TIME-DEPENDENT RELIABILITY ANALYSIS FOR TURBINE BLADE IN EXTREME WIND LOADING

Proceedings of the 5th Annual ISC Research Symposium
ISCRS 2011
April 7, 2011, Rolla, Missouri
TIME-DEPENDENT RELIABILITY ANALYSIS FOR TURBINE BLADE IN EXTREME
WIND LOADING
Zhen Hu
Department of Mechanical and Aerospace
Engineering
Missouri University of Science and
Technology,
Rolla, Missouri, US
Email: [email protected]
ABSTRACT
In order to evaluate the reliability of turbine blades over a
certain time period, a time-dependent reliability analysis model
is developed in this paper for turbine blades in extreme wind
loading. The extreme wind loading over a certain return period
and the deterioration of the blade material are considered to
investigate the time influence on the reliability of turbine
blades. Only failure in flapwise bending is taken into account.
The concept of upcrossing rate, which is based on the Poisson
approximation for first-passage problems, is employed to
address the time-dependent reliability analysis. By integrating
the first order reliability method with the upcrossing rate, the
reliability of a site-specific turbine blade over a certain time
period is computed. The results show that the MVFP (mean
value first passage) method applied in this paper is efficient and
flexible. It can quantify the degradation of reliability over time
period accurately.
1. INTRODUCTION
Using wind turbines to generate electric energy has grown
dramatically in recent years and is recognized as one of the
most successful renewable energy sources [1, 2]. Since there
are uncertainties inherent in the operation environment, such as
in wind speed, the stress response in the turbine blades and the
resistance of materials, the reliability of wind turbine systems
has motivated many researchers’ efforts in this field. Moreover,
as one of the most important components in the turbine system,
the failure modes of turbine blade have to be considered
carefully by researchers to guarantee the system’s reliability.
To reduce the high maintenance cost introduced by the
failure of turbine blade, tremendous efforts have been devoted
to analyze the failure modes of turbine blades, including the
fatigue of blade, failure of blade in extreme loading and so on.
The fatigue of the blade is analyzed to predict the lifetime of
the wind turbine and evaluate the life-cycle cost. The failure of
blade in extreme loading is considered to ensure the safety of
the turbine blade. The latter is analyzed in this paper. In the past
decades, many methods have been developed to analyze the
reliability of turbine blades in extreme wind loading. For
Xiaoping Du
Department of Mechanical and Aerospace
Engineering
Missouri University of Science and
Technology,
Rolla, Missouri, US
Email: [email protected]
example, Agarwal [3] proposes efficient extrapolation
procedures to predict the long-term extreme loads for offshore
wind turbines based on limited field data. By using inverse
reliability, Saranyasoontorn and Manuel [4] studies the
reliability of wind turbines against extreme loads. Ronold [5]
proposes a nested reliability analysis method for analysis of the
safety of a wind-turbine rotor blade against failure in ultimate
loading. It can be found that although the failure of wind
turbine blade in extreme wind loading has been studied for
many years, most of the previous researchers have not
considered the time influence on the wind loading and have not
taken the strength degradation of material in its lifetime into
account.
In practical working condition, it is well known that wind
loading fluctuates significantly with time, and the stress of
turbine blade is strongly dependent on the duration of wind
loading. The strength of turbine blade material deteriorates
inevitably during a long period. To analyze the reliability of
turbine blade in extreme wind loading over a certain time
period, a time-dependent reliability analysis method, which can
deal with time-varying wind loads, is imperative. Even though
the nested reliability method proposed by Ronold [5] can
address this kind of problem in ways of discretizing the time
period into a series of time intervals, its efficiency and accuracy
cannot be guaranteed. The Monte Carlo simulation (MCS) can
provide solutions, but is computationally expensive. Recently, a
mean value first passage (MVFP) method has been developed
by Zhang and Du [6]. It can provide the probability of failure of
a function generator over a certain time period. In this method,
the Poission approximation of the first-passage problem is
adopted to make the reliability analysis possible for the timedependent problem. As the reliability of turbine blades in
extreme wind loading is time-dependent, the MVFP method
can be employed to overcome the drawbacks of conventional
point reliability analysis method to solve the time-dependent
reliability of turbine blade.
The aim of this paper is to develop a time-dependent
reliability analysis methodology for turbine blades in extreme
wind loading. It will evaluate the reliability of turbine blades in
1
a certain time period. Similar to Ronold’s research [5], only
failure in flapwise bending during the normal operating
condition of the wind turbine is considered here. The influence
of the non-Gaussian behavior of wind loading and the strength
degradation of material are investigated first. Based on this, a
time-dependent limit state function is established. And the
MVFP method is employed to carry out the reliability analysis
eventually. The proposed model enables researchers to evaluate
the reliability of turbine blade in a certain time interval exactly
and effectively rather than just the reliability at a specific time.
In section 2, the extreme wind loading over a certain
return period, the bending moment of turbine blades and the
resistance of material are investigated to establish the limitstate function for reliability analysis. In section 3, the nonGaussian random variables are transformed into normal random
variable firstly and the limit-state function is linearized at the
mean value point after that. Based on the linearization of limitstate function, the upcrossing rate is employed to analyze the
time-dependent reliability. In section 4, a case study is carried
out to validate the proposed methodology. The results of the
case study are discussed in section 5 and section 6 provides the
future work.
Then, in order to predict the extreme wind loading in
arbitrary time, we introduce the concept of the Return Period t.
It is simply the inverse of the complementary cumulative
distribution of the extremes [10]. For example, a 20-year wind
speed has a probability of exceedence of 0.05 (i.e. 1/20) in any
one year. And the return period is expressed by
1
t=
(4)
1 − FU max (U )
where FU max (U ) is the probability of non-exceedence.
The maximum wind loading of every 10 minutes is being
considered here and so the return period here is measured in 10
minutes.
Then, we get

