083 peter tavner reliability i

An Introduction to Reliability from the Point

of View of Onshore & Offshore Wind Farms Peter Tavner

Emeritus Professor, Durham University, UK

Past President, European Academy of Wind Energy

“One has to learn to consider causes rather than symptoms of

undesirable events and avoid uncritical attitudes”

Alessandro Birolini

NTNU, EU FR7 MARE WINT Project

September 2013 1

Keynotes

• Reliability definitions;

• Random & continuous variables;

• Reliability probability distributions;

• Reliability theory;

• Wind Farm example;

• Difference between machinery and structural

reliability;

• Reliability block diagrams & wind turbine taxonomy.


September 2013 2

Definitions • Reliability: probability that a part can perform its intended function for

a specified interval under stated conditions.

• Definition breaks into four essential elements:

– Probability;

– Adequate performance;

– Time or the random variable;

– Operating conditions.

• Definition experiences difficulties as a measure for continuously operated systems that can tolerate failures, the measure for these systems is:

• Availability, probability of finding a system in the operating state at some time into the future.

• Probability of failure p(x) for continuous probability or P(X) for probability of a

discrete failure;

• Cumulative Distribution Function (CDF) of failure probabilities;

• Probability distribution function (PDF) of failures f(x or X);

• Failure intensity or hazard rate, l(t), the frequency of failures, which varies

with time, failures/unit/hr;

• Failure rate, l, the special case when l(t) = constant, hr-1;


September 2013 3

Other Definitions • Mean Time Between Failures, MTBF or q=1/ l, hr, Under

hypothesis of minimal repair, which brings machine back

to condition before failure, TBF, is time measured from

instant of installation of machine to instant after first

failure, when machine available again for operation.

Average of that and successive TBFs is MTBF and can be

averaged over a number of machines in a population.

MTBF is the sum of the MTTF and MTTR;

• Mean Time To Repair, MTTR or 1/m, hr, Time To Repair

measured from instant of first failure to instant when the

machine is available for operation again. MTTR is average

of that and successive TTRs and can be averaged over a

number of machines in a population;

• Mean Time To Failure, MTTF or 1/ l, hr, expected value

of that and successive TTFs. Does not include TTR as a

result of a failure;


September 2013 4

MTBF, MTTF, MTTR

Availability, A=MTTF/MTBF, Inherent

5

Time

Operability

100%

0%

MTTF

MTTR

MTBF


September 2013 5

MTBF, MTTF, MTTR

Availability, A=MTTF/MTBF, Operational

6

Time

Operability

100%

0%

MTTF or MTBM

MTTR

Logistic delay

time

MTBF


September 2013 6

Availability & Reliability of a

Whole Wind Turbine

• Mean Time To Failure, MTTF

• Mean Time to Repair, or downtime MTTR

• Mean Time Between Failures MTBF

MTBF≈MTTF

MTBF≈MTTF+MTTR=1/l +1/ m

MTBF=MTTF+MTTR+LogisticDelay Time

• Failure rate, l l = 1/MTBF

• Repair rate, m m=1/MTTR

• Manufacturer’s or Inherent Availability,

A=(MTBF-MTTR)/MTBF=1-(l/m) • Operator’s or Technical Availability,

A=MTTF/MTBF < 1-(l/m) • Note that these are all expressed in terms of the variable time.


September 2013 7

Cost of Energy, COE

• COE, £/kWh=

(ICC×FCR + O&M)/AEP

– ICC=Initial Capital Cost, £

– FCR=Fixed Charge Rate, interest, %

– O&M=Annual Cost of Operations & Maintenance, £

– AEP=Annualised Energy Production, kWh

• COE , £/kWh =

(ICC×FCR + O&M(l, 1/m))/AEP(A(1/l, m)} • Reduce failure rate l, Reliability MTBF ,1/l , & Availability, A, improve,

O&M cost reduces;

• Reduce Downtime MTTR, Maintainability, m, & Availability, A, improve, O&M cost reduces;

