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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. Introduction. Introduction.

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System Reliability Analysis

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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

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

### Solution

Fundamentals of Reliability

© M. Pandey, University of Waterloo

### Reliability Block Diagrams

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