 u 
 1
(5)
1 −  ≈ exp  −α N max exp  −  
 t
 2 

and
U max,t

 1
 − log e 1 − t  


= −2 log e 
 α N max





(6)
2. STATEMENT OF ROBLEM
2.2. Bending Moment of Turbine Blades
2.1. Extreme Wind Loading
The long-term distribution of the 10 min mean wind speed
can be presented as a Weibull distribution [7].
  u k 
FU10 (u ) =
1 − exp  −   
(1)
  A  
where k and A are site- and height-dependent coefficients.
Because of the cut-out speed uc of the wind turbine, the
In a turbine blade, a thin skin is glued on a box-like
structure to define the geometry of the blade. The box-like
structure behaves like a beam. So a blade can be modeled as a
simply beam when we are doing the structural analysis [11].
And the section of a turbine blade is shown in Figure 1.
upper tail of the Weibull distribution for the 10min mean wind
speed is truncated and given by [5]
FU10 (u ) =
1 − exp[−(u / A) k ]
, 0 < u < uc
1 − exp[−(u c / A) k ]
(2)
The cut-in wind speed is not considered here as it does not
have any actual meaning in a maximum wind loading analysis.
Assume that 10 min period is short enough that other local
maximum wind speed than the largest wind speed in the 10 min
will not affect the probability of failure in extreme wind loading
apparently [5]. The corresponding local maximum wind
velocity U max of the wind speed U follows a Rice distribution
[8]. And its distribution can be approximated by an extremevalue distribution, which is given as [9]

 u 
FU max (u ) ≈ exp  −α N max exp  −  
(3)
 2 

where α is the regularity factor and N max is the number of local
maxima in 10min. Both of
mean wind speed U 10 .
α and N max are related to the 10 min
Fig. 1. Section of a blade
With the simple beam theory, the stress S(x, y) in the
cross-section can be presented as [11]:
My
M
N
S=
x+
( x, y ) E ( x, y )( x y −
)
(7)
[ EI x ]
[ EI y ]
[ EA]
where MX and MY are the bending moments about the principal
axes X and Y, respectively; N is the normal force; [ EI x ] is the
bending stiffness about principal axis X; [ EI y ] is the bending
stiffness about principal axis Y and [ EA] is the longitudinal
stiffness.
Under extreme wind loading, the blades are parked or
2
idling. According to the Danish standard [12], the extreme
loads can be described as
p (r ) = q2 s C f c(r )
(8)
where
the
Cf
is
the
force
coefficient;
c(r )
is
1
2
chord; q2 s = ρU max
is the dynamic pressure from an extreme
2
wind speed time averaged over two seconds and ρ is the
density of air.
Then the root bending moment for the extreme wind load
is [11]
R
1
(9)
=
M ∫=
rF (r )dr
F (r )( R 2 − r 2 )
r
2
where R is the rotor radius, r is the root radius.
The structural model of turbine blade is simplified as
Figure 2 to calculate the moment of inertia about the flapwise
axis.
t

