviv-Slavske, Ukraine
218
Abstract - Technology for determination minimal cut sets, which are based on Markov model, is presented in the given work. This technology provides possibility for obtaining minimal cuts sets for renewable systems with limited repairs.
Keywords - Reliability, Fault tree analysis, Minimal cut sets, Markov model.
I. Introduction
For modern technical system high level of reliability and safety is required. To provide required level of reliability and safety "weak" places of system should be defined. The set of events that occurs the system elements or subsystems which cause catastrophic failure of the whole system is the "weak" place of system. This set of such events is called cut set or cross section. Minimal cut set (MCS) is the combination of events if, when any of event is removed from the set, the remaining events collectively can not cause catastrophic failure of the whole system [1]. To ensure high safety level the failure mode and effects analysis (FMEA) or its extension – failure mode, effects and criticality analysis (FMECA) is required.
Minimal cut sets could be determined by logical-probabilistic models (LBM). Fault tree (FT) is one of kinds of LBM. Fault tree analysis FTA is top down deductive failure analysis of low-level events that cause the top-level event of catastrophic system failure [2]. However, this method has a number of significant drawbacks. Manual developing of FT requires a lot of time. Also LBM does not allow analysis MCS of renewable systems with limited repairs. Modern approaches allow analysis fault-tolerant technical systems or renewable systems with unlimited repairs.
Minimal cut sets also are obtained by Markov models. However, it should be noted that this analysis requires time-consuming for manual developing of Markov model. But enhanced modeling technology [3] of discreet-continuous stochastic systems makes developing Markov model faster and easier. This technology provides possibility for analysis of renewable systems [4], system with standby doubling [5], systems with load-sharing redundancy of component [6]. Also models of the algorithm behavior of complex radio-electronic systems could be obtained by this technology [7].
Mashchak Andriy, Shkiliuk Olexandr, Ozirkovskyi Leonid, Volochiу Bohdan - Lviv Polytechnic National University Profesorska str., 2, Lviv, 79013, UKRAINE,
E-mail:
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Technique in the given paper is based on enhanced modeling technology [3] and uses methodology presented in [8-9].
II. Main part
Used methodology [8] involves the following stages:
· Developing a binary structural-automatic model (BSAM).
· Building graph of states and transitions (GST).
· Separation the sets of states, in which catastrophic system failure is occurred.
· Determination of MCS and their quantitative values.
Including in SAM vector state, which is called repairs counter, is the feature of developing and is the first stage. In this vector state number of repairs could be input. It can be any integer value. Other vector states in SAM that response of system each element can only be in one of two states – working or non-working. Absorbing catastrophic failure state is splitted.
Developed SAM model is input data for the software ASNA-1. Graph of states and transitions is built in the automatic mode in ASNA-1. System of Chapman - Kolmogorov differential equations based on GST is determined. Probability distribution being in the states is result of solution of differential equations system.
Separation the sets of states, in which catastrophic system failure is occurred, is the next stage. All sets of the GST states are compared with the condition of catastrophic failure. Separated in that way sets are called cut sets.
Using software CSD (Cut Set Definer) minimal cut sets are determined. After determining MCS the probabilities of their occurrence must be calculated. Probabilities of MCS are expressions of sum that include probabilities of the states that are included in MCS.
Obtaining the summary Qsum and general Qgen probabilities of catastrophic system failure is next stage.
Finally the results of analysis for MCS must be presented in tabular form.
III. Application example
Technique of determining MCS was approbated on the renewal system with limited repairs which consists of five modules. The first, second and fourth modules are the basic configuration that provides the implementation of the objective function of a system and two other modules are redundant. All modules has the same failure intensity λ = 0.05, recovery intensity M = 0.1 observation period is 30 hours.
Structural diagram of a system is shown in Figure 1:
Three renewal systems with two, six and thirty repairs of whole system accordingly were modeled and analyzed.
Based on SAM of the systems three GST in automatic mode were obtained by using the software ASNA-1. First graph of state and transitions consists of 96 states, second of 224 and third of 992 states. 33, 78, 371 states accordingly in which catastrophic system failure is occurred were found by using condition of catastrophic failure.
Using software CSD three MCS were obtained, when would these modules fail: fourth and fifth; first, third and fifth; second, third and fifth. It should be noted that in all three models the same MCS were found.
For all three models probabilities of all MCS were obtained and presented in Table I.
