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EI2452 Reliability Evaluation of Electrical Power Systems Lecture 2: Failure Models Niklas Ekstedt, [email protected]

EI2452 Reliability Evaluation of Electrical Power Systems · 2015. 3. 25. · System reliability theory: models, statistical methods, and applications (Vol. 396). John Wiley & Sons

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  • EI2452 Reliability Evaluation of Electrical Power Systems Lecture 2: Failure Models Niklas Ekstedt, [email protected]

  • Content

    • Introduction • Probability theory • State variable

    • Functions in Reliability Theory • Time to failure • Reliability function • Failure rate function

    • Mean time to failure • Some special distributions

    • Exponential distribution • Weibull distribution

  • Introduction – sources for this lecture

    • Course book (limited): page 14-16. • Optional course book Rausand [1]: Chapter 2.

    • Hand out print out of part of chapter, also available on course page. • Lecture loosely based on this chapter.

    • Billinton [2]: Chapter 6. • Wikipedia(!) and other online resources. For example:

    http://en.wikipedia.org/wiki/Failure_rate

    [1] Rausand, M., & Høyland, A. (2004). System reliability theory: models, statistical methods, and applications (Vol. 396). John Wiley & Sons [2] R. Billinton and R. N. Allan, Reliability evaluation of engineering systems: Springer, 1992.

    http://en.wikipedia.org/wiki/Failure_rate

  • Introduction - probability

    • Stochastic models in this course (not deterministic) • Specifically the stochastic models of failure

    Example: Consider a dice. A six means a failure. When will the component fail? Can we answer deterministic?

  • Introduction – state variable

    • State of a unit at time t is described by the stochastic variable X(t) according to:

    • Example:

    • Today: describing Time to (first) failure.

    ( )10

    unit functions at time tX t

    unit failed at time t

    =

  • Introduction

    Examples of other measures than time: • Operations (for a circuit breaker) • Distance (km for a car) • Rotations (generator) • Cycles for a periodic unit

    Some measures are discrete but can easily be approximated with continuous variables.

  • Functions in Reliability Theory – time to failure

    How do we describe the time to failure T? We start to use a component at time 0 (now). When will it fail? What is T, the time to failure? Use: • Probability density function f(t) • Distribution function F(t)

    • Draw on the white board:

  • Functions in Reliability Theory – time to failure

    F(t) describe the probability that ttf will be less than t: f(t) describe the “level” of probability that ttf will be t: In other words: f(t) x short interval is the probability that ttf will be within that interval around t:

  • Functions – Reliability function

    The Reliability function (or Survival function) of an item is defined: R(t) is the probability that an item doesn’t fail in the interval (0,t], that is; it survives the time interval. Draw on the white board:

  • Functions – Failure rate function

    The probability that a unit fails in the interval (t, t+∆t ), when we know that it functions at t: Failure rate (or hazard rate), z(t), is obtained if this probability is divided with ∆t, when ∆t -> 0:

    ( ) ( )( )( ) ( )

    ( )tRtFttF

    tTPttTtPtTttTtP −∆+=

    >∆+≤<

    =>∆+≤<

    ( ) ( ) ( ) ( ) ( )( )( )tRtf

    tRttFttF

    ttTttTtP

    tztt

    =∆

    −∆+=

    ∆>∆+≤<

    =→∆→∆

    1limlim00

  • Functions – Failure rate function Failure rates from 1960s airplane industry. Draw a failure rate function on white board.

  • Functions – Failure rate function

    If ∆t is small it follows: Compare with density function f(t): Generally: and always:

    ( ) ( ) ttztTttTtP ∆⋅≈>∆+≤<

    0

    ( ) 1z t dt∞

    ≠∫0

    ( ) 1f t dt∞

    =∫

  • Functions – Summing up

  • Functions – experiment

    Simulate failures with dices. We draw the empiric result on the white board. For example: i ii i

    i i

    n nz z tm t m

    ≈ ⇒ ∆ ≈∆

  • Mean time to failure - MTTF

    MTTF – Mean Time To Failure MTTR – Mean Time To Repair, (MDT – Mean Down Time) MTBF – Mean Time Between Failures

    TBF

  • Mean time to failure - MTTF

    Average time to failure for a unit is defined by: For repairable units where the restoration time at failure (MTTR) is much lower than time to failure (MTTF) follows that average time between failure MTBF ≈ MTTF. Also, it can be shown:

  • Mean time to failure - MTTF

    MTTF: Average failure time Median: 50% to the left, 50% to the right Mode: Failure time with highest probability

  • Mean time to failure - MTTF

    MTTF: Average failure time Median: 50% to the left, 50% to the right Mode: Failure time with highest probability What’s the MTTF, median and mode for our experiment?

  • Special Distributions (parametric)

  • Distributions – exponential distribution

    Properties: No memory. Constant failure rate. Density function: The time to failure, T, for a unit is exponentially distributed with the parameter λ, sometimes noted T~exp(λ). Survival function becomes:

  • Distributions – exponential distribution

    Average time to failure can then be calculated with: and failure rate: Thus, constant failure rate!