f (t , x) = exp − x 
(14)
T

where T is the design lifetime of the material; t is the time
period and x is a coefficient.
We treat the rate of deterioration as a function of random
variable x and time t considering the uncertainties existing in
the resistance deterioration model which cannot be obtained
exactly. And a certain distribution f (x ) is assigned to x to
represent its uncertainty [15]. Since the initial resistance
strength of the material σ max is characterized by a natural
variability and follows a certain distribution as well, the
resistance strength of material at time t is an uncertain
parameter depends on the time period t and uncertain
parameters σ max and x.
2.4. Limit State Function
Over any time period [t0, tf] in the design life T, the stress
generated by the blending moment at the blade root should not
exceed the material’s tensile strength σ F in the direction of the
fibers. Thus, a limit state function can be defined as follows:
g ( X=
, t ) S max − σ F
Fig. 2. Simplified structural model
Then, the moment of inertia about the flapwise can be obtained
2
3ρU max
C f c(r )( R 2 − r 2 )b1
(15)
t 

as follows [11]:
=
− exp  − x  σ max
3
3
8a (b1 − b2 )
 T
1
2
I = a ((2b1 )3 − (2b2 )3 ) = a (b13 − b23 )
(10)
where X = (U10 , a, x, σ max , b1 , b2 , R, r ) denotes the vector of
12
3
Because the force in the tangential direction is small, equation
random variables.
(7 ) becomes [11]:
It is apparent that the reliability analysis of turbine blade in
ultimate loading problem is a time-dependent problem. Because
M max y
(11)
S max =
the most commonly used reliability analysis methods like
I
FOSM, FORM and SORM are mainly developed for the point
and the maximum strain occurs at y = b1
reliability
are unable to deal with such kind of reliability over a
So, the maximum flapwise bending moment generated by
time
period
t, a time-dependent reliability analysis method
the extreme wind loading can be presented as
should be employed to evaluate the reliability of turbine blades
2
2
2
3ρU max C f c(r )( R − r )b1
in extreme wind loading in a certain time period. The MVFP
S max =
(12)
8a (b13 − b23 )
method is adopted in this paper.
2.3. Resistance of Material
In former researches, the strength of material has been
oversimplified to be constant in its long lifetime. In reality, the
strength of material will inevitably deteriorate over a long
period. Recent years, models have been proposed to describe
the resistance deterioration of materials such as concrete, glass
[13, 14]. In this work, the degradation of strength of turbine
blades material in its lifetime is assumed to be [15]
σ (t ) = f (t , x)σ max
(13)
where f (t , x) is the rate of deterioration, σ max is the initial
resistance strength of the material.
The rate of deterioration of the turbine blade material is
assumed to decrease exponencially from the maximum [15].
3. TIME-DEPENDENT RELIABILITY ANALYSIS FOR
TURBINE BLADES
The time-dependent reliability analysis of turbine blades is
divided into three main steps. First, the non-Gaussian random
variables are transformed into normal random variables. After
the transformation, the nonlinear limit-state function is
linearized at the mean value point. Since the mean and standard
derivation of the linearized limit-state function are both timedependent, in the third step, the upcrossing rate is introduced to
computer the reliability of turbine blade over a certain time
period.
3.1. Transformation of non-Gaussian distribution
3
In the above mentioned turbine blade reliability problem,
non-Gaussian random variables are involved, such as
lognormal, Weibull variables. Therefore, in order to make the
reliability analysis possible for non-Gaussian variables, we use
the Rosenblatt transformation method to transform these nonGaussian random variables into equivalent normal distribution.
The non-Gaussian random variable can be the transformed
as follows:
ui = Φ −1  Fx ( xi ) 
(16)
i
where Φ
−1
[⋅]
is the inverse of Φ [⋅]
Then, the Taylor series expansion of the transformation
at the mean value point µ xi is employed to get the equivalent
normal distribution [16].
∂
Φ −1 Fxi ( xi ) 
ui =
Φ −1  Fxi ( µ xi )  +
∂xi 
(
)
( xi − µ xi )
(17)
µ xi
ui can be expressed
as
xi − µ x'
φ (Φ [ Fx ( µ x )])
−1
i
i
f xi ( µ xi )
'
i
where
tf
t0
}
'
i
and
i
i
i
'
i
n
g ( U, t ) ≈ g ( U, t ) =
b0 (t ) + ∑ bi (t )U i
(19)
i =1
U = (U1 , , U n )
∂ g ( X, t )
∂X i
σ
,
b0 (t ) = g ( µ x' , t )
and
∧
(20)
n
∑b
2
i
(t ) is given as [6]
(t )
It can be found that both of µ g (t ) and σ g (t ) are function
∧
of t. Thus, g (U , t ) is time-dependent and is a nonstationary
Gaussian process. The conventional reliability analysis methods
(i.e. FORM, SORM) are unable to analyze the reliability during
a time period. Therefore, in order to analyze the reliability of
the turbine blade during a certain time period, the concept of
and β + (t ) =
ε − µ g (t )
σ g (t )
(22)
(23)
in which, b(t ) = (b1 (t ),  , bn (t )) , ε is the boundary for the
limit-state function, it is zero in the turbine blade reliability
analysis problem.
The derivations in equation (22) can be obtained by
following equations [6]
σ g (t )b ' (t ) − b(t )σ g' (t )
a ' (t ) =
(24)
σ g2 (t )
where
=
σ g' (t )
'
i
The mean and standard deviation of g (U, t ) can be
presented as
µ g (t ) = b0 (t )
i =1
b(t )
b(t )
µx'
and
+
'
+
'
a(t ) =
After transformation, the limit-sate function changes
from g ( X, t ) into g (U, t ) . Then, the linearization of the limitstate function at the mean value point is given by
∧
(21)
where v + (t ) is the upcorssing rate at time t
where
µ=
µ x − Φ −1[ Fx ( µ x )]σ x
x
3.2. Linearization of the limit-state function
σ g (t ) =
{
Re(t0 , t f ) ≈ exp − ∫ v + (t )dt
  β (t )  β +' (t )

