083 peter tavner reliability i
DESCRIPTION
Peter TavnerTRANSCRIPT
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
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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.
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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;
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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;
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MTBF, MTTF, MTTR
Availability, A=MTTF/MTBF, Inherent
5
Time
Operability
100%
0%
MTTF
MTTR
MTBF
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MTBF, MTTF, MTTR
Availability, A=MTTF/MTBF, Operational
6
Time
Operability
100%
0%
MTTF or MTBM
MTTR
Logistic delay
time
MTBF
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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.
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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
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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
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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;
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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
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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
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Continuous Variable, tT
Random Variable, l(t),
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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.
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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)
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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
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• 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
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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
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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-
= = =
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Graphical Representation of
Poisson Distributions
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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)
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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
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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
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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.
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Use of the Weibull Distribution Customer Returns to an Inverter Manufacturer
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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
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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.
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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)
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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)
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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
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Wind Farm Example
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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
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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 -=
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Useful Life
( = 1)
Bathtub Curve
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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.
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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=
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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
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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.
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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.
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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
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