TABLE 1
Minimal cut sets report
2 repairs / 6 repairs / 30 repairsN / Elem. / q mean / % / q mean / % / q mean / %
1 / 4,5 / 0,221 / 50,36 / 0,104 / 59,28 / 0,103 / 60,9
2 / 1,3,5 / 0,109 / 24,81 / 0,038 / 21,62 / 0,033 / 19,54
3 / 2,3,5 / 0,109 / 24,81 / 0,038 / 21,62 / 0,033 / 19,54
Based on Table 1 the first MCS - fail of fourth and fifth modules, is the most critical combination of system (Fig.1) because it has biggest percentage of failure probability.
The fifth module is the "crucial" place of this system. Without fifth element failing none of MCS would be happen.
With increasing of repairs quantity percentage of failure probability of two other MCS is reducing.
In Table I sensitivity of the renewal system could be analyzed. Changing quantity of repairs the probabilities of MCS were changed immediately. The optimized repairs values could be obtained by changing of repairs quantity.
This technique allows to obtain time dependencies of MCS which are presented in Figures 2, 3 and 4.
Also presented technology gives possibility to make separate MCS analysis. It means that all of obtained MCS could be analyzed in different configurations of renewable system with different quantity of repairs.
For example, sensitivity of first MCS in which fourth and fifth elements would fail is presented in Figure 5 in three configurations - with 2, 6 and 30 repairs in time dependency.
In Figure 6 sensitivity of second and third MCS are presented. Second and third MCS in analyzed system (Fig.1) has the same probabilities.
IV. Conclusion
The technique of minimal cut sets determining were presented. The usage of this technique for renewable system with limited repairs was shown. Modeling technology of discreet-continuous stochastic systems is elaborated.
Presented technology gives possibility for developers to involve not only whole system repair but repair of separated modules or system submodules. It is possible when developer involves limited repairs counter in structural-automatic model.
As result changes quantity of system repairs have an influence on probabilities of minimal cut sets and their percentage.
This methodology has low time-cost of minimal cut sets obtaining.
References
[1] Kececioglu D., Reliability Engineering Handbook, Volume 2, Prentice Hall, Inc., New Jersey, 1991.
[2] Ern. J. Henley and H. Kumamoto, Reliability engineering and risk assessment. Prentice-Hall, 1981.
[3] Volochiy B.Yu. Algorithm behavior modeling technology of information systems. Lviv: Vydavnytstvo Lvivskoi politekhniky Publ., 2004. – 220 p. [in Ukrainian]
[4] O. Lozynsky, S. Shcherbovskykh, “Determination of failure intensity using Markov reliability model for renewal of the non-redundancy systems”, Przegląd Elektrotechniczny, 85 (2009), No4, pp.89-91.
[5] S. Shcherbovskykh, O. Lozynsky, Ya. Marushchak, “Failure intensity determination for system with standby doubling”, Przegląd Elektrotechniczny, 87 (2011), No5, pp.160-162.
[6] B. Mandziy, O. Lozynsky, S. Shcherbovskykh, “Mathematical model for failure cause analysis of electrical systems with load-sharing redundancy of component”, Przegląd Elektrotechniczny, 89 (2013), No11, pp.244-247.
[7] Volochii B.Y., Ozirkovskyi L.D., Shkiliuk O.P., Mashchak A.V., "Metodyka otsinky pokaznykiv efektyvnosti radioelektronnoho kompleksu monitorynhu povitrianoho prostoru" [Method of efficiency indexes estimation for algorithm behavior of airspace monitoring radioelectronic complex system]. Visnyk Natsionalnoho universytetu "Lvivska politekhnika". Radioelektronika ta telekomunikatsii. – 2013, №766, pp. 192-201.
[in Ukrainian]
[8] Volochii B.Y., Ozirkovskyi L.D., Mashchak A.V., Shkiliuk O.P., Kulyk I.V., "Metodyka rozrahunku minimalnyh sichen’ dlia vidmovostiykyh system na osnovi strukrurno-avtomaynoii modeli" [Method of Computation of Minimal Cut Sets of Fault-Tolerant Systems Based on Structural-Automatic Model], Bulletin of National Technical University of Ukraine. Series Radiotechnique: Radioapparatus Building. ISSN 0203-6584, 2013. – №52,
pp. 38-45. [in Ukrainian]
[9] Volochiу B.Yu., Ozirkovskyi L.D., Mashchak A.V., Shkiliuk O.P. "Defining Minimal Cut Sets Based On Markov Model", 6th International Academic Conference of Young Scientists “Computer Science and Engineering 2013” (CSE-2013), Lviv, – 2013. pp. 90-91.
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TCSET'2014, February 25 - March 1, 2014, Lviv-Slavske, Ukraine