• Therefore COE, reduces

8 NTNU, EU FR7 MARE WINT Project

September 2013

Capacity Factor, Availability, Cost of

Energy, COE Energy generated in a year = C x Turbine rating x 8760

Capacity Factor, C

8760: number of hours in a year

Therefore C = Energy generated in a year / Turbine rating x 8760

C incorporates the Availability, A

Availability, A=1-MTTR/MTBF, where MTBF, q=1/l & MTTR=1/m

Capacity Factor, C Availability, A

Typical UK values

Onshore 27.3% 97%

Early offshore 29.5% 80%

Typical EU values

Vattenfall onshore

target

98%

Offshore 36% 90%

Vattenfall offshore

target

95%

COE: Onshore £30-40/MWh Offshore £69-120/MWh 9

Mission Oriented or

Repairable Mission oriented Repairable


September 2013 10

Continuous & Random

Variables in Reliability Theory

• Examples of useful Continuous or Discrete Variables

(x or X):

– Elapsed time in service , t;

– Calendar time, tc;

– Time on Test, tT;

– Energy produced, E;

• Examples of useful Random Variables:

– Continuous failure intensity or hazard rate, l(t);

– Censored failure rate, lt, , lE;

– MTBF, q;

– MTTF, m;


September 2013 11

Issues About the Variables

• The random variable in this context are the failures

recorded against a Continuous or Discrete variable x or X.

• Is it always appropriate to use Calendar Time, tc, as x?

– Calendar Time is convenient but not necessarily best;

– Time on Test, tT, or cycles may seem more appropriate;

– Turbine rotations may also be more appropriate

especially for the aerodynamic and transmission failures;

– Or GWh, E, of the turbine may also be more appropriate,

especially for electrical failures.

• Usually operators cannot measure the Time on Test, they

measure the number of failures in an interval of time

• That is censored, discrete data


September 2013 12

Reliability of a Component:

Occurrence of Failures, Non-repairable

Failures

Time on Test, tT

Total Time on Test,

Meaningful

Failures

Calendar Time, tc

Calendar Time,

Meaningless,

start times not

controlled

Example, a Gearbox High Speed Bearing

13

Failures

Revolutions, N

Meaningful,

But different

Continuous

Variable

Discrete Variable, Tc

Random Variable, lt or lE

Censored Data, same in both graphs


September 2013 14

Continuous Variable, tT

Random Variable, l(t),


September 2013 15

What Does This Show Us?

• That the method of collecting data is important

– The choice of Continuous or Discrete variable, x or X, against

which the Random Variable to be collected is important

– Should it be Calendar Time, Time of Test, GWh or rotations

– Plotting failures against different x or X reveals different

information

• Whether the component on which the data is being collected is

repairable or non-repairable

• If data collection method is good and variable chosen

appropriately then the statistical data collected should yield robust

reliability information

• If not the reliability information may be faulty

• Now we can consider the mathematics of the data distribution.


September 2013 16

Discrete Random Variables:

Probability Distribution

Function • There is a probability that a wind turbine will experience failures during

its life cycle.

• Failures are the Discrete Random Variable being counted against X the intervals, months, of the life.

• The probability of each failure during the first 5 months of operation could be determined experimentally from the field. Suppose that these probabilities are:

– P(X= 1) = 0.6561

– P(X= 2) = 0.2916

– P(X= 3) = 0.0486

– P(X= 4) = 0.0036

– P(X= 5) = 0.0001

• Note that the data is censored into 5 equal monthly periods, starting from 1st month.

• This gives the Probability Distribution Function (PDF)


September 2013 17

Graphical Representation of

Failure: Probability Distribution

Function

Definition: The Probability Distribution Function of the Discrete Random

Variable is the probability of failure in each specified censored

interval of the variable X.

For a discrete random variable f with n possible values at

x1, x2,…xn, therefore the Probability Mass Function, f(xi), is

expressed as: f(xi) = P(X= xi)

A graphical representation of

the Probability Distribution

Function (PDF) of failures f(X)

is shown in the Figure against

x. 0%

10%

20%

30%

40%

50%

60%

70%

1 2 3 4 5

Pro

bab

ilit

y o

f

fail

ure

Month


September 2013 18

• It is useful to be able to

express the cumulative

probability such as P(X ≤ x) in

terms of a formula.

• The formula for an

accumulation of probabilities is

called a Cumulative

Distribution Function (CDF).

