Quantitative Reliability Analysis

Transcription

1 Quantitative Reliability Analysis Moosung Jae May 4, 2015

2 System Reliability Analysis System reliability analysis is conducted in terms of probabilities The probabilities of events can be modelled as logical combinations or logical outcomes of other random events Graphical methods include Failure Modes and Effects Analysis (FMEA) Reliability Block Diagrams (RBD) Master Logic Diagrams etc Two main methods used include: Fault tree analysis Event tree analysis

3 Failure Modes and Effects Analysis Failure modes and effects analysis (FMEA) is a qualitative technique for understanding the behaviour of components in an engineered systems The objective is to determine the influence of component failure on other components, and on the system as a whole It is often used as a preliminary system reliability analysis to assist the development of a more quantitative event tree/fault tree analysis FMEA can also be used as a stand-alone procedure for relative ranking of failure modes that screens them according to risk i.e., as a screening tool

4 FMEA (cont d) As a risk evaluation technique, FMEA treats risk in it true sense as the combination of likelihood and consequences However, strictly speaking, it is not a probabilistic method because it does not generally use quantified probability statements Rather, failure mode occurrences are described using qualitative statements of likelihood (e.g., rare vs. frequent etc.) Consequences are also ranked qualitatively using levels or categories e.g., ranging from safe to catastrophic FMEA uses a rank-ordered scale of likelihood with respect to failure mode occurrence, so that together with the consequence categories, a rank-ordered level of relative risk can be derived for each failure mode

5 FMEA (cont d) FMEA consists of sequentially tabulating each component with all associated possible failure modes impacts on other components and the system consequence ranking failure likelihood detection methods compensating provisions Failure modes effect and criticality analysis (FMECA) is similar to FMEA except that the criticality of failure is analyzed in greater detail

6 Example Example: Consider the following water heater system used in a residential home. The objective is to conduct a failure modes and effects analysis (FMEA) for the system.

7 Solution (cont d) Define consequence categories as I. Safe no effect on system II. Marginal failure will degrade system to some extent but will not cause major system damage or injury to personnel III. Critical failure will degrade system performance and/or cause personnel injury, and if immediate action is not taken, serious injuries or deaths to personnel and/or loss of system will occur IV. Catastrophic failure will produce severe system degradation causing loss of system and/or multiple deaths or injuries The FMEA is shown in the following table

8 Component Pressure relief valve Failure Effects on Solution Mode other Jammed open Jammed closed components Increased gas flow and thermostat operation Effects on whole system Loss of hot water, more cold water input and gas Consequence Category I - Safe Failure Likelihood Reasonably probable Detection Method Observe at pressure relief valve None None I - Safe Probable Manual testing Compensating Provisions Shut off water supply, reseal or replace relief valve No conseq. unless combined with other failure modes Gas valve Jammed open Burner continues to operate, pressure relief valve opens Water temp. and pressure increase; water turns to steam III - Critical Reasonably probable Water at faucet too hot; pressure relief valve open (obs.) Open hot water faucet to relieve pres., shut off gas; pressure relief valve compensates Jammed closed Burner ceases to operate System fails to produce hot water I - Safe Remote Observe at faucet (cold water) Thermostat Fails to react to Fundamentals of Reliability temp. operate, M. Pandey, University of Waterloo rise Burner continues to pressure relief valve opens Water temp. rises; water turns to steam III - Critical Remote Water at faucet too hot Open hot water faucet to relieve pressure; pressure relief valve compensates Fails to react to temp. drop Burner fails to function Water temperature too low I - Safe Remote Observe at faucet (cold water)

9 Reliability Block Diagrams Most systems are defined through a combination of both series and parallel connections of subsystems Reliability block diagrams (RBD) represent a system using interconnected blocks arranged in combinations of series and/or parallel configurations They can be used to analyze the reliability of a system quantitatively Reliability block diagrams can consider active and stand-by states to get estimates of reliability, and availability (or unavailability) of the system Reliability block diagrams may be difficult to construct for very complex systems Fundamentals of Reliability M. Pandey, University of Waterloo

10 Series Systems Series systems are also referred to as weakest link or chain systems System failure is caused by the failure of any one component Consider two components in series 1 2 Failure is defined as the union of the individual component failures For small failure probabilities where Q denotes the probability of failure

11 Series Systems (cont d) For n components in series, the probability of failure is then Therefore, for a series system, the system probability of failure is the sum of the individual component probabilities In case the component probabilities are not small, the system probability of failure can be expressed as For n components in series

12 Series Systems (cont d) Reliability is the complement of the probability of failure For the two components in series, the system reliability can be expressed as Assuming independence For n components in series Therefore, for a series system, the reliability of the system is the product of the individual component reliabilities

