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### System Reliability Analysis

Mahesh Pandey and Mikko Jyrkama

Outline

- Introduction
- Probabilistic safety analysis (PSA)
- System reliability analysis
- Failure Modes and Effects Analysis (FMEA)
- Reliability Block Diagrams
- Series systems
- Parallel systems

Fundamentals of Reliability

© M. Pandey, University of Waterloo

Introduction

- Most engineering systems consist of many elements or components
- Need to consider multiple failure modes and/or multiple component failures
- Analysis is fairly complicated
- Need to consider

1. The contribution of the component failure events to the system’s failure

2. The redundancy of the system

3. The post-failure behaviour of a component and the rest of the system

4. The statistical correlation between failure events

5. The progressive failure of components

Fundamentals of Reliability

© M. Pandey, University of Waterloo

Probabilistic Safety Analysis (PSA)

- System reliability analysis is an integral part of probabilistic safety analysis (PSA) in a nuclear power plant
- The main objective of PSA is to provide a reasonable risk-based framework for making decisions regarding nuclear power plant design, operation, and siting
- The main task is to conduct a reliability analysis for all systems and components in the plant
- This requires
- analysis of all possible failure mechanisms and failure rates for all systems and components involved
- quantifying the interaction of the failure mechanisms and their contribution to overall plant reliability (and safety)
- PSA also involves other aspects, such as consequence analysis, uncertainty and sensitivity analyses, etc.

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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
- Two main methods used include:
- Fault tree analysis
- Event tree analysis
- Other qualitative and graphical methods include
- Failure Modes and Effects Analysis (FMEA)
- Reliability Block Diagrams (RBD)
- Functional Logic Diagrams

Fundamentals of Reliability

© M. Pandey, University of Waterloo

Failure Modes and Effects Analysis (FMEA)

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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.

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

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
- Failure is defined as the union of the individual component failures
- For small failure probabilities

1

2

where Q denotes the

probability of failure

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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
- Failure is defined by the intersection of the individual (component) failure events
- Assuming independence

1

2

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

Example Problem

Example: Compute the reliability and probability of failure for the following system. Assume the failure probabilities for the components are Q1 = 0.01, Q2 = 0.02 and Q3 = 0.03.

- Solution:
- First combine the parallel components 2 and 3
- The probability of failure is
- The reliability is

2

1

3

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

Solution (cont’d)

- The system probability of failure is equal to
- The system reliability is

which is also equal to RSYS = 1 – QSYS

- 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

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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)

Fundamentals of Reliability

© M. Pandey, University of Waterloo

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