Cumulative Distribution

Function

Definition: The Cumulative Distribution Function is an analytical method for describing the Probability Distribution Function of a Discrete Random Variable.

Therefore the Cumulative Distribution Function F(x) is expressed as: F(x) = P(X≤ x) = ∑f(xi) where xi ≤ x

0%10%20%30%40%50%60%70%80%90%

100%

1 2 3 4 5

Cu

mu

lati

ve

Pro

bab

ilit

y

of

Fail

ure

Month


September 2013 19

20 of 36

Binomial Discrete

Distribution • Consider carrying out a random experiment consisting of n

repeated and independent trials:

− Each trial results in only two outcomes, “success” or “failure”;

− The probability of a success in each trial, p, remains constant.

• An example of such a random experiment could be the

monitoring of failures (i.e. operation or non-operation) at

predefined intervals of a critical component such as a bearing

in a gearbox.

Definition: The random variable X that equals the number of trials that result in a success has a binomial distribution with parameters p and n= 1, 2, 3, … The probability mass function of X is:

RELIAWIND Training Course,

LM Wind Power Blades, Kolding Denmark

Graphical Representation of a

Binomial Distribution

Binomial distribution for selected values of n and p


September 2013 21

Poisson Discrete

Distribution • Consider the operation of a wind turbine over a period of one year:

– The number of failures of that turbine per increment of interval X is the probability of failure in successive intervals.

– If that probability of failure, P, is constant, then the turbine operation over each interval is independent of operation in previous intervals.

– Then probability P has a Binomial Distribution with respect to the discrete random variable X.

• Suppose that a constant, l, equals the average value of failures in that month. If the variance of the failures also equal l, then the random experiment is called a Poisson Process.

Definition: A Poisson Distribution can be used as an approximation of the Binomial Distribution when the number of observations is large and the probability of failure is low.

It can be represented by the equation below where x=0,1,2,3,… is the number of failures.

lim ( ) , 0,1,2,...!

x

n

eP X x x

x

ll-

= = =


September 2013 22

Graphical Representation of

Poisson Distributions


September 2013 23

Examples of Poisson

Distributions • This Figure, shows the

variation, for a large

population of Danish wind

turbines averaged over the

year for 10 years of:

−Wind Energy Index

−Turbine failures

• The histogram suggests that

they could both be

approximated by a Poisson

Distribution.

•Poisson Distributions play a special role in Reliability Theory since

under broad conditions they describe the phenomenon of catastrophic

failure in complex systems.

Average monthly Failure Rate and Wind Energy Index

for each of the 12 months over the Survey period

(1994~2004)


September 2013 24

Observation on Poisson

Distributions • Poisson Distribution defines a random variable to be a certain

interval during which a number of failures occurred.

• The continuous variable is of interest, because it could be for example Calendar Time, Time on Test , GWh produced or turbine rotations.

• If the continuous variable is Calendar Time, it could be a month or quarter or year.

• Let the continuous variable, x ,denote the duration from any starting point until a failure is detected or in general denote the duration between successive failures in a Poisson Process.

• The starting point for measuring x doesn’t matter because the probability of the number of failures in a Poisson Process depends only on the length of the interval not on the value of x.

• If the mean number of failures is l per interval, then x has an Exponential Distribution with parameter l


September 2013 25

Definition: The random variable X that equals the distance between successive counts of a Poisson Process with mean l > 0 has an exponential distribution with parameter l.

Therefore the probability density function of X is:

• Exponential Distributions play a key role

in practical computations;

• In many cases the interval between two

successive failures in a complex system,

such as a wind turbine, obeys an

Exponential Distribution.

Exponential Continuous Distribution


September 2013 26

Weibull Continuous Distribution

• The Weibull Distribution can be used to model the

time until failure of many different physical

systems.

• The parameters in the distribution provide a great

deal of flexibility to model systems in which: − Number of failures increases with time, for

example bearing wear or thermal aging; − Number of failures decreases with time, for

example early failures; − Number of failures remains constant with time,

for example random failures at the bottom of the bath tub, caused for example by random external shocks to the system.

Definition: The random variable X with probability density function has a Weibull distribution with scale parameter > 0 and shape parameter > 0.