13 Parallel Systems Parallel systems are also referred to as redundant The system fails only if all of the components fail Consider two components in parallel 1 2 Failure is defined by the intersection of the individual (component) failure events Assuming independence

14 Parallel Systems For n components in parallel, the probability of failure is then Therefore, for a parallel system, the system probability of failure is the product of the individual component probabilities The reliability of the parallel system is For n components in parallel, the system reliability is

15 Example Problem Example: Compute the reliability and probability of failure for the following system. Assume the failure probabilities for the components are Q 1 = 0.01, Q 2 = 0.02 and Q 3 = Solution: First combine the parallel components 2 and 3 The probability of failure is The reliability is

16 Solution (cont d) Next, combine component 1 and the sub-system (2,3) in series The probability of failure for the system is then The system reliability is

17 Solution (cont d) The system probability of failure is equal to The system reliability is which is also equal to R SYS = 1 Q SYS As shown in this example, the system probability of failure and reliability are dominated by the series component 1 i.e. a series system is as good as its weakest link

18 Things to Consider Reliability block diagrams can also be used to assess Voting systems (k-out-of-n logic) Standby systems (load sharing or sequential operation) Simple systems can be assessed by gradually reducing them to equivalent series/parallel configurations More complex systems would require the use of a more comprehensive approach, such as conditional probabilities or imaginary components For complex systems, great effort is needed to identify the ways in which the system fails or survives Fault trees can be used to decompose the main failure event into unions and intersections of sub-events Event trees can be used to identify the possible sequence of events (also failures)

19 Series systems Examples A series system is one which operates if and only if all of its components operate. The equivalent circuit diagram is Let R i = P(component i works) and R sys = P(system works) Then, if the components operate independently, P( system operates) P(1 works 2 works n works) P(1 works) P(2 works) P( n works) R R R R sys 1 2 n

20 Parallel systems A parallel system is one which operates if and only if any of its components operate. Q 1 R P( component i fails) i i unreliability of component i P( system fails) P(1 fails n fails) P(1 fails) P( n fails) Q Q Q Q sys 1 2 n 1 R (1 R )(1 R ) (1 R ) sys 1 2 n

21 Series/parallel systems Example Find the reliability of the system shown below, if all components have reliability 0.8. Solution. The system can be broken down into subsystems that are series or parallel.

22 Decomposition Q 1 Q Q 1 C D R R R R R 2 A B Q sys 2 E ( ) R sys Q Q 0.923

23 Conditional probability method Some complex systems cannot be broken down into series and parallel subsystems. There are several reliability analysis methods such as conditional probability, and cut sets and fault trees. Example. Find the reliability of the complex system shown below, if all components have reliability 0.8. This keystone component is chosen carefully. In this case we choose component E. Using the law of total probability gives P( system works) P( E works) P( system works E works) P( E fails) P( system works E fails) R R R Q R sys E sys E E sys E

24 Conditional probability method R sys E 2 QAB QAQB QCD R R R sys E AB CD R sys E R R R R R 2 AC A C B D Q Q Q R sys E sys E AC BD (1 0.64) R R R Q R sys E sys E E sys E

25 Cut set method Find the reliability of the complex system shown below, if all components have reliability 0.8. A cut set is a subset of the components with the property that if all components in set fail, then the system fails. For example, {A, B, E} is a cut set in the system above. Definition. A minimal cut set is a cut set for which no subset is a cut set. For example, in the system above, {A, B, E} is not a minimal cut set, but {A, B} is a minimal cut set. The list of all minimal cut sets for the system above is Define C 1 to be the event all components in cut set C 1 fail, etc. Then,,,,,,,,, C A B C C D C A D E C B C E P( system fails) Q P( C C C C ) sys

26 Standby systems One example of a standby system is of the power supply to a hospital. The primary supply is from the electricity grid. The backup might be a diesel generator. A standby system differs from a parallel system in two ways: the backup component is not in use while the primary component is operational (and therefore is not susceptible to failure) there is a switching mechanism, which detects failure of the primary component and activates the backup component. This switching mechanism may fail to operate (e.g., the diesel generator may fail to start). There may be more than one backup component, as shown below.