 β ' (t )  

=
Φ  +'
v + (t )  a ' (t )  φ [ β + (t ) ] φ 
− '

   a (t )    a (t )    a (t )   


'
i
bi (t ) =
computed by
(18)
σx
where
The Poisson approximation of the first-passage problem
assumes that the integer-valued process that counts the number
of upcrossing and downcrossings is a Poisson process [17]. The
assumption has been commonly used in structural reliability
analysis. And the reliability of turbine blade in extreme wind
loading is a upcorssing problem. Under the Poisson
assumption, Re(t0 , t f ) which represents the probability of no
The analytical equation for v
i
σx =
3.3. Mean value first passage reliability analysis
method
upcorssing event occurs in time period (t0 , t f ) can be
After substitution and transformation,
ui =
upcorssing rate which is based on the Poisson approximation
for first-passage problems is employed to make the reliability
analysis possible.
β +' (t ) =
1
b(t ) ⋅ b ' (t ) and b ' (t ) = (b1' (t ),  , bn' (t ))
σ g (t )
(25)
σ g (t ) µ g' (t ) − [bound − µ g (t )]σ g' (t )
σ g2 (t )
(26)
in which
dg ( X, t )
µ g' (t ) =
dt
µx'
(27)
Based on the above analytical equations, the upcorssing
rate v + (t ) at an arbitrary time t can be obtained and the
reliability of the turbine blade in a certain time period can be
calculated according to equation (21). And the procedure of
computing the reliability of turbine blade by using the MVFP
method can be summarized in Figure 3 [6].
4
b2
Step 1: Initialize parameters
R
r
Step 2: Linearization of limit state function
U10
0.217m
21.5m
2m
4.4 × 10−4 m
Normal
−2
2 × 10 m
−3
3 × 10 m
Determined by k, A and uc
+
Normal
Normal
Weibull-Upper
tail is truncated
Step 3: Solve for upcorssing rate v (t )
µ g (t ), µ g ' (t )
Table 3 Parameters that depend on other random variables
0.02954 arctan[1.1541(U10 − 11.701)] + 0.16636
α
N max
σ g (t ) σ g ' (t )
b(t ), b ' (t )
a (t ), a ' (t )
336.86 arctan[0.4857(U10 − 11.609)] + 2016.0
4.2. Reliability Analysis
Given the deterministic variables, the maximum bending
moment in this case study is
β + (t ), β + ' (t )
2
v + (t )
Solve for R (t0 , t f )
Fig. 3.
Flowchart of MVFP reliability analysis method
4. CASE STUDY
In this paper, a 600kW turbine with three 21.5m long rotor
blades at a specific site is considered. The reliability of wind
turbine over a 20 years design life should be evaluated. Since
the unit time is 10 minute, the time period needs to be
considered is [t0, tf] = (0, 1050055). The design lifetime of
material is 50 years (i.e. T=2625137.5). The density of the air is
1.28 kg/m3.
4.1. Data
The deterministic variables, the distributions of random
variables and parameters depend on other variables are given in
Tables 1, 2 and 3, respectively. Most of these data come from
[5] and [11].
Table 1 Deterministic variables of the turbine blade
problem
Cf
c(r )
Variable
k
A
uc
Value
1.9
9.1m/s
25m/s
1.5
1.3
Table 2 Distribution of random variables of the turbine
blade problem
Standard
Distribution
Variable
Mean
deviation
σ max
x
a
b1
518000 kPa
51800 kPa
1× 10
−3
Normal
0.1054
0.5 m
5 × 10−4 m
Normal
Normal
0.228m
4.6 × 10−4 m
Normal