September 2013 27

Use of the Weibull Distribution Customer Returns to an Inverter Manufacturer


September 2013 28

Use of the Weibull Distribution Customer Returns to an Inverter Manufacturer

The graphical calculation

of shape and scale

parameters

Example of Bi-Weibull: two

interpolating lines fit better the Weibull

chart

Date of production

move to China


September 2013 29

What are the Uses of Distributions

• Distributions can model different failure mechanisms.

• Failure mechanisms are the essential physics of failure linkage between cause and effect and in wind turbines include:

– Mechanical mechanisms:

o Fatigue due to aeroelastic behaviour;

o Fatigue due to gear meshing;

o Failure due to random shock.

– Electrical mechanisms:

o Thermal aging;

o Thermo-mechanical cycling fatigue in electromechanical components;

o Thermo-mechanical fatigue stress in power electronic components.

– Operational Factors

o Elimination of early teething problems;

o Change of component.

• Each represented by different failure probability distributions.

• The ability to distinguish between them allows faults to be detected.


September 2013 30

General Reliability Functions • The following equations and mathematical relationships between the

various reliability functions do not assume any specific failure

distribution and are equally applicable to all probability distributions

used in reliability evaluation.

• Consider N0 identical components are tested:

Ns(t) = number surviving at time t

Nf(t) = number failed at time t

• Therefore Ns(t) + Nf(t) = No

• At any time t the survivor or reliability function, R(t), is given by

R(t)=Ns(t)/No

• Similarly the probability of failure or Cumulative Distribution Function or

unreliability function , Q(t), is given by

Q(t)=Nf(t)/No

Where R(t)=1-Q(t)


September 2013 31

General Reliability Functions

• Failure Density Function f(t) is given

by:

f(t)=1/N0(dNf(t)/dt)

• Failure Intensity or Hazard Rate, λ(t),

is the Failure Density Function, f(t),

normalised to the number of survivors:

l(t)= (dNf(t)/dt)/Ns(t)

l(t)= (dR(t)/dt)/R(t)


September 2013 32

Terminology of Distributions

Hypothetical failure Probability Mass Function:

Q(t), Cumulative Failure Distribution in time t,

unreliability

R(t), Survivor Function in time t,

reliability

R(t) = 1-Q(t)

The total area under the failure

density function must be unity. Hazard rate / Failure rate

number of failures per unit time( )

number of componentsexposed to failuretl =

Fa

ilu

re D

en

sit

y F

un

cti

on


September 2013 33

Wind Farm Example


September 2013 34

Time

interval,

years

Number of

failures in each

interval, N

Cumulative

failures, Nf

Number of

survivors, Ns

Failure

density

function, f(t)

Unreliability

function or

cumulative

failure

distribution,

Q(t)

Reliability or

survivor

function, R(t)

Failure intensity

or hazard rate,

l(t)

0 240 0 1000 0.240 0.000 1.000 0.240

1 140 240 760 0.140 0.240 0.760 0.184

2 90 380 620 0.090 0.380 0.620 0.145

3 58 470 530 0.058 0.470 0.530 0.109

4 40 528 472 0.040 0.528 0.472 0.085

5 23 568 432 0.023 0.568 0.432 0.053

6 18 591 409 0.018 0.591 0.409 0.044

7 13 609 391 0.013 0.609 0.391 0.033

8 13 622 378 0.013 0.622 0.378 0.034

9 13 635 365 0.013 0.635 0.365 0.036

10 16 648 352 0.016 0.648 0.352 0.045

11 18 664 336 0.018 0.664 0.336 0.054

12 20 682 318 0.020 0.682 0.318 0.063

13 30 702 298 0.030 0.702 0.298 0.101

14 60 732 268 0.060 0.732 0.268 0.224

15 63 792 208 0.063 0.792 0.208 0.303

16 65 855 145 0.065 0.855 0.145 0.448

17 70 920 80 0.070 0.920 0.080 0.875

18 10 990 10 0.010 0.990 0.010 1.000

19 0 1000 0 0.000

Totals 1000

Wind Farm Example


September 2013 35

Bathtub Curve

36

Failure

Intensity,

l

Early Life

( < 1)