    ( )0 0

    1tMTTF R t dt e dtλλ

    ∞ ∞−= = =∫ ∫

    ( ) ( )( ) λλ

    λ

    λ

    === −−

    t

    t

    ee

    tRtftz

  • Distributions – exponential distribution

    Suppose a unit got exponential distributed time to failure T probability can then be expressed as: Demonstrating no memory! Important property for next lecture on Markov models.

    ( ) ( )( )( )

    ( )xTPee

    etTP

    xtTPtTxtTP xtxt

    >===>+>

    =>+> −−+−

    λλ

    λ

  • Distributions – exponential distribution

    The following follows for the exponential distributed time to failure: • there is no benefit in replacing a unit that works, since its

    stochastic behavior is as good as new • to estimate failure rate the number of failures and time in

    operation is enough. Note: The exponential distribution is over used due to lack of information/convenience. Only information about number of components and number of failures exist for a population with various age and application.

  • Distributions – exponential distribution

    Exp(0,5):

  • Distributions – exponential distribution

    Most used distribution for reliability analysis: • Mathematical simple expressions • Realistic models for some types of components (random

    failures) • ”resolution” in fault statistics For components with an increasing or decreasing failure rate the exponential distribution isn’t realistic and other functions are needed. (Partially constant failure rate can be used.)

  • Distributions – exponential distribution

    Example: A system with two independent components with failure rate and . Determine the probability that component 1 fails before 2!

    1λ 2λ

  • Distributions – exponential distribution

    The result can be generalized to systems consisting of n independent components with failure rates Probability that component j is the component that fails first is:

    nλλλ ,, 21

    ( )1

    jn

    i i

    P component j fails firstλ

    λ=

    =∑

  • Distributions – exponential distribution

    How do we simulate exponential distribution with the dices? Let’s try!

  • Distributions – weibull distribution

    The Weibull distribution has two parameters and can model increasing, decreasing and constant failure rates. Therefor widely used in Reliability theory. Note: Exponential is a sub-group of Weibull distribution, when failure rate is constant.

  • Distributions – weibull distribution

    Time to failure T for a unit is weibull distributed with the parameters α(>0) and λ (>0) if the distribution, density, reliability and failure rate function are given by: λ – scale parameter and α- form parameter

    ( ) ( ) ( )1 0tF t P T t e för tαλ−= ≤ = − ≥

    ( ) ( ) ( ) ( )1 0tdf t F t t e för tdt

    αα λαλ λ − −= = ≥

    ( ) ( ) ( )0 0tR t P T e för tαλ−= > = ≥

    ( ) ( )( ) ( )1 0

    f tz t t för t

    R tααλ λ −= = ≥

    t

  • Distributions – weibull distribution

    Weibull(λ,α)

  • Distributions – weibull distribution

    Weibull(λ,α)

  • Distributions – weibull distribution

    Example: A series system with n components, having independent Weibull distributed lifetime functions according to: where Calculate the survival function for the system.

    nTTT ,, 21

    ( ) niförWeibullT ii ,,2,1,~ =λα

  • Distributions – weibull distribution

    Solution: A series system fails with its first component failure. Time to system failure is given by: The survival function can be calculated to:

    sT

    { }ns TTTT ,,min 21=

    ( ) ( ) ( )( ) ( ) ( )∏

    =

    ⋅∑−∑−−

    =≤≤

    == ===

    >=>=>=

    n

    i

    ttt

    n

    iiinis

    ni i

    ni ii eee

    tTPtTPtTPR

    1

    11

    11

    min

    ααα λλαλα

  • Distributions – Weibull distribution

    Example conclusion: For a series system with independent components with a time to failure according to a Weibull distribution and the same form parameter: The system follows a Weibull distribution with the same form parameter as the components.

  • Distributions

    Other distributions: • Normal distribution • Lognormal • Poisson • Gamma • Partial constant • …. • Your own

  • The end - summary

    Understand the four functions:

    EI2452 Reliability Evaluation of Electrical Power Systems ContentIntroduction – sources for this lectureIntroduction - probabilityIntroduction – state variableIntroductionFunctions in Reliability Theory – time to failureFunctions in Reliability Theory – time to failureFunctions – Reliability functionFunctions – Failure rate function Functions – Failure rate function Functions – Failure rate function Functions – Summing upFunctions – experimentMean time to failure - MTTFMean time to failure - MTTFMean time to failure - MTTFMean time to failure - MTTFSpecial Distributions�(parametric)Distributions – exponential distributionDistributions – exponential distributionDistributions – exponential distributionDistributions – exponential distributionDistributions – exponential distributionDistributions – exponential distributionDistributions – exponential distributionDistributions – exponential distributionDistributions – exponential distributionDistributions – weibull distributionDistributions – weibull distributionDistributions – weibull distributionDistributions – weibull distributionDistributions – weibull distributionDistributions – weibull distributionDistributions – Weibull distributionDistributionsThe end - summary