 1  
− log e 1 −   



 t    (R2 − r 2 )
(28)
M max =
0.624 −2 log e 

N
α


max






From testing it has been found that the material used in
this example fails for stress larger than 0.54 times the
maximum strength [11].
Then the limit state function is
2
C f c(r )( R 2 − r 2 )b1
3ρU max
t 

=
− 0.54 exp  − x  σ max
g ( X, t )
(29)
3
3
8a (b1 − b2 )

T
where X = (U10 , a, x, σ max , b1 , b2 , R, r ) is the vector of random
variables in this problem.
After the limit state function is established, transformation
of the non-Gaussian random variables (i.e. U10 ) into standard
normal distribution is carried out by using equation (16)- (18).
Then, the limit state function is linearized as the mean
value by applying equation (19). After that µ g (t ) and σ g (t )
are derived by using equation (20). And their derivatives are
obtained by employing equation (27) and (25), respectively.
Based on these, the reliability index β + (t ) is generated using
equation (23), and its derivation is obtained using equation
(26).The upcrossing rate v + (t ) is calculated using equation (22)
after the derivation of unit vector a(t ) , a ' (t ) is obtained using
equation (24). And finally the reliability of the turbine blade in
extreme wind loading is calculated over the time period [t0, tf] =
(0, 1050055) using equation (21).
4.3. Results and Discussion
The reliability of the turbine blade was calculated in
MATLAB by following the flowchart presented in Figure 3.
Besides, in order to evaluate the accuracy of the proposed timedependent reliability analysis method for turbine blade, the
results obtained from the MVFP method is compared with its
counterpart from MCS (Monte Carlo Simulation). Table 4
shows the results generated from the MVFP and MCS over
5
different time period. The sample size of MCS is 105 for time
periods larger than 10 years and 106 for the other time periods.
Table 4 Pf of turbine blade over different time period
Time
period
(year)
[0, 20]
[0, 18]
[0, 16]
[0, 14]
[0, 12]
[0, 10]
[0, 8]
[0, 6]
[0, 4]
Pf
MVFP
MCS
−4
5.3274 × 10
3.8420 × 10−4
2.6750 × 10−4
1.7810 × 10−4
1.1181× 10−4
6.4736 × 10−5
3.3299 × 10−5
1.4171× 10−5
4.2400 × 10−6
4.9764 × 10−4
3.1696 × 10−4
2.4795 × 10−4
1.8127 × 10−4
1.2583 × 10−4
7.6171× 10−5
3.1114 × 10−5
1.3206 × 10−5
4.3905 × 10−6
Error (%)
7
21
7.8
1.7
11
15
7
7.3
3
[2]
[3]
[4]
[5]
[6]
[7]
The results in Table 4 show that the solutions of the
MVFP method are close to those of MCS.
As well as that, the MVFP method can provide us with the
reliability at any arbitrary time period, which enables us to
evaluate the reliability of the turbine blade after operating for
several years. This means that the MVFP method is much more
flexible than the nested reliability analysis method used by
Ronold [5].
Besides, the reliability over different time period of MVFP
illustrates that the MVFP method can quantify time influence
on the reliability of turbine blade in extreme wind loading
effectively.
[8]
[9]
[10]
5. CONCLUSIONS
[11]
We proposed a method for the time-dependent reliability
analysis of turbine blades in extreme wind loading. It can
evaluate the reliability of turbine blades over different time