Useful Life

( = 1)

Wear-out Period

( > 1)

Time, t

Most turbines

lie here

(years) Period Operating

Population Turbine

failures ofnumber Total

=al

te)t( l -=


September 2013

Useful Life

( = 1)

Bathtub Curve


September 2013

Time, t

Failure

Intensity,

l

Early Life

( < 1)

Wear-out Period

( > 1)

Select more reliable components

Preventive maintenance

Reliability Centred Maintenance

Condition Based Maintenance

Major sub-assembly

changeout More rigorous

pre-testing

37

To Summarise

• The Regions I, II & III can be identified in both

Failure Density Function & Hazard Rate or Failure

Intensity Function.

• Region II can be represented by an Exponential

Distribution.

• Region III can be represented by a Weibull

Distribution.

• The Hazard Rate is in the shape of the Bath Tub

Curve.


September 2013 38

Root Causes and Failure Modes

Example: Main Shaft Failure

39

How?

SCADA

analysis

CM and

diagnosis

Why?

Root

cause

analysis

Root Causes

Failure mode

Main shaft failure

Fracture Deformation

High cycle fatigue Corrosion Low cycle fatigue

or overload Misalignment

Series Systems

Consider a System consisting of two independent components A and B connected in series, for example a gear train.

Let Ra and Rb be the probability of successful operation of components A and B respectively.

Let Qa and Qb be the probability of failure of components A and B respectively.

Rs=Ra*Rb if generalised

This equation is referred as the Product Rule of reliability.

Example:

A gearbox consists of 6 successive identical gear wheels, all of which must work for system success. What is the system reliability if each gearwheel has a reliability of 0.95?

From the Product Rule: Rs=0.956=0.7350

1

n

i

i

R=

Rs=


September 2013 40

Parallel Systems Consider a system consisting of two independent components A and B,

connected in parallel, for example to lubrication oil pumps for a gearbox

connected in parallel.

From a reliability point of view the requirement is

that only one component has to be working for

system success.

Example:

A system consists of four pumps in parallel each having reliabilities of

0.99, 0.95, 0.98 and 0.97. What is the reliability and unreliability of the

system?

Qp=(1-0.99)(1-0.95)(1-0.98)(1-0.97)=3x10-7

Rp = 1- Qp =0.9999997


September 2013 41

Network Modelling and

Evaluation of Simple

Systems • Series systems:

– Components are said to be in series, from a reliability point of view, if they must all work for system success and only one needs to fail for system failure.

• Parallel systems:

– The components in a set are said to be in parallel, from reliability point of view, if only one needs to be working for system success or all must fail for system failure.


September 2013 42

Conclusions

• Definitions in reliability are important;

• Remember MTBF, MTTR & Availability

• MTTR is as important as MTBF;

• Availability definition standardisation is important;

• Note difference between Inherent & Operational

Availability;

• Definition of Cost of Energy, CoEoffshore> CoEonshore

• Reliability modelling can make use of distributions;

• Series & parallel arrangemennts and redundancy are

important;

• Wind Turbine Taxonomy is important.


September 2013 43

References • P. J. Tavner, Offshore Wind Turbines, Reliability, Availability &

Maintenance, IET Renewables Series, 2012

• D.C. Montgomery, G.C. Runger, Applied Statistics and Probability for Engineers, J. Wiley & Sons, 1999, ISBN 0-471-17027-5.

• S.E.Rigdon, A.P. Basu : Statistical Methods for the Reliability of Repairable Systems, John Wiley & Sons, New York, 2000

• R. Billinton and R.N. Allan, Reliability Evaluation of Engineering Systems, Plenum publishing corporation,1992, ISBN 0-306-44063-6.

• A Birolini, Reliability Engineering, Theory & Practice, Springer, New York, 2007, ISBN 978-3-540-49388-4

• B.V. Gnedenko, Yu.K. Belyaev, and A.D. Solovyev Mathematical Methods of Reliability Theory, Academic Press, 1969, ISBN 0-12-287250-9.

• F Spinato, Reliability of Wind Turbines, PhD, Durham University, 2008


September 2013 44

083 peter tavner reliability i

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