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

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


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:


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


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:


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


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


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.


Exp(0,5):


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.)


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

1λ 2λ


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=

=∑


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.


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


Weibull(λ,α)


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


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

Documents

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