period. The introduction of the upcrossing rate makes this
method more efficient than other random sampling methods
and nested reliability methods. However, there are some errors
existed between the results of the proposed method and its
counterparts of MCS. This may be generated from the
linearization of the limit state function and non-Gaussian
random variables. So, the improvement of the accuracy of the
method should be included in the future work. The reliability
analysis of turbine blades against fatigue should also be one of
the main parts of the future work.
[12]
6. ACKNOWLEDGMENTS
We would like to acknowledge the support of the
Intelligent Systems Center for the research presented in this
paper.
[13]
[14]
[15]
[16]
[17]
7. REFERENCES
Nick Jenkins, Ron Allan, Peter Crossley, David Kirschen,
Goran Strbac, Embedded Generation, 2000, 31-38.
[1]
6
Rajesh Karki, Po Hu, Roy Billinton, A Simplified Wind
Power Generation Model for Reliability Evaluation, IEEE
Transactions on Energy Conversion, Vol. 21, No. 2, June,
2006.
Agarwal,
Puneet,Structural reliability of
offshore wind turbines, Ph.D Thesis, The University of
Texas at Austin, 2008.
K. Saranyasoontorn, L. Manuel, Efficient models for wind
turbine extreme loads using inverse reliability, Journal of
Wind Engineering and Industrial Aerodynamics Vol. 92,
2004, 789-804.
Knut O. Ronold, Gunner C. Larsen, Reliability-based
design of wind-turbine rotor blades against failure in
ultimate loading, Engineering Structures 22 (2000) 565574.
Junfu Zhang, Xiaoping Du, Time-dependent reliability
analysis for function generator mechanisms, Journal of
Mechanical Design, [DOI: 10.1115/1.4003539].
Paul S. Veers, Steven R. Winterstein, Clifford H. Lange
and Tracy A. Wilson, User’s manual for FAROW: Fatigue
and Reliability of Wind Turbine Components, Sandia
National Laboratories, Nov. 1994.
Rice SO, Mathematical analysis of random noise. In: Wax
N, editor. Selected papers in noise and stochastic
processes. New York (NY): Dover, 1954. Reprint from
Bell System Technical Journal 1944; 23: 282-332 and
1944; 24: 46-156.
Ronold KO. Reliability of marine clay foundations in
cyclic loading. Ph.D. thesis, Stanford (CA): Stanford
University, 1993.
Holmes JD. Wind loading of structures. E & FN Spon;
2001.
Martin, O.L. Hansen, Aerodynamics of wind turbines,
Second edition, Earthscan, Sterling, VA
DS 472 (1992) Dansk Ingeniorforenings og
ingeniorsammenslutningens Norm for ‘Last og sikkerhed
for vindmollekonstruktioner’ (in Danish), Danish standard
Chan HY. System reliability analysis with time-dependent
loads and resistance. Ph.D. thesis, The University of
Newcastle; 1991.
Dalgliesh WA. Assessment of wind load for glazing
design. In: IAHR/IUTAM symposium on practical
experience with flow-induced vibrations. September 1979,
p. 686-708.
Q.S. Li, G.Q. Li., Time-dependent reliability analysis of
glass cladding under wind action, Engineering Structures,
27 (2005) 1599-1612.
Seung-Kyum Choi, Ramana V. Grandhi and Robert A.
Canfield, Reliability-based structural design, Springer,
2007.
Lutes, L.D., and Sarkani, S., 2004, Random Vibrations:
Analysis of Structural and Mechanical Systems, Elsevier,
New York.