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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