27 To Calculate the reliability function of a standby system in which, the primary component has a constant hazard rate of 0.01 per hour, while the backup component has hazard rate 0.02 per hour 1 and 2 We show the general tcase, with failure rates R ( t) R ( t) f ( s) R ( t s) ds sys s0 1t 1t 2 ( ts) e e e ds s0 1t 2t ( 2 1 ) s e e e ds Note that we need to distinguish here between two cases. If up with the result of the Example. Here we consider only the case t 1 ( 21) s 1t 2t e Rsys () t e 1e 2 1 s0 which, after some manipulation, ( 21) t 1 1t 2te e 1e t s0 1 2 t then we will end, which gives e t e t

28 Example of Standby systems The primary component has a constant hazard rate of 0.01 per hour, while the backup component has hazard rate 0.02 per hour. Compare the mean times to failure if these components are operated (a) in parallel, (b) in standby mode. (a) For the parallel system, (b) For the standby system, we have E T sys hr Thus the standby system gives the longer MTTF. This agrees with common sense, since in the standby system the standby component begins its life later. Suppose that the primary and backup components both have hazard rate 0.01 per hour. Compare reliability functions if these components are operated (a) in parallel, (b) in standby mode E T sys ET1 ET2 150hr R t e e e e R ( t) P( T 1) P(2 nd failure occurs after time t) t t t 2t sys ( ) 1 (1 )(1 ) sys sys P N 0.1 ( 1) 1.1e 0.995

29 Review on Reliability and Failure Rate

30 ET and FT IE Sys-A Sys-B Result 10-1 /yr Success Failure Success OK OK Failure CD CDF=1.1X10-7 /yr Sys-A Failure 10-4 /yr Sys-B Failure 1.1x10-2 /yr Pump-1 Failure Pump-2 Failure 10-2 /yr 10-2 /yr Pump-1 Failure Pump-2 Failure 10-2 /yr 10-3 /yr

31 Emergency Diesel Generators The probability that a device will perform successfully for the period of time intended under the operating conditions. Emergency Electric Power System ( 비상전력계통 ) E1 E2 30KVA 30KVA 30KVA G1 G2 G3 At least 60KVA

32 An Example Emergency Electric Power System ( 비상전력계통 ) E1 E2 30KVA 30KVA 30KVA G1 G2 G3 At least 60KVA 기본사건 (Basic Event) E1 E2 G1 G2 G3 기기고장률 (Demand Failure) 3.18E E E E E-06

33 FT in KIRAP 최소단절집합 (Minimal Cut Set) 총 8 가지 {G1,G2} {G2,G3} {G1,G3} {E1,E2} {E1,G2} {E1,G3} {E2,G2} {E2,G1}

34 An Example Results KCUT Version 4.8a(20) Uncertainty Boolean Equation Reduction Program + Uncertainty Copyright Han, S.H. KAERI Fri Feb 06 18:00: > > LEVEL ( 0.000e+000 ). Reporting for FAIL value = 1.053e-005 Final Cut Sets 시스템이용불능도 no value f-v acc cut sets e E1 E e E1 G e E1 G e G1 E e G2 E e G1 G e G1 G e G2 G3 Execution time 0 seconds (gen:0, exp:0, abs:0), Return Code = 1 End of CUT Run

35 KIRAP 의실행초기화면

36 KIRAP 의실행

37 KIRAP Menu

38 KIRAP 메뉴설명 이름의미 Name Type Description 현재사건의이름을입력현재사건의형태를입력현재사건에대한자세한설명 Mean, Cal. Type, Lambda, Tau EF Dist. Type Transfer Module Remark Mean 은현재사건에대한신뢰도값을의미하며, 사용자가직접입력하는것이아니라 Lambda, Tau 값들의계산을통해얻어짐 각사건에대해주어진 Mean 값의오차인자 사건에주어진신뢰도값에대한확률분포를정해줌 현재사건은전이게이트 (Transfer gate) 로만듬 현재사건을전이게이트에서해제 현재사건에대한비고나특기사항들을기록

39 KIRAP 메뉴설명 Cal. Type Lambda (λ) Tau (τ) Mean 의계산 의 미 0 Demand Failure Prob. - Mean = λ : 고장확률을바로줄경우사용. 대부분의 demand failure 가이에해당. 1 Running Failure Rate Mission Time Mean = λx τ : 사고후주어진시간동안운전하지못하는확률을표현 2 Running Failure Rate Repair Time Mean = λx τ : 항상기기를감시하다가고장이나면바로수리하는경우에이용불능도를표현 3 Standby Failure Rate Test Interval Mean = λx τ/2 : 대기상태에있으면서정기적으로점검하는기기의이용불능도를표현. 4 Failure Rate - Mean = λ : 고장율을단위로가진 Event 에사용.

40 KIRAP 메뉴설명

41 KIRAP 메뉴설명 Tree Display Option

42 KIRAP 메뉴설명 계산수행 - 이용불능도계산 - 최소단절군 (MCS) 및중요도계산 etc